compulsory macro dge approach upto12feb

7/25/2019 Compulsory Macro DGE Approach Upto12FEB

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Macroeconomics: A Dynamic General Equilibrium

Approach

Mausumi Das

Lecture Notes, DSE

February 2-12, 2016

Das (Lecture Notes, DSE) DGE Approach February 2-12, 2016 1 / 48
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Modern Macroeconomics: the Dynamic General

Equilibrium (DGE) Approach

As we have stated before, modern macroeconomics is based on adynamic general equilibrium approach which postulates that

Economic agents are continuously optimizing/re-optimizing subject to

their constraints and subject to their information set up. They optimizenot only over their current choice variables but also the choices thatwould be realized in future.All agents have rational expectations: thus their ex ante optimal futurechoices would ex post turn out to be less than optimal if and only iftheir information set was incomplete and/or there are some randomelements in the econmy which cannot be anticipated perfectly.The optimal choice of all agents are then mediated through themarkets to produce an outcome for the macroeconomy.



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Modern Macroeconomics: DGE Approach (Contd.)

This approach is dynamic because agents are making choices overvariables that relate to both present and future.

This approach is equilibrium because the outcome for the

macro-economy is the aggregation of individuals equilibriumbehaviour.

This approach is general equilibrium because it simultaneouslytakes into account the optimal behaviour of diiferent types of agents

in dierent markets and ensures that all markets clear.

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DSG Approach vis-a-vis Traditional Macroeconomics

The need to build macro-models based on internally-consistent,

dynamic optimization exercises of rational agents arose once it wasrealized that ad-hoc micro founations for the aggregative system maynot be consistent with one another.

This begs the following question: Why do we need such optimizationbased micro-founded framework at all?

Why cannot we just take the aggregative equations as arepresentation of the macro-economy and try to estimate variousparameters, using aggregative data?

After all, if we are ultimately interested in knowing how the

macroeconomy would respond to various kinds of policy shocks, allthat we need to do is to econometrically estimate the parameters ofthe aggregative system.

Then from the estimated parameter values or coecients, we canpredict the implcations of various policy changes.

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DSG Approach vis-a-vis Traditional Macroeconomics:

(Contd.)

Indeed, this is exactly how macroeconomic analysis was conductedtraditionally!

As we have already seen, traditional macroeconomics was based onsome aggregative behavioural relationship (e.g., Keyensian SavingsFunction - which postulates a relationship between aggregate income

and aggregate savings; Phillips Curve - which posits a relationshipbetween umployment rate and ination rate).

Often one would construct detailed behavioural equations for themacroeconomy and would try to estimate the parameters of theseequations using time series data.

To be sure some of these equations would be dynamic in nature.

But optimization over time was not considered to be important oreven relevant. Indeed, the concept of optimization itself - either byhouseholds or rms or even government - was rather alien in the eld

Macroeconomics.Das (Lecture Notes, DSE) DGE Approach February 2-12, 2016 5 / 48
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Lucas Critique: Optimization Comes to the Fore

The need to build macro models based explicitly on agentsoptimization exercises came from the so-called Lucas Critique.

Lucas (1976) argued that aggregative macro models which areestimated to predict outcomes of economic policy changes are useless

simply because the estimated parameters themselves may depend onthe existing policies.

As the policy changes, these coecients themsleves would change,thereby generating wrong predictions!

His solution was to build macroecnomic models with clear andspecic microeconomic foundations - models that are explicitly basedon agents optimization exercises.

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Lucas Critique: Optimization Comes to the Fore (Contd.)

Such models will enable us to dierentiate between true parameters- primitives like tastes, technology etc - which are independent of thegovernment policies, and variables that treated as exogenous by theagents but are actually endogenous and are inuenced by government

policies.Moreover such models would take into account agents expectationsabout government policies.

Predictions based on such microfounded models would be more

accurate than the aggregative models which club all the trueparameters as well as other policy-related parameters together.

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How Does Micro-foundation Help? An Example

Let us see exactly what Lucas critique means in the context of asimple example.

Consider the Keynesian savings function, specied as an aggregativerelationship:

St=1+2Yt+tAn aggregative macro model would take the above behaviouralrelationship as given and would estimate the coecients 1 and 2from data.

We have already provided a micro-foundation for this kind ofKeynesian Consumption/Savings function.

Let us re-visit that exercise.

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Micro-foundation of Keynesian Savings Function:

We assume that the economy consists of a nite number (H) ofidentical households. We can then talk in terms of a representativehousehold.

Let us dene a 2-period utility maximization problem of therepresentative household as:

Max.fct

,

ct+1 g

log(ct) + log(ct+1)

subject to,

(i) Ptct+ st = yt;

(ii) Pet+1ct+1 = (1+ ret+1)st+ y

et+1.

From (i) and (ii) we can eliminate Stto derive the life-time budgetconstraint of the household as:

Ptct+ Pet+1ct+1

(1+ ret+1)=yt+

yet+1(1+ ret+1)

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Micro-foundation of Keynesian Savings Function: (Contd.)

From the FONCs:

ct+1

ct = (1+ ret+1)

PtPet+1

.

Solving we get:

Ptct= 1

(1+) yt+ yet+1

(1+ ret+1

)Thus

st=

(1+)yt

1

(1+)

yet+1

(1+ ret+1)

Aggregating over all households:

St=

(1+)Yt

1(1+)

Yet+1

(1+ ret+1)

Notice that an aggregative model would equate (1+)

to 2 and

1

(1+)h Yet+1

(1+ret+1)i

to 1.

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Micro-foundation of Keynesian Savings Function: (Contd.)

While the coecient 2 is indeed based on true parameters(primitives) and would therefore be unaected by policy changes,coecient1 is not.

Any policy that changes the households expectation about its futureincome or future rate of interest rate would aect 1.

Thus predicting outcomes of such a policy based on the estimatedvalues of the aggregative equations would be wrong.

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Modern Macroeconomics: DGE Approach

The Lucas critique and the consequent logical need to develop aunied micro-founded macroeconomic framework which would allowus to accurately predict the macroeconomic outcomes in response toany external shock (policy-driven or otherwise) led to emergence ofthe modern dynamic general equilibrium approach.

As before, there are two variants of modern DGE-based approach:

One is based on the assumption of perfect markets (theNeoclassical/RBC school). As is expected, this school is critical of anypolicy intervention, in particular, monetary policy interventions.

The other one allows for some market imperfections (theNew-Keynesian school). Again, true to their ideological underpinning,this school argues for active policy intervention.

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Modern Macroeconomics: DGE Approach (Contd.)

However, both frameworks are similar in two fundamental aspects:

Agents optimize over innte horizon; andAgents are forward looking, i.e., when they optimize over futurevariable they base their expectations on all available information -

including information about (future) government policies. In otherwords,agents have rational expectations.

We now develop the choice-theoretic frameworks for households andrms under the DGE approach.

As before, we shall assume that the economy is populated by Hidentical households so that we can talk in terms of a representativehousehold.

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Households Choice Problem under Perfect Markets:

Innite Horizon

Let us examine the consumption-savings choices of the representativehousehold over innite horizon when markets are perfect.

To simplify the analysis, we shall only focus on the consumptionchoice of the household and ignore the labour-leisure choice (for thetime being).

At any point of time the household is endowed with one unit oflabour - which it supplies inelastically to the market.

We shall also ignore prices and the concomitant role of money andfocus only on the real variables.

Let atdenote the asset stock of the household at the beginning of

period t.Then Income of the household at time t: yt=wt+ rtat.We shall assume that savings of an household in any period areinvested in various forms of assets (all assets have the same return),

which augments the households asset stockinthenext period.Das (Lecture Notes, DSE) DGE Approach February 2-12, 2016 14 / 48
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Households Choice Problem: Innite Horizon (Contd.)

If we do not allow intra-household borrowing, then the representativehousehold hs problem would given by:

Max.

fcht gt=0,faht+1gt=0

t=0

tuchtsubject to

(i) cht 5 wt+ rtaht for all t= 0;

(ii) aht+1 = wt+ rtaht+ (1 )aht cht ; aht= 0 for all t= 0; ah0 give


( )
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Notice that that the household is solving this problem at time 0.Therefore, in order to solve this problem the households would haveto have some expectation about the entire time paths ofwt and rtfrom t=0 to t! .

We shall however assume that households have rational expectations.

In this model with complete information and no uncertainty, rationalexpectation is equivalent to perfect foresight. We shall use these twoterms here interchangeably.

By virtue of the assumption of rational expectations/perfect foresight,the agents can correctly guess all the future values of the marketwage rate and rental rate, but they still treat them as exogenous.

As atomistic agents, they belive that their action cannot inuencethe values of these market variables.


H h ld Ch i P bl I i H i (C d )
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Notice that once we choose our consumption time path

cht

t=0, the

corresponding time path of the asset level aht+1

t=0

wouldautomatically get determined from the constraint functions (and viceversa).

So in eect in this constrained optimization problem, we only have tochoose one set of variables directly. We call them the control

variables. Let our control variable for this problem bechtt=0 .

We can always treat c0, c1, c2,......as independent variables and solvethe problem using the standard Lagrangean method.

The only problem is that there are now innite number of such choicevariables (c0, c1, c2,....., c) as well as innite number of constraints

(one for each time period from t=0, 1, 2.....,) and things can getquite intractable.

Instead, we shall employ a dierent method - called DynamicProgramming - which simplies the solution process and reduces it toa univariate problem.


D i O i i i i Di Ti D i
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Dynamic Optimization in Discrete Time: Dynamic

Programming

Consider the following canonical discrete time dynamic optimizationproblem:

Max.fxt+1g

t=0,fytg

t=0

t=0

tU(t, xt, yt)

subject to

(i) yt 2 G(t, xt) for all t= 0;

(ii) xt+1 = f(t, xt, yt); xt2 X for all t= 0; x0 given.

Here yt is the control variable; xt is the state variable; U represents

the instantaneous payo function.(i) species what values the control variable yt is allowed to take (thefeasible set), given the value ofxtat time t;(ii) species evolution of the state variable as a function of previousperiods state and control variables (state transitionequation).


D i P i (C d )
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Dynamic Programming (Contd.)

It is often convenient to use the state transition equation given by (ii)to eliminate the control variable and write the dynamic programming

problem in terms of the state variable alone:

Max.fxt+1g

t=0

t=0

tU(t, xt, xt+1)

subject to

(i) xt+1 2 G(t, xt) for all t= 0; x0 given.

We are going to focus on stationary dynamic programming problems,where time (t) does not appear as an independent argument either inthe objective function of in the constraint function:

Max.fxt+1 g

t=0

t=0

tU(xt, xt+1)

subject to(i) xt+1 2 G(xt) for all t= 0; x0 given.


St ti D i P i V l F ti &
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Stationary Dynamic Programming: Value Function &

Policy Function

Ideally we should be able to solve the above stationary dyanamicprogramming problem by employing the Lagrange method. Letxt+1

t=0

denote such a solution.

We can then write the maximised value of the objective function as afunction of the parameters alone, in particular as a function ofx0 :

V(x0) Max.fxt+1 g

t=0

t=0

tU(xt, xt+1); xt+1 2 G(xt) for all t= 0;

=

t=0

tU(xt , xt+1) .

The maximized value of the objective function is called the valuefunction.

The function V(x0) represents the value function of the dynamicprogramming problem at time 0.


V l F ti & P li F ti (C td )
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Value Function & Policy Function (Contd.)

Suppose we were to repeat this exercise again the next period i.,e. att=1.

Now of course the time period t=1 will be counted as the initialpoint and the corresponding initial value of the state variable will bex1.

Let denote the new time subscript which counts time from t=1 to

. By construction then, t 1.When we set the new optimization exercise (relevant fort=1, 2....,) in terms of it looks exactly similar. In particular, thenew value function will be given by:

V(x1) Max.fx+1 g

=0

=0

U(x, x+1) ; x+1 2 G(x) for all = 0;

=

=0

U(x, x+1).


Value Function & Policy Function (Contd )
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Noting the relationship between t and , we can immediately see thatthe two value functions are related in the following way:

V(x0) =

t=0

tU(xt , xt+1)

= U(x0, x1)+

t

=1

t1U(xt , xt+1)

= U(x0, x1)+

=0

U(x, x+1)

= U(x0, x1)+V(x

1).

The above relationship is the basic functional equation in dynamicprogramming which relates two successive value functions recursively.It is called the Bellman Equation. It breaks down the ininitehorizon dynamic optimization problem into a two-stage problem:

what is optimal today (x1 );

what is the optimal continuation path (V(x1)).Das (Lecture Notes, DSE) DGE Approach February 2-12, 2016 22 / 48

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Since the above functional relationship holds for any two successivevalues of the state variable,we can write the Bellman Equation moregenerally as:

V(x) = Maxx2G(x)

[U(x, x) +V(x)] for all x2 X. (1)

The maximizer of the right hand side of equation (2) is called apolicy function:

x=(x),

which solves the RHS of the Bellman Equation above.

If we knew the value function V(.) and were it dierentiable, we

could have easily found the policy function by solving the followingFONC (called the Euler Equation):

x :U(x, x)

x +V0(x) =0. (2)


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Unfortunately, the value function is not known.

In fact we do not even know whether it exists; if yes then whether itis unique, whether it is continuous, whether it is dierentiable etc.

A lot of theorems in Dynamic Programming go into establishingconditions under which a value exists; is unique and has all the niceproperties (continuity, dierentibility and others).

For now, without going into futher details, we shall simply assumethat all these conditions are satised for our problem.

In other words, we shall assume that for our problem the valuefunction exists and is well-behaved (even though we do not know

its precise form).Once the existence of the value function is established, we can thensolve the FONC (3) (the Euler Equation) to get the policy function.

But there is still one hurdle: what is the value V0(x)?

Here the Envelope Theorem comes to our rescue.


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Recall that V(x) is nothing but the value function for the next periodwhere xis next periods initial value of the state variable (which is

given - from next periods perspective).

Since the Bellman equation is dened for all x2 X, we therefore geta similar relationship between xand its subsequent state value (x):

V(x) = Maxx2G(x)

[U(x, x) +V(x)].

Then applying Envelope Theorem:

V0(x) =U(x, x)

x . (3)

Combining the Euler Equation (3) and the Envelope Condition (4),we get the following equation:

U(x, x)

x +

U(x, x)

x =0 for all x2 X.


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Replacing x, x, xby their suitable time subscripts:

U(xt, xt+1)

xt+1+

U(xt+1, xt+2)

xt+1=0; xt given. (4)

Equation (5) is a dierence equation which we should be able to solveto derive the time path of the state variable x

t(and consequently that

of the control variable yt).

Since it is a dierence equation of order 2, apart from the initailcondition, we need another boundary condition.

Typically such a boundary condition is provided by the following

transversality condition:

limt!

tU(xt, xt+1)

xtxt=0. (5)


Stationary Dynamic Programming: Existence &
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Stationary Dynamic Programming: Existence &

Uniqueness of Value Function

We now provide some sucient conditions for the Value function ofthe above stationary dynamic programming problem to exist, to betwice continuously dierentiable, to be concave etc.

We just state the theorems here without proof. Proofs can be foundin Acemoglu (2009).

1 Let G(x) be non-empty-valued, compact and continuous in all x2 Xwhere X is a compact subset of


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Non Stationary Dynamic Programming: Existence &

Uniqueness of Value Function

Even when the dynamic programming problem is non-stationary, wecan nd analogous sucient conditions that will ensure the existence,uniqueness, concavity and dierentiability of the corresponding valuefunction.

Then we can proceed exactly as above to write down the Bellman

equation that relates the value functions of two successive timeperiods and then solve for the optimal policy function from thecorresponding Euler Equation and the Envelope condition.

All the economic problems that we would be looking at in this coursewill satisfy these suciency properties.

So we shall stop bothering about this suceny condition from now onand focus on applying the dynamic programming technique to theeconomic problems at hand.

Interested students can look up Acemoglu (2009): Introduction toModern Economic Growth, Chapter 6, for thetheorems andproofs.


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Back to Household s Choice Problem: Innite Horizon

Recall that we had specied the representative householdsoptimization problem under innite horizon as:

Max.fcht g

t=0,faht+1g

t=0

t=0

tucht

subject to

(i) cht 5 wt+ rtaht for all t= 0;

(ii) aht+1 = wt+ rtaht+ (1 )a

ht c

ht ; a

ht= 0 for all t= 0; a

h0 give

However in specifying the problem, we assumed that there is no

intra-household borrowing.This assumption of no borrowing is too strong, and we do not reallyneed it for the results that follow.

So let us relax that assumption to allow households to borrow fromone another if they so wish.


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Household s Choice Problem: Innite Horizon (Contd.)

Allowing for intra-household borrowings means that constraint (i)would no longer hold. A household can now consume beyond itscurrent income at any point of time - by borrowing from others.

Allowing for intra-household borrowings also means that a householdnow has two forms of assets that it can invest its savings into:

1 physical capital (kht);2 nancial capital, i.e., lending to other households (lht b

ht).

Let the gross interest rate on nancial assets be denoted by (1+ rt) .

Let physical capital depreciate over time at a constant rate . Thenthe gross interest rate on investment in physical capital is given by(rt+1 ).




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Household s Choice Problem: Innite Horizon (Contd.)

Arbitrage in the asset market ensures that in equlibrium two interest

rates are the same :

1+ rt=1+ rt ) rt=rt .

Thus we can dene the total asset stock held by the household in

period t as ah

t kh

t + lh

t .

Notice that lht < 0 would imply that the household is a net borrower.

Hence the aggregate budget constraint of the household is now givenby:

ch

t

+ sht

=wt+ rtah

t

, where sh

t

aht+1

aht

.

Re-writing to eliminate sht:

aht+1 =wt+ (1+ rt)aht c

ht .


Households Choice Problem: Ponzi Game
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But allowing for intra-household borrowing brings in the possibility ofhouseholds playing a Ponzi game, as explained below.

Consider the following plan by a household:Suppose in period 0,the household borrows a huge amount b- whichwould allow him to maintain a very high level of consumption at allsubsequent points of time. Thus

b0 =

b.

In the next period (period 1)he pays back his period 0 debt withinterest by borrowing again (presumably from a dierent lender). Thushis period 1 borrowing would be:

b1 = (1+ r0)b0.

In period 2 he again pays back his period 1 debt with interest byborrowing afresh:

b2 = (1+ r1)b1 = (1+ r1)(1+ r0)b0.

and so on.


Households Choice Problem: Ponzi Game (Contd.)
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( )

Notice that proceeding this way, the household eectively never paysback its initial loan b; he is simply rolling it over period after period.

In the process he is able to perpetually maintain an arbitrarily highlevel of consumption (over and above his current income).His debt however grows at the rate rt:

bt+1 = (1+ rt)bt

which implies that limt!aht ' limt!b

ht ! .

This kind scheme is called a Ponzi nance scheme.If a household is allowed to play such a Ponzi game, then thehouseholds budget constraint becomes meaningless. There is

eectively no budget constraint for the household any more; it canmaintain any arbitrarily high consumption path by playing a Ponzigame.To rule this out, we impose an additional constraint on thehouseholds optimization problem - called the No-Ponzi Game

Condition.Das (Lecture Notes, DSE) DGE Approach February 2-12, 2016 33 / 48

Households Choice Problem: No-Ponzi Game Condition
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One Version ofNo-Ponzi Game (NPG) Condition:

limt!

aht

(1+ r0)(1+ r1)......(1+ rt) = 0.

This No-Ponzi Game condition states that as t! , the presentdiscounted value of an households asset must be non-negative.

Notice that the above condition rules out Ponzi nance scheme for

sure.

If you play Ponzi game then limt!

aht ' limt!bht, when the latter term

is growing at the rate (1+ rt).For simplicity, let us assume interest rate is constant at some r. Then

b

h

t = (1+

r)

t

b.

Plugging this in the LHS of the NPG condition above:

limt!

aht(1+ r)t

' limt!

(bht)

(1+ r)t = lim

t!

(1+ r)tb

(1+ r)t =b< 0.

This surely violates the NPG condition speciedabove.


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(Contd.)

At the same time the NPG condition specied above is lenient enoughto allow for some borrowing.

In fact the condition even permits perpetual borrowing as long asborrowing grows at a rate less than the corresponding interest rate.

To see this, suppose the households borrowing is growing at some

rate g< r such thatbht = (1+ g)

tb.

Plugging this in the LHS of the NPG condition above:

limt!

aht

(1+ r)t ' lim

t!

(bht)

(1+ r)t = lim

t!

(1+ g)tb

(1+ r)t =blim

t!1+ g

1+ rt

.

Notice that g< rimplies that the term

1+ g

1+ r

is a positive

fraction and as t! , 1+g1+rt

! 0.


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(Contd.)

Since bis nite, this implies that in this case

limt

!

aht

(1+r)

t ! 0.

In other words, the NPG condition is now indeed satised - albeit atthe margin!

Economically, this kind of borrowing behaviour implies that the debt

of the agent is not exploding and the agent must have startedrepaying at least some part of it (though not all) from his own pocket!


Households Choice Problem - Revisited:
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After imposing the No-Ponzi Game condition, the householdoptimization problem now becomes:

Max.fcht g

t=0,faht+1g

t=0

t=0

tu

cht

subject to

(i) aht+1 = wt+ (1+ rt)aht c

ht ; a

ht 2 < for all t= 0; a

h0 given.

(ii) The NPG condition.

Here cht is the control variable and aht is the state variable.


Households Choice Problem - Revisited: (Contd.)
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We can now apply the dynamic programming technique to solve the

households choice problem.First let us use constraint (i) to eliminate the control variable andwrite the above dynamic programming problem in terms of the statevariable alone:

Max.faht+1g

t=0

t=0

tun

wt+ (1+ rt)aht a

ht+1

o

Corresponding Bellman equation relating V(ah0 ) and V(ah1 ) is given

by:

V(ah0 ) =Maxfah1 g

hun

w0+ (1+ r0)ah0 a

h1

o+V(ah1 )

i.


Households Problem: Bellman Equation
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More generally, we can write the Bellman equation for any two timeperiodst and t+1 as:

V(aht) = Maxfaht+1g

hun

wt+ (1+ rt)aht a

ht+1

o+V(aht+1)

i.

Maximising the RHS above with respect to aht+1, from the FONC:

u0n

wt+ (1+ rt)aht aht+1o

=V0(aht+1) (6)

Notice that V(aht+1) and V(aht+2) would be related through a similar

Bellman equation:

V(ah

t+1) = Maxfaht+2ghun

wt+1+ (1+

rt+1)ah

t+1 ah

t+2o

+V(ah

t+2)i .

Applying Envelope Theorem on the latter:

V0(aht+1) =u0

nwt+1+ rt+1aht+1 a

ht+2o .(1+ rt+1). (7)


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Combining (5) and (6):

u0n

wt+ rtaht a

ht+1

o

= u0 n

wt+1+rt+1a

h

t+1 ah

t+2o

(1+rt+1).

The above equation implicitely denes a 2nd order dierence equationis aht.

However we can easily convert it into a 2 2 system of rst order

dierence equations in the following way.


Households Problem: Optimal Solutions (Contd.)


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Noting that the terms inside the u0(.) functions are nothing but cht

and cht+1 respectively, we can write the above equation as:

u0cht

=u0

cht+1

(1+ rt+1). (8)

We also have the constraint function:

aht+1 =wt+ (1+ rt)aht c

ht ; a

h0 given. (9)

Equations (7) and (8) represents a 2 2 system of dierenceequations which implicitly denes the optimal trajectories chtt

=0

andaht+1

t=0.

The two boundary conditons are given by the initial condition ah0 , andthe NPG condition.


Optimal Solution Path to Households Problem: An
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Example

Let us look at an explicitly characterisation of the households optimalpaths for a specic example.

Supposeu(c) =log c

Let us also assume that wt= w and rt= r for all t .

Then we can immediately get two dierence equations characterizingthe optimal trajectories for the household as:

cht+1 =(1+ r)cht (10)

andaht+1 = w+ (1+ r)a

ht c

ht ; a

h0 given. (11)

The two equations along with the two boundary conditons can besolved explicitly to derive the time paths ofcht and a

ht.


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Equation (9) is a linear autonomous dierence equation, which can be

directly solved (by iterating backwards) to get the optimalconsumption path as:

cht = t(1+ r)tch0. (12)

However,we still cannot completely characterise the optimal pathbecause we still do not know the optimal value ofch0. (Recall that c

h0

is not given; it is to be chosen optimally).

Here the NPG condition comes in handy in identifying the optimal ch0.

Note that the NPG condition in this case is given by:

limt!

aht(1+ r)t

= 0.




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Now let us take the budget constraint of the household at any futuredate T > 0:

ahT+1 = w+ (1+ r)ahT chT.

Iterating backwards,

ahT+1 = w+ (1+ r)ahT c

hT

= w+ (1+ r)hw+ (1+ r)ahT1 chT1

i chT

= ....

=T

t=0

w(1+ r)Tt

T

t=0

cht(1+ r)

Tt

+ (1+ r)T+1ah0 .

Rearranging terms:

ahT+1(1+ r)T

=T

t=0

w

(1+ r)t

+ (1+ r)ah0

T

t=0

cht

(1+ r)t




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Now let T ! . Then applying the NPG condition to the LHS, weget:

t=0

w

(1+ r)t

+ (1+ r)ah0

t=0

cht

(1+ r)t

= 0

i.e.,

t=0

cht

(1+ r)t 5

t=0

w

(1+ r)t + (1+ r)ah0 . (13)

Equation (12) represents the lifetime budget constraint of thehousehold. It states that when the NPG condition is satised, thenthe discounted life-time consumption stream of the household cannotexceed the sum-total of its discounted life-time wage earnings and the

returns on its initial wealth holding.

It is easy to see that even though we have specied the NPGcondition in the form of an inequality, the households would alwayssatisfy it at the margin such that it holds with strict equality.




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Given that equation (12) holds with strict equality, we can nowidentify the optimal value ofch0.

We had already derived the optimal time path ofcht as:

cht = t(1+ r)tch0.

Using this in equation (12) above, we get:

t=0

t(1+ r)tch0

(1+ r)t

=

t=0

w

(1+ r)t

+ (1+ r)ah0

)

t=0tch0 =

" t=0

w(1+ r)t

+ (1+ r)a

h0#

) ch0 =(1 )

"

t=0

w

(1+ r)t

+ (1+ r)ah0

#.


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So for this particular example, we have been able to explicitly solvefor the optimal consumption path of the households.

But there is a problem that we still need to sort out.

Recall that while discussing the dynamic programming problem wehad specied a transversality condition (TVC) as one of our boundarycondition (Refer to equation (5) specied earlier).

Then in dening the households problem with intra-householdborrowing, we have introduced the NPG condition as anotherboundary condition.

So we now have a problem of plenty: for a 2 2 dynamic system, itseems that we have three boundary conditions!!!

Between the TVC and the NPG condition, which one should we useto characterise the solution?

As it turns out, along the optimal path the NPG condition and theTVC become equivalent.


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48/48

To see this, let us take a closer look at the TVC as had been speciedearlier in equation (5)

In the context of the current problem, this transversality conditionwould be given by (derive this yourself):

limt!

tu0(cht)(1+ rt)aht =0

For our specic example with log utility and constant factor prices,

this condition reduces to

limt!

t 1

cht(1+ r)aht =0

Now given the solution path ofcht , we can further simplify the above

condition to:

limt!

t 1

t(1+ r)tch0(1+ r)aht =0 ) lim

t!

aht(1+ r)t

=0 .

But this is nothing but our earlier NPG condition - now holding with

strict equality!Das (Lecture Notes, DSE) DGE Approach February 2-12, 2016 48 / 48
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