Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Chapter 9 A Real Intertemporal Model with Investment

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 9-2 Chapter 9 Topics Construct a real intertemporal model that will serve as a basis for studying money and business cycles in Chapters 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.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 9-3 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.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 9-4 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.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 9-5 Equation 9.1 Consumer’s current-period budget constraint:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 9-6 Equation 9.2 Consumer’s future-period budget constraint:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 9-7 Equation 9.3 Consumer’s lifetime budget constraint:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 9-8 Equation 9.4 Consumer’s current-period marginal condition: Marginal condition for work-leisure decision from Ch. 4.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 9-9 Equation 9.5 Consumer’s future-period marginal condition:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 9.6 Consumer’s intertemporal marginal condition: Marginal Condition for consumption-saving decision in Ch.8.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Tradeoff of Intertemporal Leisure Decision MRS l, l’ = w ( 1 + r ) / w’

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.1 The Representative Consumer’s Current Labor Supply Curve

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.2 An Increase in the Real Interest Rate Shifts the Current Labor Supply Curve to the Right

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.3 Effects of an Increase in Lifetime Wealth

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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 ∏, ∏’).

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 9.9 Evolution of the firm’s capital stock:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 9.8 K’ = (1 – d) K + I. I is investment made in current period, and d is depreciation rate.

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 9.11 Firm’s future-period profits: (1 – d) K’ : at the end of future period, capital can be liquidated in market.

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

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.4 The Demand Curve for Current Labor Is the Representative Firm’s Marginal Product of Labor Schedule

Copyright © 2008 Pearson Addison-Wesley. All rights reserved 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

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

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

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 9.15 The firm’s optimal investment rule, obtained by equating the marginal benefit and marginal cost of investment:

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.6 Optimal Investment Schedule for the Representative Firm

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.7 The Optimal Investment Schedule Shifts to the Right if Current Capital Decreases or Future Total Factor Productivity Is Expected to Increase

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equation 9.18 The government’s present-value budget constraint:

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

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

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.12 An Increase in Current or Future Government Spending Shifts the Y s Curve

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.13 An Increase in Current Total Factor Productivity Shifts the Y s Curve

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.14 The Output Demand Curve

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.15 A Shift in the Output Demand Curve

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.16 The Complete Real Intertemporal Model

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Experiments Using the Real Intertemporal Model G increases temporarily. G increases permanently. K decreases. z increases. z’ increases.

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.17 A Temporary Increase in Government Purchases

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.18 A Permanent Increase in Government Expenditures

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.20 The Equilibrium Effects of a Decrease in the Current Capital Stock

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.21 The Equilibrium Effects of an Increase in Current Total Factor Productivity

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

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.22 The Equilibrium Effects of an Increase in Future Total Factor Productivity

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.23 Percentage Deviations From Trend in GDP and Investment, 1990–2006

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.24 Investment as a Percentage of GDP, 1990–2006

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Figure 9.25 S and P 500 Stock Price Index, 1990–2005