the basic new keynesian model - drago bergholt

The Basic New Keynesian Model

I


January 11th 2012

Lecture notes by

Drago Bergholt, Norwegian Business School

[email protected]

mailto:[email protected]


II

Contents

1. Introduction .......................................................................................................................................... 1

1.1 Prologue ..................................................................................................................................................................... 1

1.2 The New Keynesian model – Key features ......................................................................................................... 1

2. Households ........................................................................................................................................... 3

2.1 Setup ........................................................................................................................................................................... 3

2.2 Optimal consumption vector and the aggregate price index ............................................................................ 4

2.3 Optimal allocation of consumption and labor .................................................................................................... 6

3. Firms .................................................................................................................................................... 11

3.1 Aggregate inflation ................................................................................................................................................. 11

3.2 Optimal price setting ............................................................................................................................................. 12

3.3 Log-linearization ..................................................................................................................................................... 13

4. Equilibrium ......................................................................................................................................... 18

4.1 Market clearing ........................................................................................................................................................ 18

4.2 The New Keynesian Phillips curve and the Dynamic IS equation ................................................................ 20

5. Equilibrium determinacy ................................................................................................................... 26

5.1 System representation ............................................................................................................................................ 26

5.2 Blanchard and Kahn conditions .......................................................................................................................... 27

6. Shocks .................................................................................................................................................. 30

6.1 Effects of a monetary policy shock ..................................................................................................................... 30

6.2 Effects of a technology shock .............................................................................................................................. 33

7. Distortions to the efficient allocation.............................................................................................. 36

7.1 The efficient steady state ....................................................................................................................................... 36

7.2 Distortions caused by market power .................................................................................................................. 37

7.3 Distortions caused by sticky prices ..................................................................................................................... 38

7.4 Monetary policy solutions to equilibrium distortions ...................................................................................... 39

8. The welfare loss function .................................................................................................................. 44

8.1 Introduction ............................................................................................................................................................ 44

8.2 The simplest case – A welfare loss function when real rigidities are absent ................................................ 44

8.3 Introduction of cost push shocks ........................................................................................................................ 51

8.4 A welfare loss function when real rigidities are present .................................................................................. 53

9. Welfare based evaluation of monetary policy ................................................................................ 58

9.1 Introduction ............................................................................................................................................................ 58

9.2 An efficient steady state under discretion .......................................................................................................... 58

9.3 An efficient steady state under commitment ..................................................................................................... 62

9.4 A distorted steady state under discretion ........................................................................................................... 66


III

9.5 A distorted steady state under commitment ...................................................................................................... 68

10. Wage rigidities ..................................................................................................................................... 69

10.1 Introduction ............................................................................................................................................................ 69

10.2 Firms ......................................................................................................................................................................... 69

10.3 Households .............................................................................................................................................................. 70

10.4 Inflation equations and the Dynamic IS equation ............................................................................................ 79

10.5 System representation and equilibrium determinacy........................................................................................ 82

10.6 Shocks ....................................................................................................................................................................... 83

10.7 Monetary policy design with sticky wages.......................................................................................................... 85

11. A small, open economy model ......................................................................................................... 93

11.1 Introduction ............................................................................................................................................................ 93

11.2 Households .............................................................................................................................................................. 93

11.3 Terms of trade, domestic inflation and CPI inflation ...................................................................................... 98

11.4 The real exchange rate ........................................................................................................................................... 99

11.5 International risk sharing .................................................................................................................................... 100

11.6 Uncovered interest rate parity ............................................................................................................................ 101

11.7 Firms and technologies ....................................................................................................................................... 102

11.8 Equilibrium – Aggregate demand and output ................................................................................................. 103

11.9 Equilibrium – The trade balance ....................................................................................................................... 109

11.10 Equilibrium – The supply side: Marginal cost and inflation dynamics ....................................................... 109

11.11 The New Keynesian Phillips curve and the Dynamic IS equation .............................................................. 111

11.12 Equilibrium determinacy ..................................................................................................................................... 113

11.13 Equilibrium dynamics .......................................................................................................................................... 116

11.14 Optimal monetary policy in the small open economy ................................................................................... 118

11.15 Welfare losses ........................................................................................................................................................ 123

References ................................................................................................................................................. 127

Appendix ................................................................................................................................................... 128

A. Dynare codes – A monetary policy shock with sticky prices ....................................................................... 128

B. Dynare codes – A technology shock with sticky prices ................................................................................ 129

C. Dynare codes – A monetary policy shock with sticky prices and wages .................................................... 131


1

1. Introduction

1.1 Prologue

These lecture notes take the reader through a basic New Keynesian model with utility maximizing

households, profit maximizing firms and a welfare maximizing central bank. I follow Gali’s

(2008) book as closely as possible. The notes were born during my participation at a couple of

PhD courses in monetary policy, taught by Antti Ripatti (Bank of Finland) and Krisztina Molnar

(Bank of Norway), respectively. Both courses built on the excellent book by Gali. The aim of the

notes is to provide the reader with all relevant calculations which are left out of the book. In

addition, the notes also go through equilibrium determinacy conditions in more detail, following

benchmark articles such as Blanchard and Kahn () and Bullard and Mitra (2002). Chapters 2, 3

and 4 characterize the basic New Keynesian model. I first analyze households, then firms. Results

are combined to establish general equilibrium. I derive a dynamic IS equation and a New

Keynesian Phillips curve. Determinacy and shocks are discussed in chapters 5 and 6. I perform

some welfare analysis of monetary policy in chapters 7, 8 and 9. Chapter 10 augments the basic

model with sticky wages in addition to sticky prices, following Erceg et al. (2000). Finally, the

small open economy model established by Gali and Monacelli (2005) is derived in chapter 11.

Dynare codes are provided in the appendix. A few words about notation: Variables in levels are

denoted with capital letters, logged variables with small letters. Percentage deviations are denoted

with small letters with a hat. Let us illustrate by an example: The percentage deviation in from

is presented by a first-order Taylor expansion:

( )

1.2 The New Keynesian model – Key features

So, what kind of features do the New Keynesian models possess? The most important are:

Dynamic, stochastic, general equilibrium (DSGE) modeling: Agents’ behavior today affects

future environments. Agents know this and behave accordingly. Still, uncertainty arises

because at least some processes in the economy are exposed to exogenous shocks. General

equilibrium, in the sense that it incorporates all markets in the economy, is provided.

Monopolistic competition: Prices are set by private economic agents in order to maximize

their objectives, as opposed to being determined by an anonymous Walrasian auctioneer

seeking to clear all competitive markets at once.

Nominal rigidities: At least some firms are subject to constraints on the frequency with which

they can adjust prices of the goods and services they sell. Alternatively, firms may face some


2

costs of adjusting those prices. The same kind of friction applies to workers in the presence of

sticky wages.

Short run non-neutrality of monetary policy: As a consequence of nominal rigidities, changes

in short term nominal interest rates are not matched by one-for-one changes in expected

inflation, thus leading to variations in real interest rates. The latter brings about changes in real

quantities. In the long run, however, all prices and wages adjust, and the economy reverts back

to its natural equilibrium.

While the first bullet point is a common feature in most modern macroeconomic models,

including those in the RBC literature, the last three are special ingredients in New Keynesian

models. Now it is time to present the basic model.


3

2. Households

2.1 Setup

We will study households and the implications of market power first. Consider an economy

consisting of many identically, infinitely-lived households, with measure normalized to one. The

representative household has an instantaneous (and time separable) money-in-utility function of

the form:

(

) (2.1)

The consumption level is denoted , is labor, and

is real money holdings. One can think

of as a composite of many goods. We make the following assumptions about preferences:1

, , , , ,

To simplify the analysis, we also assume that the marginal utility of one specific element in the

utility function is independent of the level of other elements, i.e. that .

A representative household maximizes lifetime utility, and discounts the future proportionally by

a factor :

{∑ (

)

} (2.2)

The consumption index is the sum of consumption of all goods , and there exists a

continuum of goods represented by the interval [ ]:

(∫

)

(2.3)

Note that utility is a nested function of , where is increasing in and (

) is

increasing in . Thus, utility is increasing in . The CES-aggregator given in (2.3) is an

assumption about preferences. Given this assumption, goods become imperfect substitutes, a

feature which equips firms with market power.2 Households’ maximization problem is subject to

a one-period budget constraint:

∫

(2.4)

In this setup, is the number of bonds purchased last period, each yielding a payoff of one,

and

is the price per bond bought today.

1 The expression represents (

)

( ) throughout the text.

2 Equation (2.3) also nests free competition as a special case. In particular, taking the limit as approaches infinity,

(2.3) becomes ∫

.


4

2.2 Optimal consumption vector and the aggregate price index

The household’s decision problem can be dealt with in two stages. First, for any given level of

consumption expenditures, it will be optimal to purchase the consumption vector that maximizes

total consumption .3 Second, given this optimal bundle of consumption goods, the household

must choose the utility maximizing combination of consumption, labor and money. Let us find

the optimal consumption vector first. For a given level of consumption expenditures, say

∫

, the consumption maximization problem is given by:

(∫

)

s.t. (2.5)

∫

This problem can be used to derive an aggregate price index in addition to the optimal

consumption vector. Let us solve the problem:

(∫

)

(∫

)

FOC:

:

(∫

)

⇒ (∫

)

[(∫

)

]

⇒

The equality must hold for all goods, so the relationship between two different goods must be:

(

)

⇒ (

)

(2.6)

Insert (2.6) into the constraint and solve for :

∫

∫ (

)

∫

3 Alternatively, one can find the consumption vector that minimizes total consumption expenditures for a given level of consumption. The two problems are equivalent and give identical results.


5

⇒

∫

Insert the result above into and evaluate the result for :

(∫

)

[∫ (

∫

)

]

[

∫

(∫

)

]

[(∫

)

]

(∫

)

Define as the expenditure needed to purchase a unit-level of , that is | . Using

this definition we can solve the above equation for :

(∫

)

(2.7)

Thus, equation (2.7) can conveniently be defined as an aggregate price index. We will use it

throughout the notes. To find the optimal consumption vector, insert (2.6) into the expenditures

level equation. Then, insert (2.7) and solve for consumption of good :

∫

∫ (

)

∫

⇒ [(∫

)

]

(

)

⇒ (

)

(2.8)

Insert (2.8) into (2.3) and rearrange:

(∫

)

(∫ [(

)

]

)

(∫ [

]

)

(∫

)

[(∫

)

]

⇒

⇒ ∫

(2.9)

Finally, we get the demand function for good by inserting (2.9) into (2.8):

(

)

(2.10)


6

Equation (2.10) is the solution to (2.5), the first stage of a representative household’s decision

problem. Once the household knows prices and has decided on , it also knows how much to

consume of each good. The next step is to decide .

2.3 Optimal allocation of consumption and labor

The problem in the second stage is established by using (2.2), (2.4) and (2.9):

{∑ (

)

}

s.t. (2.11)

Problems such as the one above are most often solved by using either Kuhn-Tucker conditions

or by dynamic programming. The results should be the same, of course. I will now show both of

these methods. First, the Kuhn-Tucker approach starts by setting up the Lagrangian. Let us go

through the steps:

∑ { (

) ( )}

(2.12)

FOC:

: (2.13)

: (2.14)

: (2.15)

: (2.16)

From (2.16):

(2.17)

From (2.13):

{

} ( ) {

}

⇒ ( ) {

} (2.18)

From (2.14) and (2.13):

(2.19)

From (2.15) and (2.13):

⇒

(2.20)


7

Equations (2.18), (2.19) and (2.20) determine the intertemporal consumption allocation (the Euler

equation), the labor-leisure choice and the money demand, respectively. Together, those

equations determine the rational, forward-looking household’s allocation decisions. An alternative

approach to derive (2.18)-(2.20) from (2.11) is to use dynamic programming. Point of departure is

the observation that the structure of the household’s optimization problem in period is identical

to the one in period , , etc. To see this, we first define total financial wealth at the

beginning of period as:

Second, rewrite the budget constraint to:

⇒ ( )

Third, assume that the budget constraint holds with equality and solve for :

( ( ) ) (2.21)

Fourth, recast (2.11) into a Bellman equation where is treated as the state variable and

as the control variable:

( ) { ( ) ( )} (2.22)

Equation (2.22) captures the core idea of dynamic programming, as it already defines a necessary

condition any solution to (2.11) has to fulfill. The Bellman equation basically states that the

highest obtainable value of the decision problem in period , ( ), is given by the

control which maximizies the sum of current period utility and the discounted value of the

decision problem next period. The Euler equation for this problem states that the marginal cost

of allocating more wealth today is equal to the marginal benefit of allocating more wealth

tomorrow. It is written as:

( )

( )

When we plug (2.21) into (2.1), this optimality condition becomes:

( ) (2.23)

The envelope theorem for the problem states that the marginal change in the value function

today from a change in total wealth must be equal to the marginal change in today’s utility. This

optimality condition is written as:

( ) ( )

When we plug (2.21) into (2.1), the envelope theorem yields:


8

( )

(2.24)

Iterate (2.24) one period forward:

( )

(2.25)

Insert (2.25) into (2.23) and we get the following consumption Euler equation:

(2.26)

Further, we characterize the remaining optimality conditions using (2.21) and (2.1):

:

(2.27)

: ( ) (2.28)

From (2.26):

(2.29)

From (2.27):

(2.30)

From (2.28):

(2.31)

Equations (2.29)-(2.31) determine the intertemporal consumption allocation (the Euler equation),

the labor-leisure choice and the money demand, respectively. Notice that they are identical to

(2.18)-(2.20), highlighting the fact that the household’s optimization problem should have the

same solutions regardless of solution method. To proceed we need to specify utility. As an

example, consider the following per-period utility function:4

( )

(

)

(2.32)

The marginal utilities of consumption, labor and money become:

(

)

The Euler equation given by (2.18) or (2.29) writes:

{(

)

} (2.33)

4 Gali (2008) excludes real money balances from the utility function, but instead imposes an ad-hoc log-linearized

money demand given by , where is the interest rate elasticity in the money demand equation.

We will see soon that this is equivalent to setting in (2.32).


9

The labor-leisure choice given by (2.19) or (2.30) writes:

(2.34)

The money demand equation given by (2.20) or (2.31) becomes:

(

)

⇒ (

)

⇒

(

)

(2.35)

Finally, it is convenient to log-linearize (2.33)-(2.35). We denote small letter variables as the log of

large letter variables. With respect to the Euler equation, define the following:

Using this, (2.33) can be rewritten to:

[ (

(

)

)] (

) (

)

It is clear from the equation above that in steady state where . Thus,

a first-order Taylor expansion of the Euler equation around steady state yields:

( ) [ ( ) ( ) ( ) ( )]

⇒ ( ) ( )

⇒

⇒

( ) (2.36)

The linearized version of the labor supply equation (2.34) is:

⇒ (2.37)

Finally, let us linearize the money demand equation given by (2.35):

[

(

)

]

⇒

(

)


10

⇒

[ (

)

( )

( )]

[ (

)

]

( )

⇒

[ (

)

]

( )

If we discard the constant term and assume an income elasticity of one, where this assumption

implies that , the money demand equation can be written as (2.38), where

( ) :

(2.38)

This ends the analysis of households in the New Keynesian model. We now turn to firms.


11

3. Firms

3.1 Aggregate inflation

Assume Cobb-Douglas technology:5

(3.1)

Here, is the output produced by firm in period , is the economy-wide technology level

and is the labor force used by the firm. One key ingredient in the New Keynesian model is

price rigidity. When firms set their prices, they can do so freely. However, they do not know a

priori when the next opportunity to price change emerges. The probability of being unable to

change the price in any given period is . Thus, this is the fraction of all firms that is stuck with

the price they had last period while the remaining firms reset their prices. The aggregate

price dynamics (inflation) in period can be calculated as follows, where is the aggregate price

level, is the optimal price set by firms who are able to reoptimize in that period, and ( )

[ ] represent the set of firms not reoptimizing their posted price:

[∫

( ) ( )

]

[

( ) ]

⇒ (

)

([ ( )

]

)

( ) (

)

⇒ ( ) (

)

(3.2)

The aggregate gross inflation is defined as

. Steady state is defined by zero inflation,

implying that and . Linearizing (3.2) around steady state yields:6

( ) [ ( ) (

)

] ( )( ) (

)

(

)

⇒ ( ) ( )( )( )

⇒ ( )( ) (3.3)

Equation (3.3) makes it clear that inflation results from the fact that firms reoptimizing in any

given period choose a price that differs from the economy’s average price in the previous period.

5 The capital stock is treated as fixed and investment is set to zero in the short run. These two specifications follow

McCallum and Nelson (1999), who argued that capital do not play a major role in most monetary policy and business cycle analyses. 6 Remember the first-order Taylor expansion: ( ) ( ) ∑

( ) where is the vector of variables

one wants to linearize around. Using this as a point of departure, it is often convenient to define a new variable as

the log deviation in from :

. This implies that , and the Taylor expansion can be

rewritten to a formula for log-linearization via Taylor series expansion: ( ) ( ) ∑ .


12

Hence, in order to understand inflation over time one needs to analyze the factors underlying

firms’ price setting decisions.

3.2 Optimal price setting

Basically, when firms are faced with the problem of setting optimal price today, they must take

into consideration that this price often determine profit in the future as well, as the probability of

being stuck with today’s price periods ahead is . Thus, a firm who reoptimizes in period

will choose the price that maximizes current market value of the profits generated while that

price remains effective. The stochastic discount factor for nominal payoffs in period is

, which is given by:7

(

)

(3.4)

The representative firm’s maximization problem is thus given by:

{∑ [ (

| | ( | ))]

}

s.t. (3.5)

| (

)

Let us spend a couple of seconds on the problem given in (3.5). | is the output in period

for a firm that last set its price in period . | ( | ) is the total cost in period

as a function of this output. The nominal, undiscounted profit in period is thus

| |

( | ). The firm’s problem is subject to a sequence of demand

constraints as given by (2.10), and market clearing in period implies that the firm produces

| (

)

. The problem can be rewritten to an unconstrained one by inserting the

constraint into the profit function. We also insert for the discount factor. This gives us:

{∑ [

(

)

(

(

)

| ((

)

))] } (3.6)

Let us find the optimal price :

∑ [ (

)

(

(

)

| ((

)

))]

FOC:

7 Go back to the Euler equation of the consumers to get the intuition.


13

∑ [ (

)

(( ) (

)

| (

)

)]

∑ [ (( ) | | | (

)

)]

∑ [ | (( ) |

)]

⇒ ∑ [ | (

|

)]

⇒ ∑ ( | )

∑ ( | |

)

Next, we insert for and | and solve for the optimal price :

∑ ( (

)

(

)

)

∑ (

(

)

(

)

| )

⇒ ∑ (

)

∑ (

| )

⇒ ∑ (

)

∑ (

|

)

⇒

∑

|

∑

(3.7)

Divide both sides by to get the optimal real price as a weighted average of future real marginal

costs:8

∑ (

)

|

∑ (

)

(3.8)

Notice that in the case with flexible prices, i.e. when , (3.6) collapses to a one period

problem and (3.7) becomes:

|

|

|

(3.9)

Thus, (3.9) gives the desired or frictionless markup.

3.3 Log-linearization

8 Note that the real marginal cost in period is denoted |

|

.


14

The next step is to log-linearize (3.7) around the steady state. In a zero inflation steady state, we

must have that:

| |

| |

|

The last three identities follow from the zero inflation definition and from market clearing.

Before log-linearizing it is convenient to divide both sides of (3.7) by :

∑

|

∑

⇒

∑

∑

|

(3.10)

A first-order Taylor expansion of the LHS of (3.10):9

9 This is just a simple first-order Taylor expansion. The first term is the LHS of (3.10) in steady state. The four last

terms contain the first derivatives with respect to , , and respectively, all evaluated in steady state.


15

∑

∑ (

)

∑ ( )

∑ ( ) ( )

∑ ( )

( )

∑

∑ ( )

∑ ( )

∑ ( )( )

∑ ( )( )

∑ [ ( ) ( ) ( )( )

( )( )]

∑ [ ( )( )

( )( )]

A first-order Taylor expansion of the RHS of (3.10):10

10

The first term is the RHS of (3.10) in steady state. The four last terms contain the first derivatives with respect to

, , and | respectively, all evaluated in steady state.


16

∑

∑

( )

∑

( )

∑ ( )

( )

∑

( |

)

∑

∑ ( )

∑ ( )

∑ ( )( )

∑ ( |

)

∑ [ ( ) ( )

( )( ) ( | )]

∑ [ ( ) ( )( )

( | )]

Finally we equate LHS with RHS and solve for :

∑ [ ( )( ) ( )( )]

∑ [ ( ) ( )( )

( | )]


17

⇒ ∑ ( )

∑ ( |

)

⇒

∑ [( |

) ]

⇒ ( ) ∑ [( |

) ]

⇒ ( ) ∑ [ |

] (3.11)

We see from (3.11) that firms will set a price that corresponds to the desired markup,11 given by

, over a weighted average of their current and expected nominal marginal costs, with

the weights being proportional to the probability of the price remaining effective at each horizon

.

11 Because

, we have that

(

( )

)

.


18

4. Equilibrium

4.1 Market clearing

Market clearing in the goods market implies:

(4.1)

Let aggregate output be defined as:

(∫

)

(4.2)

Insert (4.1) into (4.2), and then (2.10) into (4.2), to get the aggregate market clearing condition:

(∫

)

{∫ [(

)

]

}

(∫

)

(

∫

)

(∫

)

[(∫

)

]

⇒

Finally, taking logs on both sides yields:

(4.3)

Equation (4.3) is the aggregate market clearing condition. Insert (4.3) into (2.36) and get:

( ) (4.4)

Market clearing in the labor market:

∫

(4.5)

From (3.1) we see that (

)

. Insert this into (4.5) as well as the goods market clearing

condition (4.3) and the consumption demand (2.10):

∫ (

)

∫ (

)

∫ (

(

)

)

(

)

∫ (

)

(4.6)

Next we take the log of (4.6):

( ) [∫ (

)

]


19

⇒ ( ) ( ) [∫ (

)

] ( ) (4.7)

I will now show that because ∫ (

)

up to a first-order approximation

around , but first I must show that ∫

. Recall the consumer price index

(∫

)

. Rearranging gives:

(∫ (

)

)

(∫ ( )( )

)

⇒ ∫ ( )( )

(4.8)

A second order approximation of (4.8) gives us:

∫ [ ( )( )

( ) ( )

]

( )∫ ( )

( )

∫ ( )

( ) ( )∫

( )

∫ ( )

⇒ ( ) ( ) ∫

( )

∫ ( )

⇒ ∫

∫ ( )

(4.9)

From equation (4.9) it is also clear that ∫

up to a first-order approximation. Next, let

us do a second order approximation of ∫ (

)

:

∫(

)

∫[

( )]

∫

( )

∫[

(

)

( ) ]

∫( )

(

)

∫( )

∫

(

)

∫( )

Now, insert (4.9) and get:


20

∫(

)

(∫

∫ ( )

)

∫

(

)

∫( )

( )

( )∫ ( )

(

)

∫( )

[ ( )

( )

( ) ]∫ ( )

( )( )

( ) ∫ ( )

( )

( ) ∫ ( )

(4.10)

From (4.10) we conclude that up to a first-order approximation, ∫ (

)

. This

implies that:

( ) [∫ (

)

] ( )

Thus, (4.7) can be rewritten to:

( ) (4.11)

4.2 The New Keynesian Phillips curve and the Dynamic IS equation

Next, an expression for individual firms’ marginal cost as a function of the economy’s average

real marginal cost is derived. The latter is derived in (4.12), where we insert

from

(4.11):12

12 The nominal marginal cost by using labor is the wage . The nominal marginal gain is the income increase, that is the price times the marginal increase in production by adding a little more labor. Thus, the real marginal cost is the

nominal cost relative to the nominal gain, i.e.

. Linearizing gives

. It follows

from the average production function that marginal productivity is

( ) . Thus,

( ) .


21

( )

( )

⇒

( ) (4.12)

Similarly, a firm’s real marginal cost in period is:

| |

|

( ) (4.13)

Now, the market clearing condition and the demand schedule (2.10) imply that firm output is

| ( |

)

, which in linearized terms gives | ( | ) .13

Use this as well as (4.13) and (4.12) to get:

|

[ |

( )]

[

( )]

( | )

[ (

) ( ) ]

(

)

⇒ |

(

) (4.14)

Notice that the last term in (4.14) disappears if there is constant returns to scale, i.e. if .

Then |

, which implies that the marginal real cost is independent of the

production level; it is common across all firms. We shall now derive an expression for inflation.

The point of departure is (3.11), which we rewrite to:

( ) ∑ [ |

]

Insert (4.14):

( ) ∑ [

(

) ]

( ) ∑ (

)

⇒

( ) ∑ (

)

⇒

( ) ∑ (

)

13 Note that |

.


22

Define

and subtract ( ) on both sides.:

( ) ∑ (

)

( ) ∑ ( )

( ) ∑

( ) ∑ ( )

( ) ∑

( ) [ ( ) ( )

( ) ]

( ) ∑

( ) [ ( ) ( ) ]

( ) ∑

[ ( ) ( ) ]

[ ( ) ( ) ]

( ) ∑

[ ]

( ) ∑

∑

If we take out the terms of each summation operator, the equation can be written more

compactly as a difference equation:

( ) ∑

∑

( )

[( ) ∑

∑

]

( ) (

) ( )

Next we insert (3.3) and solve for inflation ( )( ):

(

) ( ) ( )(

)


23

⇒ ( ) (

) ( )

⇒ (

) ( )

⇒ ( )( ) ( ) (

) ( )( )

⇒ (4.15)

Equation (4.15) expresses inflation as the sum of (discounted) expected inflation and real

marginal costs, and we have defined ( )( )

( )( )

to ease the

notation. It is clear from (4.15) that inflation is strictly decreasing in price stickiness , in the

measure of decreasing returns , and in the demand elasticity . An alternative presentation of

inflation is found by solving (4.15) forward:

{ [ ( )

] }

∑

Equivalently, and defining the average markup in the economy as – , we see that inflation will

be high when firms expect average markups to be below their steady state or desired level – .

In that case firms that have the opportunity to reset prices will choose a price above the

economy’s average price level in order to realign their markup closer to its desired level. Thus, in

the present model, inflation results from the aggregate consequences of purposeful price-setting

decisions by firms, which adjust their prices in light of current and anticipated cost conditions.

Next, a relation is derived between the economy’s real marginal cost and a measure of aggregate

economic activity. We have derived earlier that ( ) . Insert this into

( )

, and use that ( ) :

( )

( ) [ ( ) ]

( ) [ ( )]

Insert for

and get to:

( )

( ) (4.16)

In the case with flexible prices we know from before that . Define natural output level

as the equilibrium level under full price flexibility. In this case (4.16) can be rewritten to:

( )

( ) (4.17)

Solve (4.17) for natural output:

( )

( )

⇒

( ) [

( )]

( )

( )[ ( )]

( )


24

⇒

(4.18)

If we subtract (4.17) from (4.16) we get a measure of the real marginal cost gap

as a function of the output gap from natural output, denoted :

[ ( )

( )]

[ ( )

( )]

( )

(

)

⇒

( )

(4.19)

Finally, the New Keynesian Phillips curve is established by inserting (4.19) into (4.15):

( )

⇒ (4.20)

The New Keynesian Phillips curve (NKPC) is one of the key building blocks of the New

Keynesian model, and the parameter is defined by ( )

. The second key equation

is the dynamic IS equation. If we use the definition of the real interest rate, ,

equation (4.4) becomes

( ). In a similar vein, the natural output is given as

a function of the natural interest rate:

(

) (4.21)

Subtracting (4.21) from (4.4) gives the output gap from the natural output, i.e. the dynamic IS

equation (DIS):

[

( )] [

(

)]

⇒

(

) (4.22)

Equations (4.20) and (4.22) together with an equilibrium process for the natural rate , which in

general will depend on all exogenous forces in the model, constitute the non-policy block of the

basic New Keynesian model. That block has a simple recursive structure: The NKPC determines

inflation given a path for the output gap, whereas the DIS equation determines the output gap

given a path for the exogenous natural rate and the actual real rate. To see the latter, assume the

transversality condition . Then one can solve (4.22) forward to yield:

∑ (

) (4.23)


25

Equation (4.23) emphasizes the fact that the output gap is proportional to the sum of current and

anticipated deviations between the real interest rate and its natural counterpart. To gain further

insight into the natural interest rate, note first that (4.4) implies

( ).

Second, note that the first difference of (4.18) gives

. Now, solve (4.22)

for and use these two observations to yield an expression for the natural real rate. From

(4.22):

( ) [ (

) ( )]

[ ]

[

( )

]

⇒

(4.24)

Thus, the natural real rate is a function of households’ discount rate and expected technological

progress. In some cases it is convenient to work with deviations in the natural real rate from the

discount rate, which we define as:

(4.25)

Note that if one turns off technology shocks, the real rate becomes the discount rate. Once a

process for the technological progress is specified, one can identify the real interest rate path in

(4.24). In order to close the model, we supplement (4.20) and (4.22) with one or more equations

determining how the nominal interest rate evolves over time, i.e. with a description of how

monetary policy is conducted. Observe from (4.23) that the equilibrium path of real variables

cannot be determined independently of monetary policy when prices are sticky. The output gap is

directly determined by the real interest rate gap, which is directly determined by the nominal

interest rate set by central banks. This important feature of the New Keynesian Model is in

contrast to classical models where monetary policy is neutral.


26

5. Equilibrium determinacy

5.1 System representation

Throughout this and the next sections I will look at a specific monetary policy rule, more

specifically an interest rate rule. I assume the central bank follows a rule of the form:

(5.1)

Standard reasoning implies that and are non-negative, which we assume from now on.

The first task when analyzing monetary rules is to check whether the specified policy yields a

unique and stable equilibrium. While doing this, it is convenient to work with a reduced form

representation of (4.20) and (4.22) who takes into account the policy rule under consideration.

We first derive a forward looking version of the dynamic IS equation. Insert (5.1) into (4.22), and

then (4.20) into (4.22). Solve the resulting equation for :

(

)

(

)

[ ( )

]

⇒

⇒

[ ( ) (

)] (5.2)

Equation (5.2) shows the current output gap as a function of expected output gap, expected

inflation, and shocks. We next achieve a similar representation of current inflation. Insert (5.2)

into (4.20) and get:

{

[ ( ) ( )]}

( ) ( )

( )

( )

( )

⇒

{ [ ( )] (

)} (5.3)


27

Finally, the two equations (5.2) and (5.3) can be written as a system of forward looking difference

equations:

[

]

[

( )] [

]

[ ] (

)

[

( )] [

] [

] (

)

[

] (

)

where (5.4)

[

( )]

[ ]

The system given in (5.4) is a reduced form representation of the dynamic IS curve and the New

Keynesian Phillips curve, which takes into account effects from the policy defined in equation

(5.1). The coefficient matrix represents effects from expectations on current output gap and

inflation while the coefficient vector represents effects from technology shocks in and

monetary policy shocks in . We have defined

to ease the notation.

5.2 Blanchard and Kahn conditions

In the case we consider here we have two non-predetermined variables, and .

Following Blanchard and Kahn (1980), the system (5.4) has a locally unique equilibrium if and

only if both eigenvalues of the 2x2-matrix are inside the unit circle.14 Let us characterize

necessary and sufficient conditions for this property to hold. The two eigenvalues, denoted

and , are generally solutions to the following system written in matrix form, where is an

identity matrix:

| |

In our case the system becomes:

|

[

( )] [

]|

When this is written out:

14 Consider a recursive system of the form { }

, where is a vector of predetermined and non-

predetermined variables and is a vector of exogenous variables. Blanchard and Kahn (1980) proved that there

exists a locally unique equilibrium if and only if the number of eigenvalues of inside the unit circle is equal to the number of non-predetermined variables. If the number of eigenvalues inside the unit circle is less than the number of non-predetermined variables, then an infinite number of equilibria exist. If the number of eigenvalues inside the unit circle exceeds the number of non-predetermined variables, then no equilibrium exists.


28

||

[

( )

]

||

( ( ))

( )

( )

( )

( )

( )

( )

( )

( )

Following LaSalle (1986)15 we know that the two eigenvalues of the matrix are inside the unit

circle if and only if the following two inequalities are met:

|

|

and

| ( )

|

Let us derive conditions the policy parameters and must meet for these two inequalities to

hold. From the first inequality:

|

|

⇒

⇒ ( ) (5.5)

It is clear that condition (5.5), and consequently the first inequality, are fulfilled as long as ,

which we assume. Thus, the only relevant inequality is the second one, which we rewrite to:

| ( )

|

( )

⇒ ( )

⇒

⇒ ( ) ( ) (5.6)

We see from condition (5.6) that the equilibrium is unique as long as the policy parameters

and have sufficiently high values, i.e. as long as monetary authorities respond to deviations of

15 LaSalle (1986) showed that both solutions to are smaller than one if and only if | | at

the same time as | | .


29

inflation and output with adequate strength. Note also that our assumptions about the other

parameters imply that is sufficient for (5.6) to hold. This is referred to as the Taylor

principle. Let us give (5.6) some intuition. Suppose the economy is exposed to a permanent

change in inflation; . From (4.20) we see that without any policy this leads to a permanent

change in the output gap equal to

. However, with the policy rule described here,

we can find the nominal interest rate response by inserting

into the differentiated

version of (5.1):

(

)

Furthermore, by rearranging (5.6) we get that

. This implies that the change in

inflation should be met by a larger change in the nominal interest rate. Eventually this will drive

the real rate upwards and act as a stabilizing force. Thus, from (5.6) we see that when the central

bank responds aggressively enough to changes in output gap and inflation, i.e. when and

are large enough, output is forced back to natural output and inflation back to zero.


30

6. Shocks

6.1 Effects of a monetary policy shock

Assume that the exogenous component of (5.1) follows an AR(1) process, where [ ):

(6.1)

Notice that a positive (negative) realization of is interpreted as a contractionary (expansionary)

monetary policy shock, leading to a rise (decline) in the nominal interest rate for given levels of

inflation and output gap. We want to find the contemporaneous effects of a monetary policy

shock on the output gap and inflation . One way to identify these effects is by using the

method of undetermined coefficients. Let us start by making the following guess:

(6.2)

(6.3)

The coefficients and are yet to be determined. First, insert (6.1)-(6.3) into the New

Keynesian Phillips curve given by (4.20). Find an expression for :

⇒ ( )

⇒

⇒ ( )

⇒

(6.4)

Then, insert the monetary policy rule (5.1) into the dynamic IS equation (4.22):16

(

)

[( )

]

( )

Insert (6.1)-(6.4) and solve for the coefficient :

( )

( )

16 To make the analysis as transparent as possible we set

. That is, we turn off the

technology shocks.


31

⇒ ( ) ( )

⇒ [ ( ) ] ( )

⇒ [ ( ) ]

( )

⇒ ( )[ ( ) ] ( )

⇒

( )[ ( ) ] ( ) (6.5)

Insert (6.5) into (6.4) to obtain:

( ) (6.6)

Finally, this means that the solutions to (6.2) and (6.3) are:

( ) (6.7)

(6.8)

To ease the notation,

( )[ ( ) ] ( ). It is also straight forward to show that

as long as (5.6) is satisfied. Note that if we insert (6.1) into (6.7) and (6.8), we get:

( ) ( ) ( ) ( )

( )

( )

Hence, an exogenous increase in the interest rate leads to a persistent decline in both output gap

and inflation. Because the natural level of output is unaffected by the monetary policy shock, the

response of output matches that of the output gap. Furthermore, (4.22) and (6.2) can be used to

obtain an expression for the real interest rate deviation from its steady state counterpart, the

natural real rate:

( )

( ) ( )( ) (6.9)

The response on nominal interest rate combines both the direct effect of and the indirect

effect induced by reduced output gap and inflation. From (6.8) and (6.9):

[ ( )( ) ] (6.10)

Note that if the persistence of the monetary policy shock, , is sufficiently high, the nominal

interest rate will decline in a response to a rise in . In that case, and despite the lower nominal

rate, the policy shock still has a contractionary effect on output, because the latter is inversely

related to the real rate, which goes up unambiguously. Finally, one can use (2.38) and (4.3) to

determine the change in the money supply required to bring about the desired change in the

interest rate. From , where we insert (6.8), (6.7) and (6.10):


32

( ) [ ( )( ) ]

[( )[ ( )] ( )]

⇒

[( )[ ( )] ( )] (6.11)

The sign of the change in money supply that supports the exogenous policy intervention is, in

principle, ambiguous. Note however, that

is a sufficient condition for a contraction in

the money supply. Let us simulate the effects of a monetary policy shock. Parameters are

calibrated as follows:

Table 1

0.99 1 1 1/3 6 4 2/3 1.5 0.5/4 0.5 0.0625

Setting = 0.99 implies a steady state real return on financial assets of about = ≈ 4%.

Log utility is implied by = 1 and = 1. = 2/3 implies an average price duration of

= 3

quarters. = 1.5 and = 0.5/4 is roughly consistent with observed variations in the federal

funds rate over the Greenspan era. Finally, = 0.0625 corresponds to a monetary policy shock

of 25 basis points. Simulated impulse responses are shown in the figure 1.17 Consistent with the

analytical results, it is seen that the policy shock leads to an increase in the real interest rate, and a

decrease in inflation and output. The latter two effects correspond to that of the output gap

because the natural level of output is not affected by the monetary policy shock. Under the

calibration given in table 1 the nominal interest rate goes up, though by less than its exogenous

component, as a result of the downward adjustment induced by the decline in inflation and

output gap. In order to bring about the observed interest rate response, the central bank must

engineer a reduction in the money supply. The calibrated model thus displays a liquidity effect.

Note also that the response of the real rate is higher than that of the nominal rate as a result of

the decrease in expected inflation. Overall, figure 1 shows dynamic responses which are

qualitatively similar to those estimated using structural vector auto regressive methods.

Figure 1: A monetary policy shock

17 Simulations are done with Dynare and Matlab. See Appendix A for the Dynare codes.


33

6.2 Effects of a technology shock

Assume that the technology parameter follows an AR(1) process, where [ ]:

(6.12)

Here we want to find contemporaneous effects of the technology shock on the output gap

and inflation . Given (4.25), the implied natural rate expressed in terms of deviations from

steady state, is given by:

[(

) ] ( ) (6.13)

Again we use the method of undetermined coefficients. Guess the following:

(6.14)

(6.15)

Insert (6.12) and (6.14)-(6.15) into the New Keynesian Phillips curve (4.20). Find an expression

for :

⇒ ( )

⇒

⇒ ( )

⇒

(6.16)


34

Then, insert the monetary policy rule (5.1) into the dynamic IS equation (4.22):18

(

)

[( )

]

(

)

Insert (6.12)-(6.15) and solve for the coefficient :

(

( ) )

(

( ) )

( )

⇒ ( ) ( ) ( )

⇒ [ ( ) ] ( ) ( )

⇒ [ ( ) ]

( )

( )

⇒ ( )[ ( ) ] ( )

( )

⇒

( )

( )[ ( ) ] ( )

( ) (6.17)

Insert (6.17) into (6.16) to obtain:

( ) ( )( ) (6.18)

Finally, this means that the solutions to (6.14) and (6.15) are:

( )( ) (6.19)

( ) (6.20)

To ease the notation,

( )[ ( ) ] ( ) . Note that if we insert (6.12) into

(6.19) and (6.20), we get:

( )( ) (

) ( )( )

( ) (

) ( )

Hence, a positive technology shock leads to a persistent decline in both output gap and inflation.

To find the implied equilibrium response of output, decompose output into

. Then insert for from (4.18) and from (6.19):

18 To make the analysis as parsimonious as possible we now set

. That is, we turn off the monetary policy shocks.


35

( )( ) [ ( )( ) ] (6.21)

To find the equilibrium response of employment, insert for (6.21) into (4.11):

( ) [( )

( )( ) ] (6.22)

Hence, the sign of the response of output and employment to a positive technology shock is in

general ambiguous, depending on the configuration of parameter values, including the policy

parameters and . Parameters are calibrated as follows in the simulation exercise:

Table 2

0.99 1 1 1/3 6 4 2/3 1.5 0.5/4 0.9 1

We assume that = 0.9, i.e. that it takes some time for a technology shock to die out. Simulated

impulse responses are shown in the figure 2.19 Notice that the improvement in technology is

partly accommodated by the central bank, which lowers nominal and real rates while increasing

the money in circulation. That however is not enough to close the negative output gap, which is

responsible for a decline in inflation. Output increases, but less than its natural counterpart.

Figure 2: A technology shock

19 Simulations are done with Dynare and Matlab. See Appendix B for the Dynare codes.


36

7. Distortions to the efficient allocation

7.1 The efficient steady state

I will now consider monetary policy design in the New Keynesian framework. This section

investigates distortions to the efficient allocation and how monetary authorities can cope with

these distortions. First we need to determine the efficient allocation. A natural benchmark is the

problem faced by a benevolent social planner seeking to maximize the representative households’

social welfare, given preferences and technology. Using (2.3) and (4.5), this problem reads as:20

{∑ ( )

}

{∑ [(∫

)

∫

]

}

s.t. (7.1)

The constraint is the resource constraint coming from all the firms. Notice how all goods enter

the utility function symmetrically, at the same time as utility is concave in each good. Also, all

goods are produced with identical technology. Thus, by symmetry, can never be

optimal. This gives the following efficiency conditions:

(7.2)

⇒ (7.3)

The problem therefore simplifies to:

{∑ (

) } (7.4)

Let us solve the social planner’s problem. FOC:

: ( )

⇒

( )

(7.5)

From the firm’s problem in free competition we also have the following:

{ }

{ } (7.6)

FOC:

: ( )

⇒ ( )

(7.7)

From (7.5) and (7.7):

(7.8)

20 Here we depart from the MIU-specification used previously in order to keep the analysis as simple as possible.


37

Thus, (7.8) is the relevant efficient benchmark monetary authorities should opt for. However,

there are two sources of inefficiencies built into the New Keynesian model setup. The first one is

firm’s market power, which allows firms to set prices individually instead of being price takers.

The second is staggered price setting, which prevent firms from adjusting optimally to shocks in

the economy in the short run. I will now study these two inefficiencies in turn.

7.2 Distortions caused by market power

Market power, which yields monopolistic competition, stems from the construction that each

firm perceives an imperfectly elastic demand for its differentiated product. This gives firms the

opportunity to set prices above marginal costs. Market power is unrelated to the presence of

sticky prices. To illustrate this, suppose for the moment that prices are fully flexible so that

. Firm ’s problem then becomes:

{ }

{ (

)

}

{

(

)

[(

)

]

}

⇒ { (

)

[

(

)

]

} (7.9)

FOC:

( ) (

)

[

(

)

]

(

)

⇒ (

)

[

(

)

]

[(

)

]

(

)

⇒ (

)

[

(

)

]

⇒ (

)

( )

⇒

( )

Because all firms behave in the same way, :


38

( )

⇒

( )

(7.10)

As before,

is the gross optimal markup chosen by firms and

is the marginal cost. If

we insert (7.10) into the efficient allocation (7.8), where the first equality follows from the

optimality conditions of the household, we immediately see that:

(7.11)

Thus, the presence of market power not only leads to higher prices than optimal, but also an

inefficiently low level of employment, and therefore also of output. This kind of distortion to the

efficient equilibrium can be dealt with in a simple way by means of an employment subsidy. Let

denote the rate at which the cost of employment is subsidized, and let outlays associated with the

subsidy be financed by a lump-sum tax. If the subsidy is set to

, then, by construction, the

equilibrium under flexible prices yields efficiency.21 Equation (7.10) becomes:

( )

(7.12)

7.3 Distortions caused by sticky prices

The assumed constraints on the frequency of price adjustment constitute a source of inefficiency

on two grounds. First, the fact that firms do not adjust their prices continuously implies that the

economy’s average markup will vary over time in response to shocks, and will generally differ

from the constant frictionless markup

. Denote the economy’s average markup, i.e. the ratio

of average price to average marginal cost, as . Then, from (7.12):

( )

(7.13)

The last equality follows from the assumption that the subsidy in place exactly offsets the

monopolistic competition distortion, which allows us to isolate the role of sticky prices. Insert

(7.13) into the efficient benchmark allocation (7.8):

(7.14)

Thus, (7.8) is violated whenever

. Efficiency can only be restored if policy manages to

stabilize the economy’s average markup to its frictionless level. In addition to the inefficiency

described above, staggered price setting is a source of a second type of inefficiency. The latter has

21 In much of the analysis below it is assumed that such an optimal subsidy is in place.


39

to do with the fact that relative prices of different goods will vary in a way unwarranted by

changes in preferences or technologies, as a result of the lack of synchronization in price

adjustments. Whenever we also get , and consequently , which

violates (7.2) and (7.3). To cope with distortions caused by staggered price setting, one should

therefore opt for markups that are equal across all firms at all times. I will now analyze how this

goal can be achieved by monetary authorities.

7.4 Monetary policy solutions to equilibrium distortions

To keep the analysis simple, assume that in the last period, , implying that we

had an efficient allocation at . Then this efficient allocation can be attained by a policy that

stabilizes marginal costs at a level consistent with firms desired markup, i.e.

, given the prices in place. If that policy is expected to be in place indefinitely, no firm

has an incentive to adjust its price because it is currently charging its optimal markup and expects

to keep doing so in the future. As a result, and, hence, . In other words,

the aggregate price level is fully stabilized and no relative price distortions emerge. In addition,

, and output and employment matches their counterparts in the flexible price

equilibrium allocation with a subsidy in place. From (4.19) and (4.20) we immediately see that

implies the following, where inflation is given by (3.3):

(7.15)

( )( ) (7.16)

From the dynamic IS equation (4.22) we see that once (7.15) and (7.16) are expected to take place

indefinitely, the nominal interest rate becomes the natural real rate:

(7.17)

Two features of the optimal policy are worth emphasizing. First, stabilizing output is not

desirable in and of itself. Instead, output should vary one for one with the natural level of output.

Whenever real shocks cause natural output to fluctuate a lot, so should also output. Second, price

stability emerges as a feature of the optimal policy even though, a priori, the policy maker does

not attach any weight on such objective. The next step is to analyze how to implement (7.15) and

(7.16) in practice. Because (7.15) and (7.16) imply (7.17), one could think of (7.17) as a natural

candidate for monetary policy. Although one obvious equilibrium is , we need to

whether this equilibrium is unique. Treat (7.17) as an exogenous interest rate rule and insert it

into (4.22). Combine with (4.20) to yield a system of difference equations:

(

)


40

⇒ (

) (

)

⇒ [

] [

] [

] [

]

where (7.18)

[

]

Let us calculate the characteristic equation:

| | |[

]| (

)

With respect to the two conditions necessary for smaller-than-unity eigenvalues derived by

LaSalle (1986), we get:

| |

and

|

|

Clearly, the last condition does not hold, so both eigenvalues of cannot lie inside the unit

circle. Thus, by the Blanchard and Kahn (1980) conditions, there exists a multiplicity of equilibria

because the number of eigenvalues inside the unit circle is smaller than the number of non-

predetermined variables. The zero output gap and zero inflation target is only one of them, and

there is nothing in the policy (7.17) that drives the economy back to the desired equilibrium given

by (7.15) and (7.16). The second policy rule we consider is an interest rate rule with an

endogenous component:

(7.19)

We first derive a forward looking version of the dynamic IS equation. Insert (7.19) into (4.22) and

solve for :

(

)

[ ( ) ]

⇒


41

⇒

[ ( ) ] (7.20)

Next, insert (7.20) into (4.20) and solve for :

{

[ ( ) ]}

( ) ( )

( )

⇒

{ [ ( )] } (7.21)

The two equations (7.20) and (7.21) can be written as a system of difference equations:

[

]

[

( )] [

] [

( )] [

]

[

]

where (7.22)

[

( )]

Note that the transition matrix in (7.22) is identical to the one in (6.4). Thus, whenever

condition (6.6) holds, which it does as long as one follows the Taylor principle , the

policy rule given by (7.19) yields a unique and stable equilibrium with . A last

monetary policy rule worth considering is a forward-looking interest rate rule:

(7.23)

Now, the monetary authorities adjust the nominal interest rate to variations in expected inflation

and output gap, as opposed to their current values. Insert (7.23) into (4.22):

(

)

[ ( ) ]

(

)

(7.24)

Insert (7.24) into (4.20):

[(

)

]

⇒ (

) (

) (7.25)


42

Write as a system of forward looking difference equations:

[

] [

(

)

] [

] [

]

where (7.26)

[

(

)

]

The characteristic equation:

| |

In our case:

|[

(

)

] [

]|

Written out:

|[

(

)

]|

(

) (

) (

)

(

) (

) (

)

( )( )

( ) ( )

[ ( ) ( )( )] ( )( )

[ ( )

] (

)

The inequalities from LaSalle (1986) which should be met by the two eigenvalues of :

| (

)|

and

| ( )

| (

)


43

From the first inequality:

| (

)|

⇒ |

|

⇒ | |

If :

⇒ (

)

⇒ (

) (7.27)

Condition (7.27) cannot be binding when is non-negative because (

) . If :

⇒ (

) (7.28)

From the second equality:

| ( )

| (

)

⇒ |

(

)| (

)

⇒ | [ ( ) ]| ( )

If [ ( ) ]:

[ ( ) ] ( )

⇒ ( ) ( ) ( ) (7.29)

If [ ( ) ]:

( ) [ ] ( )

⇒ ( ) ( ) (7.30)

Condition (7.30) turns out to be identical to (6.6). However, in this case the two conditions (7.28)

and (7.29) must hold in addition. We see from (7.28)-(7.30) that the policy responses should be

neither too weak nor too strong. In particular, from (7.28) and (7.30) we see that a very high

value on leads to indeterminacy, quite independently of . If the response to output is

modest, then rules with can lead to determinate equilibria. However, from (7.29) we see that

too large a value of again leads to indeterminacy. Finally, note that if , then could

be set to zero.


44

8. The welfare loss function

8.1 Introduction

I will now derive measures of the society’s welfare losses caused by deviations in output and

inflation from their steady state targets. The result will be a quadratic loss function that represents

a quadratic second-order Taylor series approximation to the level of expected utility of the

representative household in equilibrium with a given monetary policy. First I look at the simplest

case where the only distortions in the economy are the presence of monopolistic competition and

sticky prices. Then I look at an empirically more appealing case where what one refers to as cost

push shocks exist.

8.2 The simplest case – A welfare loss function when real rigidities are absent

The first case we consider is the one analyzed previously, where the government implements an

employment subsidy that removes the distortions caused by monopolistic competition. Thus, we

assume that the subsidy given by (7.12) is in place. This case will also serve as a methodological

framework for the welfare analysis conducted later. In order to lighten the notation, denote the

period utility as ( ) and the steady state utility as ( ). We will use the

following second order approximation of relative deviation in consumption from its steady state

counterpart, where logged consumption is approximated around logged steady state

consumption:

( )

( )

( )

The same kind of second order approximation is performed on labor , so that:

We need some more results as well. From (2.32) we have that

and

. From the market clearing condition we have that . Using all

these results, a second-order Taylor approximation of around steady state ( ) leads us to

the following criterion for welfare losses:22

22 From (19) we see that utility is separable in consumption and labor, i.e. . Also, we make the analysis simple and assume away money in the utility function.


45

(

)

(

)

(

) (

)

(

) (

)

⇒

(

)

⇒

(

) (8.1)

Our goal is to find a way to express (8.1) in terms of steady state deviations only, that is with the

gap in output from natural output and the gap in inflation from zero inflation. The way to such a

representation contains several steps. First, note from (4.7) that:

{ ( ) [∫ (

)

]}

( )

⇒

[( ) ( ) ( )]

( ) (8.2)

As before:

( ) [∫ (

)

] (8.3)

The next step is to get an alternative expression for . In the welfare analysis we do a second-

order approximation. Thus, while we earlier found that up to a first-order, this result can

no longer be used. The following second-order approximation of (

)

will be useful, where

is approximated around zero:

(

)

( ) ( )

( )

( )

( )

From (∫

)

, we have that (∫ (

)

)

. Thus, when taking

expectations on both sides of the above, where denotes the expectations operator with respect

to good , we get:

(

)

[ ( )

( )

]

⇒ ( )

( )


46

⇒

(8.4)

The price dispersion is denoted . Next, let us do a second-order approximation of

(

)

in :

(

)

(

)

⇒ (

)

(

)

(8.5)

Finally, insert (8.4) and (8.5) into (8.3) to get the following second-order approximation of :

( ) [∫(

)

] ( ) {∫ [

(

)

]

}

( ) {∫

∫

(

)

∫

}

( ) [

(

)

]

( ) [

(

)

]

( ) { [ ( )

( )

(

)

] }

( ) [ ( )( )

( )

]

( ) [

( )

]

( ) [

( )

]

( ) [

( )

]

( ) [

( )

] ( ) [

( )

]

( )

( )

⇒

(8.6)

As before, we have defined

. The next step is to insert (8.2) and (8.6) into (8.1),

rearrange and also get rid of non-policy terms whenever possible:


47

(

)

{

( )

[

( )]

}

{

(

)

[

(

)]

}

[

(

)

(

)

]

⇒

[

( )( )

] (8.7)

The notation is shorthand for terms independent of policy. To proceed we must find a

way to get rid of

on the RHS of (8.7). From (7.5) we see that the undistorted steady state

equilibrium implies:

(8.8)

Using the specified production function, ( ) ( )

( )

, and

the above can be rewritten to:

( )

(8.9)

Insert (8.9) into (8.8):


48

( )

[

( )( )

]

[

( )( )

]

( )( )

[

( )

( )

]

[

( )

(

)]

[

( )( )

]

[

]

[

( )

]

[

(

)

] (8.10)

From (4.18) we have:

( )

⇒

( )

( )

⇒ ( )

(8.11)

Furthermore, by the definition of and :

( ) (

) (8.12)

Insert (8.11) and (8.12) into (8.10), and write up a discounted sum of lifetime welfare losses as a

function of output gap from natural output and inflation gap from zero inflation:


49

[

(

)

( )

]

[

(

)

(

)

]

[

(

) (

)]

[

(

) (

)]

{

(

) [(

)

]}

[

(

)

(

)

]

[

(

)

]

⇒ ∑

∑ [

(

)

] (8.13)

The final step consists in rewriting the terms involving price dispersion in (8.13) as a function of

inflation. Note that because a fraction of firms are able to reset their price in period

while the remainding firms are stuck with last periods price, we can rewrite the expected price

for good to:

( )

Rewrite this:

⇒

( )

( ) (8.14)

Before proceeding, we refresh a simple, useful result from basic statistics. Consider a random

variable with expected value or mean equal to [ ]. Then the variance of is given by

( ) [( ) ]. This expression can by expanded as follows:

( ) [( ) ] [ ] [ ] [ ] [ ]

[ ] [ ] ( [ ])

Thus, the variance of is the same as the mean of the square minus the square of the mean.

Now, using this, the price dispersion measure writes as:

[( ) ] ( )

(8.15)

Furthermore, because only an exogenous draw of firms are able to reset their price:

[( ) ] [ ( )

( )( )

] (8.16)

Insert (8.16), and then (8.14), into (8.15). Then simplify:


50

[ ( ) ( )(

) ] ( )

( ) ( ) (

)

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

Iterating backward, and collecting terms for every period , yields:

{ [ (

)

]

}

( ) ∑

∑

Thus, if one takes the discounted value of these terms over all periods:

∑ ∑

( )( )∑

(8.17)

Now we can insert (8.17) into (8.13):

∑

[∑

∑ (

)

]

[

∑

∑ (

)

]

[

( )( )∑

∑ (

)

]

∑ [

( )( )

(

)

]

⇒ ∑

∑ [

(

)

] (8.18)

The parameter is defined as ( )( )

( )( )

, just as it was in equation

(4.15). Finally, we are ready to establish a quadratic welfare loss function which consists only

of those terms in (8.18) that are relevant to monetary policy:

∑ [

(

)

] (8.19)


51

Welfare losses are expressed in terms of the equivalent permanent consumption decline,

measured as a fraction of steady state consumption. The average welfare loss per period is thus

given by a linear combination of the variances of output gap and inflation:

[

( ) (

) ( )] (8.20)

From (8.19) and (8.20) we see that the relative weight of output gap fluctuations in the loss

function is increasing in , and . The reason is that larger values of those curvature

parameters amplify the effect of any given deviation of output from its natural level on the size of

the gap between the marginal rate of substitution and the marginal product of labor, which is a

measure of the economy’s aggregate inefficiency. On the other hand, the weight of inflation

fluctuations is increasing in the elasticity of substitution among goods, , and the degree of price

stickiness, . The former amplifies the welfare losses caused by any given price dispersion, the

latter amplifies the degree of price dispersion resulting from any given deviation from zero

inflation. The optimal monetary policy in the case considered here achieves, as we saw earlier, the

flexible price equilibrium with zero output gap and inflation. This is referred to in the literature as

a divine coincident. However, such an allocation is rarely seen in practice. Most of the time,

monetary authorities face a real tradeoff between stabilizing inflation and stabilizing the output

gap. Typically, reducing inflation comes at the cost of a negative output gap, given an initial

equilibrium allocation. This observation gives us a motivation for the introduction of cost push

shocks.

8.3 Introduction of cost push shocks

I will now consider a world in which the central bank has to consider policy tradeoffs between

minimizing output gap and minimizing inflation. When nominal rigidities coexist with real

imperfections, the flexible price equilibrium is generally inefficient. In that case, there is no longer

optimal for the central bank to seek that allocation. On the other hand, any deviation of

economic activity from its natural, flexible price level generates variations in inflation, with

consequent relative price distortions. Now we assume existence of some real imperfections that

generate a time-varying gap between output and its efficient counterpart, even in the absence of

price rigidities. The resulting monetary policy under this environment is referred to as flexible

inflation targeting. We will first look at two alternative ways to include real imperfections in the

New Keynesian model. One is through variations in desired price markups. Assume that the

elasticity of substitution among goods varies over time according to some stationary stochastic

process . Let the associated desired markup for firms be

. Then, the representative

firm’s optimal price setting, previously given by (3.11), becomes:


52

( ) ∑ ( |

)

⇒ ( ) ∑ ( |

) (8.21)

The only difference between (3.11) and (8.21) is that optimal prices in the former was determined

by | |

| instead of |

|

. The

resulting inflation equation previously given by (4.15) rewrites to:

(

) (

)

⇒ (

) (8.22)

Let us denote as the efficient equilibrium level of output under flexible prices and a constant

markup , whereas denotes the equilibrium under flexible prices and a time-varying markup

.23 Then, whereas (4.17) describes the real marginal cost in equilibrium with price flexibility,

derived directly from (4.16), the efficient marginal cost version of (4.16) now becomes:

( )

( ) (8.23)

Thus, whereas the real marginal cost gap previously was given by (4.19), it is now given by:

[ ( )

( )]

[ ( )

( )]

( )

(

)

⇒

( )

(8.24)

We define as the output gap from efficient output. Furthermore:

[ ( )

( )]

[ ( )

( )]

⇒

( )

(

) (8.25)

Insert (8.24) and (8.25) into (8.22) to yield the following structural equation for inflation:

( )

( )

(

)

23 We still denote

as the equilibrium output with flexible prices, but this output level is now associated with a

time-varying price markup

instead of

as in (3.9). If the labor market subsidy previously discussed is in

place, then

.


53

⇒ (8.26)

As long as ( ( ) )

(

) (

) , then it is impossible to attain

simultaneously zero inflation and an efficient level of activity. Thus, the disturbance , which is

exogenous to monetary policy, generates a tradeoff for monetary authorities. Refer to as a cost

push shock and assume it follows an AR(1) process:

(8.27)

The efficient level of output, which previously was given by (4.21), now becomes:

(

) (8.28)

Solve for the efficient real interest rate , the rate that supports the efficient allocation and is

invariant to monetary policy:

(8.29)

Subtract (8.28) from (4.4):

[

( )] [

(

)]

(

) (8.30)

Equations (8.26) and (8.30) constitute the New Keynesian Phillips curve and the dynamic IS

equation in this setting. The next step is to obtain a welfare loss function for this case such as the

one in (8.19).

8.4 A welfare loss function when real rigidities are present

At this point it is useful to distinguish between an environment with an efficient steady state and

an environment with a distorted steady state. The former is the special case in which the

inefficiencies associated with the flexible price equilibrium do not affect the steady state, which

remains efficient. Then by definition, and the flexible price steady state does not involve

any markup for firms. One way to obtain such a steady state is to implement the employment

subsidy introduced in (7.12). The latter environment is the arguably less restrictive. Here, a steady

state distortion generates a permanent gap between the actual and efficient levels of output. Let

us first look at the case where the steady state is distortive, i.e. where . The situation with

an efficient steady state follows immediately afterwards because the distortive steady state

outcome nests the efficient steady state. In the case of a steady state distortion, we introduce a

parameter which represents the wedge between the marginal product of labor and the marginal

rate of substitution between consumption and hours, both evaluated at the steady state:

( ) (8.31)


54

One example of such a steady state distortion is firms’ market power when the fiscal policy given

by (7.12) is absent. Comparing (7.11) to (8.31), this distortion would be given by:

( )

⇒

Using that the production function implies ( ) ( )

( )

when markets clear, (8.31) can be rewritten to:

( ) ( )

( ) (8.32)

Even though the economy now is subject to real rigidities, the derivation given in (8.1)-(8.27)

remains unchanged because it does not involve natural or efficient output, only actual output.

The point of departure is therefore (8.7). Insert (8.32) into (8.7):

( )

( )

[

( )( )

]

( ) [

( )( )

]

⇒

( ) [

( )( )

] (8.33)

We proceed by assuming that the steady state distortion is of the same order of magnitude as

fluctuations in the output gap and inflation. This implies that the product of and a second-

order term can be ignored as negligible. Thus, we can rewrite (8.33) to:


55

( ) [

( )( )

]

( )( )

[

( )

( )

]

[

( )

(

)]

[

( )( )

]

[

]

[

( )

]

⇒

[

(

)

] (8.34)

The efficient level of output as a function of the technology level is derived in the same way as

we did with natural output. Instead of (8.11), (8.16)-(8.18) now leads to:

( )

(8.35)

Furthermore, by the definition of and :

( ) (

) ( ) ( ) (8.36)

Insert (8.35) into (8.34). Then use (8.36). Solve out:


56

[

(

)

]

[ ]

[

(

)

( )

]

[

(

)

(

)

]

[

(

) (

)]

[

(

) (

)]

{

(

) [(

) ]}

[

(

)

(

)

]

[

(

)

]

We then take the sum over all discounted time periods:

∑

∑ {

[

(

)

]} (8.37)

Finally we split the terms and make use of the result in (8.17), which is valid even though real

rigidities are present:

∑

∑ {

[

(

)

]}

∑

[

∑

∑ (

)

]

∑

[

( )( )∑

∑ (

)

]

∑ {

[

( )( )

(

)

]}

⇒ ∑

∑ {

[

(

)

]} (8.38)

As before, ( )( )

. The last step is to normalize (8.38) (by multiplying both sides by

) and to establish a quadratic welfare loss function which consists only of those terms that

are relevant to monetary policy:


57

∑ {

[

(

)

]}

⇒ ∑ {

[

(

)

]}

⇒ ∑ {

[

]}

⇒ ∑ {

[

] }

(8.39)

Here, (

) and

as earlier and

. The period losses write as:

[

] (8.40)

Let us finally look at the case where the steady state is efficient, i.e. where and . In

this case (8.8) still holds, and therefore also (8.9). Thus, also (8.10) holds. However, the

definitions of and now implies that:

( ) ( )

(8.41)

Thus, (8.39) collapses to:

∑ (

)

(8.42)

The period losses follows directly from (8.42):

(

) (8.43)

Equations (8.39) and (8.40) also nest the case without real rigidities. That is, when the steady state

is efficient ( and ) and cost push shocks are absent (

), then:

It follows immediately that (8.39) and (8.40) then collapse to (8.19) and (8.20), respectively. Now

that we have derived the relevant welfare loss functions under different environment

assumptions, it is time to look for the optimal monetary policy.


58

9. Welfare based evaluation of monetary policy

9.1 Introduction

In this section I characterize the optimal monetary policy given the second-order approximation

to welfare losses derived above. In the first case. i.e. the case with an efficient steady state without

real rigidities, it is straightforward to show that the optimal policy is to opt for zero inflation and

zero output gap in all periods. This is done by responding sufficiently aggressively to any price

change in order to keep zero inflation. The result should come as no surprise given the analysis

conducted earlier, where it was shown that a policy that seeks to replicate the flexible price

allocation is both feasible and optimal. With zero inflation output equals its natural level, which

in turn, under the assumptions made, is also the efficient level. Thus, under that environment, the

central bank does not face a meaningful policy tradeoff and strict inflation targeting emerges as

the optimal policy. In cases with cost push shocks however, we will see that there is a short run

tradeoff between zero inflation and zero output gap. In this environment the central bank should

allow for only partial accommodation of inflationary pressures in order to avoid too large

instability of output and employment. This kind of policy is often referred to as flexible inflation

targeting. One critical point is whether the agents in the economy believe that the central bank

will commit to its stated policy or whether they think the central bank will deviate in order to

achieve short run gains. If agents trust the policy makers’ statements, we typically refer to the

policy as policy under commitment. If not, we typically refer to the policy as discretionary policy.

This section studies the cases with an efficient steady state and a distorted steady state, both

under full commitment and under full discretion.

9.2 An efficient steady state under discretion

The approximated welfare loss function in the case of an efficient steady state is derived in (8.42).

Monetary authorities want to maximize (8.42) subject to (8.26) and (8.30). The problem reads as:

{

∑ (

)

}

s.t. (9.1)

(

)

The forward-looking nature of the constraints implies that one must specify to which extent the

central bank can credibly commit in advance to future policies. With full commitment, the central

bank can credibly manipulate private sector’s beliefs, and therefore commit to a policy that


59

influences private sector’s expectations about the future. Full discretion on the other hand,

implies that the central bank cannot credibly manipulate private sector’s beliefs, and therefore

takes expectations as given. In the latter case, the problem becomes a period by period problem

in time variables only. First I show the optimality condition under full discretion. The

Lagrangian is given by:

(

) (

(

))

( )

FOC:

:

(9.2)

: (9.3)

: (9.4)

From (9.2) we get . From (9.4) we get . Insert these two into (9.3):

⇒

(9.5)

Equation (9.5) states the optimal combination of output gap and inflation in a discretionary

setting. In the face of inflationary pressures resulting from a cost push shock the central bank

responds by driving output below its efficient level, thus creating a negative output gap, with the

objective of dampening the rise in inflation. This policy goes on up to the point where (9.5) is

satisfied. To derive an expression for the equilibrium inflation under discretionary policy, first

insert (9.5) into (8.26):

⇒

⇒

Iterate forward using (8.27):


60

{

[

(

)

]

}

{

[

(

)

]

}

∑(

)

( )

⇒ (9.6)

Insert (9.6) into (9.5) to get an analogous expression for the output gap:

(9.7)

Thus, under the optimal discretionary policy, the central bank lets the output gap and inflation

deviate from their targets in proportion to the current value of the cost push shock. One might

think that (9.6) and (9.7) could be inserted into (8.30) directly to derive a monetary policy rule.

Let us do that:

(

)

⇒ [ ( ) ]

⇒ (9.8)

Combine the policy rule (9.8) with (8.26) and (8.30) to get a forward looking system. Equation

(9.8) into (8.30):

(

)

This expression into (8.26):

(

)

(

) (

)

The system representation becomes:

[

] [

] [

] [

] [

]

where (9.10)


61

[

]

[

]

Note that in (9.10) is identical to in (7.18). Thus, a policy as the one in (9.8) yields a

multiplicity of solutions, only one of which corresponds to the desired outcome given by (9.6)

and (9.7). However, one can always derive a rule that guarantees equilibrium uniqueness. The rule

can be derived by adding a term proportional to any deviation in (9.6) into (9.8). Let us see how

this works:

( )

( )

{ [ ( ) ] }

[ ( ) ( )]

⇒ (9.11)

Insert (9.11) and (8.26) into (8.30):

(

)

[ ( ) ]

⇒

⇒

This expression into (8.30):

(

)

(

( )

) (

( )

)

The system representation becomes:


62

[

]

[

] [

]

[ ( )

]

[

] [

] [

( )

] [

]

where (9.12)

[

]

[ ( )

]

Note that the transition matrix is a special case of in (5.4) where . Thus, from the

analysis of (5.4) we know that (9.12) has a stable, unique solution as long as . However,

following an interest rate rule like (9.11) is difficult because we do not observe the efficient

output level , at least not in real time.

9.3 An efficient steady state under commitment

Next we consider a central bank that is able to commit, with full credibility, to a policy plan. In

the context of the model, such a plan consists of a specification of the desired levels of inflation

and the output gap at all possible dates and states of nature, current and future. More specifically,

the monetary authority is assumed to choose a state-contingent sequence { } that solves

(9.1). Let us solve the problem:24

∑ [

(

) (

(

))

( )]

FOC:

:

(9.13)

: (9.14)

:

(9.15)

: (9.16)

From (9.13) we get . From (9.15), . From (9.16), . From

(9.14):

24 The law of iterated expectations is used to eliminate the conditional expectation that appeared in each constraint.


63

⇒

( )

(9.17)

Combining and

:

(9.18)

Equations (9.17) and (9.18) constitute the optimal combinations of output gap and inflation with

commitment. It is convenient to combine those two equations into a single expression. First,

define the log deviation between the price level and an implicit target given by the price prevailing

one period before the central bank chooses its optimal plan, as . Then, note that

inflation can be rewritten to:

( ) ( ) (9.19)

This observation together with (9.17) and (9.18) yield:

∑

∑( )

[( ) ( ) ( ) ]

( )

( )

( )

( )

⇒

(9.20)

Equation (9.20) can be viewed as a targeting rule that the central bank must follow period by

period in order to implement the optimal policy under commitment. To find a solution under

commitment, note that relation (9.19) also holds for forward-looking variables. Thus, the New

Keynesian Phillips curve (8.26) can be rewritten, using (9.19) and (9.20), to:

( ) ( )

⇒ ( )

⇒

( )

( )

( )

⇒ (9.21)

We have defined

( ) . Equation (9.21) is a second order difference equation.

Since it involves a forward-looking variable, the stability requirement is that one of the roots is

above unity, and the other is below unity. To find the solution, divide both sides of (9.21) with

and evaluate the result one period backward:


64

(9.22)

A solution should satisfy , and , where is the

root. Thus, the characteristic equation of (9.22) is:

(

)

⇒

The solutions are:

(

) √(

)

√

√

√

√

(9.23)

(

) √(

)

√

(9.24)

One can show that and ( ). To proceed we make use of the lag operator

. First, express the solution to (9.22) with lag operators:

( )( )

(9.25)

Second, define ( ) . Then shift (9.25) one period forward and solve for :

( )

⇒

⇒

Iterate forward:

[

(

)

]

∑(

)

∑(

)

∑(

)

( ) (9.26)

Then insert for ( ) and solve for :

( )

( )

Using (9.23)-(9.24) we get:


65

( )

√

√

( √ )( √ )

( )

⇒

(9.27)

Finally, insert (9.27) into (9.20) to get the output gap solution:

(

)

( )

⇒

( ) (9.28)

The response of a shock at is:

( )

The key difference between cost push shock responses under discretion and under commitment,

that is between the solutions (9.6)-(9.7) and (9.27)-(9.28), is that output gap and inflation are only

determined by current shocks in the former, while lagged variables are relevant in the latter in

addition to the shocks. Thus, under discretionary monetary policy the effect of a shock dies out

once the shock is gone, while the effect will persist even in succeeding periods in the case with

commitment. To see how this occurs, one can iterate (8.26) forward:

∑ (9.29)

We see from (9.29) that the central bank can offset the inflationary impact of a cost push shock

by lowering the current output gap, but also by committing to lower future output gaps. If

credible, such promises will bring about a downward adjustment in the sequence of expectations

{ } for . As a result, and in response to a positive cost push shock, the central

bank may achieve any given level of current inflation with a smaller decline in the current

output gap . That is in the sense in which the output gap and inflation tradeoff is improved by

the possibility of commitment. Although this strategy comes at the cost of worse tradeoff in

succeeding periods, it is still better from a welfare perspective because of the convexity in the loss

function with respect to output gap and inflation. A feature of the economy’s response under

discretionary policy is the attempt to stabilize the output gap in the medium term more than the

optimal policy under commitment calls for, without internalizing the benefits in terms of short


66

term stability that result from allowing larger deviations of the output gap at future horizons. This

characteristic is often referred to as the stabilization bias associated with the discretionary policy.

9.4 A distorted steady state under discretion

Next we characterize the optimal policy when the steady state is distorted, i.e. when (8.31) is in

place. In that case, (8.26) and (8.30) become:

(

) (9.30)

(9.31)

The cost push shock term is now given by (

). The problem is then to optimize

(8.39) subject to (9.30) and (9.31):

{ ∑ [

(

) ]

}

s.t. (9.32)

(

)

Under full discretion, the Lagrangian becomes:

[

(

) ] [

(

)]

( )

FOC:

:

(9.33)

: (9.34)

: (9.35)

From (9.33) we get . From (9.35) we get . Insert these two into (9.34):

⇒

(9.36)

Equation (9.36) states the optimal combination of output gap and inflation in this setting. In the

face of inflationary pressures resulting from a cost push shock the central bank responds by

driving output below its efficient level, thus creating a negative output gap, with the objective of

dampening the rise in inflation. This policy goes on up to the point where (9.36) is satisfied. Note

that (9.36) is similar to (9.5) except for a positive constant term. Thus, for any given level of

inflation, the policy is more expansionary than that given in the absence of a steady state


67

distortion. To derive an expression for the equilibrium inflation under discretionary policy, first

insert (9.36) into (9.31):

⇒

⇒

Iterate forward:

{

[

(

)

]

}

{

[

(

)

]

}

∑(

)

∑(

)

( )

( )

⇒

( ) (9.37)

Insert (9.37) into (9.36) to get an analogous expression for the output gap:

(

( ) )

[

( )]

( )

( )

( )

⇒ ( )

( ) (9.38)


68

Thus, under the optimal discretionary policy, the presence of a steady state distortion does not

affect the response of the output gap and inflation to shocks. However, the steady state

distortion has an effect on the average levels of inflation and the output gap in which the

economy fluctuates. In particular, when the natural level of output and employment are

inefficiently low, i.e. when , the optimal discretionary policy leads to positive average

inflation as a consequence of the central bank’s incentive to push output above its natural steady

state level. That incentive increases with the inefficiency of the natural steady state, which

explains the fact that the average inflation is increasing in , giving rise to the inflation bias

phenomenon.

9.5 A distorted steady state under commitment

TBA


69

10. Wage rigidities

10.1 Introduction

This section expands the baseline model with monopoly supply power and sticky wages in the

labor market. A continuum of differentiated labor services is assumed, all of which are used by

each firm. Each household specialize in one type of labor, which it supplies monopolistically.

Wages are sticky in an analogous way to goods prices. Each period only a constant fraction of

households can adjust their posted nominal wage. As a result, the aggregate nominal wage

responds sluggishly to shocks, generating inefficient variations in the wage markup.

10.2 Firms

We saw earlier that households face two maximization problems. First, for any given level of

consumption expenditures, they have to find the consumption vector that maximizes total

consumption. Second, and given this optimal consumption vector, they have to find the optimal

combination of consumption and labor. Now that households have monopoly power in the labor

market, firms face a two-dimensional problem as well. First, for any given level of labor costs,

they have to find the output maximizing combination of labor. Second, and given this optimal

labor vector, firms have to set prices such that profit is maximized. Firms’ output is still given by

(3.1). However, total labor used by firm , where is the quantity of type -labor

employed, is now defined by:

(∫

)

(10.1)

The elasticity of substitution among labor types is denoted . Specification (10.1) implies that

labor is an imperfect substitute as long as . Note that the assumption of individual

worker types implies an indexation [ ]. Let denote the nominal wage for type -worker

in period . In an analogous way to the optimal consumption vector derived in (2.10), firm ’s

demand for labor is given by:

(

)

(10.2)

This holds for all [ ], where:

(∫

)

(10.3)

In exactly same manner as we derived the aggregation result for consumption expenditures, one

can derive the following aggregation result for firms’ labor expenditures:

∫

(10.4)


70

Denote as the price stickiness parameter in the goods market and as the elasticity of

substitution between consumption goods. Then, the profit maximization problem given in (3.6)

writes as:

{∑

[ (

)

(

(

)

| ((

)

))] }

(10.5)

As shown in (4.15), the aggregation of the resulting price setting rules yields, to a first order

approximation and in a neighborhood of the zero inflation steady state, the following equation

for price inflation :

(10.6)

The notations are

and

( )( )

. From (10.6) we see

that when the average price markup is below its steady state value, firms that are adjusting prices

set them higher, thus generating positive inflation.

10.3 Households

Assume a continuum of households indexed by [ ]. As before, given a sequence of budget

constraints household seeks to maximize lifetime utility:25

{∑ ( ) } (10.7)

The consumption index is now given by:

(∫

)

(10.8)

We now assume that households specialize in different types of labor, and therefore face some

monopoly power in the labor market given by . They post the nominal wage at which

they are willing to supply labor services to firms who demand them, and, because of monopoly

power, this wage contains a markup. However, households also face a constraint with respect to

the frequency in which they can change wages. A constant fraction of the households is stuck

with the wage they had last period, while the remaining households can reoptimize the

price of their labor services. Thus, a household who reset its wage in period will choose in

order to maximize:

{∑ ( ) ( | | ) } (10.9)

25 We now depart from the money in utility specification we had in (2.1) to keep the analysis as simple as possible.


71

| and | denote consumption and labor at time for a household that last set the

wage at time . Because of the wage rigidity, (10.9) can be interpreted as the expected discounted

sum of utilities generated over the uncertain period during which the wage remains unchanged at

the level set the current period. Note that the utility generated under any other wage set in

the future is irrelevant from the point of view of the optimal setting of the current wage, and thus

can be ignored in (10.9). Maximization of (10.9) is subject to a sequence of labor demand

schedules and flow budget constraints that are effective while remains in place:

| (

)

(10.10)

and

∫ |

| | ∫

|

⇒ | | | | (10.11)

Equation (10.10) follows directly from (10.2), and | is the amount of labor done in time

for a household that last set its price in period . ∫

denotes the aggregate

employment at time . The integrals in (10.11) are removed by (2.9) and (10.4), respectively.

| is the market value in period of the portfolio of securities held at the beginning of

that period by households that last reoptimized in period . Thus, | is the

corresponding market value as of period of the portfolio purchased in that period, which

yields a random payoff | . Given the concavity of the utility function (10.11) holds with

equality. Furthermore, it can be solved for | :

|

( |

| | ) (10.12)

Insert (10.12), and then (10.10) into (10.9) to get an unconstrained problem:


72

{∑( ) ( | | )

}

{∑( ) ([

( |

|

| )] [(

)

])}

{∑( ) ([

( |

(

)

| )] [(

)

])}

Thus, the household’s maximization problem when it comes to wage setting is described as:

{∑( ) ([

( |

(

)

| )] [(

)

])}

(10.13)

Let us solve the problem. FOC:

{∑( ) [ ( | | )

( ) (

)

( | | ) (

)

]}

{∑( ) [ ( | | )

( ) (

)

( | | )

(

)

]}

{∑( ) [ ( | | )

( ) |

( | | )

| ]}

Multiply both sides by

:


73

{∑ ( ) [ ( | | )

|

( | | ) | ]

}

Define | ( | | )

( | | ) as the marginal rate of substitution between consumption

and hours in period for a household resetting the wage in period . Then, the equation

above can be rewritten to:

{∑ ( ) [ | ( | | ) (

| )]

} (10.14)

To solve for , first move the terms with | over to the other side and insert for

(10.10):

{∑( ) (

)

( | | )

}

{∑( ) [(

)

( | | ) | ]

}

Then take out the optimal wage of the expectation terms on both sides and solve for :

{∑( )

( | | )

}

{∑( ) [

( | | ) | ]

}

⇒

∑ ( ) ( | | ) |

∑ ( ) ( | | )

(10.15)

Notice that in the case with flexible wages, i.e. when , (10.13) becomes a one period

problem and (10.15) becomes:

( | | ) |

( | | )

|

| (10.16)

Thus,

is the wedge between the real wage and the marginal rate of substitution that prevails

in the absence of wage rigidities, i.e. the desired gross wage markup. Note also that in a perfect

foresight zero inflation steady state we have:

(10.17)

In a similar manner as we log-linearized the optimal price equation (3.7) around steady state, we

now log-linearize the optimal wage equation (10.15). First, it is convenient to rewrite it to:


74

∑( )

( | | )

∑( )

( | | ) |

(10.18)

A first-order Taylor expansion of the LHS of (10.18):

∑( ) ( )

∑( ) ( )

( )

∑( ) ( )

( )

∑( ) ( )

( | )

∑( ) ( )

( | )

∑( ) ( )

( )

∑( ) ( )

( )

Rewritten in terms of log deviations:


75

∑( ) ( )

∑( ) ( )

( )

∑( ) ( )

( )

∑( ) ( )

( | )

∑( ) ( )

( | )

∑( ) ( )

( )

∑( ) ( )

( )

( )

∑( )

[ ( ) ( )

( )

( ) ( | )

( )

( ) ( | ) ( )

( )]

( )

∑( )

[ ( ) ( )

( )

( ) ( | )

( )

( ) ( | ) ( )

( )]

The last equality follows from (10.17). A first-order Taylor expansion of the RHS of (10.18):


76

∑( ) ( )

∑( ) ( ) ( )

∑( ) ( ) ( )

∑( ) ( ) ( | )

∑( ) ( ) ( | )

∑( ) ( )( | )

( )

∑( )

[ ( ) ( )

( )

( ) ( | )

( )

( ) ( | ) ( | )]

Finally we equate LHS with RHS and solve for :

( )

∑( )

[ ( ) ( )

( )

( ) ( | )

( )

( ) ( | ) ( )

( )]

( )

∑( )

[ ( ) ( )

( )

( ) ( | )

( )

( ) ( | ) ( | )]

⇒ ∑ ( ) [(

) ( )] ∑ ( ) ( | )

⇒ ∑ ( ) [

( )] ∑ ( ) ( | )

⇒

( )

∑ ( )

( | )

⇒ ( ) ( ) ∑ ( )

( | )


77

⇒ ( ) ∑ ( )

( | ) (10.19)

We have defined ( )

. The intuition behind the wage setting rule is

straight forward. First, is increasing in expected future prices because households care about

the purchasing power of their nominal wage. Second, is increasing in the expected average

marginal disutilities of labor in terms of goods over the life of the wage, because households want

to adjust their expected average real wage accordingly, given expected future prices. Without

money in the utility equation (2.32) becomes:

( )

(10.20)

The assumed separability between consumption and hours, combined with the assumption of

complete asset markets, implies that consumption is independent of the wage history of a

household, i.e. | . Thus, from (10.20) we get:

| ( | | )

( | | )

|

⇒ | |

Also, from (10.10) we get:

| (

)

⇒ | ( )

⇒ | ( )

Let define the economy’s average marginal rate of substitution, so

that:

| ( | ) ( ) (10.21)

Insert (10.21) into (10.19) and solve for , using that

( ) is the economy’s

average wage markup:

( ) ∑( )

[ ( ) ]

( ) ∑( )

[ ]

⇒ ( ) ( ) ∑ ( )

[ ]

⇒

∑ ( )

[ ]

⇒

∑ ( )

[ ( ) ]


78

⇒

∑ ( )

[( ) ]

⇒

∑ ( )

[( ) ] ( ) (

)

⇒

( ) (

) (10.22)

We have denoted

as the deviation of the economy’s log average markup from its

steady state level. The next step is to derive the wage inflation equation. Given the wage setting

structure, the evolution of the aggregate wage index is given by:

[ ( )

]

(10.23)

A first order Taylor approximation around zero wage inflation steady state, followed by logging

the resulting equation, yields:

[ ( ) ]

( ) ( )

[ ( ) ]

( )( ) (

) ( ) ( )( )

( ) ( ) ( )

⇒

( ) ( )(

)

⇒

( ) ( )(

) ( )

⇒ ( ) (10.24)

From (10.24) we see that wage inflation is given by:

( ) ( )

( )( ) (10.25)

Finally, insert (10.22) into (10.24) and use (10.25):

( ) [ ( ) (

)]

( ) [ ( )

]

⇒ ( ) ( ) [ ( )

]

⇒ ( ) ( )

( )( )

( )

⇒

( )( )

( )

⇒

(10.26)


79

We have defined ( )( )

( ). Notice that the wage inflation equation (10.26) has a form

analogous to the gods price inflation equation (10.6). The intuition is also the same. When the

average wage in the economy is below the level consistent with maintaining (on average) the

desired markup, households readjusting their nominal wage will tend to increase the latter, thus

generating positive wage inflation. Here, (10.26) replaces condition , one of the

optimality conditions associated with the household’s problem used earlier. The imperfect

adjustment of nominal wages will generally drive a wedge between the real wage and the marginal

rate of substitution for each household, and as a result, between the average real wage and the

average marginal rate of substitution. This leads to variation in the average wage markup, and

given (10.26), also to variation in wage inflation. The last dimension worth considering with

respect to households is the Euler equation. It becomes the similar as before because |

, but because hours worked now depends on when the wage last was set, (2.18) writes as:

( ) { ( | )

( | )

}

However, given the specification (10.20), log-linearization yields:

(

) (10.27)

This expression is the exact same as before, i.e. identical to (2.36). Thus, the Euler equation does

not depend on wage rigidities in this setting.

10.4 Inflation equations and the Dynamic IS equation

I will now derive the price inflation equation, the wage inflation equation and the output gap

equation. Start with wage inflation. Let be the output gap from the level of natural

output, with the latter now being defined as the equilibrium level of output in absence of both

price and wage rigidities. Define also the real wage as , which implies that the real

wage gap is:

The natural real wage, denoted , is the real wage that would prevail in the absence of nominal

rigidities. To get an expression for , note first that the production function (3.1) implies that

output, employment and marginal productivity in their linearized versions are given by:

( ) (10.28)

⇒

(10.29)

( ) (10.30)


80

Use the fact that the natural real marginal cost is given by the identity –

and

insert (10.28)-(10.30). Solve for :

–

[ ( )

]

[ ( ) ( )

]

( ) (

)

( ) (

)

( )

( )

( )

(

)

( )

⇒ ( )

–

⇒ ( )

– (10.31)

We have defined

and

. Note also that the change in the

natural real wage is given by:

(10.32)

Equations (10.28)-(10.30) will also be used to derive an expression for the real marginal cost gap

from its steady state counterpart, where – ( ) . Insert for

(10.28)-(10.30):

( ) (

)

[ ( ( ) )] [ ( ( )

)]

[ ( ( ) )] [ ( ( )

)]

[ ( )] [ (

)]

( ) (

) ( )

( ) (

) (

)

(

) ( )

⇒

(10.33)

Finally, insert (10.33) into (10.6) to get the New Keynesian Phillips curve:

(

)

⇒

(10.34)


81

Thus, increased output and increased real wage lead to higher price inflation. Next, the wage

inflation equation is derived. Given the utility function (10.20), marginal rate of substitution in its

linearized form is given by:

(10.35)

Thus, given market clearing in the goods market and in the labor market, the log deviation in the

economy’s average wage markup from its steady state counterpart, where the wage markup writes

as ( ) , is given by:

( ) (

)

[ ( )] [ (

)]

[ ( )] [ (

)]

( ) (

) ( )

( ) (

) (

)

( ) (

) (

)

⇒ (

) (10.36)

Insert (10.36) into (10.26) to get the New Keynesian Wage Phillips curve:

[ (

) ]

(

)

⇒

(10.37)

With wage rigidity in addition to price rigidity, there is an identity relating changes in the real

wage gap to wage inflation, price inflation and changes in the natural real wage:

( ) (

)

[( ) ( )]

⇒

(10.38)

The last term in (10.38) is given by (10.32). In order to complete the non-policy block of the

model, equilibrium conditions (10.34), (10.37) and (10.38) must be supplemented with a dynamic

IS equation. Given (10.27) and market clearing in the goods market, aggregate output is given by:

(

) (10.39)

As before we subtract (4.21) and get:

[

(

)] [

(

)]

⇒

(

) (10.40)

The natural real rate and its deviation from the steady state are given as in (4.24) and (4.25):


82

(10.41)

(10.42)

10.5 System representation and equilibrium determinacy

In order to close the model, we must specify how the nominal interest rate is determined. Let us

postulate an interest rate rule of the form:

(10.43)

The five equations (10.34), (10.37), (10.38), (10.40) and (10.43) constitute the New Keynesian

model with sticky prices and sticky wages. To gather them all into a forward looking system with

period variables on the LHS and period variables on the RHS,26 we first insert (10.43)

into (10.40):

(

)

(

)

(

)

⇒

(

)

(

)

⇒ ( )

(

) (10.44)

Equation (10.44) is the first row in the system. Second, we rewrite (10.34):

(10.45)

Equation (10.45) is the second row in the system. Third, we rewrite (10.37):

(10.46)

Equation (10.46) is the third row in the system. Fourth, we rewrite (10.38):

(10.47)

Equation (10.47) is the fourth and last row in the system. Thus, (10.44)-(10.47) can be written as:

[

]

[

]

[

]

[

]

[

] [

]

or

where (10.48)

26 Note that the structure of (10.38) is backward looking, so that the element associated with (10.38) in the LHS

vector of the system will consist of and the RHS element of .


83

[

], [

], [

],

[

] and [

]

In general, (10.48) does not have a solution satisfying

, not even under

the assumption that the intercept of the interest rate rule adjusts one-for-one to variations in the

natural real interest rate, i.e. that . An implication of that is that the allocation associated

with the equilibrium with flexible prices and wages cannot be attained in the presence of nominal

rigidities in both goods and labor markets. The intuition for the previous result rests on the idea

that in order for the constraints on price and wage setting not to be binding all firms and workers

should view their current prices and wages as the desired ones. This makes adjustment

unnecessary and leads to constant aggregate price and wage levels, i.e. zero inflation in both

markets. Note, however, that such an outcome implies a constant real wage, which will generally

be inconsistent with the flexible price and wage level allocation. Only when the natural real wage

is constant, i.e. when , which according to (10.32) requires complete absence of

technology shocks, and at the same time the central bank adjusts the nominal rate one-for-one

with changes in the natural rate, i.e. , the outcome

is a solution

to (10.48). Another question of interest relates to the conditions that the rule (10.43) must satisfy

to guarantee a unique stationary equilibrium or, equivalently, a unique stationary solution to the

system of difference equations (10.48). Given that vector contains three non-predetermined

variables and one predetermined variable, local uniqueness requires that three eigenvalues of

lie inside, and one outside, the unit circle. If , the condition for

uniqueness can be shown, using numerical analysis, to be:

(10.49)

Thus, the central bank must adjust the nominal rate more than one-for-one in response to

variations in any arbitrary weighted average of price and wage inflation. This can be seen as an

extension of the Taylor principle to the case with sticky wages. Furthermore, the region

consistent with a determinate equilibrium in the ( ) parameter space becomes larger as

increases from zero.

10.6 Shocks

I will now consider a monetary policy shock within the framework of both sticky prices and

sticky wages. Parameters are calibrated as follows in the simulation exercise:


84

Table 3

0.99 1 1 1/3 6 6 2/3 3/4 1.5 0 0 0.5 0.0625

= 3/4 implies an average duration of wage spells of

= 4 quarters, consistent with

empirical evidence. Policy parameters and are set to zero as the Taylor rule (10.49) is

satisfied anyway. Simulated impulse responses are shown in the figure 2.27 Notice that the

presence of sticky wages and prices generates a much smaller inflation decline than with only

sticky prices (figure 1). The reason is that when wages are flexible, a monetary policy shock leads

to a large decline in wage inflation. Here, instead, since also wages are sticky, the inflationary

contraction is divided between prices and wages. As a result, real wage does not change much.

This in turn reduces the impact of decline in activity on the real marginal cost and, hence, the

limited size on inflation response. Thus, there is only a moderate endogenous response of

monetary authority to the lower inflation, implying higher interest rates, which in turn account

for the larger decline in output compared to figure 1.

Figure 3: A monetary policy shock

27 Simulations are done with Dynare and Matlab. See Appendix C for the Dynare codes.


85

Overall, the limited responses of price and wage inflation following a monetary policy shock

seem more in line with data than the large responses we found in figure 1. Also, the effect on the

real wage in figure 3 seems much more plausible.

10.7 Monetary policy design with sticky wages

The benevolent social planner seeks to maximize households’ utility, but this maximization is

subject to (10.8), (10.1) and (3.1). Thus, the maximization problem reads as:

{∑ ( )

}

{∑ [(∫

)

(∫

)

] }

s.t. (10.50)

The constraint is the resource constraint coming from all the firms, as in the case with flexible

wages. As before both consumption goods and labor enter the utility function symmetrically.

Thus:

(10.51)

(10.52)

The problem therefore simplifies to the one we had before, and the efficient solution becomes:

(10.53)

Thus, (10.51)-(10.53) are the relevant efficient benchmark equations monetary authorities should

opt for. However, optimal wage and price setting (when monopolistic competition is present in

both markets, but price and wage rigidities are absent) follows from (10.16) and (7.10), and

implies that:

(10.54)

(10.55)

As earlier one can get rid of the distortions caused by market power, and obtain the outcome in

(10.53), by imposing a labor subsidy financed by lump-sum taxes. In this case, the appropriate tax

is given by

, which changes (10.55) to:

( )


86

⇒

( )

(

)

(10.56)

Combining (10.56) and (10.54), we see that (10.53) is obtained, thus guaranteeing the efficiency of

the flexible price and wage equilibrium. Next I derive a second-order approximation to the

average welfare losses experienced by households in the economy with sticky wages and prices.

Point of departure is (8.1) integrated across households:

(

)

⇒ ∫

(∫

∫

) (10.57)

Next, define aggregate employment as:

∫

First, note that in terms of log deviations from a steady state and up to a second-order

approximation:

( )

( )

( )

⇒ ∫

∫

⇒ ∫

∫

(10.58)

Insert (10.58) into (10.57):

∫

(

∫

∫

)

⇒ ∫

(

∫

) (10.59)

In order to proceed, a couple of results are needed. First, a first-order approximation of (10.2)

gives:

(

)

( ) ( )

⇒

⇒

Second, and using the result above:

∫

∫ ( )

∫ ( )

⇒ ∫

∫

∫

(10.60)


87

Third, we have to take care of the term ∫

somehow. From (10.3), that is from

(∫

)

, we have that (∫ (

)

)

∫ (

)

. With

the notation

, a second-order approximation with respect to yields:

∫(

)

∫ ( )( )

∫ [ ( ) ( )

( ) ( )

]

( )∫

( ) ∫

⇒ ∫

∫

Fourth, taking expectations on both sides of the above, where denotes the expectations

operator with respect to labor for firm , we get:

∫

(10.61)

From (10.61) it is clear that ∫

is of second order. Thus, to a first order,

(10.60) can be written:

∫

∫

(10.62)

Finally, insert (10.62) into (10.59):

∫

[

(

)]

⇒ ∫

(

) (10.63)

The next step is to derive a relationship between aggregate employment and output. Using (10.2),

then (3.1), and then (2.10):


88

∫(∫

)

∫ (∫

)

∫ [∫ (

)

]

[∫(

)

]∫

[∫(

)

]∫ (

)

[∫(

)

] [∫(

)

] (

)

[∫(

)

] [∫(

)

] (

)

⇒ (

)

(10.64)

We have defined ∫ (

)

and ∫ (

)

. Taking the log of (10.64), and

then a second-order approximation to the relationship between log aggregate output and log

aggregate employment, we get an expression similar to (8.2):

( )

[( ) ]

⇒

( )

⇒ ( ) (10.65)

where

( ) [∫ (

)

] (10.66)

( ) [∫ (

)

] (10.67)

We derived in (8.3)-(8.6) that (10.67) can be written as:

Now we do a similar derivation for . Let us do a second-order approximation of (

)

in

:

(

)

( ) ( )

( )

Insert this result into (10.66) and use (10.61):


89

( ) [∫(

)

]

( ) ( ∫

∫

)

( ) (

)

( ) (

)

⇒ ( )

(10.68)

Finally, inserting for and into (10.65):

( )

[ ( )

]

⇒

( )

( ) (10.69)

Now we are ready to update the welfare loss function (10.63). Inserting for (10.69) gives:

∫

(

)

{[

( )

( ) ]

[

( )

( ) ]

}

{

( )

( )

( )

[

( )]

}

By introducing the parameter ( )( ) we get the following:


90

∫

[( )

( )( )

]

(10.70)

Throughout, stands for terms independent of policy. As earlier we take a general approach

and allow for a distorted steady state. With this framework we can insert for (8.32) into (10.70):

∫

( )

( )

[( )

( )( )

]

( ) [( )

( )( )

]

Using the assumption about a small steady state distortion, implying that we can neglect second

order moments containing , the derivation becomes similar as the one in (8.34):


91

∫

[

( )

]

[

( )

( )

]

[

( )

(

)]

[

( )

]

[

(

)

]

Note that the expression above is the same as (8.34), except for the additional term with wage

dispersion. Thus, using (8.35) and (8.36), and aggregating over time using the discounting factor

, we get an expression analogous to (8.37):

∑

∑ (∫

)

⇒ ∑

∑ {

[

(

)

]}

(10.71)

As before, ( ) ( )

is the log difference between the

output gap from efficient output and its steady state counterpart. From (8.17) we know that:

∑

( )( )∑

The equivalent derivation to (8.14)-(8.17) for wages gives us:

∑

( )( )∑

(10.72)

Using (8.17) and (10.72), (10.71) can be rewritten to:


92

∑

∑ [

(

)

]

∑

∑

∑ [

(

)

]

( )( )∑

( )( )∑

∑ [

(

)

]

∑

( )

∑

⇒ ∑

∑ {

[

( )

(

)

]}

(10.73)

Finally, the welfare loss function follows from (10.73) as:

∑ {

[

( )

(

)

] } (10.74)

The corresponding period utility losses becomes:

[

( )

(

)

] (10.75)

Note that in the particular case of an efficient steady state, and . Then the welfare

loss function and the period losses writes as:

∑ [

( )

(

)

] (10.76)

[

( )

(

)

] (10.77)

Moreover, if the subsidy suggested in (10.56) is in place, then

and where

. In that case, we get:

∑ [

( )

(

)

] (10.78)

[

( )

(

)

] (10.79)


93

11. A small, open economy model

11.1 Introduction

Existence of more than one economy makes the basic New Keynesian model more complex

because of a significant amount of new notation. One must take into account how foreign

economies affect domestic behavior. In particular, the modeler has to decide whether the

economy is large or small, the nature of international asset markets, whether they are autarkies or

complete markets, the existence of discrimination between domestic and foreign markets,

tradable versus non-tradable goods, trading costs, international policy coordination, and exchange

rate regimes. Here we model small economies, complete asset markets, existence of

discrimination between domestic and foreign goods, even though all goods can be traded

internationally, and a world economy without international policy coordination. The world

economy consists of a continuum of small open economies represented by the unit interval.

Since each economy is of measure zero, its performance does not have any impact on the rest of

the world. Different economies are subject to imperfectly correlated productivity shocks, but it is

assumed that they share identical preferences, technology and market structure. Firms are

identical across countries and have the simplest Cobb-Douglas production function with

constant returns to scale.

11.2 Households

A typical small open economy household is inhabited by a representative household who seeks to

maximize:

∑ ( ) (11.1)

The variable is a composite consumption index determined by both home and foreign goods.

Because of the complexity of the model it is convenient to set up a graphical representation of

the goods structure:

Figure 4: The consumption goods structure in an open economy

Aggregate

consumption:

Home goods:

Parameters: ,

Domestic good :

Parameter:

Imported goods:

Parameters: ,

Foreign good from

country :

Parameter:

Imported goods from

country :

Parameter:

Continuum of foreign countries

Continuum of domestic goods

Continuum of foreign goods


94

Figure 4 illustrates the goods structure which we shall specify below. The composite

consumption index is defined by:28

[( )

]

(11.2)

The parameter measures the degree of openness in the economy. Equivalently,

measures the home bias. The closer is to one, the more open is the economy. If we

get the closed economy case described earlier. Trade restrictions imposed by governments,

geographical barriers such as distance and mountainous terrain, etc, is assumed to be reflected in

. The substitutability between domestic and foreign goods from the viewpoint of domestic

consumers is denoted . is an index of consumption of domestic goods, given by the CES-

function (constant elasticity of substitution):

(∫

)

(11.3)

The parameter is interpreted as before, that is as the elasticity of substitution between varieties

produced within any given country , including the home country. is an index of imported

goods given by the CES-function:

(∫

)

(11.4)

Here, denotes the elasticity of substitution between importing countries. Finally, is an

index of the different goods imported from country , given by the CES-function:

(∫

)

(11.5)

Note that the nested system of equations (11.1)-(11.5) characterizes preferences of a

representative household. Maximization of (11.1) is subject to a sequence of budget constraints:

∫

∫ ∫

(11.6)

The domestic price on good is denoted while the price on good imported from country

is denoted . is the nominal payoff in period from a portfolio held at the end of

period . is the stochastic discount factor for one-period ahead nominal payoffs of the

28 In the particular case of , the consumption index is given by

( )

.


95

domestic household. As before the optimization problem can be dealt with in several stages.

First, for any given level of consumption expenditures on home goods, the household must

decide how much to buy of each. The utility maximizing combination of is the solution,

which determines all the elements in . An equivalent decision has to be made about imported

goods from each of the foreign countries. For instance, for any given level of consumption

expenditures on imported goods from country , the household must decide how much to

consume of each import good from that country. This determines the optimal combination of

, i.e. all the elements in . Second, for any given level of consumption expenditures on

imported goods, the household must decide how much to import from each foreign country.

This decision determines the optimal combination of , i.e. all the elements in . Third,

for any given level of total consumption expenditures, the household must decide how much to

consume of home goods relative to imported goods. This decision determines and .

Finally, the household must decide how much to consume and how much to work. This decision

determines . As before, we get an aggregate price index and optimal demand for every specific

consumption unit at each stage in the nested system. For home goods, the aggregate price index

is given by:

(∫

)

(11.7)

The price index (11.7) follows from the CES-aggregator (11.3) in exactly the same way as (2.7)

follows from (2.3). The optimal demand for home good is:

(

)

(11.8)

In a similar vein, the aggregate price index for imported goods from country is given by:

(∫

)

(11.9)

The optimal consumption of good imported from country is given by:

(

)

(11.10)

The aggregate price index for all imported goods is given by:

(∫

)

(11.11)

Optimal basket of import consumption from country is:

(

)

(11.12)

Finally, the aggregate consumption price index (CPI) in the home country is given by:


96

[( )

]

(11.13)

Optimal consumption of home goods is:

( ) (

)

(11.14)

Optimal consumption of imported goods is:29

(

)

(11.15)

Given the market equilibrium for all these aggregators, the total consumption expenditure is

derived in the same way as we derived (2.9) from (2.8) and (2.3). From the optimality condition

(11.8) and the domestic price index (11.7):

∫

(11.16)

From the optimality condition (11.10) and the import price index from country (11.9):

∫

(11.17)

From the optimality condition (11.12) and the aggregate import price index (11.11):

∫

(11.18)

Finally, from the optimality conditions (11.14) and (11.15), and from the CPI index (11.13):

(11.19)

Thus, the left hand side of the period budget constraint (11.6) can be rewritten as:

∫

∫∫

∫

⇒ (11.20)

Given the optimality conditions (11.8), (11.10), (11.12), (11.14) and (11.15)30, the household must

decide on the allocation of total consumption and labor. Analytically the problem is to maximize

(11.1) subject to (11.20). As before we specify the utility function to be:

( )

(11.21)

To derive the Euler equation, note first that the budget constraint can be rewritten, assuming that

it holds with equality, to:31

29 In the particular case of , the CPI takes the form

.

30 That is the optimal demand of good from domestic production and from country production, the optimal

demand of goods from country , the optimal demand of home goods and imported goods, respectively. 31 This setup closely follows Cochrane (2005:4-5).


97

Here, the left hand side represents consumption expenditures in period , on

the right hand side represents available gross income in period , while represents

the time investment in a portfolio with nominal payoff in period . Thus, the

constraint above tells us that whatever income is left after the portfolio investment, is used for

consumption. The intertemporal problem for the household with respect to the optimal one-

period portfolio purchase writes as:

{ ( ) ( )}

subject to (11.22)

∫

(∫ )

Here, ∫ is the market price of the one-period portfolio yielding a random payoff

, where we integrate over all possible states of nature indexed by . is the period

price of an Arrow security, i.e. a one-period security that yields one unit of domestic currency if a

specific state of nature is realized in period , and zero otherwise. is the probability

that a given state of nature is realized in period . Equivalently, the price can be written as

. Thus, the stochastic discount factor can be defined as

To solve (11.22), insert the constraints into the maximum. Then take the first order conditions

and find the optimal intertemporal allocation:

{ [(

∫

) ]

[(

) ]}

FOC:

(

)

⇒

⇒

(

)

⇒ (

)

Taking conditional expectations on both sides, the Euler equation becomes the same as in (2.18):

{ (

)

} (11.23)


98

The first order condition which summarizes the labor-leisure choice becomes identical to (2.19):

( )

( )

(11.24)

As before we can log-linearize equations (11.23) and (11.24) and get:

(11.25)

( ) (11.26)

11.3 Terms of trade, domestic inflation and CPI inflation

Bilateral terms of trade between the domestic economy and country is defined as the price of

country ’s goods in terms of home goods:

(11.27)

The effective terms of trade are thus given by:

(∫

)

(11.28)

A first order approximation around a symmetric steady state satisfying gives us:

(∫

)

(∫( )

( )

) ∫( )

⇒

∫

∫

(11.29)

Similarly, log-linearization of the CPI (11.13) around the same symmetric steady state where

:

[( )

]

[( )

]

[( )( )

( )

( ) ( )]

[( ) ( )

( )]

[( )( ) ( )]

⇒

( )

⇒ ( ) (11.30)

Domestic inflation is given by:

(11.31)

Thus, using (11.30) CPI inflation is given by:

( ) ( ) (11.32)


99

We see from (11.32) that the gap between domestic inflation and CPI inflation is proportional to

the percentage change in terms of trade, with the coefficient of proportionality given by the

openness index .

11.4 The real exchange rate

The next step is to look at exchange rates. Define as the bilateral nominal exchange rate, i.e.

the price of country ’s currency in terms of domestic currency. Thus, measures how many

domestic currency units one country currency unit is worth. As an example, the bilateral

nominal exchange rate between Norway and US could be

. Define

as the

price of country ’s good in terms of its own currency, for example the price of an iPhone ( ) in

terms of US dollars ( ). Assume that the law of one price holds for individual goods at all times

for both import and export prices.32 Thus, for all goods [ ] in every country [ ]:

(11.33)

Suppose the price of an iPhone ( ) in the US ( ) in terms of US currency is

.

Then, the law of one price implies that the Norwegian price on iPhone in terms of Norwegian

currency is

. Aggregation across all goods using

(11.9) gives:

(∫

)

[∫( )

]

(

∫

)

(∫

)

⇒ (11.34)

Here, (∫

)

is defined as the aggregate price level in country in terms of

country currency, i.e. country ’s domestic price index. Thus, (11.34) is the law of one price at

the country level where represents the domestically produced goods in country , in contrast

to which represents all goods in country . Insert (11.34) into (11.11):

32 This law loosely states that the relative price on a good is equal to the nominal exchange rate, i.e. that

.

For example, if an Iphone costs twice as many in Norway as in US, the nominal exchange rate must be 2 for the law of one price to hold.


100

(∫

)

(∫ ( )

)

Then, log-linearize around a symmetric steady state:

∫ ( )

∫

∫

(11.35)

The (log) domestic price index for country expressed in terms of it own currency is denoted

∫

, the effective nominal exchange rate is denoted ∫

, and the (log)

world price index is denoted ∫

. Notice that for the world as a whole, there is no

distinction between CPI and domestic price level, nor between their corresponding inflation

rates. Insert (11.35) into (11.29):

(11.36)

Equation (11.36) expresses the terms of trade as a linear function of the effective nominal

exchange rate, the world price and the price on domestically produced goods. Next, define the

bilateral exchange rate between the home country and country , i.e. the ratio of the two

countries CPI’s, both expressed in terms of domestic currency, as:

(11.37)

In logs:

(11.38)

Then, let the (log) effective real exchange rate be:

∫

(11.39)

Insert (11.38) into (11.39), using (11.36) and (11.30):

∫( )

∫

∫

∫

( )

⇒ ( ) (11.40)

Notice that the last equality holds only up to a first order approximation when .

11.5 International risk sharing

Under the assumption of complete markets for securities traded internationally, a condition

analogous to (11.23) must also hold for the representative household in any other country, say

country :

{ (

)

} {

(

)

} (11.41)


101

Divide (11.23) by (11.41) and solve for :

{

(

)

}

{ (

)

}

{(

)

}

{(

)

}

{(

)

(

)

}

⇒ {(

)

} {(

)

}

⇒ {(

)

}

{

} {

}

⇒

(11.42)

{

} is some constant that will generally depend on initial conditions regarding

relative net asset positions. Without loss of generality, assume symmetric initial conditions, i.e.

zero net foreign asset holdings and an ex-ante identical environment. This implies

. Taking logs of both sides of (11.42):

(11.43)

Equation (11.43) is at the household level. Note that world consumption is given by

∫

Integrating (11.43) over all and using (11.39) and (11.40) yields:

∫ (

)

(11.44)

Thus, the assumption of complete markets at the international level leads to a simple relationship

linking domestic consumption with world consumption and the terms of trade, where relative

home consumption to world consumption is given by .

11.6 Uncovered interest rate parity

Allow households to invest both in domestic and foreign bonds; and . The budget

constraint may be written as:


102

(11.45)

The optimality conditions with respect to these assets are:

{ (

)

} (11.46)

{ (

)

} (11.47)

Divide (11.46) by (11.47) to obtain:

{

(

)

}

{

(

)

}

{ }

{

} {

}

⇒

{

} (11.48)

Log-linearizing (11.48) gives:

{ }

⇒ (11.49)

This is the familiar uncovered interest rate parity equation, which states that the nominal interest

rate at home is equal to the world nominal interest rate plus expected rate of depreciation of the

home currency. Now, from (11.36) we have that:

⇒

Thus, using (11.49) we get the following stochastic difference equation:

( )

⇒ (

) ( ) (11.50)

Given that the terms of trade are pinned down uniquely in the perfect foresight steady state, and

given the assumptions of stationarity in the models driving forces and unit relative prices in

steady state, it follows that Hence, (11.50) can be solved forward to obtain:

(

) ( )

{(

) ( ) (

)

( ) }

⇒ {∑ [(

) ( )] } (11.51)

Equation (11.51) expresses the terms of trade as the expected sum of real interest rate

differentials between the world market and the home market.

11.7 Firms and technologies

Now we turn to the supply side of the economy. Because domestic firms take the business

environment as given, including state of affairs in foreign economies, the individual firm still only


103

takes into account its own marginal cost. Assume that a typical firm in the home economy

produces a differentiated good with linear technology represented by the production function:

(11.52)

Assume that an employment subsidy identical to the one in (7.12) is in place. From (4.11), using

CRS, we get and . Thus, (7.12) now becomes:

(11.53)

The employment subsidy is captured in the term ( ). The firms optimal price

setting behavior is identical to the one described in the closed economy case. As in (3.11) the

optimal price is:33

( ) ∑ [ | ]

(11.54)

The log of the gross markup, or equivalently, the equilibrium markup in the flexible price

economy, is denoted

.

11.8 Equilibrium – Aggregate demand and output

Market clearing for good in the home economy implies:

∫

(11.55)

The supply of domestically produced good is denoted , the domestic demand is denoted

, and country ’s demand for good produced in the home economy is denoted

. Due to

the nested structure one can express demand in sub-markets in terms of total demand by

combining all demand functions from each level. For instance, insert (11.14) into (11.8) and get:

(

)

( ) (

)

(

)

(11.56)

Furthermore, the demand for domestically produced good in country is expressed by nesting

up across different demand layers in country . First, note that the consumption of domestically

produced good in country is a function of country ’s consumption of goods produced in the

home economy, given as in (11.8):

(

)

Second, note that country ’s consumption of goods produced in the home economy is a

function of country ’s consumption of foreign goods, given as in (11.12):

(

)

33 Now denotes the optimal price, instead of

, which denotes the world price.


104

Third, note that consumption of imported goods in country is a function of total consumption

in that country, given as in (11.15):

(

)

Combining all these yields the demand for domestically produced good in country as a

function of total consumption in that country:

(

)

(

)

(

)

(11.57)

Thus, we can insert (11.56) and (11.57) into (11.55) and get:

( ) (

)

(

)

∫ (

)

(

)

(

)

⇒ (

)

[( ) (

)

∫ (

)

(

)

] (11.58)

To aggregate, start with the definition of aggregate domestic output:

(∫

)

(11.59)

Insert (11.58) into (11.59) and solve out:


105

{

∫((

)

[( ) (

)

∫(

)

(

)

])

}

{

[∫(

)

] [( ) (

)

∫(

)

(

)

]

}

[∫(

)

]

[( ) (

)

∫(

)

(

)

]

(∫

)

[( ) (

)

∫(

)

(

)

]

[

(∫

)

]

[( ) (

)

∫(

)

(

)

]

[( ) (

)

∫(

)

(

)

]


106

⇒ ( ) (

)

∫ (

)

(

)

(11.60)

Next, factorize out the elements in the integral and insert for (11.37):

( ) (

)

∫

( ) (

)

∫

( ) (

)

∫ (

)

(

)

( ) (

)

(

)

∫(

)

(

)

⇒ (

)

[( ) ∫ (

)

] (11.61)

Define the effective terms of trade for country as:

(11.62)

Use this, and also insert for the bilateral terms of trade between the domestic economy and

country from (11.27), and for

from (11.42):

(

)

[( ) ∫(

)

]

(

)

[( ) ∫(

)

]

(

)

[( ) ∫(

)

]

⇒ (

)

[( ) ∫ ( )

] (11.63)

Log-linearization of (11.63) around a symmetric steady state yields the following:

(

)

[( ) ∫ (

)

]

⇒ (

) (11.64)


107

Insert for (11.40):

(

) ( )

( ) ( )

[ ( ) ( )]

[ ( )( )]

⇒

(11.65)

Note that ( )( ) is reasonable because [ ]. A condition

analogous to (11.65) will hold for all countries. Thus, for a generic country it can be rewritten as

. By aggregating over all countries, a world market clearing condition can be

derived as:

∫

∫ (

)

∫

∫

∫

(11.66)

This result follows from the fact that ∫

. Inserting (11.44) and (11.66) into (11.65)

yields:

( )

⇒

(11.67)

Note that

( ) . Finally, insert for from (11.65) into the Euler equation (11.26)

to get the IS equation:

{

}

( )

⇒

( )

(11.68)

Thus, the IS equation is similar to the one in a closed economy except that now there is an

additional term linking domestic output to the international environment. An alternative

representation including domestic goods inflation is found by inserting (11.32) into (11.68):

( { } )

( )

( )

⇒

( )

(11.69)

Note that

( )( )


108

if and are sufficiently high. Yet another representation is found by inserting for from

(11.67):

( )

{(

) ( )}

(

)

( )

⇒ (

) (

)

( )

⇒

(

)

( )

(

)

⇒

( )

Use that and

( ):

( )

( )

( )

( )

( )

[ ( )] ( ) ( )

( )

[ ( )] ( ) ( )

( )

( )

( )

( )

⇒

( ) (11.70)

The last term, , is exogenous to domestic allocations. Note that in general, the degree of

openness influences the sensitivity of output to any given change in the domestic real rate

. Also note from (11.69) that if ( ) ( )( )

, i.e. if and are sufficiently high, we have that

( ) , and output is more

sensitive to real rate changes than in the closed economy case. The reason is the direct negative


109

effect of an increase in the real rate on aggregate demand and output is amplified by the induced

real appreciation and the consequent switch of expenditure toward foreign goods. This will be

partly offset by any increase in CPI inflation relative to domestic inflation induced by the

expected real depreciation, which would dampen the change in the consumption based real rate,

, which is the one ultimately relevant for aggregate demand, relative to .

11.9 Equilibrium – The trade balance

Next, we can define net exports as the difference between total domestic production and

total domestic consumption, relative to steady state output :

(11.71)

A first-order approximation around a symmetric steady state with price level and

output level , i.e. zero net export, yields:

[( )

( )

( )

( )]

( ) ( ) ( ) ( )

The last equality follows from (11.30). To get an even simpler expression, insert for from

(11.65):

(

) (11.72)

In the special case with , , though the latter property will also hold

for any configuration satisfying ( )( ) . More generally,

the sign of the relationship between the terms of trade and net export is ambiguous, depending

on the relative size of , and .

11.10 Equilibrium – The supply side: Marginal cost and inflation dynamics

As before market clearing in the labor market requires that:

∫

(11.73)

Thus, using (11.52) and the market clearing condition:

⇒ ∫

∫

(

)

∫ (

)

(11.74)


110

As shown in (4.6)-(4.11), log-linearization gives, up to a first order:

⇒ (11.75)

Domestic inflation is derived as in (4.15), and is given by:

(11.76)

where

( )( )

and

.

The real marginal cost is now derived from (11.53). Insert (11.25) and (11.30):

( ) ( )

Next, insert (11.44) and (11.75) and make use of world market equilibrium:

(

) ( )

( ) ( )

⇒

( ) (11.77)

Thus, we see that the real marginal cost is increasing in terms of trade and the world output.

These variables end up influencing the real wage through the wealth effect on labor supply

resulting from their impact on domestic consumption. In addition, changes in the terms of trade

have a direct effect on the product wage for any given level of consumption wage. The influence

of technology (through its direct effect on labor productivity) and of domestic output (through

its effect on employment and, hence, the real wage for given output) is analogous to what we get

in the closed economy setting described in (4.16). Finally we can use (11.67) to insert for :

( ) ( )

( ) (11.78)

From (11.78) we see that domestic output affects marginal costs through its impact on

employment (captured by ) and the terms of trade (captured by , which is a function of the

degree of openness and the substitutability between domestic and foreign goods). World output

on the other hand, affects marginal costs through its effect on consumption (and hence, the real

wage as captured by ) and the terms of trade (captured by ). The effect of world output on

marginal costs is positive if

, that is if ( )( ) .

This is because with sufficiently high and , the size of the real appreciation needed to

absorb the change in relative supplies is small, with its negative effect on marginal costs more

than offset by the positive effect from a higher real wage. What about the natural level of output


111

, i.e. the output when prices are flexible? We know from earlier that in this case, .

Thus, the flexible price version of (11.78) is simply:

( ) ( )

( ) (11.79)

Solve (11.79) for natural output and use that

:

( ) ( ) ( )

⇒

( )

⇒

⇒

(11.80)

Here,

,

, and

. Again the effect of world output on

natural output is ambiguous, depending on the effect of world output on domestic marginal

costs, which in turn depends on the relative importance of the terms of trade effect discussed

above.

11.11 The New Keynesian Phillips curve and the Dynamic IS equation

In this section I set up a canonical representation of the small open economy version of the basic

New Keynesian model. First we denote as the domestic output gap from flexible

price output. Second, if we subtract (11.79) from (11.78) the real marginal cost gap emerges:

( ) ( )

( )

[ ( ) ( )

( ) ]

⇒ ( ) (11.81)

Then insert (11.81) into (11.76) to get the New Keynesian Phillips curve for the small open

economy:

( )

⇒ (11.82)

Note that (11.82) nests the special case of a closed economy because implies that

and then (11.82) becomes identical to (4.20). In general, the relation between the degree

of openness parameter , an increase in the output gap, and domestic inflation, depends on the

sign on . This is because

. If (i.e. if and are sufficiently high), an

increase in the openness will make domestic inflation less responsive to a change in the output

gap. On the other hand, if , then more openness will make domestic inflation more


112

responsive to output gap changes. To derive the open economy dynamic IS equation (DIS) we

have to do some additional steps. First, note that the real interest rate is defined as:

Using this, (11.70) can be written as:

( )

( )

( )

In a similar vein, the natural output is given as a function of the natural real interest rate:

( )

(11.83)

The DIS equation emerges by subtracting (11.83) from (11.70):

[

( ) ]

[

( )

]

⇒

( ) (11.84)

Equations (11.82) and (11.84), together with an equilibrium process for the natural real rate ,

constitute the non-policy block of the small open economy version of the New Keynesian model.

The natural real rate can be extracted from (11.84), but first one should note that (11.70) implies

that:

( )

Second, (11.80) implies that:

Third, (11.84) implies:

( )

Using these results, one can solve for :


113

[ ( ) (

)]

[

]

[(

( ) ) (

)]

( )

( ) (

)

( )

( )

⇒ ( )

(11.85)

Thus, we see that the New Keynesian Phillips curve and the DIS equation in the small open

economy equilibrium is similar to the counterparts in the closed economy. A couple of

differences appear however. First, the degree of openness influences the sensitivity of the output

gap to interest rate changes. Second, openness generally makes the natural real interest rate

depend on expected world output growth, in addition to domestic productivity. Finally, it is

convenient to define the real rate gap as:

( )

(11.86)

As in the closed economy case the real rate converges to the discount rate once technology

shocks and world output growth is turned off. Note however, that the real rate will typically by

higher than the discount rate because the world experiences a positive growth on average.

11.12 Equilibrium determinacy

In order to close the model one must specify a monetary policy rule. Suppose the central bank

follows an interest rate rule of the form:

(11.87)

where is a monetary policy shock. To set up the equilibrium system we first insert (11.87) and

(11.82) into (11.84) and solve the resulting equation for :


114

( )

( )

[ ( ) ]

[ ( ) ]

( )

⇒ (

)

( )

⇒

(

) (11.88)

Equation (11.88) is the reduced form version of the DIS equation, and shows the current output

gap as a function of expected output gap, expected domestic inflation, and shocks. We next

achieve a similar representation of current inflation. Insert (11.88) into (11.82) and solve the

resulting equation for :

[

( )]

( )

( )

( )

( )

( )

⇒

(

)

(

) (11.89)


115

The equilibrium dynamics above is represented as a system of difference equations, and is written

in matrix form as:

[

]

[

( )] [

]

[

] (

)

[

( )] [

] [

] ( )

[

]

( )

where (11.90)

[

( )]

[

]

We have defined

to ease the notation. The system (11.90) is a reduced form

representation of the dynamic IS curve and the New Keynesian Phillips curve, which takes into

account effects from the monetary policy defined in (11.87). The Blanchard and Kahn (1980)

conditions state that the system (11.90) has a locally unique equilibrium if and only if both

eigenvalues of the 2x2-matrix are inside the unit circle. To check this we derive the

characteristic equation:

| |

When this is written out:

|

[

( )] [

]|

( ( ))

( )

( )

( )

( )

( )

( )

( )

( )


116

Following LaSalle (1986), the two eigenvalues of lie inside the unit circle if and only if:

|

|

and

|

(

)

|

The first condition holds because and all elements in the denominator are positive.

The second condition can be rewritten to:

( )

(

)

( ) ( )

Thus, the second condition holds as long as the policy parameters and are sufficiently

high. In fact, whenever , then can actually be zero. Therefore the Taylor principle

holds in this open economy setting as well.

11.13 Equilibrium dynamics

In order to get an analytical expression for the dynamic processes one must specify how the

monetary policy shock evolves. Suppose it follows an AR(1)-process:

(11.91)

For simplicity and without loss of generality we now set . We use the method of

undetermined coefficients and guess that the solution takes the form:

(11.92)

(11.93)

To proceed we insert (11.91)-(11.93) into (11.82):

⇒

⇒ ( )

⇒

(11.94)

Then, insert the monetary policy rule into (11.84):


117

( )

( )

( )

⇒

( )

⇒ [ ( ) ] ( )

⇒ [ ( ) ]

( )

⇒ [

( ) ]( ) ( )

⇒

( )[ ( ) ]

( ) (11.95)

We have defined

( )[ ( ) ]

( ) to ease the notation. Insert (11.95)

back into (11.94) to obtain:

( ) (11.96)

Using (11.95)-(11.96) the guessed solutions (11.92)-(11.93) become:

( ) (11.97)

(11.98)

To get an AR(1) representation we insert (11.91) into (11.97) and (11.98):

( ) ( ) ( ) ( )

⇒ ( ) (11.99)

( )

⇒ (11.100)

Thus, a positive monetary policy shock gives a decline in both the output gap and domestic

inflation. Further, it can be shown that is increasing in the degree of openness, implying that a

given monetary policy shock will have a larger impact in the small open economy than its closed

economy counterpart. The response on the real interest rate is found by inserting (11.97)-(11.98)

into the policy rule (11.87):34

( ) { [ ( )] } (11.101)

34 Note that we look at the effect on the nominal interest rate, not at the interest rate level. Thus, the constant is not part of the expression.


118

The sign of the response of the nominal interest rate is ambiguous and depends on parameter

values. The response of the real interest rate:

( [ ( )] )

⇒ ( [( ) ( )] ) (11.102)

Using (11.67) we find the effect on the terms of trade:

( ) (11.103)

Thus, a monetary contraction leads to an improvement in the terms of trade, i.e. a decrease in the

relative price on foreign goods. Using (11.36) we find the effect on the change in the nominal

exchange rate:

( ) ( )

⇒ [( ) ( ) ] (11.104)

Thus, a monetary contraction leads to a nominal exchange rate appreciation. Using (11.40) we

find the effect on the effective real exchange rate:

( ) ( ) ( ) (11.105)

Thus, the effective real exchange rate appreciates as well. Using (291) we find the effect on net

exports:

(

) (

)

( ) (11.106)

The sign on the response of net exports to a monetary contraction is negative whenever .

11.14 Optimal monetary policy in the small open economy

In the following I will characterize the optimal monetary policy for the small open economy. In

order to get analytical results I make some assumptions regarding parameter coefficients. These

assumptions are as follows:

First we characterize the optimal allocation from the viewpoint of the social planner. The optimal

allocation maximizes household utility (11.1) subject to the technological constraint (11.52), a

consumption/output possibilities set implicit in the international risk-sharing conditions (11.42),

and the market clearing condition (11.63). The latter condition is somewhat changed under the

parameter restrictions above. First, from (11.44), implies that:

( )

⇒

⇒ (

)

(11.107)


119

Furthermore, when , the CPI given by (11.13) takes the Cobb-Douglas form:

(11.108)

When we rewrite (11.108), and then insert (11.28):

(

)

Thus, (11.63) becomes:

(

)

[( ) ∫(

)

]

(

)

[( ) ∫(

)

]

⇒ (11.109)

The period optimization problem of the social planner follows as:

{ ∑ ( )

}

subject to (11.110)

It is useful the make the problem simpler by getting rid of some constraints. Insert (11.107) into

(11.109) and combine with (11.66), which states that

. The result is an equilibrium

identity linking domestic consumption to domestic and world output:

(

)

⇒

⇒

(11.111)

Finally, to achieve an consumption expression useful to the social planner we insert (11.52) into

(11.111) and use that the optimal allocation implies just as in the closed economy case:

( )

(11.112)

The period optimization problem of the social planner now becomes a problem in only:

{ ∑ ( )

}

{ ∑ [( )

] } (11.113)

FOC:


120

: ( )( )

⇒ ( )( )

⇒

( )

( ) (11.114)

Using the specified utility with , which implies that ( )

, the LHS of

(11.114) becomes:

( )

⇒

⇒ ( )

(11.115)

Thus, the optimal employment is constant. From home firms optimization problem in flexible

price, free competition, we also have the following:

{ }

{ } (11.116)

FOC:

:

⇒

(11.117)

From (11.114) and (11.117) we get the optimal allocation of domestic quantities in the economy:

( ) ( )

(11.118)

As a comparison, let us first study the distortion in a market equilibrium where firms have

monopolistic power, but where prices are flexible. This is what we refer to as the natural

equilibrium (illustrated by top script ). Home firms’ maximization problem follows from firm

production (11.52), the demand for home good , given by (11.8), the aggregated version of

(11.52), and finally market clearing conditions. We know from the closed economy case that

monopolistic competition yields a distorted equilibrium which, in the absence of sticky prices,

can be fixed by a labor subsidy. Thus, we also add the labor subsidy with size yet to be

determined:


121

{

( )

}

{

( )

}

{

( )

}

{

(

)

( )

(

)

}

⇒ {

(

)

( )

(

)

} (11.119)

FOC:

( ) (

)

( )

(

)

⇒ ( ) ( )

⇒

( )

The LHS can be rewritten by inserting for (11.24) and (11.109):

( )

( )

( )

( )

⇒

( )

(11.120)

If we insert (11.115) to get the social planner’s solution, the optimal subsidy is found as:

( ) [( )

]

( )( )

⇒

( )

( )

( )

(

) (11.121)

Note that (11.121) nests the closed economy case where . In this case, (11.121) collapses

to

. In addition, and because , a sufficiently open economy (and )

implies a wage tax as the optimal fiscal policy instead of the subsidy. As in the closed economy,

the optimal monetary policy requires stabilizing the output gap, i.e. . The New Keynesian

Phillips curve given by (11.82) the implies that . Thus, in the special case under

consideration, (strict) domestic inflation targeting (DIT) is indeed the optimal policy. From the

dynamic IS equation (11.84) we see that implies in equilibrium, with all

variables matching their natural levels at all times. As discussed earlier, an interest rate rule of the


122

form is associated with an indeterminate equilibrium, and hence, does not guarantee that

the outcome of full price stability is attained. However, the central bank can get to the desired

outcome if it commits to a rule of the form:

(11.122)

where

( ) ( )

Under strict domestic inflation targeting, the behavior of real variables in the small open

economy corresponds to the one that would be observed in the absence of nominal rigidities.

Hence, from (11.80), that is

(where

,

, and

), we see that domestic output always increases in response to a positive technology

shock at home. The sign of the response to a rise in world output depends on the sign of ,

however. The natural level of the terms of trade is found by inserting (11.80) into the natural

level (flexible price) version of (11.67):

(

) (

)

(

)

[ ( )

( )

( )

]

{

( )

[ ( ) ]

}

(

)

⇒

( ) (11.123)

Because , an improvement in domestic technology leads to a real depreciation through its

expansionary effect on domestic output. On the other hand, and since

, an

increase in world output generates an improvement in the terms of trade, (i.e. a real appreciation),

given domestic technology. Because domestic prices are fully stabilized under DIT, it follows

from (11.36) that it can be written as:

(11.124)


123

Thus, the nominal exchange rate moves one for one with the natural terms of trade and inversely

with the price level. Assuming constant world prices, the nominal exchange rate will inherit all

the statistical properties of the natural terms of trade. Accordingly, the volatility of the nominal

exchange rate under DIT will be proportional to the volatility of the gap between the natural level

of domestic output, which in turn is related to productivity, and world output. In particular, the

nominal exchange rate volatility will tend to be low when domestic natural output displays a

strong positive comovement with world output. When that comovement is low or even negative,

possibly because of a large idiosyncratic componenent in domestic productivity, the volatility of

the terms of trade and the nominal exchange rate under DIT will be enhanced. The implied

equilibrium process for the CPI can also be derived, by substituting (11.124) into (11.30):

(

) (11.125)

Thus, it is seen that under the DIT regime, the CPI level will also vary with the natural terms of

trade and will inherit its statistical properties. If the economy is very open, and if domestic

productivity and hence, the natural level of domestic output, is not much synchronized with

world output, CPI prices could potentially be highly volatile, even if the domestic price level is

constant. One lesson from this analysis is that potentially large and persistent fluctuations in the

nominal exchange rate, as well as in some inflation measures like the CPI, are not necessarily

undesirable, nor do they acquire a policy response aimed at dampening such fluctuations. Instead,

and especially for an economy that is very open and subject to large idiosyncratic shocks, those

fluctuations may be an equilibrium consequence of the adoption of an optimal policy, as

illustrated by the model above.

11.15 Welfare losses

In the following I will derive a welfare loss function for the special case with log utility and unit

elasticity of substitution between goods of different origin, i.e. for:

With log consumption, a derivation similar to the one who previously lead to (8.1) now gives:

(

)

(

)

(

) (

)

(

) (

)


124

⇒

(

) (11.126)

A few results are needed to proceed. First, notice that in the special case considered here, (11.67)

can be rewritten to:

where I have used that the parameter restrictions above implies:

( )

( ( )( ) )

( ( )( ) )

Thus, (11.44) becomes:

( )

( )( ) ( )

(11.127)

Insert (11.127) into (11.126) and use that

in the log consumption case:

( )

(( )

)

(

)

⇒

( )

(

) (11.128)

stands for terms independent of policy as usual. The next step is to rewrite as a function

of the output gap and price dispersion. From the production function (11.52),

. Thus,

using (11.8), market clearing in the labor market and the goods market requires:

∫

∫

∫(

)

∫(

)

⇒ [∫ (

)

] (11.129)

Using the same derivation as in (8.3)-(8.6):

(11.130)

Here, I have used that

in (8.6) because of the CRS assumption. By combining

(11.129) and (11.130), the employment gap from steady state employment writes as:

( ) ( ) ( )

(11.131)

The next step is to insert (11.131) into (11.128):


125

( )

[(

)

(

)

]

( )

[

( )

]

Notice that the steady state version of (11.114) becomes:

( )

When we insert this:

( )

( )

[

( )

]

( ) ( ) [

( )

]

[ ( )( )

]

[ ( )(

)]

⇒

[ ( )

( ) ] (11.132)

To proceed, note that with CRS and the parameter restrictions above, (4.18) becomes:

( )

( )[ ( )]

( )

[ ]

Thus, from the definition of the natural output gap from its steady state counterpart, the

technology level, which previously was specified in (8.11), now satisfies:

(

) (

) (11.133)

Insert (11.133) into (11.132):

[ ( ) ( )

]

[ ( )(

)]

[ ( )(

)]

[ ( )(

) ( )

]

[ ( )(

) ]

⇒

[ ( )

] (11.134)

The last line follows from (8.12). When we take the sum over all discounted periods and make

use of (8.17):


126

∑

∑ [ ( )

]

[ ∑

( )∑

]

[

( )( )∑

( )∑

]

∑ [

( )( )

( ) ]

⇒ ∑

∑ [

( ) ]

(11.135)

The parameter ( )( )

is defined as in (11.76). Thus, we can write the second-order

approximation to the utility losses of the domestic representative consumer resulting from

deviations in optimal policy, expressed as a fraction of steady state consumption, as:

∑ [

( ) ]

(11.136)

Taking unconditional expectations on (11.136) and letting , the expected welfare losses for

any policy that deviates from strict inflation targeting can be written in terms of the variances of

inflation and the output gap:

[

( ) ( ) ( )] (11.137)

Text


127

References

Blanchard and Kahn ()

Bullard and Mitra (2002)

Erceg et al. (2000)

Gali, Jordi (2008), Monetary Policy, Inflation and the Business Cycle. Oxford: Princeton University

Press.

Gali, Jordi and Monacelli Tommasso (2005), Monetary Policy and Exchange Rate Volatility in a

Small Open Economy. Review of Economic Studies 72(707-734)

Hamilton (1994)

LaSalle (1986)

Molnar, (2011)

Ripatti, Antti (2011)

Sydsæter (2006)

Woodford ()


128

Appendix

A. Dynare codes – A monetary policy shock with sticky prices

// The basic New Keynesian model - A monetary policy shock // Gali ch. 3, fig. 3.1 // Modified by Drago Bergholt //------------------------------------------ // Preamble //------------------------------------------ // Variables var pi y Y rn i m_r n a v; varexo eps_v eps_a; //

// Parameters parameters beta epsilon theta sigma rho phi alpha phi_pi phi_y eta PSI_yan

THETA lambda kappa rho_v rho_a LAMBDA_v LAMBDA_a; beta = 0.99; sigma = 1; phi = 1; alpha = 1/3; epsilon = 6; eta = 4; theta = 2/3; phi_pi = 1.5; phi_y = 0.5/4; PSI_yan = (1+phi)/(sigma*(1-alpha)+phi+alpha); THETA = (1-alpha)/(1-alpha+alpha*epsilon); lambda = (1-theta)*(1-beta*theta)*THETA/theta; kappa = lambda*(sigma+(phi+alpha)/(1-alpha)); rho = 1/beta-1; rho_v = 0.5; rho_a = 0.9; LAMBDA_v = 1/((1-beta*rho_v)*(sigma*(1-rho_v)+phi_y)+kappa*(phi_pi-

rho_v)); LAMBDA_a = 1/((1-beta*rho_a)*(sigma*(1-rho_a)+phi_y)+kappa*(phi_pi-

rho_a)); //------------------------------------------ // Model //------------------------------------------ model(linear); // Taylor-Rule i = rho+phi_pi*pi+phi_y*y+v; // eq'n. (25), p. 50 // IS-Equation y = y(+1)-1/sigma*(i-pi(+1)-rn); // y is output gap (22) rn=rho+sigma*PSI_yan*(a(+1)-a); // natural rate of interest (23) Y = PSI_yan*(1-sigma*(1-rho_a)*(1-beta*rho_a)*LAMBDA_a)*a; // actual

output; 3rd eq'n from bottom, p. 54 // Phillips Curve pi = beta*pi(+1)+kappa*y; // (21) // Money Demand m_r = y-eta*i; // ad hoc money demand; m_r = m-p // Employment n = (((PSI_yan-1)-sigma*PSI_yan*(1-rho_a)*(1-beta*rho_a)*LAMBDA_a)/(1-

alpha))*a; // bottom p. 54 // Autoregressive Error a = rho_a*a(-1) + eps_a; // technology shock (28) v = rho_v*v(-1) + eps_v; // shock to i (bottom p. 50) end; //


129

//------------------------------------------ // Steady State //------------------------------------------ check; //------------------------------------------ // Shocks //------------------------------------------ shocks; var eps_v = 0.0625; var eps_a = 0; end; tech = 0; policy = 1; //------------------------------------------ // Computation //------------------------------------------ stoch_simul(irf=12); //stoch_simul(periods=1000,irf=12); //------------------------------------------ // Plots //------------------------------------------ if policy==1; // Gali's figure 3.1 figure(2); clf; subplot(3,2,1); plot(y_eps_v, '-o'); title('Output gap'); subplot(3,2,2); plot(4*pi_eps_v, '-o'); title('Inflation'); subplot(3,2,3); plot(4*i_eps_v, '-o'); title('Nominal interest rate'); subplot(3,2,4); plot(4*i_eps_v(1:end-1)-4*[pi_eps_v(2:end)], '-o');

title('Real interest rate'); subplot(3,2,5); plot(4*(m_r_eps_v-[0;m_r_eps_v(1:end-1)]), '-o');

title('Real money growth'); subplot(3,2,6); plot(v_eps_v, '-o'); title('v'); end; if tech==1; // Gali's figure 3.2 figure(2); clf; subplot(4,2,1); plot(y_eps_a); title('Output gap'); subplot(4,2,2); plot(4*pi_eps_a); title('Inflation'); subplot(4,2,3); plot(Y_eps_a); title('Output'); subplot(4,2,4); plot(n_eps_a); title('Employment'); subplot(4,2,5); plot(4*i_eps_a); title('Nominal interest rate'); subplot(4,2,6); plot(4*rn_eps_a); title('Real interest rate'); subplot(4,2,7); plot(4*(m_r_eps_a-[0;m_r_eps_a(1:end)])); title('Real money

growth'); subplot(4,2,8); plot(a_eps_a); title('a'); end;

B. Dynare codes – A technology shock with sticky prices

// The basic New Keynesian model - A technology shock // Gali ch. 3, fig. 3.2 // Modified by Drago Bergholt //------------------------------------------ // Preamble //------------------------------------------ // Variables var pi y Y rn i m_r n a v; varexo eps_v eps_a; // // Parameters


130

parameters beta epsilon theta sigma rho phi alpha phi_pi phi_y eta PSI_yan

THETA lambda kappa rho_v rho_a LAMBDA_v LAMBDA_a; beta = 0.99; sigma = 1; phi = 1; alpha = 1/3; epsilon = 6; eta = 4; theta = 2/3; phi_pi = 1.5; phi_y = 0.5/4; PSI_yan = (1+phi)/(sigma*(1-alpha)+phi+alpha); THETA = (1-alpha)/(1-alpha+alpha*epsilon); lambda = (1-theta)*(1-beta*theta)*THETA/theta; kappa = lambda*(sigma+(phi+alpha)/(1-alpha)); rho = 1/beta-1; rho_v = 0.5; rho_a = 0.9; LAMBDA_v = 1/((1-beta*rho_v)*(sigma*(1-rho_v)+phi_y)+kappa*(phi_pi-

rho_v)); LAMBDA_a = 1/((1-beta*rho_a)*(sigma*(1-rho_a)+phi_y)+kappa*(phi_pi-

rho_a)); // //------------------------------------------ // Model //------------------------------------------ model(linear); // Taylor-Rule i = rho+phi_pi*pi+phi_y*y+v; // eq'n. (25), p. 50 // IS-Equation y = y(+1)-1/sigma*(i-pi(+1)-rn); // y is output gap (22) rn=rho+sigma*PSI_yan*(a(+1)-a); // natural rate of interest (23) Y = PSI_yan*(1-sigma*(1-rho_a)*(1-beta*rho_a)*LAMBDA_a)*a; // actual

output; 3rd eq'n from bottom, p. 54 // Phillips Curve pi = beta*pi(+1)+kappa*y; // (21) // Money Demand m_r = y-eta*i; // ad hoc money demand; m_r = m-p // Employment n = (((PSI_yan-1)-sigma*PSI_yan*(1-rho_a)*(1-beta*rho_a)*LAMBDA_a)/(1-

alpha))*a; // bottom p. 54 // Autoregressive Error a = rho_a*a(-1) + eps_a; // technology shock (28) v = rho_v*v(-1) + eps_v; // shock to i (bottom p. 50) end; // //------------------------------------------ // Steady State //------------------------------------------ check; // //------------------------------------------ // Shocks //------------------------------------------ shocks; var eps_v = 0; var eps_a = 1; end; // tech = 1; policy = 0;


131

//------------------------------------------ // Computation //------------------------------------------ stoch_simul(irf=12); //stoch_simul(periods=1000,irf=12); // //------------------------------------------ // Plots //------------------------------------------ if policy==1; // Gali's figure 3.1 figure(2); clf; subplot(3,2,1); plot(y_eps_v, '-o'); title('Output gap'); subplot(3,2,2); plot(4*pi_eps_v, '-o'); title('Inflation'); subplot(3,2,3); plot(4*i_eps_v, '-o'); title('Nominal interest rate'); subplot(3,2,4); plot(4*i_eps_v(1:end-1)-4*[pi_eps_v(2:end)], '-o');

title('Real interest rate'); subplot(3,2,5); plot(4*(m_r_eps_v-[0;m_r_eps_v(1:end-1)]), '-o');

title('Real money growth'); subplot(3,2,6); plot(v_eps_v, '-o'); title('v'); end; if tech==1; // Gali's figure 3.2 figure(2); clf; subplot(4,2,1); plot(y_eps_a, '-o'); title('output gap'); subplot(4,2,2); plot(4*pi_eps_a, '-o'); title('inflation'); subplot(4,2,3); plot(Y_eps_a, '-o'); title('output'); subplot(4,2,4); plot(n_eps_a, '-o'); title('employment'); subplot(4,2,5); plot(4*i_eps_a, '-o'); title('nominal interest rate'); subplot(4,2,6); plot(4*i_eps_a(1:end-1)-4*[pi_eps_a(2:end)], '-o');

title('real interest rate'); subplot(4,2,7); plot(4*(m_r_eps_a-[0;m_r_eps_a(1:end-1)]), '-o');

title('real money growth'); subplot(4,2,8); plot(a_eps_a, '-o'); title('a'); end;

C. Dynare codes – A monetary policy shock with sticky prices and wages

// The basic New Keynesian model - A monetary policy shock // Gali ch. 6, fig. 6.3 // Modified by Drago Bergholt

//------------------------------------------ // Preamble //------------------------------------------

// Variables var y Y n yn i pip piw rn w wn v a; varexo eps_v eps_a;

// Parameters parameters alpha gammap gammaw sigma beta phi thetap thetaw psinya psinwa

kappap kappaw lambdap lambdaw rho phip phiw phiy rhoa rhov ew ep; beta = 0.99; sigma = 1; phi = 1; alpha = 1/3; ew = 6; ep = 6; thetaw = 3/4; thetap = 2/3;


132

phip = 1.5; phiw = 0; phiy = 0;

rho = 1/(beta-1); psinya = (1+phi)/(sigma*(1-alpha)+phi+alpha); psinwa = (1-alpha*psinya)/(1-alpha); lambdap = ((1-thetap)*(1-beta*thetap)/thetap)*((1-alpha)/(1-

alpha+alpha*ep)); lambdaw = ((1-beta*thetaw)*(1-thetaw))/(thetaw*(1+ew*phi)); kappap = lambdap*alpha/(1-alpha); kappaw = lambdaw*(sigma + phi/(1-alpha));

rhoa = 0.9; rhov = 0.5;

//------------------------------------------ // Model //------------------------------------------

model(linear);

y = y(+1) - (1/sigma)*(i - pip(+1) - rn); y = Y - yn; yn = psinya*a; Y = a + (1-alpha)*n; rn = sigma*psinya*(rhoa - 1)*a; pip = beta*pip(+1) + kappap*y + lambdap*(w-wn); piw = beta*piw(+1) + kappaw*y - lambdaw*(w-wn); w = w(-1) + piw - pip; wn = psinwa*a; i = rho + phip*pip + phiw*piw + phiy*y + v; v = rhov*v(-1)+eps_v; a = rhoa*a(-1)+eps_a; end;

//------------------------------------------ // Steady State //------------------------------------------

check;

//------------------------------------------ // Shocks //------------------------------------------

shocks; var eps_v = 0.0625; var eps_a = 0; end;

tech = 0; policy = 1;


133

//------------------------------------------ // Computation //------------------------------------------

stoch_simul(order=1, irf=12); //stoch_simul(periods=1000,irf=12);

//------------------------------------------ // Plots //------------------------------------------

if policy==1; // Gali's figure 6.1 figure(2); clf; subplot(3,2,1); plot(y_eps_v, '-o'); title('Output gap'); subplot(3,2,2); plot(4*pip_eps_v, '-o'); title('Price inflation'); subplot(3,2,3); plot(4*piw_eps_v, '-o'); title('Wage inflation'); subplot(3,2,4); plot(w_eps_v, '-o'); title('Real wage gap'); subplot(3,2,5); plot(n_eps_v, '-o'); title('Employment'); subplot(3,2,6); plot(v_eps_v, '-o'); title('v'); end;

if tech==1; // Gali's figure 3.2 figure(2); clf; subplot(4,2,1); plot(y_eps_a); title('Output gap'); subplot(4,2,2); plot(4*pi_eps_a); title('Inflation'); subplot(4,2,3); plot(Y_eps_a); title('Output'); subplot(4,2,4); plot(n_eps_a); title('Employment'); subplot(4,2,5); plot(4*i_eps_a); title('Nominal interest rate'); subplot(4,2,6); plot(4*rn_eps_a); title('Real interest rate'); subplot(4,2,7); plot(4*(m_r_eps_a-[0;m_r_eps_a(1:end)])); title('Real money

growth'); subplot(4,2,8); plot(a_eps_a); title('a'); end;

the basic new keynesian model - drago bergholt

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