Marginal Adjustment Costs and Tobin's q |
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Marginal Adjustment Costs and Tobin's q |
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It has been recognized, as early as Wicksell (1898, 1901), that there is a stock-flow difficulty with the theory of capital and investment. Specifically, as identified by Friedrich Hayek (1941), Abba Lerner (1944, 1953) and Trygve Haavelmo (1960), it is virtually impossible to allow marginal productivity theory to determine the "optimal" level of capital, and then have marginal efficiency of investment theory determine the optimal level of investment without thereby eliminating the flow investment term entirely.
The dilemma arises precisely because we can not take the Fisherian-Austrian concept that the rate of interest will be determined by a flow term (I) and combine it with the Clarkian Neoclassical theory that has interest related to a stock term (K). This dilemma has led some, such as Haavelmo (1960), to conclude that investment demand does not really exist and that the system is merely overdetermined.
"What we should reject is the naive reasoning that there is a `demand schedule' for investment which would be derived from a classical scheme of producers' behavior in maximizing profits. The demand for investment cannot simply be derived from the demand for capital. Demand for a finite addition to the stock of capital can lead to any rate of investment, from almost zero to infinity, depending on the additional hypothesis we introduce regarding the speed of reaction of capital-users." (T. Haavelmo, 1960: p.216).
Lerner's (1944, 1953) proposed resolution was to appeal to some form of increasing costs to investment. In effect, Lerner argued, rising supply price in capital goods industries implies that firms approach the optimal capital stock only gradually. This gradualness is governed by increasing marginal costs which are, in turn, the reason for a falling marginal efficiency of investment (MEI) function. Thus, as investment increases, the cost of new capital goods rise, and thus MEI falls so that MEI = r before the optimal capital stock is reached. These rising costs would therefore slow down adjustment and allow for both optimal capital and optimal investment to be defined.
To see the problem and the solution more clearly, let us define the marginal efficiency of investment (MEI), the "return on a new project", as:
MEI = ｦ K - MAC
where ｦ K is the marginal product of capital and MAC are convex "marginal adjustment costs" (MAC can be thought of as the supply price of capital goods). The argument must now proceed with caution in sequential form - we must divide between what happens in the short-run (single period) and what happens in the long-run (several periods). Let us argue that ｦ K stays fixed within a single period but changes between periods. Now, by the Fisherian argument, firms will invest in any single period of time until MEI = r. The MEI has a downward sloping relationship with investment in a single period because as investment increases, the marginal adjustment costs (MAC) will rise (think of it as the supply price of capital goods rising as a bunch of firms waving their investment plans bear down on the capital goods industry - the more firms invest in any period, the greater will every additional unit of investment cost because the capital goods industry, facing short-run supply constraints, will raise the price of its goods). Thus, in any single period, the optimal level of investment, I*, will be found where MEI = r.
However, we know from standard microeconomic theory as developed by J.B. Clark (1899) that the optimal level of capital (K*) is where ｦ K = r. But if ｦ K = r, then capital is at its optimal level and thus no investment at all should occur! Why invest (i.e. increase the capital stock) if we are the optimal capital level already? The standard marginal adjustment cost solution we are outlining in terms of sequences of periods here was initiated by Hayek (1941) and Lerner (1944, 1953) and formally incorporated in two forms in modern economics. The first, and more simplistic form, is to follow Eisner and Strotz (1963), Lucas (1967), Gould (1968), Treadway (1969) and Uzawa (1969) and allow for convex marginal adjustment costs which are internal to the firm, although why these costs exist are left largely unexplained. The stricter version of Foley and Sidrauski (1970, 1971) and Witte (1968) follows the rising supply price story - thus, in this case, marginal adjustment costs are external to the firm.
In either case, we are describing investment in terms of sequences of periods where ｦ K is fixed between periods. If there were no marginal adjustment costs (MAC), then we can see that MEI = ｦ K at all times - within and between every period - so that no investment would ever occur and, if it did (because, say, r rose), then there would be an instantaneous jump to the new optimal capital stock (as is implied in the Jorgenson model). In order to have investment flows drawn out over time, we need marginal adjustment costs in order to delay the movement to the optimal level of capital, or rather to extend it over several periods so that, within any period, we can talk of "investment" with a straight face. Marginal adjustment costs is what permits us to reconcile the Neoclassical microeconomic theory of capital (Clark) with the macroeconomic theory of investment (Fisher).
So, let return to our main argument, using Figure 6 as an illustration. At the beginning of any period t, there is a stock of capital, Kt, which yields us a marginal product of capital, ｦ Kt. If, for some reason, r is below ｦ Kt, then a higher optimal level of capital, K*, is called for by the Clarkian marginal productivity argument. This is shown on the right hand side of Figure 6 where the marginal product of capital schedule ｦ k and r meet. If there were no marginal adjustment costs, we would just move there immediately.
Figure 6 - Marginal Efficiency of Investment and Marginal Product of Capital
With marginal adjustment costs, things are different. At the beginning of time period t, when we have zero investment (I = 0), we have MEI = ｦ Kt > r - thus firms have an incentive to invest. But as investment increases, MAC increase (ｦ K stays constant though), and thus we can draw out an MEI schedule for period t (labelled here MEIt) which falls as investment rises. This decline reflects the rising marginal adjustment costs. Firms will continue increasing their investment in period t until MEI = r. Obviously, as the optimal investment level is positive, It* > 0, then marginal adjustment costs are positive, so that it is still true that ｦ Kt > r. We have invested some amount at time t so, at the beginning of the next period (t+1), capital stock will increase by the amount invested in the previous period and thus the new initial marginal product of capital, ｦ K will fall. If it does not fall all the way to r, then new investment is still required, and thus we invest some amount in the second period.
We can see the next step in Figure 7. When we add It* to the previous Kt, we obtain the new capital stock Kt+1 = Kt + It* which is greater than Kt but not yet at K*. This level of capital stock Kt+1 will generate a new but lower marginal product ｦ kt+1 which is still above r. Thus, we can draw a new MEI curve for period t+1 (labelled in Figure 7 as MEIt+1) which starts from ｦ kt+1 and thus lies below our old MEIt. However, the same marginal adjustment costs apply, so it is also downard sloping - and, once again, during t+1, firms will invest until MEIt+1 intersects r at It+1*. This amount of investment is then added to Kt+1 to yield Kt+2 = Kt+1 + It* (as in Figure 7), which is still below the optimal capital stock K*, but yet closer to it. We can thus generate a new marginal product of capital ｦ Kt+2 for the next period, derive a new MEI schedule, and so on.
Figure 7 - Approaching the Optimal Capital Stock, K*
As we see, we have investment every period increasing the capital stock slowly until we reach the optimal K*. Notice also that since the MEIt curves are consecutively lower for higher and higher t, this implies that every investment increment is smaller (e.g. It* is greater than It+1*), thus we not only approach K* slowly, we approach it asymptotically. At K*, the corresponding marginal product of capital is merely r - thus the MEI curve (represented in Figure 7 as MEI･ ) will be such that MEI = r at zero investment (I = 0), and thus the process ends. We have achieved the optimal capital level, K*. Thus, it is only at this final point that the Fisherian story (MEI = r) and the Clarkian marginal productivity story (ｦ K = r) will both be true - but investment will be zero. As Lerner writes:
"The difference between the marginal productivity of capital and the rate of interest is the force which makes the stock of equipment grow or decline. Only in a stationary society [when net investment is zero] does this difference disappear, and then the marginal productivity of capital and MEI are both equal to the rate of interest." (A. Lerner , 1944 :p.335).
The system, then, depends upon the presence of adjustment costs to prolong the disequilibrium so that investment occurs every period. The speed of adjustment is entirely dependent upon the elasticity of the MEI curve which, in turn, is entirely dependent (if we conceive of marginal adjustment costs as rising supply price of capital) to the elasticity of the supply curve of the capital goods industries. If supply is highly inelastic, then the MEI curve will be steeper and thus adjustment towards the desired capital level will take longer. If the supply of capital goods is entirely elastic (i.e. the capital goods industries are capable of meeting an any increase in demand for capital goods), then the MEI will be similarly elastic and adjustment will be instantaneous.
As a consequence, we can posit the amount of (net) investment as a function of MEI and r, or:
I = I(MEI/r - 1)
but where if MEI > r, then MEI/r > 1 and investment rises. If MEI/r = 1, then we have a determinate I*. The term MEI/r has a particularly famous connotation as one of many versions of Tobin's marginal "q" (see Tobin, 1969; Brainard and Tobin, 1968), i.e. q = MEI/r = (ｦ K - MAC)/r. If q = 1, then investment does not change whereas if q is greater than or equal to 1, then investment will rise and fall accordingly. So, we can posit a function of the type, I = I(q - 1) and argue accordingly, that I'(.) > 0. Note the important result that if r rises, then q falls and so investment falls. Thus, dI/dr = Ir < 0.
Attempts were made to incorporate marginal adjustment costs into Jorgenson's theory in order to obtain a proper theory of investment in an optimization context. To this end, the works of Robert Eisner and Robert H. Strotz (1963), Robert E. Lucas (1967) and John P. Gould (1968) were instrumental. Specifically, to the term sI in Jorgenson's equation, Lucas-Gould modification was to replace it with sI + C(I)sI where C(I) reflects convex marginal adjustment costs, thus C(I) is a convex function centered around zero, where C｢ > 0 when I > 0 (positive investment) and C(0) = 0 (no costs if no investment) and C｢ < 0 when I < 0 (negative investment) and C｢ ｢ > 0 (convexity). These adjustment costs can be due to "intrinsic" factors (i.e. costs of installation) or "extrinsic" factors (rising supply price, as Foley and Sidrauski (1970) propose). Thus, firms now face the problem:
max V = ・/font> 0･ [ptYt - stIt - C(It)stIt - wtNt] e-rt dt
s.t. Yt = F(Kt, Nt)
dKt/dt = It - d Kt
Thus, setting up the current-value Hamiltonian:
H = pF(K, N)- sI - C(I)sI - wN + l [It - d K]
so:
dH/dN = pFN - w = 0
dH/dI = -s - sCII - sC + l = 0
- dH/dK = dl /dt - rl = - pFK + l d
dH/dl = dK/dt = I - d K
The first solution yields, once again, FN = w/p. The second can compactly be rewritten as
l = s(1 + C + CII)
and the second is merely:
dl /dt = (r+d )l - pFK
Now, define marginal "q" as q = l /s. Then, dq/dt = [(dl /dt)s - (ds/dt)l ]/s2 or dq/dt = (dl /dt)/s - (ds/dt)l /s2. Now, as l = sq, then dq/dt = (dl /dt)/s - (ds/dt)q/s, so:
s(dq/dt) + (ds/dt) = dl /dt
substituting in for dl /dt we obtain:
s(dq/dt) + (ds/dt) = (r+d )l - pFK
so:
dq/dt = (r+d )q - (ds/dt)/s - pFK/s
where, recall, q = l /s = 1 + C + CII. Thus, we can define implicit function, I = Y (q), with y (1) = 0 and y ｢ > 0. Then, for our last equation we obtain:
dK/dt = It - d Kt = y (qt) - d Kt.
thus we have a two differential equation system in K and q where, if we had explicit terms for C(I) we could solve for the path K(t) and q(t). Thus, the marginal adjustment cost model does not yield an "optimal capital" level but rather an optimal adjustment path. The q defined in the first equation is, as noted earlier, James Tobin's "q" (which, in this model, is defined as q = l /s where l is a costate variable representing the shadow value of capital). Notice that in our net investment equation, dK/dt, we have the implicit function y (q) - i.e. a Keynesian "investment" schedule y (q) where y (1) = 0 and y ｢ > 0.
James Tobin's "q" theory of investment was presented in Brainard and Tobin (1968) and Tobin (1969). Effectively, Tobin's q theory proposes that a firm will invest until q = 1 where q is defined as the ratio between the stock-market valuation of existing real capital assets and its current replacement cost. In Keynes's (1936: p.135) language, q = V/C where V is what Keynes defined as present value of the prospective yield of the capital asset while C is what he defined as the supply price of the capital asset. Consequently, the margin, q can be seen as the ratio of the marginal efficiency of investment to the rate of interest, i.e. q = MEI/r so that the Keynesian investment function can be rewritten as I(q - 1) where firms invest until q = 1 (or, equivalently, MEI = r). As we can see immediately from above, this is equivalent to y (q). Thus, as commentators such as Abel (1979) and Hayashi (1982) have noted, the Eisner-Strotz-Lucas-Gould theory of investment with marginal adjustment costs is formally equivalent to Tobin's "q" theory of investment - and, of course, logically equivalent to what Abba Lerner (1944, 1953) had already long proposed.