Chapter 3B: Profit Maximization Problem

Transcription

1 Econ 3821 Labor Economics Chapter 3B: Profit Maximization Problem Instructor: Hiroki Watanabe Fall 212 Watanabe Econ B PMP 1 / 57 1 Introduction 2 Short-Run Profit Maximization Problem 3 Short-Run Labor Demand 4 Comparative Statics 5 Now We Know Watanabe Econ B PMP 2 / 57 1 Introduction Assumptions 2 Short-Run Profit Maximization Problem 3 Short-Run Labor Demand 4 Comparative Statics 5 Now We Know Watanabe Econ B PMP 3 / 57

2 Assumptions Question 1.1 (Agenda Where does labor demand function come from? Watanabe Econ B PMP 4 / 57 Assumptions Recall that labor demand is a derived demand. It is a result of profit maximization. Watanabe Econ B PMP 5 / 57 Assumptions Assumption 1.2 (Price Taker Assumption A competitive firm is a price taker, i.e., it takes output price p and factor price = ( C, K as given. Watanabe Econ B PMP 6 / 57

3 Assumptions Assumption 1.3 (Time Frame 1 In the short run, Jack cannot revise at least one of his input levels. 2 In the long run, Jack is unrestricted in its choice of input levels. Watanabe Econ B PMP 7 / 57 Assumptions Rent on kitchen does not affect Jack s short-run decision making because it is Definition 1.4 (Sunk Cost Sunk cost is the amount Jack has already paid or promised to pay. It cannot be recouped regardless of future actions. Rent on kitchen is not sunk in the long run. Watanabe Econ B PMP 8 / 57 1 Introduction 2 Short-Run Profit Maximization Problem Short-Run Profit Maximization Problem Positive Profit 3 Short-Run Labor Demand 4 Comparative Statics 5 Now We Know Watanabe Econ B PMP 9 / 57

4 Short-Run Profit Maximization Problem Problem 2.1 (Short-Run Profit Maximization Problem Given p, C, K, K, Jack max C π( C, K = pƒ ( C, K C C K K = pƒ ( C, K C C FC. Watanabe Econ B PMP 1 / 57 Short-Run Profit Maximization Problem How does Jack go about Problem 2.1? Try random C s? Recall marginal condition. Watanabe Econ B PMP 11 / 57 Short-Run Profit Maximization Problem If Jack hires one more chef (Liz, does she increase his profit? 1 If she does, go ahead. 2 If she decreases the profit, downsize. 3 If she doesn t change, then that s the C he s looking for. Watanabe Econ B PMP 12 / 57

5 Short-Run Profit Maximization Problem Change in profit (marginal profit from an additional chef has two sources: 1 Additional cost (marginal cost that Jack has to pay for Liz: C. 1 2 Additional revenue (marginal revenue that Liz will bring in: pmp C( C. 2 1 in a perfectly competitive labor market. 2 in a perfectly competitive cheesecake market. Watanabe Econ B PMP 13 / 57 Proposition 2.2 (Marginal Condition The output level that maximizes Jack s profit C satisfies: C = pmp C ( C, provided that π(. Watanabe Econ B PMP 14 / 57 Example 2.3 (Short-Run Profit Maximization Problem (Discrete If ( C, K = (1, 15 and K is constant at K = 1, p =.5, Jack s short-run profit is given by: C y py K K C C TC π MC MP MR Mπ Watanabe Econ B PMP 15 / 57

6 8 TC(x C TR(x C 6 π(x C TC, TR, π ($ Watanabe Econ B PMP 16 / 57 Fact 2.4 (Profit-Maximizing Scale The gap between total revenue and total cost grows largest when their slopes are same to each other. The slope of production function marks MP C ( C. 3 The slope of total revenue marks pmp C ( C. The slope of total cost marks MC C ( C = C. translates to: The slope of total revenue coincides with the slope of total cost at C.4 Proposition C.f., Example 2.7 in previous lecture. 4 provided π( >. Watanabe Econ B PMP 17 / MC, MR, Mπ ($ MC(x C MR(x C Mπ(x C Watanabe Econ B PMP 18 / 57

7 Note that exclusion of fixed cost does not change the solution. Watanabe Econ B PMP 19 / 57 8 VC(x C 7 TR(x C 6 Sπ(x C VC, TR, Sπ ($ Watanabe Econ B PMP 2 / 57 Question 2.5 (Short-Run Profit Maximization Problem (Continuous Jack s production function is ƒ ( = 2 C K. Given p = 2, C = 12, K = 1, K = 5, and marginal product K MP C ( C, K = 1, C 1 set up SPMP, 2 what is the marginal condition, and 3 how many chefs should Jack hire? Watanabe Econ B PMP 21 / 57

8 TC, TR, π ($ TC(x C TR(x C π(x C / Watanabe Econ B PMP 22 / MC(x C MR(x C Mπ(x C MC, MR, Mπ ($ / Watanabe Econ B PMP 23 / 57 Positive Profit Caveat with Proposition 2.2. Watanabe Econ B PMP 24 / 57

9 Positive Profit 2. VC(x C TR(x C 1.5 Sπ(x C VC, TR, Sπ ($ Watanabe Econ B PMP 25 / 57 Positive Profit Proposition 2.2 just gives you the largest gap. It does not guarantee that Jack will get positive profit. It might actually pick up the worst C. Watanabe Econ B PMP 26 / 57 Positive Profit 2. MC(x C MR(x C 1.5 Mπ(x C MC, MR, Mπ ($ Watanabe Econ B PMP 27 / 57

10 Positive Profit Fact 2.6 (Positive Profit If the marginal condition Proposition 2.2 picks up the point C where MP C ( C is increasing, Jack earns the smallest profit with C. a a This is known as a second-order condition. Watanabe Econ B PMP 28 / 57 1 Introduction 2 Short-Run Profit Maximization Problem 3 Short-Run Labor Demand Short-Run Labor Demand Shut-Down Point 4 Comparative Statics 5 Now We Know Watanabe Econ B PMP 29 / 57 Short-Run Labor Demand Definition 3.1 (Short-Run Labor Demand Function Jack s short-run labor demand function SD( C indicates the profit-maximizing number of chefs C for each C, given p, K, and K. Watanabe Econ B PMP 3 / 57

11 Short-Run Labor Demand In Example 2.3 and Question 2.5, we fixed C (1 and 12 each. To derive short-run labor demand function, we switch the value of C and see how Jack will revise his decision, in particular, on C. Watanabe Econ B PMP 31 / 57 Short-Run Labor Demand Question 3.2 (Short-Run Profit Maximization Problem Jack s production function is ƒ ( = 2 C K. Given p = 2, C = 12 and 16, K = 1, K = 5, and marginal K product MP C ( C, K = 1, C 1 How many chefs should Jack hire when C = 12 and 16? Watanabe Econ B PMP 32 / 57 Short-Run Labor Demand 1 8 TC, TR, π ($ TR(x C TC(x, w =12 C C π(x C, w C =12 TC(x, w =16 C C π(x, w =16 C C 25/16 25/ Watanabe Econ B PMP 33 / 57

12 Short-Run Labor Demand The next graph is really important. Watanabe Econ B PMP 34 / 57 Short-Run Labor Demand 2 16 MC, MR, Mπ ($ MR(x C MC(x, w =12 C C Mπ(x C, w C =12 MC(x, w =16 C C Mπ(x, w =16 C C 4 25/16 25/ Watanabe Econ B PMP 35 / 57 Short-Run Labor Demand Notice the difference: 1 MR C ( C defines the additional revenue at each C. 2 C = MR C ( C only at C. For each C, we can find corresponding C, and that s how we find short-run labor demand function SD( C. Watanabe Econ B PMP 36 / 57

13 Shut-Down Point Say C = 1K. Does it pay to hire? If C gets too high, Sπ sinks below zero with certain types of technology. Watanabe Econ B PMP 37 / 57 Shut-Down Point 2. TR(x C VC, TR, Sπ ($ VC(x C 1.5 Sπ(x C SD VC(x C : w C 1. SD Sπ(x : w C C Watanabe Econ B PMP 38 / 57 Shut-Down Point Definition 3.3 (Shut-Down Point Shut-down point is a point ( SD C earns zero profit in the short run. a, SD C where Jack a Long-run profit is already below zero at this point because rent for kitchen is due. Watanabe Econ B PMP 39 / 57

14 Shut-Down Point 2. MC(x C SD MR(x C : w C 1.5 SD Mπ(x : w C C MC, MR, Mπ ($ Watanabe Econ B PMP 4 / 57 Shut-Down Point SD( C is 1 if C > SD C 2 any C between and SD C if C = SD C 3 according to marginal condition C = pmp C ( C from SD C onwards. Proposition 2.2 Watanabe Econ B PMP 41 / 57 Shut-Down Point What if Jack insist on hiring according to the marginal condition Proposition 2.2 even when C hits above SD C? 1 He is still maximizing his short-run profit. 5 2 He is actually minimizing his loss (=negative profit. 3 There is a better alternative than doing ➋. 5 as long as MP C( C is decreasing. C.f., Fact 2.6. Watanabe Econ B PMP 42 / 57

15 1 Introduction 2 Short-Run Profit Maximization Problem 3 Short-Run Labor Demand 4 Comparative Statics 5 Now We Know Watanabe Econ B PMP 43 / 57 When and how does SD( C shift? Consider when 1 p 2 K Watanabe Econ B PMP 44 / 57 Question 4.1 (Short-Run Profit Maximization Problem Jack s production function is ƒ ( = 2 C K. Marginal K product is MP C ( C, K = 1 and K = 5. Currently, C p = 2 and K = 1. 1 If K = 1 4, how does SD( C respond? 2 If p = 2 4, how does SD( C respond? Watanabe Econ B PMP 45 / 57

16 15 TC, TR, π ($ 1 TR(x C, x K =1 TC(x, w =12, x =1 C C K 5 π(x C, w C =12, x K =1 TR(x, x =4 C K TC(x, w =12, x =4 C C K π(x, w =12, x =4 C C K 25/ /9 16 Watanabe Econ B PMP 46 / 57 MC, MR, Mπ ($ MR(x, x =1 C K MC(x, w =12, x =1 C C K Mπ(x, w =12, x =1 C C K MR(x, x =4 C K MC(x, w =12, x =4 C C K Mπ(x, w =12, x =4 C C K 25/ /9 16 Watanabe Econ B PMP 47 / MC, MR, Mπ ($ 1 5 MR(x C, x K =1 MC(x, x =1 C K Mπ(x C, x K =1 MR(x, x =4 C K MC(x, x =4 C K Mπ(x, x =4 C K 25/ /9 16 Watanabe Econ B PMP 48 / 57

17 15 TC, TR, π ($ 1 TR(x C, p=2 TC(x, w =12, p=2 C C 5 π(x C, w C =12, p=2 TR(x, p=4 C TC(x, w =12, p=4 C C π(x, w =12, p=4 C C 25/ /9 16 Watanabe Econ B PMP 49 / 57 MC, MR, Mπ ($ MR(x, p=2 C MC(x, p=2 C Mπ(x, p=2 C MR(x, p=4 C MC(x, p=4 C Mπ(x, p=4 C 25/ /9 16 Watanabe Econ B PMP 5 / MC, MR, Mπ ($ 1 5 MR(x C, x K =1 MC(x, x =1 C K Mπ(x C, x K =1 MR(x, x =4 C K MC(x, x =4 C K Mπ(x, x =4 C K 25/ /9 16 Watanabe Econ B PMP 51 / 57

18 Exercise 4.2 (Short-Run Profit Maximization Problem Jack s production function is ƒ ( = K log( C + 1. Marginal product is MP C ( C, K = K C + 1 and K =.1. Currently, p = 2 and K = 1 or 1 Sketch SD( C : K = 1 and SD( C : K = 2. 2 If K = 1 2, how does SD( C respond? Watanabe Econ B PMP 52 / 57 Cheesecakes y (slices f(x, x =1 C K f(x C, x K =2 f(x, x =3 C K f(x C, x K = Watanabe Econ B PMP 53 / 57 pmp, MC(=w ($ C C pmp (x, x =1 C C K pmp C (x C, x K =2 pmp (x, x =3 C C K pmp C (x C, x K = Watanabe Econ B PMP 54 / 57

19 1 Introduction 2 Short-Run Profit Maximization Problem 3 Short-Run Labor Demand 4 Comparative Statics 5 Now We Know Watanabe Econ B PMP 55 / 57 Profit maximization problem Marginal condition Short-run labor demand function Watanabe Econ B PMP 56 / 57 Index competitive firm, 6 ƒ (, see production function factor price, 6 marginal condition, 11, 21, 41 marginal cost, 13 marginal product, 21, 45, 52 marginal profit, 13 marginal revenue, 13 MC C, see marginal cost MP C, see marginal product output price, 6 p, see output price price taker, 6 production function, 17, 21, 45, 52 SD, see short-run labor demand function short-run labor demand function, 3, 31, 36, 52 shut-down point, 39 sunk cost, 8 total cost, 17 total revenue, 17, see factor price SD, see shut-down point C SD, see shut-down point C Watanabe Econ B PMP 57 / 57