I..SJTls                                               LSM -73
AUG. 1990
Living Standards
Measurement Study
Working Paper No. 73
Shadow Wages and Peasant Family Labor Supply
An Econometric Application to the Peruvian Sierra
Hanan G. Jacoby



LSMS Working Papers
No. 4      Towards More Effective Measurement of Levels of Living, and Review of Work of the United Nations
Statistical Office (UNSO) Related to Statistics of Levels of Living
No. 5      Conducting Surveys in Developing Countries: Practical Problems and Experience in Brazil, Malaysia,
and the Philippines
No. 6      Household Survey Experience in Africa
No. 7      Measurement of Welfare: Theory and Practical Guidelines
No. 8      Employment Data for the lMeasurement of Living Standards
No. 9      Income and Expenditure Surveys in Developing Countries: Sample Design and Execution
No. 10     Reflections on the LSMS Group Meeting
No. 11     Three Essays on a Sri Lanka Household Survey
No. 12     The ECIEL Study of Household Income and Consumption in Urban Latin America: An Analytical
History
No.13      Nutrition and Health Status Indicators: Suggestions for Surveys of the Standard of Living in
Developing Countries
No. 14     Child Schooling and the Measurement of Living Standards
No. 15     Measuring Health as a Component of Living Standards
No. 16     Procedures for Collecting and Analyzing Mortality Data in LSMS
No. 17     The Labor Market and Social Accounting: A Framework of Data Presentation
No. 18     7ime Use Data and the Living Standards Measurement Study
No. 19     The Conceptual Basis of Measures of Household Welfare and Their Implied Survey Data Requirements
No. 20     Statistical Experimentation for Household Surveys: Two Case Studies of Hong Kong
No. 21     The Collection of Price Data for the Measurement of Living Standards
No.22      Household Expenditure Surveys: Some Methodological Issues
No.23      Collecting Panel Data in Developing Countries: Does It Make Sense?
No.24      Measuring and Analyzing Levels of Living in Developing Countries: An Annotated Questionnaire
No.25      The Demand for Urban Housing in the Ivory Coast
No.26      The C6te d'Ivoire Living Standards Survey: Design and Implementation
No.27      The Role of Employment and Earnings in Analyzing Levels of Living: A General Methodology with
Applications to Malaysia and Thailand
No.28      Analysis of Household Expenditures
No. 29     The Distribution of Welfare in Cote d'Ivoire in 1985
No. 30     Quality, Quantity, and Spatial Variation of Price: Estimating Price Elasticities from Cross-Sectional
Data
No.31      Financing the Health Sector in Peru
No.32      Informal Sector, Labor Markets, and Returns to Education in Peru
No.33      Wage Determinants in C6te d'Ivoire
No.34      Guidelines for Adapting the LSMS Living Standards Questionnaires to Local Conditions
No. 35     The Demand for Medical Care in Developing Countries: Quantity Rationing in Rural C6te d'Ivoire
No.36      Labor Market Activity in C6te d'Ivoire and Peru
No.37      Health Care Financing and the Demand for Medical Care
No.38      Wage Determinants and School Attainment among Men in Peru
No.39      The Allocation of Goods within the Household: Adults, Children, and Gender
(List continues on the inside back cover)



Shadow Wages and Peasant Family Labor Supply
An Econometric Application to the Peruvian Sierra



The Living Standards Measurement Study
The Lving Standards Measurement Study (LSMS) was established by the
World Bank in 1980 to explore ways of improving the type and quality of house-
hold data collected by statistical offices in developing countries. Its goal is to foster
increased use of household data as a basis for policy decisionmaking. Specifically,
the L1MS is working to develop new methods to monitor progress in raising levels
of living, to identify the consequences for households of past and proposed gov-
ernment policies, and to improve communications between survey statisticians, an-
alysts, and policymakers.
The LSMS Working Paper series was started to disseminate intermediate prod-
ucts from the LSMS. Publications in the series include critical surveys covering dif-
ferent aspects of the LSMS data collection program and reports on improved
methodologies for using Living Standards Survey (LsS) data. More recent publica-
tions recommend specific survey, questionnaire, and data processing designs, and
demonstrate the breadth of policy analysis that can be carried out using LSs data.



LSMS Working Paper
Number 73
Shadow Wages and Peasant Family Labor Supply
An Econometric Application to the Peruvian Sierra
Hanan G. Jacoby
The World Bank
Washington, D.C.



Copyright © 1990
The International Bank for Reconstruction
and Development/THE WORLD BANK
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First printing August 1990
To present the results of the Living Standards Measurement Study with the least possible delay, the
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ISSN: 02534517
Hanan G. Jacoby is assistant professor in the Department of Economics at the University of Rochester.
Library of Congress Cataloging-in-Publication Data
Jacoby, Hanan.
Shadow wages and peasant family labor supply: an econometric
application to the Peruvian Sierra / Hanan G. Jacoby.
p. cm.-(LSMS working paper, ISSN 0253-4517; no. 73)
Includes bibliographical references.
ISBN 0-8213-1637-0
1. Peasantry-Peru-Econometric models. 2. Self-employed-Peru-
Econometric models. 3. Agricultural laborers-Peru-Supply and
demand-Econometric models. 4. Agricultural wages-Peru-
Econometric models. 5. Households-Peru-Econometric models.
I. Title. II. Series.
HD1531.P4J33 1990
331.12'5'0985-dc2O                                   9043877
CIP



ABSTRACT
A striking feature of developing economies is the large proportion of the
work force that is self-employed. Lack of widespread labor market participation
in the agricultural sector can pose a major obstacle to the empirical
implementation of economic models of the peasant household, whether because wage
data are simply not available, or because the assumptions required to make use
of wage data stretch the bounds of credulity. This paper develops a methodology
for estimating a structural labor supply model for primarily self-employed
peasant households, which holds under a general agricultural technology and set
of labor market conditions.   The unique feature of the approach is that the
opportunity cost of time, or shadow wage, of household workers is explicitly
estimated from an agricultural production function and is subsequently used to
identify a set of structural labor supply parameters. Recent household survey
data form rural Peru is employed to estimate and perform various diagnostic tests
on the model. The empirical findings lend support to the hypothesis that peasant
households allocate their members' time as if to maximize a family utility
function, aiid, moreover, demonstrate the tractability of the shadow wage
methodology and its usefulness in estimating more elaborate time allocation
models.
v



ACKNOWLEDGENENTS
I wish to thank my thesis advisors James Heckman and Joe Hotz, and the
Welfare and Human Resources Division of the World Bank for their support.
vi



INTRODUCTION
I.      Introduction  . . . . . . . . . . . .1
II.      The Theoretical Framework ....... .   .      ... 5
A.  The basic recursive model . . . . . . . . . . . . . 5
B.  A shadow wage model . . . . . . . . . . . . . . . . 8
III.      The Empirical Strategy       .      .      ....... . 13
A. The estimating model and the stochastic
environment .................. . 13
B.  Identifying the model . . . . . . . . . . . . . . . 15
IV.      Estimation of the Agricultural Technology . . . . . . . . 17
A.  The data  ....  .  .  .  .  .  .  .  .  .  .  .  .   i .  .  .   . 17
B.  Estimates of Cobb-Douglas production function . . .21
C.  Estimates of a translog production function . . . . 25
V.      Labor Supply Estimation ....  .  .  .  .  .  .  .  .  .  .  .  .   . 28
A.  The specification of labor supply . . . . . . . . . 28
B.  Labor supply estimates with the Cobb-Douglas
technology .................. . 33
C.  Labor supply estimates with the translog
technology  ....  .  .  .  .  .  .  .  .  .  .  .  .  .  .   . 38
VI.      Conclusions ....  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   . 41
References            .................. . 43
LIST OF TABLES
Table 1    Definitions, Means and Standard Deviations of Production
Function Variables and Additional Instruments . . . . . 18
Table 2    Production Function Estimates   . . . . . . . . . . . . . 22
Table 3    Tests of the Equality of Wages and Marginal Product
for Labor Market Participants: Instrumental Variables
Estimates .29
Table 4    Characteristics of Men and Women Working on Family
Farms .31
Table 5    Estimates of Log-Linear Male and Female Labor Supply
Functions .34
Table 6    Estimates of Labor Supply Parameters with Varying
Instrument Sets and Tests of Overidentifying
Restrictions    .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  38
vii



I. INTRODUCTION
A striking feature of developing economies is the typically large
proportion of the work force that is not primarily engaged in wage labor.
Self-employment is particularly pervasive in agriculture, where the dominant
unit of production is the family farm.   Lack of widespread  labor market
participation can pose a major obstacle to the empirical implementation of
economic models of the peasant household.   Such models are  important  in
evaluating the effects of policies directed at this poorest segment of the
population.    Schooling,  migration  and  fertility  choices,  and  even  the
allocation of food among household members, are thought to depend upon current
or future opportunity costs of time (see, e.g., Rosenzweig and Evenson 1977
and Rosenzweig and Schultz 1982). For labor market participants this cost is
just their wage rate, which is usually taken as exogenous to the individual in
empirical studies.   The absence of wages for the self-employed complicates
even a basic study of their labor supply behavior, not to mention the
estimation of more elaborate time allocation models.
This paper develops a general methodology for estimating a structural
labor  supply  model  for  an  agricultural  household  whose  members  are
self-employed, and applies it to household survey data from rural Peru, where
self-employment is the norm.  The unique feature of the approach is that the
opportunity cost of time, or "shadow wage", of individuals working exclusively
on their own farm is determined from within the household, rather than by
market forces.   The endogeneity of shadow wages is presumed to result either
from the imperfect substitutability of various labor inputs in the production
process, or from transaction costs or other frictions in rural labor markets.
1



Thus, the methodology proposed here is not burdened by the unpalatable
assumptions of previous studies of the peasant household.
The methodology is closely related to the treatment of labor supply with
progressive income taxes (e.g., Hall 1973).   A concave budget constraint,  in
the form of an agricultural production function, is "linearized" at the
household's optimum to obtain a set of shadow wages for different workers. At
the equilibrium point, the shadow wage of a particular type of worker is just
the marginal product of their labor in agriculture.   Along with equilibrium
farm profits, these shadow wages map out the structural labor supply
parameters of each worker.   The empirical strategy,  therefore,  is to first
estimate an agricultural production function, which has different types of
labor (that of men, women, children and hired workers) as distinct inputs.
Since this study focuses on the labor supply of men and women, the marginal
product of their farm labor is calculated for each household.   Using these
marginal products in place of wages, labor supply functions for men and women
are estimated which are comparable to those of the neoclassical family labor
supply model (e.g., Ashenfelter and Heckman 1974). The set of restrictions on
the parameters of these labor supply functions implied by utility theory are
then tested.
Rosenzweig  [1980]  is  the  only  previous  attempt  to  confront  the
neoclassical family labor supply model with data from a developing country.
Under the assumption that the labor market in rural India is efficient,
Rosenzweig uses the wages of men and women to estimate their joint supply of
time to the market.   One criticism of his approach is that in testing the
signs of market wage elasticities, the controversial hypothesis of efficient
rural labor markets is inevitably being tested along with the implications of
2



utility maximization. But, perhaps more seriously, labor market participation
in rural India, especially among women, is quite limited.  So, even if labor
markets are efficient, more assumptions are required before sex-specific
market wages can be equated to the opportunity cost of time of all men and
women in the sample.
Rosenzweig's  model  is  essentially  an  elaboration  on  the  highly
influential consumer-producer agricultural household models exemplified in
the studies by Lau, Lin and Yotopoulos [19781 and Barnum and Squire [1979].
The empirical tractability of these models derives from their invocation of
the Separating Hyperplane Theorem, giving them a "recursive" property.   The
recursive property says that farm profit can be maximized independently of
preferences, and utility can then be maximized taking profit as given.   This
two-stage maximization problem implies an aggregate labor supply function for
the family that depends only on a single market wage and on farm profit,
regardless of the household's position in the labor market. The cost of this
convenience, however, beyond maintaining labor market efficiency, is that the
labor of all family workers, and that of family and hired workers, must be
assumed perfectly substitutable  in agricultural  production.    Rosenzweig's
model implicitly allows the labor of adult males and females to be imperfectly
substitutable, but it still requires that perfect substitutes for the labor of
these family workers be available on the market.
A model of peasant family labor supply is called for that retains the
structure and tractability of these recursive agricultural household models,
but that eschews their stringent assumptions.1  An advantage of the proposed
ILopez [1984] is the only previous attempt to estimate a nonrecursive
3



model is that the implications of utility theory and the hypothesis of
efficient rural labor markets do not have to be tested jointly.  In addition,
labor is not restricted to be perfectly substitutable in agricultural
production, allowing relative variation in the shadow wages (i.e., marginal
products of labor) of different workers across households, so that the labor
supply parameters of these different workers can be compared.   Finally, with
the present approach, pure income effects on labor supply can be obtained
because farm profits capture the returns to quasi-fixed productive assets,
such as land.  In Rosenzweig's study no adequate measure of non-labor income
for poor households is available.
The next section of the paper fixes ideas by briefly spelling out the
assumptions and implications of the standard recursive agricultural household
model, before the more general shadow wage model is presented.   In section
III, the latter model is made operational and the identification issue is
addressed.    The next  two sections  report  the empirical  work,  with the
production function estimates in section IV and the labor supply estimates in
section V. Conclusions drawn from this study appear in the final section.
agricultural household model, where work on the family farm and market work
are  imperfectly substitutable  in utility and production.    His  model  is
estimated using aggregate Canadian data and relies on some strong flnctional
form assumptions,  including constant returns to scale in production.   Lopez
ignores heterogeneity of labor by sex and age of worker.
4



II. THE THEORETICAL FRAMEWORK
The basic recursive model
To understand the recursive nature of the agricultural household models
most often found in the literature, consider the following schematic version.
The household produces agricultural output, Y, according to the strictly
d
concave function F such that Y = F(L , A), where A is the fixed quantity of
land and L  is the total amount of labor employed (other variable inputs are
ignored without loss of generality).   Embodied in this production technology
is the restriction that all types of labor are perfectly substitutable; i.e.,
the marginal rate of transformation between these inputs is unity.
Normalizing the price of the consumption commodity, C, and setting it
equal to that of farm output, and letting W be the market wage, the
household's one-period budget constraint is written as
(1) C = Y - W Ld + W LS = f + W Ls.
where L  is the family's supply of both on and off-farm labor and E is farm
profit. Implicit in this constraint is the assumption that the price at which
the household can hire labor is equivalent to the price at which it can sell
2All asset accumulation decisions are presumed to have been made at an earlier
stage of the family's multi-stage budgeting process.
5



labor; i.e., the labor market is competitive and free of transactions costs.
The single price of time, W, means that the leisure of various household
members can be aggregated to form a Hicks composite commodity. Household
utility can thus be expressed as a function of total family consumption and
total family leisure, 2 = T - L , where T is the family's endowment of time.
In addition, a vector of taste shifters Z is introduced so that utility is
written  as  u=U(C,Y;Z).      The  familiar  first  order  conditions  for  the
household's utility maximization problem imply,
(2a)  U  / Uc =W
(2b) FL =W .
Conditions (2a) and (2b) describe the standard neoclassical separation
result or recursive property.   The production, or labor demand, decision is
determined solely by real factor costs (W), resource endowments (A), and the
technology,  entirely independently of preferences.    Meanwhile,  consumption
depends on production only through the level of profits; that is, via the
budget constraint. The recursive property becomes even clearer by setting up
the equivalent two-stage maximization problem,
(3)  Tn(W, A) = Max [F(L , A) - W L]
Ld
Max U(C,T-L ) s.t. C = n* + W Ls.
Ls
Here the maximized level of farm profits, 1I , plays the role of nonlabor or
property income in the pure consumer choice problem.
6



Solving the first order conditions yields explicit expressions for the
demand and supply of labor as functions of the exogenous variables,
d    d
(4)  L  =L (W, A)
(5)  L  = L (W, EI , Z).
Again, the essence of the recursive property is the absence of the taste
shifters, Z, from the labor demand equation, and the fact that no variables
from the production side of the model (fixed inputs or prices of variable
inputs, if any) enter the labor supply equation separately. An equation based
on (5) can be estimated straightforwardly by ordinary least squares, once the
profit function has been estimated.
Clearly,  the  assumption  that  labor  is  perfectly  substitutable  is
extremely useful, since it means that in an efficient labor market the going
wage is the price of time of all workers in the household, whether they are
selling their labor or not.  Recent work, however, has begun to question the
homogeneity of  labor  in peasant  agriculture.    Deolalikar  and  Vijverberg
[1987], for example, reject the hypothesis that family and hired labor are
perfect substitutes in a production function using two different LDC data
sets, and suggest that differences in the nature and timing of tasks performed
or in work incentives may be at the root of the finding.3
3Laufer [1985] finds evidence of a division of labor between male and female
workers in estimates of a flexible production function for different crops in
rural India, indicating imperfect substitutability of family labor.
7



A shadow wage model
This section considers a more general model in which there are two types
of workers (e.g., male and females, as in Rosenzweig 1980) whose time inputs
are imperfectly substitutable in production, and where hired labor is not a
perfect  substitute  for  family  labor.    Labor  heterogeneity  of  t -i     '
destroys the recursive property.   Of course, even if perfect substitutes for
each type of family worker are available for hire, transactions costs or other
frictions in the labor market may drive a wedge between market wage rates and
the marginal products of farm labor, and lead to a breakdown in recursiveness.
The latter eventuality is explored empirically in section V. In what follows,
the assumption of access to an efficient labor market is maintained, although
it is important to understand that the analysis applies equally well to a
household that is completely isolated from the labor market.
With labor heterogeneity, the general form of the agricultural production
function is Y = F(L1,L2,H1,H2,A), where H1 and H2 are the quantities of pac ,/h
type of labor hired by the household at wage rates WH and W1, respectively.2
Each family worker (i=1,2) allocates Ti hours between work on the family farm
(Li), work in the labor market (M,), housework (SI) and leisure (tI).  Workers
can choose not to participate in the labor market, so that M Z0,  but if they
I
do participate they earn a wage Wi, which is not necessarily equal to
Since off-farm wages W1 and W2 need not always move in unison, the family's
leisure no longer forms a composite commodity, so that u = U(C, 21P  2; Z).
4Another interpretation of this inequality constraint is that, in a poorly
functioning labor market, sometimes a worker is "rationed out"--unable to find
any work off the farm at all at the going wage.
8



In Peru, for example, women who work on the family farm also spend a
great deal of time in housework activities, such as cooking and cleaning (see
table 4 below). To account for this phenomenon, a concave production function
for housework services, Q, is postulated, of the form Q = t(Si, S2, I), where
I is a vector of fixed characteristics which affect housework production.
Each worker is presumed to engage in at least some housework, which is not
produced jointly with farm output.   It is also assumed that C and Q are
perfect substitutes in utility (Gronau, 1977, makes an identical assumption),
so that Q can be absorbed into the composite consumption good. Let r = C + eQ
be this new composite commodity, where 6 is the marginal rate of substitution
in utility between C and Q.
The family is assumed to solve
Max      U(    11 2  Z)   subject to
C,Q,Li,9Mi.,SiH IH
1  1  1  111                      2  1   
C  t' FLL29H1,H2,A) - WH1H  - W2H2 + WlM1+ W2M2
Q = t(ZS1,S2,I)
T        .+ Li+ Mi+ S.  and  M.2 0  i=1,2.
The corresponding Lagrangian is given by
(7)  U(C + eQ, T-L1-M --S1, T-L2-M2-S2  Z) +
.5[F(L ,L  H ,H  A) -WHH  2WHC +W
1  2' 1' 2'      1 1    2   + W1M1+ W2M   - C] +
p[C(sivs2,I) - Q] + AI1  +XM
U ( 2  wuL'  ac2L
2tq   bfAL>t L~(L



where 8, yi and A. are Lagrange multipliers.  Maximization of (7) yields
(8a)   U2i U= W + A /a = Wi    i=1,2,  6>0
(8b)   FLi   W.i    i=1,2
(8c)   e4   = W.      i=1,2
(8d)   F       i      i=1,2
Hi     1
Equations (8a-c) show that the allocation of time to leisure, farm work and
housework is determined by equating the returns to each activity in terms of
the numeraire to what can be interpreted as a shadow wage, 1.i.   When the
individual participates in the labor market, so that AC=O, condition (8a) says
that the shadow wage is just the off-farm wage.   Otherwise, A i/  represents
the amount of compensation above and beyond this market wage required to
induce the individual to spend his or her marginal hour in market work or any
other non-leisure activity. The shadow wage, defined by conditions (8a-c), is
a function of Z, A, I and depends on the forms of the preference function and
both of the household's technologies; i.e., it is endogenously determined.
The violation of the recursive property brought about by the decision of
various family members not to work off the farm leads to a household budget
constraint that is generally nonlinear in the leisure goods.  At the optimum,
however, the gradient of the budget constraint is just the shadow wage vector
{Wi}.  Thus, at this point the constraint is linear.   In particular, define
full income at the household optimum by
(9)   A = n (W             +                 T    2 '
where
t   'W' f >vC'ttyj WS- C   10
- V\9 S 9 K-    1



1 =  Max  F( 1L  H 1H 2A)   < HH  - <WHH  - W1L1 -~ 2L   and
1'2'1' 2'       1 1    2 2    11      22
Li.,Hi
' =   Max  e4(S1,S2I) -                   =122'
S.
(see Strauss 1986 for a similar analysis). IT again represents farm profits,
with the opportunity cost of family labor properly deducted. Likewise, if is
the "profit" from housework.   The household full income constraint evaluated
at the optimum--i.e, the "linearized" budget constraint--can be written as
(10)  A = s + W 2  + W2     -
11    2 2              4- i'J) -'-'
Maximization of the utility function subject to (10) yields conditions
(8a-d),  but  looks  exactly  like  a  pure  consumer  choice  problem.    The
linearization of the budget constraint is equivalent to turning (7) into a
two-stage maximization problem as in the recursive model of II.A, except that
full income is conditioned on the demand for leisure and, conversely, the
optimal leisure choices are conditioned on the production decisions.
The solution to this revised utility maximization problem is a standard
set of Marshallian labor supply functions
(11)  hi= T.   2  = hi(, W2, A; Z)   i=1,2
Observe that in contrast to the model with homogeneous labor, the labor supply
of each worker would in general respond to variation in the shadow wages of
both workers.
If both workers participate in the labor market (i.e., Ai=O, i=1,2), then
11 
jit//  L



(11) reduces to the two worker analog of recursive labor supply function (5)
in II.A.   If either one of the workers fails to participate in the labor
market  (e.g.,  A1=0, ½2>0), then immediately the recursive property breaks
down. Although each labor supply function would still depend on the exogenous
W1,  they would also each depend on W2 and A,  which are determined by
preferences and the technology.   When neither worker participates  in the
market, all the arguments of the labor supply functions are endogenous.
To get standard comparative static results with respect to changes in the
shadow wages, notice that the dual to the utility maximization problem, the
constrained minimization of (10) with respect to  t, £1 and £22  gives the
expenditure function,  A (W1, W , u).   The Marshallian and Hicksian labor
1'2'
supply functions can then be equated,
(12)  h.(W1* W2, u) = hI(w1,  2, A(             u))  i,2
and differentiated with respect to the shadow wages to give the Slutsky
equations for i=1,2 and j=1,2
(13)  ahI/8Wj= ahI/88 j   + (T-   )8hIA.
When the household cannot hire perfectly substitutable workers from the
outside, it can be thought of as participating in "shadow markets" for the
labor of its members.   The household's net position in a particular shadow
market, or its point of compensation, is just its own supply of that type of
labor.   Utility theory implies that when i=j the first term on the right hand
5Thus, if the family has several type 1 workers, say, the point of
12



side of (13), the compensated own wage effect, is unambiguously positive, and
that  the  compensated  cross  wage  effects  are  equal;  i.e.,   h 1/8W21  =
8h2/8W1 | U.
III. THE EMPIRICAL STRATEGY
The estimating model and the stochastic environment
Labor supply functions (11) can be made operational by substituting the
marginal product of farm labor for the corresponding shadow wage, using
condition (8b), and by replacing full income with farm profits to give
i=                         _' W~~*H 1-            LJ
(14)   h   h (FL, FL.  ' (FL  F           W_); Z)      i=1,2.
Li' FLI,   L2' 1.              2
Both  the  marginal  products  and  farm  profits  are  evaluated  at  the
optimal--i.e.,  the observed--labor supply choices of each household.   Note
that in a sample containing part-time wage earners it may be desirable to use
the market wage, when available, rather than the marginal product of farm
labor  in equation  (14).   But,  before  imposing the restriction that  the
marginal product equal the wage for these workers, that restriction should be
tested (see below).
The general empirical model consists of a production function and a pair
compensation for each individual worker is the sum of the labor supply of all
type 1 workers. Note also that the derivative of labor supply with ;espect to
A could just as well be replaced by the derivative with respect to IT
13
IS  -          vA
94a 7tSQ5s



of labor supply equations,
(15)  Y =  [f(L1, L2, X, A, 13), £)
(16)  hi= gI(r, Z ', a'. uI)   i=1,2
where  r =(f     fL, f - fL L  -f L  -P X)
and  where  e  and  ui  are  the  production  and  labor  supply  disturbances,
respectively, X is the vector of variable inputs with prices Px, A are the
fixed inputs, and 1 and 7i are the production and labor supply parameters,
respectively.   r contains the marginal products and farm profit expressions
and depends, in particular, on Li and g.  The functions 0 and gi permit either
additive or multiplicative disturbances.
The  interpretation of the error  term  in an agricultural  production
function is an old question.   Zellner, Kmenta and Dreze [1966] regard the
disturbance (e.g., a weather shock) as orthogonal to the variable inputs, as
long as it is not revealed to the farmer in advance of his input decision. If
the shock is anticipated, or if the error contains unobservables, such as
managerial ability, which are known to the household, simultaneity bias in the
production function estimates could result. Although it is important to test
for the presence of such bias, it should be noted that the endogeneity of the
variable inputs is unrelated to the question of whether the underlying
agricultural household model is recursive or not.
Whereas nothing  in the theoretical  model says that £ is necessarily
correlated with the regressors in the production function, that is certainly
not the case of ui in the labor supply functions (16). These disturbances can
be interpreted as unobserved differences in the preference for leisure.   The
14



correlation between uI and the shadow wages is a direct consequence of the
nonrecursive nature of the model. Thus, in order to get consistent estimates
of the i., instruments must be found for the elements of r.
Rather than estimate the system formed by (15) and (16) jointly, a more
pragmatic two-step method is adopted.    First the production function is
estimated and r is calculated using the estimates of f.  Then instrumental
variable estimates of the labor supply parameters are obtained and their
covariance matrices are adjusted for the fact that r is a function of
pre-estimated regressors.
Identifying the model
Ideal instruments for variable inputs in a production function are the
prices of those inputs.   In the absence of such data, variables which proxy
indirect costs (such as remoteness of the village or the degree of modernity)
can be employed. Instruments for the family's own labor input are those
variables which covary with the returns to other uses of time, such as
housework and leisure; the housework inputs, I, and taste shifters, Z, are
natural choices.
In addition, the variation across households in the number of people
working on the farm can be exploited. Given that the labor inputs of family
6If the production and labor supply disturbances are correlated, then greater
efficiency might be achieved by employing a full-information estimation method
(as in Lopez 1984). The difficulty is that even if the production function is
linear in 3, and the labor supply functions are linear in TV the latter
functions will generally not be linear in (13ti)   The approach adopted here
is similar to that used by MaCurdy and Pencavel [1986] in an entirely
different context.
15



workers of a particular type are perfectly substitutable, output is just a
function of the total labor input of each worker type. The number of wcorkers
is obviously highly  correlated with their total labor input.   At the same
time, if that number is a quasi-fixed asset of the household, It is
uncorrelated with the disturbance  in the production function.   Thus,  the
number o  farm workers of each type in the household should make excellent
Instruments for their various f     lab     ns_       {    cQ4>-& /V4i-   2
Interpreting  the  number  of  workers                  tegory  as  resource
endowments of the household also allows them to be used as instruments fo r the
marginal products or shadow wages in the labor supply functions. Intuitively,
the presence of another adult male in the household should affect the labor
supply of adult males only by lowering their marginal product and raisirg the
family's profit from farming.   If, however,  demographic variables are also
taste shifters, they cannot be legitimately excluded from the labor supply
equation. Moreover, if the number of household members allocated to farmn work
Is part of the short run labor supply decision, then using such variables as
instruments  may be  inappropriate.    The  validity  of  these  instruments  is
ultimately an empirical question, which can only be resolved by testing the
overidentifying restrictions supplied by the theoretical model.
Additional exclusion restrictions for the labor supply functions are the
fixed farm inputs and characteristics, the effects of which should already be
captured in the shadow wages and farm profits via the production function.
Also, condition (8c), equating the marginal returns from housework to the
shadow wage, implies that the housework production shifters, I, are valid
instruments for the shadow wages.
dX&2.ktil4Xw  es ;   , 6,(
At&.AY  ',    :I  I ° 
a~~~~L



IV. ESTIMATION OF THE AGRICULTURAL TECHNOLOGY
The data
The Peruvian Living Standards Survey (PLSS) was administered by the World
Bank between 1985 and 1986 throughout the whole of Peru, and provides detailed
information on the activities of about five thousand households and the time
use of their members.  Only households from the highlands region, or Sierra,
that worked land and reported harvesting some crops are selected for this
analysis.   It is in the Sierra where most of Peru's subsistence farming is
concentrated, and this region is ecologically distinct from both the coast and
eastern jungle. Since this study focuses on the productivity and joint labor
supply decisions of adult males and females, the final sample consists of the
1,034 households in which at least one adult male and female worked on the
family farm  (thus  dropping  about  500  agricultural  households).    Sample
statistics are reported in table 1.
Because most farm households in the Sierra grow several crops and raise
livestock as well, and since data on all inputs (particularly labor and land)
are not broken down by crop or activity, the different crop outputs are
aggregated using the medians of their reported prices within each village as
weights.   Obtaining the value of output from livestock production  (e.g.,
cattle  or sheep herding)  poses a problem  because  of  the difficulty  of
measuring the appreciation in the value of the herd over the year.  Thus, in
what follows, livestock output is taken to be the sales of dairy, wool and
other animal products, plus some fraction of the value of the household's
17



TABLE 1
DEFINITIONS, MEANS AND STANDARD DEVIATIONS OF PRODUCTION FUNCTION
VARIABLES AND ADDITIONAL INSTRUMENTS
Standard
Definition of production function variables                    Mean  deviation
Value of output      Value of all crops harvested +            8885       553109
sales of animal by-products +
.2 x value of livestock (see text)
Land                 Land area in hectares, owned and          4.6        20
worked by the household, or rented
or sharecropped
Equipment            Value   of    farm    equipment           580        8880
Insecticide          Expenditures on insecticide               71         215
Fertilizer           Expenditures on fertilizer                142        567
Transportation       Expenditures on transportation            66         342
Livestock            Expenditures on livestock inputs          173        568
Hired labor          (Days of labor hired + days received    292          1235
in labor exchange) x 10
Adult male           Hours farm work, all male                 2354       1491
labor                household members, age>19
Adult female         Hours farm work, females age>19           1940       1311
labor
Teenager labor       Hours farm work, ages 12-19               1050       1609
Child labor          Hours farm work, ages 6-11                541        963
Farm animals         Value of oxen, horses and mules           1541       3414
Irrigation           Proportion of land irrigated              .30        .40
Head's age           Age of household head                     48.1       13.5
Head's schooling    Years schooling of head                    2.9        2.9
Permanent crops?    Dummy: 1 if had perennial                   .41       .49
(e.g., tree) crops (O otherwise)
Harvest season?      Dummy: 1 if interviewed during             .23       .42
harvest season
Planting season?    Dummy: 1 if interviewed during             .28        .45
planting season.
Off-season?          Dummy: 1 if interviewed between            .25       .43
harvest and planting season
North?               Dummy: 1 if live in northern Sierra        .21       .41
Central?             Dummy: 1 if live in central Sierra         .33       .47
18



TABLE 1--Continued
Definition of additional instruments
I.   Household  composition  variables:    Number  of  males  age>19,  females
age>19, males 15<age<=19, females 15<age<=19, males 11<age<=15, females
11<age<=15, all children 5<age<=11, all children age<=5.
II.  Household characteristics and housework production shifters:  Fraction of
land area owned, village size dummy (1 if  2,000 inhabitants, 0 otherwise),
home ownership dummy (1 if own, 0 otherwise), water source dummy (0 if from
river, 1 otherwise), light source dummy (1 if electric, 0 otherwise), cooking
fuel dummy (0 if use wood, 1 otherwise), sewage disposal dummy (0 if none, 1
otherwise).
III. Community level wages and prices:   Adult male daily field wage,   adult
fe-male daily field wage, price of kerosene, price of rice.
Notes: All variables refer to the twelve month period prior to the date
of interview. Monetary values are expressed in June 1985 Intis.
herd,   under the assumption that the appreciation is proportional  to this
value.   The constant of proportionality is set, rather arbitrarily,  at .2
throughout the analysis, though other values are tried to assess the
robustness of the estimates to this choice.
The input of land is measured as the amount at the household's disposal
in the survey year, whether owned (which constitutes about ninety percent of
the land area in this sample),  rented or sharecropped.   It is not known,
70xen, horses and mules are excluded from this calculation since animal
traction is an input into the production of crops, leaving only cows, sheep,
pigs, goats and llamas.
19



however, if all this land was actually cultivated. Some of the land may have
been left fallow, though in Peru such land is commonly used to graze
livestock. Furthermore, the proportion of land irrigated is known. Irrigated
land is more likely to have been cultivated and is usually more fertile than
dry land. Land and irrigation along with the value of farm equipment (mainly
animal plows), the value of farm animals used for traction (oxen, horses and
mules) and a dummy variable for whether or not perennial crops are grown, are
all taken as exogenous relative to the one-year planning horizon. Dummies for
the agricultural season in which the household was Interviewed, and a set of
regional dummies for the northern and central parts of the Sierra are also
included in the production function. In addition, two characteristics of the
household head, his or her age and years of schooling, enter as proxies for
the management input.
Only expenditure data are available for the variable physical inputs:
Insecticides, fertilizer, transportation, and livestock inputs (mainly food
and medicine). The use of such data in place of quantities in the production
function can lead to biased estimates if input price variation is substantial.
Yet, taking this route seems preferable to ignoring these inputs altogether
and suffering an omitted variables problem.
Farm labor is divided into five categories: adult (age twenty and oLder)
male and female,  teenage (twelve to nineteen),  child (under twelve),   and
The questionnaire uses a seven day recall period, asking for both the actual
and the "usual" hours per week, and weeks per year, spent working in the main
jobs. If the individual did not work in the week prior to the interview, then
a 12 month reference period is used.   If the person identifies himself as a
farmer who is a "self-employed or unpaid family worker," his usual hours in
that job, calculated on an annual basis, are taken as the farm labor input.
20



non-family labor.   The latter category includes both hired labor and labor
received from other households  in labor exchanges.   The survey does not
differentiate such labor by sex, but in the Peruvian Sierra non-family
agricultural workers are predominantly male.
Estimates of a Cobb-Douglas production function
The   Cobb-Douglas   (CD)   production   function   imposes   imperfect
substitutability  between  the  labor  inputs.9    Despite  its  parsimony  in
parameters and log-linearity, the CD has the drawback that it does not allow
inputs to take on a value of zero.  None of the variable physical inputs are
employed by every farm in the present sample, and many households report zero
labor inputs for teenagers and children, either because they have no members
of that age, or because they choose not to work them on the farm.  Following
MaCurdy and Pencavel [1986], for example, the constant one is added to all the
inputs, except land and adult male and female labor, which are always positive
by construction of the sample. 10
But a CES production function is estimated in Jacoby [1989], and the nested
hypothesis that adult male family labor and hired labor are perfect
substitutes is rejected at the 5% level.
10The choice of the additive constant is arbitrary and the CD parameter
estimates are fairly robust to different choices within an order of magnitude
or so around unity, except for the coefficient on the hired labor variable,
which is substantially more sensitive.
21



TABLE 2
PRODUCTION FUNCTION ESTIMATES
(DEPENDENT VARIABLE:  LOG VALUE OF OUrPUT)
Cobb-Douglas               Translog (OLS)
Independent variable        OLS             IVa           I              II
Log land                     .221          .217          .221           .217
(.020)        (.029)        (.020)         (.020)
Log equipment                .004          .001          -.006          -.006
(.013)        (.015)        (.013)         (.013)
Log insecticideb             .062          .301          .000           .006
(.016)        (.079)        (.024)         (.024)
Log fertilizerb              .023          -.122         .009           .007
(.015)        (.075)        (.015)         (.015)
Log transportationb          .093          .117          .068           .074
(.016)        (.062)        (.016)         (.016)
Log livestock inputs         .054          .074          .048           .049
(.013)        (.044)        (.012)         (.012)
Log hired laborb             .107          .081          .125           .127
(.012)        (.040)        (.015)         (.015)
Log adult male laborb        .150          .167          .202           .209
(.033)        (.096)        (.045)         (.045)
Log adult                    .069          .077          .099           .108
female laborb                (.036)        (.103)         (.041)        (.042)
Log teenager laborb          .015          .026          .014           .014
(.008)        (.017)        (.008)         (.008)
Log child labor              .009          .013           .010          .011
(009)         (.016)        (.009)         (.009)
Log farm animals             .042          .027          .051           .053
(.009)        (.012)        (.009)         (.009)
Irrigation                  .186           .235          .215           .197
(.071)        (.089)        (.069)         (.070)
Head's age                   .000          .001           .001          .001
(.002)        (.003)        (.002)         (.002)
Head's schooling             .043          .047          .041           .041
(.011)        (.015)        (.011)         (.011)
22



TABLE 2--Continued
Independent                  Cobb-Douglas              Translog (OLS)
variable                   OLS            IV           I             II
Permanent crops?           .166          .144         .185          .185
(.059)       (.076)        (.058)        (.058)
Log insecticide squared                               .030          .029
(.009)       (.009)
Log hired labor squared                               .016          .015
(.006)       (.006)
Log adult male                                        .044          .045
labor squared                                         (.018)        (.018)
Log adult female                                      .057          .047
labor squared                                         (.023)        (.023)
Log farm animals                                      .017          .017
squared                                               (.004)        (.004)
Log adult male labor                                  -.018         -.024
X log farm animals                                    (.009)        (.009)
Log adult female labor                                -.076          ....
X log land                                             (.020)
R                          .48           .36          .51           .49
Marginal Products:c
Male labor                 .47           .59          ...           .44
(1.23)       (1.47)                     (.62)
Female labor               .26           .35          ...           .28
(.60)        (.90)                      (.44)
Notes:   Standard errors in parentheses.   Season of interview dummies,
regional dummies and a constant are included in all regressions.
'The value of the Wu-Hausman statistic for the joint exogeneity test is
19.0, while X 2    = 16.9 and X 2     = 19.0.
b      X(9,*.05)           (9.025)
Endogenous under the alternative hypothesis.
These are calculated using the formula MPLY= piY/Li  where P i is the
coefficient on log(Li) and Y is predicted value of output.   Means over the
sample of 1,034 households are reported, except in the last column, where
negative marginal products are thrown out leaving a sample of 1,015.   Units
are June 1985 Intis per hour.  The standard errors do not take into account
the sampling distribution of the estimate.
23



The first column of Table 2 displays OLS estimates of the CD production
function, and shows that it explains about half of the variation in the value
of output.   Land has the largest and most significant coefficient,  while
irrigation and the other physical inputs have positive coefficients as well,
with all but fertilizer and equipment achieving significance at the 5% level.
The coefficients on the labor inputs also seem to make sense; adult males
contributing the most to output, followed by hired workers (who are mostly
adult males) and adult females.   The coefficient on this latter category of
labor verges on 5% statistical significance.    The contribution of teenagers
and children appears to be relatively small, though only 42% of the households
in this sample put any children under 12 years old to work on their farms.
To test the hypothesis that the variable inputs are exogenous, tlhe CD
technology  is estimated by the instrumental  variables  (IV) method.   Nine
inputs  are  endogenous  under  the  alternative  hypothesis:   insecticide,
fertilizer, transportation, the livestock inputs, and the five categories of
farm labor. Based on the discussion in section III, variables included in the
instrument set (summarized in table 1) are the number of family members in
various age-sex categories, characteristics of the household, community level
When livestock appreciation is set at either .3 or .1 times the value of the
herd, most parameters are not significantly affected.   Not surprisingly, the
coefficient on livestock input expenditures increases with this fraction, but
the change is not very dramatic. Also, when the fraction is set at .3, adult
female labor is significant, but when set at .1, that coefficient is only
weakly positive.   This change is probably due to the fact that women spend
relatively more of their time than men in tending livestock, an indication of
labor heterogeneity.
24



agricultural wages and prices,12 and quadratic terms in the instruments (twenty
are used here).
IV estimates of the CD production function are reported in the second
column of table 2.   A comparison of the OLS and IV estimates reveals no
appreciable differences between the coefficients on the labor inputs, though
the fertilizer and insecticide coefficients differ quite substantially. Based
on a Wu-Hausman test, the joint exogeneity hypothesis is just rejected at the
5% significance level, however not at the 2.5% level.   Marginal products of
male and female labor derived from these two production functions are reported
at the bottom of table 2.  The correlation coefficient between the marginal
products across columns is very high, about .8 for male labor and .95 for
female labor, suggesting that the OLS and IV estimates should not lead to
perceptibly different labor supply parameter estimates in the next stage of
the analysis, though the ones based on the OLS estimates would be more
efficient.
Estimates of a translog production function
Identification of all the elements of X  in equation (16) requires that
the shadow wages of men vary relative to those of women.  Since the relative
shadow wage is just the marginal rate of transformation (MRT) between male and
female labor (i.e., the ratio of the marginal products of these two inputs),
12A community questionnaire appended to the PLSS provides village level
information on only about half of the present sample of households. Wage and
price data can be extrapolated to the whole sample, however, if it is assumed
that prices vary less within a geographical division, e.g. a province, than
across them. Thus, households in communities with missing prices are assigned
the means of community level prices for that province.
25



the restrictions that the chosen functional form of the technology imposes on
the MRT must be carefully considered.   The CD production function imposes
strong separability, the restriction that the MRT depends only on (the ratio
of) the male and female labor inputs and not on the level of any other input.
In order not to impose separability a priori, a flexible functional form such
as the translog must be estimated.  Shadow wages derived from a nonseparable
technology might lead to estimates of the labor supply parameters that are
quite different from those based on the CD technology.
Using a procedure, based on Denny and Fuss [1977], that interprets the
translog as a second order approximation to some unknown arbitrary production
function, restrictions implied by strong and weak separability are tested and,
if not rejected, imposed sequentially to obtain a parsimonious translog
specification.    Column 3 of table 2 shows the OLS estimates of the resulting
production function, with the inputs scaled at their geometric means.   The
last two terms of translog I indicate that the underlying technology exhibits
nonseparability of male and female labor with respect to both land and the
value of farm animals.
Both of these negative interaction terms are plausible consequences of
the sexual division of labor in the Peruvian Sierra.  Oxen, horses and mules
are used in plowing and transport, which are tasks performed primarily by men.
Hence, animal power would tend to be relatively more substitutable with male
labor than with female labor, though it may be complementary with both.   As
for the interaction between land and female labor in translog I, Deere [19821
Details of this procedure can be found in Jacoby [19891.
26



argues that in households of the Sierra with little land, men spend more time
off the farm in market work, leaving women to do the heavier chores Involving
agricultural implements.   Thus, the kinds of tasks performed by women, and
consequently the slope of the isoquant between male and female farm labor,
changes with farm size. 14
A well known feature of translog production functions, and other flexible
functional forms, is that they do not guarantee positive or diminishing
marginal products at every data point. The strong negative coefficient on the
female labor-land interaction term in translog I is a source of a large number
of violations of the regularity conditions.15  Dropping this term from the
translog alleviates the problem to a great extent, but at the loss of a source
of relative shadow wage variability. The correlation coefficients between the
marginal products of labor derived from the resulting production function,
translog II in column 4 of table 2, and those derived from their counterparts
in column 1 are not significantly different from zero, indicating that
translog II still yields quite a different shadow wage structure than the CD.
14See Rosen [19781 for an elaboration of the relationship between the division
of labor and substitutability.   Clearly,  data on the hours of each task
performed by males and females would be informative in this regard.
15While in all but five households the male marginal products are positive, two
hundred or nearly 20% of the households show a negative marginal product of
female labor.
27



V. LABOR SUPPLY ESTIMATION
The specification of labor supply
The samples used in the labor supply analysis are composed of the men and
women (age twenty and above) who work on the 1,034 family farms selected in
the previous section.   The relatively few individuals who worked exclusively
for wages or in nonfarm self-employment are omitted from consideration,
leaving a sample of males numbering 1,229, and a sample of females numbering
1,233.   Among the male farmers there are still 287 whose primary job is for
wages  (virtually  none  of  the  females  are  in  the  labor  market).    The
theoretical model states that for these workers the wage should equal the
marginal product on the farm, a restriction that can now be tested.
Under the assumption that the estimated marginal products of labor are in
fact the pre-harvest expected marginal products, subject to a random
measurement error, they can be regressed on wages for the subsample of labor
market participants as follows,
(17) MPLi = a + b WI + ei.
The  disturbance  term  ei  consists  of  errors  in  measurement  and  random
optimizations  errors.    The  implications  of  utility  maximization  and  an
efficient labor market are embodied in the null hypothesis (a,b)=(O,1). Since
the wage is calculated as reported yearly cash and value of in-kind payments
divided by yearly hours in that job, it is likely to be correlated with the
28



measurement  error  in the marginal  product.    Hence,  in the estimates of
equation (17) in table 3, the wage is instrumented using the worker's age,
years of schooling and quadratic terms in these variables.   Regardless of
which production function is used to derive the marginal products, and in
spite of a strong positive relationship between wages and marginal products,
an F-test emphatically rejects the null hypothesis.
TABLE 3
TESTS OF THE EQUALITY OF WAGES AND MARGINAL PRODUCT FOR LABOR MARKET
PARTICIPANTS: INSTRUMENTAL VARIABLES ESTIMATES
(N = 287)
MPL derived from:    a          b         R         F-test    Wu-Hausman
Cobb-Douglas         -.06      .37        .10        136           7.5
(OLS)               (.17)      (.07)
Translog II          -.41      .58        .14        62            31
(.22)     .08)
Note: Standard errors in parentheses.   The R2 for the first stage wage
equations is .34.
Null hypothesis: (a,b)=(0,1). The 5% critical point for F(2,287) = 3.
bNull hypothesis:  No measurement error in wages.  x1 2    )= 3.8.
29



Two explanations for this rejection, beyond simple irrationality on the
part of farmers, come to mind.  First, the estimated marginal products may in
fact be systematically biased so that the IV method does not lead to
consistent estimates of (a,b).   Alternatively,  as alluded to earlier,  the
equality of marginal products and wages may fail to hold in reality because of
some transaction cost or imperfection in the labor market.   For example, a
commuting cost or disutility associated with off-farm work, or a 'binding
restriction on hours worked in the market, would drive a wedge between the
shadow wage and the market wage.  The marginal product, however, would still
theoretically equal the shadow wage under these circumstances.   Therefore,
(under the assumption that they are unbiasedly estimated) the marginal
products of labor are used even for part-time wage earners in the labor supply
estimation to follow.
To estimate the labor supply function derived in section II, the hours
worked by each adult are calculated by summing the yearly hours spent in farm
work, off-farm self-employment,  wage employment   and housework.   The first
part of table 4 breaks down these non-leisure hours of men and women into the
different categories. Notice especially the extent to which women who work on
the farm engage in housework, raising their total labor supply far above that
of men on average.   Although women spend much less time in wage employment
than men do, they work more in rural family enterprises, e.g., selling farm
produce in local markets.
30



TABLE 4
CHARACTERISTICS OF MEN AND WOMEN WORKING ON FAMILY FARMS
Adult males              Adult females
(N = 1229)               (N = 1233)
Variable definition        Mean      Std. dev.       Mean      Std. dev.
Yearly hours in:
Family farming             1980      994             1627      915
Wage employment            317       718             31        225
Non-farm self-             217       592             346       608
employment
Housework                  359       439             1534      832
Total work                 2863      1032            3537      1396
Age                        43.1      15.3            41.5      14.6
Years schooling            3.5       3.2             1.7       2.6
Married? (1=yes, O=no)    .63       .48              .62       .49
Household head?            .78      .42              .05       .21
The individual level variables listed in the second half of Table 4 are
likely to capture some of the variation in hours worked that the household
level marginal products and profit variable16 cannot account for.   Age and
To construct farm profits, two types of expenses are subtracted from the
predicted value of farm output in addition to the shadow value of adult male
and female farm labor (the shadow value of child and teenager labor is
considered exogenous income and is thus not netted out): (a) the shadow rental
payments on the land not owned by the household (i.e.,the estimated marginal
product of land times the hectares rented or sharecropped), and (b) actual
expenses on the purchased inputs (insecticide, fertilizer, transportation,
livestock inputs and hired labor).  With the CD parameters there are 25 cases
of negative profits, which are assigned the value of fifty to avoid numerical
problems.
31



schooling pick up idiosyncratic productivity differences as well as possible
cohort and taste effects.   Also included in the labor supply functions are
some family composition variables; the number of children in various
categories and the number of adults not working on the farm. Village size and
regional dummies along with community level prices for rice and cooking fuel
are used to proxy geographic variation in labor supply behavior and the cost
of living.   Season of interview dummies may help redress any biases caused by
seasonality in reported hours worked. As discussed in section III, the number
of adult male and female farm workers are excluded from the labor supply
equations, as are housework production shifters (availability of electricity,
plumbing, etc.) and the exogenous agricultural production variables.
A log-linear version of labor supply function (16) is estimated below
(quadratic and linear functions are found to yield similar labor supply
elasticities  and are not reported here).    The disturbances  in  (16)  are
presumed   to   possess   all   of   the   classical   properties,   including
homoscedasticity, except that they are correlated with the marginal products
and farm profit variable.  No account is taken of the covariance between the
disturbances of members of the same household, an oversight which at worst
leads to a loss of efficiency.  Finally, as pointed out above, the covariance
matrices of the labor supply parameters must be adjusted for the fact that
marginal products and farm profits are estimated quantities; conventional
standard errors would understate the true sampling variation.           A general
formula for the covariance matrix of a sequential estimator can be found in
Newey [1984].  The particular formula used here is somewhat simpler owing to
32



the linearity of the labor supply functions in parameters and the
homoscedasticity assumption. 17
Labor supply estimates with the Cobb-Douglas technology
Estimates of log-linear labor supply functions for men and women based on
shadow wages derived from the OLS estimates of the Cobb-Douglas production
function are presented in table 5.  Initial estimates, with a larger set of
regressors, reveal that many of the nonbudget constraint variables do not
significantly affect labor supply.18  Thus, to obtain more efficient estimates
of the variables of primary interest, those nonbudget constraint variables
(except the age and schooling terms) not significant at least at the 10% level
are omitted, leaving the specifications in table 5.
17Suppressing the subscript,  write  (16) as  log(h)  =     =V  + u,  where
V=(log(r),Z). The standard instrumental variables covariance matrix for 7 is    ( a'  
given by ;2X= S2(V'W(W'W) 'W'V) , where W is the matrix of instruments and s 2
u    u                                                                u
is a consistent estimate of the variance of u.  If the error terms u and e are
independent, then the adjusted covariance matrix is
Var(7) = uX + ZV'W(W'W) tW'. gVar(~)4,'W(W'W) W'VE,
where 4  = E ik alog(rk)18g| , k indexes the columns of r, and $ is the
estimated production function parameter vector.
18Particularly interesting is that the number of children five years old and
under does not depress women's hours worked, which includes housework, though
not child care per se. Rosenzweig [19801 finds no impact of young children on
the supply of market hours by rural Indian women.
33



TABLE 5
ESTIMATES OF LOG-LINEAR MALE AND FEMALE LABOR SUPPLY FUNCTIONS
Cobb-Douglas                       Translog II
Independent            Males    Females                   Males        Females
variable               (N=1229)  (N=1233)                 (N=1203)      (N=1211)
Log male               .102       .006                   .030           .062
shadow wage            (.039)    (.031)                  (.029)         (.037)
Log female            -.010      .079                    -.020          .157
shadow wage            (.034)    (.036)                  (.046)         (.088)
Log farm profit       -.058      -.082                   .004          -.109
(.033)    (.029)                  (.034)         (.085)
Age                    .025       .021                   .028           .020
(.005)    (.005)                  (.005)        (.006)
Age squared            -.00029   -.00027                 -.00032       -.00026
(.00006)  (.00005)                (.00005)       (.00005)
Years schooling        .009       .005                    .018          .007
(.014)    (.016)                  (.011)         (.028)
Years schooling        -.0024    -.0028                  -.0025        -.0027
squared                (.0009)   (.0015)                 (.0008)        (.0035)
Married?                ....     .076                     ....          .084
(.029)                  ....          (.041)
Household               ....      .162                    ....          .191
head?                             (.064)                  ....          (.072)
Number non-farm        -.108       ....                  -.070           ....
adult males            (.053)                             (.049)
Number males          -.060        ....                  -.060           ....
age 12-15              (.027)                             (.024)
F-testa               8.0         8.6                      10.6         8.6
GMM x2 testb          22.4        43.4                    23.3          37.8
[30.1]      [38.91                 [30.11        [38.9]
34



TABLE 5--Continued
Cobb-Douwlas                       Translog II
Males    Females                 Males        Females
(N=1229)  (N=1233)                (N=1203)     (N=1211)
Compensated effects:
Own shadow wage       901         1578                   187          2646
(338)      (598)                   (217)       (1625)
Cross shadow wage    203          486                    -218         85
(492)      (338)                   (625)       (520)
Notes:   Standard errors in parentheses,  x2 critical values in square
brackets.   Male equations includes a constant, village size dummy, prices of
kerosene and rice, off-season dummy and five regional dummies based on
department of residence.   Female equations include a constant, village size
dummy and the kerosene price.   All instruments in table 1 are used in the
estimation, except the number of adult farm workers replaces number of adult
household members. Polynomials in age are also used as instruments.
aJoint significance test, based on adjusted covariance matrix.
bJoint exogeneity test, degrees of freedom equal to number of
overidentifying restrictions.
Uncompensated own wage elasticities (the own wage coefficients in table
5) are positive for both men and women.    In contrast,  Rosenzweig  [1980]
estimates a backward bending market labor supply function for Indian males.
In fact, gross own wage elasticities here are slightly higher for men than for
women.   U.S. labor supply studies generally find women have a greater gross
labor supply response than men.(see Killingsworth 1983 pp. 103-104), and argue
it is because women have more margins of substitution out of market time.
But, since the definition of labor supply used here includes housework, a
35



greater response by women to changes in their shadow wage should not be
expected, a priori. Both the male and female coefficients on farm profits are
negative, hovering around significance.   One finding in agreement with many
U.S. studies is that the income elasticities are greater for women than men.
There is, however, the possibility of omitted variable bias here, in that
households may have other sources of nonlabor income which are correlated with
their farm profits, and with the instruments as well.
Compensated own and cross shadow wage effects--i.e., the Slutsky matrix--
are reported at the bottom of table 5, evaluated at the sample means of the
shadow wages, farm profits and compensation points.          Standard errors are
calculated  using  the  corrected  covariance  matrices.    The  positive  and
significant compensated own wage effects are highly supportive of the utility
maximization hypothesis.    Because of their  larger  income effect,  women's
compensated own wage effect is larger than the male counterpart. Compensated
cross shadow wage effects in both the male and female equations are positive,
though small relative to the own wage effects, and not significant. A t-test
cannot reject the equality of the compensated cross wage effects impli(ed by
utility maximization (in contrast to U.S. studies, where it is often
resoundingly rejected). Still, the judgment about whether symmetry holds for
this data must be reserved until a more flexible underlying agricultural
production function is considered.20
19Means of the points of compensation are 3,649 hours for men and 4,495 hours
for women, and the mean of farm profits is 2,373 Intis.
20 Another implication of utility maximization is that the determinant of the
Slutsky  matrix  is  positive  (see  Ashenfelter  and  Heckman,  1974).    This
restriction is satisfied at the sample means, though the standard error of the
determinant is not calculated here.
36



A Generalized Method of Moments (GMM) specification test (Hansen 1982)
cannot reject the joint null hypothesis that the instruments in the male labor
supply equation are uncorrelated with the residual at the 5% level.
Meanwhile, the same test in the female case yields a weak rejection, but since
exogeneity of the instruments was not rejected when a larger set of regressors
was included in the female labor supply equation, it appears that the
rejection now is due to some omitted variables.
The exogeneity of certain subsets of the instruments can be assessed
using a more powerful test.  A variant of the Wu-Hausman test   is employed to
test whether the number of adult male and female farm workers   are valid
exclusion restrictions in each of the labor supply equations. As can be seen
in table 6 this exogeneity hypothesis cannot be rejected in either the male or
female labor supply equation. Similarly, the joint exogeneity of these latter
instruments and the housework production shifters is not rejected.  It can be
concluded, therefore, that the decision about how many adult household members
to send out into the fields in a given year is not correlated with their taste
for  leisure.    Furthermore,  the orthogonality of the housework production
shifters lends support to the time allocation model presented above.
21 To tell whether two IV estimates, based on alternative instrument sets, are
significantly different from each other, the variance of their difference must
be calculated, which, unlike Wu-Hausman, must take into account the covariance
between the estimates (see Mroz 1987).
37



TABLE 6
ESTIMATES OF LABOR SUPPLY PARAMETERS WITH VARYING INSTRUMENT SETS
AND TESTS OF OVERIDENTIFYING RESTRICTIONS
log male       Log female      Log farm    Exclusions
shadow wage    shadow wage    profits       tested         x test
Males:
(1)       .201            -.109          -.04         Z2             1.08
(.098)         (.090)          (.061)                      [16.91
(2)       .167            -.107           -.021       z1             2.47
(.074)         (.072)          (.040)                      [6.0]1
Females:
(1)       -.028           .082            -.064       Z2             .38
(.057)         (.044)          (.037)                      [16.91
(2)       -.036           .075            -.053       Z              2.64
(.045)         (.042)          (.030)        1             [6.0]1
Note:   Standard errors in parentheses and chi-square critical values
(degrees of freedom equal to number of overidentifying restriction tested) in
square brackets.   Labor supply specifications are identical to those in the
first two columns of table 5.
a
aZ= {# of adult male farm workers, # of adult female farm workersl,
Z={Z ' the six housework production shifters}.
bTest of the significance of the difference between these estimates and
those using the full set of instruments in table 5.
Labor supply estimates with the translog technology
Calculating the budget constraint variables from translog II in table 2
allows shadow wages of men and women to vary relative to one another as a
function of the value of the stock of farm animals.   The robustness of the
38



estimated wage effects to this change can now be evaluated.   But first, to
avoid dealing with negative shadow wages, those few observations with either a
negative male or female marginal product are dropped (twenty-six cases in the
male sample and twenty-two in the sample of women).
The new labor supply estimates are displayed in the third and fourth
columns of table 5.  Although the overall fit of the male equation is better
than in the CD case, the budget constraint variables are estimated with
conspicuously less precision.   Farm profit,  in fact,  has a very slightly
positive coefficient.  Meanwhile, the results for females are quite striking.
While the coefficients on shadow wages and farm profit all have the same signs
as in the CD case, the male shadow wage coefficient, which is just the
uncompensated cross wage elasticity, is much larger in magnitude. It is also
interesting that, while for males the adjustments to the standard errors to
correct for pre-estimated regressors are rather small, the same corrections in
the female translog case are extremely large.   For example, the uncorrected
standard errors for the male wage, female wage and farm profit coefficients in
the female equation are .026, .035 and .025, respectively.   Finally, observe
that with the budget constraint variables derived from translog II the GMM
test does not find evidence of any misspecification in the female equation,
as it did in the CD case.
Notwithstanding the large gross effect of the male shadow wage in the
female equation, the compensated cross effect for women shown in column 4 of
table 5 is essentially zero. So to is the cross effect in the male equation,
making it impossible  once again, to reject the symmetry restriction.   Note
also that the compensated own wage effect of women is larger and that of men
is smaller with the translog than with the CD.   Overall, while some of the
39



quantitative results appear to be sensitive to the functional form of the
agricultural   technology,   the  broad  qualitative   conclusions  are   not
dramatically altered by it. Although cross wage effects were of no importance
in the CD case, the translog case shows that it may be inappropriate to ignore
the Joint maximization framework and restrict such effects to be zero in
individual labor supply equations.
40



VI. CONCLUSIONS
The statistical evidence just presented suggests that Peruvian farm
households indeed allocate their members' time as if to maximize a family
utility function. In particular, work effort is higher among peasants who are
more productive at the margin and who thus face a higher opportunity cost of
time. Beyond lending support to peasant rationality, this paper demonstrates
the tractability of the shadow wage methodology and its usefulness in
estimating time allocation models when wage data are simply not available, or
when the assumptions required to make use of wage data stretch the bounds of
credulity.
In developing countries, typically very few women and children
participate in the wage labor market.   Yet, many of the important issues in
development--most  notably,   fertility  demand--hinge  precisely  upon  the
opportunity cost of time of these agents. Rosenzweig and Schultz [1982], for
example, argue that differences in infant mortality rates by sex might be a
parental response to differing lifetime earning capacities of males and
females. They test this hypothesis using data on employment rates of men and
women  in rural  India as proxies for earnings  potentials.    A much more
convincing empirical case could be made, though, if the actual productive
contribution of men and women in agriculture were estimated, as they have been
in this paper.
41



I



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44



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LSMS Working Papers (continued)
No. 40     The Effects of Household and Community Characteristics on the Nutrition of Preschool Children:
Evidence from Rural C6te d'Ivoire
No. 41     Public-Private Sector Wage Differentials in Peru, 1985-86
No. 42     The Distribution of Welfare in Peru in 1985-86
No. 43     Profits from Self-Employment: A Case Study of C6te d'Ivoire
No. 44     The Living Standards Survey and Price Policy Reform: A Study of Cocoa and Coffee Production in
C6te d'Ivoire
No. 45     Measuring the Willingness to Payfor Social Services in Developing Countries
No. 46     Nonagricultural Family Enterprises in C8te d'Ivoire: A Descriptive Analysis
No. 47     The Poor during Adjustment: A Case Study of C6te d'lvoire
No. 48     Confronting Poverty in Developing Countries: Definitions, Information, and Policies
No. 49     Sample Designs for the Living Standards Surveys in Ghana and Mauritania/Plans de sondage pour
les enqu2tes sur le niveau de vie au Ghana et en Mauritanie
No. 50     Food Subsidies: A Case Study of Price Reform in Morocco (also in French, 50F)
No. 51     Child Anthropometry in C6te d'Ivoire: Estimates from Two Surveys, 1985 and 1986
No. 52     Public-Private Sector Wage Comparisons and Moonlighting in Developing Countries: Evidence from
C6te d'Ivoire and Peru
No. 53     Socioeconomic Determinants of Fertility in C6te d'Ivoire
No. 54     The Willingness to Payfor Education in Developing Countries: Evidencefrom Rural Peru
No. 55     Rigidite des salaires: Donnees microeconomiques et macro&conomiques sur l'ajustement du marche du
travail dans le secteur moderne (in French only)
No. 56     The Poor in Latin America during Adjustment: A Case Study of Peru
No. 57     The Substitutability of Public and Private Health Carefor the Treatment of Children in Pakistan
No. 58     Identifying the Poor: Is "Headship" a Useful Concept?
No. 59     Labor Market Perfonrance as a Determinant of Migration
No. 60     The Relative Effectiveness of Private and Public Schools: Evidencefrom Two Developing Countries
No. 61     Large Sample Distribution of Several Inequality Measures: With Application to C6te d'Ivoire
No. 62     Testing for Significance of Poverty Differences: With Application to C6te d'Ivoire
No. 63     Poverty and Economic Growth: With Application to C6te d'Ivoire
No. 64     Education and Earnings in Peru's Informal Nonfarm Family Enterprises
No. 65     Formal and Informal Sector Wage Determination in Urban Low-Income Neighborhoods in Pakistan
No. 66     Testing for Labor Market Duality: The Private Wage Sector in C6te d'Ivoire
No. 67     Does Education ray in the Labor Market? The Labor Force Participation, Occupation, and Earnings
of Peruvian Women
No. 68     The Cornposition and Distribution of Income in C6te d'lvoire
No. 69     Price Elasticities from Survey Data: Extensions and Indonesian Results
No. 70     Efficient Allocation of Transfers to the Poor: The Problem of Unobserved Household Income
No. 71    Investigating the Determinants of Household Welfare in C6te d'Ivoire
No. 72     The Selectivity of Fertility and the Determinants of Human Capital Investments: Parametric and
Semniparametric Estimates



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