A Real Intertemporal Model with Investment Part 1 - …A Real Intertemporal Model with Investment...

A Real Intertemporal Model with InvestmentPart 1

Chapter 9

Topics in Macroeconomics 2

Economics DivisionUniversity of Southampton

April 2010

Chapter 9, Part I 1/29 Topics in Macroeconomics

The Representative ConsumerThe Representative Firm

Government, Mkt Clearing and Competitive Equilibrium

Goals in this Chapter

I Construct a real intertemporal model that will serve as abasis for studying money and business cycles in Chapters10-12.

I Understand the investment decision of the firm.

I Show how macroeconomic shocks affect the economy.

I Focus on the implications of future expectations for currentmacroeconomic performance, and the difference betweentemporary and permanent shocks.




Model Ingredients

I Current and future periods.

I Representative Consumer - labour/leisure choiceconsumption/savings decision

I Representative Firm - hires labour and invests in currentperiod, hires labour in future

I Government - spends and taxes in present and future, andborrows on the credit market.




Consumer’s Budget ConstraintCurrent labour supplyCurrent demand for consumption goods

Outline

The Representative ConsumerConsumer’s Budget ConstraintCurrent labour supplyCurrent demand for consumption goods

The Representative FirmProduction and InvestmentProfits and current labour demandRepresentative Firm’s Investment Decision

Government, Mkt Clearing and Competitive EquilibriumGovernmentMarket clearingCompetitive equilibrium





Consumer’s budget constraints:

1. Consumer’s current-period budget constraint:

C + Sp = w(h − l) + π − T

2. Consumer’s future-period budget constraint:

C′ = w ′(h − l ′) + π′− T ′ + (1 + r)Sp

3. Consumer’s lifetime budget constraint:

C +C′

1 + r= w(h − l) + π − T +

w ′(h − l ′) + π′− T ′

1 + r





Marginal conditions for Consumer:

1. Consumer’s current-period marginal condition:

MRSl ,C = w

2. Consumer’s future-period marginal condition:

MRSl ′,C′ = w ′

3. Consumer’s intertemporal marginal condition:

MRSC,C′ = 1 + r





Outline








Current labour supply

Current labour supply determined by current real wage, w ,interest rate, r , and lifetime wealth, we

I Current labour supplied increases with the real wage(substitution effects are assumed to dominate incomeeffects).

I Current labour supplied increases with the real interestrate, through intertemporal substitution of leisure.

I Current labour supplied decreases with lifetime wealth(e.g. taxes fall).





Representative Consumer’s Current Labour Supply





An Increase in the Real Interest Rate Shifts theCurrent Labour Supply Curve to the Right





Outline








Current demand for consumption goods

Current demand for consumption goods determined by interestrate, r , and lifetime wealth, we, which is affected by presentvalue of current and future income and taxes

I Current demand for consumption goods as a function ofcurrent income, Y , is increasing, with a MPC less than 1.

I Current demand for consumption goods as a function ofcurrent income shifts down if the interest rate increases(assuming again that substitution effect dominates)

I Current demand for consumption goods as a function ofcurrent income shifts up if lifetime wealth increases(e.g. decrease in present value of taxes; increase in futureincome)




Production and InvestmentProfits and current labour demandRepresentative Firm’s Investment Decision

Outline








Firm’s production and investment:

1. Firm’s current-period production function:

Y = zF (K , N)

2. Firm’s future-period production function:

Y ′ = z′F (K ′, N ′)

3. Evolution of the firm’s capital stock:

K ′ = (1 − d)K + I





Firm’s objective: present value profits

1. Firm’s current-period profits:

π = Y − wN − I

2. Firm’s future-period profits:

π′ = Y ′

− w ′N ′ + (1 − d)K ′

3. The firm maximizes the present value of profits:

V = π +π

′

1 + r

by choosing current labour demand, N, future labourdemand, N ′, and current investment, I.





Outline








Intra-temporal decisions for firm:

I Firm’s intra-temporal marginal conditions:

MPN = w MPN′ = w ′

I Current labor demand is decreasing in the current wage.

I Current labor demand shifts out (right) if TFP, z, or Kincrease because the MPN increases for a given N.

How does the firm choose K ′ or I?





Representative Firms’s Current Labour Demand Curve





Outline








Representative firm’s inter-temporal decision

1. Marginal cost: MCI = 1

I The firm has to give up one unit of output today for oneextra unit of capital tomorrow/investment today.

2. Marginal benefit (present value):

MBI =MP ′

K + (1 − d)

1 + r

I The firm produces MP′

K units of output more tomorrow forone extra unit of capital tomorrow/investment today.

I The firm can liquidate the remaining stock of capital.





Representative firm’s inter-temporal decision

3. Firm’s intertemporal marginal condition:

MBI =MP ′

K + (1 − d)

1 + r= 1 = MCI

Simplifying, we get net marginal product of K equals r

(opportunity cost → bonds)

MP ′

K − d = r





Representative Firm’s Investment Schedule





The Optimal Investment Schedule

The optimal investment schedule shifts to the right if currentcapital decreases or future total factor productivity is expectedto increase. (see numerical example in Williamson)




GovernmentMarket clearingCompetitive equilibrium

Outline








The government’s present-value budget constraint:

G +G′

1 + r= T +

T ′

1 + r

Recall:I The budget constraint in the future period implies that

B =T ′

− G′

1 + r

I Replace this expression for B in the current period budget constraint to get

G = T +T ′

− G′

1 + r︸︷︷︸

B

I Rearranging gives the government present value budget constraint above. The LHS is the present value ofspending, which must be equal to the present value of taxes collected on the RHS





Outline








All Markets clear

Labour markets:Period 1: N = Ns = Nd

Period 2: N ′ = Ns′ = Nd ′

Goods markets:Period 1: C + G + I = Y

Period 2: C′ + G′ = Y ′ + (1 − d)K ′

Credit market:Sp = B + I





Outline








Definition: Competitive Equilibrium

A competitive equilibrium is a set of endogenous quantities(C,C′,N,N ′,Y ,Y ′,T ,T ′,B) and endogenous prices (w ,w ′,r ) suchthat, given exogenous variables (z,K ,G,G′), the following holds:

1. For the representative consumer, given (w , w ′, r , T , T ′, π, π′),

the bundle (C, C′, Ns

, Ns′, Sp) maximizes the consumer’s utility

subject to his/her present value budget constraint

2. For the representative firm, given (w , w ′, r , K ), labour demand(Nd

, Nd′) and investment (I) maximizes present value profits.

3. The government present value budget constraint holds.

4. Markets clear:Labour markets: N = Ns = Nd and N ′ = Ns′ = Nd′

Goods markets: C + G + I = Y and C′ + G′ = Y ′ + (1 − d)K ′

Credit market: Sp = B + I


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