In this chapter we present the main neoclassical model of investment, under convex adjustment costs. This model is known as the “q model of investment”, where q is the ratio of the market value of installed capital relative to its replacement value. We discuss the model both under conditions of certainty and under uncertainty.
So far we have been assuming that firms choose their capital stock so that the marginal product of capital equals the user cost of capital, as determined by the real interest rate and the rate of depreciation. This theory in fact determines the optimal capital stock and not the amount of investment. Investment flows determine how quickly a firm moves from its current to the optimal capital stock. When firms can adjust their capital stock immediately and without cost, the flow of investment is not defined, as the capital stock jumps immediately to its optimal level.
In fact, however, the change of the capital stock involves adjustment costs. A firm that chooses to raise its stock of productive capital should rent or buy additional space, buy and install new equipment, and train employees to use the extra equipment. In additional there are delivery lags and installation costs, making it more costly to adjust the capital stock quickly. All these costs are beyond the cost of buying additional capital goods. In addition, it is to be expected that these adjustment costs will be convex, i.e. they will depend on the size of the new investment. The higher the size of new investment, the greater will be the average adjustment cost of installing (or de-installing) an additional unit of capital.
In the presence of adjustment costs, the investment decisions of firms will thus not only depend on present conditions, such as the relation of the user cost of capital to the marginal product of capital, but also on past and expected future decisions. The problem of the firm becomes truly dynamic. Jorgenson (1963) assumed that, precisely because of the existence of adjustment costs, firms are not immediately but only gradually adjusting their stock of capital towards its “optimal” level, as determined by the user cost of capital and the marginal product of capital. He thus postulated an investment function which determined current investment as a fraction of the difference between the current and the “optimal” (desired) capital stock.
However, Jorgenson did not derive the speed of adjustment, and thus the flow of investment, from a fully dynamic optimization problem. This was accomplished later by Lucas (1967), Gould (1968) and Treadway (1969), who, instead of postulating the investment function, as Jorgensen had done, solved for the optimal investment function from the dynamic problem of a firm maximizing the present value of its profits, subject to convex costs of adjusting its capital stock. Soon afterwards, Lucas and Prescott (1971) extended this framework to examine the determination of investment under uncertainty.
A second approach to the problem of investment was that of Tobin (1969), who compared the ratio of the market value of installed capital of a firm, to the replacement cost of capital, naming this ratio q. Tobin argued that if the already installed capital stock of a firm has higher value than the cost of replacing the capital goods that compose it, i.e. if q is greater than one, then it will be profitable for the firm to invest, i.e. purchase and install new capital goods. Tobin argued that the rate of investment will be an increasing function of q. The ratio of the value of the already installed capital stock to its replacement cost has since been called “Tobin’s q”. However, much like Jorgenson, Tobin did not derive his investment function from a dynamic optimization problem either.
Several years later, Abel (1982) and Hayashi (1982) showed that Tobin’s “q theory” and the theory of “adjustment costs” for investment of Jorgenson, as modeled by Lucas, Prescott, Gould and Treadway, can be combined in a unified framework. This synthesis of the two theories is now considered as the main neoclassical dynamic model of investment.
The firm does not choose the level of its capital stock, by equating at any time the marginal product of capital to the sum of the real interest rate and the depreciation rate, but it chooses the amount of investment, taking into account the adjustment costs of the capital stock. Since marginal adjustment costs increase with the amount of investment, investment results in a gradual adjustment of the capital stock towards its steady state value. On the adjustment path, the firm takes into account both the current and future effects of its investment decisions. Thus, investment depends on both current and expected future developments in the value of the marginal product of capital and the user cost of capital.
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Dynamic Macroeconomics 