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

Two rival investment theories – Jorgenson’s neoclassical theory and Tobin’s q – were each incomplete in a way the other seemed to repair, and the paper sets out to show the connection rigorously. Jorgenson’s earlier neoclassical approach “derives the optimal capital stock under constant returns to scale and exogenously given output. To make the rate of investment determinate, the model is completed by a distributed lag function for net investment.” The author lists two drawbacks: “the assumption of exogenously given output (which makes the optimal capital stock determinate) is inconsistent with perfect competition,” and “the theory itself cannot determine the rate of investment; rather, it relies on an ad hoc stock adjustment mechanism,” with adjustment costs entering only implicitly through the lag function. Lucas, Gould, Uzawa and Treadway repaired this by introducing the cost of installing new investment goods, so that “capital stock is given to the firm at each moment of time because of the adjustment costs in changing capital stock. What the firm can control at each moment of time is the rate of investment, not the stock of capital.” Jorgenson himself had recommended the modification, and the paper quotes him: “A derivation of this model incorporating installation costs explicitly with constant returns to scale in both production and installation is obviously much more satisfactory than the original derivation.” Tobin’s theory, meanwhile, made the rate of investment a function of q but left the production function’s role obscure.

The equivalence had been noticed, but only in special cases or without the connection to q being drawn. “Lucas and Prescott were the first to recognize this, although they never indicated the connection to the ‘q’ theory. Later Abel has shown that the optimal rate of investment is the rate for which q-1 is equal to the marginal cost of installment. However, his discussion is focused primarily on the Cobb-Douglas technology. Yoshikawa arrived at the same conclusion as Abel did, but his model is characterized by static expectations.” What Section 2 adds is generality: it “integrates the two theories of investment in a very general model of the firm’s present value maximization and derives the optimal rate of investment as a function of q.”

Maximise the present value of after-tax net receipts, with the tax code written in explicitly, subject to a capital accumulation constraint containing an installation function. Net receipts are “profits after tax plus depreciation tax deductions minus purchases of investment goods plus investment tax credits,” with the depreciation allowance per dollar of investment depending on the asset’s age and on the tax code in effect when it was acquired. The author flags an assumption embedded in this formulation: it “implicitly assumes that changes in the depreciation formula do not apply retrospectively to past investments, which is the case in the postwar U.S. except for the 1962 change.” Two adjustment-cost formulations are discussed – the Lucas-Gould-Treadway one, where an installation cost is subtracted inside the profit function, and Uzawa’s, where gross investment converts into capital at a rate given by the installation function – and the paper focuses on Uzawa’s “since the two formulations of adjustment costs give similar results concerning the optimal investment rule,” noting in a footnote the corresponding modifications if the other is used. The firm may be a price-taker or a price-maker; if it is a price-maker, the output price depends on output.

It is the present value of tax deductions owed to past investments – “often neglected in the literature” – and because it is independent of current and future decisions it drops out of the optimisation entirely, yet it is exactly what must be subtracted from average q later. Rewriting the objective splits it into a term the firm optimises over and this second term; “since A(0) is independent of current and future decisions by the firm, the optimization problem is equivalent to maximizing the first term.” Its counterpart for new investment, the present discounted value of tax deductions per unit of current investment, “corresponds to uz in the notation in Hall and Jorgenson which assumes static expectations about future tax rate.”

As equating the marginal benefit of installing one more unit of new investment goods to its marginal cost, with three pieces on the cost side. Rewritten in economic terms, the condition has “the acquisition price of new investment goods from the viewpoint of the firm. Because of the investment tax credits, it is less than the market price”; a second term representing “(implicit) adjustment costs associated with investment” – if there were no adjustment costs the firm’s market value would rise by the shadow price for each unit of investment, “but the capital stock actually increases by only [the marginal product of the installation function],” so the difference “represents the market value foregone due to the concave installation function”; and on the other side, the present value of tax deductions due to one unit of current investment. The companion condition on the shadow price “states that [it] is the present discounted value of additional future (after-tax) profits that are due to one additional unit of current investment,” and the author notes there are two such benefits – increased productive capacity, and the fact that “the installation function as a function of I shifts downward as K increases.”

Investment depends only on modified q, the capital stock and time – so the entire influence of the production function and the demand curve is summarised in one number. Defining marginal q as the shadow price of capital divided by the price of investment goods, and modified q as marginal q divided by one minus the investment tax credit rate minus the present value of new-investment tax deductions, the first-order condition inverts to give the investment rule. The author underlines the point: “the remarkable information content of q should be noted. Once q is known (along with [the tax terms] and K), the firm can decide the optimal rate of investment through the knowledge of the installation function alone. All the information about the demand curve for the firm’s output and the production function that are relevant to investment decision is summarized by q. Expectations about future course of the rate of investment tax credits are also incorporated in q and do not affect the form of the investment function.” He adds a precise condition for the familiar form: the rule “reduces to” an investment rate I/K depending on modified q “if and only if the installation function is linearly homogeneous in I and K,” and notes that this linear homogeneity “will pay an important role in the discussion of marginal q and average q.”

Because marginal q cannot be observed and average q can, and the empirical literature was already substituting one for the other. “If we knew marginal q, then econometric implementation of the ‘q’ theory would be quite straightforward. Unfortunately, however, marginal q is not directly observable. What we can (in principle) observe is average q. There have been increasing efforts to measure average q for the U.S. corporations, and people are busying themselves regressing investment on average q. The researchers should feel uneasy about doing this, unless they are sure that average q and marginal q are practically the same thing.”

Marginal q equals average q less the scaled present value of past-investment tax deductions, if and only if both the installation function and the production function are linearly homogeneous – given that the firm is a price-taker and the transversality condition holds. The proposition is stated with both directions: after proving the forward direction by integrating a differential equation along the optimal path and applying transversality, the author writes “the converse is now obvious.” Three remarks follow: “the proposition holds at any point in time along the optimal path”; “since the installation function is concave in I, the optimal path is unique if it exists”; and the integral relationship used in the proof “has already been noted in a different context by Blinder and Weiss.”

Under constant returns and price-taking, average q does not depend on how much capital the firm starts with, so an extra unit of capital is worth exactly the average. Setting aside taxes and depreciation, consider a firm with capital K0 following an optimal policy, and a second firm with identical technology but capital K0’. “It is clear that this second firm’s optimal policy is [the first firm’s policy scaled by K0’/K0] if the production and installation functions are characterized by constant returns to scale. Hence the expected future profits of the second firm are equal to K0’/K0 times those of the first firm, implying that the first firm’s average q is equal to the second firm’s average q. In other words, average q is independent of the initial capital stock if the production and installation functions are linearly homogeneous and if the firm is a price-taker.” It follows that when a firm invests and adds capital, “the market value of these additional units of capital… is [the increment] times average q, because the average market value of the firm with K is equal to that of the firm with K + [the increment].”

The intuition breaks, but a simple relationship survives: average q exceeds marginal q by the monopoly rent. “The economic intuition behind Proposition 1 does not carry over to a price-making firm, since if the firm expands its output, the output price will fall. The market value of additional units of capital is therefore less than the average market value of the existing capital stock. However, a fairly simple relationship between marginal q and average q still exists.” Proposition 2 states it for a price-maker with linearly homogeneous production and installation and a satisfied transversality condition: marginal q equals average q, less the scaled past-deduction term, less a discounted integral of after-tax revenue weighted by the inverse elasticity of demand for the firm’s output. “The last term has the clear interpretation of monopoly rent,” which the author makes precise: a price-taking firm with the same technology and capital stock would choose the same investment and labour if its output price were the price-maker’s price times one minus the inverse demand elasticity, so the difference between the two firms’ profits is exactly that wedge times revenue.

An estimable investment equation: the investment rate as a function of average q net of the past-deduction term, scaled by the tax factor. If the optimal rule takes the I/K form, the production function is linearly homogeneous, and the firm is a price-taker, then the investment rate depends on average q less the scaled past-deduction term, divided by one minus the investment tax credit rate minus the present value of new-investment deductions. A footnote records the special case that ties the result back to the earlier literature: “if taxes and subsidies are all ignored, then [this] reduces to the investment function derived by Lucas and Prescott.”

Static expectations about the discount rate, a constant expected corporate tax rate, and a stylised depreciation practice the author is candid about. The computation covers “the period 1952-78 for the U.S. corporate sector as a whole assuming the actual U.S. postwar tax system.” Expectations about the future tax rate and the future nominal discount rate must be specified: “We assume static expectations about r(t). Since we will use the long rate – the Baa corporate bond rate plus 4% – for r(t), this assumption seems innocuous. We also assume that future corporate tax rate is expected to be constant at 48%.” On depreciation, “it is assumed that the straight line depreciation formula was adopted by the corporations prior to 1954 and that the sum-of-years-digits formula was adopted after 1954,” justified by the fact that under the Revenue Act of 1954, “for a wide range of the tax life and the interest rate, the sum-of-years-digits formula dominates the other two depreciation formulas in that it gives the highest value of z.” The author immediately qualifies: “actual depreciation practice by the U.S. corporations are much more complicated than our assumption presumes. More fact-finding empirical research on this is highly desirable.” Tax lives and corporate investment for three asset types – producers durable equipment, nonresidential structures and residential structures – come from Christensen and Jorgenson, as does the replacement cost of capital, which “does not include land and inventories” and is built by perpetual inventory, a method that “does not take the installation costs into account”; the author offers the reading that “the installation costs are submerged in physical depreciation.” Average q is taken from von Furstenberg.

The tax terms rise through the mid-sixties and then fall, and modified q moves less than average q. “Both [the past-deduction term and the new-investment deduction term] increase during the period ending the mid-sixties. This is because the U.S. tax law has been allowing faster write-offs (shorter tax lives). From the late sixties the nominal interest rate began to increase noticeably. This explains the downward trend in [the new-investment deduction term] since 1967.” For the scaled past-deduction term, rising rates “also partly explains the drastic downward trend… in the late sixties and the seventies, but the main reason… is that [the numerator] is evaluated at acquisition prices. The denominator (i.e., the capital stock at replacement costs) immediately reflects changes in the price of investment goods, while the numerator depends on the prices at which the existing assets were acquired in the past. Therefore, in a period of sustained inflation, the denominator grows faster than the numerator.” Comparing the two q measures, “the movement in modified q is less pronounced than that in average q. This is explained by the downward trend in [the scaled past-deduction term] after the mid-sixties and by the upward trend in the rate of investment tax credits.”

The author says explicitly that it does not, in this channel. In a footnote attached to the inflation observation: “this is not a channel through which inflation depresses investment. To see this, suppose a change occurs in the corporate tax law which now allows a firm to evaluate the existing assets at current prices in calculating depreciation allowances. Then, [the past-deduction term] will immediately change to reflect current asset prices, but that change will be exactly offset by a re-evaluation of the firm in the stock market, leaving modified q unchanged.”

Very little – it is framed as a rough check and reported with a diagnostic that undercuts it. The section opens by saying the estimation is “to get a rough idea about how much [modified q] can explain aggregate investment,” and the linear form is estimated by ordinary least squares. The result is an intercept of 0.0980 (standard error 0.00840) and a slope on modified q of 0.0423 (standard error 0.00912), with R-squared 0.46, over 1953-1976, the mean of the dependent variable being 0.136. The author’s only comment on fit is the caveat: “the Durbin-Watson statistic (D.W.) is very low and indicates a strong positive serial correlation in the error term.” No claim about the magnitude of the estimated adjustment speed, or about the theory being confirmed, is made.

That the celebrated equality of marginal and average q rests on a conjunction of assumptions that the paper states as necessary and sufficient, not as approximations. Price-taking in the output market, linear homogeneity of the production function in capital and labour, linear homogeneity of the installation function in investment and capital, and a satisfied transversality condition are all required; the homogeneity pair is an if-and-only-if. Drop price-taking and a monopoly-rent term appears; drop homogeneity and the equality fails. And even under the full set of conditions, what average q equals is not marginal q itself but marginal q plus the scaled present value of tax deductions on past investment – a term the author notes had been “often neglected in the literature,” and which his own data show trending sharply over the sample.