In equation (11) it is implicitly assumed that wages respond equally to transitory and
permanent shocks to the firm’s performance, i.e. that the b coefficient is the same for the
two shock components Ujt and Δ¾⅛. Yet we can test whether the amount of insurance varies
with the temporary or permanent nature of the shock. Let bu and bυ denote respectively the
different response of wages to permanent and transitory shocks. We can distinguish various
insurance regimes depending on the values of bu and bv. The contemporaneous covariance
between shocks to performance and shocks to wage growth has the following structure:
b (σu + 2σ2υ)
e Wy,t∆ωil-t) - < btiσ2 + 2bυσ2
buσl
2bυσ1υ
full insurance
homogeneous partial insurance
heterogeneous partial insurance
transitory full insurance
permanent full insurance
(13)
where we have assumed that E ^ξijUljs) — E (ζijtvjs^
— 0 for all s,t, and similarly for μij∙t,
— 0 for all s,t. For simplicity, we have also assumed covariance
stationarity. If workers are fully insured against fluctuations in the performance of the firm,
the contemporaneous covariance between shocks to performance improvement and shocks
to wage growth is zero and full insurance obtains against shocks of any nature. On the other
hand, if workers share part of the fluctuations, without distinguishing between short-lived
and durable shocks, equation (13) equals b (σ^ + 2σ2), where b — bυ — bu. We call this case
“homogeneous partial insurance”. This is likely to arise if firms are unable to distinguish be-
tween transitory and permanent shocks. Three other cases may arise. The optimal contract
may result in a different reaction to shocks of different nature. For instance, workers may
bear a substantial portion of the firm’s permanent shocks but a limited portion of transitory
shocks: in this case, which we call heterogeneous partial insurance, the contemporaneous
covariance equals buσ%l + 2bυσ2. Two special cases occur when workers bear only transitory
shocks but are insulated from permanent shocks ( “permanent full insurance”, characterized
by E (ΔεjΔωω>ij't) — 2bυσ2) or bear permanent shocks but are insured against transitory
of the permanent shock σ%l.
11