Principal-Agent Problem: Managers vs. Owners and Overproduction in the Presense of Risk

As individuals and collectively people engage in principal-agent relationships where one person is delegated the task of performing in the interest of another person or company.  Economist have done a great deal of scholarly work in the area of incentive design to understand what are some of the problems that can arise when agents are pursuing their own interest and not those that they were delegated.  Examples of this phenomenon abound:

  • Real Estate agents concerned about a quick sale even if it means underpricing their principals home
  • Bank executives taking on larger risk than necessary on behalf of their shareholders
  • Employees not working to their full potential while on the job

The common denominator in the principal-agent problem is the disconnect of incentives.  One could hope for some sort of global awakening where agents decided they would ignore their interest and act on behalf of principal in all circumstances, but I wouldn’t hold my breath waiting for that kind of Utopia to manifest on this Earth in our lifetimes.  Instead, the solution lies in the structure of the compensation packages, this would lead to more efficient outcomes.  Instead of diving into the complex compensation structures available, this entry will demonstrate mathematically the principal-agent problem using a simple profit-maximization model under uncertainty.

Suppose that there is a farmer who decides to hire an economist to tell him how much he should produce in order to maximize his profit subject to his agricultural production cost.  The farmer tells the economist that he want to know how much to produce because overproducing or under producing will be sub-optimal.  The price of wheat, which is the farmers only product, goes down if there is a lot of rainfall because of abundant supply.  A year with heavy rain will mean wheat will sell for $20 a bushel and in a drought it will sell for 30$ a bushel. The probability of rain next year is 1/2 hence the probability of drought is 1/2.  The farmer has a certain cost of production which increases for every quantity produced.  The formulation and solution are as follows:

From Microeconomic Theory, Thie

The graphical representation of the revenues under different scenarios and the cost curve can be illustrated in this graph:

The expected profit would be maximized if the farmer produced 20 units of wheat and the expected profit would be 100.  The only problem with this formulation is that the farmer typically is more risk adverse than the economist solely because he has more to loose if the economist is wrong.  The economist who is advising him losses a percentage of the profits where the farmer could lose his livelihood.  In order to capture this kind of risk aversion we can model the farmers utility as a function of profit.  This function should have decreasing marginal utility to capture the saturation of profit as a function of utility.  The function should also be increasing since more profit is preferred to less, given these sensible restrictions one can postulate the square root of profits to be a good candidate. The expected utility for the farmer from the economists recommendation would be:

The next question would be, is this level of output maximizing the farmers utility?  Do these recommendations provide the maximum profit given the wheat farmers risk aversion as opposed to the recommendation made by the economic consultant whose objective is maximization of fees?  Is the incentive structure of percentage of profit strong enough to discourage sub-optimal production in the form of excessive risk taking?  The answer to all of these questions is no and we can prove that a lower rate of production would be optimal for the farmer.  Maximization of expected utility for the farmer gives the following result:

The following table shows how the profit maximizing quantity might not be the best quantity for this risk adverse farmer.  Producing 20 units of wheat maximizes both expected profits and expected commissions, BUT it is not the optimal quantity that maximizes the wheat farmers utility since he exhibits risk aversion.  This is a case where the incentive structure imposed on the agent sounds like it would create a link between the principal and the agent, but in fact there is a tendency to overproduce which places more emphasis on the favorable outcome of a drought then on the lower price generating event of heavy rainfall:

Firm Short-Run Output Decisions Under Different Taxation Schemes: An Application of Multivariable Calculus

When firms are deciding in the short-run on how much output to produce in a given period they implicitly or explicitly equate marginal cost to marginal benefits.  In the short-run different types of taxation schemes can change the optimum production levels while other forms of taxation do not change the output decisions of firms.  In the long-run when all inputs are flexible, Lump Sum Taxes and Proportional Taxes on Profits  reduces the number of firms in the market, but Lump-sum taxes on profits and a proportional tax on profits do not affect the firms output decision in the short-run.  A Tax on Output or an Input Tax on a variable input does reduce the output that the representative firm decides to produce in the short-run as well as increasing the likelihood of a firm to exit this industry.  The mathematical proof for these statements are outlined using partial derivatives and multivariable calculus in conjunction with the microeconomic theory of profit maximizing firms.