19.4 Multi-tasking without Homogeneity

In the previous section we saw, for example, that if the agent has quadratic costs, the principal pays the agent half the value of each activity. Moreover, the more rapidly costs rise in scale, the lower are the payments to the agent.

This remarkable theorem has several limitations. The requirement of homogeneity is itself an important limitation, although this assumption is reasonable in some settings. More serious is the assumption that all of the incentives are set optimally for the employer. Suppose, instead, that one of the incentives is set too high, at least from the employer’s perspective. This might arise if, for example, the agent acquired all the benefits of one of the activities. An increase in the power of one incentive will then tend to spill over to the other actions, increasing for complements and decreasing for substitutes. When the efforts are substitutes, an increase in the power of one incentive causes others to optimally rise, to compensate for the reduced supply of efforts of that type.Multi-tasking (and agency theory more generally) is a rich theory with many implications not discussed here. For a challenging and important analysis, see Bengt Holmstrom and Paul Milgrom, “The Firm as an Incentive System,” American Economic Review 84, no. 4 (September 1994): 972–991.

We can illustrate the effects of cost functions that aren’t homogeneous in a relatively straightforward way. Suppose the cost depends on the sum of the squared activity levels:

$$c(x)=g\left({\displaystyle \sum _{i=1}^n{x}_i^2}\right)=g(x•x)\text{.}$$

This is a situation where vector notation (dot-products) dramatically simplifies the expressions. You may find it useful to work through the notation on a separate sheet, or in the margin, using summation notation to verify each step. At the moment, we won’t be concerned with the exact specification of g, but instead we will use the first-order conditions to characterize the solution.

The agent maximizes

$$u=p•x-g(x•x)\text{.}$$

This gives a first-order condition

$$0=p-2g^{\prime }(x•x)x\text{.}$$

It turns out that a sufficient condition for this equation to characterize the agent’s utility maximization is that g is both increasing and convex (increasing second derivative).

This is a particularly simple expression because the vector of efforts, x, points in the same direction as the incentive payments p. The scalar that gives the overall effort levels, however, is not necessarily a constant, as occurs with homogeneous cost functions. Indeed, we can readily see that x • x is the solution to $p•p={(2g^{\prime }(x•x))}^2(x•x)\text{.}$

Because x • x is a number, it is worth introducing notation for it: S = x • x. Then S is the solution to $p•p=4S{(g^{\prime }(S))}^2\text{.}$

The principal or employer chooses p to maximize $\pi =v•x-p•x=v•x-2g^{\prime }(x•x)(x•x)\text{.}$

This gives the first-order condition $0=v-4\left(g^{\prime }(x•x)+(x•x)g^{″}(x•x)\right)x\text{.}$

Thus, the principal’s choice of p is such that x is proportional to v, with constant of proportionality $g^{\prime }(x•x)+x•x{g}^{″}(x•x)\text{.}$ Using the same trick (dotting each side of the first-order condition $v=4\left(g^{\prime }(x•x)+x•x{g}^{″}(x•x)\right)x$ with itself), we obtain

$$v•v=16{\left(g^{\prime }(S*)+S*g^{″}(S*)\right)}^{2}S*\text{,}$$

which gives the level of x • x = S* induced by the principal. Given S*, p is given by $p=2g^{\prime }(x•x)x=2g^{\prime }(S*)\frac{v}{4(g^{\prime }(S*)+S*g^{″}(S*))}=\frac{1}{2}\left(\frac{1}{1+\frac{S*g^{″}(S*)}{g^{\prime }(S*)}}\right)v\text{.}$

Note that this expression gives the right answer when costs are homogeneous. In this case, g(S) must be in the form S^r/2, and the formula gives $p=\frac{1}{2}\left(\frac{1}{1+r-1}\right)v=\frac{v}{r}$ as we already established.

The natural assumption to impose on the function g is that ${\left(g^{\prime }(S)+S{g}^{″}(S)\right)}^{2}S$ is an increasing function of S. This assumption implies that as the value of effort rises, the total effort also rises.

Suppose $\frac{S{g}^{″}(S)}{g^{\prime }(S)}$ is increasing in S. Then an increase in v_i increases S, decreasing p_j for j ≠ i. That is, when one item becomes more valuable, the incentives for performing the others are reduced. Moreover, because $p•p=4S{(g^{\prime }(S))}^2\text{,}$ an increase in S only occurs if p • p increases.

These equations together imply that an increase in any one v_i increases the total effort (as measured by S* = x • x), increases the total incentives as measured by p • p, and decreases the incentives for performing all activities other than activity i. In contrast, if $\frac{S{g}^{″}(S)}{g^{\prime }(S)}$ is a decreasing function of S, then an increase in any one v_i causes all the incentives to rise. Intuitively, the increase in v_i directly causes p_i to rise because x_i is more valuable. This causes the agent to substitute toward activity i. This causes the relative cost of total activity to fall (because $\frac{S{g}^{″}(S)}{g^{\prime }(S)}$ decreases), which induces a desire to increase the other activity levels. This is accomplished by an increase in the incentives for performing the other activities.

This conclusion generalizes readily and powerfully. Suppose that c(x) = g(h(x)), where h is homogeneous of degree r and g is increasing. In the case just considered, h(x) = x • x. Then the same conclusion, that the sign of $\frac{d{p}_i}{d{v}_j}$ is determined by the derivative of $\frac{S{g}^{″}(S)}{g^{\prime }(S)}\text{,}$ holds. In the generalization, S now stands for h(x).