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118

Chapter 8 INVESTMENT

This chapter investigates the demand for investment. As described at the beginning of Chapter 7, there are two main reasons for studying investment. First, the combination of firms investment demand and households saving supply determines how much of an economys output is invested; as a result, investment demand is potentially important to the behavior of standards of living over the long run. Second, investment is highly volatile; thus investment demand may be important to short-run fluctuations.

Section 8.1 presents a baseline model of investment where firms face a perfectly elastic supply of capital goods and can adjust their capital stocks costlessly. We will see that this model, even though it is a natural one to consider, provides little insight into actual investment. It implies, for example, that discrete changes in the economic environment (such as discrete changes in interest rates) produce infinite rates of investment or disinvestment.

Sections 8.2 through 8.5 therefore develop and analyze the q theory model of investment. The models key assumption is that firms face costs of adjusting their capital stocks. As a result, the model avoids the unreasonable implications of the baseline case and provides a useful framework for analyzing the effects that expectations and current conditions have on investment.

Sections 8.6 and 8.7 introduce two important extensions of the model: Section 8.6 considers uncertainty, and Section 8.7 investigates financial-market imperfections. Finally, Section 8.8 presents some empirical tests and applications of the models.

8.1 Investment and the Cost of Capital

The Desired Capital Stock

Consider a firm that can rent capital at a price of . The firms profits at a point in time are given by rr(K,Xi,X2, ...,X) - , where is the amount of capital the firm rents and the Xs are variables that it takes as given. In the



Solving this expression for I yields ( . 1,..., )

7TKK(K,Xi,...,Xn)

(8.3)

Since TTKK is negative, (8.3) implies that is decreasing in r. A similar analysis can be used to find the effects of changes in the Xs on K.

The User Cost of Capital

Most capital is not rented but is owned by the firms that use it. Thus there is no clear empirical counterpart of . This difficulty has given rise to a large literature on the user cost of capital.

Consider a firm that owns a unit of capital. Suppose the real market price of the capital at time f is Pxit), and consider the firms choice between selling the capital and continuing to use it. Keeping the capital has three costs to the firm. First, the firm forgoes the interest it would receive if it sold the capital and saved the proceeds; this has a real cost of r(f )p(f) per unit time, where r{t) is the real interest rate. Second, the capital is depreciating; this has a cost of Spic(t) per umt time, where 8 is the depreciation rate. And third, the price of the capital may be changing. This increases the cost of

case of a perfectly competitive firm, for example, the Xs include the price of the firms product and the costs of other inputs. 7 (») is assumed to account for whatever optimization the firm can do on dimensions other than its choice of K. For a competitive firm, for example, tt{K,Xi,...,X) - gives the firms profits at the profit-maximizing choices of inputs other than capital given and the Xs. We assume that > 0 and < 0, where subscripts denote partial derivatives.

The first-order condition for the profit-maximizing choice of is

7 ( , 1,..., ) = . (8.1)

That is, the firm rents capital up to the point where its marginal revenue product equals its rental price.

Equation (8.1) imphcitly defines the firms desired capital stock as a function of and the Xs. We can differentiate this condition to find the impact of a change in one of these exogenous variables on the desired capital stock. Consider, for example, a change in the rental price of capital, . By assumption, the Xs are exogenous; thus they do not change when changes. K, however, is chosen by the firm. Thus it adjusts so that (8.1) continues to hold. Differentiating both sides of (8.1) with respect to shows that this requires

.MXir...,X) = 1. (8.2)



) = r(t)pK{t) + SpKit) - PK(t)

(8.4)

PK(t).

PKit)

This analysis ignores the existence of taxes. In practice, however, the tax treatments of investment and of capital income have large effects on the user cost of capital. To give an idea of these effects, consider an investment tax credit. Specifically, suppose the firms income that is subject to the corporate income tax is reduced by fraction f of its investment expenditures; for symmetry, suppose also that its taxable income is increased by fraction f of any receipts from selling capital goods. Such an investment tax credit implies that the effective price of a unit of capital to the firm is (1 - ( ) ( ), where is the marginal corporate income tax rate. The user cost of capital is therefore

(1 - fr)pK(t). (8.5)

Thus the investment tax credit reduces the user cost of capital, and hence increases firms desired capital stocks. One can also investigate the effects of depreciation allowances, the tax treatment of interest, and many other features of the tax code on the user cost of capital and the desired capital stock.

Difficulties with the Baseline Model

This simple model of investment has at least two major failings as a description of actual behavior. The first concerns the impact of changes in the exogenous variables. Our model concerns hrms demand for capital, and it implies that firms desired capital stocks are smooth functions of the exogenous variables. As a result, a discrete change in one of the exogenous variables leads to a discrete change in the desired capital stock. Suppose, for example, that the Federal Reserve reduces interest rates by a discrete amount; as the analysis above shows, this discretely reduces the cost of capital, . This in turn means that the capital stock that satisfies (8.1) rises discretely.

The problem with this implication ts that, since the rate of change of the capital stock equals investment minus depreciation, a discrete change

The seminal paper is Hall and Jorgenson (1967). See also Problems 8.2 and 8.3.

using the capital if the price is falling (since the firm obtains less if it waits to sell the capital), and decreases the cost if the price is rising. This has a cost of - ) per unit time. Putting the three components together yields the real user cost of capital:



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