Week 10: Theory of the Firm (Jehle and Reny, Chapter 3)

Transcription

1 Week 10: Theory of the Firm (Jehle and Reny, Chapter 3) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China November 22, 2015 First Last (shortinst) Short title November 22, / 34

2 3.1 Primitive Notions The second important actor on the microeconomic stage is the individual firm. A firm is an entity created by individuals for some purpose. This entity will typically acquire inputs and combine them to produce output. Inputs are purchased on input markets and these expenditures are the firm s costs. Output is sold on product markets and the firm earns revenue from those sales. Similar to utility maximization, a firm is assumed to pursue profit maximization.the owners of a firm are consumers, the more profit the firm can make, the greater will be its owner s command over goods and services. Of course, profit maximization may not be the only motive behind firm behavior. Sales, market shares, or even prestige maximization are also possibilities. However, in the long run, it is not sustainable for a firm without profit. So we keep the assumption of profit maximization. First Last (shortinst) Short title November 22, / 34

3 3.2 Production Production is the process of transforming inputs into outputs. The firm has a production possibility set, Y R m, where each vector y = (y 1,, y m ) Y is a production plan whose components indicate the amounts of the various inputs and outputs (where m are the numbers of output and input factors). A common convention is to write element of y Y so that y i < 0 if resource i is used up in the production plan, and y i > 0 if resource i is produced in the production plan. For simplicity, we consider firms producing only a single product from many inputs. For that, it is more convenient to describe the firm s technology in terms of a production function. When there is only one output produced by many inputs, we shall denote the amount of output by y, and the amount of input i by x i, so that with n inputs, the entire vector of inputs is denoted by x = (x 1,, x n ). Of course, the input vector as well as the amount of output must be nonnegative. First Last (shortinst) Short title November 22, / 34

4 The production function, f, is therefore a mapping from R n + into R +. When we write y = f (x), we mean that y units of output can be produced using the input vector x. We make the further assumption on the production function f. Assumption 3.1 Properties of the Production Function The production function, f : R n + R +, is continuous, strictly increasing, and strictly quasiconcave on R n +, and f (0) = 0. Continuity of f ensures that small changes in the vector of inputs lead to small change in the amount of output produced. We require f to be strictly increasing to ensure that employing strictly more of every input results in strictly more output. The strictly quasiconcave says that any convex combination of two input vectors can produce at least as much as one of the original two. The last condition states that a positive amount output requires positive amount of some of the inputs. First Last (shortinst) Short title November 22, / 34

5 When the production function is differentiable, its partial derivative, f (x)/ x i, is called the marginal product of input i and gives the rate at which output changes per additional unit of input i employed. The assumption of strictly increasing production function implies f (x)/ x i > 0 for all i. For any fixed level of output, y, the set of input vectors producing y units of output is called the y-level isoquant, a level set of f. We denote this set by Q(y), Q(y) {x 0 f (x) = y}. An analog to the marginal rate of substitution in consumer theory is the marginal rate of technical substitution (MRTS) in production theory. This measures the rate at which one input can be substituted for another without changing the amount of output produced. Formally, the MRTS between inputs i and j when the current input vector is x, denoted MRTS ij (x), is defined as the ratio of marginal products, First Last (shortinst) Short title November 22, / 34

6 MRTS ij (x) = f (x)/ x i f (x)/ x j This show the rate at which input i can be substituted for input j when holding output constant. In the two-input case, as depicted in Figure 3.1, MRTS 12 (x 1 ) is the absolute value of the slope of the isoquant through x 1 at the point x 1. Figure 3.1. The marginal rate of technical substitution First Last (shortinst) Short title November 22, / 34

7 In general, the MRTS between any two inputs depends on the amount if all inputs employed (see x in the formula). However, if it is independent of all inputs, then we call the production is separable. The MRTS measures how easy an input can be substituted for another input at a point in the isoquant, sometimes we are more interested in how easy an input can be substituted for another input in the whole isoquant. This measure is called the elasticity of substitution. Intuitively, If the MRTS ij does not change at all for change in x j /x i, we might say that substitution is easy, because the ratio of the marginal productivities of the two inputs does not change as the input mix changes. Alternatively, if the MRTS ij changes rapidly for small change in x j /x i, we would say that substitution is difficult, because minor variations in the input mix will have a substantial effect on the inputs relative productivities. (Nicholson, 2004). Formally, Definition 3.2 The Elasticity of Substitution For a production function f (x), the elasticity of substitution between inputs i and j at the point x is defined as First Last (shortinst) Short title November 22, / 34

8 Definition 3.2 (Continued) inputs i and j at the point x is defined as σ ij = percent change in x j/x i = d ln(x j/x i ) percent change in MRTS ij d ln(f i (x)/f j (x)) = d(x j/x i ) f i (x)/f j (x) x j /x i d(f i (x)/f j (x)), where f i and f j are the marginal products of input i and j. When the production function is quasiconcave, the elasticity of substitution can never be negative, so σ ij 0. See Figure 3.2. (the next slide) as an example. In (a), the isoquant is linear so d MRTS 12 = 0, or σ 12 is infinite. The inputs 1 and 2 are perfect substitutable. In (c), the two inputs are productive only in fixed proportions with one another, d MRTS 12, or σ 12 = 0. In (b), the production function is an intermediate case where σ is neither zero nor infinite, and the isoquant are neither straight lines nor right angles. First Last (shortinst) Short title November 22, / 34

9 Figure 3.2. (a) σ is infinite and there is perfect substitutability between inputs. (b) σ is finite but larger than zero, indicating less than perfect substitutability. (c) σ is zero and there is no substitutability between inputs. First Last (shortinst) Short title November 22, / 34

10 Example 3.1 Consider the CES production function y = (x ρ 1 + x ρ 2 )1/ρ for 0 ρ < 1. To calculate the elasticity of substitution, we first note that ln(x 2 /x 1 ) = ln(x 2 ) ln(x 1 ), so that when we take the total differential in the numerator of σ, we get ( ) x2 d ln = d ln(x 2 ) d ln(x 1 ) = 1 dx 2 1 ( 1 dx 1 = dx 1 1 ) dx 2. (E.1) x 2 x 1 x 1 x 2 x 1 Calculating the partials of f, forming the required ratio, and taking logs, we have ( ) ( ) f1 x ρ 1 1 ln = ln f 2 x ρ 1 = (ρ 1)[ln(x 1 ) ln(x 2 )]. 2 Taking the total differential gives the denominator of σ: ( ) ( f1 1 d ln = (ρ 1) dx 1 1 ) dx 2. (E.2) x 1 x 2 f 2 First Last (shortinst) Short title November 22, / 34

11 Example 3.1 (Continued) Divide (E.1) by (E.2), we get σ = 1 1 ρ which is a constant: hence, the initials CES, which stand for constant elasticity of substitution. The closer ρ is to unity, the larger is σ; when ρ is equal to 1, σ is infinite and the production function is linear. Other popular production functions can be derived from specific CES forms. Such as Cobb-Douglas production function, with σ ij = 1. And the Leotief form, with σ ij = 0. y = n i=1 x α i i, where n i=1 α i = 1. y = min{a 1 x 1,, a n x n }. First Last (shortinst) Short title November 22, / 34

12 All CES production functions (including the cases of Cobb-Douglas and Leontief) are members of the class of linear homogeneous production functions. One of the important properties of linear homogeneous production functions is they are always concave. Theorem 3.1 (Shephard) Linear Homogeneous Production Functions are Concave Let f (x) be a production function satisfying Assumption 3.1 and suppose it is homogeneous of degree one. Then f (x) is a concave function of x. We frequently want to know how output responds as the amounts of different inputs, or returns, are varied. For instance, in the short run, the period of time in which one input is fixed, output can be varied only by changing the amounts of some inputs but not others. Figure 3.3.(see the next slide) shows how output behaves as we move through the isoquant map along the horizon at x 2, keeping x 2 constant and varying the amount of x 1. In the long run, each input is flexible First Last (shortinst) Short title November 22, / 34

13 to change. Returns to scale refer to how output responds when all inputs are varied in the same proportion, so they exist in the context of long run. In Figure 3.3., returns to scale have to do with how output behaves as we move through the isoquant map along a ray such as OA, where the levels of x 1 and x 2 are changed simultaneously, always staying in the proportion x 2 /x 1 = α. Figure 3.3. Returns to scale and varying proportions First Last (shortinst) Short title November 22, / 34

14 The common measures of returns includes the marginal product MP i (x) f i (x), and the average product, AP i (x) f (x)/x i, of each input. The output elasticity of input i, measuring the percentage response of output to a 1 percent change in input i, is given by µ i (x) = df (x)/f (x) dx i /x i = df (x)/dx i f (x)/x i = MP i(x) AP i (x) Each of these is a local measure, defined at a point, i.e. at different points, the values differ. The scale properties of the technology may be defined either locally or globally. A production function is said to have globally constant, increasing, or decreasing returns to scale according to the following definitions. Definition 3.3 (Global) Returns to Scale A production function f (x) has the property of (globally): 1. Constant returns to scale if f (tx) = tf (x) for all t > 0 and all x. 2. Increasing returns to scale if f (tx) > tf (x) for all t > 1 and all x. 3. Decreasing returns to scale if f (tx) < tf (x) for all t > 1 and all x. First Last (shortinst) Short title November 22, / 34

15 Many technologies exhibit increasing, constant, and decreasing returns over only certain range of output. It is therefore useful to have a local measure of returns to scale. One such measure, defined at a point, tells us the instantaneous percentage change in output that occurs with a 1 percent increase in all inputs. The measure is known as the elasticity of scale or the elasticity of output, and is defined as follows. Definition 3.4 (Local) Returns to Scale The elasticity of scale at the point x is defined as n d ln[f (tx)] i=1 µ(x) lim t 1 = f i(x)x i. d ln(t) f (x) Returns to scale are locally constant, increasing, or decreasing as µ(x) is equal to, greater, than, or less than one. The elasticity of scale and the output elasticities of the inputs are related as follows: µ(x) = n i=1 µ i(x) First Last (shortinst) Short title November 22, / 34

16 Example 3.2 Let s examine a production function with local returns to scale: y = k(1 + x α 1 x β 2 ) 1 where α > 0, β > 0, and k is an upper bound on the level of output, so that 0 y < k. Calculating the output elasticities for each output, we obtain µ 1 (x) = α(1 + x α 1 x β 2 ) 1 x α 1 x β 2, µ 2 (x) = β(1 + x α 1 x β 2 ) 1 x α 1 x β 2, Adding these two gives the elasticity of scale: µ(x) = (α + β)(1 + x α 1 x β 2 ) 1 x α 1 x β 2 = (α + β)(1 y k ) which also varies with x 1 and x 2, or output y. If (α + β) > 1, the production function could exhibit increasing and decreasing to scale for low and high levels of output, respectively. First Last (shortinst) Short title November 22, / 34

17 3.3 Cost For a given level of output, there are various input vectors that can achieve it. The firm must decide which of the possible production plans (input vectors) it will use. If the object of the firm is to maximize profits, it will necessarily choose the least costly, or cost-minimizing, production plan for every level of output. We will assume throughout that firms are perfectly competitive on their input markets and that therefore they face fixed input prices. Let w = (w 1,, w n ) 0 be a vector of prevailing market prices at which the firm can buy inputs x = (x 1,, x n ). Because the firm is a profit maximizer, it will choose to produce some level of output while using that input vector requiring the smallest money outlay. One can speak therefore of the cost of output y - it will be the cost at prices w of the least costly vector of inputs capable of producing y. First Last (shortinst) Short title November 22, / 34

18 Definition 3.5 The Cost Function The cost function, defined for all input prices w 0 and all output levels y f (R n +) is the minimum-value function, c(w, y) min x R n + w x s.t. f (x) y. If x(w, y) solves the cost-minimization problem, then c(w, y) = w x(w, y). Because f is strictly increasing, the constraint will always be binding at a solution. Consequently, the cost minimization problem is equivalent to min x R n + w x s.t. y=f (x). Let x denote a solution to the minimization problem and f is differentiable at x. By Lagrange s theorem, there is a λ R such that First Last (shortinst) Short title November 22, / 34

19 w i = λ f (x ) x i, i = 1,, n. Because w i > 0, i = 1,, n, we may divide the preceding ith equation by the jth to obtain f (x )/x i f (x = w i. )/x j w j Thus, cost minimization implies that the marginal rate of technical substitution between any two inputs is equal to the ratio of their prices. From the first-order condition, it is clear the solution depends on the parameters w and y, i.e. x (w, y). Given Assumption 3.1 and assume w 0, the solution is unique. Figure 3.4. (see the next slide) illustrates the solution to the cost-minimization problem with a two-input case. The solution is point where the y-level isoquant Q(y) and an isocost line if the form w x = α, α > 0 tangent. If x 1 (w, y) and x 2 (w, y) are solutions, then the minimum cost is c(w, y) = w 1 x 1 (w, y) + w 2 x 2 (w, y). First Last (shortinst) Short title November 22, / 34

20 Additionally, x (w, y) is referred to as the firm s conditional input demand, because it is conditional on the arbitrary level of output y. So at this point, cost minimization may or may not be profit maximizing. Figure 3.4. The solution to the firm s cost-minimization problem. In the figure, α < α. First Last (shortinst) Short title November 22, / 34

21 Example 3.3 There is a cost minimization problem with the CES production function, min x1 0,x 2 0w 1 x 1 + w 2 x 2 s.t. (x ρ 1 + x ρ 2 ) 1 ρ y Assuming the interior solution exists, the first-order Lagrangian conditions reduce to the two conditions w 1 w 2 = ( ) ρ 1 x1 x 2 y = (x ρ 1 + x ρ 2 ) 1 ρ Solve these two equations for x 1 and x 2, x 1 = yw 1/(ρ 1) 1 (w ρ/(ρ 1) 1 + w ρ/(ρ 1) 2 ) 1/ρ, x 2 = yw 1/(ρ 1) 2 (w ρ/(ρ 1) 1 + w ρ/(ρ 1) 2 ) 1/ρ. First Last (shortinst) Short title November 22, / 34

22 Example 3.3 (Continued) To obtain the cost function, just substitute x 1 and x 2 into the objective function for the minimization problem. c(w, y) = w 1 x 1 (w, y) + w 2 x 2 (w, y) = y(w ρ/(ρ 1) 1 + w ρ/(ρ 1) 2 ) ρ 1/ρ. The cost function and the expenditure function in consumer theory is identical, so they share the same properties. We state them here without proofs for their proofs are identical to the those given for the expenditure function. Theorem 3.2 Properties of the Cost Function If f is continuous and strictly increasing, then c(w, y) is Zero when y = 0. Continuous on its domain. For all w 0, strictly increasing and unbounded above in y. First Last (shortinst) Short title November 22, / 34

23 Theorem 3.2 (Continued) Increasing in w. Homogeneous of degree one in w. Concave in w. Moreover, if f is strictly quasiconcave we have Shephard s lemma: c(w, y) is differentiable in w at (w 0, y 0 ) whenever w 0 0, and Example 3.4 c(w 0, y 0 ) w i = x i (w 0, y 0 ), i = 1,, n. Consider a cost function with the Cobb-Douglas form, y = x α 1 x β 2, where α + β = 1. We can obtain the cost function c(w, y) = Aw α 1 w β 2 y where A = β β α α. First Last (shortinst) Short title November 22, / 34

24 Example 3.4 (Continued) By Shephard s lemma, x 1 (w, y) = x 2 (w, y) = c(w, y) w 1 c(w, y) w 2 = αaw α 1 1 w β 2 y = βaw α 1 w β 1 2 y As solutions to the firm s cost-minimization problem, the conditional input demand functions possess certain general properties. These are analogous to the properties of Hicksian demands, so once again it is not necessary to repeat the proof. Theorem 3.3 Properties of Conditional Input Demands Suppose the production function satisfies Assumption 3.1 and that the associated cost function is twice continuously differentiable. Then First Last (shortinst) Short title November 22, / 34

25 Theorem 3.3 (Continued) 1 x(w, y) is homogeneous of degree zero in w. 2 The substitution matrix, defined and denoted x 1 (w,y) x w 1 1 (w,y) w n σ (w, y)..... x n(w,y) x w 1 n(w,y) w n is symmetric and negative semidefinite. In particular, the negative semidefiniteness property implies that x i (w, y)/ w i 0 for all i. Definition: Homothetic Function A real-valued function h on D R n is called homothetic if it can be written in the form, h(x) = g(f (x)), where g : R R is strictly increasing and f : D R is homogeneous of degree 1. First Last (shortinst) Short title November 22, / 34

26 The cost and conditional input demands associated with homothetic production technologies have some special properties collected in what follows. Theorem 3.4 Cost and Conditional Input Demands When Production is Homothetic When the production function satisfies Assumption 3.1 and is homothetic, the cost function is multiplicatively separable in input prices and output and can be written c(w, y) = h(y)c(w, 1), where h (y) > 0 and c(w, 1) is the unit cost function, or the cost of 1 unit of output; the conditional input demands are multiplicatively separable in input prices and output and can be written x(w, y) = h(y)x(w, 1), where h (y) > 0 and x(w, 1) is the conditional input demand for 1 unit of output. First Last (shortinst) Short title November 22, / 34

27 The general form of the cost function that we have been considering until now is most properly viewed as giving the firm s long-run cost because the firm may freely choose the amount of every input it uses. In the short run, the firm is stuck with fixed amounts of certain inputs. We should expect its cost in the short run to differ from its costs in the long run. The followings define the short-run cost function. Definition 3.6 The Short-Run, or Restricted, Cost Function Let the production function be f (z), where z (x, x). Suppose that x is a subvector of variable inputs and x is a subvector of fixed inputs. Let w and w be the associated input prices for the variable and fixed inputs, respectively. The short-run, or restricted, total cost function is defined as sc(w, w, y; x) min x w x + w x s.t. f (x, x) y. If x(w, w, y; x) solves this minimization problem, then First Last (shortinst) Short title November 22, / 34

28 Definition 3.6 (Continued) sc(w, w, y; x) = w x(w, w, y; x) + }{{}}{{} w x total variable cost total fixed cost It should be clear that for a given level of output, long-run costs, where the firm is free to choose all inputs optimally, can never be greater than short-run costs, where the firm may choose some but not all inputs optimally. This point is illustrated in Figure 3.5. (see the next slide) where x 2 is fixed in x 2 in the short run. For the level of output y 1, the optimal choice of x 2 is x 2 < x 2 and the corresponding x 1 (the input combination B) where the isoqant tangent to the long-run cost c(y 1 ). But in the short run, x 2 is fixed in x 2, to minimize the cost given y 1, the firm has to use the input combination A. And the cost is sc(y 1 ) > c(y 1 ). For the case where the level of output y 3 is produced, in the short run the level of input x 2 is smaller than the long-run optimal level x 2, so sc(y 3 ) > c(y 3 ). The only exception is the case where the short-run constraint of x 2 is exactly equivalent to the First Last (shortinst) Short title November 22, / 34

29 long-run choice, like the case that sc(y 2 ) = c(y 2 ) in the center of Figure 3.5. Figure 3.5. sc(w, w, y; x) c(w, w, y) for all output levels y. First Last (shortinst) Short title November 22, / 34

30 To explore the relationship between the long-run and short-run cost functions a bit further, let x(y) denote the optimal choice of the fixed input to minimize short-run cost of output y at the given input prices. Then we ve argue that c(w, w, y) sc(w, w, y; x(y)) (eq.1) must hold for any y. Differentiate this equation with respect to y, dc(w, w, y) dy = sc(w, w, y; x(y)) y + i sc(w, w, y; x(y)) x i (y) x i y (eq.2) But since x(y) is the optimal choice of the fixed input at the level y, sc(w, w, y; x(y)) x i 0 for all fixed input i. Therefore, (eq.2) becomes First Last (shortinst) Short title November 22, / 34

31 dc(w, w, y) sc(w, w, y; x(y)) = dy y (eq.3) The short-run cost-minimization problem involves more constraints on the firm than the long-run problem, so we know that sc(w, w, y; x) c(w, w, y) for all levels of the fixed input. (eq.1) shows that for some levels of output y, sc(w, w, y ; x) is equal to c(w, w, y ). (eq.3) further shows that when sc(w, w, y ; x) = c(w, w, y ), the short-run and long-run cost coincides at a point where their slopes are equal in the cost-output plane, illustrated in Figure 3.6. (see the next slide) where at y 1, y 2 and y 3, the short-run cost is equal to the long-run cost. First Last (shortinst) Short title November 22, / 34

32 Figure 3.6. Long-run total cost is the envelope of short-run total cost. First Last (shortinst) Short title November 22, / 34

33 3.4 Duality in Production Given the obvious structural similarity between the firm s cost-minimization problem and the individual s expenditure minimization problem, there is also a duality between production and cost just as there is between utility and expenditure. It is possible to recover the production function through the conditional input demand functions. And if the conditional input demand function x(w, y) is homogeneous of degree zero in w, and its substitution matrix is negative semidefinite and symmetric, then we can ensure it is consistent with the cost-minimization behavior. First Last (shortinst) Short title November 22, / 34

34 2nd Midterm Exam Date/Time: Monday Nov. 30th, 2015 / 9:00 am -11:30 am Location: The meeting room on the 2nd floor, School of Economics Coverage: Consumer Theory (Week 6- Week 9) Rules: Same as the 1st midterm First Last (shortinst) Short title November 22, / 34