Perloff (2014, 3e, GE), Section

Transcription

1 Part 3B. Cost Minimization 3. Cost Minimization Problem 成本最小化問題 C: Input Choice Cost Minimization Problem Output Maximization Problem (Optional) Perloff (2014, 3e, GE), Section

2 C: Input Choice Input Choice From among the technologically efficient combinations of inputs, a firm wants to choose the particular bundle with the lowest cost of production, which is the economically efficient combination of inputs. To do so, the firm combines information about technology from the isoquant with information about the cost of labor and capital. 2

3 Iso-cost ine ( 等成本線 ) All the combinations of inputs that require the same (iso-) total expenditure (cost cost). K C wvk C w K the isocost line C v v C 0 v slpoe w v 0 C 0 w 3

4 Figure: Cost-Minimizing Input Choices Given output q 0, we wish to find the least costly point on the isoquant. K per period Costs are represented by parallel lines with a slope of -w/v C 1 < C 2 < C 3 q 0 C 1 C 2 C 3 per period Nicholson and Snyder (2012, 11e), Figure 10.1, p

5 Figure: Cost-Minimizing Input Choices The minimum cost of producing q 0 is C 2. K per period This occurs at the tangency between the isoquant and the total cost curve. K* The optimal choice is *, K* q 0 C 1 C 2 C 3 per period * Nicholson and Snyder (2012, 11e), Figure 10.1, p

6 Figure: Cost Minimization This firm is seeking the least cost way of producing 100 units of output. C 24 C K the isocost lines Perloff (2014, 3e, GE), Figure 7.3, p

7 Cost Minimization Problem The C To minimize costs for a given level of output, q 0. Min C w vk { K, } st s.t. f (, K) q 0 Step 1: Setting up the agrangian Min ( w vk ) [ q f (, K )] { K,, } 0 7

8 Step 2: F.O.C.s (for an Interior Minimum) f ( ): w 0 ff ( K ): v 0 K K ( ): q0 f(, K) 0 w v K w MRTS v K K S.O.C. holds if the PF is strictly quasi-concave. 8

9 Implications of F.O.Cs: w f / v f / K K MRTS ( for K ) Cost is minimized where the factor price ratio equals the ratio of marginal productivity, i.e., the MRTS. K w v K For costs to be minimized, the marginal productivity yper dollar spent should be the same for all inputs. ( 邊際生產力均等法則 ) 9

10 Interpreting the agrangian Multiplier w v K w d dc v dk dc or dq dq dq dq It measures how much the cost increase if we produce one more unit of output. the marginal cost of production Note: By the Envelope Theorem, q C * q * * MC 10

11 Step p3 3: Solution to the C Choice Functions * c ( w, v, q) conditional on q Conditional i l(contingent) Demand dfor * c K K w v q (,, ) Conditional (Contingent) Demand for K * ( wvq,, ) Minimum i Value Functions * * * C w vk C w v q (,, ) Cost Function 11

12 Factor Price Change When the wage, w, falls, the firm minimizes its new cost by substituting away from the now relatively more expensive input, capital, toward the now relatively less expensive input, labor. The change in the wage does not affect technological efficiency. Thus, it does not affect the isoquant. 12

13 Figure: Change in Factor Price: w ike the result of E, the impact of change in wage has only the Substitute Effect. Perloff (2014, 3e, GE), Figure 7.5, p

14 Step p5 5: Comparative Statics Conditional Factor Demand Functions c ( w, v, q) + + K K c ( w, v, q) + + ( wvq,, ) ++ + Cost Functions C C( w, v, q)

15 Q: What Is the Dual Problem of C? 15

16 Output Maximization Problem O: The Dual Problem of C To maximizing output for a given level of cost, C 0. Max q f ( K, ) { K, } s.t. w vk C Step 1: Setting up the agrangian { K,, } 0 Max f ( K, ) [ C wvk] D 0 16

17 Figure: Output Maximization This firm is seeking the maximum output way of spending $2,000. Perloff (2014, 3e, GE), Figure 7.4, p

18 Step 2: F.O.C.s (for an Interior Minimum) D f ( ): w 0 f D f ( K): v 0 K K D ( ): C0 wvk 0 K w v w v K 1 S.O.C. holds if the PF is strictly quasi-concave. 18

19 Step p3 3: Solution to the O Choice Functions * wvc (,, ) conditional on C K * Conditional i ldemand dfor K ( w, v, C ) Conditional Demand for K * ( wvc,, ) Maximum Value Function * * * q f K f w v C (, ) (,, ) IPF (Indirect Production Function) 19