econ 3020 intermediate macroeconomicsxiaohuihuang.weebly.com/.../econ3020_ch11_slides.pdf ·...

ECON 3020 Intermediate MacroeconomicsChapter 11 A Real Intertemporal Model with Investment

Instructor: Xiaohui Huang

Department of EconomicsUniversity of Virginia

c© Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 11 1 / 61

Main Topics

Complete the real intertemporal modelConsumersFirmsGovernment

Study the competitive equilibrium: determination ofReal aggregate output, consumption, investment, andemploymentReal wage, and real interest rate.

Experiments: shocks and their effects on CEMacro shocksFuture expectations


A Brief Overview of the Model

Two periods: current period and future period.Representative consumer:

Consumes, saves and works in the current period;Consumes and works in the future period.

Representative firm:Hires labor and invests in the current period;Hires labor in the future period.

Government:Spends and taxes in current and future periods;Borrows from the credit market.


A Brief Overview of the Model

Competitive Equilibrium: Interactions in Two Markets


Representative Consumer

Time endowment in each period: h units of time.The consumer is a price-taker:

Real wage rate in current period: wReal wage rate in future period: w ′

Real interest rate: r

Exogenous lump-sum taxes: T and T ′.Making choices of:

Consumptions: C and C′

Leisure: ` and `′

Savings in current period: Sp


Representative Consumer: Budget Constraints

Budget constraint in current period:

C︸︷︷︸consumption

+ Sp︸︷︷︸savings

= w(h − `)︸︷︷︸labor income

+ π − T︸︷︷︸non-labor income

Budget constraint in future period:

C′︸︷︷︸consumption

= w ′(h − `′)︸︷︷︸labor income

+ π′ − T ′︸︷︷︸non-labor income

+ (1 + r)Sp︸︷︷︸payoffs on savings

Lifetime budget constraint:

C +C′

1 + r︸︷︷︸PV of lifetime consumption

= w(h − `) + π − T +w ′(h − `′) + π′ − T ′

1 + r︸︷︷︸PV of lifetime wealth


Representative Consumer: Optimal Choices

Consumer’s optimization problem:

maxC,C′,`,`′

U(C,C′, `, `′)

s.t. C +C′

1 + r= w(h − `) + π − T +

w ′(h − `′) + π′ − T ′

1 + rConsumer’s optimizing decision rules are described bythe following marginal conditions:

Current work-leisure decision:MRS`,C = w

Future work-leisure decision:MRS`′,C′ = w ′

Current consumption-savings decision:MRSC,C′ = 1 + r

Consumer’s choice determines Ns and Cd .


Representative Consumer: Current Labor Supply Ns

Consumer’s labor supply in current period, Ns, isdetermined by three factors:

Current real wage, w .Real interest rate, r .Lifetime wealth, we.



Figure 11.1 The Representative Consumer’s Current LaborSupply Curve

w ↑ ⇒ opposing SE and IE⇒ ` ambiguous.Assume SE > IE.Then w ↑ ⇒ ` ↓ and Ns(r) ↑.Current labor supply curve isupward-sloping.



Figure 11.2 An Increase in the Real Interest Rate Shifts theCurrent Labor Supply Curve to the Right

Consider: r1 ↑ to r2.Relative price of current leisure interms of future leisure isp`

p`′=

w(1 + r)

w ′. Why?

r ↑ ⇒ current leisure becomesrelatively more expensive⇒ ` ↓, `′ ↑ and Ns(r) ↑.(Assume SE > IE.)



Figure 11.3 Effects of an Increase in Lifetime Wealth Shifts theCurrent Labor Supply Curve to the Left

Hold r constant.we ↑ ⇒ ` ↑ ⇒ Ns(r) ↓.(Current leisure is a normal good.)


Representative Consumer: Current ConsumptionDemand Cd

Consumer’s demand for consumption goods in currentperiod is determined by three factors:

Current aggregate income, Y .Real interest rate, r .Lifetime wealth, we.

Other demands for consumption goods come from:Firm’s demand for investment goods.Government purchases.



Figure 11.4 The Representative Consumer’s Current Demandfor Consumption Goods Increases with Income

Hold r constant.Y ↑ ⇒ Cd (r) ↑ and Sp ↑⇒ Cd (r) is upward-sloping withrespect to Y .Marginal Propensity to Consume(MPC): slope of Cd (r)

MPC =∂C∂Y

.

Y ↑ ⇒ 0 < ∆C < ∆Y⇒ 0 < MPC < 1.



Figure 11.5 An Increase in the Real Interest Rate Shifts theCurrent Demand for Consumption Goods Down

Consider: r1 ↑ to r2.Recall: MRSC,C′ = 1 + r .r ↑ ⇒ C is relatively more expensive⇒ C ↓, C′ ↑ (SE).r ↑ ⇒ higher return on savings andthus lifetime wealth⇒ C ↑, C′ ↑ (IE).Assume SE > IE .r ↑ ⇒ Cd (r) ↓.



Figure 11.6 An Increase in Lifetime Wealth Shifts the CurrentDemand for Consumption Goods Up

Hold r and Y constant.we ↑ ⇒ Cd (r) ↑.(C is a normal good.)


Representative Firm: Production Functions

The firm produces in both periods using capital and labor.The firm is endowed with K units of capital at thebeginning of current period.Firm’s production function in current period:

Y = zF (K ,N)

Then the firm accumulates for K ′ through investment.Firm’s production function in future period:

Y ′ = z ′F (K ′,N ′)


Representative Firm: Capital Accumulation/Investment

Investment goods are produced from output.Firm’s investment decision: a tradeoff between currentoutput and future production capacity.Accumulation of capital stock:

K ′ = (1− d)K︸︷︷︸capital left at the end of current period

+ I︸︷︷︸investment in capital

d : depreciation rate of capital.


Representative Firm: Profits

Firm’s profit in current period:

π = Y︸︷︷︸current output

− wN︸︷︷︸wage pauyment

− I︸︷︷︸investment

Firm’s profit in future period:

π′ = Y ′︸︷︷︸future output

− w ′N ′︸︷︷︸wage payment

+ (1− d)K ′︸︷︷︸value of capital stock left

Present value of firm’s profits:

V = π +π′

1 + r


Representative Firm: Profit Maximization

Firm’s optimization problem:

maxN,N′,I

V = π +π′

1 + r=

= Y − wN − I +Y ′ − w ′N ′ + (1− d)K ′

1 + r

s.t. K ′ = (1− d)K + I

Firm’s optimizing decision rules are described by thefollowing marginal conditions:

Labor demand choice:

MPN = w and MPN′ = w ′

Investment choice: marginal benefit = marginal cost.


Representative Firm: Investment Decision Id

The optimal level of an economic activity is determined by:Marginal Benefit = Marginal Cost

This is also the case how the representative firm makesthe investment decision.



Marginal benefit from investment:I ↑ ⇒ K ′ ↑ ⇒ π′ ↑.Marginal benefit in terms of the present value of profits:

MB(I) =1

1 + r∂π′

∂I=

11 + r

( MPK ′︸︷︷︸more output produced

+ 1− d︸︷︷︸more capital left

)

Marginal cost of investment:I ↑ ⇒ less output sold⇒ π ↓.Marginal cost in terms of the present value of profits:

MC(I) = −∂π∂I

= 1



Optimal investment rule: MB(I) = MC(I).

MPK ′ + 1− d1 + r

= 1

orMPK ′ − d︸︷︷︸

net return on capital

= r︸︷︷︸real interest rate, return on bonds

r is also the opportunity cost of investment in capital.Intuitition:

There are two assets in the economy, bonds and capital.In equilibrium, return on both assets must be the same.

Note: optimal investment rule depends on MPK ′ , not MPK .



Another way to look at firm’s investment choice:Firm invests until the marginal contribution of investment tothe present value of profits is zero:

∂V∂I

=∂π

∂I︸︷︷︸negative of marginal cost

+1

1 + r∂π′

∂I︸︷︷︸marginal benefit

= 0

This also gives the same optimal investment decision rule.


Representative Firm: Current Labor Demand Nd

Firm’s current labor demand Nd is affected by:Real wage rate: w .Current TFP: z.Initial capital stock: K .



Figure 11.7 The Demand Curve for Current Labor Is theRepresentative Firm’s Marginal Product of Labor Schedule

Firm’s rule of current labor demand:

MPN = w

Diminishing MPN :w ↑ ⇒ MPN ↑ ⇒ Nd ↓Nd is downward-sloping.



Figure 11.8 The Demand Curve for Current Labor Shifts Due toChanges in Current TFP and in the Current Capital Stock

MPN =∂Y∂N

= z∂F (K ,N)

∂N.

z ↑ ⇒ MPN ↑ ⇒ Nd ↑.K ↑ ⇒ MPN ↑ ⇒ Nd ↑.Firm’s choice of future labor demandin the future period is similar to itschoice of current labor demand. Weignore that for simplicity.



Optimal investment Id can be affected by:Real interest rate: r .Future efficiency of capital: MPK ′ .Depreciation rate of capital: d .



Figure 11.9 Optimal Investment Schedule for theRepresentative Firm

Optimal investment rule:MPK ′ − d = r .r ↑ ⇒ MPK ′ ↑ ⇒ K ′ ↓ ⇒Id = K ′ − (1− d)K ↓.Id is downward-sloping.



Figure 11.10 The Optimal Investment Schedule Shifts to theRight if Current Capital Decreases or Future TFP is Expectedto Increase

K ↓ ⇒ given I, K ′ = (1− d)K + I ↓⇒ MPK ′ ↑ ⇒ Id ↑z ′ ↑ ⇒ MPK ′ ↑ ⇒ Id ↑


Variability of Aggregate Investment

In data, aggregate investment expenditures tends to bemore variable than aggregate output or aggregateconsumption.Aggregate consumption is less variable due toconsumption-smoothing behavior.Investment behavior is about firm’s responses to perceivedmarginal rates of return to investment, which is affected byTFP and real interest rate.Real interest rate and anticipated future TFP varysufficiently over the business cycle, so aggregateinvestment is also more variable.


Investment with Asymmetric Information

Reading assignment: textbook pp.392 - pp.396.


Government

Exogenous government purchase in both periods:In current period: G.In future period: G′.

Government finances its purchases through taxation andissuing government bonds.

In current period: T and B.In future period: T ′.

Government’s present-value budget constraint:

G +G′

1 + r= T +

T ′

1 + r


Competitive Equilibrium

Competitive equilibrium consists ofEquilibrium prices: w∗ and r∗.Equilibrium quantities: N∗, Y ∗, C∗, I∗.

Given the prices, consumers optimize⇒ Ns and Cd .Given the prices, firms optimize⇒ Nd and Id .Government holds balanced budget.All markets clear:

Current goods market clears: Y = Cd + Id + G.Current labor market clears: Ns = Nd = N∗.Credit market clears: Sp = B.Future goods market clears.Future labor market clears.



For simplicity, future markets for labor and goods areneglected here.Credit market automatically clears if the current goodsmarket clears (same as Ch9).So only consider two markets: current labor market andcurrent goods market.Construct CE in diagrams:

Current labor market clears⇒ output supply curve Y s.Current goods market clears⇒ output demand curve Y d .Then Y s = Y d means both markets are cleared⇒ Y ∗, r∗.Determine w∗ and N∗ accordingly.


Current Labor Market and Output Supply Curve Y s

On current labor market:Supply side: the representative consumer Ns.Demand side: the representative firm Nd .In equilibrium, two curves intersect.

Construct the output supply curve in (Y , r) plane:Consists of all combinations of current output and realinterest rates, (Y , r).Every point (Y , r) on output supply curve represents anequilibrium in current labor market.



Figure 11.14 Determination of Equilibrium in the Current LaborMarket Given the Real Interest Rate r

Labor demand curve: Nd .Labor supply curve: Ns(r).Labor supply, Ns(r), depends on thereal interest rate r .

r ↑ ⇒ Ns ↑.Equilibrium of current labor market:

Employment: N∗

Real wage: w∗

Aggregate output: Y s



Figure 11.15 Construction of the Output Supply Curve

Output supply curve reflectsrelationship between Y s and r .Consider: r1 ↑ to r2.r ↑ ⇒ Ns(r) shifts out⇒ N∗ ↑ and w∗ ↓ ⇒ Y s ↑Y s is upward-sloping.


Shifts in Output Supply Curve

Reasons for a shift in output supply curve:A shift in current labor demand curve;A shift in current labor supply curve;A shift in production function.

Consider effects of following exogenous changes on theoutput supply curve:

we ↓ ⇒ shift in labor supply curve.z ↑ ⇒ shift in labor demand curve and production function.K ↑ ⇒ shift in labor demand curve and production function.


Shifts in Output Supply Curve: we ↓ (by G ↑ or G′ ↑)Figure 11.16 An Increase in Current or Future GovernmentSpending Shifts the Y s Curve to the Right

G ↑ or G′ ↑ ⇒ T +T ′

1 + r↑

⇒ we ↓ ⇒ Ns ↑⇒ Ns shifts to the right⇒ N∗ ↑ and w∗ ↓, given r⇒ Y s ↑


Shifts in Output Supply Curve: z ↑ (or K ↑)Figure 11.17 An Increase in Current TFP Shifts the Y s Curve tothe Right

z ↑ ⇒ Y s ↑ in two ways:z ↑ ⇒ MPN ↑ ⇒ Nd shifts to right⇒ N∗ ↑ and w∗ ↑ ⇒ Y s ↑z ↑ ⇒ production function shifts up⇒ Y s ↑.

Effects of K ↑ are the same as z ↑.


Current Goods Market and Output Demand Curve Y d

On current goods market:Supply side: output of representative firm Y (currentincome).Demand side: Cd (r), Id (r), G.In equilibrium, Y = Y d = Cd (r) + Id (r) + G.

Construct the output demand curve in (Y , r) plane:Consists of all combinations of current output and realinterest rates, (Y , r).Every point (Y , r) on the output supply curve represents anequilibrium in the current goods market.



Figure 11.18 Determination of Equilibrium in the Current GoodsMarket Given the Real Interest Rate r

Y d = Cd (r) + Id (r) + GOnly Cd (r) is depend on Y .Y ↑⇒ Cd (r) ↑.Slope of Y d is the same as Cd (r),which is the MPC.

In equilibrium, Y d = Y , intersectsthe 45◦ line.



Figure 11.19 Construction of the Output Demand Curve

Output demand curve reflectsrelationship between Y d and r .Consider: r1 ↑ to r2.r ↑ ⇒ Cd (r) ↓ and Id (r) ↓⇒ demand curve on current goodsmarket shifts down⇒ Y d ↓Y d is downward-sloping.


Shifts in Output Demand Curve

Reasons for a shift in output demand curve:A shift in current demand for consumption goods Cd (r);A shift in demand for investment goods, Id (r);A shift in current government purchase, G.

Consider effects of following exogenous changes on theoutput demand curve:

G ↑Present value of taxes ↓Y ′ ↑z ′ ↑K ↓


Shifts in Output Demand Curve: G ↑

Figure 11.20 An Increase in Current Government SpendingShifts the Output Demand Curve to the Right

G ↑ ⇒ Y d = Cd (r) + Id (r) + G ↑⇒ Y ↑ in equilibrium of currentgoods market⇒ Y d shifts to the right.


Shifts in Output Demand Curve: Other Changes

Present value of taxes T +T ′

1 + r↓

T +T ′

1 + r↓ ⇒ we ↑ ⇒ Cd (r) ↑ ⇒ Y d shifts to the right.

Future income Y ′ ↑Y ′ ↑ ⇒ we ↑ ⇒ Cd (r) ↑ ⇒ Y d shifts to the right.

Future TFP z ′ ↑z ′ ↑ ⇒ Id (r) ↑ ⇒ Y d shifts to the right.

Current capital stock K ↓K ↓ ⇒ Id (r) ↑ ⇒ Y d shifts to the right.



Figure 11.21 The Complete Real Intertemporal Model

Two markets clear in competitive equilibrium:In the current labor market: Nd = Ns(r∗)⇒ w∗, N∗, Y s.In the current goods market: Y = Cd (r) + Id (r) + G⇒ Y d .Y d = Y s ⇒ r∗, Y ∗.


Equilibrium Effects

Consider the following experiments:A temporary increase in current government purchase: G ↑.A decrease in current capital stock: K ↓.A temporary increase in current TFP: z ↑.An expected increase in future TFP: z ′ ↑.More severe credit market frictions.Sectoral shock and labor market mismatch.

Study the effects of these shocks on: w∗, N∗, Y ∗ and r∗.


Equilibrium Effects: G ↑

In one-period model in Chapter 5, G ↑ ⇒ C ↓, N ↑ and Y ↑.What’s different in the real intertemporal model?

Additional crowding-out through intertemporal substitution.Both private consumption and investment are crowded out.

Government expenditure multipliers.Now consider a temporary increase in G:

G ↑G′ unchanged



Assume MPC is constant⇒ demand for current goods is astraight line.Effects on demand for currentgoods:

Direct effect:G ↑ ⇒ Cd (r) + Id (r) + G ↑.Y ↑ by ∆G.

Indirecet effect:G ↑ ⇒ we ↓ ⇒ Cd (r) ↓Y ↓ by MPC ·∆G.

Total effect: ∆Y = (1−MPC)∆G.

Y d (in (Y , r) plane) shifts to theright. ∆Y d is the distance AB.



We can calculate equilibrium ∆Y d as follows:

MPC =length(BD)length(AB)

=length(BF )− length(DF )

length(AB)

=Y d

2 − Y d1 − (1−MPC)(G2 −G1)

Y d2 − Y d

1

Solve for Y d2 − Y d

1 :

Y d2 − Y d

1 = G2 −G1

Partial Expenditure Multiplier: ratio of total increase indemand for goods to increase in government spending.

∆Y d

∆G=

Y d2 − Y d

1G2 −G1

= 1


Equilibrium Effects: G ↑Figure 11.22 A Temporary Increase in Government Purchases

On current labor market:G ↑ ⇒ we ↓⇒ Ns(r) shifts to right, given r1.Nd unchanged.

On current goods market:G ↑ ⇒ Y d shifts to the right by ∆G.G ↑ ⇒ N ↑ ⇒ Y s shifts to the right.Y ∗ ↑ for sure.r∗ may rise or fall, but there isstrong theoretical support for r∗ ↑.

r ↑ leads to a further rightward shiftin Ns(r).Total equilibrium effects:

N∗ ↑, w∗ ↓, Y ∗ ↑, r∗ ↑r∗ ↑ ⇒ C∗ ↓ and I∗ ↓


Government Expenditure Multipliers

Partial expenditure multiplier:∆Y d

∆G= 1.

Y s shifts to the right too, but dominated by shift in Y d .In equilibrium, ∆Y ∗ < ∆G.

Total government expenditure multiplier:∆Y ∗

∆G< 1.

Total expenditure multiplier will become smaller ifRightward shift in Y s becomes smallere.g. wealth effect on labor supply falls.Y s becomes steepere.g. intertemporal substitution effect of r on labor supplyfalls.

Keynesians:∆Y ∗

∆G> 1. What’s the problem?

Ignore the negative effect of G ↑ on demand forconsumption goods.Ignore production and intertemporal substitution.



A temporary increase in government spending stimulatesaggregate output.But it comes at a cost:

Consumer consumes less, takes less leisure, and faces alower real wage rate.Current investment spending is lower, which implies lowerfuture capital stock and lower future capacity of production.


Equilibrium Effects: K ↓

Usually, capital stock accumulates through investment, andthis occurs slowly over time.Under some extreme situation, such as war or naturaldisaster, a large decrease in K may occur.


Equilibrium Effects: K ↓

Figure 11.23 A Decrease in Current Capital Stock

On current labor market:K ↓ ⇒ Y = zF (K ,N) ↓⇒ Y s shifts left.K ↓ ⇒ MPN ↓ ⇒ Nd shifts left⇒ Y s shifts left further.

On current goods market:K ↓ ⇒ K ′ ↓ ⇒ MPK ′ ↑ ⇒ Id (r) ↑⇒ Y d shifts rightr∗ ↑ for sure.Y ∗ is ambiguous.

r ↑ leads to a rightward shift of Ns(r).Total equilibrium effects:

w∗ ↓, N∗ ambiguous.r∗ ↑, Y ∗ ambiguous.r∗ ↑ ⇒ C∗ ↓, Id (r) ↓.In equilibrium, I∗ ↑. Why?


Equilibrium Effects: z ↑Figure 11.24 An Increase in Current TFP

On current labor market:z ↑ ⇒ Y = zF (K ,N) ↑⇒ Y s shifts right.z ↑ ⇒ MPN ↑ ⇒ Nd shifts right.Nd ↑ ⇒ Y s shifts right further.

On current goods market:Y d unchanged.In equilibrium, Y ∗ ↑, r∗ ↓.

r ↓ leads to a leftward shift of Ns(r).Total equilibrium effects:

w∗ ↑, N∗ ambiguous. Empirically,effect of r on labor supply is small.Shift in Nd dominates and N∗ ↑.r∗ ↓, Y ∗ ↑.r∗ ↓ ⇒ C∗ ↑, I∗ ↑.

Consistent with business cycle facts.c© Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 11 57 / 61

Equilibrium Effects: z ′ ↑

Expectation of future TFP can be influenced by:News about future events."Animal spirits".

Recent empirical research in macroeconomics supportsthat news about future events is a key determinant ofaggregate economic activity in the present.


Equilibrium Effects: z ′ ↑Figure 11.26 An Increase in Expected Future TFP

On current goods market:z ′ ↑ ⇒ Id (r) ↑ ⇒ Y d shifts right.Y s unchanged.In equilibrium, Y ∗ ↑, r∗ ↑.

On current labor market:r ↑ ⇒ Ns(r) shifts right.Nd unchanged.In equilibrium, N∗ ↑, w∗ ↓.

Effects on consumption C∗:r∗ ↑ ⇒ Cd (r) ↓.Y ∗ ↑ and possibly Y ′ ↑ ⇒ Cd (r) ↑.In equilibrium, C∗ is ambiguous.

Effects on investment I∗:z ′ ↑ ⇒ Id (r) ↑.r∗ ↑ ⇒ Id (r) ↓.In equilibrium, I∗ must rise. Why?


Equilibrium Effects: Other Changes

Reading assignment: pp.421 - pp.430.


Summary

Y r N w C I

G ↑ ↑ ↑ ↑ ↓ ↓ ↓K ↓ ? ↑ ? ↓ ↓ ↑z ↑ ↑ ↓ ↑ ↑ ↑ ↑z ′ ↑ ↑ ↑ ↑ ↓ ? ↑


econ 3020 intermediate macroeconomicsxiaohuihuang.weebly.com/.../econ3020_ch11_slides.pdf ·...

Documents