Irving Fisher's theory of capital
and investment was introduced in his *Nature of Capital and Income* (1906) and *Rate
of Interest* (1907), although it has its clearest and most famous exposition in his *Theory
of Interest* (1930). We shall be mostly concerned with what he called his "second
approximation to the theory of interest" (Fisher,
1930: Chs.6-8), which sets the investment decision of the firm as an intertemporal
problem.

In his theory, Fisher *assumed* (note carefully) that all capital was
*circulating* capital. In other words, all capital is used up in the production
process, thus a "stock" of capital K did not exist. Rather, all
"capital" is, in fact, investment. Friedrich Hayek
(1941) would later take him to task on this assumption - in particular, questioning how
Fisher could reconcile his theory of investment with the Clarkian theory of production
which underlies the factor market equilibrium.

Given that Fisher's theory output is related not to capital but rather to
investment, then we can posit a production function of the form Y = ｦ
(N, I). Now, Fisher imposed the condition that investment in any time period yields output
only in the next period. For simplicity, let us assume a world with only two time periods,
t = 1, 2. In this case, investment in period 1 yields output in period 2 so that Y_{2}
= ｦ (N, I_{1}) where I_{1} is period 1
investment and Y_{2} is period 2 output. Holding labor N constant (and thus
striking it out of the system), then the investment frontier can be drawn as the concave
function where ｦ ｢ > 0 and ｦ ｢ ｢ <
0. The mirror image of this is shown in Figure 1 as the frontier Y_{2} = ｦ (I_{1}). Everything below this frontier is technically
feasible and everything above it is infeasible.

Letting r be the rate of interest then total costs of investing an amount
I_{1} is (1+r)I_{1}. Similarly, total revenues are derived from the sale
of output pY_{2 }or, normalizing p = 1, simply Y_{2}. Thus, profits from
investment are defined as p = Y_{2} - (1+r)I_{1}
and the firm faces the constraint Y_{2} = ｦ (I_{1})
(we have omitted N now). Thus, the firm's profit-maximization problem can be written as:

max p = ｦ (I

_{1}) - (1+r)I_{1}

so that the optimal investment decision will be where:

ｦ ｢ = (1+r)

In Fisher's language, we can define ｦ ｢ -1 as the "*marginal rate of return over cost*",
or in more Keynesian language, the "*marginal
efficiency of investment*", so MEI = ｦ ｢ - 1. Thus, the optimum condition for the firm's investment
decision is that MEI = r, i.e. marginal efficiency of investment is equated with rate of
interest. Obviously, as ｦ (I_{1}) is a concave
function, then as I_{1} rises, ｦ ｢
declines. As the rate of interest rises, then to equate r and MEI, it must be that
investment declines - thus the negative relationship between investment and interest rate.
Succinctly, I = I(r) where I_{r} = dI/dr < 0.

Figure 1- Fisher's Investment Frontier

In Figure 1, we have drawn Fisher's investment frontier Y_{2} = ｦ (I_{1}) where the concave nature of the curve reflects, of
course, diminishing marginal returns to investment. Suppose we start at initial endowment
of intertemporal output E - where E_{1} > 0 and E_{2} = 0, so we only
have endowment in period 1. Then the amount of "investment" involves allocating
some amount of period 1 endowment to production for period 2. The output left over for
period 1 consumption, let us call that Y_{1}*, is effectively the amount of
intitial endowment that investment has not appropriated, i.e. Y_{1}* = E_{1}
- I_{1}*. The investment decision will be optimal where the investment frontier is
tangent to the interest rate line, i.e. where ｦ ｢ = (1+r). At this point, intertemporal allocation of income becomes
Y* = (Y_{1}*, Y_{2}*) where Y_{2}* = ｦ
(I_{1}*) and Y_{1}* = E_{1} - I_{1}*. It is obvious, by
playing with this diagram, that as r increases (interest rate line becomes steeper), then
I_{1}* declines; whereas as r declines (interest line becomes flatter), then I_{1}*
increases. Thus, dI/dr < 0, so investment is negatively related to the interest rate.

So far, we have said nothing about the ownership structure of the firm or how this theory can be grafted into a wider macroeconomic theory. There might be potential modifications in this regard. There are two main questions that arise here. Firstly, if we suppose that firms are owned by entrepreneurs, might not the investment decision of the firm be affected by the owner's desired consumption-savings decision? Secondly, what exactly is the relationship between the firm's investment decision, its financing decision and wider financial markets?

As Jack Hirshleifer (1958,
1970) later noted, we can answer these questions by reworking Fisher's full theory of
investment into a "two-stage" budgeting process. Specifically, Hirshleifer noted
that *if *we consider firms to be owned by entrepreneurs, then we must integrate Fisher's (1930) consumption-savings decision (the
"first approximation") of the owner-entrepreneur with the investment decision
(the "second approximation") of the firm which that entrepreneur owns.

If we consider an entrepreneurial firm, i.e. a firm owned by a person,
then we must endow the firm with a utility function U(.). Now, if we have the entrepreneur
maximize utility with respect solely to the intertemporal investment frontier, we achieve
a solution akin to point G* in Figure 2. In this case, then, it seems that the optimal
investment decision of the firm is affected by owner's preferences. However, by realizing
that firms have, in fact, a two-stage budgeting process by which firms *first*
maximize present value as before (point Y*) and *then* borrow/lend their way to the
entrepreneur's optimal solution (such as at point C* or F* in Figure 2, depending on the
preferences of the firm's owner) we realize that the original point G* was not optimal.
Hirshleifer refers to "investment", then, as incorporating both the
"productive opportunities" implied at point Y* and the "market
opportunities" offered up by points C* or F*.

Figure 2- Fisher's Separation Theorem

The two central results of this two-stage budgeting has become known as
the *Fisher Separation Theorem*:

(i) the firm's investment decision is

independentof the preferences of the owner;(ii) the investment decision is

independentof the financing decision.

We can see the first by noting that *regardless* of the preferences
of the owner, the firm's investment decision will be such that it will position itself at
Y*, thus making the maximization of present value *the* objective of the firm (which,
of course, is equivalent to Keynes's "internal rate of return" rule of
investment).

The second part of the separation theorem effectively claims that the
firm's financing needs are independent of the production decision. To see why more
clearly, we can restate this in terms of the Neoclassical
theory of *"real" loanable funds* set out by Fisher (1930). The demand for "loanable
funds" equals desired investment plus desired borrowing of borrowers whereas the
supply of "loanable funds" equals desired savings minus desired investment of
savers. In Figure 2, suppose we have two entrepreneurs with identical firms, both of which
start with endowment E and one invests and saves to achieve point F* while another invests
and then borrows to achieve point C*. Looking carefully at Figure 2, we see that the first
agent's desired investment is I_{1} = E_{1} - Y_{1} while his
desired saving is equal to E_{1} - F_{1}*. In contrast, the second agent
has desired investment equal to I_{1} = (E_{1} - Y_{1}) as well,
but desires to borrow the amount (C_{1}* - E_{1}). Thus, the total demand
for loanable funds is D_{LF} = (E_{1} - Y_{1}) + (C_{1}* -
E_{1}) = C_{1}* - Y_{1 }while the total supply of loanable funds
is S_{LF} = (E_{1} - F_{1}*) - (E_{1} - Y_{1}) = Y_{1}
- F_{1}*. Now, *if *there is equilibrium in the market for loanable funds,
then:

S

_{LF}= Y_{1}- F_{1}* = C_{1}* - Y_{1}= D_{LF}

but by plugging in the details for these terms:

S

_{LF}= (E_{1}- F_{1}*) - (E_{1}- Y_{1}) = (E_{1}- Y_{1}) + (C_{1}* - E_{1}) = D_{LF}

and rearranging:

2(E

_{1}- Y_{1}) = (E_{1}- F_{1}*) - (C_{1}* - E_{1})

Now, each agent invested E_{1} - Y_{1}, thus total
investment is I = 2(E_{1} - Y_{1}). Simultaneously, the first agent saved
(E_{1} - F_{1}*) and the second agent dissaved (E_{1} - C_{1}*)
so total saving is S = (E_{1} - F_{1}*) - (C_{1}* - E_{1}).
Thus, the equation for loanable funds equilibrium can be rewritten simply as:

I = S

i.e. total investment equals total savings.

Note the condition that for total investment to be equal to total savings,
then the demand for loanable funds must equal the supply for loanable funds and this is *only*
possible if the rate of interest is appropriately defined. If the interest rate was such
that the demand for loanable funds was not equal to the supply of it, then we would also *not*
have investment equal to savings. Thus, in Fisher's "real" theory of loanable
funds, the rate of interest that equilibrates supply and demand for loanable funds will
also equilibriate investment and savings. This is effectively *the* story in Neoclassical macroeconomic theory.

[Note: our expression is slightly different from Fisher's original 1930 formulation as, instead of netting out as we have done, Fisher had the supply for loanable funds defined as savings plus disinvestment and demand for loanable funds defined as investment plus dissaving; nonetheless, the equilibrating interest rate is unchanged by whichever definition we choose.]