Insurance within the firm

standard, one-period model. Firm performance is given by:

y = z + ^f (e) + ε                                  (1)

where ε ~ N(0,σ²) is a random shock, z denotes a predictable component that depends
on observable characteristics of the firm (such as location, industry, size, etc.), e is an
unobservable (to the firm) component that depends on the worker’s effort or action, and f
represents the sensitivity of performance to effort.

Following Holmstrom and Milgrom (1987), assume that risk-averse workers receive a
stochastic compensation that is the sum of a fixed component, a, that may vary with
observable workers’ characteristics (education, experience, etc.) and a variable component
that depends on the performance of the firm, i.e. a bonus tied to output y:

w = a + by

Firms maximize profits π = y — w, while workers maximize utility u(w — c(e)), where
c(e) is the disutility of effort in wage units. Holmstrom and Milgrom show that if utility is
of the CARA type, i.e. u(x) = —ɪ exp {—px}, the optimal contract is linear and specifies:

b =    ^f (e)

1 + pc' ^z(e)σ²

where p = — is the coefficient of absolute risk aversion and c^zz (.) the curvature of the
agent’s effort function, which is assumed to be convex; thus c^zz (.) > 0. Consider the special
case in which f (e) = 1. If workers are risk-neutral, p = 0 and they bear all risks. In
this case b = 1. Risk aversion makes it worthwhile to reduce the impact of risk on wages
(b < 1); under full insurance, b = 0. In the general case, the greater the marginal response of
performance to effort f (e), the higher b. Note that if a firm employs workers with different
preferences (p and c(.)) or different impact on performance (f), then we should expect
different contracts to be offered accordingly. Finally, note that the sensitivity of workers’
compensation to firm performance declines with output variability:     < 0.

The predictions of this model are the basis of our empirical tests. The model suggests

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