To this point, I have developed a function to deal with the quantitative relationship between factor inputs and final commodity outputs (a production function), and I have developed a separate function to deal with the cost from renting/hiring diverse quantities of production factors at objective factor market prices (a cost function). Now, I am going to put the two together in order to analyze how Walrasian/Paretian firms maximize profits relative to factor cost constraints.
Conceptually, all firms deal continuously with the problem of how to maximize output relative to a particular level of investment in factor inputs. In the theoretic world of a general equilibrium economy, however, we will assume that this task is rigorously mathematical and, hence, yields a definite, knowable result for every level of total cost and for every desired level of output. Mathematically, in reference to differential calculus, our task amounts to a constrained maximization of the production function in relation to the objective limitations imposed by the cost function. This procedure finds a definitive maximum output level that can be realized for any given set of total factor input costs, a point that I intend to prove mathematically in the next section relative to Cobb-Douglas-type production functions. Approaching from the opposite direction, for every conceivable level output, the procedure enables us to find a particular combination of factor inputs at objective factor market prices that will minimize production costs.
Graphically, we will be superimposing, in three-dimensional xlk-space, the production function and the cost function, even if, in practical terms, we will analyze level curves in two-dimensional lk because, graphically, it is just easier to draw and explain. On the one hand, in terms of the cost function, we are asking what is the highest possible level of production that we can reach, measured by the intersection of isoquant and isocost curves. On the other hand, in terms of the production function, we are asking what is the lowest conceivable level of cost that we can reach, measured by the intersection of isoquant and isocost curves, for any given output level. Intuitively, the results here, demonstrated in figures 9 and 10, will be represented by points of tangency between isoquant and isocost functions. The remainder of my explanation in this section will seek to explain why these points of tangency represent profit maximizing/cost minimizing combinations of labor and capital.
Elaborating on my graphical demonstration in figure 9, a given firm is faced with a particular situation. It maintains a fixed quantity of financing to rent labor and capital, enabling it to expend up to but not over some unidentified 0-level of total costs (my choice of 0 for the total cost level should not be construed to mean zero-total costs - I was just trying to find some arbitrary identifying subscript for the level of total cost incurred by the firm and selected the term 0 by chance!). Faced with a tangible limit on its financing, the firm seeks, rationally, to maximize its profits in the production of the good it offers to the market (expressed by its production function). If, under the assumption that the firm is a price taker incapable of affecting the output price for its good in final commodity markets, then the sole conceivable sales limit faced by the firm might be the scale of its market and its share of total sales within the market. Here, however, we will continue to assume, in the framework of general equilibrium thinking, that any limits of this nature have already been resolved in the determination of the firm's financing. That is to say, households possessing money capital will only lend to the firm up to the point where it will be able to liquidate all of its marketable goods and not incur any costs from accumulating inventories. Therefore, whatever the firm produces it will sell at the objective final commodity output price. With this in mind, a profit maximizing firm is going to attempt to maximize its output relative to a given level of total costs.
Analyzing points of intersection between isocost curve IC0, denoting all possible combinations of labor and capital that the firm can rent at total cost 0, and the production function generating isoquant curves 1, 2, 3, and 4, certain insights should be readily apparent. First, at total cost level 0, the firm is simply incapable of realizing output level 4. The totality of the isoquant curve I4 denoting this output level lies above isocost curve IC0. No combination of labor and capital available at total cost 0 will enable the firm to produce at this level. Second, two unidentified combinations of labor and capital, denoted by intersections between I1 and IC0, will enable the firm to produce at output level 1. I have not bothered to identify these points because, as it should be pretty obvious, under the assumption that the firm will be capable of selling all that it produces, it can do better by selecting combinations of labor and capital that will produce larger quantities of output than those denoted by level 1. By the same reasoning, the points B and C on isoquant I2 can be produced with given combinations of labor (lb and lc) and capital (kb and kc) for a total cost 0, but, if the firm can sell every unit of output it produces at the same output market price, it would be worthwhile for it to see if there is some larger quantity of output that it can reach at the same level of total cost. This higher level of output exists with the combination of labor (la) and capital (ka) denoted by the point A, where isocost curve IC0 intersects with isoquant curve I3 at a single point of tangency. Under the strict set of assumptions that the firm is a price taker constrained by a definite maximum level of financing and that it can sell every unit of output it produces at a known, fixed, objective output price, the firm must maximize profits where it is maximizing output for any given total cost level, denoted here by a single point of intersection between an isocost curve and the highest attainable isoquant curve.
Figure 10 offers us an opportunity to graphically approach the problem of profit maximization from another direction. If, in terms of our analysis of figure 9, firms facing objective output prices rationally attempt to maximize output relative to cost constraints, the same firms may conceive their profit maximization problem from the framework of cost minimization given a target level of output. Such an analysis proceeds from the designation of desired production level to the determination of an appropriate combination factors to minimize costs in the production of such a level. In these terms, however, we are simply reversing the direction of our inquiry relative to figure 9.
As such, in figure 10, the firm has selected the output level 1, designated by the level isoquant curve I1. An infinitely large range of possible factor input combinations will enable the firm to realize output level 1, but, at least for our purposes, only one combination will minimize the firm's costs for renting production factors. I have simplified matters by showing just three total cost levels (11, 12, and 13), designated by isocost curves, and five available combinations, represented at point D, E, F, G, and H, to produce level 1 output. To proceed through the same sort of logic used above, points D and H on isocost curve IC13 are both possible choices for the firm, but, among the identified combinations here, they are the most costly. At point D, the firm is using a lot of capital and very little labor under conditions where the substitution of more labor for capital would reduce overall costs without affecting the firms output. The opposite is true for point H, where the firm could add more capital and reduce both its labor costs and its total costs without affecting its output. Moving from each of these points to more progressively balanced combinations of labor and capital, we can reach total cost level 12, with the two separate factor combinations E and G. Again, at combination E, the firm could reduce its total costs from producing level 1 output by substituting more labor for capital. At combination G, it could reduce its total costs by substituting more capital for labor. Only when the firm reaches factor combination F on isocost curve IC11 does it minimize its costs. This cost minimization outcome arises at a single point of tangency between the isoquant I1 representing the target output level and the lowest possible isocost curve IC11, representing the minimum possible level of total costs for producing output level 1.
Proceeding to the underlying mathematics/calculus of these outcomes, the previous sections/posts on the theory of the firm introduced two formal mathematical formulations that enable us (and our theoretic firms) to determine a unique profit maximizing/cost minimizing combination of production factors:
x1 = f(l1, k1) (Production Function)
TC1 = Pll1 + Pkk1 (Cost Function)
Mathematically, the profit maximization procedure operates as maximization of the production function, constrained by the objective factor market pricing conditions of the cost function. The easiest way to derive a solution to this problem, when we are dealing with a simple, two-variable factor (labor and capital) model, is to determine the factor combination(s) where the instantaneous rate of change in output from the production function given a change in one or the other factor is equal to the ratio of factor costs from the cost function. I will first show that this result replicates the outcome of our graphical analysis and, subsequently, explain the conceptual relevance of the result in regard to the larger logic of general equilibrium thinking.
Repeating a procedure that I undertake in the first section, if I differentiate the production function with respect to each of the factors of production, I get the following first order partial derivative, defining the marginal products of capital and labor, respectively:
δx1/δk1 = fk(k1, l1) = MPk (Marginal Product of Capital)
δx1/δl1 = fl(k1, l1) = MPl (Marginal Product of Labor) Moving forward, I want to develop a formula approximating the total rate of change of the production function as we apply infinitely small change in for each of the production factors. Conceptually, this involves the specification of a total differential for the production function, defined as:
δx1 = fk(k1, l1)δk1 + fl(k1, l1)δl1 = MPkΔk1 + MPl Δl1
This formulation states that we can approximate the total change in the production function as a sum of the rates of change for each independent variable, defined by their partial derivatives (i.e. the marginal products of capital and labor), times the change in each independent variable. In differential calculus, such a procedure is really only valid for very small changes to a multivariate equation, in the limit as changes to the independent variables approach zero. In this case, I am going to do something relevant in defining the graphical contours of our production function. I am going to set the change in the function value δx1 equal to zero. In this regard, we will redefine the terms of the total differential so that we can ask how the quantities of each production factor must change as we increase or decrease the other production factor under the condition that the total change in output remains constant (i.e. a change equal to zero). Setting δx1 equal to 0 and manipulating algebraically, I get:
0 = fk(k1, l1)δk1 + fl(k1, l1)δl1 = MPkΔk1 + MPl Δl1
-(fl(k1, l1)δl1) = (fk(k1, l1)δk1)
-(fl(k1, l1)/fk(k1, l1)) = δk1/δl1 = Δk1/Δl1
The far left hand term in the last equation tells us something important about the slope of the production function, as we hold the function value constant. Namely, it tells us that this slope is equal to a ratio of the marginal productivities of the two factors of production. I will specifically label this ratio the marginal rate of technical substitution (MRTS), and, in the above case, we are specifically identifying the marginal rate of technical substitution of labor for capital (i.e. the rate for which the quantity of capial rented by the firm must be increased/decreased in order to maintain a constant level of output if we decrease/increase the quantity of labor services rented by the firm by one unit). The MRTS is important, in itself, because it specifically identifies the slope of a level isoquant curve for a production function, where the function value is being held constant as we substitute factors of production. The MRTS is negative, reflecting the negative slope of isoquant curves. The negativity of the MRTS also implies that the marginal products for both factors of production, as the constituent arguments of the MRTS, must be positive along the surface of the isoquant curves.
Knowing the slope of an isoquant curve is important, moreover, because our graphical explanation of profit maximization/cost minimization requires that the firm produce at a point of tangency between an isoquant curve representing the target production quantity and the lowest available cost level represented by an isocost curve (or, alternately, a point of tangency between an isocost curve representing a target level of total costs and the highest available quantity of output represented by an isoquant curve). I could generalize this condition to include more complex production functions, with continuous identifiable function values but piecewise ranges with extended linear segments. I attempt to convey this idea in figure 11.
In figure 11, the production function containing isoquant curve I1 is a piecewise continuous function with one segment, at output level 1, defining the relationship between capital and labor with capital-labor ratios greater than (kA/lA), a second segment defining this relationship in a discrete range between capital-labor ratios (kA/lA) and (kB/lB), and a third segment defining this relationship for capital-labor ratios less than (kB/lB). The particular set of relative prices embodied in the cost function in figure 11, represented by isocost curve IC11 at total cost level TC11, enables the production function to intersect with the cost function not at a single point of tangency but for an extended range of factor combinations, all of which are profit maximizing/cost minimizing. Any infinitessimally small change in the factor price ratio here would disrupt this extended range of profit maximizing combinations for this level of output. If the price of labor increased relative to capital, for example, a factor combination on the range to the left of point A (inclusive) would be selected. Conversely, if the price of capital increased relative to capital, a factor combination to the right of point B (inclusive) would be selected.
Complex production functions like the one represented in figure 11 share an important feature with simple production functions defined by continuous, smooth, non-linear indifference curves with tangent intersections to cost functions. The points in which they intersect with cost functions are characterized by an equality of slopes between the cost function, represented by an isocost curve, and the production function, represented by an isoquant curve. Therefore, if we are attempting to locate a profit maximizing/cost minimizing level of output/factor combination, then our task involves finding the slopes of the production function for a given level of output (i.e. its MRTS) and equate this slope to that of the cost function defined by the objective factor market prices faced by the firm.
Proceeding, thus, in the other direction, deriving the slope of the cost function, as a linear combination of factor costs, is comparatively simpler. If I proceed with the same sort of procedure that I adopted to evaluate the production function (i.e. define the total differential), I start from the basic definition of the cost function and obtain the first order partial derivatives for its two factors:
TC1 = Pll1 + Pkk1
δTC1/δk1 = Pk
δTC1/δl1 = Pl
Taking our two partial derivatives, I define the total differential as:
δTC1 = Plδl1 + Pkδk1
Rounding out an overly complicated procedure to determine the slope of the isocost curve, if we set δTC1, the rate of change of total cost, equal to 0, accounting for the fact that total cost is constant on the isocost surface, then we get:
0 = Plδl1 + Pkδk1
Manipulating algebraically, we get:
-Plδl1 = Pkδk1
-(Pl/Pk) = δk1/δl1 = Δk1/Δl1
Where the far left side of the last equation, the negative relative price of labor in terms of capital, is the slope of the isocost curve. Therefore, adopting the mathematical resolution of our graphical evaluation of profit maximization/cost minimization, a firm's profit maximizing/cost minimizing level of output/factor combination occurs under the specific condition:
-(Pl/Pk) = -(fl(k1, l1)/fk(k1, l1))
Where the left hand side is the negative relative price of labor (the slope of an isocost curve) and the right hand side is the MRTS of capital into labor. Thus, for any given level of output and/or any given level of total cost, where a production function features continuous substitutability along its entire range (and, thus, has smooth, downward sloping isoquant curves), a single, unique point of tangency can be found where the above equality applies.
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