Contents
(A) The Paretian Producer
(B) Cost-Minimization
(C) Output-Maximization and Duality
(D) Profit-Maximization and Indeterminacy
We now turn to one of the main elements of Neoclassical theory - the producer's decision. A "producer" is any entity which transforms factors of production into finished goods. A firm is an example of a producer, but we do not restrict ourselves to only analyzing firms as understood in modern Western society. For instance, a producer could be an independent artisan such as a village blacksmith, who may not be officially "incorporated" as a firm.
Furthermore, "Paretian" production theory restricts its definition of producer to an entity over which one person makes a decision (cf. Kaldor, 1934). Many modern corporations are either partnerships or have hundreds of shareholders all of whom, ostensibly, have decision-making power. Viewed in this light, the decisions made in modern multi-owner corporations are the outcome of the often conflicting desires of the different owners, and thus has to be treated as something of a public goods problem. This, however, is a different line of research (cf. Drèze, (1985) and the references therein). For our purposes, the "Paretian" firm ought to be conceived as a firm owned by a single entrepreneur who has an unchallenged power of decision over all aspects of production process: what his firm will produce, the technique of production, the hiring of factors of production, etc.
One element upon which the Paretian producer does not have power of decision is in the pricing of the output and the payment to factors. Prices and factor returns are forced upon him by that amorphous entity, the competitive market. In other words, Paretian producers are price-takers both on the output and input side. Given output and factor prices, the producer makes decisions on what, how much and by what means to produce, but he cannot set the prices.
The assumption that makes producers price-takers is known as perfect competition. We will have much more to say on how to characterize "perfect competition" later, but suffice it here to point out that Paretian producers take prices as parameters for their decisions, and not as variables. In the Marshallian theory of production, which we cover elsewhere, price-setting is one of the central aspects of the producer's decision, but in the Paretian theory, producers consider prices as "given".
We should note that Léon Walras (1874) took the perfect competition assumption to the extreme: indeed, he had everything set by the market and allowed the producer almost no leeway in terms of output or factor-hiring decisions. Of course, to end up there, he had to assume fixed proportions (i.e. Leontief) technology: when there is only a single technique to produce each output, there are few "choices" to be made by a producer anyway. Indeed, Vilfredo Pareto (1898, 1906) himself, who we seem to credit with the introduction of the "Paretian firm" into Neoclassical economic theory, resisted the introduction of fully variable techniques, although he also emphasized the unreality of completely fixed coefficients.of production (cf. Hicks, 1934).
The idea of a motivated producer making an active production decision required the introduction of variable techniques of production. The idea of a profit-maximizing producer is due originally to Augustin Cournot (1838). The idea of a profit-maximizing producer making choices among various techniques of production was shared by many early Neoclassical economists, notably Knut Wicksell (1893), Philip H. Wicksteed (1894, 1910) and Vilfredo Pareto (1898, 1906).
However, great credit for what follows here must be given to the inter-war "Paretian" economists, notably Jacob Viner (1931), Harold Hotelling (1932, 1935), John Hicks (1932, 1939), Abba Lerner (1944), Paul A. Samuelson (1947) and Ragnar Frisch (1965). Later developments, particularly the extensive exploitation of duality theorems, are due in great part to Ronald W. Shephard (1953), Hirofumi Uzawa (1962), Daniel McFadden (1966) and Werner E. Diewert (1974). Thus we refer to the theory in this section as the analysis of the Paretian producer. We use this term to distinguish not only from the Walrasian theory (where firms are not really decision-makers), but also to distinguish it from the Marshallian firm. In Marshallian theory, choice of technique is largely ignored, and the main decisions of a profit-maximizing firm is the scale of production and (if the firm can), the pricing of output. However, in Paretian theory, both these decisions are usually out of the hands of individual producers: both the pricing of output and the scale of production are usually determined by the "market" in a general equilibrium context. But this will become clearer as we go on.
Having defined a producer as an entrepreneur-owned entity which transforms factors of production into output, we may summarize him entirely by a production function such as:
y = ｦ (x_{1}, x_{2}, .., x_{m})
where y is output and x_{1}, x_{2}, .., x_{m} are factors of production. Notice that we have assumed that only one output is produced. Naturally, producers often produce many types of output, what is known as "joint production". However, for simplicity we shall assume that a producer produces merely one type of output with multiple factors.
Notice that we do not characterize a producer by "preferences" of any sort. The entrepreneur-owner, indeed, may very well have personal preferences of some sort, but his enterprise does not. As far as his enterprise is concerned, the objective is merely to maximize profits. As Pareto put it, "A producer endeavors to ascend as far as he can on the hill of profit. That is, he strives the have the greatest possible residue." (Pareto, 1906: p.129). How the entrepreneur-owner then puts that profit to use into satisfying his preferences falls in the scope of consumer, not producer, theory.
There have been many challenges to the idea that firms, in reality, actually attempt to maximize profits as opposed to, say, maximize their size, or simply seek to achieve a comfortable (not necessarily maximal) level of profits or, even more simply, just follow traditional routines of production with little rhyme or reason. These objections have engered something of a debate in economics, which we summarize elsewhere. For now, we will not attempt to justify the profit-maximization hypothesis. We will simply accept it axiomatically.
A producer's profit, which we denote as p , is defined as the total revenue from the sale of its output minus the total costs in producing that output. Let us consider the canonical two-input, one-output production context, where output Y is produced with capital and labor, summarized by the production function Y = ｦ (K, L). Let us now define p as the price of the output, w as the wage paid to the owner of a of unit of labor and r be the rate of return paid to the owner of a unit of capital. Consequently, total revenue is:
TR = pY
and total costs are:
TC = wL + rK
consequently profit is:
p = TR - TC = pY - wL - rK
As noted, producers take prices as "given", thus p, w and r are assumed to be set by the market. They simply have decision-making power over Y, L and K. As is obvious, the more output is produced, the more revenue the producer receives. However, as this greater output must be made by hiring more factors of production, then the more output it produces, the greater its costs. Consequently, at least at this point, it is not clear whether increasing the scale of operation will actually increase or decrease profits.
There is a second opening involving the choice of the technique of production. As we saw, with variable techniques of production, the producer can choose various capital-labor combinations all of which yield the same output. Thus, simply by changing the proportions of factors K and L, it may be able to decrease total costs without affecting total revenue - and thus increase profits. Thus, we first begin with analyzing the cost-minimizing choice of technique for a firm which seeks to produce a particular level of output.
It is important to note that this story is not the whole story behind the producer's profit-maximizing decision. In cost-minimization, we leave the decision on the level of output out of the picture. The full, profit-maximization story would require that output level enter as a variable and not as a given. We shall consider this later.
Surprisingly, the "cost-minimization" exercise for choice of inputs was only developed quite late: specifically, it was really only laid out by Paul Samuelson (1947: Ch. 4) and explored more exhaustively by Ronald W. Shephard (1953). Let us assume the producer seeks to produce a given level of output, Y*. Now, the cost-minimizing decision is the following: at the given factor prices, what are the amounts of the factors of production which will produce this most cheaply? For the canonical two-input case, where we have the production function Y = ｦ (K, L) and K is capital and L is labor, the costs the producer faces are:
C = rK + wL
where C is total factor costs and w and r are the rental rates for labor and capital respectively. Notice that this cost equation defines a function in L-K space of the following linear form, L = C/w - (r/w)K. This is known as an isocost curve and is depicted as a downward-sloping straight line, as we see in Figure 7.1. Any factor input combination on a particular isocost curve has the same total costs, C. The vertical intercept of the isocost curve is C/w and the slope of the isocost curve is -(r/w). The horizontal intercept is obviously C/r. Thus, for different levels of costs, C, there will be different (but parallel) isocost curves. The isocost curves closest to the origin represent relatively low total costs, those furthest away represent relatively high total costs. Thus, referring to the isocost curves depicted in Figure 7.1, C < C* < C｢ . Note that a change in the factor prices, r or w, will change the slopes of all the isocost curves.
Figure 7.1 - Cost-Minimization
Now, our program, for the cost minimizing firm, is then to minimize cost given that a certain level of output must be reached. This is stated as follows:
min C = wK + wL
s.t.
Y* = ｦ (K, L)
In other words, as the output level Y* is given, then the consraint is the isoquant Y* depicted in Figure 7.1. What the firm seeks to do is thus find the factor combination, (K, L), which has the lowest total cost and yet still produces output Y*. Diagramatically, this is represented as the tangency of the given isoquant Y* and the lowest isocost curve, C*. This will be at point e* in Figure 7.1, which represents factor input combination K*, L*.
Notice that point e is unattainable: although the factor combination corresponding to e yields a lower cost (as it lies on isocost curve C which is below isocost curve C*), it does not produce Y* and thus will not be considered. In contrast, point e｢ is attainable as it lies on the isoquant Y*, but obviously, as it also lies on the higher isocost curve C｢ , it is more costly than the combination e*. Thus, point e* - and thus capital employment K* and labor employment L* - will be the optimal choice of inputs for a firm which seeks to minimize the costs of producing Y*.
We can obtain this tangency solution from the cost-minimization problem via simple mathematical programming. Setting up in Lagrangian form, we have:
E = rK + wL + l (Y* - ｦ (K, L))
where l is the Lagrangian multiplier. This yields the following first order conditions for a maximum are (we are assuming an interior solution):
ｶ E/ｶ K = r - l ｦ _{K} = 0
ｶ E/dL = w - l ｦ _{L} = 0
ｶ E/dl = Y* - ｦ (K, L) = 0
Combining the first two yield:
r/w = ｦ _{K}/ｦ _{L}
Thus, for a minimum, the (negative of the) slope of the lowest isocost curve must equal the ratio of marginal products, or MRTS, which, as we know, is the (negative of the) slope of the isoquant Y*. Thus, minimum cost is achieved by finding the factor combination which yields a tangency between the isoquant and the lowest isocost curve.
The second-order condition for a minimum requires that the relevant bordered Hessian of the optimization problem be negative, in other words:
0 |
ｦ _{K} |
ｦ _{L} |
|||
|B| |
= |
ｦ _{K} |
-l ｦ _{KK} |
-l ｦ _{KL} |
< 0 |
ｦ _{L} |
-l ｦ _{LK} |
-l ｦ _{LL} |
or:
l (ｦ _{KK}ｦ_{ L}^{2} + ｦ _{LL}ｦ_{ L}^{2} - 2ｦ _{KL}ｦ_{ K}ｦ_{ L}) < 0
as l > 0, then all that is necessary is that the term inside the parenthesis is negative. Notice that strict quasi-concavity of the production function is enough to gaurantee that this will be true.
In greater generality, suppose we have m factors. Let x_{i} be the employment of the ith factor by the producer. Consequently, an input configuration is a vector x = [x_{1}, x_{2}, ..., x_{m}]｢ . The production function can thus be written as y = ｦ (x). We also have m factor prices, which we shall denote by the vector w = [w_{1}, w_{2}, ..., w_{m}], where w_{i} is the rental rate of the ith factor. The cost-minimizing exercise is then:
min C = wx = ・/font> _{i=1}^{m} w_{i}x_{i}
s.t.
y* = ｦ (x) = ｦ (x_{1}, x_{2}, .., x_{m})
and the typical first order condition for a minimum is:
w_{i} = l ｦ _{i}
which holds for all factors i = 1, .., m, where ｦ _{i} = ｶ y/ｶ x_{i} is the marginal product of the ith factor and l is the Lagrangian multiplier. The second-order conditions for a minimum are analogous.
There are two principles that can be derived from the first order conditions. To understand these, one must first recall the meaning of the Lagrangian multiplier l . In general, l represents the gain in the value of the optimized objective function given a marginal change in the constraint. In our canonical case, the objective is total cost C and the constraint is the given output, Y*. Thus, l = ｶ C/ｶ Y*, i.e. the change in minimum cost from a marginal change in output. A Marshallian would recognize this immediately as the very definition of marginal cost of increasing output. Consequently, the first order conditions, which establish that r = l ｦ _{K} and w = l ｦ _{L}, can be interpreted as saying that, at an optimum, factor prices are proportional to marginal products, where the factor of proportionality is the marginal cost of output, l .
[To prove that indeed l = ｶ C/ｶ Y*, where C is minimum costs and Y* is the desired output level, we can undertake a simple application of the envelope theorem. Recognize that at an optimum C = wL* + rK*, where L* and K* are the optimal factor levels. Consequently, differentiating with respect to Y*, we obtain ｶ C/ｶ Y* = w(ｶ L*/ｶ Y*) + r(ｶ K*/dY*). By the first order conditions, r = l ｦ _{K} and w = l ｦ _{L}, this becomes ｶ C/ｶ Y* = l ｦ _{L}(ｶ L*/ｶ Y*) + l ｦ _{K}(ｶ K*/ｶ Y*), or simply ｶ C/ｶ Y* = l [ｦ _{L}(ｶ L*/ｶ Y*) + ｦ _{K}(ｶ K*/ｶ Y*)]. To show that the term in the square brackets is equal to 1, consider the constraint Y* = ｦ (K*, L*) which will be met at the optimum. Differentiating with respect to Y*, we obtain 1 = ｦ _{L}(ｶ L*/ｶ Y*) + ｦ _{K}(ｶ K*/ｶ Y*), thus ｶ C/ｶ Y* = l , i.e. l is indeed marginal cost.]
We also see from the first order conditions that ｦ _{K}/r = ｦ _{L}/w = 1/l . This has a particularly interesting interpetation: namely, we can think of 1/l = dY/dC as the extra output yielded by spending an extra dollar, or, as Samuelson calls it, the "marginal productivity of the last dollar" (Samuelson, 1947: p.60). If this dollar is spend on labor, then the labor bill increases by wdL, while if it is spent on capital, then the capital bill increases by rdK. But notice that ｦ _{L}/w = dY/wdL, which can be thought of as the marginal productivity of an extra dollar spent on labor; similarly, ｦ _{K}/r = dY/rdK, which can be thought of as the marginal productivity of an extra dollar spend on capital. Consequently, the condition ｦ _{K}/r = ｦ _{L}/w = 1/l effectively claims that "the marginal productivity of the last dollar (1/l ) must be equal in every use." (Samuelson, 1947: p.60)
Now, the factor price represents the cost of increasing employment of that factor by a unit. Thus, we can think of w as the marginal cost of employing labor and r as the marginal cost of employing capital. Note the elementary yet important conclusion that our tangency conditions imply that firms will attempt to equate marginal products to marginal costs of factors. At a point such as e｢ in Figure 7.1, ｦ _{K}/ｦ _{L} < r/w, thus we can interpret this as saying that at e｢ , the marginal product of capital is below its marginal cost of capital, or equivalently, that the marginal product of labor exceeds the marginal costs of capital. Consequently, firms will attempt to reduce labor employment and increase capital employment while attempting to maintain the same output level. This is represented by a move along the isoquant from point e｢ towards point e.
(C) Output-Maximization and Duality
In our cost-minimization problem, we assumed that the producer was trying to find the input combinations that would produce a given level of output at minimum cost. An alternative way of thinking about the producer's decision on factor inputs is to ask him to find the highest amount of output he can produce for a given total cost. Such an exercise could be expressed as the following:
max Y = ｦ (K, L)
s.t.
C* = wL + rK
Notice that now the constraint is now a given level of costs. This is analogous to a consumer case: the entrepreneur is now given a "budget" (the maximum amount of costs he is allowed to incur, C*) and will thus try to achieve as much output as he can out of this by choosing the appropriate factor input combinations.
Diagramatically, we impose on the producer a particular isocost curve (e.g. C* in Figure 7.2) and then ask him to to choose factor inputs such that output is maximized. In Figure 7.2, we see that the maximimum output is represented by the isoquant Y* which is tangent to the given isocost curve, C*, at point e*. Point e｢ is not output-maximizing as that factor combination produces a lower level of output Y｢ (< Y*), while point e｢ ｢ is not available, as costs at e｢ ｢ would be greater than C*, thus our constraint would be violated.
Figure 7.2 - Output-Maximization
The tangency condition depicted in Figure 7.2 is easily derived. Setting up a Lagrangian:
O = ｦ (K, L) + m (C* - wL - rK)
where m is the Lagrangian multiplier. The first order conditions for a maximum are:
ｶ O/ｶ K = ｦ _{K} - m r = 0
ｶ O/ｶ L = ｦ _{L} - m w = 0
ｶ O/ｶ m = C* - wL - rK = 0
the third first order condition guarantees that our solution will be on the given isocost function. The first two can be combined to yield the tangency condition:
ｦ _{K}/ｦ _{L} = r/w
which effectively equates the slope of the maximum isoquant with the slope of the isocost constraint. This is precisely the same tangency condition as we had in our cost-minimization exercise. Note also that ｦ _{K} = m r and ｦ _{L} = m w. It is a simple matter to note, therefore, that m = 1/l , so the Lagrangian multiplier of the output-maximization problem is the inverse of the Lagrangian multiplier of the cost-minimization problem, i.e. m is precisely the "marginal productivity of a dollar" spoken of earlier.
As we have seen time and time again, to any given optimization problem, there is usually a dual optimization problem. Intuitively, the duality relationship states that if, in the primal problem, we minimize an objective function with respect to a given constraint, then in the dual problem, we maximize that constraint subject to a given objective function. The cost-minimization and the output-maximization problem fulfill this duality relationship. In the primal problem, cost-minimization, the objective is C = wL + rK and the constraint is Y* = ｦ (K, L). In output-maximization, the objective is Y = ｦ (K, L) and the constarint is C* = wL + rK. Thus the objective of the primal is the constraint of the dual and the constraint of the dual is the objective of the primal.
There is a duality theorem which states something interesting: namely, that if we choose as the constraint of the dual the optimized value of the objective in the primal, then the solution to the dual will be the solution the primal and the optimized value of the dual will be precisely the constraint of the primal. This is obvious in our example. Compare Figures 7.1 and 7.2. In the cost-minimizing exercise in Figure 7.1, we began with Y* as the constraint and obtained the value of the objective C* as a result, with the solution given by e* = (K*, L*). Now, suppose that the C* we use as the constraint in the output-maximization exercise in Figure 7.2. is exactly the same as the C* we obtained as a result in Figure 7.1 Consequently, maximizing output we obtain Y* as a result with solution given by e* = (K*, L*). What the duality theorem claims is that this Y* in Figure 7.2 is precisely the Y* in Figure 7.1 and the e* in Figure 7.2 is the same e* we had in Figure 7.1. This is so evident diagramatically that we will not bother to prove this mathematically here, but simply refer to our section in mathematical programming.
(D) Profit-Maximization and Indeterminacy
However, it turns out that profit-maximization yields a degree of indeterminacy when considered alone. It is important to recall that Pareto (1906) couched his profit-maximizing producer in a general equilibrium context and, indeed, the determination of the profit-maximizing level of output of the producer hinges crucially on what happens in other markets and what other producers do. Consequently, we postpone the profit-maximizing story until later and turn first to the simpler cost-minimization exercise.
We began our discussion of the production decision with the notion of profit-maximization. However, we subsequently swept that under the rug and began talking about cost-minimization and then output-maximization. However, in both those cases, we could say that we were always really talking about profit maximization all along, the question was which variables were assumed to be fixed and which were not.
To understand this, let us define our profit maximization problem as follows (for our canonical case):
max p = pY - C
s.t.
Y = ｦ (K, L)
C = rK + wL
The variables in this equation are output prices (p), factor prices (w, r), output quantity (Y), factor quantities (K, L) and total costs (C). In cost-minimization exercise, we chose factor inputs to minimize cost given that we wanted to achieve a particular output level. Consequently, the cost-minimization exercise is equivalent to the profit-maximization given above if p, w, r and Y are fixed while K, L and C are flexible. In output-maximization exercise, we chose factor inputs to maximize output levels while fulfilling a particular cost budget. Thus, we see immediately that the output-maximization exercise is equivalent to profit-maximization if one holds p, w, r, and C fixed while letting K, L and Y vary.
Let us now loosen things a bit a consider a more traditional form of profit-maximiation: namely, holding only p, w and r fixed, while letting K, L, C and Y be flexible. This is the traditional Paretian profit-maximization hypothesis: namely, only prices are "given". This means that both output level and total costs are variable. Thus, in the "profit-maximization" exercise, the producer seeks to determine factor inputs which maximize profits without any output constraint or cost constraint. The optimization problem of the producer is thus, for the canonical case:
max p = pｦ (L, K) - wL - rK
The solution the is system is readily available. The first order conditions for a maximum are:
ｶ p /ｶ K = pｦ _{K} - r = 0
ｶ p /ｶ L = pｦ _{L} - w = 0
Combining these we see that:
ｦ _{K}/ｦ _{L} = r/w
which is precisely the tangency condition we obtained earlier in our output-maximization and cost-minimization exercises. Notice that pｦ _{K} and pｦ _{L} can be defined as the marginal value products (MVP) of capital and labor respectively. Thus, the profit-maximization exercise tells us that factors will be employed up until their marginal value products are equal to their respective rental rates.
But can the first order conditions tell us more? Unfortunately, not very much more. There is already a good degree of indeterminacy in the air. To see why, consider Figure 7.3, where we have our traditional canonical, homothetic production function. A particular factor price ratio, r/w, will determine an entire series of isocost curves, C_{1}, C_{2}, C_{3}, all of which have the same slope, but represent different total costs. All that we have been told from the profit-maximization exercise so far is that at any particular isocost line, the producer will choose input combinations (K, L) and a level of output (Y) such that the slope of the corresponding isoquant, -ｦ _{K}/ｦ _{L}, is equal to the slope of the isocost line, -r/w. In other words, every one of the points e, e｢ and e｢ ｢ in Figure 7.3 (and all points in between and beyond them on the ray E) are solutions to the profit-maximizing problem. Thus, output levels are indeterminate.
Figure 7.3 - Indeterminacy of Profit-Maximization
Or are they? The Marshallian solution to the problem, and one followed by many Paretians, notably Harold Hotelling (1932), to whom much of the analysis is due, is to proceed by assuming diminishing returns/increasing costs. As we shall argue later, and have hinted already earlier, this is a very questionable maneouvre, but one which nonetheless remains quite popular and necessary if we are to obtain a determinate output level from profit-maximization. Note that the original Paretian solution, is to embed the question in a general equilibrium system: they are perfectly willing to keep indeterminacy of output for an independent, price-taking producer, but they will be able to determine what the levels of output for each good will be from general equilibrium.
Before we proceed, we ought to see, formally, why constant or increasing returns will not yield a determinate solution. To see this directly, turn to the second order conditions for a maximum. The relevant condition is that the Hessian be negative definite. The Hessian in our case is:
pｦ _{KK} |
pｦ _{KL} |
||
|B| |
= |
pｦ _{LK} |
pｦ _{LL} |
Thus, in order for it to be negative definite, then pｦ _{KK} < 0, pｦ _{LL} < 0 and p^{2}(ｦ _{KKｦ LL} - ｦ _{KLｦ LK}) > 0. Assuming p ｳ 0, then the first set of conditions, that ｦ _{KK} < 0 and ｦ _{LL} < 0, implies that, at the optimum, marginal products of each of the factors must be declining - thus the profit-maximizing producer will only produce where there are diminishing marginal productivities. This is, of course, reasonable.
The interesting case is the requirement that (ｦ _{KK}ｦ_{ LL} - ｦ _{KL}ｦ_{ LK}) > 0. Notice that quasi-concavity of the production function does not guarantee this. Rather, this condition is precisely equivalent to stating that the production function is strictly concave, i.e. that it exhibits diminishing returns to scale at the optimum. In other words, solutions to the profit-maximization problem are only possible at places on the production function where there are diminishing returns to scale. Thus, the constant returns to scale case and the increasing returns to scale case do not yield a determinate solution for the profit-maximization problem.
Given this greater restriction, why study profit-maximization? The main reasoning, as already indicated above, is that output price enters explicitly into the problem. The supply curve we normally use in demand and supply theory relates output price to output supply. So far, we do not have this relationship. Furthermore, the imputation theory argues that the value of factors depends on the value of outputs, thus we would like to relate the effects of output markets on factor markets. So far, factor demands have been determined by firms independently of output prices. We need to connect these two again, and profit maximization (unlike cost-minimization) attempts this explicitly.
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