Dale W. Jorgenson (1963, 1967, 1971) proposed a different investment theory which was derived, in part, from the Fisher-Hirshleifer theory. Let output at any particular time be related via some production function to labor employed (N) and existing capital stock:

Y = F(N, K)

Letting total revenues be pY then costs are wN and sI where s is the price of a unit of capital. Net proceeds at time t are (suppressing time subscripts henceforth):

R = pY - sI - wN

so that the value of the firm is merely the discounted stream of earnings:

V = ・/font>

_{0･ }Re^{-rt}dt = ・/font>_{0･ }[pY - sI - wN] e^{-rt}dt

Gross investment is defined as: I = dK/dt + d K. Following the Fisherian lead, the objective of the firm is to maximize net present value, thus the optimization problem is:

max V = ・/font>

_{0･ }[pY - sI - wN] e^{-rt}dt

s.t. Y = F(K, N) = 0

dK/dt = I - d K

for some initial capital stock, K_{0}. Forming a current-value
Hamiltonian:

H = pF(K, N) - sI - wN + l (I - d K)

where l is the current-value costate variable representing the shadow price of capital. The first order conditions for the optimal capital path, using employment N and investment I as control variables and K as our state variable are:

dH/dI = -s + l = 0

dH/dN = pF

_{N}- w = 0

- dH/dK = dl /dt - rl = - pF

_{K}+ l d

dH/dl = dK/dt = I - d K

Obviously, the second condition implies that F_{N} = w/p, i.e.
labor is employed until the marginal product of labor is equal to the wage. From the first
equation, we see that, s = l , thus ds/dt = dl /dt. Thus:

ds/dt - rq = -pF

_{K}+ sd

or:

pF

_{K}= s[d + r - (ds/dt)/s]

But recall that F_{K} = dY/dK, thus the marginal product of
capital is:

F

_{K}= s[d + r - (ds/dt)/s]/p

Now, it might be useful to define Jorgenson's "real user cost of capital" as c = s[d + r - (ds/dt)/s] thus:

pF

_{K}= c

i.e. the firm reaches an equilibrium level of capital stock when the value
of the marginal value product of capital (pF_{K}) is equal to its implicit rental
rate (real user cost, c). The logic of c is simple: the rent on a unit of capital must
cover the opportunity cost of lending it out (r), depreciation per unit (d ) minus the expected capital gains (ds/dt)/s. If we had an explicit
and invertible production function, then the optimal capital stock K* could be determined
easily from F_{K} = c/p. For instance, suppose we had a Cobb-Douglas function Y =
K^{a} L^{(1-a ) }so
that F_{K} = a (Y/K), then K* = pa
Y/c. Thus, in general K* = ｦ (Y, p, r, d
, s, ds/dt, p) or simply K* = ｦ (Y, p, c) where K* depends
positively on Y and p and negatively on c.

This is fine for optimal capital stock, but how does one obtain
investment? Investment is defined as the instantaneous change in the *optimal* stock
of capital, thus, in principle, there is no investment unless there is some reason to *change*
the optimal stock of capital (by say, imposing exogenous some rate of technical change or
some population growth rate) or, alternatively, investment is derived from the adjustment
path *towards* the optimal capital stock, K*. Following the first case, suppose that
K_{t}* ｹ K_{t+1}* for some reason. Then, in
principle, moving to continuous time, from any given K, then investment is defined as I =
dK* + d K, thus:

I = ｦ (dY, dp, dc) + d K

so investment is a function of changes in the real user cost of capital
(dc) (which in turn depend positively on changes in r, s, p and negatively on changes in
(ds/dt)/s), changes in output (dY), changes in the price of output (dp) and the *level *of
capital (K).

Of course, Jorgenson's (1963)
theory is less about investment and more about optimal capital. If investment, as
Friedrich Hayek (1941) and Trygve Haavelmo (1960) argued, is seen as the *adjustment* from a given level of capital to the optimal level
of capital stock, then, in Jorgenson, investment is instantaneous. To obtain investment,
then, Jorgenson subsequently added controversial delivery lags. Jorgenson proposed that
only a fraction l _{0} of the investment goods ordered
in the current period are delivered in a current period, fraction l
_{1} delivered in the next period, fraction l _{2}
in the period after that and so on. As a result, actual investment (or actual change in
capital stock) at any time period t is the sum of the proportion of past desired
investment that will be delivered at t, specifically:

I

_{t}= l_{0D }K_{t}* + l_{1D }K_{t-1}* + .... + l_{nD }K_{t-n}*

so that actual investment I_{t} at time t is some weighted
function of desired movements to the optimal capital stock in the previous n periods, D K*_{t-i} for i = 0, .., n which are delivered at time t.
This structure proved highly controversial in the literature, e.g. Eisner and Strotz (1963), Eisner and Nadiri (1968), not only for its *ad
hoccery* but also because it makes Jorgenson's theory untestable empirically because
not only does it lead to autocorrelated error terms but also it becomes virtually
intistinguishable from accelerator theories of investment. As a result, Jorgenson's theory
of investment, which relies heavily on prices as independent variables, performs usually
extraordinarily poorly at the empirical level.