Tobin's Marginal q and Average q: A Neoclassical Interpretation

By the late 1970s the investment literature was dominated by two theories – Jorgenson’s neoclassical theory, which starts from the firm’s optimisation but, in its earlier form, could determine the optimal capital stock only under constant returns and exogenously given output and had to be completed by an ad hoc distributed-lag function to pin down the rate of investment; and Tobin’s q theory, in which investment depends on the ratio of the market value of new capital goods to their replacement cost, but in which “the role of the production function is never clear.” That the two are equivalent was increasingly recognised – Lucas and Prescott saw it first though without drawing the connection to q, and Abel showed the optimal investment rate equates q minus one to the marginal installation cost, though mostly for Cobb-Douglas technology, while Yoshikawa reached the same conclusion under static expectations. This paper does the derivation in full generality. A firm maximises the present value of after-tax receipts, with net receipts written to include profits after tax, the tax deductions generated by past and current investment, investment tax credits and the purchase of investment goods; capital accumulates through Uzawa’s installation function, which is increasing and concave in investment and “drops sharply as I changes from 0 to negative, reflecting the irreversibility of investment.” Applying Pontryagin’s maximum principle yields an optimal investment rule in which investment depends on a tax-adjusted “modified” q and the capital stock alone – and the striking feature is what it does not depend on: “the form of investment function is independent of both the production function and the demand curve for the firm’s output.” Everything about demand and technology that bears on the investment decision is summarised in q, and so are expectations about future investment tax credits. The paper’s central difficulty is then that marginal q is unobservable while average q – the ratio of the firm’s market value to the replacement cost of its existing capital – is, and researchers “are busying themselves regressing investment on average q.” Proposition 1 gives the exact condition under which that is legitimate: if the firm is a price-taker in its output market and the transversality condition holds, marginal q equals average q less the present value of tax deductions on past investment scaled by the replacement cost of capital, if and only if both the installation function and the production function are linearly homogeneous. The intuition is that under constant returns and price-taking, a firm with a different initial capital stock scales its entire optimal policy proportionally, so its average q is unchanged – average q is therefore independent of the initial capital stock, and the market value of an increment of capital is just that increment times average q. Proposition 2 generalises to a price-maker, where the same relationship holds with an additional term “which has the clear interpretation of monopoly rent,” equal to the discounted stream of after-tax revenue scaled by the inverse elasticity of demand; economically, because expanding output lowers the price, “the market value of additional units of capital is therefore less than the average market value of the existing capital stock,” so average q exceeds marginal q. Section 4 puts the theory to work on U.S. corporate-sector data for 1952-78, computing the tax terms under the actual postwar tax system with static expectations about the discount rate (the Baa corporate bond rate plus 4 percent) and a corporate tax rate expected to stay at 48 percent, straight-line depreciation before 1954 and sum-of-years-digits after, with tax lives and investment for three asset classes taken from Christensen and Jorgenson and average q from von Furstenberg. Two patterns emerge: both tax terms rise through the mid-sixties as the tax law permitted faster write-offs, then the deduction term turns down as nominal interest rates rise – though the main reason for the sharp fall in the scaled past-deduction term is that its numerator is evaluated at acquisition prices while its denominator is at replacement cost, so “in a period of sustained inflation, the denominator grows faster than the numerator.” Consequently “the movement in modified q is less pronounced than that in average q.” A simple OLS regression of the investment rate on modified q for 1953-76 gives a slope of 0.0423 with a standard error of 0.00912 and an R-squared of 0.46, but with a Durbin-Watson statistic of 0.43 that “indicates a strong positive serial correlation in the error term” – an exercise the author frames as giving only “a rough idea about how much [modified q] can explain aggregate investment.”

Fumio Hayashi | Macro Paper Warehouse

Tobin's Marginal q and Average q: A Neoclassical Interpretation