Chapter 9

Chapter 9

A Real

Intertemporal

Model with

Investment

Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Chapter 9 Topics

• Construct a real intertemporal model that will serve as a basis for studying money and business cycles in

Chapters 10-12.

• Understand the investment decision of the firm.

• Show how macroeconomic shocks affect the economy.

• Focus on the implications of future expectations for current macroeconomic performance, and the difference between temporary and permanent shocks.


9-2

Real Intertemporal Model

• Current and future periods.

• Representative Consumer – consumption/savings decision

• Representative Firm – hires labor and invests in current period, hires labor in future

• Government – spends and taxes in present and future, and borrows on the credit market.


9-3

Model - Consumer

• Time endowment in each period , h.

• Real wage in current and future period, (w, w’).

• Lump-sum taxes (T, T’)

• Dividend incomes (∏, ∏’).

• Price takers: w, w’ and r are taken as given.

• Two decisions:

– Saving in current period.

– (C, l) in current period and (C’, l’) in future period.


9-4

Equation 9.1

Consumer’s current-period budget constraint:


9-5

Equation 9.2

Consumer’s future-period budget constraint:


9-6

Equation 9.3

Consumer’s lifetime budget constraint:


9-7

Equation 9.4

Consumer’s current-period marginal condition:

Marginal condition for work-leisure decision from Ch. 4.


9-8

Equation 9.5

Consumer’s future-period marginal condition:


9-9

Equation 9.6

Consumer’s intertemporal marginal condition:

Marginal Condition for consumption-saving decision in Ch.8.


9-10

Tradeoff of Intertemporal Leisure

Decision

• MRS l , l’

= w ( 1 + r ) / w’


9-11

Consumer’s Current Labor

Supply Behavior

• Current labor supplied increases with the real wage (substitution effects are assumed to dominate income effects).

• Labor supply increases with an increase in the real interest rate, through an intertemporal substitution effect.

• An increase in lifetime wealth (e.g. taxes fall) reduces labor supply.


9-12

Figure 9.1 The Representative

Consumer’s Current Labor Supply

Curve


9-13

Figure 9.2 An Increase in the Real

Interest Rate Shifts the Current Labor

Supply Curve to the Right


9-14

Figure 9.3 Effects of an Increase in Lifetime Wealth


9-15

Equation 9.7

Firm’s current-period production function:

N is choice variable in current period.

That is, firms choose N to maximize current profits.

K is stock of capital in current period and firms take it as given.


9-16

Investment

• Def: goods and service that are produced in current period, yet not consumed.

• Investment in current period.

– One unit of consumption good can be transformed into one unit of investment good.

• Since I uses some of c, c drops. Since I enhances future period’s productive capacity (recall that MP

K

>0), c’ increases.

– I represents tradeoff between c and c’ (or between ∏, ∏’).


9-17

Equation 9.9

Evolution of the firm’s capital stock:


9-18

Equation 9.8

K’ = (1 – d) K + I.

I is investment made in current period, and d is depreciation rate.


9-19

Equation 9.10

Firm’s current-period profits:

I: capture the foregone profits in current period since investing in I units of capital requires I units of consumption goods in current period. The profit drops by I.


9-20

Equation 9.11

Firm’s future-period profits:

(1 – d) K’ : at the end of future period, capital can be liquidated in market.


9-21

Equation 9.12

The firm maximizes the present value of profits,

Firms profit maximizing problem: given w, w’ and r, maximizing PV of profits by choosing N, N’, and I.


9-22

The Firm’s Labor Demand

As in Chapter 4, the firm’s labor demand schedule is the marginal product of labor for the firm, which is downward sloping.

MP

N

= w.

That is, the marginal product of N equals real wage.


9-23

Figure 9.4 The Demand Curve for Current

Labor Is the Representative Firm’s Marginal

Product of Labor Schedule


9-24

Figure 9.5 The Current Demand Curve for Labor

Shifts Due to Changes in Current Total Factor

Productivity z and in the Current Capital Stock K


9-25

The Representative Firm’s

Investment Decision

• The firm invests to the point where the marginal benefit from investment equals the marginal cost

(MB=MC).

• This decision process is similar to the optimal demand for labor in current period.


9-26

Equation 9.13

The marginal cost of investment is 1, as the firm gives up one unit of current profits for each unit it invests, so:


9-27

Equation 9.14

The marginal benefit of investment is the marginal product of future capital plus the quantity of capital that will be left in the future after depreciation, all discounted back to the present:


9-28

Equation 9.15

The firm’s optimal investment rule, obtained by equating the marginal benefit and marginal cost of investment:


9-29

Equation 9.16

Simplified optimal investment rule:

Investing in K until the net marginal product of K equals real interest rate.

Intuition: opportunity cost of investing in K is real interest rate, the rate of return on other forms of assets, such as bond, in economy.

If firms do not invest, but trade output for bond. Then 1 unit of output in current period gives firms (1 + r) units of future period goods.


9-30

Figure 9.6 Optimal Investment

Schedule for the Representative Firm


9-31

Figure 9.7 The Optimal Investment Schedule Shifts to the Right if Current Capital Decreases or Future Total

Factor Productivity Is Expected to Increase


9-32

Equation 9.18

The government’s present-value budget constraint:


9-33

Derive Competitive Equilibrium

• Labor market: supply = demand.

• Derive the relation between Y s and r.

• Derive the relation between Y d and r.

• Good market: supply = demand.


9-34

Figure 9.10

Determination of Equilibrium in the Labor Market Given the Real

Interest Rate r

N s is upward-sloping under the assumption that sub. effect dominates income effect.

Position of N s is affected by r and we.

Given r, equilibrium N is determined by

N d and N s .

Given z and K, the equilibrium N determines equilibrium value of Y.


9-35

Figure 9.11

Construction of the Output Supply Curve

Y s : output net of current period I.

IS curve: relation between Y s and r.

IS curve is upward-sloping.

At each point on Y s , the labor market is in equilibrium. That is, for a given r, LM is in equilibrium when firms produce Y.


9-36

Figure 9.12

An Increase in Current or

Future Government Spending Shifts the Y s Curve


9-37

Figure 9.13

An Increase in Current

Total Factor Productivity Shifts the Y s

Curve


9-38

Equation 9.19

Total demand for output equals demand for consumption goods plus demand for investment goods, plus demand for goods from government:

So I d and G are both independent of Y d .

C d and I d are both negatively related to r.

As a result, when r increases, Y d decreases.


9-39

Figure 9.14

The Output

Demand Curve


9-40

Figure 9.15

A Shift in the

Output Demand Curve


9-41

Figure 9.16

The Complete

Real Intertemporal Model


9-42

Define of Competitive Equilibrium

CE is a set of endogenous quantities (c*, I*, N d *, N s *, Y*) and endogenous prices, such as w and r, such that

• Given (w*, r*, y*), (c*, l* or N s *) solves consumers’ utility max problem in current period.

• Given (w*, r*), (N d *, I*) solves firms’ profit max problem in current period.

• Labor market clears where N s = N d =N*, and the real wage w* in current period is determined by intersection of N d and N s (r).

• Good market clears where Y s = Y d =Y*, and the real interest rate r* is determined by intersection of Y d and Y s .


9-43

Experiments Using the Real

Intertemporal Model

•

G increases temporarily.

•

G increases permanently.

•

K decreases.

• z increases.

• z’ increases.


9-44

Temporary Increase in

Government Purchases

• Output increases, real interest rate increases, real wage falls, consumption and investment decrease, employment rises.

• Government spending crowds out both consumption and investment.


9-45

Figure 9.17

A Temporary Increase in

Government Purchases


9-46

Permanent Increase in

Government Purchases

• Output increases by the amount of the increase in government spending.

• No change in the real interest rate, investment and consumption unchanged, real wage falls, employment rises.

• No crowding out of

C and I , but leisure falls for the consumer.


9-47

Figure 9.18

A Permanent Increase in

Government Expenditures


9-48

A Decrease in the Current

Capital Stock

• This could arise due to a war or natural disaster.

• Output may rise or fall, depending on how large the output demand effect is relative to the output supply effect.

• The real interest rate rises, the real wage falls, employment may rise or fall.


9-49

Figure 9.20 The Equilibrium Effects of a

Decrease in the Current Capital Stock


9-50

Current Total Factor

Productivity Increases

• Real interest rate falls, consumption and investment rise, employment rises, real wage rises.

• Productivity shocks are a potential explanation for business cycles – see Chapter 11.


9-51

Figure 9.21

The Equilibrium Effects of an

Increase in Current Total Factor

Productivity


9-52

Total Factor Productivity

Expected to Increase in Future

• Output demand curve shifts right.

• Real interest rate rises, investment increases, consumption may rise or fall, employment rises, real wage falls, output rises.

• Important in explaining investment boom in the

1990s.


9-53

Figure 9.22

The Equilibrium Effects of an Increase in Future Total Factor

Productivity


9-54

Figure 9.23

Percentage Deviations

From Trend in GDP and Investment,

1990 –2006


9-55

Figure 9.24

Investment as a

Percentage of GDP, 1990 –2006


9-56

Figure 9.25

S and P 500

Stock Price Index, 1990 –2005


9-57

Chapter 9

Chapter 9

A Real

Intertemporal

Model with

Investment

Model - Consumer

Tradeoff of Intertemporal Leisure

Decision

Consumer’s Current Labor

Supply Behavior

Figure 9.3 Effects of an Increase in Lifetime Wealth

Investment

The Representative Firm’s

Investment Decision

Derive Competitive Equilibrium

Construction of the Output Supply Curve

The Output

Demand Curve

A Shift in the

Output Demand Curve

The Complete

Real Intertemporal Model

Define of Competitive Equilibrium

Experiments Using the Real

Intertemporal Model

Temporary Increase in

Government Purchases

Permanent Increase in

Government Purchases

A Decrease in the Current

Capital Stock

Current Total Factor

Productivity Increases

Total Factor Productivity

Expected to Increase in Future

Figure 9.24

Investment as a

Percentage of GDP, 1990 –2006

S and P 500

Stock Price Index, 1990 –2005

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