Estimating the choice behavior of self-employed business proprietors: an application to dairy farmers.

• Author: Thornton, James

1. Introduction

   The vast majority of business concerns operating in contemporary market oriented economies are owned and controlled by a single individual and perhaps family members. Yet little work has been performed in the area of estimating the choice behavior of self-employed business proprietors, particularly under conditions where production and consumption decisions are interdependent.

   Lau, Lin, and Yotopolous |16~, Barnum and Squire |2~, and others have developed models to analyze and estimate the complete choice behavior of farm households in developing nations. These analyses invariably separate production and consumption decisions so that the sole link between the two sides of the model lies in the impact of farm profit on the household budget constraint. This type of separability implicitly assumes the existence of perfect markets for all goods and inputs.(1) While this assumption may be a valid approximation for many farm households in developing economies, it is not generally applicable to self-employed proprietors, especially those operating in more advanced economies |8; 19; 21; 26~.

   To my knowledge, only two prior empirical studies have been performed which attempt to analyze the choice behavior of self-employed business proprietors under conditions where production and consumption decisions are interdependent. Brown and Lapan |4~ employ aggregate U.S. time series data to estimate physician labor and output supply responses to changes in output price. Their estimation procedure centers on a single reduced form output supply equation. Parameter estimates are interpreted in the context of the theoretical framework assuming constant returns to scale technology and homothetic preferences. They conclude that while the physician labor supply curve is backward bending, the output supply curve is positively sloped.

   Lopez |19~ formulates a model analyzing the interdependent production and labor supply decisions of households that own and operate a farm. His framework integrates output, input, and labor supply choices, and allows for differences in preferences between on-farm and off-farm work. Imposing constant returns to scale and using cross-section Canadian data aggregated by census division, he jointly estimates output supply, input demand, on-farm, and off-farm labor supply equations. Several elasticity estimates are reported which are consistent with economic theory, however, standard errors are not provided for these estimates.

   In this paper, I analyze the choice behavior of self-employed dairy farmers who also work off the farm. This makes a substantial contribution to the existing small body of empirical literature examining proprietor choice behavior in developed economies. The distinguishing features of this study are the following. First, the model estimated postulates interdependent production and consumption decisions and permits the proprietor's choice behavior to be based on utility maximization. In the present context, this approach is superior to the estimation of a separable model and/or one that employs ad hoc reduced form regression analysis. Second, the model is estimated using individual household level cross-sectional data obtained from a random sample of Utah dairy farmers. This differs from the two previous empirical studies of this nature which employ aggregated data. Finally, the model does not impose the restriction of constant returns to scale technology and allows for the existence of fixed factors of production. This results in a more general analysis as well as allowing for an explicit test of the constant returns to scale hypothesis. Both prior studies in this area assume no fixed inputs and production technology homogeneous of degree one.

   The model and findings presented in this paper have interesting policy implications for self-employed dairy farm households who have opportunities for outside employment. The results suggest that alterations in lump-sum subsidies and those that generate price effects have differential impacts on output supply behavior and the allocation of work effort and other resources to dairy farm activities. Moreover, these types of self-employed business proprietors may well respond to income shocks arising from exogenous price changes and policy actions largely through adjustments in wage employment rather than adjustments in self-employment agricultural activities.

   The remainder of this paper is organized in the following manner. Section II presents a theoretical model of proprietor choice behavior for farm households that engage in both self-employment and wage employment. In section III, the econometric model employed to estimate dairy farm production and consumption responses is described. Section IV discusses estimation methods and results. The paper is summarized in section V.

2. A Model of Proprietor Behavior

   Following Lopez |19~, it is assumed that the farm household engages in both on-farm self-employment and off-farm wage employment. In addition, the farm household has different preferences for on-farm and off-farm work and provides input services to the dairy operation that are unique. Consequently, hired labor is not a perfect substitute in production. The absence of a perfect labor market implies that the price of on-farm input services provided by the household is not an exogenous market price, but rather an endogenous virtual price. This virtual price is defined as the price that would induce the farm household to make the same choices in a perfect labor market as it actually makes in the present environment.(2) It is this virtual price, not an observable market wage, that is relevant to the household when making the on-farm work choice.

   The farm household wishes to maximize a continuous, monotonic, quasi-concave utility function

   U = U(X, T - |L.sub.f~, T - |L.sub.o~) (1)

   where X is an n-dimensional vector of consumption goods, T is farm household total time endowment, and |L.sub.f~ and |L.sub.o~ are on-farm and off-farm labor services. It is assumed that the farm household does not consume its own output. Utility function (1) is maximized subject to time, technology, and budget constraints.

   The time constraint is given by

   |L.sub.f~ + |L.sub.o~ |is less than or equal to~ T. (2)

   Technology is given by a continuous, monotonic, quasi-concave production function

   Q = Q(V, Z, |L.sub.f~) (3)

   where Q is farm output, V is an m-dimensional vector of variable inputs, and Z is a k-dimensional vector of fixed inputs. The household budget constraint is

   |P.sub.X~X = (|P.sub.Q~Q - |W.sub.V~V - |W.sub.Z~Z) + |W.sub.o~|L.sub.o~ + M (4)

   where |P.sub.X~ is an n-dimensional vector of consumption good prices, |W.sub.V~ is an m-dimensional vector of variable input prices, |W.sub.Z~ is an k-dimensional vector of fixed input prices, |P.sub.Q~ is farm output price, |W.sub.o~ is the off-farm wage rate, and M is exogenous income, henceforth called asset income. It is assumed that |P.sub.X~, |W.sub.V~, |W.sub.Z~, |W.sub.o~, |P.sub.Q~, and M are exogenous. The expression inside parentheses represents household net farm income defined as the difference between total receipts and total outlays on variable and fixed inputs. Note that this expression does not measure profit since it excludes the imputed cost of farm household input services. Substitution of (3) into (4) yields the production function augmented budget constraint

   |P.sub.X~X = ||P.sub.Q~Q(V, Z, |L.sub.f~) - |W.sub.V~V - |W.sub.Z~Z~ + |W.sub.o~|L.sub.o~ + M. (5)

   The problem of the farm household is to maximize utility function (1) subject to budget constraint (5) and the time constraint. Assuming interior solutions for all choices, the first-order necessary conditions for utility maximization are

   |Delta~U/|Delta~|X.sub.i~ = |Lambda~|P.sub.Xi~ i = 1,..., n (6a)

   |Delta~U/|Delta~(T - |L.sub.f~) = |Lambda~|P.sub.Q~|Delta~Q/|Delta~|L.sub.f~ (6b)

   |Delta~U/|Delta~(T - |L.sub.o~) = |Lambda~|W.sub.o~ (6c)

   |P.sub.Q~|Delta~Q/|Delta~|V.sub.j~ = |W.sub.Vj~ j = 1,..., m (6d)

   and time and budget constraints. The Lagrangian multiplier |Lambda~ gives the marginal utility of income. Equations (6b) and (6c) indicate that the prices to which the farm household responds when making utility maximizing on-farm and off-farm work decisions differ. The price of off-farm labor supply is the parametric market wage |W.sub.o~, however, the price of on-farm work effort is an endogenous virtual price. This virtual price is given by the farm household's value of marginal product associated with on-farm input services. Given the assumed properties of the utility and production functions, the second-order conditions for a maximum are satisfied and therefore the first-order conditions yield a set of utility maximizing demand and supply functions for production, consumption, and labor supply choices.

   The theoretical model presented above is not sufficient to guide econometric specification. However, necessary conditions (6) suggest a more useful way to reformulate the model and conceptualize the problem of the farm household. Equation (6d) implies that for any quantity of |L.sub.f~ chosen, amounts of variable inputs will be selected that maximize net farm income conditional on that value of |L.sub.f~. This idea can be formalized by defining the net income function

   N(|P.sub.Q~, |W.sub.V~, |W.sub.Z~, Z; |L.sub.f~) = G(|P.sub.Q~, |W.sub.V~, Z; |L.sub.f~) - |W.sub.Z~Z (7)

   where

   G(|P.sub.Q~, |W.sub.V~, Z; |L.sub.f~) = max{|P.sub.Q~Q(V, Z, |L.sub.f~) - |W.sub.V~V}. (8)

   The net income function gives the solution to the farm household's net farm income maximization problem for predetermined |L.sub.f~, fixed Z, and variable V. It is easily demonstrated that the function G( ) possesses all the properties of the variable profit function.(3)

   The farm household's budget constraint is now rewritten as

   |P.sub.X~X = N(|P.sub.Q~, |W.sub.V~, |W.sub.Z~, Z; |L.sub.f~) + |W.sub.o~|L.sub.o~ + M. (9)

   Intuitively, the farm household can be...