Lecture-Q_Theory_Investment(Intro)

Chapter 3
In v estm en t
This chapter deals with the theory of investment. Investment refers to the purchase by
fi rms of investment g oods for use in future production. The investment function is derived
intuitively and its main determinants are ex amined closely. W e provide further insig hts
about adjustment costs and also discuss another related approach k nown as Tobin’s q.
This chapter concludes with a simple model of investment in the case of irreversibility.
3.1
Capital in the pro d u c tio n fu n c tio n
Investment is the purchase of investment g oods by fi rms for use in future production.
In other words, investment is the process throug h which the stock of physical capital in
the economy is chang ed. P hysical capital is an input which is necessary for production of
consumption g oods. W e could therefore conclude that there are really two types of g oods in
the economy, consumption g oods and investment g oods. Y et, we retain the assumption of a
sing le g ood in the economy. Therefore, households (which are composite units of consumers
and fi rms) demand both consumption g oods and investment g oods, and yt = ct + it , so
that output is eq ual to the sum of consumption demand and investment demand. In the
previous chapter, we have look ed at the determinants of consumption demand. W e now
turn to the determination of investment demand.
S ince capital is now necessary for the production of consumption g oods, we aug ment
the production function and write it as a function of both labour and physical capital.
yt = f (kt−1 , lt )
where fk (.) > 0, fkk (.) < 0, fl (.) > 0, fll (.) < 0. These assumptions on the shape of
the production function mean that both the marg inal product of labour and the marg inal
product of capital are positive. A n additional unit of input will increase output by an
amount g iven by these marg inal products. M oreover, both marg inal products are diminishing , so that as labour or capital increase, their respective marg inal products decrease.
G raphically, the slope of each tang ent to the production function is positive but declines
as capital increases.
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Figure 3.1: The production function with capital and labour
yt
6
yt = f (kt−1 , lt )
s
-
kt−1
C apital enters the production function with one lag. Such an assumption refl ects the
fact that capital takes time to become fully operational. Therefore, production at time t
relies on a given stock of capital determined at time t − 1.
C apital will also depreciate over time. This decaying will be captured by the rate of
depreciation δ, so that
dt = δkt−1
where dt denotes the total amount of the depreciation. We can now compute the total
stock of capital in the economy at some given time t, starting with the previous stock
of capital at time t − 1, adding investment, and subtracting the amount of depreciation.
H ence,
kt = kt−1 + it − δkt−1
(3.1)
kt = it + (1 − δ)kt−1
(3.2 )
or
Gross investment is the quantity of investment goods which are bought, equal to it .
N et investment refers to changes in the capital stock, which is equal to gross investment,
it , minus the amount of depreciation, δkt−1 .
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3.2
Investment demand
What are the determinants of investment? Investment is an intertemporal decision. Given
some capital and labour supply, households could consume a lot of units of output as
consumption and few as investment, so current consumption would be high and future
consumption would be low. Alternatively, they could save more and invest more today, so
that current consumption would remain relatively low and future consumption would be
high.
E ach household decides how much to invest by comparing the benefits and the costs
of this investment. There are two components to the return to investment. First, an
additional unit of investment raises capital, kt , by one unit. Holding the amount of labour
input fixed, output in the next period will increase by the marginal product of capital,
M P K t . As households sell this output at price Pt+1 in the next period the additional
dollar revenue is Pt+1 M P K t . Second, households can sell their old capital on the goods
market at time t + 1 (they can always buy back this capital during time t + 1 to use for
production at time t + 2), so they would receive an amount of Pt+1 (1 − δ) per unit of
capital. Clearly, capital has depreciated between time t and time t + 1, so this must be
taken into account.
An additional unit of investment must be bought in the goods market at price Pt .
Hence, Pt is the dollar cost of an extra unit of investment. Consequently, the net dollar
return of an extra unit of investment is equal to
Pt+1 (M P K
t
+ (1 − δ)) − Pt
or, dividing by Pt and noting that Pt+1 = (1 + πt )Pt ,
(1 + πt )(M P K
t
+ 1 − δ) − 1
This rate of return looks good or bad depending on how it compares to other rates of
return. Households can earn the nominal interest rate, Rt by holding bonds, or borrow
at rate Rt to finance investment. If the nominal return of return on investment, given by
(1 + πt )(M P K t + 1 − δ) − 1, is greater than the nominal interest rate, Rt , then households
will invest in order to raise capital. They will do so until the nominal rate of return on
investment is equal to the nominal interest rate, so that
(1 + πt )(M P K
t
+ 1 − δ) − 1 = Rt
Since (1 + Rt ) = (1 + rt )(1 + πt ), where rt is the real interest rate, we get
1 + Rt
(M P K
1 + rt
M PK
t
t
+ 1 − δ) = 1 + Rt
+ 1 − δ = 1 + rt
or
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(3.3)
Figure 3.2: Optimal choice of the capital stock
M P Kt − δ
6
rt
-
kt∗
M P Kt − δ = rt
kt
(3.4 )
The left-hand side is the real rate of return on investment, the gross return minus the
rate of depreciation, and households act so as to equate this return to the real rate of
return on bonds. Graphically,
Given the schedule of the marginal product of capital, the optimal stock of capital
depends on the real interest rate, rt , and the depreciation rate, δ. So,
kt∗ = k ∗ (rt , δ, ...)
(3.5 )
where ... reflect the factors that aff ect the position of the schedule of the marginal
product of capital. Once that we have derived the optimal stock of capital, we can derive
the investment demand function. It is given by
i∗t = kt∗ − (1 − δ)kt−1
(3.6 )
For given values of kt−1 and depreciation, gross investment changes one-for-one with
the desired stock of capital. Therefore,
i∗t = k ∗ (rt , δ, ...) − (1 − δ)kt−1
= i∗ (rt , δ, kt−1 , ...)
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(3.7 )
Therefore, investment demand depends on several factors.
1. An increase in the real interest rate will reduce investment demand.
2. An increase in labour supply will raise the marginal product of capital, thereby raising
investment demand.
3. Technological progress will also raise the marginal product of capital, thereby raising
investment demand.
4. A greater level of the previous period’s capital will tend lower investment demand.
5. A higher depreciation rate carries an ambiguous effect on gross investment demand.
A higher δ reduces the optimal stock of capital but tends to increase investment
demand through the need of replacing decaying capital.
3.3
Extensions
The classical theory of investment has been extended in several ways. We discuss three
important contributions hereafter.
3.3.1
A djustment costs
The standard model predicts that investment purchases bring capital to its optimal level
within a single period. In other words, households adjust the capital stock to its optimal
level instantaneously. However, such an adjustment ignores a variety of costs that arise
when households install new capital goods. In general, it is likely that a faster installation
generates higher costs than a slower installation. Models of investment with adjustment
costs assume that investment brings about costs which are increasing in the amount of
investment. The presence of adjustment costs will make it optimal to smooth increases in
the capital stock over several periods of time.
3.3.2
Investment and the stock mark et: T ob in’s q
In general, economic activity is related to stock market prices. In particular, economic
activity affects stock prices, while stock prices are important determinants of economic
activity. One channel is through wealth effects; when stock prices increase, households
become richer and consumption demand increases in turn. Moreover, another channel is
investment.
Stock prices reflect the present value of current and expected future profits of firms.
Stock prices tend to be higher when firms face profitable opportunities because these
opportunities mean higher future profits for shareholders. J ames Tobin proposed that
investment depends on the ratio of the market value of installed capital to the replacement
cost of installed capital:
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q=
market value of installed capital
replacement cost of installed capital
When the market value of installed capital is greater than the replacement cost of
installed capital, i.e. when q > 1, investment is positive. For example, when q = 1.3, a
firm that spends 100 on investment goods will increase its market value by 130, thereby
giving a value of 30 to uninstalled capital. As the marginal product of capital decreases
as capital increases, investment will reduce the return on capital over time, and Tobin’s q
will return to unity.
The q theory of investment is related to our previous result that investment depends
negatively on the real interest rate. The real interest rate is used by stock market participants to discount future profits of firms. A higher real interest rate means that future
profits are more heavily discounted, thereby reducing stock prices. L ower stock prices
would lead to lower investment. Therefore, the negative effect of the real interest rate on
investment is incorporated into the q theory of investment. However, Tobin’s q also incorporates two other determinants of investment. First, gains in the productivity of capital
which raise future profits will be reflected in stock prices and lead to a higher q, thereby
increasing investment. Moreover, the q theory embodies the role of expectations. Investment is intimately affected by uncertainty. Firms buy investment goods today for use in
future production and future demand remains uncertain at the time at which investment
is made. Stock market assess the extent and effects of uncertainty in a continuous manner.
Indeed, the relatively high volatility of stock prices partly explains why investment is the
most volatile component of gross domestic product.
3.3.3
Irreversible investment under uncertainty
B oth the classical theory of investment and Tobin’s q theory abstract from three important
aspects of investment decisions. Firstly, most investment decisions are irreversible. Once
a factory is built, it is hard to get rid of it within a reasonable period of time. Secondly,
investment decisions must deal with uncertainty about the future returns to investment.
Suppose that demand for goods increases today, thereby leading to an increase in the capital
stock through investment. What happens if preferences change afterwards and consumer
demand switches to other goods? Thirdly, investment is an intertemporal decision and
therefore, there is an issue as to when the investment should be made. Today, tomorrow?
These three features modify the standard theories of investment in an important manner. Consider the following example of a firm trying to decide whether to make an irreversible investment today (at time 0) or tomorrow (at time 1). For convenience, we shall
assume that the factory can be built instantaneously at a fixed cost equal to 1600. V ariable costs are z ero. The price at which the good is sold is equal to 200 today (known with
certainty), and the price at which the good is sold tomorrow and in any future period is
either 300 if markets conditions are good, or 100 if market conditions are bad. Market
conditions are good, respectively bad, with a probability of 0.5. The firm discounts future
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profits using an interest rate of 0.10. Importantly, the interest rate will be assumed to be
constant over time.
Should this firm invest today or tomorrow? Suppose that the firm invest today. It will
pay a cost of 1600. What is the present value of the profits from this investment? The
good produced will be sold at a price of 200 at time 0, and the expected price at which
the good will be sold in all future periods is 0.5 × 300 + 0.5 × 100 = 200. Therefore, the
present discount value of all profits will be equal to
PV of current and future profits =
∞
X
200
t=0
(1.1)t
This calculation involves the computation of the sum of a geometric series. R ewriting
the sum yields
∞
X
200
t=0
(1.1)t
1
1
+
= 200 1 +
+ ...
1.1 (1.1)2
1
= 200
1
1 − 1.1
= 2200
!
Therefore, the net present value of this investment is equal to 2200 − 1600 = 600. Since
the current value of the firm, captured by the present value of current and future profits,
is greater than the cost of the investment, it seems that the firm should invest at time 0.
However, such a conclusion remains incorrect. Suppose now that the firm waits one period
and decides to invest tomorrow. The firm will invest only if the price of the good goes up
relative to today. In this case, the net present value of the investment is even greater than
600. It is obtained as
∞
300
−1600 X
+
0.5 ×
t
1.1
t=1 (1.1)
!
=
850
= 773
1.1
To derive this, we focus again on the computation of the sum of a geometric series.
R ewriting the sum yields
∞
−1600 X
300
+
0.5 ×
t
1.1
(1.1)
t=1
!
∞
X
1
−1600
+ 300
0.5 ×
t
1.1
t=1 (1.1)
!
"
1
1
1
−1600
+ 300
+
1+
+ ...
0.5 ×
1.1
1.1
1.1 (1.1)2
"
1
−1600
1
+ 300
0.5 ×
1
1.1
1.1 1 − 1.1
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#!
#!
1 1.1
−1600
+ 300
0.5 ×
1.1
1.1 0.1
1
−1600
0.5 ×
= 772.72
+ 300
1.1
0.1
Clearly, the firm should better wait and invest in the second period (at time 1) instead
of investing at time 0 even if investing at time 0 would bring about a positive net present
value. It must be noted that this result would not emerge if (a) the firm could fully reverse
its investment, in the case that market conditions turn out to be bad, (b) the firm did not
have the possibility to wait.
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