Insurance within the firm

We generalize equation (2) as:

^wij t = a'_ijtδ + by_{j t} + ψ_ijt

where the subscript i stands for the i-th individual and w⅛_t is the logarithm of worker
compensation.⁵ The term ⅛_z∙_t denotes a vector of systematic factors that affect individual
i's compensation, which can vary across workers, firms and time, while ψ_ijt is the stochastic
component of earnings, which is unrelated to the firm’s fortunes. These idiosyncratic shocks
are meant to capture unanticipated fluctuations in individual ability, shocks to productivity
(such as illness), idiosyncratic changes in labor supply (child-raising, family labor supply
effects, etc.).

Let us assume that idiosyncratic earnings shocks are in turn the sum of a permanent
random walk component η_ijt = η_ijt-1 + ξ_ijt and a serially uncorrelated transitory shock

μ_ijt.⁶ Again to save on notation, it is maintained that covariance stationarity holds, i.e.

uncorrelated with each other at all leads and lags.

Taking first differences of (5) and (7), and using the stochastic structure outlined above,
we get the following system of two equations:

A(L,p)Δy∙_t = ∆^θ + u_{j t +} ∆υ_jt                         (8)

A(L,p)Δw_ijt = ∆a^l_ijtλ + ∆z'_jtbθ + bu_{j t} + b∆v_jt

+A(L,p)ξ_{ij t} + A(L,p)Δμ_ijt                    (9)

where λ = A (L,p) δ. For the purpose of this paper it is more convenient to define firm per-
formance and earnings growth after adjusting for observable firm and worker characteristics,

^aWe let earnings to depend on contemporary firm performance, i.e. assume that wages adjust immediately
to changes in performance. In practice, wages might adjust with a lag (think of overtime or bonus decisions,
which are usually taken at the end of the calendar year). Nevertheless, if adjustments are made at a frequency
higher than a year (say, a quarter), annual data of the type used here will not detect deviations from the
contemporaneous adjustment assumption.

⁰In this context, an idiosyncratic shock to earnings is a purely individual innovation, i.e. it is not shared
by co-workers within the firm.

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