

1 Introduction This text presents the perfect capital market model and the associated Fisher separation theorem. This model demonstrates that utility maximizing and perfectly rational owners will agree on forcing the managers of the firms they own to pursue the profitmaximizing strategy. This is so for diverse timepreferences among the owners as long as capital markets are perfect. 2 The Model Model
assumptions: 1)
Capital
markets are perfect ·
Agents are perfectly rational and they pursue
utility maximization. ·
There are no direct transaction costs,
regulation or taxes, and all assets are perfectly divisible. ·
Perfect competition in product and securities
markets. ·
All agents receive information simultaneously
and it is costless. The information is either certain or risky. 2)
An
arbitrary number of agents are endowed with some initial resources (N_{0}=7
mill. $) of a good (C_{0}). This good may either be consumed today (P_{0})
or be invested today (I_{0}) and transformed into consumption
tomorrow (P_{1}). 3)
The
agents in the economy may choose to buy stocks in a firm that has four
investment projects at its disposal. The outlays and returns on these
projects are displayed in figure I below. A manager is hired by the agents to
run the firm. From the numbers, it is clear that there is a decreasing return
to scale on investments. In other words, the marginal rate of return falls as
investments rise. 4)
The agents have different but monotonous
preferences, and they exhibit decreasing marginal utility. Notation: ·
N_{0} is the aggregated initial
resources held by all agents. ·
r_{m} is the marketdetermined
interest. ·
C_{0} is today’s consumption or investment
good. ·
C_{1} is tomorrow’s consumption good.
C_{1} = W*_{1}  (1+ r_{m})C_{0} is the
capital market line. ·
P*_{0} is the efficient consumption
today by all agents. ·
P*_{1} is the efficient production =
consumption tomorrow by all agents. ·
W*_{0} is the present value of
efficient production and consumption: W*_{0} = P*_{0} + P*_{1}/(1+r_{m}). ·
W*_{1} is future value of efficient
production and consumption: W*_{1 }= W*_{0}(1+r_{m}). Model illustrated 



Analysis Efficiency requires
MRS = (1+r_{m}) = MRT. In words, the marginal rate of substitution
must equal the marginal rate of transformation between consumption today and
tomorrow. In the above figure this is illustrated by the simultaneous
tangency of the capital market line to the production possibility curve and
the agents’ time preferences of consumption. As may be observed the agents
will maximize their utility when they invest a total of 4 million $ of their
initial endowment (7 mill. $) in stocks and then order the manager to
maximize profit. The manager does that by undertaking project D and B and
produce at point B. The manager cannot pursue his own interest because the
agents have full information about the projects and thereby about the profit
maximizing strategy. In other words there are no agency problems in this
world. Very importantly, the production at point B is strongly efficient
because it maximizes the agents’ total life income. The figure shows a situation with
two agents. Agent I is patient and prefers to consume tomorrow, and agent II
is impatient and prefers to consume today. Imagine that agent I is the only
capital owner in the economy. His initial endowment is $7 million, and he
owns the entire firm. If a perfect capital market existed, he would maximize
his utility by ordering his manager to maximize profit and produce at point
B. However, he would be able to consume at point E1 by lending
money in the capital market. Without capital markets his utility and profit
maximizing decision would imply production and consumption at point X
and his welfare would be reduced. Now, replace agent I with agent II. If
perfect capital market existed he would consume at point E2 by borrowing, but
as agent I he would also instruct his manager to produce at point B. Without
capital markets he would consume and produce at point Z. The above results
would still hold for many agents and dispersed ownership. Imagine an economy
with 100 firms each with the same production opportunity curve as the firm
above. Furthermore, imagine 100 agents each with 7 mill. $ in initial
endowment but with different time preferences. As long as perfect capital
market existed these agents would be indifferent to having their own firm or
having say 1/100 of each firm. The reason is, that they would agree to order
the manager to produce at point B, because this would maximize the value of
their ownership stakes. Now, the Fisher separation
theorem can be stated: Fisher separation theorem: Given perfect and complete capital markets, the
production decision (P_{0}, P_{1}) is governed solely by the
profitmaximization objective (max. present value of production, P_{0},
P_{1}), and the decision is separated from the consumption decision
that is governed solely by utilitymaximization (max. utility of consumption,
C_{0}, C_{1}). This theorem is also known as the unanimity principle because it unites
the shareholders in agreeing on the profit maximization strategy. The theorem
is part of the general microeconomic theory that demonstrates the welfare
gains from specialization and trade. In this model, the gain only comes from
trade on differences in preferences. Alternatively, we could demonstrate the
gain from specialization by assuming different investment opportunities but
identical preferences. A caveat Unfortunately, the above model is not a
general equilibrium model. It is a short run model. To see why, consider the
following. Note that the average rate of return
on capital would always be equal to or higher than the marginal rate of
return when the latter is decreasing. Furthermore, the optimal marginal rate
of return is equal to or higher than the market determined interest rate. The
model does not say much about the cost
of capital. However, the four projects’ cost of capital have to be the
same because the agents in this model only focus on return, not on risks.
Besides, the agents determine the optimal production by ordering projects
initiated that have a higher return than the market interest rate. This imply
that the project cost of capital must be equal to the market rate. So, the
cost of capital is less than the average return of capital, and firms will
earn excess profits. This is not sustainable in the long run. In a general
long run equilibrium model, the cost of capital have to equal the return of
capital. 

 Copyright 19972019, H. Mathiesen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Legal notice. 