The Entrepreneurial Critique of the Optimization Paradigm

 

ROBERT F. MULLIGAN

mulligan@wcu.edu

828-227-3329

Associate Professor of Economics, Department of Business Computer Information Systems & Economics, College of Business, Western Carolina University, Cullowhee NC 28723; Research Associate, Department of Economics, State University of New York at Binghamton

 

Abstract.  The optimization paradigm in neoclassical economics is defined and criticized.  Although the optimization paradigm provides the foundation for most mathematical formalism in modern technical economics, the paradigm rest in turn on extreme and extremely unrealistic information assumptions.  No scope for entrepreneurial activity is admitted.  This paper argues that entrepreneurial change is one of the most basic ingredients for understanding the functioning of markets and the coordination of economic activity.  Several extensions of the standard model are presented and discussed.  Alternative views of market equilibrium offered by Schumpeter and Kirzner are developed, and the relationship between the two competing views of equilibrium is examined.

 

Key words: optimization paradigm, entrepreneur, resource allocation, productive activity

 

JEL classification: B41, C61, C62, D11, D40.

 

"What we cannot speak about we must pass over in silence."

Ludwig Wittgenstein[1]

 

This paper criticizes the optimization paradigm in neoclassical microeconomics with respect to its treatment of entrepreneurship.  Conventional cost minimization-profit maximization exercises central to modern neoclassical economics virtually ignore entrepreneurial action.  Entrepreneurial managers should be understood as seeking to (a) adjust the production process to extract more output from each input or more output from the same level of inputs, (b) switch inputs or combine new inputs with those already used to produce more or cheaper output, (c) devise strategies to sell their output or some part of it for a higher price, (d) attract new buyers and devise new uses for the output, and (e) lower the costs of their inputs.  In the theory of the consumer, the optimization exercise ignores entrepreneurial innovations which imply changes in the (a) number of utility function arguments, (b) form of the utility function, (c) consumer income, (d) prices of consumed goods, and (e) quality of consumed goods.  More fundamentally, the optimization paradigm ignores entrepreneurial consumers' alertness to these opportunities to costlessly improve the satisfaction of their wants. 

 

In addition to developing an Austrian critique of the prevailing orthodoxy, this paper also offers some modest extensions which partially address the critique, though they fail to overcome it.  It is hoped that as the optimization paradigm continues to be central to what is taught in graduate economics programs and much economic research, its practitioners will come to accept a more mature understanding of its limitations.

 

1.  Introduction

 

The optimization paradigm in neoclassical microeconomics is the modeling of economic behavior as a series of constrained and unconstrained optimization problems.  Market participants are thought of as knowingly optimizing known objective functions: minimizing cost, maximizing output, maximizing profit, or maximizing utility, subject to known constraints (Robbins 1935).  However, analytical functions can never be more than imperfect analogues of the processes of producing output or obtaining satisfaction through consumption[2].  In a sense, considered as mathematical models, they do not purport to be anything more.  The optimization paradigm is based on fundamental assumptions precluding the possibility of entrepreneurial activity. The uncertainty entrepreneurs overcome and exploit to earn entrepreneurial profits is removed by assumption. 

 

The optimization paradigm is usually presented through graphical analysis at the undergraduate level, while graduate courses are typically more formalized (Samuleson 1947, Debreu 1959, Silberberg 1978, Henderson and Quandt 1980, Chiang 1984, Varian 1984, Takayama 1985).  The optimization paradigm assumes away the entrepreneur and treats all the data of the market as perfectly known in advance.  This approach fundamentally misrepresents the nature of market competition.  The Austrian school has always been skeptical of mathematical formalism: "in the imaginary construction of an evenly rotating system nobody is an entrepreneur and speculator.  In any real and living economy every actor is always an entrepreneur and speculator...." (Mises 1949: 252).  Entrepreneurial action sometimes improves market coordination and moves the market toward a hypothetical equilibrium, but also sometimes increases the level of discoordination, moving the market further away from idealized equilibrium.  Thus entrepreneurship cannot be defined exclusively in terms of its relation to either equilibria or coordination among economic planners. 

 

<<Table 1 about here.>>

 

Entrepreneurial action is necessarily multifaceted and defies simple quantification.  Blaug (1998: 227) cites several different historical views of entrepreneurship.  Entrepreneurial action includes arbitrage (Cantillon 1755), coordination (Say 1803; Kirzner 1973), innovation (Schumpeter 1934), uncertainty-bearing (Knight 1921), and most recently (Casson 1982, 1985) increasing the range of available judgments on resource allocation. There is no simple way to incorporate all modes of entrepreneurship in an optimization problem.  Entrepreneurs change objective and constraint functions, subverting the analytic character of the optimization paradigm. 

 

The basic problem with the optimization paradigm may be formulated as an unwarranted assumption that all information summarized in market prices, including the preferences of others, is known to agents in advance of their decision.  Thus the optimization model of economic decision-making is perfectly deterministic. However prices cannot be known with certainty until after an exchange occurs.  In fact entrepreneurs do not optimize with respect to any objectively knowable information set. 

 

A more meaningful way to consider entrepreneurial behavior is that it proceeds experimentally and so generates the information of the market, which can never be known in advance of a voluntary exchange, but only after one takes place (Buchanan 1986), and only as having no necessary implications for potential subsequent exchanges.  Once an individual has bought or sold a good or service at a certain price, special to the time, place, and parties to the transaction, unalterable historical information of the market has been created.  But this information does not necessarily have any significance for the future, and thus objective knowledge cannot form the basis for market participants' optimizing behavior. 

 

Real behavior is better captured through less precise "rules of thumb" than through the supposedly rigorous optimization paradigm.  It seems clear to even casual observers that human behavior is not based on mathematical optimization, either underlying or explicit.  Because much human knowledge is tacit, and thus inherently decentralized and subjective, it becomes particularly difficult to justify assumptions of perfect knowledge and foresight which underlie the optimization paradigm.

 

The rest of this paper is organized as follows.  Section two discusses the entrepreneur emphasizing the Austrian tradition.  Section three presents a discussion of cost minimization. Section four presents a parallel discussion of profit maximization.  Section five introduces a theory of the entrepreneurial consumer.  Section six discusses the distinction between Schumpeter's and Kirzner's definitions of entrepreneurial action.  Finally, section seven presents concluding remarks.

 

 

2.  The Entrepreneur's Function in Society

 

Schumpeter (1934) identifies five types of entrepreneurial innovation (1) introducing new outputs or improving the quality of existing outputs, (2) introducing new methods of production, (3) opening new output markets, especially new export markets, (4) finding new sources of supply of raw materials or intermediate inputs, and (5) creating new kinds of industrial organizations.  To Schumpeter, entrepreneurs are not inventors but decision makers who allocate resources to exploit inventions;  they are not risk-bearing capitalists but borrow funds from the capitalists to finance their innovations.  Casson (1987: 151) notes that Schumpeterian entrepreneurs are defined by their managerial or decision-making role.

 

Mises (1949: 253) defines the entrepreneur as "acting man exclusively seen from the aspect of the uncertainty inherent in every action."  The entrepreneur is not merely a particular human archetype, but a category of catallactic action (ibid. 251).  To Mises uncertainty is everywhere and consequently, everyone is an entrepreneur.  But in the optimization paradigm, uncertainty has been assumed away along with the entrepreneur.  Entrepreneurial action always responds to an uncertain future, facing uncertainties of future consumer demand, changes in demand and supply in factor markets, and differences in the ability of other entrepreneurs to foresee with a compatible vision, the prerequisite for mutually compatible plans (Mises 1949: 212-214).  The entrepreneur always faces uncertainty because production takes time, so any entrepreneurial action necessarily involves "speculation in the anticipation of future events" (Greaves 1974: 39).  Lachmann (1986: 65-70, 116-117, ) also emphasizes that entrepreneurial action responds to uncertainty about the future. 

 

Much of Kirzner's (1973, 1979, 1984a, 1997) work on entrepreneurship can be reformulated as a detailed critique of the optimization paradigm.  Kirzner and Hayek stress the problem of dispersed knowledge given to no one in its entirety (Hayek 1949: 77-78), a problem whose "extent and seriousness cannot be known in advance," "arising out of unawareness of one's ignorance" (Kirzner 1984b: 162).  Mises emphasizes the entrepreneurial response to, and exploitation of, market information: "entrepreneur means acting man in regard to the changes occurring in the data of the market" (Mises 1949: 254).  The Misesian entrepreneur is alert to arbitrage opportunities which occur when prices in resource markets are not adjusted to prices in the product markets (Kirzner 1973: 85).  This is a special case of Hayek's (1949) problem of dispersed knowledge.

 

In contrast, in some areas Hayek's (1973: 27) understanding of the entrepreneur is particularly sterile and defective[3].  In his view, business firms are designed artifacts like radios, toasters, or telescopes.  Once the laws of economics are put in place, entrepreneurs exploit them to construct business organizations much the same way engineers employ physical laws to construct machines.  Hayek's understanding of firm organization proceeds according to a military model, where the manager issues orders that are followed without question or exception.  Hayek recognizes that the entrepreneur is endogenous to the firm, but he fails to perceive that once a business enterprise is set up, its employees enjoy some degree of autonomy, and the firm in effect takes on a life of its own.

 

 

3.  The Entrepreneurial Producer I: Cost Minimization

 

In the optimization paradigm, the theory of the firm provides dual and equivalent cost minimization and profit maximization problems.  Managers exploit a given and static production technology and a given and static vector of resource and output prices to transform a set of given inputs into a given output.  Furthermore, the market structure is also given, when in fact it results from the action of entrepreneurial innovators.  Entrepreneurial action ignored by the optimization paradigm includes seeking to (a) adjust the production process to extract more output from each input or more output from the same level of inputs, (b) switch inputs or combine new inputs with those already used to produce more or cheaper output, (c) devise strategies to sell their output or some part of it for a higher price, (d) attract new buyers and devise new uses for the output, and (e) lower the costs of their inputs.  

 

It might be argued that theories of market segmentation and product differentiation allow for item (c), but this is only possible if the market is structured in a way that prevents firms from facing perfectly elastic demand.  In theory, market structure is predetermined by assumption;  in reality it results from the behavior of market participants.  In reformulating the Walrasian theory of perfect competition, Makowski and Ostroy (2001: 480) recognize "(1) prices are not exogenously given, they arise from bargaining," and "(2) the set of active markets is not exogenously given, it results from innovation."  Since the conventional view is that entrepreneurs work primarily through firms, the neoclassical theory of the firm's inability to accommodate entrepreneurial action is a major shortcoming.

 


3.1  The Standard Cost Minimization Problem

 

One way to formulate the firm's conventional optimization problem is to maximize output subject to a fixed cost constraint.  The objective function for optimization is the production function relating the inputs x1 and x2 to the quantity of output q, and is given as

 

q = f(x1, x2)

 

though, unlike the consumer's utility function, the production function always has a cardinally-measurable physical or value output.  Nevertheless, it remains debatable whether the production process can ever be meaningfully captured by an analytic function purporting to be, at best, an approximation.  The possibility of using additional inputs x3 through xn is ignored.  The key Austrian insight that production occurs over time (Menger 1871: 152[4]; Hayek 1931, 1933, 1935, 1939, 1941) is ignored.  Non-Austrian economists are hardly ignorant of this fact, but theorize as if they were. 

 

The constraint function is the producer's cost constraint which is conventionally given as

 

C0 = r1x1 + r2x2 + b

 

where r1 and r2 are the resource prices of input resources X1 and X2, b is the cost of fixed inputs, and C0 is the fixed total cost.  Again, the cost constraint is assumed given and known in advance.  No scope for entrepreneurial discovery is permitted (Kirzner 1984b: 154).  Mises notes

 

in the imaginary construction of the evenly rotating economy there is no room left for entrepreneurial activity, because this construction eliminates any change of data that could affect prices.  As soon as one abandons this assumption of rigidity of data, one finds that action must needs be affected by every change in the data.  As action necessarily is directed toward influencing a future state of affairs, even if sometimes only the immediate future of the next instant, it is affected by every incorrectly anticipated change in the data occurring in the period of time between its beginning and the end of the period for which it aimed to provide (period of provision).  Thus the outcome of action is always uncertain.  Action is always speculation. (1949: 252)

 

The contrasting view of neoclassical microeconomics is that action is always optimization, and that the "correct" outcome is always predetermined by the unalterable data of the market.  The optimization paradigm fails to recognize the scope for uncertainty, ignorance, and risk.  The cost constraint also suffers from omission of the same arguments x3 through xn as the production function, as well as their prices r3 through rn.  One means for exercising entrepreneurial action is in obtaining lower input prices[5] r1 and r2, or r1 through rn, or a combination of different inputs that minimizes total cost (Schumpeter 1934: 133-135).  Entrepreneurial alertness might also be exercised in seeking to reduce the "fixed" cost b.  Since b is only fixed by construction, alert entrepreneurs should seek ways of reducing it whenever possible. The producer, having chosen C0 or had it imposed externally, seeks to produce as much output q as possible. 

 

The constrained output maximization problem is constructed as

 

V = f(x1, x2) + μ(C0 - r1x1 - r2x2 - b)

 

where μ ≠ 0 is an undetermined Lagrange multiplier.  Finding the first partial derivatives of V with respect to x1, x2, and μ, and setting them equal to zero:

 


∂ V

=

f1 – μ r1 = 0

∂ x1

 

∂ V

=

f2 – μ r2 = 0

∂ x2

 

∂ V

=

C0 - r1x1 - r2x2 - b = 0

∂ μ

 

Algebraic manipulation yields the familiar relation[6]

 

f1

=

r1

f2

r2

 

which states that, at a cost minimum, the ratio of the marginal products of resource inputs X1 and X2 must be equal to the ratio between their prices.  The minimum found through optimization is subject to the constraint that the quantities of other inputs X3 through Xn have been arbitrarily set at zero.  These ratios between the marginal products and the resource prices define the rate of technical substitution (RTS) between X1 and X2.  The first-order conditions can also be written as

 

μ =

f1

=

f2

r1

r2

 

indicating the contribution to output of the last dollar spent on each input must equal the Lagrange multiplier μ, which is thus the first derivative of output with respect to cost C with prices r1 and r2 constant and resource quantities x1 and x2 allowed to vary. 

 

The cost minimization problem can be made more realistic by allowing total cost to vary.  When C is allowed to vary, the differential of the cost equation is

 

dC = r1dx1 + r2dx2

 

The fixed cost b drops out, though if entrepreneurial search aims at lowering b, it should not drop out of the first-order conditions. 

 

The firm's problem may also be formulated as minimizing a variable cost function subject to the constraint that a prescribed quantity of output must be produced with the given technology described by the production function.  The constrained cost minimization problem is expressed as

Z = r1x1 + r2x2 + b + λ[q0 - f(x1, x2)]

 

and the partial derivatives of Z are set equal to zero to find the first-order conditions:

 

∂ Z

=

r1 – λ f1 = 0

∂ x1

 

∂ Z

=

r2 – λ f2 = 0

∂ x2

 

∂ Z

=

q0 – f(x1,x2) = 0

∂ λ

 

The first-order conditions allow us to solve for the following relations,

 

f1

=

r1

 

or

 

1

=

f1

=

f2

 

or

 

RTS =

r1

f2

r2

 

 

λ

r1

r2

 

 

r2

 

The Lagrange multipliers λ and μ are reciprocals.  λ gives the first derivative of cost with respect to output, that is, the marginal cost[7].

 

3.2  Limitations of Standard Cost Minimization

 

The optimization paradigm is based on the principle of knowingly optimizing a known objective function subject to known constraints, removing much of the uncertainty with which real-world market participants have to contend.  The standard model of perfect competition imposes extreme, and extremely unrealistic, information requirements (Makowski and Ostroy 2001: 480). Furthermore, analytical functions can never be more than imperfect analogues of the processes of producing output or obtaining satisfaction through consumption, and the Austrian school has always criticized excessive mathematical formalism. Limitations of the optimization paradigm should be recognized in that the true objective function may not be known with certainty to market participants, and may not be expressible as an analytic function, even if "known" (Kirzner 1984b: 154). 

 

An alert entrepreneur would always seek to discover ways to adjust the production process to extract more output from each input, or equivalently, extract the same level of output from fewer inputs (Schumpeter 1934: 129-133).  Instead, the optimization problem assumes a static, given technology.  The optimization problem is necessarily untenable unless the technology has been optimized in advance.  In reality, production technology results from the creative process of human action and entrepreneurial discovery.  Technology is created, not given.[8] 

 

(a) Static Input Set

 

Entrepreneurial alertness might also be applied in discovering new inputs X3 and X4 to either substitute or complement X1 and X2 in producing a given output.  Because the optimization paradigm arbitrarily excludes inputs X3 through Xn, a more realistic optimum is defined by

 

μ =

f1

=

f2

=

...

=

fn

r1

r2

rn

 

This more global optimum still assumes a pre-determined and static technology, and can never be fully realized, as the scope for entrepreneurial alertness in discovering new inputs is inexhaustible.  In neoclassical microeconomics the optimization problem is constrained by arbitrarily setting the quantities of potential inputs X3 through Xn equal to zero.  Even in the context of the optimization paradigm, imposing this indefinite number of constraints, though analytically necessary, is also necessarily unrealistic. Also this exercise, even as modified, ignores the possible choice of different outputs (Schumpeter 1934: 134).

 

Entrepreneurial managers may switch inputs, use new inputs, or both to produce cheaper output.  This is not an optimization process.  If the production technology A which is assumed uses inputs a and b, and a different production technology B uses c and d, and can produce the same output at a lower unit cost, obviously technology B should be used, but the optimization paradigm assumes this is known in advance.  In fact it can only be discovered through experience, trial-and-error, entrepreneurial alertness, discovery, etc.  These less tractable transactions of economic behavior cannot be modeled as optimization processes because they cannot meaningfully be represented as having predetermined outcomes. 

 

Commitment to the more expensive, less efficient, technology A may preclude discovery of benefits potentially offered by technology B.  There are always better versus worse ways to combine inputs, but never a final best, the optimum, which forever precludes discovery of improved technology.  This holds even when the input set is fixed.

 

(b) Information Constraints

 

One fault of the optimization paradigm is that it assumes as given both the objective function and constraints.  Provided this information is attainable, it can only be uncovered through extensive search and entrepreneurial alertness.  Perhaps the most severe criticism against the optimization paradigm is that an entrepreneurial producer can carry out a valid optimization exercise based on unique, subjective, and fallible expectations of prices at which the inputs can be bought and output sold (Kirzner 1990: 167).  If these expectations are not realized, the producer may lose money in spite of having optimized, because market participants often fail to optimize with respect to their true circumstances.  True circumstances can only be known ex post and can only be discovered through experience, but by then it is too late.  This first kind of knowledge problem (Kirzner's Knowledge Problem A) causes planned exchanges to be impossible to fulfill.  Kirzner notes that these kinds of problems are self-correcting (Kirzner 1990: 169-171), as market participants either adjust their plans to recognize the realities of the market, or withdraw from the market.

 

It is also possible for the optimization exercise to lead to the erroneous conclusions that inputs cannot be obtained, or output sold, at sufficiently low or high prices, or that production technology, input quality, or consumer demand for the output, are actually better than anticipated by entrepreneurial planners (Kirzner 1990: 168-9).  In these cases, (Kirzner's Knowledge Problem B), exchanges which are theoretically feasible, and could be seen ex post to have been feasible, are never planned or undertaken, because market participants are unaware of the feasibility of the potential exchanges.  Entrepreneurs always seek to discover such opportunities, but many must go undiscovered.  These kinds of problems are not self-correcting, and await entrepreneurial discovery before anyone can be aware of them.  The entrepreneur can profit by uncovering, and remedying, instances of Knowledge Problem B.

 

These objections based on Kirzner's two knowledge problems can be given an alternative formulation, drawing mainly from Hayek (1949) and Kirzner (1984a, 1984b, 1990): the information set assumed by the optimization paradigm does not exist in reality.  No person possesses it in its totality, but the optimization paradigm assumes perfect information shared by all market participants.  In reality, it is more accurate to suggest that each market participant possesses some relevant information, much of which is purely subjective.  Much of this information is held exclusively by a certain individual, for example, that individual's subjective preferences or his or her plans for future consumption and production.  Individuals also differ in their alertness, both in terms of intensity and application (Kirzner 1979: 170).  Entrepreneurs overcome the social problem of information dispersal whenever they generate flows of information that stimulate revision of uncoordinated decisions toward greater mutual coordination (Kirzner 1984a: 147) moving the market toward a never-realized equilibrium state.  The very concept of market equilibrium is merely an analytical convenience with little practical relevance (Nelson and Winter 1982; Makowski and Ostroy 2001).

 

Prices summarize relevant information which would otherwise be useless to market participants in satisfying their wants, but the inadequacies in market prices also create the profit-and-loss incentives for entrepreneurs to adjust prices (Kirzner 1984a: 149).  A price may summarize economic information regarding the supply and demand conditions in the relevant markets, without signaling whether the price represents an equilibrium or a disequilibrium.  Entrepreneurs compete in adjusting prices in a "competitive process which digs out what is in fact discovered" (Kirzner 1984a: 150).  The competitive process Hayek (1978: 180) describes, where "competition is valuable only because, and so far as, its results are unpredictable and on the whole different from those which anyone has, or could have, deliberately aimed at," is utterly incompatible with the optimization paradigm.

 

Kirzner's (1984b: 160) view is that disequilibrium prices offer pure profit opportunities for alert entrepreneurs who can arbitrage among different prices temporarily prevailing in different markets.  This incentive allows entrepreneurs to effect adjustment of the disequilibrium price vector towards equilibrium.  Although, by construction, an equilibrium is never reached, and if it were reached, is never persistent, the Kirznerian entrepreneur always acts to lessen the extent of the disequilibrium. The equilibrium which is approached constitutes a spontaneous order, a level of coordination which results from human action but not from human design (Hayek 1967). In contrast, a Schumpeterian entrepreneur moves the market price vector away from equilibrium, by introducing new production technologies, marketing and distribution media, and creating new plans which increase the social dispersion of knowledge.  This distinction is explored more fully in section 6.  Both kinds of entrepreneurship are ignored by the optimization paradigm. 

 

The cost-minimization problem also fails to recognize the mode of entrepreneurial action which occurs when a firm incurs additional production or selling costs to make the product more desirable to the consumer (Kirzner 1973: 24).  Schumpeter (1934: 135) mentions a kind of entrepreneurship consisting of a search for new markets for an existing product.  In the optimization paradigm, output quantity is the decision variable, and selling price is fixed along with production technology.  Increased consumer preference can (a) justify increased production or selling costs, (b) enable the firm to increase product price, and (c) be engineered by improving product quality or marketing, but these issues are ignored.  This category of entrepreneurial action includes both increased production costs incurred to improve the subjective quality of the product in consumers' eyes, and increased selling costs which also aim at improving subjective quality.

 

Table 2 summarizes some of the limitations of the optimization paradigm and cites some instances of more realistic, less restrictive, though also less mathematically formalized, treatment of entrepreneurship given by the Austrian school.

 

<<Table 2 about here.>>

 

3.3  Extensions of Standard Cost Minimization

 

The production function can be adapted to demonstrate entrepreneurial innovation, but only at the cost of excessive delimitation of the choice faced by entrepreneurs. 

 

(a) Additional Inputs

 

Within the optimization paradigm, additional arguments can be introduced to capture some, though not all, potential for entrepreneurial action.  At least three inputs must be included, each of which can be considered a composite commodity: X1, the bundle of inputs already used in producing output Q; X2, the bundle of inputs subject to entrepreneurial discovery; and X3, the bundle of inputs remaining undiscovered.  The constrained cost minimization problem is then

 

Z = r1x1 + r2x2 + r3x3 + b + λ[q0 - f(x1, x2, x3)]

 

and the partial derivatives of Z are set equal to zero to find the first-order conditions.  The entrepreneur introduces the innovation of violating the constraint that x2 = 0, but the constraint remains that x3 = 0.  Before entrepreneurial action, the first order conditions are:

 

∂ Z

=

r1 – λ f1 = 0

∂ x1

 

∂ Z

=

q0 – f(x1, x2, x3) = 0

∂ λ

 

After entrepreneurial innovation, the first order conditions become:


∂ Z

=

r1 – λ f1 = 0

∂ x1

 

∂ Z

=

r2 – λ f2 = 0

∂ x2

 

∂ Z

=

q0 – f(x1, x2, x3) = 0

∂ λ

 

Because a linear restriction has been removed, the cost minimum after entrepreneurship must always be below the before-entrepreneurship minimum, provided the constraint was initially binding.  Although this optimization exercise, constructed to show the optimum before and after entrepreneurial innovation, does capture one highly delimited, highly stylized species of entrepreneurial action, many other kinds are not amenable to analysis or representation in the optimization paradigm.  The kind of predefined innovation modeled here thus could not have been particularly valuable.  No one learns anything terribly profound about the process of entrepreneurial innovation from conducting such augmented optimization exercises.

 

(b) Technological Improvement

 

Suppose the Cobb-Douglas production function q = f(x1, x2) = Cx1ax2b, represents the production technology understood by managers and engineers prior to an innovation.  The prior, restricted production function has the logarithmic form

 

ln q = ln C + a ln x1 + b ln x2.

 

It is nested in the more general transcendental-logarithmic (translog)[9] form

 

ln q = ln C + a ln x1 + b ln x2 + c ln x12 + d ln x1x2 + e ln x22.

 

Suppose a particular entrepreneurial discovery consists of learning how to make use of the second-order combinations of inputs in the translog.  The resulting, more flexible, optimization exercise will realize lower costs and higher profits, because restrictions on the second order terms in the production function have been lifted.[10] 

 

The difficulty faced by the approach of modeling entrepreneurship as a removal of known constraints is that if the less restrictive technology were known to the entrepreneur in advance, they would have made full use of it.  This construction can only be set up in an environment where the before and after optima are both fully determined, and only the entrepreneur is ignorant of the difference.  This is a particularly sterile way of modeling entrepreneurship, because it constrains the entrepreneur exclusively to realizing artificial productivity gains only as defined in advance by the theorist.  Real world entrepreneurs recognize no such constraints.  Thus one way of expressing the limitations the optimization paradigm imposes on modeling entrepreneurial behavior is that it reduces entrepreneurship to an Easter egg hunt, where the only prizes which can be uncovered by entrepreneurial discovery, must always be defined in advance by the theorist.  This is a reductio ad absurdum against the optimization paradigm.  The real state of affairs is that entrepreneurs often uncover advances undreamed of by anyone else, or even by themselves prior to the discovery.

 

4.  The Entrepreneurial Producer II: Profit Maximization

 

4.1  Standard Profit Maximization

 

The firm's cost minimization problem is often represented by its dual.  Since the entrepreneur is free to vary cost along with output, the unconstrained profit maximization problem is written as

 

π = pq – C

 

where p is the output price, assumed to be given.  Substituting in the production and cost functions, the profit function is conventionally written as

 

π = p f(x1, x2) – r1x1 - r2x2 - b

 

which is maximized with respect to the choice variables x1 and x2.  Again, limitations imposed by assumptions of static technology and limited inputs invalidate the exercise.  It should be emphasized that this algebraically unconstrained optimization problem yields an economically constrained solution, as a consequence of the assumptions which are imposed.  The first-order conditions[11] are

 

∂ π

=

pf1 – r1 = 0

∂ x1

 

 

 

∂ π

=

pf2 – r2 = 0

∂ x2

 

which can be written as

 

pf1 = r1  and  pf2 = r2

 

The partial derivatives of the production function with respect to the inputs are the marginal products.  Multiplied by price, they provide the marginal revenue products, which are set equal to the input prices r1 and r2 at the profit maximum. This is the familiar condition that an input is used as long as its marginal revenue product exceeds or equals its cost to the producer.  However, the constraints that the quantities of additional potential inputs X3 through Xn are fixed at zero, guarantee this is a constrained optimum.

 

4.2  Limitations of Standard Profit Maximization

 

Because all search activity is ignored, important features of economizing action are disregarded and assumed away (Kirzner 1984: 156) For example, one forum for entrepreneurial action is to seek monopoly ownership of a resource.  This broad category of human action includes both cornering the market for a particular input, and creation of any form of intellectual property.  Schumpeter (1934: 152) discusses how temporary monopoly profits always accrue to innovators.  Entrepreneurs pursue such strategies all the time.  Because the optimization paradigm assumes input and output prices as given, this mode of entrepreneurial action is necessarily ignored.  Entrepreneurs should always desire to be monopolist resource owners because that allows them to charge a higher price for their output than non-monopolists (Kirzner 1973: 21).  Entrepreneurs also seek to increase output price through becoming monopoly suppliers of the output, or monopolistic-competitive suppliers, for example, through creating intellectual property in their output, or through advertising.  Recognizing that market structure is determined by the competitive process of the market is part of Makowski and Ostroy's (2001) reformulation of the Walrasian general equilibrium model of perfect competition.

 

The price of output is assumed given, thus the vector of prices is not permitted to adjust to realize efficient allocation.  Producers are not permitted to adjust the type of output, which is also assumed given.  One obvious area for entrepreneurial innovation is offering different kinds of output for sale to consumers (Schumpeter 1934: 134-135).  The role flexible prices and their adjustments play in overcoming Hayek's (1949) problem of dispersed knowledge and coordinating the plans of producers and consumers, is similarly ignored (Kirzner 1984a: 139-140).  Kirzner notes that equilibrium prices, if they could persist, would signal market participants' plans to each other, and thus guide future planning.  In contrast, disequilibrium prices signal to alert market participants how revised plans may benefit market participants in the future.  In Kirzner's view, disequilibrium prices predominate, thus entrepreneurial opportunities are everywhere.  Entrepreneurial consumers and producers who take advantage of these disequilibrium prices move the market toward dynamic equilibria which are generally never realized. 

 

4.3  An Expanded Profit Maximization Exercise

 

The profit maximization exercise could be expanded to realize a more global optimum, which would be represented by the n conditions

 

pf1 = r1,  pf2 = r2, ... , pfn = rn

 

but it should be emphasized that this optimum can never be realized.  In reality, most of the n conditions for a global profit maximum are necessarily invalidated.  Economic agents' ignorance sets all but a few of the xis equal to zero.  Thus the realized profit maximum is constrained. The solution also assumes a fixed output price p and fixed input prices ri, and the assumption of static production technology ensures the first and second derivatives (fis and fijs) of the production function are also constant.  Entrepreneurial activity can be described analytically as removing constraints on p, and the xis, ris, fis, and fijs, however, this fails to capture all entrepreneurial action, as entrepreneurs also seek to produce those outputs which will yield the highest profits.

 

5.  The Entrepreneurial Consumer

 

We do not generally think of entrepreneurship outside firms.  However, given Mises' definition, "acting man exclusively seen from the aspect of the uncertainty inherent in every action," (1966: 253) it is clear the consumer can be an entrepreneur in the act of consumption.  In the neoclassical theory of the consumer, consumers maximize an unknown and unobservable utility function, which must be specified in a finite number of arguments, subject to a budget constraint.  In a formal sense, an entrepreneurial consumer would always be (a) trying out new arguments in the utility function, as well as (b) trying out new ways of extracting additional utility from old arguments.  For example, entrepreneurial consumers would experiment with new preference functions in an effort to shift their indifference curves inward.  The neoclassical conception ignores both kinds of entrepreneurial behavior, by assuming they have been carried out in advance. 

 

To account for the possibility of entrepreneurial consumption, the number of arguments in the utility function and budget constraint must be made indefinite, as with the cost and profit functions.  In addition, the entrepreneurial consumer is always trying to increase his or her income, as well as searching for lower prices and higher quality arguments for consumption.  Entrepreneurial consumers are alert to improving (a) the number of utility function arguments, (b) the form of the utility function, (c) consumer income, (d) the prices of consumed goods, and (e) the quality of consumed goods.  More fundamentally, the optimization paradigm ignores entrepreneurial consumers' alertness to these opportunities to costlessly improve satisfaction of their wants.

 

5.1  Conventional Utility Maximization

 

Kirzner (1973: 18) notes the convention of treating consumers as merely passive price-takers is only an analytical convenience.  The consumer's optimization problem is usually defined as maximizing a preference or utility function subject to a budget constraint.  In the two-good case the budget constraint is

 

y0 = p1q1 + p2q2

 

where y0 is the consumer's fixed income and p1 and p2 are the prices of commodities Q1 and Q2 respectively.  Thus unreality is introduced by arbitrarily setting the quantities of commodities Q3 through Qn at zero.  Entrepreneurial alertness which might be exercised through discovering additional arguments is assumed away and ignored.  The price vector facing the consumer is assumed to be given.  Thus any entrepreneurial activity aiming at improving the price vector from the consumer's perspective is also ignored. The consumer seeks to maximize the subjective, unobservable, ordinal utility function

 

U = f(q1, q2)

 

where q1 and q2 are the quantities of Q1 and Q2 the consumer consumes, subject to the budget constraint (Debreu 1959: 55-58, Silberberg 1979: 214-233, Henderson and Quandt 1980: 8-18, Chiang 1984: 400, Varian 1984: 113-115, Takayama 1985: 179-183.)  For generality, no explicit functional form (such as Ut = Acta ltb) is assumed.  (Varian 1984: 128-130 discusses explicit functional forms, including Cobb-Douglas and constant-elasticity-of-substitution utility.) 

 

Because the utility function is assumed to be fixed, no scope is allowed for the entrepreneurial consumer to extract additional well-being from equal amounts of the same consumption goods.  However, imagine a household budgeting for a fixed amount of fixed-price consumption goods over a period of time.  A household which experimented with new food recipes could enjoy higher well-being by discovering recipes they prefer, (or recipes they dislike and avoiding them).  This would shift the utility function closer to the origin.  The optimization paradigm ignores this possibility of entrepreneurship in consumption.  Entrepreneurial consumers would also seek to negotiate lower prices for consumer goods, and higher household incomes (Rothbard 1962: 183-200).  This entrepreneurial behavior is all ignored by convention.  Conventionally, the Lagrangean is

 

V = f(q1, q2) + λ(y0 - p1q1 - p2q2)

 

where λ is the as yet undetermined Lagrange multiplier.  First-order conditions for an optimum are obtained by setting the first partial derivatives with respect to q1, q2, and λ equal to zero:

 

∂ V

=

f1 – λp1 = 0

∂ q1

 

∂ V

=

f2 – λp2 = 0

∂ q2

 

∂ V

=

y0 - p1q1 - p2q2 = 0

∂ λ

 

Transposing the second terms in the first two equations and dividing yields the condition that the marginal utilities must be equal to the price ratio for a maximum[12],

 

f1

=

p1

f2

p2

 

The ratio of the first partial derivatives of the utility function, f1/f2 is the rate of commodity substitution or marginal rate of substitution between Q1 and Q2. The first-order condition for a maximum expresses the equality between the rate of commodity substitution and the price ratio.  The first two equations can also be written as,

 

f1

=

f2

 = λ

 

p1

p2

 

5.2  Limitations of Utility Maximization

 

Hayek notes the dispersal of knowledge as an economic problem, "a problem of the utilization of knowledge which is not given to anyone in its totality" (Hayek 1949: 77-78).  Dispersed knowledge provides one forum for entrepreneurial activity.  Entrepreneurs seek out, or at least remain alert to, the dispersed character of this knowledge, hoping to profit through offering information to those who can benefit from it, but do not yet possess it.  Thus, entrepreneurs act to overcome the dispersal of knowledge.  This kind of entrepreneurial activity overcomes disequilibria based on information asymmetries, acting to move the market toward equilibrium and symmetric information.  This equilibrium can never be realized, furthermore, in the act of overcoming asymmetric information disequilibria, entrepreneurs upset the entrepreneurial plans of others, preventing coordination of individual plans, ensuring a disequilibrium state even as they move the market toward equilibrium. This interpretation of the entrepreneurial function leaves room for both Schumpeter's and Kirzner's views of the entrepreneur.  Entrepreneurs profit from removing information imbalances and bring about a movement toward information equilibrium, where everyone enjoys equal, and more complete, information (Kirzner 1973: 66-67).

 

4.  Market Coordination, Equilibrium, and the Entrepreneur: Schumpeter's and Kirzner's Views

 

Kirzner and Schumpeter give competing views of the role of the entrepreneur.  For Schumpeter the entrepreneur seeks to shift the production function and cost function (Triffin 1940: 168, Schumpeter 1962: 104-105, 132).  For Kirzner the entrepreneurial function consists of noticing that the cost and revenue functions have shifted: "entrepreneurship for me is not so much the introduction of new products or of new techniques of production as the ability to see where new products have become unexpectedly valuable to consumers and where new methods of production have, unknown to others, become feasible" (Kirzner 1973: 81).

 

Kirzner and Schumpeter offer diametrically opposed views of the role of the entrepreneur in relation to market equilibria.  Schumpeterian entrepreneurs destroy equilibria (Schumpeter 1934: 64), whereas Kirznerian entrepreneurs effect adjustment toward new equilibria (Kirzner 1973: 72-73).  Although a new attempt at reconciling these competing views risks oversimplifying them, it seems hopeful to suggest the two views of the entrepreneurial function arise from differing concepts of the underlying reference equilibrium.  In Schumpeter's scheme, the hypothesized equilibrium is associated with the ideal state of an evenly rotating economy.  This hypothetical reference equilibrium is unrealized because entrepreneurial action prevents the economy from settling in an evenly rotating state.  Schumpeter defines entrepreneurial action with reference to an ex ante equilibrium which was never actually realized; the hypothesized equilibrium is ex ante to the entrepreneurial action, and provides alert entrepreneurs with profit opportunities to exploit. 

 

Kirzner views entrepreneurship as moving the market toward a hypothesized equilibrium.  This equilibrium is never realized but can be seen as the goal of the entrepreneurial action ex post; Kirzner's equilibrium is moved toward after the entrepreneurial action.  In Kirzner's view the "final" equilibrium is the goal of entrepreneurial action, a goal that is never reached.  Schumpeterian entrepreneurs create information asymmetries.  The actual dispersion of the asymmetric information created by Schumpeterian entrepreneurs lessens the coordination of economic plans.  Kirznerian entrepreneurs exercise alertness to discover already-existing information asymmetries.  The Schumpeterian and Kirznerian entrepreneur may well be two different persons, but can equally well be the same.  Kirznerian entrepreneurs exploit information asymmetries to earn entrepreneurial profits. 

 

Imagine an initial state of affairs, where information is dispersed, but market participants, ignorant of their own ignorance, are unaware of the asymmetric character of the information they possess.  In this context, are market prices equilibrium prices?  Not in the Schumpeterian sense, but certainly in the Kirznerian sense.  A Kirznerian entrepreneur can exploit this opportunity by brokering information about resource and product prices, but only after discovering the information asymmetry.  Then the extent of the entrepreneur's success eliminates the asymmetry, ultimately eliminating the entrepreneurial profit opportunity. 

 

In Kirzner's view, prices remain in equilibrium until entrepreneurial alertness discovers information dispersal and asymmetry, and acts to take advantage of the arbitrage opportunity, forcing prices to adjust.  Price adjustments effected by Kirznerian entrepreneurs act to reestablish a new equilibrium where market participants' plans are better coordinated and their wants can be better satisfied (Kirzner 1984b: 160)[13]. A Schumpeterian entrepreneur exploits the asymmetric information in such a way that they might seek to extend or maintain the dispersal of knowledge.  The Schumpeterian entrepreneur disturbs and destroys the old equilibrium; the Kirznerian entrepreneur moves the market toward a new one.  Kirznerian entrepreneurs avoid risk and cost (Blaug 1998: 223) because they move the market toward a new equilibrium which could only exist hypothetically after the entrepreneurial action.  Schumpeterian entrepreneurs move the market away from the pre-established, though equally hypothetical equilibrium which is destroyed, or at least left behind, by the entrepreneurial action.  The Schumpeterian entrepreneur may succeed at moving the market out of equilibrium, or at least away from a hypothesized equilibrium state; the Kirznerian entrepreneur can never really succeed in establishing a new equilibrium.

 

6.  Conclusion

 

Entrepreneurship is not amenable to limits imposed by an arid mathematical formalism.  This is partly due to the facts that (a) the concept of entrepreneurship remains vague and ill-defined, (b) several competing concepts, e.g., Schumpeterian versus Kirznerian entrepreneurship, are floating around, (c) these concepts are often used interchangeably, and most importantly, (d) entrepreneurs are so innovative their behavior often defies categorization.

 

The neoclassical optimization paradigm largely ignores entrepreneurial activity, thus ignoring much of what is interesting in economic behavior.  Although some kinds of entrepreneurial action can be described within the optimization paradigm, many cannot.  For example, production technology, created through one species of entrepreneurial discovery, and implemented through another, is taken as given for analytical convenience.  The nature of the entrepreneurial function is treated as beyond the scope of the discipline because it does not result from optimization.  Entrepreneurial discovery results from something other than optimization because entrepreneurs act to remove optimization constraints.  There is no global optimum to an entrepreneur, only constraints which eventually fall before the onslaught of the human action of inquisitive, profit-seeking entrepreneurs.

 

Successful entrepreneurs should always move the market from a less optimal to a more optimal state of affairs by removing constraints imposed by ignorance and uncertainty[14].  However, in the optimization paradigm, agents purposely optimize with respect to known objective and constraint functions.  Their understanding of these analytic functions may be imperfect or uncertain, resulting in imperfectly implemented human action on which Kirznerian entrepreneurs can always improve.  Naturally entrepreneurs profit from providing this service.

 

It may also happen that Schumpeterian entrepreneurs can enter the scene and change the objective and constraint functions.  Apart from the fact that this species of human action automatically creates an opportunity for Kirznerian entrepreneurship, as market participants may not be immediately aware that their objective and constraint functions have changed, initially imperfect human action is driven even further off the mark by changing the analytic functions market participants assumed when they made their production and consumption plans, or were analytically modeled as assuming.

 

Acknowledgement

 

Robert F. Mulligan is associate professor of economics in the Department of Business Computer Information Systems and Economics at Western Carolina University and a research associate of the Department of Economics at SUNY Binghamton.  Financial support in the form of a research fellowship at the American Institute for Economic Research is gratefully acknowledged.  Thanks are due to Sanford Ikeda for helpful comments.  The author remains responsible for errors.

 

 

Notes


 

References

 

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Casson, Mark. (1982). The Entrepreneur: an Economic Theory, Oxford: Martin Robertson.

 

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Casson, Mark (1995). Entrepreneurship and Business Culture, vol. 1, Aldershot: Edward Elgar.

 

Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics, (3rd ed.), New York: McGraw-Hill.

 

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Greaves, Percy L. (1974). Mises Made Easier, Dobbs Ferry NY: Free Market Books.

 

Harper, David A. (1998). "Institutional Conditions for Entrepreneurship," Advances in Austrian Economics, 5: 241-275, Stamford CT: JAI Press.

 

Hayek, F.A. (1931). Prices and Production, (1st ed.)

 

Hayek, F.A. (1933). Monetary Theory and the Trade Cycle, reprint, New York: Augustus M. Kelley, 1966.

 

Hayek, F.A. (1935). Prices and Production, (2nd ed.), reprint, New York: Augustus M. Kelley, 1967.

 

Hayek, F.A. (1939). Profits, Interest, and Investment, and Other Essays on the Theory of Industrial Fluctuations, reprint, New York: Augustus M. Kelley, 1969.

 

Hayek, F.A. (1941). The Pure Theory of Capital, Chicago: University of Chicago Press.

 

Hayek, F.A. (1949). "The Use of Knowledge in Society," in Individualism and Economic Order, London: Routledge and Kegan Paul, originally published in the American Economic Review 35(4): 519-530.

 

Hayek, F.A. (1967). Studies in Philosophy, Politics, and Economics, Chicago: University of Chicago Press.

 

Hayek, F.A. (1973). Law, Legislation, and Liberty, Volume 1, Rules and Order, Chicago: University of Chicago Press.

 

Hayek, F.A. (1978). New Studies in Philosophy, Politics, Economics, and the History of Ideas, Chicago: University of Chicago Press.

 

Henderson, James M., and Quandt, Richard E. (1980). Microeconomic Theory: a Mathematical Approach, 3 ed., New York: McGraw-Hill.

 

Khalil, Elias L. (1995). "Organizations versus Institutions," Journal of Institutional and Theoretical Economics 151(3): 445-466.

 

Khalil, Elias L. (1997). "Friedrich Hayek's Theory of Spontaneous Order: Two Problems," Constitutional Political Economy 8: 301-317.

 

Kirzner, Israel M. (1973). Competition and Entrepreneurship, Chicago: University of Chicago Press.

 

Kirzner, Israel M. (1979). Perception, Opportunity, and Profit, Chicago: University of Chicago Press.

 

Kirzner, Israel M. (1984a). "Prices, the Communication of Knowledge, and the Discovery Process," in Leube, K.R. and Zlabinger, A.H., (eds.), The Political Economy of Freedom: Essays in Honor of F.A. Hayek, Amsterdam: Philosophia Verlag, reprinted in Kirzner, The Meaning of Market Process, London: Routledge, 1992: 139-151.

 

Kirzner, Israel M. (1984b). "Economic Planning and the Knowledge Problem," Cato Journal 4(2), reprinted in Kirzner, The Meaning of Market Process, London: Routledge, 1992: 152-162.

 

Kirzner, Israel M. (1990). "Knowledge Problems and their Solutions: Some Relevant Distinctions," Cultural Dynamics reprinted in Kirzner, The Meaning of Market Process, London: Routledge, 1992: 161-179.

 

Kirzner, Israel M. (1992). The Meaning of Market Process, London: Routledge.

 

Kirzner, Israel M. (1997). "Entrepreneurial Discovery and the Competitive Market Process: An Austrian Approach," Journal of Economic Literature 35: 60-85.

 

Knight, Frank H. (1921). Risk, Uncertainty and Profit, New York: Houghton Mifflin.

 

Lachmann, Ludwig M. (1986). The Market as an Economic Process, Oxford: Basil Blackwell.

 

Lewin, Peter. (1997). "Hayekian Equilibrium and Change," Journal of Economic Methodology 4(2): 245-266.

 

Makowski, Louis, and Ostroy, Joseph M. (2001). "Perfect Competition and the Creativity of the Market," Journal of Economic Literature 39: 479-535.

 

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Mises, Ludwig von. (1949 [1998]).  Human Action, (5th ed.), Auburn: Ludwig von Mises Institute.

 

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Table 1

The Optimization Paradigm

 

Samuelson

(1983[1947])

Silberberg

(1978)

Henderson

& Quandt

(1980[1958])

Varian

(1984[1978])

Chiang

(1984[1967])

Takayama

(1985[1974])

Cost minimization

57-69

179-185

75-78

60-64

418-421

134-135

Profit maximization

76-78

263-275

78-80

47-52

247-250

142

Utility maximization

98-100

223-233

13-16

115-120

400-409

135-137

 


 

Table 2

Some Modes of Entrepreneurial Activity

 

Optimization paradigm

Austrian school

Theory of the Consumer: Utility maximization

Add arguments in utility function

Ignored but can be accommodated analytically

 

Replace existing arguments in utility function

Ignored but can be accommodated analytically

 

Attempt to extract additional utility from old arguments in utility function

Utility function assumed given

 

Alter preference function

Utility function assumed given

 

Negotiate lower prices for consumer goods

Price vector assumed given

Kirzner 1973: 18

Negotiate increase in income

Income assumed given

Rothbard 1962: 183-200

Seek improved quality of consumer goods

Quality ignored for analytical simplicity

 

Theory of the Firm: Cost minimization - profit maximization

Improve input productivity

Production technology and input productivity assumed given

Schumpeter 1934: 129-131, 1962: 104-105, 132; Kirzner 1973: 81

Substitute new, less expensive inputs

Inputs and input prices assumed fixed

Schumpeter 1934: 133-134; Kirzner 1984b: 154

Substitute new, more productive inputs

Production technology and input productivity assumed given

 

Introduce new complementary inputs

Inputs assumed fixed

 

Increase output price through monopoly

Can be accommodated analytically

Schumpeter 1934: 152

Increase output price through market segmentation

Can be accommodated analytically

 

Increase output price through advertising

Output price assumed fixed; advertising costs zero at cost minimum

Kirzner 1973: 24

Attract new buyers

Demand assumed given

Schumpeter 1934: 135-136

Lower input costs

Input prices assumed fixed

 

Lower fixed costs

Fixed costs assumed fixed

 

Seek monopoly ownership of inputs

Input prices assumed fixed

Kirzner 1973: 21

Switch outputs

Output assumed fixed

Schumpeter 1934: 134-135

 

 

 

 



[1] Wittgenstein 1922: proposition 7.

 

[2] It might be suggested that the true objective function is an unknowable transcendental function which theorists crudely attempt to mimic with algebraic functions.  Even when the objective function of an optimization exercise is rendered as a transcendental function, it cannot be correctly rendered as the underlying reality is, at best, some other, unknown transcendental function.

 

[3] Hayek is not particularly explicit about how he views firms.  See particularly Khalil 1997: 302 for an explanation of why Hayek's distinction between designed and spontaneous orders supports this interpretation.  See also Khalil 1995 and Dupuy 1996.

 

[4] "The transformation of goods of higher order into goods of lower order takes place, as does every other process of change, in time."

 

[5] Makowski and Ostroy (2001: 483) note the lack of realism of the standard Walrasian model of perfect competition, in that it treats perfectly competitive firms as price-takers.  As they note, prices arise through the competitive process.  It can be extremely misleading to model prices as predetermined before competition.

 

[6] The second-order condition for a cost minimum is that the bordered Hessian of second derivatives be positive:

 

f11

f12

-r1

 

 

 

 

 

f21

f22

-r2

> 0

 

 

 

 

-r1

-r2

0

 

 

 

 

 

 

 

 

 

[7] Because this is a minimum, the second-order condition requires that the bordered Hessian be negative:

 

- λ f11

- λ f12

-f1

 

 

 

 

 

- λ f21

- λ f22

-f2

< 0

 

 

 

 

-f1

-f2

0

 

 

Manipulation of this condition demonstrates it is equivalent to the second-order condition for constrained output maximization. 

 

[8] More precisely, technology is created by innovators and given to imitators.  Imitation of technological improvements is an important mode of entrepreneurial behavior.

 

[9] Before taking logarithms, the translog is q = f(x1, x2) = Cx1ax2b(x12)c(x1x2)d(x22)e.  The Cobb-Douglas form imposes the constraints that the logarithmic second-order terms are all zero, and it is convenient, though not necessarily realistic, to assume that innovative entrepreneurs discover more efficient production technology equivalent to removing the zero constraints on the second-order terms.  The second-order translog form can be thought of as nested within third or higher-order, but as yet unknown, forms.  Thus entrepreneurial discovery can be modeled as an indefinitely ongoing process.  Nevertheless, potential improvements realizable by changing to any other new technologies, unknown ex ante, are explicitly excluded.

 

[10] This model can be rendered open-ended by realizing that the second-order translog is nested in third and higher order functions.  However, the additional precision offered by higher order terms rapidly becomes negligible (Todd 1963).  Thus this approach to modeling open-ended technological progress becomes analytically futile as well as computationally cumbersome, if not completely intractable.

 

[11] Second-order conditions for profit maximization require that the principal minors of the unbordered Hessian alternate in sign:

2 π

2 π

=

p2

 

 

>0

∂ x12

∂ x1∂ x2

f11

f12

 

 

 

 

2 π

2 π

f21

f22

∂ x2∂ x1

∂ x22

 

 

and

2 π

=

pf11

<0

and

2 π

=

pf22

<0

∂ x12

∂ x22

 

The full-rank conditions ensure profit is decreasing with respect to further applications of both inputs simultaneously, while the second set of conditions ensure profit is decreasing for further applications of either input.

 

[12] First-order conditions are necessary but not sufficient for a maximum.  The optimum must occur at a local convexity to represent a maximum.  The second order condition is that the bordered Hessian must be positive definite:

 

f11

f12

-p1

 

 

 

 

 

f21

f22

-p2

> 0

 

 

 

 

-p1

-p2

0

 

 

The second-order condition for a maximum is satisfied by the assumption that the utility function is regular strict quasi-concave, which also guarantees a global maximum.

 

[13] Kirzner defines equilibrium as a static condition which can only be arrived at by removal of all unexploited profit opportunities.  Given this definition, market prices are never equilibrium prices because there are always unexploited profit opportunities. In this paper, the situation considered is the discovery and exploitation of a hypothetical and discrete item of information.  Kirzner might not agree with the formulation used in this paper that the market is actually in equilibrium until some agent discovers the previously unknown information.  He would agree that as agents exploit new profit opportunities, the market moves toward a new equilibrium which may never actually be reached.

 

[14] This is not a normative statement.  It merely asserts that when entrepreneurs are successful, they improve resource allocation and satisfy previously unmet consumer wants, or allow previously unmet want to be satisfied.  Improving allocation and satisfying unmet wants allows them to collect entrepreneurial profits.