Virtual prices and a general theory of the owner operated firm.

• Author: Thornton, James

1. Introduction

   Theories of entrepreneurial behavior for owner operated firms have a long history in economics. These models treat the owner operator as both a producer and consumer of goods. Production and consumption choices result from utility maximization. A preponderance of the work performed in this area has addressed two theoretical issues, the consistency of utility and profit maximization and comparative static behavior of the entrepreneur.

   Scitovszky's seminal paper [29] demonstrates that under certain circumstances the entrepreneur may trade-off income for leisure giving rise to the possibility of income effects on production and inconsistency of utility and profit maximization. Building on Scitovszky's work, Graaff [14], Clower [6], Auster and Silver [3], and Lapan and Brown [18], analyze the comparative static behavior of the utility-maximizing owner operator and how this differs from the traditional neo-classical profit-maximizing firm. Emphasis is on the possibility of income effects on production and how they my result in seemingly uneconomic behavior. A shortcoming of this literature is the focus on special cases based on strong restrictions on technology (constant returns to scale) and the market for the entrepreneurial input (no such market exists). As of yet, a general analysis of the comparative static behavior of the entrepreneur has not been undertaken.

   Scitovszky's work also initiated a vigorous debate concerning the consistency of utility and profit maximization for owner operated firms. This debate centers on two principal questions. Does utility maximization imply profit maximization? If not, can a utility-maximizing owner operated firm survive through time? Piron [25], Olsen [23; 24], and Hannon [15] conclude that in competitive long-run equilibrium utility-maximizing entrepreneurs necessarily maximize profit and must do so to remain viable. Ladd [17], Auster and Silver [3], Feinberg, [8; 9; 10], and Schlesinger [28] conclude that utility and profit maximization may diverge. Moreover, these writers maintain that non-profit-maximizing owner operated firms can survive through time. More recently, Formby and Millner [11] argue that utility and profit maximization necessarily converge for the marginal firm only.

   The questions raised concerning the consistency of utility and profit maximization have yet to be resolved. Different conclusions have been deduced from a diversity of assumptions concerning markets for commodities and inputs, technology, preferences, and measurement of profit. What remains to settle this debate is a general framework of entrepreneurial behavior that is capable of incorporating the variety of assumptions present in the literature and yielding theoretically consistent behavioral measure of profit.

   In this paper, we present a general model of entrepreneurial behavior for owner operated firms based on the notion of virtual prices. Our model has three specific advantages: (1) it has existing theories as special cases which result from restrictions on the entrepreneur's choice set, preferences, and/or technology; (2) it provides framework for conducting a general analysis of the comparative static behavior of the owner operator and (3) it resolves the debate concerning the consistency of utility and profit maximization.

   The remainder of this paper is organized as follows. Section II presents the perfect markets model of entrepreneurial behavior for owner operated firms. We show that utility maximization implies profit maximization when perfect markets exist for all goods and inputs. Section III formulates the imperfect markets model based on the notion of virtual prices and demonstrates that perfect markets are a special case of this more general analytical framework. Section IV examines the comparative static behavior of the entrepreneur. We conclude that utility and profit-maximizing firm behavior are indistinguishable in an environment of perfect markets; however, the presence of market imperfections gives rise to the possibility of income effects on production and ill-behaved substitution effects. Section V makes the important distinction between economic profit, observable profit and accounting profit. A measure of economic profit, incorporating virtual prices, is derived from our general framework and employed to analyze the consistency of utility and profit maximization and firm viability. Section VI argues that non-cost minimizing owner operator behavior results from market failures rather than X-inefficiency. Section VII summarizes.

2. The Perfect Markets Model

   The decision making unit of interest is the entrepreneur. The entrepreneur is defined as a single individual who owns and controls a firm where "control" implies ultimate decision making authority. By assumption, all goods and inputs that pertain to the entrepreneur under investigation are traded on perfect markets. A perfect market exists when a good or input is exchanged on a competitive market and constitutes a perfect substitute for a good or input supplied and/or demanded by the entrepreneur. The assumption of perfect markets places the lower bound of restrictions on the entrepreneur's choice set and implies that market prices reflect true opportunity costs and benefits in the decision making process.

   The entrepreneur wishes to maximize a twice continuously differentiable, monotonic, quasi-concave utility function (1) [Mathematical Expression Omitted] where [X.sub.1], . . . ,[X.sub.n] are commodities and [Y.sub.1], . . . ,[Y.sub.m] are inputs. The set of arguments [X..sub.1], . . . ,[X.sub.n], [Y.sub.1], . . . ,[Y.sub.m] are goods where a good is defined broadly as any object that is a direct source of utility. We permit the entrepreneur to experience utility directly from consumption of inputs that can alternatively be employed to produce commodities. The most prominent example of such an input is the entrepreneur's time input. Henceforth, the consumption of the time input will be called leisure and designed [Y.sub.m].

   The entrepreneur has at his disposal an initial endowment of resources [Mathematical Expression Omitted], called self-owned inputs, and given technology. Self-owned inputs must satisfy the constraint (2) [Mathematical Expression Omitted] where [Mathematical Expression Omitted], and [Y.sub.i] are the initial endowment, factor supply, and consumption of the jth input, respectively. Production technology is given by (3) [Mathematical Expressions Omitted] where F is a twice continuously differentiable quasi-concave function, [Q.sub.1], . . . ,[Q.sub.n] are outputs, [V.sub.1], . . . ,[V.sub.m-1] are typical production inputs and [V.sub.m] is the entrepreneurial input.

   The general form of the entrepreneur's budget constraint is (4) [Mathematical Expression Omitted] where [P.sub.i] is the ith commodity price and [W.sub.j] is the jth input price. By assumption of perfect markets, all prices are given parametrically. The budget 8 constraint indicates that total money outlays must equal total receipts and allows for net purchases and sales of commodities and inputs.

   Constraint (4) is not independent of constraint (2) since self-owned inputs can be transformed into commodities by consuming less and employing more of these inputs. Substituting (2) into (4) and rearranging yields the single constraint (5) [Mathematical Expression Omitted] where [Mathematical Expression Omitted]. The left hand side gives total expenditures, both explicit and implicit, on all commodities an inputs consumed whereas the right hand side represents a modified version of full-income [5]. The entrepreneur's full-income is obtained from two sources: the value of his initial endowment of resources (I) and observable profit. By definition, observable profit is the difference between total revenue and total opportunity cost of the firm's operations measured in terms of observable market prices. When perfect markets exist for all commodities and inputs, observable profit and economic profit coincide. In this case, obtaining a proper measure of economic profit is a straightforward and unambiguous exercise.

   The problem of the entrepreneur is to maximize utility function (1) subject to full-income constraint (5) and technology (3). Assuming interior solutions, the first order necessary conditions for a constrained maximum are (6a) [Mathematical Expression Omitted] (6b) [Mathematical Expression Omitted] (6c) [Mathematical Expression Omitted] (6d) [Mathematical Expression Omitted] along with the budget and technology constraints, where [Lambda] and [Mu] are the Lagrangian multipliers attached to constraints (5) and (3), respectively. The Lagrangian multiplier [Lambda] gives the marginal utility of full-income. Given the assumed properties of the utility and production functions, the second-order conditions for a maximum are satisfied. The first order conditions can then be expressed as a set of implicit utility-maximizing demand and supply functions for goods, inputs, and outputs. By analyzing these first-order conditions, the economic behavior of the utility-maximizing entrepreneur operating in an environment of perfect markets can be deduced.

   To find the entrepreneur's utility-maximizing choices, the system of equations given by the first-order conditions can be decomposed into two sets and solved recursively. The first set of equations to be solved represents the production side of the model and is given by (6c), (6d), and (3). The solution yields optimal values for inputs and outputs of the form (7a) [Mathematical Expression Omitted] (7b) [Mathematical Expression Omitted] The second set of equations...