Advanced Microeconomics

Transcription

1 Advanced Microeconomics Ivan Etzo University of Cagliari Dottorato in Scienze Economiche e Aziendali, XXXIII ciclo Ivan Etzo (UNICA) Lecture 3: Cost Minimization 1 / 3

2 Overview 1 The Cost Minimization problem The Conditional Factor Demand Functions 3 Cost minimization and specific technologies 4 The average cost function and the Returns to Scale 5 Long-Run and Short-Run Costs Ivan Etzo (UNICA) Lecture 3: Cost Minimization / 3

3 Cost Minimization and Profit Maximization If a firm maximizes its profit by producing the output level y, then the firm is also minimizing the cost to produce y. The profit maximization problem can be split in two parts: 1 How to produce any desired level of output y at the minimum cost. To choose the specific level of output y which maximizes the profits. The point 1 does not depend on the specific output market environment (e.g. competitive, monopoly, oligopoly,..). Ivan Etzo (UNICA) Lecture 3: Cost Minimization 3 / 3

4 The Cost Minimization problem Suppose the production function of the firm is y = f (x 1, x ) and w 1 and w are the factors prices. We want to find out the cheapest way to produce a given level of output y. the cost minimization problem can be written as follows: such that min x 1,x w 1 x 1 + w x f (x 1, x ) = y The solution to this problem is a function of w 1, w and y, that is the cost function c(w 1, w, y) The Cost Function measures the minimal costs of producing output y when factor prices are (w 1, w ) Ivan Etzo (UNICA) Lecture 3: Cost Minimization 4 / 3

5 The Cost Minimization problem The geometric solution How to find the solution for the Cost Minimization Problem? Three possible ways: 1 Geometric solution; Lagrangian multiplier; 3 By substitution (see the example with the Cobb-Douglas technology). The geometric solution We can write the objective function as follows w 1 x 1 + w x = C which represents all the combinations of inputs for a specific level of cost C, that is the isocost line or equivalently x = C w w 1 w x 1 Ivan Etzo (UNICA) Lecture 3: Cost Minimization 5 / 3

6 The Cost Minimization problem The geometric solution The Isocost Line Ivan Etzo (UNICA) Lecture 3: Cost Minimization 6 / 3

7 The Cost Minimization problem The geometric solution The Isocost Lines Map where C > C > C > C Ivan Etzo (UNICA) Lecture 3: Cost Minimization 7 / 3

8 The Cost Minimization problem The geometric solution Holding the output level fixed, we can draw the related isoquant curve. Ivan Etzo (UNICA) Lecture 3: Cost Minimization 8 / 3

9 The Cost Minimization problem The geometric solution Solution Ivan Etzo (UNICA) Lecture 3: Cost Minimization 9 / 3

10 The Cost Minimization problem The solution using the Lagrange multipliers Recall the constrained-minimization problem such that we can set up the Lagrangian min x 1,x w 1 x 1 + w x f (x 1, x ) = y L = w 1 x 1 + w x γ(f (x 1, x ) y) Taking the partial derivatives with respect to x 1, x and γ gives the FOCs: w 1 γ f (x 1, x ) x 1 = 0 w γ f (x 1, x ) x = 0 f (x 1, x ) y = 0 Ivan Etzo (UNICA) Lecture 3: Cost Minimization 10 / 3

11 The Cost Minimization problem The solution using the Lagrange multipliers Rewrite the first two FOCs as follows w 1 = γ f (x 1, x ) x 1 w = γ f (x 1, x ) x And divide the first equation by the second to get: w 1 w = f (x 1, x )/ x 1 f (x 1, x )/ x or, equivalently w 1 w = TRS(x 1, x ) Ivan Etzo (UNICA) Lecture 3: Cost Minimization 11 / 3

12 The Conditional Factor Demand Functions For each factor price (w 1, w ) and output level y there will be some choice of factor inputs (x1, x ) that minimizes the cost of producing y units of output. The functions that give the optimal choices are called conditional factor demand functions or derived factor demands. x 1 (w 1, w, y); x (w 1, w, y) the factor demand choices are conditional on the firm producing a given level of output y. Ivan Etzo (UNICA) Lecture 3: Cost Minimization 1 / 3

13 Cost minimization and specific technologies Perfect complements Suppose f (x 1, x ) = min{ax 1, bx } The firm must operate at point where y = ax 1 = bx Accordingly, in order to produce y units of output, the firm must use y/a units of factor 1 and y/b units of factor. Hence, the cost function is y c(w 1, w, y) = w 1 a + w y ( b = y w1 a + w ) b Ivan Etzo (UNICA) Lecture 3: Cost Minimization 13 / 3

14 Cost minimization and specific technologies Perfect substitutes technology f (x 1, x ) = ax 1 + bx The factors are perfect substitutes, thus the firm will use the cheapest one. We will have a boundary solution where y = ax 1 or y = bx Hence, the cost function will be { y c(w 1, w, y) = min w 1 a, w y } b { w1 = min a, w } y b Ivan Etzo (UNICA) Lecture 3: Cost Minimization 14 / 3

15 Cost minimization and specific technologies Cobb-Douglas technology Consider the cost minimization problem min x 1,x w 1 x 1 + w x such that f (x 1, x ) = x a 1 x b We can solve the constraint for x and substitute in the cost function such that the cost-minimization problem can be written as follows min x 1 w 1 x 1 + w y 1 a b x b 1 The FOC is w 1 a b w y 1 b x b 1 = 0 Ivan Etzo (UNICA) Lecture 3: Cost Minimization 15 / 3

16 Cost minimization and specific technologies Cobb-Douglas technology Solving the FOC for x 1 we get the conditional demand function for factor 1: ( ) b aw 1 x 1 (w 1, w, y) = y bw 1 Similarly, by repeating the same substitution procedure for the other factor we get the following conditional demand function for factor : ( ) a aw 1 x (w 1, w, y) = y bw 1 The cost function is c(w 1, w, y) = w 1 x 1 (w 1, w, y) + w x (w 1, w, y) Ivan Etzo (UNICA) Lecture 3: Cost Minimization 16 / 3

17 Cost minimization and specific technologies Cobb-Douglas technology ( ) b w ( a ) b ( ) a c(w 1, w, y) = w 1 y 1 w ( a ) a + w y 1 w 1 b w 1 b = w a 1 w b = ( a b ) b y 1 + w b [ ( a ) b ( a ) a ] + b b w a 1 w a ( a b ) a y 1 1 w b y 1 we can also write c(w 1, w, y) = Kw a where, K = 1 w b y 1 [ ( a ) b ( a ) a ] + b b Ivan Etzo (UNICA) Lecture 3: Cost Minimization 17 / 3

18 Cost minimization and specific technologies Cobb-Douglas technology c(w 1, w, y) = Kw a 1 w b y 1 Note that the relationship between the output changes and the costs depends on the returns to scale. If there are constant RtS (i.e. a + b = 1) then the costs increase linearly with output: c(w 1, w, y) = Kw a 1 w b y If there are increasing RtS (i.e. a + b > 1) then the costs increase less than linearly with output. If there are decreasing RtS (i.e. a + b < 1) then the costs increase more than linearly with output. Ivan Etzo (UNICA) Lecture 3: Cost Minimization 18 / 3

19 The average cost function and the Returns to Scale The average cost function is the cost per unit of output when the firm produces a specific level of output y: AC(y) = c(w 1, w, y). y For example let us consider the Cobb-Douglas technology: In case of constant RtS the average cost function is AC(y) = Kw a 1 w b y y = Kw a 1 w b. That is, the average cost function is constant in y. Ivan Etzo (UNICA) Lecture 3: Cost Minimization 19 / 3

20 The average cost function and the Returns to Scale If the technology exhibits increasing RtS (i.e. a + b > 1) the average cost function is AC(y) = Kw a 1 w b y 1 y = Kw a 1 w b y 1 () where 1 () < 0 Thus, the AC(y) will be decreasing in y. If the technology exhibits decreasing RtS (i.e. a + b < 1) the average cost function is AC(y) = Kw a 1 w b y 1 y = Kw a 1 w b y 1 () where 1 () > 0 Thus, the AC(y) will be increasing in y. Ivan Etzo (UNICA) Lecture 3: Cost Minimization 0 / 3

21 Long-Run and Short-Run Costs The cost function gives the minimum cost of production for a given level of output. The Short-run cost function is the minimum cost of production for a given level of output when at least one factor cannot be adjusted (i.e. is fixed). The Long-run cost function is the minimum cost of production for a given level of output when all factors can be adjusted (i.e. they are all variable). Hence, if factor is fixed at x, the short-run cost function is: c(y, x ) = min x 1 w 1 x 1 + w x such that f (x 1, x ) = y Thus, the minimum cost to produce the output level y depends also on the fixed factor x and its price. Ivan Etzo (UNICA) Lecture 3: Cost Minimization 1 / 3

22 Long-Run and Short-Run Costs From the solution of the short-run cost minimization problem are derived the short-run factor demads: x 1 = x s 1(w 1, w, x, y) x = x Accordingly, the short run demand of factor 1 depends on the fixed level of factor as well. Moreover, by definition, the short-run cost function can be written as follows: c s (y, x ) = w 1 x s 1(w 1, w, x, y) + w x. Ivan Etzo (UNICA) Lecture 3: Cost Minimization / 3

23 Long-Run and Short-Run Costs The long-run cost function is defined by: c(y) = min x 1,x w 1 x 1 + w x such that f (x 1, x ) = y which depends on the level of output and the factor prices The long-run factor demads are x 1 = x 1 (w 1, w, y) x = x 1 (w 1, w, y). And the long-run cost function can be written as. c(y) = w 1 x 1 (w 1, w, y) + w x (w 1, w, y) Ivan Etzo (UNICA) Lecture 3: Cost Minimization 3 / 3