Equal_Compensation_Principle - Atlas of Economic Models
 

Equal Compensation Principle

Introduction

$$ P $$ often required $$ A $$ to undertake multiple tasks. By providing an incentive contract, $$ P $$ risks that $$ A $$ will concentrate solely on the activity which yields the highest marginal return (i.e. the activity associated with the highest $$ \beta $$), thus if $$ P $$ values multiple outputs equally then in order for agents to split their time equally between tasks they must get the same marginal return for each activity.

Example

In the simplest example there are now two outputs, $$ x_1 $$ and $$ x_2 $$ where $$ x_1 = a_1+\epsilon_1 $$ and $$ x_2 = a_2+\epsilon_2 $$ such that $$ \epsilon_1 , \epsilon_2 \sim (0, \sigma^2) $$ . Now agents are paid on the basis of two outputs so $$ w = \alpha + \beta_1x_1 + \beta_2x_2 $$ . $$ E[w] = \alpha + \beta_1a_1 + \beta_2a_2 $$ and $$ Var[w] = (\beta_1^2 + \beta_2^2 + \beta_1\beta_2)\sigma^2 $$ .

The agent's 'cost of effort' function now includes both $$ a_1 $$ and $$ a_2 $$ : $$ c(a_1,a_2) = c[a_1^2 + a_2^2 + 2ma_1a_2] $$ . If $$ m>0 $$ then the activities are substitutes, if $$ m<0 $$ then the activities are complements since $$ \frac{\partialc}{\partiala_1} = 2c(a_1+ma_2) $$ , i.e. if $$ m<0 $$ then putting effort into $$ x_2 $$ actually reduces the marginal cost of $$ x_1 $$ .

The principal's utility function is now $$ U(x,w) = k_1x_1 + k_2x_2 -w $$ but since $$ P $$ cares about both outputs equally $$ k_1=k_2 $$ so this reduces to $$ U(x,w) = k(x_1+x_2) - w $$ .

Solving

$$ V(a,w) = E[w] + \frac{rVar[w]}{2} - c(a) $$ thus:

  • $$ a*_1 = \frac{\beta_1}{2c} - ma_2 $$
  • $$ a*_2 = \frac{\beta_2}{2c} - ma_1 $$

Using the same steps as when solving the 'simple' Principal-Agent model, $$ \beta*_1 = \beta*_2 = \frac{k}{1+2(1+m)rc\sigma^2} $$ . When the activities are complements ($$m<0$$) then $$ \beta* $$ is higher so incentives are stronger and agents will increase their effort levels, thus reducing the marginal cost of producing both goods. When the activities are substitutes then incentives are weaker since it may be optimal for different agents to specialise in different tasks .

This model implies that the marginal return to activities must be equal or else one task will receive no attention. If the output of some important activity cannot be monitored then there should not be an incentive scheme which pays directly upon the output of some other task because agents will not put any effort into the former.

Applications and Extensions