We now discuss how the model maps into the empirical decomposition of labor reallo- cation presented in Section 3. We start by considering the frictionless case in order to build intuition, and then turn to the general case.
In absence of mobility frictions (i.e for i = f = 0), the log agricultural employment at time t of any cohort c can be written as
31Assumption 2 guarantees that frictions are sufficiently small to generate positive reallocation. Trivially,
if f ! 1 or i ! 1, there would be no reallocation. Reallocation is either zero, or does not depend on f and i. Our parametric restrictions exclude the case in which reallocation is zero.
log lA,t,c = ˆ + ⌫ 1 log ptztX ↵L ↵ A,t | {z } year effects ⌫ 1 log hc | {z } cohort effects
where ˆ is a cohort- and time-invariant function of parameters. This equation maps into the empirical specification (1). Through the lens of the model, cohort effects are proportional to cohort-level human capital, while year effects are proportional to the relative wage across sectors, which depends on aggregate prices and quantities. The age effects introduced in specification (5) are redundant in this case; once time and cohort effects are accounted for, age does not play any independent role.
Consider the year and cohort components, as defined in (2)-(4), of the rate of labor reallocation between t and t + 1. The year component captures the difference between the year effects associated to t and t+1, which is identified by the average change in agricultural employment for a given cohort. In the frictionless model the rate of reallocation is common across all cohorts, so that - for any c - the year component is given by
log t= log lA,t+1,c log lA,t,c= v
1 ((1 ⇥D) log g✓z+ ⇥S log gh)
where the second equality follows from plugging the expression for log gLA from Proposition 1 into equation (14). The cohort component captures the change over time in the average cohort effects for the active cohorts. Given that in our model cohort effects change across cohorts by a constant amount, this corresponds to the difference between the cohort effects of any two consecutive cohorts, which in turn is given by the cross-cohort agricultural employment gaps averaged across all time periods. In absence of frictions these cross-cohort gaps are constant over time, so that - for any t - the cohort component is
log t = Cc+1 Cc = log lA,t,c+1 log lA,t,c= ⌫
1 log gh. (15)
These two quantities correspond to different aspects of the process of labor reallocation. Notice from equation (St) that (15) represents the magnitude of the shift in the agricultural labor supply driven by human capital growth. As displayed in Figure XII, the cohort component captures the partial equilibrium effect of the change in supply, i.e. the decrease from log LA,tto log LP EA,t+1. The year component captures the residual part of reallocation, i.e. the difference between log LP E
A,t+1and log LA,t+1. Intuitively, gaps in agricultural employment between different cohorts at a given point in time (i.e. the cohort component) identify the extent to which changes in human capital shift the supply curve, keeping wages fixed; on the other hand, changes over time for a given cohort (i.e. the year component) identify the
movement along a given supply curve driven by changes in relative wages.
Consider now the general case with mobility frictions. Under specification (1), the struc- tural interpretations of the cohort and year components discussed above would not apply. Lemma 1 shows that the rate of reallocation across t and t + 1 is cohort-specific, with con- strained cohorts not reallocating at all; the year component would therefore pick up the reallocation rate of unconstrained cohorts, scaled down by the share of constrained cohorts. By the same logic, the cohort component would be larger than ⌫
1 log gh, as it would combine the cross-cohort employment gaps for both unconstrained cohorts and constrained cohorts, with the latter being larger than the former. This is where the age controls intro- duced in specification (5) become important. Under the identification restriction of a zero age effect for any young (unconstrained) cohort, age controls capture the reallocation be- havior of old (constrained) cohorts, so that the resulting year and cohort components retain the structural interpretations illustrated in Figure XII. The following proposition formalizes this result.
Proposition 2: Decomposition of Labor Reallocation Consider the specification
log lA,t,c | {z }
agr share of cohort c at time t
= |{z}Y˜t year effects + |{z}C˜c cohort effects + A˜t c |{z} age dummies + "t,c
estimated with model-generated data under the restriction that ˜Aa = ˜Aa 1, where a 2 [1, ˆa]. Define the year and cohort components of labor reallocation between t and t + 1 as
log ˜t ⌘ ˜Yt+1 Y˜t
log ˜t = log LA,t+1 log LA,t log ˜t+1. Then, for all t,
log ˜t= log ˜ = ✓ v 1 ◆ (1 ⇥D) log g✓z+ ⇥S log gh ! log ˜t= log ˜ = ✓ v 1 ◆ log gh.
Proof. See Appendix.
the two components. Since mobility frictions limit the reallocation of older workers, not controlling for age results in an overstatement of the cohort component and an understate- ment of the year component. The difference between the year components estimated with and without age controls is proportional to the share of constrained cohorts, (f), a natural measure of the severity of reallocation frictions.
Corollary 1: Bias in the Basic Decomposition
Consider specification (1) estimated with model-generated data. The estimated year and cohort components would be
log = 1 (f ) !
log ˜ log = log ˜ + (f )
1 (f )log ˜ where (f) 2⇥0, 1 is the share of constrained cohorts,
(f ) = N + 1 ˆa N + 1
which is increasing in the fixed cost f and does not depend on the iceberg cost i. Proof. See Appendix.
Implications for Wage Data. The mapping between the model and the decomposition developed in this section implies that both human capital growth and reallocation frictions can be quantified without relying on the measurement of wages, which is notoriously diffi- cult for developing countries and the agricultural sector. The model does have predictions on wages that are in line with the limited available evidence; in particular, we show in Appendix E.4 that it is consistent with the observational wage gains for workers moving from agriculture to non-agriculture being smaller than the corresponding cross-sectional gaps (Hicks et al., 2017; Herrendorf and Schoellman, 2018; Alvarez, 2020). However, our analysis in Appendix E.4 also shows that wage data, even if with a panel dimension, would not be enough to infer the magnitude of the frictions; the fixed cost makes the sectoral de- cision dynamic, and to estimate mobility costs one would need the hypothetical wage paths in agriculture in non-agriculture for both movers and non-movers. Corollary 1 provides an alternative way of quantifying these costs.