Consider now a dynamic framework. As in the static model, at the beginning of the first period, neither the worker or the firm knows the worker’s productivity. The firm, however,
9The terms ‘skill,’ ‘ability,’ and ‘productivity’ will be used synonymously here.
10Oettinger assumes workers are paid a weighted average of expected marginal productivity and piece rate wages. This specification produces the same average wage (m) but is not used here because of the nature of
receives a noisy signal about the worker’s skill, where
st = α+vt . (4.25)
Consistent with the static model, α ∼ N(m, σ2
α), and independent of α, v ∼ N(0, σv2) and
is i.i.d. The worker’s (unconditional and conditional (on α)) signal is distributed normally, where
s ∼ N(m, σ2
α+σv2) (4.26)
s|α ∼ N(α, σ2v) . (4.27)
At the beginning of the first period, the state’s beliefs about a worker’s ability can only be conditioned on the first observed signal. That is,
α|s ∼ N¡(1−λ2)m+λ2s, σα2(1−λ2)¢, (4.28)
where λ is defined as in Equation 4.24.11
At the end of the first period (and at the end of every subsequent employment year), the state evaluates the worker’s performance in his current position. Like a worker’s signal, observed performance, pt, is also a noisy measure of the worker’s true ability. Thus,
pt = α+κt, (4.29)
where, independent of α,κ∼N(0, σ2
κ). Similar to a worker’s signal, performance evaluation
is distributed normally, both conditionally and unconditionally.
p ∼ N(m, σ2
α+σκ2) (4.30)
p|α ∼ N(α, σ2κ). (4.31)
Additionally, given that α and v are uncorrelated, as are α and κ, it must be the case that
s and p are orthogonal, conditional on α.
At the beginning of the next period, after evaluating a worker’s performance, the state updates its beliefs about a worker’s ability by conditioning on both the current signal and last period’s performance evaluation.12 Using Bayes’ Rule, it can be shown that, conditional
on both the signal and previous evaluation, the variance of worker ability is13
var(α|s, p) = σ 2 ασv2σκ2 σ2 sσp2−σα4 . (4.32)
Comparing the period two prior (period one posterior) belief about the variance of ability to the period one prior information, it is clear that the additional information about a worker’s performance decreases the conditional variance of ability. Specifically, var(α|s, p)<var(α|s) if σ2 ασv2σκ2 σ2 sσp2−σα4 < σ2 α(1−λ2), or σα2σv2σκ2 σ2 sσp2−σα4 < σ2ασ2v σ2 α+σv2 0 < σ4 ασv4 (4.33) Since σ4
ασv4 is always positive, the conditional variance of ability will decrease with the ad-
dition of new information (the worker’s performance evaluation) each period. The posterior information at the end of periodt will become the prior belief of the state at the beginning of periodt+ 1. As a worker’s tenure with the state increases, beliefs about ability will evolve in this manner according to Bayes’ Rule.
12For examples of Bayesian updating in models with dynamic learning, see Crawford and Shum (2005), Hamilton and Chan (2005), and Mira (2005).
13For probability density function,f(·),f(α|s, p) =f(α)f(s,p|α)
f(s,p) =
f(α,s,p)
f(s,p) since, by Bayes Rule,f(s, p|α) =
f(α,s,p)
In addition to examining the evolution of the conditional variance of ability, the state’s conditional expectation of ability will yield information regarding how changing beliefs influ- ence wage, promotion, performance evaluation, and promotion decisions, as reflected in the equations estimated in the model. Specifically, the posterior mean of ability (conditioned on both the original signal and the worker’s performance evaluation) will be larger than the prior (conditioned on only the signal) if the following is true:
pσ2 ασv2+mσ2κσ2v+sσα2σκ2 σ2 sσp2−σα4 > (1−λ2)m+λ2s, or pσ 2 ασv2+mσ2κσ2v+sσα2σκ2 σ2 sσp2−σα4 > mσ 2 v +sσα2 σ2 α+σ2v ,
which can be shown to simplify to
(p−s)σα2 > (m−p)σv2 . (4.34)
While nothing definitive can be said about the theoretical relationships betweenp and
s or between m and p, one can conclude that the larger is the variance of the signal error (σ2
v), ceteris paribus, the more likely it is that the posterior belief about ability will exceed
the prior. In fact, one may even assume that p, s, σ2
α, and m do not differ by race. Thus,
since the underlying statistical discrimination assumption is that σ2
v,b > σv,w2 , this result
suggests that blacks are more likely than whites to benefit from the added information. This is consistent with the conclusions drawn by Oettinger (1996) and Goldsmith, et al. (2006), who suggest that black workers benefit from continuing to work for their present employer, while whites can often better themselves by seeking outside wage offers.
By allowing the state’s expectation of a worker’s productivity in a potentially new po- sition to be conditioned on his current signaland known (perceived) performance in previous periods, Bayesian updating describes how the state’s productivity beliefs evolve. If the state is observed to statistically discriminate, one might expect more favorable outcomes for a discriminated against worker as his tenure with the state increases, as the new information
As this new information is used in future employment decisions, a black worker or woman should be observed to be more likely to experience outcomes which are positively correlated with expected ability.