Adding Another Factor to Principal-Agent Model
Background
In a traditional principal-agent model, firm output is a function of the agent’s effort and the principal observes only the output not agent’s effort. The principal carefully designs the agent’s compensation package, especially the sensitivity of the agent’s pay to firm output, to maximize the firm value. Now, what if we add another factor to the relationship between firm output and agent’s effort? How would the optimal pay sensitivity change?
My earlier paper (Gao, Leung, and Qiu 2021) studied this issue by assuming such a factor, organization capital, that substitutes agent’s effort in improving firm output. I find that if firm output is a function of two substituting factors (of which one is agent’s effort), the optimal sensitivity of agent’s pay to firm output can be both higher or lower, depending on the principal’s choice.
To yield this two-way prediction, let’s see a simple extension to the standard principal-agent model following Holmstrom and Milgrom (1987), where the principal hires an agent (CEO) to run the firm. We added organization capital (OC) as an additional determinant of firm outcomes, but in fact we can assume any factor, e.g., intellectual property, IT infrastructure, etc., that either strengthens or weakens the relation between firm output and executive effort.
Model
The production function is given by
is an increasing function in both and . The substitutability of OC and executive effort implies that , i.e., OC reduces the marginal effect of any agent action on firm outcomes.- Without loss of generality, we assume
, where both and follow a continuous uniform distribution from 0 to 1.
The agent is paid a wage
- The increasing and weakly convex function
represents the cost of effort. - The function
represents his utility function and represents his felicity function (i.e., his utility from cash), both increasing and weakly concave. - The functions
, and are all twice continuously differentiable.
The risk-neutral principal chooses the effort level
subject to the individual rationality or participation constraint (IR) and incentive compatibility constraint (IC) as follows:
We first consider the case where the optimal effort is determined endogenously. Under the Holmstrom and Milgrom (1987) framework, the following assumptions are made:
- an exponential utility function of the agent,
, where is the coefficient of constant absolute risk aversion, , such that cost of effort is pecuniary and can be viewed as a subtraction to cash pay,- a normal noise,
, and - that the agent chooses his effort continuously in a multiperiod model and the optimal contract is linear, i.e.,
, where is the fixed wage and represents the proportion of firm outcomes shared with the agent via compensation (i.e., represents the agent’s pay-for-performance sensitivity).
Further, Holmstrom and Milgrom (1987) show that the problem is equivalent to a single-period static problem under these assumptions. For simplicity, we also assume a quadratic cost of effort,
subject to
Substituting in
Since
Moreover, his chosen effort is independent of the fixed wage
This optimal level of pay-for-performance sensitivity is derived as follows. Substituting
The first-order condition (FOC) of the agent with respect to
Since we assume
Setting the participation constraint to bind, we have
The above equation implies:
where
Thus, by substituting
The principal’s FOC with respect to
Other things equal, we can see that the optimal pay-for-performance sensitivity
On the other hand, fixing
Implications
The relation between OC and executive pay-for-performance sensitivity depends critically on the optimal level of effort the principal wants to implement:
- If the principal wants to implement a fixed target action (e.g.,
in our case, or to induce full effort in general), the optimal is increasing in the firm’s organization capital. - If the principal trades off the benefits and costs of high effort, the optimal
is decreasing in the firm’s organization capital.
Therefore, the model offers two empirical predictions. On the one hand, high OC firms may offer higher pay-for-performance sensitivity to induce executive effort. On the other hand, pay-for-performance sensitivity may be reduced in high OC firms as a result of efficiency gains from the substitution of OC for executive effort.
Now, coming back at the question at the beginning, adding another factor to the principal-agent model may cause the optimal pay structure to change in either direction, even if such factor has a directional impact on the relation between firm output and agent’s effort. In our case, such factor reduces the marginal effect of agent’s effort on firm output. But one can easily find many other factors that may increase the marginal effect of agent’s effort and yield similar predictions.
Perhaps, what’s also interesting is that, if we know the directional effect of a factor while observing both pay-for-performance sensitivity and the level of such factor, we may be able to infer whether the principal elicits full executive effort at all costs. Paired with firm performance, could this be some indicators of governance or board ability? Seems like some future research questions.
This post is adapted from the online appendix of Gao, Leung, and Qiu (2021).