Productivity and Agricultural Out-Migration
in the United States
Benjamin N. Dennisa
Department of Economics
University of the Pacific
3601 Pacific Avenue
Stockton, CA 95211, USA
Talan B. İşcan∗
Department of Economics
Dalhousie University
Halifax, Nova Scotia
B3H 3J5, Canada
May 23, 2003
Preliminary and incomplete – do not quote or cite
Abstract
In the U.S. the average annual labor reallocation from agriculture to non-agriculture accelerated considerably in the twentieth century, reaching about 5 percent per year between
1940 and 1980, a period that also coincides with the acceleration of farm productivity
growth. Despite a tremendously rapid labor reallocation from the farm to the non-farm
sector, there were nevertheless: (i) large and sustained wage gaps between the farm and
non-farm sectors, and (ii) episodes of on-farm migration (as seen in the 1930s). We examine this acceleration in structural change within a general equilibrium context that allows
for both absolute and relative farm productivity growth. Our analysis integrates fixed
costs of labor reallocation, and stochastic fluctuations in farm–non-farm relative productivity growth in a way that also accords with the stylized facts of a persistent wage gap
and reverse migration.
JEL Classification: O15, N22
Keywords: sectoral reallocation of labor, fixed costs, migration, U.S.
∗ Corresponding
a E-mail:
author: Tel.: (902) 494-6994. E-mail: Talan.Iscan@dal.ca.
bdennis@uop.edu.
We thank Sheena Starky for research assistance.
1
Introduction
Throughout the world, gradual but massive migration from agriculture to non-agriculture
has accompanied industrialization and capitalist development. The U.S. experience is
especially remarkable. Between the early nineteenth and late twentieth centuries, the
average annual labor reallocation from agriculture to non-agriculture was approximately
3 percent of the farm population. This reallocation process accelerated considerably in
the twentieth century, reaching approximately 5 percent per year between 1940 and 1980,
a period that also coincides with the acceleration of farm productivity growth. Despite a
tremendously rapid labor reallocation from the farm to the non-farm sector, there were
nevertheless: (i) large and sustained wage gaps between the farm and non-farm sectors,
and (ii) episodes of on-farm migration (as seen in the 1930s). Figure 1 presents these
trends.1 In this paper, we seek to explain this acceleration in structural change within a
general equilibrium context that allows for both absolute and relative farm productivity
growth in a way that also accords with the stylized facts of a persistent wage gap and
reverse migration.
The traditional approach to off-farm migration has emphasized absolute farm productivity growth in conjunction with subsistence consumption of agricultural goods.2 This
approach is quite intuitive. As productivity in agriculture rises, due to low income elasticity of demand for these goods, supply greatly outstrips demand, and resources need to
be reallocated out of agriculture. It also has an empirical appeal. As stated above, absolute farm productivity growth did accelerate at the same time that farm out-migration
accelerated, and the low income elasticity of demand for farm goods is one of the undisputed facts in economics. Note, however, that this approach pays no explicit attention to
productivity growth in the farm sector relative to the rest of the economy. Yet, a striking
aspect of the most of the twentieth century U.S. data is that productivity growth in the
farm sector has also exceeded that of the non-farm sector.
We show that, in addition to absolute farm productivity growth, relative farm–nonfarm productivity growth can be an (equally) important driving mechanism behind U.S.
net off-farm migration. This observation hinges on a crucial departure from most of the
literature on this topic: a non-unitary elasticity of substitution between farm and nonfarm products. The seemingly innocuous assumption of unitary elasticity of substitution
between farm and non-farm goods assigns no role to relative productivity growth. In this
special case a mere productivity growth differential across sectors does not lead to any
1
See Gardner (2002, pp.9̃8–99) for a recent account of the off-farm migration trends. Note that rate
of migration out of the farm sector was about 2 percent annually during the 1920s and 1930s, despite two
years of net movement into agriculture during the Depression (1932 and 1933). In fact, during the 1930’s
the New England states and the Pacific U.S. had a net increase in farm population (Hathaway 1964,
Table 1). Hatton and Williamson (1992) also document and examine the short-lived farm in-migration
during the Depression America. We discuss our data sources underlying this figure and our estimates
throughout the paper in a detailed Data Appendix.
2
See, e.g., Nurkse (1953), Lewis (1954), Timmer (1988), and Kongsamut et al. (2001).
1
change in relative prices, and hence would have no impact on the relative wage, which
drives sectoral labor flows. However, in the empirically relevant case of low demand
elasticity of substitution between farm and non-farm goods (i.e., they are gross complements), relative technological progress in the farm sector results in unfavorable shifts in
the agricultural terms-of-trade and exerts pressure on labor to move out of farming.3
To gauge the quantitative importance of both absolute and relative productivity
growth to structural change, we provide a detailed reinterpretation of the U.S. net migration flows and sectoral labor shares in the twentieth century. In particular, we use a
baseline general equilibrium model to calibrate sectoral labor shares. The baseline model
allows for instantaneous sectoral reallocation of labor in response to a change in relative
prices. We decompose the changes in these calibrated series into the contributions of absolute and relative farm productivity growth. The calibration exercise requires estimates
of (i) subsistence to consumption ratio, and (ii) relative productivity levels. These in turn
determine the weights assigned to absolute and relative productivity growth rates in the
decomposition analysis. Since we lack precise estimates of either of these variables, we
impute them using consumption expenditure data on food. We find that in general the
baseline model delivers a farm share of labor that is considerably lower than the actual
share in the data. This excess labor in the farm sector disappears by the mid-1980s.
We also find that relative contributions of absolute and relative productivity growth to
labor reallocation in the calibrated data are very sensitive to the initial choice about
food subsistence consumption to total food consumption ratio. In this context we examine several cases.4 A low initial value of subsistence-consumption ratio tends to reduce
the discrepancy between the calibrated series and actual data. A high initial value of
subsistence-consumption ratio tends to increase the discrepancy. Thus, the postwar acceleration in off-farm migration is especially notable in that it took place when declining
influence of subsistence consumption should have attenuated it. We attribute this to the
acceleration in the farm productivity rate.
Given the significant discrepancy between actual and calibrated series in the baseline
model, we consider an extension in which workers have to incur fixed and sunk costs
when they switch sectors. This creates a certain amount of inertia in sectoral allocations
of labor. We show that relocation decisions follow simple (S, s) rules whereby relocation
is triggered only when sectoral wage gaps exceed certain S or s percentage points. These
wage gaps can be sustained for long periods of time.
Although our focus is related to a voluminous literature on structural change, which
3
The main predecessor of the point we are making in a general equilibrium context is the so-called
“Baumol’s cost disease” (Baumol, 1986) in which sectors with slower productivity growth tend to have
higher cost inflation.
4
According Bairoch (1975, pp. 38–40), the calorie intake in the U.S. had surpassed the minimum
subsistence requirement before the turn of the twentieth century. However, we have no obvious way
of mapping such information onto our theoretical construct. We thus have to entertain alternative
initializations.
2
is surveyed in Syrquin (1988), let us briefly mention here two of the most recent studies.
Kongsamut et al. (2001) model very long-run economic growth with three sectors in
which relative employment shares vary over time. They assume unitary elasticity of
substitution across goods, and invoke identical productivity growth rates across sectors.
As a consequence of these two assumption, in their theoretical model, all structural change
is driven by absolute productivity growth. Caselli and Coleman (2001) model regional
convergence over the medium- to long-run, and allow for differential productivity growth
rates across farm and non-farm sectors. The issue that we raise is potentially more relevant
for their analysis. However, as we discuss in detail below, the economic significance of
this assumption is ultimately an empirical matter.
Our emphasis on modelling frictions that lead to protracted periods of off-farm migration with sustained sectoral wage gaps is also related to an earlier literature on the
so-called “transfer problem” of moving labor out of agriculture and into industry.5 It is
striking that many contemporary commentators felt that the off-farm migration rate was
too low, despite the substantial transfer already taking place, and their analysis was primarily driven by the protracted size of the wage gap in favor of industry.6 Our calibration
results of the baseline model are thus entirely consistent with this view of “shortfall” in
migration, and attempts to alleviate costs of migration in the form of calls for government
intervention in providing information and assistance to potential migrants.
The rest of our paper is organized as follows. Section 2 outlines the basic two-sector
model. Section 3 discusses the baseline frictionless model, presents its quantitative implications for changes over time in sectoral labor shares, and links these to relative and
absolute and relative productivity growth rates. Section 4 integrates equilibrium wage
gaps to the two-sector model with fixed costs. Data and some technical and computational details are contained in a series of appendices.
2
A Basic Model
As we mentioned in the Introduction, any model of off-farm migration sector must address the stylized fact of farm–non-farm wage gaps via endogenous labor immobility. To
achieve this, we adopt the elegant two-sector model advanced by Dixit and Rob (1994).
5
See, e.g., Shultz (1945), Johnson (1956), Heady (1962), the 1960 AEA session on “Facilitating Movements of Labor Out of Agriculture” in the Proceedings of the American Economic Association Meetings,
and a more recent assessment by Mundlak (2000, p.2̃64): “The off-farm labor migration is a universal
phenomenon that continues over a long time. Why does it take such a long time before it comes to
an end? If nonagriculture is more attractive then all farm labor should leave it at once.” Rural–urban
migration models address related issues. See Williamson (1988) for a survey.
6
Perhaps unlike earlier episodes, these real wage gaps were significant even after they were adjusted to
take into account purchasing power and income tax differences; see, e.g., Johnson (1956), and Hatton and
Williamson (1992). Farm and non-farm wage differential has apparently became economically insignificant
in the 1990’s.
3
Their framework involves a combination of dynamic uncertainty and (exogenous) fixed
costs, and allows us to obtain closed form solutions. The advantage of our analytic solutions is that they help us disentangle the independent contributions of: (i) relative and
absolute sectoral productivity growth, and (ii) subsistence consumption of food. In the
next section, we describe this basic environment.
2.1
Preferences and Production
We work with continuous time, t ∈ [0, ∞). There is a continuum of workers indexed by
i, and the measure of the entire set of workers in the economy is normalized to unity, i ∈
[0, 1], with each worker of measure zero. Each worker lives forever and the set of workers
is fixed over time. (Population growth and differential fertility rates are discussed below.)
Workers have preferences over a composite consumption good (C) and inelastically supply
one unit of indivisible labor. There are only two sectors, farm (or “agriculture”) (A) and
non-farm (“manufacturing”) (M ), and each individual works in either the A sector or the
M sector. For expositional purposes, we think of non-farm jobs as being located in the
city, and agricultural jobs on the farm.
When there are frictions, workers can change sectors at any time, but they incur a
fixed and sunk cost when they relocate. The frictionless model will abstain from such
costs. Since we wish to allow for reversals in migration flows (as we see in the data), we
work with fluctuating (stochastic) incomes.
To jointly determine the sectoral employment and consumption decisions, we assume
that each worker maximizes the following expected utility function:
#
"Z
∞
X
e−ρt ln C(t) dt −
e−ρtj c ,
E
0
j
subject to (omitting the time indices):
C =
h
iν/(ν−1)
(ν−1)/ν
η 1/ν cM
+ (1 − η)1/ν (cA − γA )(ν−1)/ν
,
w ≥ pM cM + pA cA ,
where cM and cM represent consumption and the unit price of the non-farm good; cA
and pA represent consumption and the unit price of the farm good; and w represents
nominal earnings, which depends solely on the sectoral wage rate of wM in the M sector
or wA in the A sector. E is the conditional expectations operator. The parameters are
interpreted as follows: γA ≥ 0 is the subsistence consumption of food, η ∈ (0, 1) measures
the consumption weight of the manufactured good; ν > 0 is the elasticity of substitution
between manufacturing and agricultural goods; and 0 < ρ < 1 is the subjective discount
rate. If ν < 1, goods are said to be “gross complements,” and otherwise they are “gross
substitutes.”
4
The second term in the utility function captures the fixed cost c ≥ 0 of changing the
sector of employment (i.e., the cost of migration). The relocation cost is modelled as
an instantaneous psychic cost or moving and training costs proportionate to per period
income.7 To reduce the burden on notation we also assume that costs are not sector
specific. We discuss the significance of asymmetric costs below. When c > 0, job switches
take place in discrete instances tj . When c = 0 we have the frictionless case.
Demand .—Since workers can neither smooth consumption using storable assets, nor
share risk among fellow workers, the following first-order conditions (which correspond
to a period-by-period maximization of instantaneous utility) characterize the optimal
resource allocation by each worker:
µ
¶ µ ¶ν
cM
η
pA
=
.
(1)
c A − γA
1−η
pM
Henceforth, we will work with relative
assume that (cA − γA ) > 0 is
R 1 price, p = pM /pA , Rand
1
always satisfied. Further, let CA = 0 cA (i)di and CM = 0 cM (i)di denote aggregate consumption of farm and non-farm goods, respectively. Of course, the optimal consumption
ratios in equation (1) must hold in the aggregate, as well as for each worker.
Supply.—A fraction LA of the labor force is employed in the agricultural sector, leaving
LM = 1 − LA employed in the manufacturing sector. We use constant returns to scale
technology with labor as the sole factor of production in both sectors:
YM = zM LM ,
YA = zA LA ,
(2)
where zM and zA measure labor productivity in manufacturing and agriculture. To ensure
subsistence consumption, we assume that the economy is always sufficiently productive:
γA < zA . In what follows, we find it convenient to work with manufacturing productivity
relative to that of agriculture, z = zM /zA , which we assume is determined exogenously.
Stochastic variability in z is the ultimate source of uncertainty, but plays no interesting
role in this static model when there are no relocation costs.
2.2
Static Equilibrium
We assume that factor and product markets are competitive, and that the wage rate in
each sector is equal to its sectoral marginal revenue product. Net labor flows between
7
Note that we could have used the following expected utility function:
·Z ∞
¸
C(t)
−ρt
E
e
ln
dt ,
I(t)
0
where the indicator function I would take value ec when the worker reallocates, and 1 otherwise. In this
case, relocation cost would act as a “tax” on current consumption net of subsistence, when reallocation
occurs. Dixit and Rob (1994) label these costs as psychic in more general utility specifications.
5
agriculture and manufacturing depend on the sectoral wage differential. It is useful to
represent this differential as a ratio:
wM
≡ w = z · p.
wA
(3)
In our general equilibrium framework, w is endogenous and depends on the exogenously
given value for relative productivity, z, and on the endogenously determined value of the
relative price, p. When w > 1, A sector workers will have an instantaneous incentive
to migrate to the M sector. Historically, this has been the dominant tendency. When
w < 1, however, there will be an instantaneous incentive to migrate to the A sector, or
an incentive for “on-farm migration”.
This distinction between per worker and aggregate variables is particularly important
when relative wages depart from 1. The market clearing conditions in this case are:
CA = YA ,
C M = YM ,
LA + LM = 1.
(4)
Note that, when w = 1, there is an inverse relationship between z and p; i.e., z = 1/p.
When w 6= 1, we can find an expression for p using equations (1), (2), and (4):
·µ ¶ µ
¶µ
¶µ
¶¸1/ν
1
η
1 − LM
γA
p(z, LM ) =
1−
.
z
1−η
LM
CA
(5)
Our task is to find equilibrium values of LM for given realizations of z. In the next two
sections, we impute the values of z and γA /CA , and calculate the corresponding labor
shares under different assumptions about market frictions. We consider the baseline
“frictionless” case first, and discuss why it fails to explain the actual labor reallocation in
the twentieth century U.S.
3
Frictionless Equilibrium
When relocation decisions do not entail any costs to the individual, job switches across
sectors can be frequent and will eliminate any relative nominal wage differentials that may
arise due to fluctuations in z. In this case w = 1. Equations (1)–(4), together with this
frictionless equilibrium condition, determine the frictionless labor allocation share LfM in
terms of z and the model’s parameters:
¶
¶
¸−1 µ
·
µ
γA
1−η
f
1−ν
.
(6)
z
1−
LM = 1 +
η
zA
6
3.1
Structural Change
Our measure of structural change depends only on the distribution of labor across sectors,
i.e., LM .8 This measure is distinct from the off-farm migration rate M , which has also
been used to capture the pace of structural change:
M (t) =
NM (t) − NM (t − 1)
,
NA (t)
(7)
where Ni is population in sector A or M . Appendix A discusses the differences between
the population and employment based estimates of the off-farm migration rate, and shows
that both of these two series are consistent with the rapid structural change observed in
the twentieth century U.S.9 In what follows, we focus on the employment based data and
the structural change measure is defined as:
µ
¶
¸
·
%
dzA
dz
G(z)
f
d ln LM =
+ (ν − 1)
.
1 − % zA
1 + G(z) z
where:
γA
%=
,
zA
µ
and
G(z) =
1−η
η
¶
z 1−ν .
Let µA dt = dzA /zA be the “trend” productivity growth in the farm sector, and µdt = dz/z
be the relative-productivity growth rate.10 Then we have the rate of change over time in
the share of non-farm farm employment given by two contributions:
d ln LfM
dt
=
+
[ % / ( 1 − % ) ] × µA
{z
}
|
absolute farm prod. growth
(ν − 1) [G(z)/(1 + G(z))] × µ .
|
{z
}
relative farm prod. growth
(8)
In what follows we maintain the assumption that farm and non-farm goods are gross
complements; i.e., ν < 1. Note that in the absence of subsistence food consumption,
γA = 0, a higher farm-sector labor-productivity growth rate is the only determinant of
8
An alternative measure is to use (1 − LM )/LM which would be more appropriate if we allow for
population growth, and differential fertility rates. We use the simpler measure here for ease of exposition,
but derive the relevant expressions in Appendix B.
9
We are currently working on expanding our model to allow for differential fertility rates between the
farm and non-farm workers.
10
We use the term “trend” rather informally here in that we allow for trend shifts, and do not necessarily think of it as the very long-run productivity growth rate (as typically assumed in models with a
constant growth path). Our’s is not a steady-state analysis. Our view is that a century long structural
transformation can involve long but transitional periods of relative and absolute productivity growth that
differ from the steady state rate. It is also possible that a sequence of such transitory dynamics can lead
to a constant overall economic growth (Jones, 2002).
7
labor reallocation and off-farm migration. Even in the case of subsistence consumption
of agricultural goods, i.e., γA 6= 0, as long as % → 0 over time (which occurs eventually
since γA is constant and zA grows over time), only relative productivity growth matters
in the long run.
Of course, in the intermediate case, γA 6= 0 and % 6= 0, in which the rate of labor
reallocation into the non-farm sector depends on the relative strengths of these two factors.
In the early stages of industrialization absolute productivity growth in agriculture is likely
to be the dominant factor behind structural change (because %/(1 − %) would be large).
However, in periods when µA is positive and µ is negative, its relative contribution to
labor reallocation depends on which of the two terms, % or G(z), goes zero at a faster
rate. As well, if µ = 0, then LM increases at a decreasing rate with rising zA , and because
% → 0, the share of labor in each sector will come to stabilize at a steady-state as both
components of equation (8) will be equal to zero.
3.2
Engel’s Law
Engel’s law emerges from this framework whereby, as incomes rise, the share of income
spent on agricultural goods falls (even in the absence of subsistence consumption). This
occurs because faster productivity growth in agriculture reduces its relative price which,
in turn, reduces the expenditure share on agricultural output (given a low elasticity of
substitution between farm and industrial output).
Specifically, the expenditure share of farm goods, θA can be calculated as follows (see
also Appendix B). Use equations (1), and (3):
·
µ
¶
µ
¶¸−1
CA
η
γA
ν−1
θA =
= 1+
z
1−
.
(9)
CA + pCM
1−η
CA
With economic development one would expect γA /CA to decrease at a diminishing rate
over time. As long as this is the case and ν < 1, this formulation establishes an inverse
relationship between the share of expenditure on agricultural output and the productivity
ratio. As z falls (i.e., farm productivity increases relative to non-farm productivity),
the relative price of agricultural goods falls and the share of expenditure on agriculture
declines as well. Figure 2 shows the share of food in total expenditures, which exhibits
a secular decline. When ν = 1 (as is sometimes assumed), Engel’s Law emerges from
the combination of subsistence consumption of food and increased food consumption due
to absolute productivity growth in agriculture. However, this has the (controversial)
implication that the agricultural terms of trade should remain constant over time.
3.3
Quantitative Performance of the Baseline Model
When we examine the quantitative performance of the baseline frictionless model, two important issues emerge. First, the calibrated model requires particular assumptions about
8
subsistence–consumption ratio to generate a sufficiently high trend rate of relocation.
Second, the model fails to account for persistent (and fluctuating) wage gaps. We discuss
the benchmark model’s performance in light of these two shortcomings.
The frictionless model with γA = 0 is illustrative and we consider it first. Figure 3
shows that this model does a very poor job in accounting for structural transformation in
the twentieth century U.S. Because this version of the model assigns no role to subsistence
consumption, the overall rate of structural transformation is determined solely by ν and
µ. To see this, set % = 0 in equation (8). Our calibrated value of µ is −1.44 percent for
1900–91, and −1.53 percent for 1920–91. See Appendix section B.3 for computational
details. Clearly, for ν = 0.1 and µ = −1.53 percent, the model’s estimate of off-farm rate
of labor reallocation (LM /(1 − LM )) is 1.4 percent, considerably smaller than the actual
rate of 3.6 percent (see Appendix A).
Figure 3 also shows how subsistence consumption and especially assumptions about
initial subsistence–consumption (γA /CA ) ratio can have an impact on the calibrated rate
of labor reallocation out of farm sector. Two examples are considered: γA /CA = 0.5
and 0.7 at the beginning of the calibration period. Both series also imply convergence to
actual labor shares by 1990s. for initial γA /CA = 0.5 the rate of structural transformation
between 1900–91 would be 2.4 percent, and for initial γA /CA = 0.7 it would be about 3.4
percent. Thus, the discrepancy between actual and calibrated series declines with higher
values of initial subsistence–consumption ratio. In any case, taken literally, our imputed
labor shares suggest that there were relatively “too many” workers in the farm sector in
the second half of the twentieth century.
The second issue is that a simple and intuitive general equilibrium model fails to
account for the large and persistent farm and non-farm wage earnings. As mentioned
in the Introduction farm–non-farm wage gaps might suggest that actual labor flows were
actually lower than what they would have been if “transfer” from farm to industry followed
the route of a frictionless equilibrium. Our calibration results are broadly consistent with
this observation. We now turn our attention to the discussion of basic model with real
frictions.
4
Equilibrium with Frictions
In the frictionless case, the critical assumption is real wages are continuously equalized
across sectors, w = 1. Clearly this assumption is untenable given the recorded real wage
differentials between farm and non-farm employment. Prior to formal modelling of these
wage gaps, to illustrate how departures from this assumption might affect the allocation
of labor across sectors, we recalculated the implied employment shares without imposing
sectoral wage parity. In particular, we used the actual wage ratio w, and equations (1)–(4)
9
to solve for implied sectoral labor shares (see also Appendix B):
·
µ
¶
¸−1 µ
¶
1−η
γA
w
ν 1−ν
LM = 1 +
w z
1−
.
η
zA
(10)
The impact of wage gap in favor of industry on implied labor shares is as follows. Throughout the entire period, farm wages have lagged behind non-farm wages, so w > 1. Given
relative productivity, in order for this to be compatible with our stated equilibrium conditions, non-farm price had to be relatively high. This can happen when there are relatively
“few” workers in the non-farm sector. Clearly, these shares lie below the corresponding
series shown in Figure 3. This is not surprising because when we assumed w = 1, the
actual share of non-farm labor fell short of the calibrated LM . With w > 1, calibrated
series impute less non-farm employment and therefore reduce the gap between actual and
calibrated series.11
So far this discussion of frictions is agnostic about the sources of these wage gaps,
and “disequilibrium” sectoral allocation of labor. One reason for why these might arise
is costly labor relocation, and next we analytically solve our model with fixed relocation
costs to establish an equilibrium framework.
4.1
Relocation with Fixed Costs
Since workers cannot divide their time between on- and off-farm employment, in each
period they must choose a sector of employment. When workers incur a fixed cost to
migrate, there will be periods during which some workers will choose not to migrate (i.e.,
periods of “inaction”), despite fluctuations in productivity, z. For a given LM , the zone
of inaction corresponds to a range of z values for which wage gaps can be sustained up to
a maximum value. Once one of these wage-gap maxima is reached or exceeded, there will
be migration. Henceforth, we refer to these maxima as “thresholds.” For a given LM , the
thresholds encapsulate each worker’s evaluation of benefits relative to costs. Benefits are
always measured against real income in the destination sector. Thus, our first objective
is to compare the benefits of staying versus moving, and determine the desired direction
of migration. We start with the definition of relative wages (w = z · p), use equation (5),
and compare the instantaneous utilities entailed by employment in the M sector versus
the A sector:
·µ
¶µ
¶µ
¶¸
ν−1
1
1 − LM
η
γA
ln w =
ln z + ln
1−
.
(11)
ν
ν
LM
1−η
zA (1 − LM )
11
We should mention two issues at this point. First, we have not made any cost of living and income tax
adjustments to the relative wage series. However, as mentioned in the Introduction, such adjustments
(which are only available for a limited number of years) are unlikely to change the basic conclusions.
Second, while the imputed series are not very sensitive to the particular choice of η, they are affected
by ν: higher values of the elasticity coefficient attenuate the gap between actual and implied non-farm
employment shares.
10
The relevance of equation (11) for our purposes is that it can be different from zero for
sustained periods of time because the size of relocation costs relative to perceived gains
may be large.
4.2
Dynamic Equilibrium
In the absence of fixed costs, uncertainty plays no role in the analysis, because workers
can respond to instantaneous incentives. With non trivial costs, however, they have to
consider the current as well as future wage gaps, and value waiting before engaging in
a costly move. In order to study equilibrium under uncertainty, we thus have to specify
the stochastic processes that workers condition their decisions upon. We assume that
unpredictable variability in z is the ultimate source of uncertainty, and it is given by a
geometric Brownian motion:
dz
= µ dt + σ dω,
z
(12)
where µ is the trend of the diffusion process, σ is the standard deviation, and dω is a
standard Weiner increment.
The second important reason for considering fluctuations in relative productivity is to
allow for reverse migration flows. While off-farm migration has been the secular tendency,
there were episodes of on-farm migration. Short-run fluctuations in z allows us to present
a unified framework within which medium- to long-run trends as well as short-run labor
net flows can be consistent with our model.
Our objective is to determine the maximum sectoral wage differentials (thresholds)
that are admissible given preferences, technology, the fixed costs of reallocation and the
relative productivity process. A more detailed solution of the model with frictions is
contained in Appendix C. Here we outline the basic results, some of which are already
contained in Dixit and Rob (1994).
Each worker’s decision involves (i) pricing the net option value of waiting (UO ), (ii) calculating the present value of consumption differentials (U∆ ) that arise when the worker is
employed in one sector rather than the other, and (iii) comparing these to c to compute
relocation thresholds that mark the boundaries of zone of inaction.
Of course, both U∆ and UO are determined by z and LM (or LF ). Jointly they
determine the desired direction of migration. Let positive values of U∆ correspond to
higher M sector wages in present value terms. Then, workers relocate from the farm
sector A to non-farm sector M when:
U∆ (z, LM ) + UO (z, LM ) > c.
And workers relocate from sector M to A when:
U∆ (z, LM ) + UO (z, LM ) < −c.
11
The values of relative productivity at which the above expressions are satisfied with
equality are the relocation thresholds, ZM and ZA . Once these thresholds are crossed,
workers migrate until narrowing of wage gaps no longer warrants any relocation.
4.3
Equilibrium Wage Gaps
In this section we calculate the relation thresholds and relate them to equilibrium wage
gaps. We start with the analytically tractable case in which the subsistence consumption
is zero, and then discuss the implications of γA > 0 for our analysis.
4.3.1
No Subsistence Food Consumption
In the case of γA = 0, the relocation thresholds turn out to be remarkably intuitive:
µ
¶
ν−1
move from A to M if: ln w =
s > 0,
ν
µ
¶
ν−1
move from M to A if: ln w =
S < 0.
ν
In words, migrants rule simple rules: When the (adjusted) wage gap between non-farm
and farm wages, ν/(ν − 1) · ln w, reaches or exceeds s percentage points, there is farm outmigration. As a result, the non-farm share of employment increases. Conversely, when
this gap reaches S percentage points, there is farm in-migration and agriculture’s share
of employment rises. Within the (s, S) bands, workers are immobile despite current and
expected wage gaps. What makes this rule simple is the fact that relocations decisions
do not depend on the sectoral distribution of labor; i.e., the level of LM , so workers
need only worry about the current wage gaps (given the underlying stochastic process).
The expressions for (s, S) are given in Appendix C, and are functions of the underlying
parameters.
An important feature of equation (11) is that they can be expressed as function of
sectoral labor imbalance:
"
#
1
1 − LM
1 − LfM
ln w =
ln
− ln
.
ν
LM
LfM
Thus, along the relocation thresholds corresponding to moves from A to M , and from M
to A, respectively, we have:
ln
1 − LfM
1 − LM
− ln
LM
LfM
= (ν − 1)s > 0
ln
1 − LM
1 − LfM
− ln
LM
LfM
= (ν − 1)S < 0.
12
4.3.2
Subsistence Consumption
When γA > 0, then the above analysis must be modified but only slightly. In particular,
under the assumption that γA /zA (1 − LM ) exhibits steady decline over time, we show
in the Appendix that the thresholds discussed above require a minor adjustment: The
subsistence terms increase the absolute value of off-farm migration threshold, and reduce
the on-farm migration threshold, ceteris paribus. With subsistence consumption, relative
price of farm goods falls more slowly compared to the case in which there is no subsistence
consumption. This in turn tapers workers’ desire to relocate out of the farm sector for a
given non-farm labor share and relative productivity. Note that a steady decline in the
subsistence–consumption ratio is in our context similar to a rising LM /(1 − LM ) ratio:
Both affect the levels of relative productivity that trigger relocation, but neither has any
direct impact on the maximum permissible equilibrium wage gaps. Hence, the decision
rules and the relocation thresholds are independent of LM and zA .
In the next section, we illustrate how the intuition formalized by this model can help
interpret the farm–non-farm wage gaps and agricultural out-migration in the U.S.
5
Calibrated Wage Gaps and Costs
TO BE COMPLETED
13
Table A.1:
Data Sources for the Farm Share of Employment
Period (frequency)
1800–1860 (Decennial)
1870–1890 (Decennial)
1900–1947 (Annual)
1948–1970 (Annual)
1972–1991 (Annual)
LM
Explanation
emp. in ag.
labor force
emp. in non-ag.
emp. in ag. + non-ag.
non-farm emp.
total emp.
emp. in non-ag.
total emp.
emp. in ag.
1−
labor force
1−
Series
10 years old and over
D167–181
16 years old and over
D11–25
14 years old and over
D1–10
16 years old and over
D1–10
10 years old and over
D167–181
Note: All series are from the Historical Statistics.
Appendix
A
Data Sources and Variables
All of our data, unless otherwise stated, come from the Historical Statistics of the United
States (Statistical History supplemented by Datapedia of the United States). The definitions of variables used and data sources are as follows.
A.1
The Farm share of employment
In our model, the farm share of employment corresponds to 1 − LM . Unfortunately, we
do not have consistent time series data on the share of non-farm employment, LM and so
we formed the farm employment data underlying Figure 1 as shown in Table A.1.
Note that in our empirical work we primarily use data from 1900 to 1991 and there is
a change in definition in 1948. However, broader trends in the relative employment share
should not be affected. Based on these data, we first calculate
long-run rate
³ the actual
´
1−LM
follows a diffusion
of labor reallocation in the U.S. In particular, assuming that LM
process with drift µ̃ and standard deviation σ̃, the maximum likelihood estimates of µ̃
and σ̃ are:
µ
¶
T
X
1
(1
−
L
(t))/L
(t)
M
M
ˆ =
µ̃
ln
,
T 1
(1 − LM (t − 1))/LM (t − 1)
·µ
¶
¸2
T
X
1
(1
−
L
(t))/L
(t)
M
M
ˆ =
ˆ .
σ̃
ln
− µ̃
T 1
(1 − LM (t − 1))/LM (t − 1)
14
Table A.2:
Estimates of long-run rate of off-farm labor reallocation
Period
1800–1991
1900–1966
1900–1991
1920–1966
1920–1991
Drift (µ̃)
Standard Deviation (σ̃)
0.0292
0.0377
0.0358
0.0396
0.0368
0.0455
0.0429
0.0497
0.0451
Source: Authors’ calculations as explained in the text.
ˆ gives our measure of the long-run rate of structural change. Table A.2
The term −µ̃
presents these estimates of the rate of reallocation towards the non-farm sector for different periods in our sample and suggests that the rate of structural transformation has
accelerated in later periods.12
A.2
The Off-farm migration rate
Off-farm migration rate M using model based variables is defined as:
M (t) =
LM (t) − LM (t − 1)
.
1 − LM (t)
(A.1)
and is based on changes in the farm share of employment. Most of the earlier off-farm
migration studies use different measures. For comparison, we also examine net off-farm
migration and farm population data covering the period 1920 to 1970. These migration
data, however, pertain to the entire farm population, and cover the period from April 1st
of one year to March 31st of the next. In the source material, the migration numbers
given for, say, 1921, refer to the period from April 1, 1920 to March 31, 1921. This
makes it difficult to align the migration data with the rest of our variables. Given these
constraints, we compute the off-farm migration rate in two alternative ways. In the first
method (M 1), our migration rate for 1920 is net off-farm migration from April 1920 to
March 1921 divided by the total farm population in April 1920, and so on. In the second
method (M 2), the starting year for the series is 1921, and it measures off-farm migration
12
Since the earlier data are irregularly sampled, we simply fit the following regression:
ln
1 − LM
= constant + slope × time,
LM
where “time” varies from 1800 to 1970, and the estimated value of the “slope” coefficient is the sample
counterpart of µ̃ in the model for this extended sample.
15
Table A.3:
Rural-Urban Fertility Differentials, USA
A. Number of Children Under 5 Yrs Old per 1,000 White Women 20 to 44 Yrs Old
Area
1910
1920
1930
1940
1950
Urban
Rural
469
782
471
744
388
658
311
551
479
673
B. Number of Children Under 5 Yrs Old per 1,000 Women All Races 15 to 49 Yrs Old
Area
1910
1920
1930
1940
1950
Urban
Rural nonfarm
Rural farm
NA
NA
NA
NA
NA
NA
NA
NA
NA
230
361
473
363
495
584
Source: Grabill et al. (1958), panel A from Table 7, p. 17, and panel B from Table 23, p. 70.
from April 1920 to March 1921 divided by the total farm population in April 1921, and
so on.
Aside from the problem of overlapping observations, there are two other issues regarding the calculated off-farm migration rates. First, since farm and city fertility rates are
not identical, the two sets of measures differ by definition. Table A.3 shows the difference
in fertility rates across these two sectors.
Second, our model concerns labor flows, but these alternative data pertain to persons
who reside on farms. The data may overstate the net off-farm migration rate if the number
of household members under working age in the out-migration population exceeds that
of those involved in in-migration.
In any event, we have computed the correlation coefficients between these three definitions, and they are given in Table A.4. In summary, the correlations are relatively low,
but there is a closer match between the M and M 1 definitions.
A.3
The Relative wage
In calculating the relative wage, the theoretically appropriate variable to use is labor
earnings in the farm sector relative to labor earnings in the non-farm sector. Because this
data does not exist, we compute the relative wage as follows.
The Farm wage rate.—In the absence of reliable labor earnings from farm production,
16
Table A.4:
Correlation Matrix for M, M1 and M2
M
M1
M1
M2
.475
.376
.424
Source: Authors’ calculations as explained in the text.
we used the “farm wage rate per month with house” (series K179). There are several
issues involved with these data. First, farm operators heavily rely on own and unpaid
family labor, and only about a fourth of total farm labor is hired. Due to well-known
monitoring problems in agriculture, the wage rates for farm workers are likely to be lower
than the return to family labor. An alternative would be to infer the return to family labor
from the net earnings of farm operators. However, since farm income includes the return
on land and farm machinery and equipment, and measuring the revenue share of these
inputs is very difficult, we did not pursue this approach. Schultz (1953, pp.101–02) reports
that during the first half of the twentieth century, the ratio between net farm income per
family worker and wage income per hired farm worker has stayed fairly constant, except
during the war years (when the net farm income increased relative to the wage income of
hired farm workers), and during the periods 1920–1923 and 1930–1933 (when the relative
net farm income fell).
Second, wage data, especially from the earlier part of the twentieth century is highly
unreliable. There are numerous measurement problems. First, during our sample period
agricultural wages typically included either room and board or house, as well as some “in
kind” payments. A comparison of data on (daily) wage rates across different payment
schemes – i.e., with room and board, with house (no meals) and with no room and
board – suggests that implied valuation of meals is about 15 to 25 percent of actual wage
(especially in the early periods) and that of housing is about 20 percent.
The Non-farm wage rate.—In the absence of comprehensive non-farm wage data, most
authors (including us) use manufacturing wages to compute the relative farm wage rate.
Three problems stand out. First, average skill levels across the agricultural and manufacturing sectors may differ. Using manufacturing wages for “lower skilled” labor may
only partly address some of these problems. Second, job creation and destruction rates
may vary across sectors, leading to sectoral variations in (frictional) unemployment, which
is something we do not model. Third, a manufacturing job, even with the lowest skill
requirements, is not the only alternative to a farm job.
The non-farm wage data are constructed as follows. From 1890 to 1920, we used “lower
skilled labor, full time weekly earnings” (series D778), and from 1921 to 1970, we used
“average manufacturing wage per week” (series D804), both multiplied by four to convert
17
them to monthly earnings. As an alternative we have also computed the manufacturing
wage rate by “average weekly hours worked” times “average hourly pay” (series D802,
D803). These series are only available after 1914. We find that these three series have a
very high correlation (above .98) during the period (1914–1920) when the data overlap.
Another difficulty involves converting these nominal wage gaps into real wage gaps
by adjusting for the farm-urban cost of living differential. Willamson and Lindert (1980,
p.1̃21) observe that the standard benchmark estimate, by N. Koffsky, is about 25 percent
for the year 1941. Furthermore, their own estimates (Appendix H) suggest that this
differential has remained relatively constant over our sample period.
Caselli and Coleman (2001) further discuss the problems encountered in estimating
farm-city wage gaps and conclude that different data sources yield conflicting results. Our
empirical analysis should thus be interpreted under the caveat that we lack good quality
high-frequency relative labor earnings data.
A.4
Expenditures on Food and Non-Food
Personal consumption expenditures on all items and food (current dollars), and implicit
price deflator for final sales of domestic product for 1929–91 are obtained from the website
of Bureau of Economic Analysis (www.bea.org, downloaded on 5/21/03). All these series
are based on national income and product accounts (NIPA). To estimate the growth rate
of expenditures on food for the period 1900-1929, we used the labor productivity growth
rate in the farm sector (see below) minus the growth rate in non-farm employment share.
A.5
The Share of Food Expenditure
The share of food in total expenditures is taken from D. Costa (2001), and is based
on NIPA (for 1929–91). Note that given our stylized theoretical model, these series are
preferable to alternatives such as the share of farm products in national income, which
includes investment in fixed capital and government services. Schultz (1953, Tables 5–6
and 5–7) also provides estimates for expenditure share of farm products which share the
same downward trend but are not in fact comparable to Costa’s series (see Table A.5).
A.6
Relative Farm Productivity Growth
Our primary variable of interest is labor productivity because our model does not include
other fixed or quasi-fixed factors of production. Section B.3 discusses our method of imputing relative productivity. Here we discuss the alternative measures that are available.
First, we used indices of employee output in the total private economy based on “farm
and non-farm output per man-hour” (Series D683–688, columns 684, 686) to compute the
relative sectoral productivity growth rate (defined as the productivity growth rate in the
non-farm sector minus that of the farm sector). There are two disadvantages associated
18
Table A.5:
Expenditures on Farm Products as a Percent of National Income, %
1870
1880
1890
1900
1910
1922
1925
1929
1934
1937
1939
34
32
22
17
19
16.1
15.4
13.4
12.8
13.7
11.6
Note: Expenditures on farm products are adjusted for agricultural exports and imports.
Source: Schultz (1953), Table 5–6 and Table 5–7, p.66, and p.67.
with these series: (i) they are indices and both productivity series are set to 100 in 1958,
and (ii) they end in 1966. Therefore, to estimate the trend productivity growth rate, we
first took the log ratio of these series.
Second, to estimate the parameters of equation (12) we specified its empirical counterpart:
d log z = αdt + σdω.
We estimated the mean α (α̂) and standard deviation σ (σ̂) of the log of relative productivity using maximum likelihood estimates:
µ
¶
T
1X
z(t)
α̂ =
ln
,
T 1
z(t − 1)
·µ
¶
¸2
T
1X
z(t)
σ̂ =
ln
− α̂ .
T 1
z(t − 1)
Note that α̂ ≡ µ̂ − 12 σˆ2 , where the µ̂ is the empirical counterpart of the drift parameter
in equation (12). Table A.6 shows the results. Although the estimates are subject to
qualifications, they show the acceleration of productivity growth in the agricultural sector
after the 1920s relative to productivity growth in the non-agriculture sector. Furthermore,
the estimates are broadly consistent with out imputed series. Evidently, both the increased
mechanization of U.S. agriculture starting in the early 1920s and the “chemical revolution”
of the 1950s-60s partly account for these trends.
19
Table A.6:
Estimates of Non-farm versus Farm Relative Productivity Growth
Period
1889–1966
1900–1966
1920–1966
Mean (α)
Standard Deviation (σ)
Drift (µ)
-0.0007
-0.0026
-0.0111
0.0782
0.0789
0.0645
0.0023
0.0005
-0.0090
Source: Authors’ calculations based on data from the Historical Statistics of the United States,
Series D683-688.
Admittedly, there is some uncertainty surrounding the exact magnitude of long-run
agricultural productivity growth relative to non-farm productivity growth. However, the
findings of the existing total factor productivity (TFP) literature are comparable to our
estimates. For example, in their calibrations of the U.S. economy from 1880 to 1990,
Caselli and Coleman (2001, p.6̃14) use double the value of the non-agricultural TFP
growth rate as an estimate of the agricultural productivity growth rate (which roughly
corresponds to a .8 percentage point gap in productivity growth per annum). Their
numbers are in fact smaller than those estimated by Jorgenson and Gallop (1992), which
are about 1.2 percentage points in favor of agriculture between 1947 and 1985. For the
period 1949–79 Jorgenson, Gallop, and Fraumeni (1987, Table 9.3 and Table D.1) also
give estimates for aggregate and agricultural TFP growth, which are respectively 1.5 and
0.8 percent.
In addition, according to data reported by Mundlak (2000, Figure 1.11), from 1960 to
1992, the growth rate of labor productivity in agriculture exceeded that of non-agriculture
in about 80 percent of the countries in a sample of 88 observations. The median value
by which the growth rate in average agricultural labor productivity exceeded that of
manufacturing was 1.58 percentage points [see also Mundlak (2000, p.3̃88)].13
A.7
Agriculture’s Share in National Income
For the period 1890 to 1959, we used Kendrick (1961, p. 298-301, table AIII), and Series
F126–128 in Historical Statistics. These refer to gross domestic product originating from
private farm sector divided by the sum of farm and non-farm sectors. For 1960–1989 we
used Series F226-227, which correspond to the share of agriculture, forestry, and fisheries
in national income.
13
Syrquin (1988), on the other hand, argues that the long-run total factor productivity trend has
favored industry, at least in some industrial countries. But, he does not discuss the restrictions this
imposes on demand for agricultural products and on sectoral labor flows.
20
A.8
The Relative Price of Food to Non-Food Items
The Bureau of Labor Statistics (BLS) website (http://www.bls.gov/data/home.htm) contains a CPI series of food prices (Series CUUR0000SAF1) for the period 1913 to 2002.
However, the CPI series for All Items less Food (Series CUUR0000SA0L1) given by the
website only covers the period 1935 to 2002. To extend the data for this category back
to 1913, we first obtained the series that comprise all non-food items and their weights in
the All Items less Food series.
The weights are given in Table 113 of the 1983 Handbook of Labor Statistics published
by the BLS. We use the earliest available weights (for the period 1935-39) which are as
follows for the All Items series: Food and Beverages (35.4%), Housing (33.7%), Apparel
and Upkeep (11.0%), Transportation (8.1%), Medical Care (4.1%), Entertainment (2.8%),
and Other Goods and Services (4.9%). The 1950 Handbook of Labor Statistics provides
the CPI data for Housing and Apparel back to 1913, but does not provide the other
series. However, the series Miscellaneous is defined by the 1950 Handbook (p. 97) as
including: transportation, medical care, household operation, recreation, personal care,
etc. We thus impute a weight for the category, Miscellaneous, of 30.8% by adding the
weights (given above) for Transportation, Medical Care, Entertainment, and Other Goods
and Services. Using the 1950 Handbook series for Miscellaneous (from Table D-1), the
1983 Handbook series for Residential and Apparel and Upkeep (from Table 110), and
the weights for Housing, Apparel and Upkeep, and Miscellaneous described above, we
calculated a series for All Items less Food from 1913 to 1950 (note that all series were
re-calibrated to the same base year). For the period of overlap (1935-1950) between this
series and the one given on the BLS website, the correlation coefficient was .9986. Using a
minor scaling factor, we spliced the constructed series (through 1934) to the BLS website
series (1935-2002) to obtain a full time series from 1913 to 2002 for All Items less Food.
B
B.1
The Solution of Frictionless Model
Expenditure and labor shares
To derive the expenditure and labor share equations given in the text, we first distinguish between aggregate consumption CA , CM and the quantity demanded by worker i,
cA (i), cM (i). Then using equation (1):
µ
¶ µ ¶ν
η
cM (i)
1
=
.
cA (i) − γA
1−η
p
Let:
½
j
w (i) =
wM if i ∈ [0, LM )
.
wM if i ∈ (LM , 1]
21
The budget constraint is:
wj (i) = pA cA (i) + pM cM (i).
Solving for the demand for farm goods gives:
³ ´
η
wj /pA + γA 1−η
p1−ν
³ ´
cA (i) =
.
η
1−ν
1 + 1−η p
The market clearing equations are:
Z 1
cA (i)di = CA = zA (1 − LM ),
0
Z 1
cM (i)di = CM = zM LM ,
0
zA (1 − LM ) + pzM LM = pLM wM + (1 − LM )wA = Y.
The last line is real output measured in farm-goods prices. Noting that:
³ ´
³ ´




η
η
wM /pA + γA 1−η
p1−ν
wA /pA + γA 1−η
p1−ν
 + (1 − LM ) 
,
³ ´
³ ´
CA = LM 
η
η
p1−ν
p1−ν
1 + 1−η
1 + 1−η
and using the definition of real output measured in units of farm goods Y , as well as the
expenditure share of farm goods:
θA =
we obtain:
CA
CA
=
CA + pCM
Y
³
η
1−η
´
Y + γA
p1−ν
³ ´
CA =
.
η
1 + 1−η
p1−ν
Consequently, we can rewrite the expenditure share of farm goods:
·
µ
µ
¶¸−1
¶³ ´
η
γA
w 1−ν
.
θA = 1 +
1−
1−η
z
CA
(B.1)
(B.2)
To determine LM , we start with equation (B.1) for aggregate farm good consumption:
·
µ
¶¸
¶
µ
η
γA
1−ν
Y = CA 1 +
p
1−
1−η
CA
22
We use the market clearing condition CA = zA (1 − LM ), the definition of Y = zA (1 −
LM ) + pzM LM , and w = zp to obtain the expression for LM given in equation (10):
·
µ
¶
¸−1 µ
¶
1 − η ³ w ´ν
γA
LM = 1 + z
1−
.
(B.3)
η
z
zA
When w = 1, we obtain equation (6).
B.2
The Rate of Structural Change
Here we derive the rate of labor reallocation using our alternative measure LM /(1 − LM ),
which we used in empirical analysis. In the case of frictionless equilibrium, equation (6)
implies:
³ ´
z 1−ν
(γA /zA ) + 1−η
η
f
³ ´
1 − LM =
,
1−ν
1 + 1−η
z
η
and therefore:
LfM
1−
LfM
=
1 − γA /zA
³ ´
.
1−ν
z
γA /zA + 1−η
η
Consequently,
Ã
d ln
LfM
1 − LfM
!


³
1−η
η
´
(1 − ν)
z −ν
γA 
1
1

³ ´
³ ´
=
dzA −
dz
+
γA
1−η
zA zA − γA γ + z 1−η z 1−ν
1−ν
+
z
A
A
η
zA
η
³ ´
³ ´


γA
1 + 1−η
z 1−ν
(1 − ν) 1−η
z 1−ν dz
η
η
dz
A
zA
´  h³
´
i
³ ´
= ³
−
γA
γA
1−η
γA
z
1−ν
A
1 − zA
+ η z
+ 1−η
z 1−ν z
zA
zA
η
µ
¶·
¸
·
¸
%
1 + G(z) dzA
G(z)
dz
=
− (1 − ν)
.
1−%
% + G(z) zA
% + G(z) z
where:
γA
%=
,
zA
µ
and
G(z) =
1−η
η
¶
z 1−ν .
The rate of structural transformation over time is again made up of two contributions:
[%/(1 − %)] [(1 + G(z))/(% + G(z))] × µA − (1 − ν) [G(z)/(% + G(z))] × µ .
|
|
{z
}
{z
}
absolute farm prod. growth
relative farm prod. growth
23
This measure has the following nice properties: if γA = 0, then the rate of structural
change only depends on µ(1 − ν). Even in the case of subsistence consumption of agricultural goods, i.e., γA 6= 0, as long as % → 0 over time (which occurs eventually since γA
is constant and zA grows on a positive trend), only relative productivity growth matters
for structural change.
B.3
Calibrating Labor Shares
To calibrate labor shares implied by different versions of our model we need (i) estimates
of ν, η, and γA /CA ; and (ii) an estimate of the level of relative productivity, z. We discuss
our choices in turn.
ν: We chose a low elasticity of substitution, ν = .1. This estimate is consistent
with that reported in Brown and Heien (1972). Also, we are currently working
on estimating this parameter.
η: We set this parameter equal to .8.
γA /CA : Our calibration results are sensitive to the value of γA /CA at the beginning
of our sample period. So, we initialized it at either 0, 0.5 or 0.7. The first
choice corresponds to no subsistence consumption. In the latter two choices,
this ratio varies over time. We calculated its growth rate by (minus) the
growth rate of real food consumption.
z values: Our calibration results are sensitive to the level of relative productivity
z. Although index-number-based relative productivity data (see Appendix A)
can be used to obtain the drift and standard deviation of the productivity
growth rate, these series arbitrarily set z = 1 in the base year. In order to
approximate the level of z, we begin with the assumption that, by 1991, farm
income and labor shares (Fig. 1) and farm–non-farm wages had all converged
to their long-run levels (see Caselli and Coleman, 2001). Thus, we use the
actual LM at the end of our sample period to “back out” the corresponding
equilibrium level of z (z ∗ ):
·µ
¶µ
¶µ
¶¸1/(1−ν)
γA
1 − LM
η
∗
z =
1−
.
CA
LM
1−η
Using the share of food in total expenditure, θA , in equation (B.2), we computed the implied relative productivity series (z̃) from for the baseline model
with γA = 0:
¶¸1/(1−ν)
·µ
¶µ
η
θA
.
z̃ =
1−η
1 − θA
24
We scaled the series to ensure that it converged to z ∗ at the end of the sample,
and used them in equations (6) and (10) to calculate the non-farm labor shares.
In sum, our calibrated series of LM only use actual growth rate of expenditures on
food and share of food in total expenditures.
C
Solution with Fixed Costs
This section presents the derivations for the relocation thresholds. Our arguments are
essentially identical to those in Dixit and Rob (1994, especially pp. 60–66), with the
difference being that they consider the case θ 6= 1 and ν = 1, whereas we solve for θ = 1
and ν 6= 1 (see also their Fig.1̃).
In what follows, for ease of exposition, we refer to the non-farm sector as “city” and
the farm sector as “farm”, and consider U∆ and UO in turn.
First consider U∆ (z, LM ), which is the present discounted utility of consumption differentials assuming that the worker will never reallocate in the future. If this expression is
positive the worker has an instantaneous incentive to switch from F to N . For an initial
level of productivity, z0 , and for a given LM (since each worker takes this as a constant
in competitive equilibrium), we have:
¸
·Z ∞
−ρt
e ln w(z, LM ) dt .
(C.4)
U∆ (z, LM ) = E
0
The expression for ln w is given in equation (11).
As given by Harrison (1985, pp. 44–45), the expected discounted value of ∆ is the
solution U∆ (z, LM ) to the following differential equation:
ρU∆ (z, LM ) − µz
h
Let γ = 1 −
γA
zA (1−LM )
∂U∆ (z, LM ) σ 2 2 ∂U∆ (z, LM )2
− z
= ∆(z, LM ).
∂z
2
∂ 2z
i
, and assume that:
dγ
= µγ dt + σγ dωγ ,
γ
where µγ > 0. Then, solving equation (C.4) for the function U∆ , we obtain:
µ ¶·
¸ µ
¶µ
¶
1
1
ν−1
σ2
ν−1
U∆ (z0 , LM ) =
ln z0 + (ln B(LM ) + Γ) +
µ−
,
ρ
ν
ν
νρ2
2
where z0 is the fixed initial level of productivity, and
µ
¶µ
¶
¶
µ
1 − LM
η
1
1 2
B(LM ) =
, Γ = ln γ +
µγ − σγ .
LM
1−η
ρ
2
25
(C.5)
Here we have assumed that γ and w are uncorrelated. Of course, this assumption is
unrealistic in our case but simplifies the algebra. Note that γ in the last expression is
evaluated at the fixed initial values of zA and LM , and the (absolute) value of Γ tends to
decline over time.14 When the subsistence consumption parameter γA = 0, Γ = 0, and we
obtain the results given in the case with no subsistence consumption.
Note that U∆ is calculated on the basis of a permanent relocation to the other sector.
However, farm return migration is a well-documented phenomenon which we wish to allow
for. This entails a tradeoff of options. When migration actually takes place, the worker
gives up the “option to stay” in the sector of origin and in return acquires the “option to
stay” in the destination sector. The first derives its value from waiting before engaging
in a costly move, and the second derives its value from the possibility of switching back.
In each period, a rational worker considers the net value of these two options, as well as
U∆ , so that the thresholds are actually higher than they otherwise would be if the worker
acted in a myopic way and ignored the value of waiting.
We define UO as the option of staying in the non-farm sector minus the option of
staying in the farm sector, and we value it as a non-dividend paying “asset” measured
in utility terms, whose value depends purely on the “capital gains” that may result from
fluctuations in z. Over a time interval dt, the expected return on this net option (ρUO )
is equal to the expected capital gain on the option:
ρUO (z, LM ) =
1
E [dUO (z, LM )] .
dt
We use Itô’s Lemma to expand the right hand side of this equation and obtain:
ρUO (z, LM ) − µz
∂UO (z, LM ) σ 2 2 ∂UO (z, LM )2
− z
= 0.
∂z
2
∂ 2z
The general solution to this equation takes the form [see, e.g., Dixit and Pindyck (1996,
pp.1̃40–144)]:
UO (z, LM ) = K1 (LM )z β1 + K2 (LM )z β2 ,
(C.6)
where K1 and K2 are constants to be determined, and β1 > 0 and β2 < 0 are the roots of
the quadratic equation:
Q(β) ≡
σ2
β(β − 1) + µβ − ρ.
2
(C.7)
Together equations (C.5) and (C.6) give the total utility gains associated with migration. Our next task is to evaluate these terms at the relocation thresholds ZM and ZA ,
and find analytic expressions for them.
14
To ensure economically plausible results, we will assume that two times the trend decline in
subsistence–consumption ratio exceeds its volatility, µγ > .5σγ .
26
C.1
Migration from Farm to City
For a worker considering whether to migrate from the farm to the non-farm sector, the
present discounted value of the wage gap between the non-farm and farm sector (ln(wN )−
ln(wA )) plus the net option value of migrating at ZM is:
U∆ (ZM ) + UO (ZM ) = c.
The first term is given by:
¶
µ ¶·
¸ µ
¶µ
ν−1
1
σ2
1
ν−1
.
U∆ =
ln ZM + (ln B(LM ) + Γ) +
µ−
ρ
ν
ν
νρ2
2
For the option value of migration our choice of which root to use depends on the value of
ν. Consider the plausible case where ν < 1. For farm-to-city migration the net options
value is the value of the option to stay in the city (destination) sector minus the value of
waiting in the farm (origin) sector (which is forfeited in the event of a move).
For ν < 1, within the zone of inaction, a higher value of z should reduce the option
value of waiting to switch to the non-farm sector: the combination of higher non-farm productivity and low elasticity of substitution between farm and non-farm products depresses
the prices of the non-farm sector, hence relative non-farm wages, and makes relocation
less likely. As the value of z approaches the threshold ZM , the option of waiting, which
is to be given up, increases in value, so:
lim UO < 0.
z↓ZM
Setting K1 = 0, and using the negative root, β2 , satisfies this.
The two optimality conditions can now be stated. Value-matching:
¸ µ
¶µ
¶
µ ¶·
ν−1
ν−1
1
1
σ2
β2
ln ZM + (ln B(LM ) + Γ) +
µ−
+ K 2 ZM
= c.
2
ρ
ν
ν
νρ
2
and smooth-pasting:
µ
ν−1
νρZM
¶
β2 −1
+ K 2 β 2 ZM
= 0.
Solving these for ZM , the threshold value for off-farm migration gives:
¾
½
1
[ln B(LM ) + Γ] ,
ZM = exp s +
1−ν
where K2 < 0, and
¶
1
σ2
−µ +
< 0,
2
β2
sµ
¶2
µ
1
2ρ
1
µ
− 2−
−
+
< 0.
=
2 σ
σ2 2
σ2
νρc
1
s =
+
ν−1 ρ
β2
µ
27
C.2
Migration from City to Farm
The relevant move is from city to farm. We can use the same logic as above to find:
U∆ (ZA ) + UO (ZA ) = −c.
At the relocation threshold, the present discounted value of the wage gap (ln w) is:
µ ¶·
¶
¸ µ
¶µ
1
ν−1
1
σ2
ν−1
U∆ =
.
ln ZA + (ln B(LM ) + Γ) +
µ−
ρ
ν
ν
νρ2
2
We continue to consider the case ν < 1. The relevant move is from city to farm, and the
net options value is the value of waiting in the city (origin) minus the value of staying in
the farm (destination).
For ν < 1, within the zone of inaction, as z decreases, the value of waiting in the city
should decrease due to a less favorable relative farm wage. As the value of z approaches
the threshold ZA , the option of waiting, which is to be given up, increases in value:
lim UO > 0.
z↑ZA
Setting K2 = 0 and using the positive root β1 satisfies this.
The relevant threshold ZA can be determined using the two optimality conditions.
The value-matching condition:
µ ¶·
¸ µ
¶µ
¶
1
ν−1
ν−1
1
σ2
ln ZA + (ln B(LM ) + Γ) +
µ−
+ K1 ZAβ1 = −c.
2
ρ
ν
ν
νρ
2
and smooth-pasting:
µ
ν−1
νρZA
¶
+ K1 β1 ZAβ1 −1 = 0.
Combining the two gives the threshold value for on-farm migration as:
½
¾
1
ZA = exp S +
[ln B(LM ) + Γ] ,
1−ν
where K1 > 0 and
¶
1
σ2
−µ + ,
2
β1
sµ
¶2
µ
1
2ρ
1
µ
− 2+
−
+ 2 > 1.
=
2
2 σ
σ
2
σ
νρc
1
S =
+
1−ν ρ
β1
µ
Substituting the expressions for ZM and ZA in the wage gap expression (11) gives the
rules shown in the text.
28
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30
Non-farm Relative Productivity Growth Rate
Farm Employment and Income Share, %
1
0.20
0.8
0.10
Employment
0.6
0.00
0.4
0
1800
-0.10
Income
0.2
1820
1840
1860
1880
1900
1920
1940
1960
1980
-0.20
1890
1900
1910
1920
1930
1940
1950
1960
Off-farm Migration Rate, %
Non-farm to Farm Relative Wage
15
3
10
2.5
M, population based
5
2
0
1.5
-5
On-farm migration
1
1893
1903
1913
1923
1933
1943
1953
1963
-10
1901
M1, employment based
1911
Fig. 1: US Farm Data
1921
1931
1941
1951
1961
Fig. 2: Expenditure Share of Food, %
40
All food
35
30
25
20
Food at home
15
10
5
0
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
Fig. 3: Share of Non-farm Labor, w =1
1.0
0
0.9
0.8
.5
.7
Actual
0.7
initial gammaA/CA= 0, .5, .7
0.6
0.5
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990