Perfect capital market - Fisher separation theorem

Model: Perfect capital market - Fisher separation theorem

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 profit-maximizing strategy. This is so for diverse time-preferences 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 (N₀=7 mill. $) of a good (C₀). This good may either be consumed today (P₀) or be invested today (I₀) and transformed into consumption tomorrow (P₁).

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:

· N₀ is the aggregated initial resources held by all agents.

· r_m is the market-determined interest.

· C₀ is today’s consumption or investment good.

· C₁ is tomorrow’s consumption good. C₁ = W*₁ - (1+ r_m)C₀ 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 (P₀, P₁) is governed solely by the profit-maximization objective (max. present value of production, P₀, P₁), and the decision is separated from the consumption decision that is governed solely by utility-maximization (max. utility of consumption, C₀, C₁).

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 micro-economic 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.