On Price Risk and the Inverse Farm Size-productivity Relationship

The oft-observed inverse relationship between farm size and productivity is generally explained by labor market imperfections. Although other explanations exist (e.g., size-sensitive cropping patterns and variable soil quality), the literature ignores uncertainty as an explanation. Using a simple two-period model of an agricultural household that both produces and consumes under price uncertainty at the time labor allocation decisions are made, this paper demonstrates analytically that an inverse relationship may exist, even absent any of the more common explanations. A simple data exercise suggests the plausibility of temporal price risk as an explanation for this phenomenon. JEL Classification: D13, O13, Q12 ___________________________ This research was supported by an International Predissertation Fellowship from the Social Science Research Council and the American Council of Learned Societies with funds provided by the Ford Foundation. Michael Carter, Jean-Paul Chavas and participants in the International Development Workshop at Wisconsin provided valuable comments, but remaining errors are the author’s sole responsibility. ON PRICE RISK AND THE INVERSE FARM SIZE-PRODUCTIVITY RELATIONSHIP The existence of an inverse relationship between farm size and farm productivity has been observed by agricultural and development economists for some time. Crop yields per unit area cultivated that decline with size of holdings follow from regular observations of more intensive labor use on small farms than large (Berry and Cline 1979, Cornia 1985, Sen 1981). This common empirical finding has drawn several different explanations, with important implications for both policy and theory. The most popular rely on labor market dualism with a distinctly Chayanovian flavor. Small, peasant households are believed to face a lower opportunity cost of labor than large, commercial farms. Consequently, small farms apply their own labor in such quantities that the expected marginal value product of household labor applied to homestead cultivation is less than a market wage-based measure of the opportunity cost of labor (Carter and Wiebe 1990, Chayanov 1966, Hunt 1979, Sen 1966, 1975). A closely related but distinct theoretical approach assumes that principal-agent problems and the associated labor supervision costs render effective labor costs higher on farms large enough to hire labor than on peasant plots (Carter and Kalfayan 1989, Feder 1985). Finally, a third, less formally developed labor markets explanation has been posited by Binswanger and Rosenzweig (1986). They note that imperfect information in labor search results in a positive probability of misallocation of labor. Labor-selling households that fail to find casual labor reallocate the time they had planned for wage labor to work on their own farm instead, up to the point where the marginal utility of home production equals the marginal utility of leisure. Since the household wanted work, the marginal utility of the wage (and thus of production) necessarily exceeded that of leisure, so at least some windfall labor time goes to home farming. Exactly the reverse happens for labor-hiring households who fail to hire casual labor; they fall short of planned labor applications. In each of the above models, the inverse relationship between farm size and productivity per unit land cultivated depends on imperfect labor markets. Such explanations imply the criticality of incorporating some flavor of labor 1 Amartya Sen (1962) launched contemporary inquiry into the issue with his analysis of India’s Studies in the Economics of Farm Management. Chayanov (1966) had identified the issue with respect to Russian agriculture much earlier this century. 2 Feder (1985) demonstrates further that multiple market failures are necessary. In other words, farmers can cope with a single market failure (e.g., for wage labor) without the shadow wage having to diverge from the 2 market failure into theoretical analysis of low-income agriculture. Moreover, if labor market imperfections inevitably render small farmers more productive, then it might follow that land redistribution can stimulate agricultural productivity, adding an efficiency argument to the equity aims of such policy. These inferences were influential for some years. In addition to smaller farms’ more intensive application of labor inputs, Bharadwaj (1974) cites the sensitivity of cropping patterns to farm size as an explanation of the inverse relationship, with smaller farms allocating a higher proportion of area to more lucrative and labor-intensive crops. In particular, Bharadwaj notes that differences in cropping patterns resolve an apparent aggregation bias evident in empirical studies: inverse relationships which seem to hold at farm level often seem not to hold for individual crops. While Bharadwaj’s findings ratify the implications for theory of the more mainstream, labor market failures explanations, they yield more ambiguous policy implications. Yet another view is presented by Bhalla (1988), Bhalla and Roy (1988), and Benjamin (1991), among others, who posit a size-sensitivity in the quality of factor endowments, especially soil fertility. These authors claim that more intensive labor use on small farms reflects primarily the superior fertility of soils on smaller plots, thereby warranting more intensive labor per unit area. These results conflict with those of Bharadwaj (1974) and Carter (1984), each of whom considered soil quality and other inputs (e.g., irrigation, draught animals) and found that these accounted for only a small portion of the observed variation in productivity across farm sizes. But if correct, the soil quality variability argument undermines efficiency arguments for land reform and implies that parsimonious neoclassical theory might not be inappropriate for analyzing low-income agriculture. The stark differences in implications for policy and theory of the imperfect labor markets and variable soil quality explanations are especially troubling because each is exceedingly difficult to prove empirically. Reliable data market wage. But if another market (e.g., for land or credit) also is imperfect, then the inverse relationship emerges. 3 It should be noted that size-sensitivity of cropping patterns of the opposite nature (i.e., with larger farms cultivating a greater proportion of area in higher-value crops) is often posited, as in Fafchamps (1992). 4 Sen (1975) raises this point more casually. 3 on labor allocation and soil quality are rare. Moreover, to the best of my knowledge no existing data set encompasses both, thereby precluding simultaneous direct testing of the competing hypotheses. An ideological stalemate results. This paper offers an alternative explanation of the inverse relationship based on three empirically sound stylized facts. First, farmers in low-income countries cannot fully hedge uncertain staple crop prices through futures or insurance contracts, nor by forward sales at the time labor allocation decisions are made. The connection between risk aversion and labor allocation is well known (Block and Heineke 1973, Roe and Graham-Tomasi 1986, Srinivasan 1972). Although risk aversion has typically been defined in the narrow Arrow-Pratt sense of income risk aversion, it is a straightforward extension to consider the case of price risk aversion (i.e., uncertainty regarding both income and consumer prices). Second, land is unevenly distributed across the agricultural population. Even under if technologies and preferences are uniform among agents, differences in land endowments create a heterogeneous society. Third, households’ net agricultural purchases are inversely related to landholdings. Small farms tend to be net produce purchasers while large farms tend to be net sellers. These three common features of low-income agriculture combine to induce an endogenous agrarian class structure, characterized by heterogeneous labor allocation behavior across classes of farmers, much as in Roemer (1982). This next section proves analytically that price risk and distinct agrarian classes suffice in explaining the oft-observed inverse relationship between farm size and productivity, even with competitive labor markets and uniform soil quality. The following section offers an informal empirical demonstration of the plausibility of this explanation. The concluding section draws out some of the implications of this explanation for both policy and theory. An Agricultural Household Model 5 Srinivasan explains the inverse relationship by yield risk. He defines utility over income alone, imposes restrictions on the coefficients of risk aversion and on how risk enters production, and assumes constant returns to scale in production. Although highly restrictive, his is an important and much under-appreciated contribution to the literature on the inverse relationship. The present work complements his paper, exploring the other major source of agricultural uncertainty: prices. 4 Assume an agricultural household exhibits Von Neumann-Morgenstern utility defined over consumption of leisure(L) and two goods: a staple food (S), and a non-staple (N). The staple can either be produced or purchased, the nonstaple is available only through market purchase. The household has an endowment of land (T) and of labor time (L). Production is strictly increasing in land and labor, and concave in labor. Labor used in production is a function of household labor (L) and hired labor (L), but might not be just the sum of the two if there exists a labor supervision problem (Feder 1985, Stiglitz 1986). Labor markets are competitive by assumption but, to isolate the variables of interest, land and credit markets are assumed not to exist. Just as the household can hire labor in, so can it hire out its time (L) at a known exogenous wage (w). The household faces a time constraint, L + L + L ≤ L. Its income comes from wage labor, agricultural production and exogenous transfers (I). This is a two-period model. All product prices are unknown when production decisions (i.e., labor allocation decisions) are made but are revealed before consumption decisions are made. The household’s utility maximization problem can be expressed as (1) Max L , L , L L E Max N, S U(L , N, S) s.t. P S P N ≤ Y Y w[L S L ] P F(L̂,T) I L̂ e(L ) L H L 0 ≥ L H L L L S where E is the mathematical expectation operator, P is the staple price, P is the non-staple price, and Y is endogenous income. The effective labor function, e(.), converts hired labor units into household labor equivalents such that e(L)/L takes values over the closed interval [0,1]. This is a perfectly general formulation, permitting hired labor to be a perfect substitute or non-substitutable for family labor (or any intermediate substitutability). Thus the present model nests within it as special cases those explanations which rely on labor supervision problems. Farm 6 Superscripts distinguish among goods across subcategories and time. Subscripts denote derivatives. 5 size-dependent patterns of labor use are robust to variations in labor market specification, as will become clear shortly. The household allocates labor conditional on ex-post optimal choice of consumption quantities. A variable indirect utility function, V(L,P,P,Y) which is the dual of U(.) can be derived (Epstein 1975). V(.) is homogeneous of degree zero in (P,P,Y) and, therefore, invariant to unit of measurement. So set P=1 and let P=P/P and Y=Y/P. Since Y is itself a function of P, V(.) has multiple stochastic arguments. The household exhibits Arrow-Pratt income risk aversion (VYY<0). The labor allocation problem can be represented as (2) Max L , L , L L EV(L , P, Y) : Y w[L 0 L L L H L ] PF(e(L ) L H , T) I where V(L , P, Y) Max N, S U(L , N, S) : P[S F(e(L ) L H ,T)] N w[L 0 L L L H L ] I The first-order necessary conditions for an optimum are (3) (4) (5) w.r.t. hired labor : E VY[PFL D w] ≤ 0 ( 0 if L D > 0 ) w.r.t. household labor : E VY[PFL H w] ≤ 0 ( 0 if L H > 0 ) w.r.t. leisure : E VL L VYw ≤ 0 ( 0 if L L > 0 ) As pointed out already, this analysis is very general with respect to labor in that the effective labor function allows for hired and household labor to be either perfect or imperfect substitutes. In the case where they are perfect substitutes, (3) and (4) are identical. If hired labor is not a perfect substitute for household labor (i.e., if e(L)<L), there are segments of the land distribution for which border solutions to (3) prevail (Carter and Kalfayan 1989). Therefore we concentrate on (4), which will likely have an interior solution if the household has land, since household labor is at least as efficient as hired labor. Under price uncertainty, solution of the first-order condition (4) yields the following: 7 Formally, F(.),P, and w must be such that . Even under rudimentary technology and fixed, suboptimal prices in low-income agriculture, wages are unlikely to be so high as to violate that condition. 6 Proposition 1: If an agricultural household faces a stochastic price for a commodity that it both produces and consumes, then the household will apply labor in excess of the quantity that would equilibrate labor’s marginal value product with the prevailing wage rate if and only if that price is positively correlated with the marginal utility of