1. A Real Intertemporal Model with Investment
Chapter 11
A Real Intertemporal Model with Investment
Macroeconomics
6th Edition
Stephen D. Williamson
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2. Learning Objectives, Part I
11.1 Explain the decisions made by the representative consumer in the real intertemporal model.
11.2 Explain the decisions made by the representative firm in the real intertemporal
model.
This chapter takes what we have learned about labor market behavior from Chapters 4 and 5, and about consumption-savings behavior from Chapters 9 and 10, to put together a working model of the macroeconomy that will be used in later chapters. We add one thing, though, which is investment behavior.
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3. Learning Objectives, Part II
11.3 Show how the firm’s investment decision is structured, and determine how
changes in the environment faced by the firm affect investment.
11.4 Construct the output supply curve.
11.5 Construct the output demand curve.
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4. Learning Objectives, Part III
11.6 Show how a competitive equilibrium is determined in the real intertemporal model.
11.7 Use the real intertemporal model to explain the effects of particular shocks to the economy.
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5. 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
This is a two period model with a representative firm, a representative consumer and a government. The economy is closed – not international trade. Also, no monetary exchange. Money gets added in Chapter 12, so we can think about monetary policy.
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6. Representative Consumer’s Budget Constraints
Consumer’s current-period budget constraint:
Consumer’s future-period budget constraint:
Start with the consumer, who has budget constraints for the current and future period. The consumer makes labor supply decisions in the current and future periods, and a consumption/savings decision in the current period.
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7. Consumer’s Lifetime Budget Constraint
As in Chapter 9, we can collapse the consumer’s two constraints into a single lifetime budget constraint.
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8. Marginal Conditions for the Consumer
Current Period:
Future Period:
Intertemporal Choice:
The consumer makes three choices: current labor supply, future labor supply, current savings. Each can be summarized by a standard marginal condition.
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9. 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.
We will leave out the future labor supply decision for simplicity. Current labor supply is increasing in the real wage, increasing in the real interest rate (intertemporal substitution effect), and decreasing in lifetime wealth.
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10. Figure 11.1 The Representative Consumer’s Current Labor Supply Curve
The figure shows the current labor supply curve, assuming that the substitution effect dominates the income effect.
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11. Figure 11.2 An Increase in the Real Interest Rate Shifts the Current Labor Supply Curve to the Right
Labor supply shifts right if the real interest rate increases – this is important. This is the intertemporal substitution effect. Just as current consumption falls when the real interest rate rises, due to intertemporal substitution, so does current leisure.
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12. Figure 11.3 Effects of an Increase in Lifetime Wealth
An increase in lifetime wealth reduces labor supply. For example, an increase in current or future taxes reduces lifetime wealth.
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13. The Current Demand for Consumption Goods
MPC = marginal propensity to consume – the increase in demand for consumption goods induced by a one-unit increase in current real income.
Intertemporal substitution: the demand for consumption goods decreases with an increase in the real interest rate (Chapter 9)
An increase in lifetime wealth (e.g. taxes fall) increases the demand for consumption goods.
For the current demand for consumption goods, three things matter: current income, the real interest rate, and lifetime wealth (taxes for example).
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14. Figure 11.4 The Representative Consumer’s Current Demand for Consumption Goods Increases with Income
The figure shows the demand for consumption goods as a function of current income. The slope of the curve is the MPC (marginal propensity to consume).
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15. Figure 11.5 An Increase in the Real Interest Rate Shifts the Demand for Consumption Goods Down
Due to intertemporal substitution, the real interest rate causes the demand for current consumption goods to fall.
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16. Figure 11.6 An Increase in Lifetime Wealth Shifts the Demand for Consumption Goods Up
An increase in lifetime wealth increases the demand for consumption goods.
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17. Firm’s Current and Future Production Technology
The representative firm produces using labor and capital in the current and future periods. In principle, TFP can be different in the current and future periods. We’ll assume that z’ is known in the current period.
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18. Evolution of the Firm’s Capital Stock
As in the Solow growth model, there is depreciation of capital and new investment.
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19. Firm’s Current and Future Profits
For the firm, current profits are the firm’s output minus wage payments, minus investment. Future profits are output minus wage payments plus the undepreciated quantity of capital left at the end of the future period.
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20. The Firm Maximizes the Present Value of Profits
The firm maximizes the present value of profits over the two periods – it maximizes the value of the firm at the first date.
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21. 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.
As we learned in Chapter 4, the marginal product schedule is the same as the firm’s demand curve for labor.
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22. Figure 11.7 The Demand Curve for Current Labor Is the Representative Firm’s Marginal Product of Labor Schedule
The demand for labor is downward-sloping.
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23. Figure 11.8 The Current Demand Curve for Labor Shifts Due to Changes in z and K
As in Chapter 4, the demand for labor schedule shifts right if productivity or capital stock is larger.
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24. The Representative Firm’s Investment Decision
The firm invests to the point where the marginal benefit from investment equals the marginal cost.
As with any economic decision, the firm invests so as to maximize net benefits – i.e. it sets marginal benefit equal to marginal cost.
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25. Marginal Cost of Investment
The marginal cost of investment is 1, as the firm gives up one unit of current profits for each unit it invests, so:
In this case, the marginal cost of investment is 1, which is the number of units of present-value profit the firm gives up to invest one unit.
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26. Marginal Benefit of Investment
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:
The marginal benefit of investment has two components: (i) the firm can produce more in the future period; (ii) the firm has more capital at the end of the future period, which it can liquidate.
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27. Optimal Investment Rule for the Firm
The firm’s optimal investment rule, obtained by equating the marginal benefit and marginal cost of investment:
In setting the marginal cost of investment equal to marginal benefit, the firm derives the rule that the marginal product of future capital, net of depreciation, should equal the real interest rate. That is, the real interest rate plays the role of the opportunity cost of investing.
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28. Figure 11.9 Optimal Investment Schedule for the Representative Firm
The figure shows the optimal investment schedule for the firm, which is decreasing in the real interest rate. If the interest rate is lower, the firm invests more, since the opportunity cost of investment has fallen.
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29. Figure The Optimal Investment Schedule Shifts to the Right if K Decreases or z’ Is Expected to Increase
If the current quantity of capital were lower, this increases the marginal benefit of investing today. As well, higher future TFP increases the future marginal product of capital and increases investment for each r.
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30. Investment with Asymmetric Information and the Financial Crisis
Suppose many firms in the economy – some good, some bad.
Bad firms borrow in the credit market, consume the proceeds as executive compensation, then default.
Lending rate of interest is r, loan rate is rl
Asymmetric information: Lenders cannot distinguish good from bad firms.
Suppose an asymmetric information setup like the one in Chapter 10, except for firms rather than consumers.
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31. Asymmetric Information and Investment
rl-r = x, where x is a default premium to compensate for borrowers who default.
x is an interest rate spread that reflects the severity of the asymmetric information problem.
For firms that borrow to invest,
Letting x denote the margin between borrowing and lending rates, we include x in the firm’s optimal investment rule. Then, given r, higher x implies less investment.
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32. Asymmetric Information and Investment
When the asymmetric information problem worsens, as in the financial crisis, x increases.
Given the interest rate, this reduces the borrowing firm’s optimal quantity of investment.
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33. Figure 11.11 The Effect of an Increased Default Premium on a Firm’s Optimal Investment Schedule
Higher credit market friction acts to shift the investment schedule to the left.
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34. Figure 11.12 Investment and the Interest Rate Spread
In the figure, we can see how a higher interest rate spread tends to be associated with lower investment in the aggregate, just as in the theory.
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35. Figure 11.13 Scatter Plot: Investment vs. Interest Rate Spread
This is a scatter plot of the same data as in the previous figure.
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36. The Government’s Present-Value Budget Constraint
The last element of our model is the government’s present-value budget constraint, just as in Chapter 9.
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37. Figure 11.14 Determination of Equilibrium in the Labor Market Given the Real Interest Rate r
The real interest determines the position of the labor supply curve. Then, demand and supply in the labor market determine the real wage and the quantity of labor. Then, the production function determines output, given N.
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38. Figure 11.15 Construction of the Output Supply Curve
Part of our model will be an output supply relation, which is an upward sloping curve in the right-hand panel. A higher interest rate shifts the labor supply curve right, which increases the quantity of labor in equilibrium, which increases output. So, higher real interest rates are associated with higher output, due to the intertemporal substitution effect.
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39. Figure 11.16 An Increase in Current or Future Government Spending Shifts the Ys Curve
We want to understand how the output supply curve shifts. First, if government spending goes up, in the present or the future, this increases the present value of taxes, reduces lifetime wealth and increases labor supply, shifting the output supply curve right.
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40. Figure 11.17 An Increase in Current Total Factor Productivity Shifts the Ys Curve
An increase in z shifts the labor demand curve to the right and shifts up the production function. Output is higher for any interest rate, so the output supply curve shifts right.
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41. Figure 11.18 The Demand for Current Goods
Determining the output demand curve is a bit tricky. Consumption demand depends on income, but consumption demand is also part of income – there is a multiplier effect on demand. Where the curve intersects the 45 degree line is determines the level of income that just supports total demand for that level of income.
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42. Figure 11.19 Construction of the Output Demand Curve
A higher real interest rate reduces the demand for consumption and investment goods, shifting the curve down in the left-hand panel. This determines a lower level of total demand for output. Therefore, the output demand curve is downward sloping.
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43. What Shifts the Output Demand Curve to the Right?
A decrease in the present value of taxes.
An increase in future income.
An increase in future total factor productivity.
A decrease in the current capital stock.
The slide lists the set of factors that shift the output demand curve. Now we know everything we need to do experiments in this model.
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44. Figure 11.20 The Output Demand Curve Shifts to the Right if Current Government Spending Increases
The effect of an increase in government spending on the output demand curve is a bit tricky, as we have to account for taxes. G goes up, so taxes go up in the present and/or in the future to finance this. As a result, the demand for consumption goods goes down, but not by as much as the increase in G. Therefore, the output demand curve shifts right.
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45. Figure 11.21 The Complete Real Intertemporal Model
The model shows the complete real intertemporal model. The left panel is the labor market, and the right panel is the goods market. The model determines output, the real interest rate, the real wage, and employment.
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46. The Equilibrium Effects of a Temporary Increase in G
Question: What is the rightward shift in the output demand curve due to an increase in G?
Answer: The curve shifts to the right one-for-one with the increase in G.
The demand multiplier is one.
Suppose G increases temporarily. How far to the right does the output demand curve shift? We can determine that the shift is just the increase in government spending. Therefore, the demand multiplier is one.
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47. Government spending crowds out both consumption and investment.
Equilibrium Effects
Output increases, real interest rate increases, real wage falls, consumption and investment decrease, employment rises.
Government spending crowds out both consumption and investment.
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48. Figure 11.22 A Temporary Increase in Government Purchases
When G goes up, the output demand curve shifts right, and the output supply curve also shifts right, because of the increase in taxes, and the effect on labor supply. In the right panel, we show the supply curve shifting less than the demand curve. This is because the wealth effect on labor supply is small. Then in the right panel, output must go up and the real interest rate goes up. In the left panel, we then need to account for the effect of the real interest rate on labor supply. The real wage must fall, and employment goes up. Investment goes down (r goes up) and consumption must fall given what we know about the rightward shift in the output demand curve. Government spending crowds out investment and consumption.
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49. Important Points
The total multiplier – the ratio of the equilibrium increase in Y, to the increase in G – must be less than 1.
In rudimentary Keynesian analysis, the multiplier is greater than 1. Why? That type of analysis does not take account of: (i) how the extra output is to be produced; (ii) the effect of taxes (present and future) on consumption.
But Keynesian analysis also has sticky prices or wages, typically – see Chapter 14.
The total multiplier is less than one – output increases by less than the increase in government spending. In Keynesian analysis, we can sometimes get multipliers greater than one.
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50. A Decrease in the Current Capital Stock, K
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.
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51. Figure 11.24 The Equilibrium Effects of a Decrease in the Current Capital Stock
If some capital stock is destroyed, there are two countervailing effects on output. Investment demand goes up because of the fall in K, so output demand shifts right. But output supply shifts left because of the fall in K. The real interest rate must rise, but output could rise or fall. The real wage falls, but employment could rise or fall.
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52. 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 13.
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53. Figure 11.25 The Equilibrium Effects of an Increase in Current Total Factor Productivity
This is a key experiment, as TFP shocks are important for business cycles. An increase in z shifts output supply right, reducing the real interest rate and increasing output. Consumption must rise, and investment must rise. There is a leftward shift in labor supply (r falls), and labor demand shifts right. On net, the real wage and employment go up, as the intertemporal substitution effect on labor supply is assumed to be small.
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54. Output demand curve shifts right.
Total Factor Productivity Expected to Increase in Future – A News Shock
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 in the United States.
News shocks are potentially important. Firms are forward-looking in making their investment decisions. In this case, suppose firms anticipate that TFP will increase in the future.
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55. Figure 11.27 The Equilibrium Effects of an Increase in Future Total Factor Productivity
Investment demand goes up, and the output demand curve shifts right. The real interest rate increases, and output increases. In the left panel the labor supply curve shifts right. The real wage falls and employment rises. Thus, events anticipated for the future matter for current economic activity.
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56. Figure 11.28 Percentage Deviations from Trend in Investment and a Relative Stock Price Index
People make bets on the future productivity of firms in the stock market. Here, we can see an effect consistent with our news shock, in that booms in the stock market are associated with booms in investment.
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57. Credit Market Frictions and the Financial Crisis
One feature of the financial crisis was more severe credit market frictions – asymmetric information and limited commitment (as studied in Chapter 10).
More severe credit market frictions reduce the demand for goods by consumers, and increase labor supply.
Output demand curve shifts to the left, output supply curve to the right.
Real interest rate falls, output could rise or fall (but demand effect should be stronger).
This is an experiment which can capture some of what was going on during the financial crisis.
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58. Figure 11.29 The Effect of More Severe Credit Market Frictions
With more severe credit market frictions investment demand declines, and labor supply goes up. This shifts the output demand curve left and the output supply curve right. The supply effect is assumed to be smaller, so the safe real interest rate falls, and output contracts. Employment goes down and the real wage rises.
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59. Sectoral Shocks and Labor Market Mismatch
“Jobless recoveries” a feature of the last 3 recessions.
Could be due to sectoral reallocation – changes in labor markets and the structure of production giving rise to movement of factors of production across sectors.
Adds friction to the labor market – labor demand and labor supply curves shift to the left. Output supply shifts to the right.
Sectoral reallocation is the process by which factors of production migrate from declining sectors of the economy to growing sectors. In the United States, such reallocation has occurred across industries (manufacturing to services, for example), and across geographical areas (northeast and midwest to south and west).
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60. Figure 11.30 The Effects of a Sectoral Shock
A sectoral shock acts to shift the demand and supply curves for labor to the left. The output supply curve shifts left so output falls and the real interest rate rises. Consumption and investment fall, and employment falls.
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