Informativeness_Principle - Atlas of Economic Models
 

Informativeness Principle

Introduction

The informativeness principle states that any signal that is correlated with the unobserved action of the agent should be used to condition the wage contract (also known as the Sufficient Statistic Theorem). $$ P's $$ payoff is decreasing in the variance of $$ A's $$ action (i.e. output), so any factor which reduces the error with which $$ A's $$ choices are estimated should be included in the optimal contract.

Example

Continuing from the example laid out in the Principal_Agent_Model, there is some observable signal $$ s $$ which is uncorrelated to the effort of the agent $$ a $$ (thus has no effect on expected output $$ E[x] $$) but is correlated with the noise term $$ \epsilon $$ . $$ s $$ is also a stochastic shock with mean $$ 0 $$ and variance $$ \sigma_s^2 $$, and $$ Cov(\epsilon,s) = \delta $$ .

Now let $$ w = \alpha + \beta(x+\gammas) $$ .

  • $$ E[w] = \alpha + \beta a $$
  • $$ Var[w] = \beta^2\sigma_{\epsilon}^2 + \beta^2\gamma^2\sigma_s^2 + 2\beta^2\gamma\delta $$

Solving

Maximising $$ V(a,w) $$ with respect to $$ a $$ still gives $$ a* = \frac{\beta}{c} $$ since including $$ s $$ does not change the payoff to $$ a $$ , which gives $$ V(a*,w) = \alpha + \frac{\beta^2}{2c} - \frac{r(\beta^2\sigma_{\epsilon}^2 + \beta^2\gamma^2\sigma_s^2 + 2\beta^2\gamma\delta)}{2} $$ .

$$ P $$ now maximises their expected utility $$ E[U(x,w)] = \frac{(k-\beta) \beta}{c} - \alpha $$ subject to $$ V(a^{*},w(a^{*})) \geq 0 $$ .

  • If $$ V(a*,w) = \alpha + \frac{\beta^2}{2c} - \frac{r(\beta^2\sigma_{\epsilon}^2 + \beta^2\gamma^2\sigma_s^2 + 2\beta^2\gamma\delta)}{2} = 0 $$ then $$ \alpha = \frac{r(\beta^2\sigma_{\epsilon}^2 + \beta^2\gamma^2\sigma_s^2 + 2\beta^2\gamma\delta)}{2} - \frac{\beta^2}{2c} $$ .

  • Therefore $$ P $$ now $$ \max_{\beta} \frac{k\beta}{c} - \frac{\beta^2}{2c} - \frac{r(\beta^2\sigma_{\epsilon}^2 + \beta^2\gamma^2\sigma_s^2 + 2\beta^2\gamma\delta)}{2} $$

    • $$ \gamma* = \frac{-\delta}{\sigma_s^2} $$
    • $$ \beta* = \frac{k}{1+rc[\sigma_{\epsilon}^2 + \gamma^2\sigma_s^2 + 2\gamma\delta]} = \frac{k}{1+rc\left[\sigma_{\epsilon}^2 - \frac{\delta^2}{\sigma_s^2}} \right] $$

$$ \gamma* $$ shows that payment conditional upon signal $$ s $$ should be negatively related to the covariance between $$ \epsilon $$ and $$ s $$ . For example, if $$ s $$ is total industry demand then this is positively correlated with the noise term ($$ \delta>0 $$). If $$ s>0 $$ and output $$ x = a+\epsilon $$ appears to be increasing then this is likely to be due to the noise term $$ \epsilon $$ rather than the agent's effort $$ a $$, so the agent should not be rewarded with part of this increased output ($$ \gamma*<0 $$) . However, as the (absolute) covariance between $$ \epsilon $$ and $$ s $$ (i.e. $$ |\delta| $$) increases, the optimal $$ \beta $$ also increases because the signal allows output (or rather $$ \epsilon $$) to be estimated more accurately, thus a higher $$ \beta $$ gives agents the incentive to increase their effort input $$ a $$ if they know that they will be rewarded for it.

These have conflicting effects on the value of $$ \alpha* $$ . A higher $$ \beta $$ increases the expected wage, which increases $$ V(a,w) $$ . However, the effect on $$ Var[w] $$ is ambiguous since $$ \beta $$ is higher but the inclusion of $$ s $$ reduces the variance. Since $$ V(a,w) = \alpha + \frac{\beta^2}{2c} -\frac{rVar[w]}{2} $$ , if $$ \beta $$ is higher and $$ Var[w] $$ is (for illustrative purposes) roughly the same as in the 'simple' Principal-Agent model, $$ P $$ can choose a lower value of $$ \alpha $$ to keep $$ V(a,w)=0 $$ .

Comparisons to 'Simple' P-A Model

Parameter

'Simple' Model

Informativeness Principle Model

Comments

$$w$$

$$\alpha + \betax$$

$$\alpha + \beta(x+\gammas)$$

Signal $$ s $$ included as part of the wage contract

$$E[w]$$

$$\alpha + \betaa$$

$$\alpha + \betaa$$

$$ E[s] = 0 $$ so $$ E[w] $$ does not change

$$Var[w]$$

$$\beta^2\sigma_{\epsilon}^2$$

$$\beta^2\left[\sigma_{\epsilon}^2-\frac{\delta^2}{\sigma_s^2} \right]$$

Including $$ s $$ reduces the variance with which $$ \epsilon $$ is estimated, but since $$\beta$$ is also greater the overall effect is ambiguous

$$a*$$

$$\frac{\beta}{c}$$

$$\frac{\beta}{c}$$

Including the signal $$ s $$ in the wage contract does not change the direct payoffs to $$ a $$ (but the values of $$ \beta $$ differ between the two models so the optimal $$ a* $$ under the 'simple' model is higher)

$$\beta*$$

$$\frac{k}{1+rc\sigma_{\epsilon}^2}$$||$$\frac{k}{1+rc\left[\sigma_{\epsilon}^2-\frac{\delta^2}{\sigma_{\epsilon}^2} \right]$$

Including $$ s $$ reduces the variance with which $$ x $$ is estimated and thus increases the optimal amount of risk that the agent should bear by increasing their incentives to put in effort $$ a $$

$$\alpha*

$$-\frac{\beta^2}{2c}+\frac{rVar[w]}{2}$$||$$-\frac{\beta^2}{2c}+\frac{rVar[w]}{2}$$

Including $$s$$ increases $$\beta$$ and has an ambiguous effect on $$Var[w]$$ so it is likely that it will reduce $$\alpha*$$

Applications and Extensions

Relative performance evaluation uses a signal of the average performance of other agents to condition the wage contract of a particular agent. $$ P $$ knows that output $$ x = a+\epsilon $$, but $$ \epsilon $$ is actually composed of two random variables: one idiosyncratic shock and one shock common to all agents (e.g. market demand conditions as used above). Therefore, the output of other agents is correlated with the output of a particular agent and should be included in the optimal wage contract. As explained, since average output is positively correlated with $$ \epsilon $$ then $$ \gamma*<0 $$ , meaning that agents have to put in more effort $$ a $$ to get above-average sales and therefore be rewarded with a higher $$ w $$ . However, if the shocks experienced by agents are uncorrelated (i.e. $$\delta=0$$) then the average performance of other agents should not be included since it cannot reduce the error with which $$ A's $$ choices are estimated.

ToDo

  • Sort out table!