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chapter

11

A Real Intertemporal Model with Investment
This chapter brings together the microeconomic behavior we have studied in previous
chapters, to build a model that can serve as a basis for analyzing how macroeconomic
shocks affect the economy, and that can be used for evaluating the role of macroeconomic policy. With regard to consumer behavior, we have examined work–leisure
choices in Chapter 4 and intertemporal consumption–savings choices in Chapters 9
and 10. From the production side, in Chapter 4 we studied a firm’s production technology and its labor demand decision, and then in Chapter 5 we showed how changes
in total factor productivity affect consumption, employment, and output in the economy as a whole. In Chapters 9 and 10, we looked at the effects of choices by the
government concerning the financing of government expenditure and the timing of
taxes. While the Solow growth model studied in Chapters 7 and 8 included savings
and investment, in this chapter we examine in detail how investment decisions are
made at the level of the firm. This detail is important for our understanding of how
interest rates and credit market conditions affect firms’ investment decisions.
In this chapter, we complete a model of the real side of the economy. The real
intertemporal model we construct here shows how real aggregate output, real consumption, real investment, employment, the real wage, and the real interest rate are
determined in the macroeconomy. To predict nominal variables, we need to add money
to the real intertemporal model, which is done in Chapter 12. The intertemporal aspect
of the model refers to the fact that both consumers and firms make intertemporal
decisions, reflecting trade-offs between the present and the future.
Recall from Chapter 2 that the defining characteristic of investment—expenditure
on plants, equipment, and housing—is that it consists of the goods that are produced
currently for future use in the production of goods and services. For the economy
as a whole, investment represents a trade-off between present and future consumption. Productive capacity that is used for producing investment goods could otherwise
be used for producing current consumption goods, but today’s investment increases
future productive capacity, which means that more consumption goods can be produced in the future. To understand the determinants of investment, we must study the
microeconomic investment behavior of a firm, which makes an intertemporal decision
regarding investment in the current period. When a firm invests, it forgoes current
profits so as to have a higher capital stock in the future, which allows it to earn higher
future profits. As we show, a firm invests more the lower its current capital stock, the
higher its expected future total factor productivity, and the lower the real interest rate.
375

376

Part IV Savings, Investment, and Government Deficits

The real interest rate is a key determinant of investment as it represents investment’s opportunity cost. A higher real interest rate implies that the opportunity cost
of investment is larger, at the margin, and so investment falls. Movements in the real
interest rate are an important channel through which shocks to the economy affect
investment, as we show in this chapter. Further, monetary policy may affect investment
through its influence on the real interest rate, as we show in Chapters 12 to 14.
In addition to the effect of the market interest rate, the investment decisions of
firms depend on credit market risk, as perceived by lenders. That is, firms may find it
more difficult to borrow to finance investment projects if lenders, including banks and
other financial institutions, perceive lending in general to be more risky. Perceptions
of an increase in the degree of riskiness in lending were an important factor in the
global financial crisis. In this chapter, we will show how credit market risk can play a
role in investment behavior, by incorporating asymmetric information, a credit market
imperfection. The role of asymmetric information in a firm’s investment decision will
turn out to be very similar to its role in a consumer’s consumption–savings decision, as
studied in Chapter 10.
A good part of this chapter involves model building, and there are several important steps we must take before we can use this model to address important economic
issues. This requires some patience and work, but the payoff arrives in the last part of
this chapter and continues through the remainder of this book, where this model is the
basis for our study of monetary factors in Chapter 12, business cycles in Chapters 13
and 14, and for other issues in later chapters.
This chapter focuses on the macroeconomic effects on aggregate output, investment, consumption, the real interest rate, and labor market variables of aggregate
shocks to government spending, total factor productivity, the nation’s capital stock, and
credit market risk. Although we studied elements of some of these effects in Chapters 5,
9, and 10, there are new insights in this chapter involving the effects on the interest rate
and investment of these shocks, and the effect of the anticipation of future shocks on
current macroeconomic activity. For example, including intertemporal factors shows
how credit markets play a role in the effects of government spending on the economy.
As well, we will be able to use the real intertemporal model to analyze aspects of the
impact of the financial crisis on aggregate economic activity.
As in Chapters 4 and 5, we work with a model that has a representative consumer,
a representative firm, and a government, and, for simplicity, ultimately we specify this
model at the level of supply and demand curves. We are able to capture the essential
behavior in this model economy by examining the participation of the representative
consumer, the representative firm, and the government in two markets: the market
for labor in the current period, and the market for goods in the current period. The
representative consumer supplies labor in the current labor market and purchases consumption goods in the current goods market, while the representative firm demands
labor in the current labor market, supplies goods in the current goods market, and
demands investment goods in the current goods market. The government demands
goods in the current goods market in terms of government purchases.

The Representative Consumer
The behavior of the representative consumer in this model brings together our knowledge of the consumer’s work–leisure choice from Chapter 4 with what we know about

Chapter 11 A Real Intertemporal Model with Investment

intertemporal consumption behavior from Chapter 9. In the model we are constructing here, the representative consumer makes a work–leisure decision in each of the
current and future periods, and he or she makes a consumption–savings decision in
the current period.
The representative consumer works and consumes in the current period and the
future period. He or she has h units of time in each period and divides this time
between work and leisure in each period. Let w denote the real wage in the current
period, wœ the real wage in the future period, and r the real interest rate. The consumer
pays lump-sum taxes T to the government in the current period and Tœ in the future
period. His or her goal is to choose current consumption C, future consumption Cœ ,
leisure time in the current and future periods, l and lœ , respectively, and savings in the
current period, Sp , to make himself or herself as well off as possible, given his or her
budget constraints in the current and future periods. The representative consumer is a
price-taker who takes w, wœ , and r as given. Taxes are also given from the consumer’s
point of view.
In the current period, the representative consumer earns real wage income w(h - l),
receives dividend income p from the representative firm, and pays taxes T, so that his
or her current-period disposable income is w(h - l) + p - T, just as in Chapter 4. As in
Chapter 9, disposable income in the current period is then split between consumption
and savings, and savings takes the form of bonds that earn the one-period real interest rate r. Just as in Chapter 9, savings can be negative, in which case the consumer
borrows by issuing bonds. The consumer’s current budget constraint is then
C + Sp = w(h - l) + p - T.

(11-1)

In the future period, the representative consumer receives real wage income wœ (h - lœ ),
receives real dividend income pœ from the representative firm, pays taxes Tœ to the
government, and receives the principal and interest on savings from the current period,
(1 + r)Sp . Because the future period is the last period and because the consumer is
assumed to make no bequests, all wealth available to the consumer in the future is
consumed, so that the consumer’s future budget constraint is
Cœ = wœ (h - lœ ) + pœ - Tœ + (1 + r)Sp .

(11-2)

Just as in Chapter 9, we can substitute for savings Sp in Equation (11-1) using
Equation (11-2) to obtain a lifetime budget constraint for the representative consumer:
C+

wœ (h - lœ ) + pœ - Tœ

= w(h - l) + p - T +
.
1+r
1+r

(11-3)

This constraint states that the present value of consumption (on the left-hand side of
the equation) equals the present value of lifetime disposable income (on the righthand side of the equation). A difference from the consumer’s lifetime budget constraint
in Chapter 9 is that the consumer in this model has some choice, through his or her
current and future choices of leisure, l and lœ , over his or her lifetime wealth.
The representative consumer’s problem is to choose C, Cœ , l, and lœ to make himself
or herself as well off as possible while respecting his or her lifetime budget constraint,
as given by Equation (11-3). We cannot depict this choice for the consumer conveniently in a graph, as the problem is four-dimensional (choosing current and future
consumption and current and future leisure), while a graph is two-dimensional. It is

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Part IV Savings, Investment, and Government Deficits

straightforward, however, to describe the consumer’s optimizing decision in terms of
three marginal conditions we have looked at in Chapters 4 and 9. These are as follows:
1. The consumer makes a work–leisure decision in the current period, so that when

he or she optimizes, we have
MRSl,C = w,

(11-4)

that is, the consumer optimizes by choosing current leisure and consumption so
that the marginal rate of substitution of leisure for consumption is equal to the
real wage in the current period. This is the same marginal condition as in the
work–leisure problem for a consumer that we considered in Chapter 4. Recall
that, in general, a consumer optimizes by setting the marginal rate of substitution of one good for another equal to the relative price of the two goods. In
Equation (11-4), the current real wage w is the relative price of leisure in terms
of consumption goods.
2. Similarly, in the future the consumer makes another work–leisure decision, and
he or she optimizes by setting
MRSlœ ,Cœ = wœ ,

(11-5)

that is, at the optimum, the marginal rate of substitution of future leisure for
future consumption must be equal to the future real wage.
3. With respect to his or her consumption–savings decision in the current period,
as in Chapter 9, the consumer optimizes by setting
MRSC,Cœ = 1 + r,

(11-6)

that is, the marginal rate of substitution of current consumption for future consumption equals the relative price of current consumption in terms of future
consumption.

Current Labor Supply

Our ultimate focus is on interaction between the representative consumer and the representative firm in the markets for current labor and current consumption goods, and
so we are interested in the determinants of the representative consumer’s supply of
labor and his or her demand for current consumption goods.
First, we consider the representative consumer’s current supply of labor, which is
determined by three factors—the current real wage, the real interest rate, and lifetime
wealth. These three factors affect current labor supply as listed below.
1. The current quantity of labor supplied increases when the current real wage increases.

The consumer’s marginal condition, Equation (11-4), captures the idea that substitution between current leisure and current consumption is governed by the
current real wage rate w. Recall from Chapter 4 that a change in the real wage has
opposing income and substitution effects on the quantity of leisure, so that an
increase in the real wage could lead to an increase or a decrease in the quantity
of leisure, depending on the size of the income effect. Here, we assume that the
substitution effect of a change in the real wage is always larger than the income
effect, implying that leisure decreases and hours worked increases in response to

Chapter 11 A Real Intertemporal Model with Investment

an increase in the real wage. This might seem inconsistent with the fact, pointed
out in Chapter 4, that over the long run, income and substitution effects on labor
supply appear to cancel. However, the model we are building here is intended
mainly for analyzing short-run phenomena. As we argued in Chapter 4, the canceling of income and substitution effects in the long run can be consistent with
the substitution effect dominating in the short run, as we assume here.
2. The quantity of current labor supplied increases when the real interest rate increases.
The consumer can substitute intertemporally not only by substituting current
consumption for future consumption, as we studied in Chapter 9, but also by
substituting current leisure for future leisure. In substituting leisure between the
two periods, the representative consumer responds to the current price of leisure
relative to the future price of leisure, which is w(1w+œ r) . Here, w is the price of
current leisure (labor) in terms of current consumption, wœ is the price of future
leisure in terms of future consumption, and 1+r is the price of current consumption in terms of future consumption. Therefore, an increase in the real interest
rate r, given w and wœ , results in an increase in the price of current leisure relative to future leisure. Assuming again that the substitution effect is larger than
the income effect, the consumer wants to consume less current leisure and more
future leisure. An example of how this intertemporal substitution of leisure
effect works is as follows. Suppose that Paul is self-employed and that the market interest rate rises. Then, Paul faces a higher return on his savings, so that if
he works more in the current period and saves the proceeds, in the future he can
both consume more and work less. It may be helpful to consider that leisure,
like consumption, is a good. When the real interest rate increases, and substitution effects dominate income effects for lenders, current consumption falls (from
Chapter 6), just as current leisure decreases when the real interest rate increases
and substitution effects dominate.
3. Current labor supply decreases when lifetime wealth increases. From Chapter 4,
we know that an increase in current nonwage disposable income results in an
increase in the quantity of leisure and a decrease in labor supply for the consumer, as leisure is a normal good. Further, in Chapter 9, we showed how income
effects generalize to the intertemporal case where the consumer chooses current
and future consumption. An increase in lifetime wealth increases the quantities
of current and future consumption chosen by the consumer. Here, when there
is an increase in lifetime wealth, there is an increase in current leisure and, thus,
a decrease in current labor supply, because current leisure is assumed to be normal. The key wealth effect for our analysis in this chapter is the effect of a change
in the present value of taxes for the consumer. Any increase in the present value
of taxes implies a decrease in lifetime wealth and an increase in current labor
supply.
Given these three factors, we can construct an upward-sloping current labor supply curve as in Figure 11.1. In the figure, the current real wage w is measured along the
vertical axis, and current labor supply N is on the horizontal axis. The current labor
supply curve is labeled Ns (r) to indicate that labor supply depends on the current real
interest rate. If the real interest rate rises, say from r1 to r2 , then the labor supply curve

379

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Part IV Savings, Investment, and Government Deficits

Figure 11.1 The Representative Consumer’s Current Labor Supply Curve

w = Current Real Wage

The current labor supply curve slopes upward, under the assumption that the substitution effect of an increase in the
real wage outweighs the income effect.

N s (r)

N = Current Labor Supply

shifts to the right, as in Figure 11.2, because labor supply increases for any current real
wage w. In Figure 11.3, an increase in lifetime wealth shifts the labor supply curve to
the left from N1s (r) to N2s (r). Such an increase in lifetime wealth could be caused by a
decrease in the present value of taxes for the consumer. In Figure 11.3 the real interest
rate is held constant as we shift the current labor supply curve to the left.

The Current Demand for Consumption Goods

Now that we have dealt with the determinants of the representative consumer’s current
labor supply, we can turn to his or her demand for current consumption goods. The
determinants of the demand for current consumption goods were studied in Chapter 9,
where we showed that the primary factors affecting current consumption are lifetime
wealth and the real interest rate. Further, lifetime wealth is affected by current income,
and by the present value of taxes.
Given our analysis of the consumption–savings behavior of consumers in
Chapter 9, it proves useful here to construct a demand curve which represents the
quantity demanded of current consumption goods by the representative consumer,
as a function of current aggregate income, Y, as shown in Figure 11.4. Recall from
Chapter 9 that if the real interest rate is held constant and current income increases for
the consumer, then current consumption will increase. In Figure 11.4, we graph the
quantity of current consumption chosen by the representative consumer, for each level

Chapter 11 A Real Intertemporal Model with Investment

Figure 11.2 An Increase in the Real Interest Rate Shifts the Current Labor Supply Curve to the Right

w = Current Real Wage

This is because the representative consumer consumes less leisure in the current period and more leisure in the future
when r increases.

N s (r1)

N s (r2)

N = Current Labor Supply

of real income Y, holding constant the real interest rate r. In the figure, the demand for
consumption goods is on the vertical axis, and aggregate income is on the horizontal
axis. We let Cd (r) denote the demand curve for current consumption goods, indicating
the dependence of the demand for consumption on the real interest rate. Recall from
Chapter 9 that, if current income increases for the consumer, then consumption and
savings both increase, so that the quantity of consumption increases by less than one
unit for each unit increase in income. In Figure 11.4, the slope of the curve Cd (r) is
the MPC or marginal propensity to consume, which is the amount by which current
consumption increases when there is a unit increase in aggregate real income Y.
When there is an increase in the real interest rate, assuming again that the substitution effect of this increase dominates the income effect, there will be a decrease in
the demand for current consumption goods because of the intertemporal substitution
of consumption (recall our analysis from Chapter 9). In Figure 11.5, if the real interest
rate increases from r1 to r2 , the demand curve for current consumption shifts down
from Cd (r1 ) to Cd (r2 ). Also, holding constant r and Y, if there is an increase in lifetime
wealth, then, as in Figure 11.6, the demand curve for current consumption shifts up
from C1d (r) to C2d (r). Such an increase in lifetime wealth could be caused by a decrease
in the present value of taxes for the consumer, or by an increase in future income.
The demand for current consumption goods is only part of the total demand in
the economy for current goods. What remains for us to consider are the demands for
current goods coming from firms (the demand for investment goods) and from the

381

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Part IV Savings, Investment, and Government Deficits

Figure 11.3 Effects of an Increase in Lifetime Wealth

w = Current Real Wage

More leisure is consumed in the present, due to an income effect, and the current labor supply curve shifts to the left.

N 2s(r)

s

N 1(r)

N = Current Labor Supply

government (government purchases). Total demand for current goods will be summarized later in this chapter by the output demand curve, which incorporates the behavior
of the representative consumer, the representative firm, and the government.

The Representative Firm
Now that we have covered the important features of the consumer’s current labor supply and current consumption demand decisions, we can turn to the key decisions of
the representative firm for the current labor market and the current goods market.
The representative firm, as in Chapter 4, produces goods using inputs of labor
and capital. The key differences here are that output is produced in both the current
and future periods, and that the firm can invest in the current period by accumulating
capital so as to expand the capacity to produce future output. In the current period,
the representative firm produces output according to the production function
Y = zF(K, N),

(11-7)

where Y is current output, z is total factor productivity, F is the production function,
K is current capital, and N is current labor input. Here, K is the capital with which the
firm starts the current period, and this quantity is given. The production function F is
identical in all respects to the production function we studied in Chapter 4.


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