Our baseline model implies that the average wage of workers in group l is the same
across all equipment-occupation pairs. This implication is rejected by the data. In Section F.1we argue that these differences in average wages across(k,w)do not drive our results.
In Section F.2 we show that incorporating preference shifters for working in different occupations generates differences in average wages across occupations within a labor group and we show how to conduct a decomposition in this case.
F.1 Between-within decomposition
Here, we conduct a between and within decomposition of changes in the average wage of group l, wt(l)/wt, wherewt is the composition-adjusted average wage across all labor groups. We consider variation in average wages within a labor group across occupations but not across equipment types, wt(l,w), because the MORG CPS contains wage data for only a subset of observations (those respondents in the Outgoing Rotation Group).
The following accounting identity must hold at eacht, wt(l)
wt =
Â
wwt(l,w)
wt pt(l,w). The previous expression implies
Dwt(l) wt =
Â
w D wt(l,w) wt p¯t(l,w) +Â
w wt(l,w) wt Dpt(l,w) (20) where Dxt = xt1 xt0 and ¯xt = (xt1 +xt0)/2. The first term on the right-hand sideof the equation (20) is the within component whereas the second term is the between component. According to the model, the contribution to changes in wages of the within component should be 100% for each labor group.37 We conduct this decomposition using the MORG CPS data between 1984 and 2003 for each of 30 labor groups and find that the median contribution across labor groups of the within component is above 86%. Hence, while in practice there are large differences in average wages for a labor group across occupations, these differences do not appear to be first order in explaining changes in labor group wages over time.
F.2 Compensating differentials
Here we extend our model to incorporate preference heterogeneity in addition to produc- tivity heterogeneity. This simple extension implies that the average wage of workers in group l varies across equipment-occupation pairs. We show how to use data on aver-
age wages across equipment-occupation pairs to identify the parameters of the extended model.
Environment and equilibrium. The indirect utility function of a workerz 2 Z(l)earn-
ing income It(z)and employed in occupationwwith equipmentkis
U(z,k,w) = It(z)ut(l,k,w) (21) where ut(l,k,w) > 0 is a time-varying preference shifter.38 We have normalized the price index to one. We normalize ut(l,k1,w1) = 1 for all land t. This model limits to our baseline model whenut(l,w,k) =1 for alltand(l,k,w).
A occupation production unit hiring k units of equipment kand l efficiency units of
laborlearns profits pt(w)ka[Tt(l,k,w)l]1 a pt(k)k vt(l,k,w)l. The profit maxi- mizing choice of equipment quantity and the zero profit condition yield
vt(l,k,w) = a¯pt(k)1 aa pt(w)11a Tt(l,k,w)
if there is positive entry in (l,k,w). Facing the wage profile vt(l,k,w), each worker
z 2Z(l)chooses(k,w)to maximize her indirect utility,#t(z,k,w)ut(l,k,w)vt(l,k,w). In our extended model, preference parameters ut(l,k,w) and productivity parame- ters,Tt(l,k,w), affect worker utility in the same way, as shown by the previous expres-
37Of course, this does not mean that changes in occupation shifters do not drive changes in wages in our
model.
38In this extended environment it is straightforward to allow for u
t(l,k,w) = 0, in which case no
sion. Hence, they also affect worker allocation in the same way: the probability that a randomly sampled worker,z2 Z(l), uses equipmentkin occupationwis
pt(l,k,w) = h ut(l,k,w)Tt(l,k,w)pt(k) a 1 a pt(w)11a iq(l) Âk0,w0 h ut(l,k0,w0)Tt(l,k0,w0)pt(k0)1 aa pt(w0)11a iq(l). (22) On the other hand, preferences and productivities affect wages differently. The average wage of workersz2 Z(l)teamed with equipmentkin occupationwis now given by
wt(l,k,w) = u g(l) t(l,k,w) k
Â
0,w0 ut l,k0,w0 Tt(l,k0,w0)pt k0 a 1 a pt w0 11a q(l)!1/q(l) (23) Ifut(l,k,w) > ut(l,k0,w0), then the average wage of grouplis lower in(k,w) than in (k0,w0)in periodt.The general equilibrium condition is identical to our baseline model and is given by equation (5), although total labor income in occupationw is now given by
zt(w)⌘
Â
l,k
wt(l,k,w)Lt(l)pt(l,k,w).
Parameterization. Here, we focus on measuring preference shifters and shocks under the restriction thatq(l) =qfor alll, takingqas given.
From equation (23), we have
wt(l,k,w)
wt(l,k1,w1) =
1
ut(l,k,w). (24)
Hence, we measure preference shifters directly from average wages. Equations (6) and (22) give us,
pt(l,k2,w) pt(l,k1,w) = ut(l,k2,w)q ut(l,k1,w)q qt(k2)q qt(k1)q which, together with equation (24), gives us
qt(k2)q qt(k1)q = pt(l,k2,w) pt(l,k1,w) wt(l,k2,w)q wt(l,k1,w)q
Hence, we obtain log qˆ(k2)q ˆ qk(k1)q = log pt1(l,k2,w) pt1(l,k1,w) logpt0(l,k2,w) pt0(l,k1,w) +qlogwt1(l,k2,w) wt1(l,k1,w) qlogwt0(l,k2,w) wt0(l,k1,w)
We can then average over all(l,w)and then exponentiate, exactly as in our baseline, to
obtain a measure of changes in equipment productivity (to the power q). We obtain a
measure of changes in transformed occupation prices to the power q similarly and use
this to measure changes in occupation shifters using equation (11), as in our baseline approach. Finally, we have
wt(l,k1,w1) =Tt(l)g(l)
Â
k,w [ut(l,k,w)qt(k)qt(w)]q !1/q so that ˆ w(l,k1,w1) = Tˆ(l)Â
k,w (uˆ(l,k,w)qˆ(k)qˆ(w))qpt0(l,k,w) !1/qHence, given measures of changes in transformed occupation prices (to the power q),
changes in equipment productivity (to the power q), and changes in preference shifters
(obtained above) as well as observed changes in wages and observed allocations in period t0, we can measure changes in relative labor productivities using the previous expression for grouplrelative to groupl1.