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

Eakin, B. Kelly

I. 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.

II. 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 represents the consumption side of the model and is given by (6a), (6b), and (5). Substituting solutions (7) into budget constraint (5) and solving the consumption equations yields optimal values for goods of the form (8a) [Mathematical Expression Omitted] (8b) [Mathematical Expression Omitted]

Two important conclusions emerge from this model. In a world of perfect markets where a single entrepreneur can be identified for each firm, utility maximization implies profit maximization. The intuition is as follows. To maximize utility the entrepreneur must necessarily maximize full-income. This is accomplished by choosing those quantities of inputs and outputs that maximize observable profit. Maximization of observable profit results in maximization of economic profit since the two are identical when markets exist and function smoothly. This is a generalization and rigorous affirmation of the conclusion of Feinberg [8] and provides the foundation for a more general analysis of the consistency of utility and profit maximization.

Since profit maximization is a necessary condition for utility maximization, the utility-maximizing entrepreneur will make the same input and output choices as the profit-maximizing entrepreneur. However, given the nature of the budget constraint the behavior of the entrepreneur as consumer will differ somewhat from the traditional case. This assertion is demonstrated formally in section IV of this paper.(1)

III. The Imperfect Markets Model

The assumption of perfect markets is clearly very stringent. For certain types of owner-controlled firms the entrepreneurial input may be heterogeneous or perform a firm specific function and therefore a perfect market will not exist for this factor. Alternatively, the entrepreneur may derive utility from nonpecuniary goods which must be obtained within the firm since markets do not exist for these goods. Several additional examples of imperfect markets can be cited.

The existence of one or more imperfect markets can be interpreted as placing additional restrictions on the entrepreneur's production and consumption choices relative to an environment of perfect markets. In analyzing the behavior of the entrepreneur in these instances, it is necessary to incorporate these added constraints into the perfect markets model. These market environment constraints make explicit the nature of markets for goods and inputs. That is, if perfect markets exist for all goods and inputs so that they can be freely traded, then the market environment constraints are not binding and the results are the same as those in the perfect markets model. Alternatively, if one or more imperfect markets prevail, then the pertinent market environment constraints might be binding. If such a constraint is binding, the good or input to which it relates cannot be freely exchanged. This places additional restrictions on the entrepreneur's feasible choice set resulting in a divergence between virtual and market prices. Since virtual prices reflect opportunity costs and benefits that are relevant when making utility-maximizing choices, the entrepreneur will respond directly to virtual prices rather than market prices.(2)

The general expressions for the market environment constraints for the ith commodity and jth input are (9a) [Mathematical Expression Omitted] (9b) [Mathematical Expression Omitted] where [G.sub.j] and [R.sub.j] are market restriction parameters and all other variables have been defined previously. To make explicit the assumption about the nature of the market for the ith commodity and jth input, it is necessary to specify an equality or inequality constraint and the magnitude of restriction parameters. If either (9a) or (9b) are binding, then inclusion in the constrained utility maximization problem will influence the entrepreneur's effective choice set and consequently his behavior relative to the perfect markets case.

Models of the owner operated firm typically assume a market imperfection for a good or input. The nature of this imperfection varies among models. For example, Graaff [14] suggest that the entrepreneur possesses an extreme form of firm-specific human capital so that his services can be neither acquired from outside the firm nor provided to other firms. Olsen [23; 24], Feinberg [8], and others [11; 15; 28] cite monitoring costs, shirking, preferences for self-employment, nepotism, and discrimination as sources of market imperfections. All of these assumptions can be translated into market environment constraints of the form (9) and the implications of these market imperfections on entrepreneurial choice can then be readily deduced.

The problem of entrepreneur is (10a) max U ([X.sub.1], . . . ,[X.sub.n]; [Y.sub.1], . . . ,[Y.sub.m]) subject to(3) (10b) [Mathematical Expression Omitted] (10c) F([Q.sub.1], . . . , [Q.sub.n]; [V.sub.1], . . . , [V.sub.m]) = 0 (10d) [Q.sub.i] - [X.sub.i] = [G.sub.i] i = 1, . . . , n (10e) [V.sub.] + [Y.sub.j] - [Mathematical Expression Omitted] = [R.sub.j] j = 1, . . . , m. The Lagrangian function for this problem is (11) [Mathematical Expression Omitted] where [Lambda], [Mu], [[Sigma].sub.i], and [[Phi].sub.j] are Lagrangian multipliers. For generality, the constrained utility maximization problem of the entrepreneur is written in a form that allows for the possibility of a binding market environment constraint for any commodity or input. If such a constraint is not binding, it merely vanishes. Assuming interior solutions, the first-order conditions can be expressed as (12a) [Mathematical Expression Omitted] (12b) [Mathematical Expression Omitted] (12c) [Mathematical Expression Omitted] (12d) [Mathematical Expression Omitted] along with constraints (10b) through (10e).

The parenthetic expressions in first-order equations (12) suggest the notion of a utility-maximizing price that may diverge from the observable market price. This idea can be formalized by employing the construct of a virtual price. The notion of virtual prices was first introduced by Rothbarth [27] and subsequently extended by Graaff [13] and Neary and Roberts [22] who employ virtual prices to analyze household consumption behavior under rationing. We, however, apply this concept more generally to an entrepreneur who owns and controls a firm.

In our model, virtual prices are defined as the set of prices that would induce the entrepreneur to make the same choices in a perfect markets environment as he actually makes when there exists one or more imperfect markets. To make this definition explicit, consider the ith commodity consumed. The first-order condition for this commodity is (13) [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is the virtual price. If this commodity is traded on a perfect market the virtual price equals the market price, [P.sub.i]. However, suppose that the ith commodity is exchanged on an imperfect market for which the market environment constraint binds. In this case, the first order condition is (14) [Alpha] U/ [Alpha] [X.sub.i] = [Lambda] [P.sub.i] - [[Sigma].sub.i] where the Lagrangian multiplier [[Sigma].sub.i] is the shadow price of the market environment constraint.(4) Solving (13) and (14) for the virtual price yields (15) [Mathematical Expression Omitted] Equation (15) reveals that the virtual price of the ith commodity is equal to the sum of the market price and the ratio of the shadow price of the market environment constraint to the marginal utility of income. If in perfect market the entrepreneur would maximize utility by consuming more (less) of the ith commodity than he consumes in an imperfect market, then the shadow price would be negative (positive and consequently the virtual price would be greater (less) than the market price. If in a perfect market the entrepreneur would maximize utility by consuming the same amount of the ith commodity as he consumes in an imperfect market, then the shadow price would be zero and hence the virtual price would equal the market price. However, this is merely the case where the market environment constraint in not binding.

Similarly, the virtual price of the jth input consumed [Mathematical Expression Omitted] is (16) [Mathematical Expression Omitted] The virtual prices of the ith output produced and jth input employed assume forms identical to (15) and 16), respectively. Since [Sigma] i , [Phi] j, and [Lambda] are determined by the solution to the constrained utility maximization problem, virtual prices are endogenous variables. Moreover, it can be easily shown that the market price divergence terms are given by [[Sigma].sub.i]/[Lambda] = [[Delta] [[Pi].sup.obs]/[Delta] [G.sub.i] and [[Phi].sub.j]/[Lambda] = [Delta] [[Pi].sup.obs]/[Delta] [R.sub.j], where [[Pi] [sup.obs] is observable profit, and therefore depend on the amount by which observable profit would change in response to an infinitesimal relaxation of the market restriction parameter.

Analytically, the perfect markets model can be viewed as a special case of the more general imperfect markets model since the latter reduces to the former only if virtual and market prices coincide.

An example concerning the nature of the market for the entrepreneurial input will help to elucidate the intuition underlying the model. Following Olsen [24], let us suppose that the entrepreneur perceives hired management to be non-substitutable for managerial services which he himself provides to his firm.(5) This assumption is made explicit by defining the market environment constraint [V.sub.m] - [Z.sub.m] [is less than or equal to] O where [V.sub.m] is employment of managerial services and [Z.sub.m] is the amount of input services supplied by the entrepreneur. This constraint indicate that [V.sub.m] > [Z.sub.m] are perceived as irrelevant alternatives and therefore reside outside of the entrepreneur's effective choice set. Should the entrepreneur wish to be a net supplier of input services at the prevailing market wage [W.sub.m], the market environment constraint is non-binding and hence market and virtual prices coincide. However, if the entrepreneur desires to be a net hirer of a homogeneous managerial input, this is not possible. As a result, the market environment constraint is strictly binding and [Mathematical Expression Omitted] which implies [Mathematical Expression Omitted]. That is, the virtual price exceeds the observable market price and does so by the amount by which observable profit would change as a result of a marginal relaxation of the market environment constraint which when binding compels the entrepreneur to equate his own work effort with the quantity of managerial services that he employs. Relaxation of the constraint i.e., d [R.sub.m] > O, would result in an increase in observable profit by an amount equal to the difference between the marginal revenue product of the entrepreneurial input and observable market wage, the former evaluated at the point of constrained utility maximization.(6)

Making assumptions explicit through the use of market environment constraints can expose conceptually incorrect measures of cost. For instance, implicit in the analyses of Scitovszky [29], Ladd [17], and Piron [25] is the supposition that the entrepreneur performs a firm specific function such that the services which he renders cannot be purchased or sold on a market. This assumption is made explicit by the market environment constraint [V.sub.m] - [Z.sub.m] = O.(7) In the context of the theoretical framework employed by these writers, it is easily demonstrated that the virtual price is given by the value of the entrepreneur's marginal product evaluated at the at the point of constrained utility maximization. However, following Scitovszky all of these writers measure the cost of the entrepreneurial input using the income/leisure indifference curve that "represents the minimum satisfaction that will keep the entrepreneur in his profession" [29, 58].

IV. Comparative Static Properties

To investigate the comparative static behavior of the entrepreneur, we analyze the partial derivatives of the utility-maximizing choice functions. Total differentiation of the first-order equations (12), (10b), (10c), (10d), and (10e) yields the matrix equation (17) Hs = d (17) where H is a square, symmetric, bordered Hessian matrix of dimension 3n + 3m + 2, d is a vector of constants, and s is a solution vector s = [Mathematical Expression Omitted] for i = 1, . . . , n, j = 1,..., m. Using Cramer's rule, we write all price derivatives in the form of the following Slutsky type equations(8) (18a) Mathematical Expression Omitted] (18b) [Mathematical Expression Omitted] (18c) [Mathematical Expression Omitted] (18d) [Mathematical Expression Omitted] (18e) [Mathematical Expression Omitted] (18f) [Mathematical Expression Omitted] (18g) [Mathematical Expression Omitted] (18h) [Mathematical Expression Omitted] where [Lambda] is the marginal utility of full-income, ~H~ is the determinant of H, and [C.sub.ab] is the cofactor of the element in the ath row and bth column of ~H~.

Derivative properties (18) indicate that a change in the price of the ith commodity or jth input initiates two potential effects on the optimal amount of an activity chosen. The first term on the right hand side is a substitution effect, the second term is an income effect. Analogous to ordinary consumer theory, utility-maximizing behavior in general places no restrictions on the sign of the income effect.

Since the matrix H is symmetric, the cofactor matrix of H is also symmetric so that [C.sub.ab] = [C.sub.ba] for all a and b. Thus, if there exists only a single cofactor in the substitution term, then a set of symmetry conditions characterize cross substitution effects. Moreover, by the second order conditions for utility maximization, H must be negative definite so that all direct substitution effects are well-behaved and unambiguous in sign. However, the general expressions for the substitution effects in (18) contain multiple cofactors. This raises the possibility of asymmetric and ill-behaved substitution effects on production and consumption choices. This possibility was noted by Graaff [14] for the case where the owner operator provides an "entrepreneurial service which can be made available to no firm but his own". We find that this result holds generally.

The case of perfect markets is nested within the more general imperfect markets model. If we impose the restriction of perfect markets for all goods and inputs substitution effects become symmetric and income effects in production disappear. If in addition, no commodities are both produced and consumed by the owner, then the income effect on consumption is the same as results found in Slutsky [30]. However, income effects on consumption may differ from Slutsky income effects if any goods are both produced and consumed by the owner. Inspection of derivative property (19c) reveals that when the kth commodity is both consumed and produced by the entrepreneur, the expression for the income effect includes the additional term [Mathematical Expression Omitted]. This term, which represents the effect of a change in the price of the ith commodity on profit and hence entrepreneurial income, could theoretically swamp the traditional substitution and income effects resulting in a positively sloped demand curve for a normal good.(9)

Assume that a binding market environment constraint exists for the ith commodity or jth input both supplied and demanded by the entrepreneur and in addition that the kth commodity (hth input) is either supplied or demanded. In this situation, the asymmetric cofactor in the substitution term vanishes for all kth commodity (hth input) price derivatives, except the ith/jth price; the income effect term will generally be non-zero. Thus, price disturbances generate income effects on kth output (hth input) production choices and substitution effects are symmetric and well-behaved for all choices except those pertaining to price changes for the ith commodity or jth input.

Finally, let us assume that a binding market environment constraint exists for the ith commodity (jth input) both supplied and demanded, and the entrepreneur both supplies and demands the kth commodity (hth input). In this case, neither the income effect nor asymmetric element in the substitution term vanish. Optimal kth (hth) choices are subject to income effects and asymmetric cross substitution effects. Moreover, the possibility of ill-behaved direct substitution effects now exists because the second-order conditions place no restrictions on the sign of the off-diagonal cofactors.

To summarize, binding market environment constraint for commodities or inputs both supplied and demanded result in income effects on production as well as consumption choices. Furthermore, these commodities and inputs have substitution effects which are asymmetric and ambiguous in sign. These are important results because they provide the basis for the theoretical possibility of downward sloping output supply curves and upward sloping input demand curves. Furthermore, these results may prove useful in understanding other apparent irregularities and suggest generalization of theoretical implication such as Roy's Identity and Shephard's and Hotelling's lemmas.

V. Economic Profit, Observable Profit and Accounting Profit

Consistency of utility and profit maximization has been an ongoing debate [17; 25; 23; 24; 3; 8; 9; 10; 28; 15; 11]. The meaningful questions are 1) does utility maximization imply maximization of observable profit? and 2) if the firm does not maximize observable profit, can it survive in the long run? The debate has been unclear on what is meant by profit maximization because there has not been an explicit distinction between economic, observable and accounting profit. Once these different profit concepts are clearly defined, the debate is easily resolved.

Since optimal choices of the entrepreneur are based on virtual prices, these prices are the prices relevant for a behavioral profit measure. Therefore, we define economic profit, [Pi]*, as the difference between revenue and cost calculated using virtual prices. That is, (19) [Mathematical Expression Omitted] The prices in (19) are virtual prices given in (15) and (16). This measure of [Pi]* is not an exact measure of profit because virtual prices are marginal rather than average prices. However, we believe the measure given by (19) is appropriately called economic profit because it directly embodies the optimizing behavior of the entrepreneur and consequently assumes the resource allocation role played by profit in traditional theory.

Defining an exact behavioral measure of profit is neither possible nor necessary in analyzing entrepreneurial behavior when utility maximization is postulated. An exact measure is not possible because virtual prices are only defined when the entrepreneur is in equilibrium.(10) An exact measure is not needed because economic profit as defined by (19) answers the meaningful questions regarding entrepreneurial behavior.

Observable profit is the difference between total revenue and total cost measured by observable market prices. To calculate observable profit, implicit revenue and cost from self-produced consumption and self-owned inputs are evaluated using market prices. The measure of observable profit is (20) [Mathematical Expression Omitted]

Accounting profit is defined in the traditional way as the difference between explicit revenue and explicit cost. That is, (21) [Mathematical Expression Omitted] Accounting profit represents cash flow and differs from observable profit if there are any self-produced consumption commodities or self-owned inputs.

Observable profit is equal to economic profit if perfect markets exist for all commodities and inputs relevant to the entrepreneur under investigation because then there is a coincidence of virtual and market prices. However let us suppose that kth commodity and lth input are not traded on perfect markets. In this case, economic profit becomes (22) [Mathematical Expression Omitted] which can be rewritten as (23) [Mathematical Expression Omitted] It follows directly from (23) that (24) [Mathematical Expression Omitted] That is, for the utility-maximizing entrepreneur, economic profit can be greater than, equal to, or less than observable profit.

It is clear that employing observed profit, equation (20), as a measure of economic profit can be in error when dealing with a utility-maximizing entrepreneur who makes choices in an environment of imperfect markets. However, Olsen [23] and Feinberg [8;9] implicitly impose binding market environment constraints on at least one commodity or input but then employ a measure equivalent to (20) for profit. Olsen argues that this is an "objective" measure of economic profit while Feinberg maintains that using any magnitude other than an observable market price reduces the notion of economic profit to a tautology. We disagree with both these views. Virtual prices are the prices determining individual behavior. These prices are derived from the individual's subjective utility function and embody market environment constraints. Only in a world of perfect markets can objective market prices be used to calculate a behavioral measure of profit.

Economic profit as defined above embodies entrepreneurial behavior and explains resources allocation. In this form it is easy to verify that economic and observable profit in general do not coincide and that utility maximization does not imply maximization of observable profit.

The second question of the debate is whether firms that fail to maximize observable profit can survive in the long run.(11) Our measure of economic profit based on virtual prices further illuminates this question. There are two aspects to the survivability questions: i) under what conditions would the firm choose to stay in the industry? and ii) under what conditions will the firm be forced to exit the industry? Assuming no irregularities such as non-convexities, the firm chooses to remain in the industry if economic profit is non-negative. The owner is deriving greater utility from operating in the industry than he would derive from using self-owned inputs in their best alternative areas of employment. The utility-maximizing firm nevertheless faces a cash flow constraint. This constraint requires non-negative accounting profit given by (21). If accounting profits are continually negative, then the firm would eventually be forced from the industry.

Consider the following scenario. Let us suppose that we have two types of firms selling a homogenous product in a competitive commodity market and purchasing inputs in competitive factor markets. Type A firms are comprised of traditional neoclassical profit maximizers. Type B firms are utility-maximizing owner operated firms whose owners derive satisfaction from at least one good not traded a perfect market. Given the assumption of competitive commodity and input markets, type A firms face exogenously given prices and therefore economic profit is correctly measured by observable profit.

Now suppose that in long-run competitive equilibrium type A firms are experiencing zero observable and hence zero economic profit whereas type B firms are recording negative observable profit. One might hasten to conclude that type B firms cannot survive. However, if economic profit as measured by (19) is non-negative, then these firms may indeed survive and choose to maintain their self-owned inputs, including entrepreneurial services, in their present area of employment. Intuitively, condition (24) suggests that the utility-maximizing entrepreneur may be willing to sacrifice observable profit for sources of utility that cannot be acquired in the marketplace. Only if type B firm experience negative economic profit as measured by (19) or negative accounting profit as measured by (21) will they exit the industry and reallocate self-owned inputs to other uses.

The results that negative economic profit implies exit from the industry is dependent upon the absence of non-convexities in preferences and technology. However, if such non-convexities do exist, economic profit measured in terms of virtual prices may not be a consistent indicator of entry/exit decisions and hence resource allocation.(12) Consider a simple economy with two goods, output and leisure, and a single input, labor. Figure 1 depicts the choice problem for a representative entrepreneur. Distance OT on the horizontal axis measures total time endowment with point T the point of maximum leisure and zero work. The slope of line TC measures the market wage in terms of output when the entrepreneur engages in non-entrepreneurial activity. If the entrepreneur chooses to work for himself, the production possibilities set is given by TBAO, which indicates that TB hours of work effort is required before positive production is possible. If a perfect market exists for entrepreneurial labor, then a feasible consumption set for the entrepreneur is given by OANLT. To maximize utility, the entrepreneur produces at point N where the marginal product equals the real market wage thereby maximizing observable profit, the vertical distance between point N and point P. He then consumes at point F where the marginal rate of substitution equals the real market wage thereby working TS hours at his own firm and employing SR hours of hired labor, a perfect substitute.

Now assume that an imperfection exists in the market for entrepreneurial labor so that the entrepreneur cannot be a net purchaser at the prevailing market wage. In this case, the entrepreneur must equate his own labor demand and supply so that point E depicts the utility-maximizing position. The slope of line JK measures the virtual wage, which clearly exceeds the market wage. Economic profit evaluated in terms of the virtual wage is negative and equal to the vertical distance between point E and point H. If the entrepreneur could sell his labor services at this virtual wage, he would maximize utility (point G on indifference curve IV) by closing down his firm and working for another firm. However, given the market imperfection the wage rate available in the outside market is below the virtual wage. Therefore, even though the entrepreneur is experiencing negative economic profit measured in terms of virtual prices he will still choose to operate in the industry. This is because operating the firm results in a higher utility level (point E on indifference curve II) than shutting down the firm and supplying labor at the market wage (point D on indeferrence curve I). This paradox arises because the virtual wage is a marginal rather than average value and varies with the level of work effort. However, this type of inconsistency between utility-maximizing choices and economic profit would not exist if the production possibilities set were convex.

In the absence of non-convexities, utility-maximizing firms experiencing non-negative economic profit will choose to remain in the industry while those experiencing negative economic profits will choose to exit. Long-run feasibility requires that accounting profit be non-negative. Thus, long-run survival requires that both economic an accounting profit be non-negative, but puts no restrictions on the largely irrelevant observed profit.

VI. Non-Cost Minimizing Behavior

The model of the power operated firm we develop in this paper suggests that the entrepreneur may trade-off observable profit for goods that cannot be acquired on perfect markets. Possibilities include such "goods" as nepotism, discrimination, against certain types of inputs, and charitable contributions in the form of excessive wages paid to employees. The existence of these non-market goods result in the divergence of virtual prices for inputs from their observed prices. The result is non-minimization of observed costs. One might be tempted to label this phenomenon "X-inefficiency" and conclude that it is a firm specific problem.(13) However, we believe our model offers advantages over the X-inefficiency theory. We maintain a framework where seemingly inefficient behavior actually from rational decision making. Our model of the owner operated firm provides more structure and explanation and forces attention on market failures. We have shown that if perfect markets exists for all commodities and inputs, then utility-maximizing firms would make the same choices as competitive profit-maximizing firms and achieve economic efficiency from a social view. In our model, the explicit source of seemingly inefficient behavior is the existence of an imperfect or absent market instead of a more nebulous "firm specific inefficiency.

Our model of the owner operated firm also provides greater theoretical foundations for models of non-cost minimizing behavior such as those presented by Toda [31], Atkinson and Halvorsen [2] and Eakin and Kniesner [7]. These models relax the cost-minimization assumption by letting perceived input prices systematically differ from observed market prices. The term capturing the divergence is typically modelled as a single parameter to be estimated. Our theory provides an explanation for this deviation, but also makes explicit the endogeneity of the shadow price. While we have bolstered the theoretical foundations of the none-minimum cost function, we have also exposed an inconsistency in these empirical models which needs to be rectified.

VII. Summary and Conclusions

We have presented a general theory of the owner operate firm in which market imperfections result in deviations between virtual and observed prices. When markets imperfection are not binding constraints, the model reduces to one of perfect markets and there is a convergence of virtual and observed prices and economic an observed profit. The comparative static properties for the owner operated firm show that binding market constraints result in income effects in production and asymmetric substitution effects. These non-traditional effects originate solely from market imperfections and not from restrictive assumptions, about technology, preferences or behavior. The existence of these effects gives rise to the theoretical possibility of downward sloping output supply curves and upward sloping input demand curves.

We resolve the debate on the consistency of utility maximization and profit maximization by demonstrating that in the presence of imperfect markets economic profit diverges from observed profit. Observed profit is not directly relevant to either the firm's objective or survival. We also have shown that market imperfections may be the source of seemingly uneconomic behavior, thereby providing a theoretical foundation for models of X-inefficiency and systematic allocative inefficiency. (1)The perfect markets model is structurally equivalent to the variety of recursive farm household models for developing economies [16; 19; 4]. In this sort of model estimating the production and consumption sides separately greatly facilitates the estimation procedure. (2)The concept of virtual prices is explained later in this section. (3)Equations (2) and (9b) imply that [Mathematical Expression Omitted]. This is the form of the last constraints facing tthe entrepreneur given by equation (10e). (4)This usage of the term shadow price is consistent with that of Neary and Roberts [22]. (5)Theoretical and empirical evidence supporting this assumption is in Olsen [24]. (6)In the case of one output, the first-order condition imply [Mathematicall Expression Omitted] where P is output price ad [Delta] F/[Delta] [V.sub.m] is the marginal product of the entreprenuer. (7)This assumption is also implicit in the short-run analysis of Olsen [23]. (8)When the jth input price changes the term [W.sub.j] [Z.sub.j] is separated from I in order to capture the full effect of the alteration in price. The net effect is to change the weight attached to the income derivative from [Mathematical Expression Omitted]. This is true for j = I, ..., m. (9)This non-traditional income effect is widely recognized in the development literature and has been found principally to generate sloped demand curves for farm households in these economies [4; 19]. (10)Of course in a world of perfect markets economic profit and observable profit coincide and constitute an exact measure. (11)In addition to those papers cited in section I, see Reder [26], Alchian [1], Friedman [12], and Williamson [32]. (12)We thank the referee for pointing out this possibility. (13)For a discussion of the notion of X-inefficiency see Leibenstein [20; 21].

References

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