Selection, Agriculture and Cross-Country Productivity Differences
David Lagakos and Michael E. Waugh
Online Appendix
A. Proofs of Propositions and Corollary
1.1. Proof of Proposition 1
Let p
P
a
, Y
P
a
and Y
P
n
be the equilibrium relative price and quantities in an economy with economy-
wide e fficiency A
P
. Denote by p
R
a
, Y
R
a
and Y
R
n
the equilibrium of an economy with efficiency
A
R
.
Suppose that p
R
a
= p
P
a
, and that p
R
a
clears the output market in the rich economy. Then by
(3), each worker i would choose to work in the same sector in the two economies. Thus out-
put in each sector would be scaled up by a factor equal to the ratio of the efficiency terms:
Y
R
a
/Y
P
a
= Y
R
n
/Y
P
n
= A
R
/A
P
. But by the demand functions, we know that workers must de -
mand a higher fraction of non-agriculture goods in economy A
R
than A
P
. But this implies that
Y
R
n
/Y
P
n
> Y
R
a
/Y
P
a
, which is a contradiction. Thus p
R
a
6= p
P
a
.
The only way to be consistent with the worker solutions, the demand functions, is for more
workers to supply labor in the non-agriculture sector in economy A
R
than economy A
P
. By (3),
this occurs if and only if p
R
a
< p
P
a
.
1.2. Proof of Proposition 2
Assume that E(z
a
|z
a
/z
n
> x) is increasing in x. By (3) we know that for any worker i with
individual productivities z
i
a
and z
i
n
, if i chooses to work in agriculture in country P then z
i
a
/z
i
n
>
1/p
P
a
, and if i chooses to work in agriculture in country R then z
i
a
/z
i
n
> 1/p
R
a
. By Proposition 1
we know that p
P
a
> p
R
a
. H ence, by our assumption, E(z
a
|z
a
/z
n
> 1/p
P
a
) < E(z
a
|z
a
/z
n
> 1/p
R
a
).
Thus
Y
R
a
/N
R
a
Y
P
a
/N
P
a
=
A
R
A
P
·
E(z
a
|z
a
/z
n
> 1/p
R
a
)
E(z
a
|z
a
/z
n
> 1/p
P
a
)
>
A
R
A
P
.
A similar result holds when E(z
n
|z
n
/z
a
> x) is increasing in x.
1
1.3. Proof of Corollary 1
It suffices to prove that the E(z
a
|z
a
/z
n
> 1/p
a
) is decreasing in p
a
and E(z
n
|z
n
/z
a
> p
a
) is
increasing in p
a
. To obtain closed-form expressions for the conditional expected productivities
in question, one must derive Prob{z
n
≤ p
a
z
a
}. To do so, note that this probability is represented
by
π
a
=
Z
∞
0
exp{− (p
a
z
a
)
−θ
}g(z
a
)dza,
where the first term in the integral is the cumulative distribution function for productivity in
non-agriculture evaluated at random variable p
a
z
a
, and the second term g(z
a
) is the individual
productivity distribution function in agriculture. The anti-derivative for this integral is given
by
1
p
−θ
a
+ 1
× exp{−(p
−θ
a
+ 1)z
θ
a
}.
Evaluating the integral yields
π
a
=
1
p
−θ
a
+ 1
,
and similar arguments yields
π
n
=
p
−θ
a
p
−θ
a
+ 1
.
To compute the conditional average individual productivity in each sector, we make the fol-
lowing argument. First notice that the conditional productivity distribution for workers in
non-agriculture is
Prob {z
n
< z|z
n
> p
a
z
a
} =
Prob {z
n
< z, z
n
> p
a
z
a
}
Prob {z
n
> p
a
z
a
}
.
Then computing the probabilities in the numerator and the denominator we have
Prob {z
n
< z, z
n
> p
a
z
a
}
Prob {z
n
> p
a
z
a
}
= exp{−(p
θ
a
+ 1)z
−θ
n
}.
Notice that the conditional productivity distribution of workers in non-agriculture is itself
Fr´echet distributed with centering parameter (p
θ
a
+ 1). Using this insight we can now com-
pute the average individual productivity of non-agriculture workers conditional on working in
2
non-agriculture to be
E(z
n
|p
a
z
a
< z
n
) = (p
θ
a
+ 1)
1
θ
γ.
where the constant γ is the gamma function evaluated at
θ −1
θ
. Similar arguments imply that
average individual productivity of agriculture workers conditional on working in agriculture
is
E(z
a
|p
a
z
a
> z
n
) = (p
−θ
a
+ 1)
1
θ
γ.
B. The Role of Capit al i n Explaining Sector Productivity Differences
To study the role of sector differences in capital per worker across countries, we use data on
agricultural capital stocks constructed by Butzer, Mundlak, and Larson (2010). The capital
stocks they construct represent estimates of the total value of machine ry, structures, treestock
and livestock used in agricultural production. They h a ve estimates for a set of 30 countries from
all levels of the world income distribution. One strength of this study is the effort to which the
authors go to construct measures that are internationally comparable, which is no easy task
given the data challenges inevitable in calculations of this na ture. The main limitation is, as the
authors point out, that there are still reasons to be skeptical of the international comparability
of the data.
For our accounting calculations, we make use of their agricultural capital stock estimates from
1985, the year corresponding with the sector productivity da ta analyzed by
Caselli (200 5), and
we express the capital stocks in international prices using the investment price deflators from
the PWT. We construct the non-agricultural capital stocks by subtracting the agriculture capital
from the total capital stocks used by
Caselli (2005). We end up with estimates of both output
and capital per worker, by sector, for 28 countries.
Table
1 reports our findings for the role of capital per worker differences in accounting for sector
productivity differences. Here we employ
Caselli (2005) preferre d metrics for the “success” of
capital per worker differences. The first, success
1
, is defined as the ratio of log variance in
output per worker in a world with only capital per worker differences, divided by the a ctual
log variance. The second, suc c ess
2
, is defined as the 90-10 ratio of output per worker in a world
with just capital per worker differences compared with the actual 90-10 ratio. The idea behind
both of these metrics is that the lower they are, the larger is the role for TFP differences in
explaining output per worker differences. For comparison, we also reproduce the results of
Caselli (2005) (Table 5).
3
Table 1: Role of Capital in Accounting for Sector Productivity Differences
Source Sector success
1
success
2
Our calculations Agriculture 0.22 0.12
(n=28) Non-agriculture 0.29 0.50
Caselli (2005) Agriculture 0.15 0.09
(n=65) Non-agriculture 0.59 0.63
Note: A uthors’ calculations using data from Butzer, Mundlak, and Larson (2010) and Caselli
(2005).
Our calculations suggest that TFP differences are the key component of output per worker
differences and they seem to play an even larger role in explaining agriculture productivity
differences across countries than in non-agriculture. As one can see in Table
1, by either met-
ric, cap ital per worker differences far from fully account for sector productivity differences in
either sector. For success
1
, we find a ratio of 0.22 in agriculture and 0.29 in non-agriculture. For
success
2
, we find an even lower 0.12 in a griculture and 0.50 in non-agriculture. These calcula-
tions paint a very similar picture to those of
Caselli (2005), even though we employ different
methodology and a different set of countries.
C. Estimation of the N on-Transitory Component of Wa ges
In this section we discuss how we estimate the variance of the non-transitory component of
wages by sector to which we calibrate the model. The rationale for calibrating the model to
match variation in the non-transitory component of wages, rather than all wage variation,
is that wa ge variation in the model arises only from productivity differences across workers,
whereas wage variation in the data may include other factors unrelated to productivity. This
distinction is important because transitory effects may be relatively more prevalent in agricul-
ture , for example, as a result of we ather shocks.
3.1. CPS Data
To estimate the variance of the non-transitory component of wages, we make use of micro-level
data from the March Current Population Survey (CPS). We use data from 1996 to 2010, which
are the most recent years available which allow for consistent matching of workers across years.
We calculate each individual’s wage as total labor income in the previous year divided by hours
worked in the previous year. We define total labor income as the sum of salary income plus
4
Table 2: Summary Sta tistics of CPS Data: 20 03-2010
Statistic Value
Percent of Workers in Agriculture 1.55
Ratio of Average Wage in Agriculture / Non-agriculture 0.701
Variance of Log Wages, Agriculture 0.355
Variance of Log Wages, Non-Agriculture 0.380
0.66 of business income plus 0. 46 of farm income, where the fractions of business and farm
income assigned to labor are those estimated for the U.S. non-agricultural and agricultural sec-
tors found by
Valentinyi and Herrendorf (200 8). We exclude all individuals who have missing
hours or income data or whose wage is lower than the Federal minimum wage. We express all
wages in 2010 dollars using the U.S. Consumer Price Index.
We make use of the short pane l dimension of the CPS, which allows a subset of individuals to
be matched in two consecutive years. We follow exactly the criteria of Madrian an d Lefgren
(2000) in eliminating any potentially spurious matches. We end up with 202,677 individuals
total that can be matched in two consecutive years. We define agricultural workers to be those
whose primary industry of employment in both years is agriculture, forestry or fishing. We
define non-agricultural workers to be those in any other sector in both years.
Table
2 presents some summary statistics of the data. Agricultural workers constitute 1.55%
of all workers, which is in line with estimates of agriculture’s share in employment from other
sources, e.g.
Herrendorf and Schoellman (2011). The a verage hourly wage in a griculture is
0.701 times as high as in non-agriculture. The variances of log wages are 0.355 in agriculture,
and slightly higher at 0.380 in non-agriculture. These values are consistent with those reported
in
Heathcote, Perri, and Violante (2010) from the CPS in their study of cross-sectional inequality
in the United States using various micro-level data sources.
3.2. Specification and Estimation of Non-Transitory Components
To estimate the fraction of wage variance arising from the non-transitory component of wages,
we assume that log wages for an individual in sector j at time t are given by
log(w
j,t
) = log(z
j
) + ǫ
j,t
5
