- November 30, 2020

**Chapter 9**

**A Real Intertemporal Model with Investment**

# Teaching Goals

At first glance, students may think that much of this chapter is simply a repetition of the material in the previous chapter. Students may also get impatient with the many intricate steps of model building. I like to emphasize that the possibility of accumulating capital represents a fundamental difference to an economy.

It also explains one crucial component of GDP that was characterized with its very volatile behavior:

investment. In the previous chapter, average consumption must equal per capita total output less per capita government spending. For a given amount of government savings, aggregate private savings is fixed. One consumer may only reallocate consumption across time if another consumer is willing to make the complementary reallocation. Borrowing and lending can improve economic outcomes only to the extent that it can help bridge differences in consumers’ allocations of income over time. In particular, if there is a

single representative consumer, this consumer is stuck with consuming her gross income net of government spending in the current period. The investment process allows the whole economy to effectively reallocate consumption across time. Try also to tie this in with the previous chapter on growth where investment was crucial.

Many students try to get by with this material by rote memorization of a great many curve shifts and laundry lists of the effects of specific disturbances. However, to really understand this material, students must be able to work out the effects of disturbances on their own. I therefore encourage the students to put a good deal of effort into solving problems with the model. Often there may be a specific current political issue to which the model of this chapter may be put to use. When presenting the impact of a particular shock, always go back to the micro-foundations to explain why particular curves shift.

# An Alternative Graphical Presentation

Chapter 9 in the text adopts the traditional approach of analyzing the effects of disturbances with the help of output supply and demand curves. Another way of presenting the same results is based on the application of the second fundamental theorem of welfare economics. This approach is developed in some detail in Chapter 5 for the case of the one-period economy.

This approach is also applicable to the two-period economy. The solution to the two-period planner problem will also be a competitive equilibrium. To keep things relatively simple, consider the case in which first- and second-period employment is fixed at *N *and *N’*, respectively. First-period gross output, including the scrap value of the first-period capital stock, (1−*d K*) , is equal to *Y *= *zF K N*( , ) + −(1 *d K*) . Gross first-period output can be used for first-period consumption, gross investment, and government purchases. For a given amount of first-period consumption, available second-period consumption is given by:

*C’ *= *z’F Y*( − −*C G N*, )+ −(1 *d*)(*Y** *− − −*C G*) *G’*.

This equation represents a production possibilities frontier for the economy in *C*, *C*’ space. The slope of this *PPF* is equal to −[1+ (*MP’** _{K }*−

*d*)]. Therefore, the slope can be computed from the net marginal product of capital. A Pareto optimum is at a point of tangency of this

*PPF*with an indifference curve like those developed in Chapter 6. The slope of these indifference curves is equal to −

*MRS*

*C C’*, . The competitive real rate of interest is then given by (1+

*r*) =

*MRS*

_{C C}_{, ‘ }=1+(

*MP’*

*−*

_{K }*d*). The real rate of interest is equal to the net marginal product of capital, which represents the solution of the firm’s optimization problem. The marginal rate of substitution is equal to one plus the real interest rate, which is the solution of the consumer’s optimization problem. The economy is at equilibrium at the point

*E*

^{*}in the figure below.

and *K* only affect *Y*, and these changes generate parallel horizontal shifts in the *PPF*, as do changes in *G*.

Changes in *G*’ generate parallel vertical shifts in the *PPF*. All of these changes are pure income effects and *C* and *C*’ move in the same direction. Changes in *z*’ involve a substitution effect as well as an income effect. An increase in *z*’ unambiguously increases *C*’, but has an ambiguous effect on *C*.

The graphical apparatus of Chapter 4 may now be used to infer the likely effects of these disturbances on *N*, and to help confirm their effects on *C*. Other than changes in *z*, all disturbances generate parallel shifts in the first-period *PPF*. The only new wrinkle is that we need to remember that investment is another possible use of first-period output. For a given amount of *I*, the *PPF* is given in the figure below. Changes in *z* were covered in some detail in this setting in Chapter 5. The new considerations are those changes that have intertemporal substitution implications. Changes in investment, as well as changes in first-period government spending, shift the *PPF* in this diagram. However, such shifts are simply parallel vertical shifts.

A change in the equilibrium level of investment signals a desire to rearrange utility over the consumer’s lifetime. Those changes that increase equilibrium investment (and therefore savings) are accomplished both through a reduction in first-period consumption and a reduction in first-period leisure. We are assured that these changes are in the same direction as long as first-period consumption and first-period leisure are normal goods.

As one possibility, consider the case of a temporary increase in government spending. Manipulation of the first figure indicates equilibrium reductions in both first-period and second-period consumption. The reduction in *C*’, with no change in *G*’, indicates that investment must also decline, but by less than the increase in *G*. Therefore, *G *+* I* must increase. In the second figure, this shows up as reductions in both first-period consumption and first-period leisure. Therefore, first-period employment must also increase.

# Outline

## I. The Representative Consumer

- Consumer Choices
- Current Work-Leisure Decision 2. Future Work-Leisure Decision

3. Consumption-Savings Decision

- Current Labor Supply
- Current Real Wage Effects

- Real Interest Rate Effects—Intertemporal Substitution of Leisure

- Lifetime Wealth Effects

- Current Labor Supply Schedule

- Current Demand for Consumption Goods
- Real Interest Rate Effects

- The Effect of Current Income

- Changes in the Present Value of Taxes

- The Current Consumption Demand Schedule

## II. The Representative Firm

- Firm Choices
- Current Production

- Future Production

- Investment and Capital

- Depreciation of Capital

- Profits and Current Labor Demand
- Current Profits

- Future Profits

- The Present Value of Profits

- Current Employment Choice

- The Investment Decision
- The Marginal Cost of Investment

- The Marginal Benefit of Investment

- The Net Marginal Product of Capital

- The Optimal Investment Rule

- The Optimal Investment Schedule
- Real Interest Rate Effects

- Future Total Factor Productivity

- The Current Capital Stock

**III. Government**

## IV. Competitive Equilibrium

- The Current Labor Market and the Output Supply Curve
- Slope of Output Supply—Real Interest Rate Effects

- Shifts in Output Supply
- Lifetime Wealth

- Current Total Factor Productivity

- Current Capital Stock

- The Current Goods Market and the Output Demand Curve
- Slope of Output Demand—Real Interest Rate Effects

- Marginal Propensity to Consume and the Multiplier

- Shifts in Output Demand
- The Present Value of Taxes

- Future Income

- Future Total Factor Productivity

- Current Capital Stock

- The Complete Real Intertemporal Model
- Equilibrium in the Goods Market

- Equilibrium in the Labor Market

- Comparative Statics Experiments

## V. A Temporary Increase in Government Purchases

- Impact Effects
- Labor Supply

- Output Supply

- Output Demand

- Equilibrium Effects
- Goods Market:
*Y*↑,*r*↑

- Labor Market:
*N*↑,*w*↓

- The Composition of Output:
*C*↓,*I*↓

- Goods Market:

## VI. A Permanent Increase in Government Purchases

- Impact Effects
- Labor Supply

- Output Demand

- Equilibrium Effects
- Goods Market:
*Y*↑,*r*↓

- Labor Market:
*N*↑,*w*↓

- The Composition of Output:
*C*?,*I*↑

- Goods Market:

## VII. A Reduction in the Current Capital Stock

- Impact Effects
- Labor Supply

- Labor Demand

- Output Supply

- Output Demand

- Equilibrium Effects
- Goods Market:
*Y*?,*r*↓

- Labor Market:
*N*?,*w*↓

- The Composition of Output:
*C*↓,*I*↑

- Goods Market:

## VIII. An Increase in Current Total Factor Productivity

- Impact Effects
- Labor Supply

- Labor Demand

- Output Supply

- Equilibrium Effects
- Goods Market:
*Y*↑,*r*↓

- Labor Market:
*N*?(likely increases),*w*↑

- The Composition of Output:
*C*↑,*I*↑

- Goods Market:

## IX. An Increase in Future Total Factor Productivity

- Impact Effects
- Labor Supply

- Output Demand

- Equilibrium Effects
- Goods Market:
*Y*↑,*r*↑

- Labor Market:
*N*↑,*w*↓

- The Composition of Output:
*C*↓,*I*↑

- Goods Market:

# Textbook Question Solutions

## Questions for Review

- An increase in the real interest rate makes future leisure less expensive than current leisure. Therefore, current labor supply increases when the real interest rate increases. An increase in the real interest rate also makes future consumption less expensive than current consumption. Therefore, current consumption falls when the real interest rate increases. Alternatively, an increase in the real interest rate raises the rate of return on savings. The consumer can save more both by working more and consuming less in the current period.
- The primary determinants of current labor supply are the current real wage, the real interest rate, and lifetime wealth.
- Current labor demand is exclusively governed by the current-period marginal product of labor schedule. Therefore, labor demand depends on total factor productivity and the current-period capital stock.
- The representative firm maximizes the present value of its profits.
- The representative firm should invest until the point at which the net marginal product of capital equals the real rate of interest.
- The current capital stock does not affect the optimal amount of capital that the firm wants in place next period. However, if the firm has more capital in the current period, a lower amount of investment is needed to achieve the desired amount of capital.
- An increase in future total factor productivity raises the net marginal product of capital, and therefore investment increases.
- The government must equate the present value of government spending with the present value of taxes. The government may run a deficit in the current period, but if it does so it must run a surplus in the future period.
- Changes in the lifetime wealth of consumers, changes in current-period total factor productivity, and changes in the current capital stock all shift the output supply curve.
- Changes in current and future government spending and the present value of taxes shift the output demand curve. Changes in expected future income, future total factor productivity, and the current capital stock also shift the output demand curve.
- In a competitive equilibrium, output supply is equal to output demand and labor supply is simultaneously equal to labor demand. The point of intersection of the output supply curve and the output demand curve allows us to find the competitive equilibrium levels of aggregate output and the real rate of interest.
- A temporary increase in government purchases increases the real interest rate, increases aggregate output, increases employment, decreases the real wage, decreases consumption, and decreases investment.
- A permanent increase in government purchases does not modify the real interest rate, increases aggregate output, increases employment, decreases the real wage, and does not change investment and consumption.
- A decrease in the current capital stock increases the real interest rate, decreases the real wage, decreases consumption, and increases investment. However, the effects on output and employment are ambiguous. The decrease in the capital stock lowers the marginal product of labor and shifts labor demand to the left. The decrease in lifetime wealth shifts labor supply to the right. According to which of these effects is strongest, employment may either decrease or increase. If employment falls, then we have fewer workers and less capital, so output must decline. However, if there is a sufficiently large increase in employment, the increase in employment more than makes up for the lost capital, and output may rise. In terms of output supply and demand, less capital shifts output demand to the right due to the increase in investment, while the reduction in capital also shifts output supply to the left. The effect on output depends on which of these competing forces is strongest.
- A temporary increase in total factor productivity decreases the real interest rate, increases aggregate output, increases employment, increases the real wage, increases consumption, and increases investment. These results are consistent with the facts that employment, real wages, investment, and consumption are all procyclical.
- An increase in future total factor productivity increases the real interest rate, increases aggregate output, increases employment, decreases the real wage, decreases consumption, and increases investment. There are two principle differences between changes in current and future total factor productivity. First, an increase in future total factor productivity does not help with current production. Therefore, if the productivity effect is in the future, the increase in current output requires an increase in current employment and employment may only increase if the current real wage decreases. Second, an increase in future total factor productivity affects current investment demand, while an increase in current total factor productivity does not. This accounts for the fact that the current increase in
*z*lowers the real interest rate, while the future increase in*z*increases the real interest rate. This rise in the real interest rate also accounts for the fact that consumption rises when current*z*rises and falls when future*z*rises.

## Problems

- There are two effects of an increase in the depreciation rate. First, there is the direct effect, which implies that, given the marginal product of capital in period two,
*MP*′,the net marginal product of capital,_{K}*MP’*−_{K }*d*, will decrease when the depreciation rate increases. For any given real interest rate, this effect lowers investment demand, and so the investment demand schedule shifts to the left. This direct effect is the result of the fact that a higher depreciation rate implies that the scrap value of the capital the firm invests in will be lower at the end of period two.

In addition to this direct effect, there is also an indirect effect of the depreciation rate on investment. Since *K’ *= (1− *d K*) + *I*, given the initial capital stock, *K*, the quantity of capital in period two will be smaller, for any *I*, if the depreciation rate is higher. Therefore, when *d* increases, the investment schedule shifts to the right. The direct and indirect effects work in opposite directions, and so, given the real rate of interest, investment may either rise or fall with an increase in the depreciation rate.

- The problem supplies the following production function, where future output only depends on the level of second-period capital, in this case the number of trees.

__Future Trees Future Output__

- 155.0
- 162.0

- 168.0

- 173.0

- 177.0

- 180.0

- 182.0

- 183.8

- 184.8

- 185.2

- 185.4

- The production function is depicted below.

- The marginal product of capital schedule is computed from the previous table. In table form:

Future Trees | Future Output | MP′_{K} |

15 | 155.0 | — |

16 | 162.0 | 7 |

17 | 168.0 | 6 |

18 | 173.0 | 5 |

19 | 177.0 | 4 |

20 | 180.0 | 3 |

21 | 182.0 | 2 |

22 | 183.8 | 1.8 |

23 | 184.8 | 1.0 |

24 | 185.2 | 0.4 |

25 | 185.4 | 0.2 |

These data are plotted in the figure below.

- Tom’s first-year profits are equal to π=
*Y*−*I*. The present value of second-year profits is equal

_{to }π*‘ *= * ^{Y’ }*−

^{(1}−

^{d K’}^{) }=

*−*

^{Y’ }^{(1}−

^{d K’}^{) }. These calculations are given in the column

*V*, below.

(1+ *r*) 2

- The net marginal product of capital is equal to
*MP*′ −_{K}*d*=*MP*′ −0.1. These calculations are also included in the table below._{K}

Future Trees | Future Output | Required I | V | MP d′ −_{K} |

15 | 155.0 | –3 | 267.25 | — |

16 | 162.0 | –2 | 270.20 | 6.9 |

17 | 168.0 | –1 | 279.65 | 5.9 |

18 | 173.0 | 0 | 274.60 | 4.9 |

19 | 177.0 | 1 | 276.05 | 3.9 |

20 | 180.0 | 2 | 277.00 | 2.9 |

21 | 182.0 | 3 | 277.45 | 1.9 |

22 | 183.8 | 4 | 277.80 | 1.7 |

23 | 184.8 | 5 | 277.75 | 0.9 |

24 | 185.2 | 6 | 277.50 | 0.3 |

25 | 185.4 | 7 | 276.95 | 0.1 |

- Tom’s optimal level of
*V*is equal to 277.80. To earn this amount of profit, Tom needs to plant 4 new trees. Note that at*I*= 4,*MP’*−_{K }*d*=1.7 >*r*=1.0. Planting the 4th tree is therefore profitable. However, at*I*= 4,*MP’*−_{K }*d*= 0.9 <*r*=1.0. Planting the 5th tree is not profitable. The maximum*V*is therefore attained at the last tree for which*MP’*−_{K }*d*>*r*. - The costs of the output subsidy and the investment subsidy would each require an increase in other (lump-sum) taxes to satisfy the government budget constraint with unchanged government purchases. This increase in taxes reduces consumer wealth and so labor supply shifts to the right and output supply also shifts to the right. This effect tends to increase output and decrease the real interest rate.

In the case of the output subsidy, the decrease in the real interest rate increases both consumption spending and investment spending to match the increase in output. In the case of the subsidy to investment, there is also a shift to the right in the output demand curve. This effect provides an additional increase in output. Also the increase in the real interest rate (or the smaller-sized decrease in the real interest rate) reduces consumption spending so that more of the increase in output goes to investment spending and less goes to consumption spending. Therefore, the investment subsidy is likely to be more effective in increasing investment.

- The new second-period profits of the firm are now π′=
*Y*′*– w*′*N*′+(1 −*d*)*p*′._{K}K- The new marginal benefit from investment is now

*MB*(I) =(*MP*′* _{K }*+ (1 −

*d*)

*p*′

*)/(1 +*

_{K}*r*)

As the marginal cost from investment remains at one, the new investment rule is then *MP*′* _{K }*= (1 +

*r*) − (1 −

*d*)

*p*′

_{K}- With an increase in
*p*′, the marginal product of future capital needs to be reduced, thus more future capital is needed and investment rises. Indeed, as the liquidation value of capital goes up, you want to invest more in capital. Thus investment is positively correlated with stock prices._{K}

- Slope of the output demand curve.
- A reduction in the real interest rate increases consumption and investment spending. This is the primary reason for the downward slope of the output demand curve. However, as output rises, there is a further increase in consumption spending according to the size of the marginal propensity to consume. The larger the marginal propensity to consume, the flatter is the aggregate demand curve.

- The intertemporal substitution effect on consumption is one of the primary reasons why demand rises at lower interest rates. The larger the sensitivity of consumption spending to the real rate of interest, the flatter is the output demand curve.

- The responsiveness of investment demand to the real rate of interest is one of the primary reasons why demand rises at lower interest rates. The larger the responsiveness of investment demand to the real rate of interest, the flatter is the output demand curve.

- Slope of the output supply curve.
- The figure below depicts the effect of an increase in labor supply, due to an increase in the real interest rate, on the equilibrium level of employment. The diagram shows two alternate labor demand curves with differing slopes. Note that the equilibrium level of employment increases more when the marginal product of labor declines at a slower rate with increases in the level of employment. Therefore, when the marginal product of labor declines at a faster rate as the quantity of labor used in production increases, there is a smaller increase in employment and therefore a smaller increase in output supply. The output supply curve is steeper in this case.

- When the substitution effect of an increase in the real rate of interest decreases, there is a smaller effect on equilibrium employment of an increase in the real interest rate. Therefore there is a smaller increase in output supply. The output supply curve is steeper in this case.

- A future increase in government spending generates a negative income effect. Therefore, current-period consumption declines and current-period labor supply increases. The increase in currentperiod labor supply shifts the output supply curve to the right. The real interest rate falls, and the levels of employment and output likely increase. The results are summarized in the figures below. The equilibrium level of output increases from
*Y*to^{*}*Y*, and the level of employment rises from^{**}*N*to^{*}*N*. The equilibrium rate of interest unambiguously declines. This decline in the real rate of interest is responsible for the second, leftward shift in the labor supply curve. If, on net, employment and output increase, then it must be the case that the real wage falls. (If, on the other hand, output falls on net, then employment must fall and the real wage must rise.) The reduction in the real interest rate assures us that investment increases. The income effect tends to lower consumption, while the decline in the real rate of interest tends to increase consumption. Most likely, consumption falls, although consumption may also increase.^{**}

- Labor supply shifts to the right, so output supply also shifts to the right. Consumption demand also increases, so the output demand curve must also shift to the right. Output must increase although the real rate of interest may rise or fall. In light of the increase in output, equilibrium employment must increase. A higher level of employment, in the absence of a shift in the labor demand curve, assures us that the real wage rate must also fall. Investment rises if the real rate of interest declines, and investment falls if the real rate of interest increases. Because output has increased, consumption will rise as long as investment remains the same or declines. Consumption falls only in the case of a decline in the rate of interest of sufficient size to increase investment by more than the increase in output.
- To summarize:
*Y*↑,*N*↑,*w*↓, ?, ?,*r I C*?, but most likely increases.

- As one possibility, at low levels of nutrition, it may be infeasible for the consumer to work very much (a very high
*MRS*_{l C}_{, }). In this case, an increase in nutrition would make the consumer more willing (and able) to work more and consume more. One could also imagine some change in the technology of using leisure that is more goods intensive. In this case the value of leisure is low without a lot of consumption goods.

- To summarize:
- A temporary increase in
*z*increases output and employment, raises the real wage, and lowers the real rate of interest. Consumption and investment both increase. An increase in future total factor productivity,*z*’, shifts the current-period output demand curve to the right. Current output and employment increase, and the real interest rate increases. Since the current-period labor demand curve does not shift, the shift in labor supply due to the lower real interest rate causes the real wage rate to decline.

A permanent increase in total factor productivity simply combines the effects of the temporary and permanent changes in *z*. Current output and employment unambiguously increase. The real wage rate may either rise or fall. The real interest rate may either rise or fall. As long as the direct effect of the increase in *MP’** _{K}* outweighs any indirect effect due to a possible increase in the real interest rate, then investment will increase. As long as the direct effects of the increases in current and future income dominate any indirect effect of a possible rise in the real interest rate, then consumption will also increase.

- The increase in
*z*’ shifts the output demand curve to the right, but has no effect on the output supply curve. The increase in*K*shifts the output demand curve to the left, and shifts the output supply curve to the right. The combined effects shift the output supply curve to the right. The shift in the output demand curve is uncertain. An increase in the current capital stock lowers investment spending. An increase in future total factor productivity increases investment spending. As one possibility, suppose that the effect of the prospective increase in total factor productivity is that investment increases. In this case, both the output supply curve and the output demand curve shift to the right. Output rises unambiguously, but the effect on the real interest rate is uncertain.

If a lack of capital were the only reason for low output in poor countries, then we would expect that the real interest rates in poor countries would be higher than the real interest rates in rich countries. This is not the case. Alternatively, if poor countries are poor both because they have less capital and because they have worse prospects for future investment, then this explanation of the difference between poor and rich countries need not be in conflict with observed differences in real interest rates.

- A temporary increase in the price of energy is best modeled as a reduction in current-period total factor productivity. Such a disturbance shifts output supply to the left. Therefore, output falls and the real interest rate increases. In question 3, above, we showed that a larger value for the marginal propensity to consume implied a flatter output demand curve. In the figure below, we show the shift in output supply with two alternative output demand curves. When the marginal propensity to consume is high, the output demand curve is flat and the reduction in
*z*results in a large reduction in output and a small increase in the real interest rate. When the marginal propensity to consume is smaller, there is a smaller reduction in output, and a larger increase in the real interest rate.

- A hurricane destroys a significant amount of capital. This disturbance may be analyzed as an exogenous decrease in the stock of capital. The production function shifts downward. Labor demand shifts to the left. These effects result in a leftward shift in the goods supply curve. The loss in capital also increases the expected marginal product of capital, and so the goods demand curve shifts to the right. The figures below depict the case in which equilibrium output decreases.

- The analysis of the effects of the hurricane suggests that it is reasonable to expect a decrease in national income. However, because the model is based upon maximizing principles, it is likely that the reduction in national income represents an optimal response to the reduction in the capital stock. There is therefore no presumption that policy will improve the situation.
- An appropriate-sized increase in government spending can restore the economy to the original level of output,
*Y*_{1}^{*}. A temporary increase in government spending generates a negative wealth effect shifting the labor supply curve to the right. The temporary increase in government spending also shifts the output demand curve to the right. The figure below depicts a case in which the increase in government spending exactly returns the economy to the original level of output.

- An appropriate-sized increase in government spending can restore the economy to the original level of output,

Output is unchanged. The real interest rate increases. Employment must increase. In order to produce the same amount of output an increase in employment is needed to substitute for the lost capital.

- A more sensible rationale for an increase in government spending would be based upon needs to replace government-provided capital. That is, the government might want to increase spending to replace roads, sewer systems, and other infrastructure.
- A short war is best modeled as a temporary increase in government spending. Such a disturbance shifts the output demand curve to the right because the increase in current-period government spending will be larger than the reduction in consumption demand due to the decline in consumers’ lifetime wealth. The output supply curve also shifts to the right because of the reduction in consumers’ lifetime wealth. Output and employment unambiguously increase. Because the increase in government spending is only temporary, the effect on lifetime wealth is likely to be small, so the demand curve shifts farther than the supply curve. Therefore, the interest rate most likely increases.

Williamson • *Macroeconomics,* Third Edition

In order to more clearly see how the size of the intertemporal substitution effect on consumption comes into play, let us assume that the lifetime wealth effect is small enough to be ignored. In this case we need only be concerned with the shift in output demand and not the shift in output supply. The flatter output demand curves correspond to the case in which the interest rate effect on consumption is stronger. As the figure below depicts, the increase in output in this case is smaller. The intuition is as follows. When consumption is very sensitive to changes in the interest rate, it takes a smaller increase in the interest rate to crowd out demand to fit the increased *G*. With a smaller increase in the real interest rate, there is a smaller shift in labor supply, and so there is a smaller increase in employment and output.