Chapter 19: Profit Maximization Problem

Econ 23 Microeconomic Analysis Chapter 19: Profit Maximization Problem Instructor: Hiroki Watanabe Fall 2012 Watanabe Econ 23 19 PMP 1 / 90 1 Introduction 2 Short-Run Profit Maximization Problem 3 Comparative Statics 4 Long-Run Profit Maximization Problem 5 Factor Demand 6 Returns to Scale & Profit Mazimization Problem 7 Summary Watanabe Econ 23 19 PMP 2 / 90 1 Introduction Question Firm s Objective Definitions Price-Taker Assumption 2 Short-Run Profit Maximization Problem 3 Comparative Statics 4 Long-Run Profit Maximization Problem 5 Factor Demand 6 Returns to Scale & Profit Mazimization WatanabeProblem Econ 23 19 PMP 3 / 90 7 Summary

Question Question 1.1 (Agenda for Today) 1 What is Jack s hiring and investment plan? In the short run and long run. How do they differ? 2 How does Jack respond to environmental change? 3 What is his labor demand? 4 What would be the limitation of profit maximization problem? Watanabe Econ 23 19 PMP 4 / 90 Firm s Objective Assumption 1.2 (Firm s Objective) The firm s objective is to maximize their profit. Bake as many as you can? Watanabe Econ 23 19 PMP 5 / 90 Firm s Objective Hiring too many chefs will reduce the productivity eventually. How does Jack find the right production plan? 1 Dumb Jack: trials and errors. 2 Smart Jack: tangency condition Watanabe Econ 23 19 PMP 6 / 90

Definitions = ( C, K ) denotes the factor price (unit price of inputs). = (10, 1) means hourly wage is $10 and rental rate is $1. Definition 1.3 (Total Cost) Total cost associated with the input bundle ( C, K ) is TC( C, K ) =. If Jack hires 20 chefs and purchases 10 stand mixers under = (1, 1), TC( C = 20, K = 5) = 1 20 = 1 5 = 25. Watanabe Econ 23 19 PMP 7 / 90 Definitions All the costs are measured in terms of opportunity cost. Jack s financial capital (interest rate =10%): A unit of kitchen Self-financed Borrowed Out-of-pocket 10K 0K Loan 0K 10K Accounting Cost 10K 11K Opportunity Cost 11K 11K K is not 10K but 11K. Watanabe Econ 23 19 PMP 8 / 90 Definitions Definition 1.4 (Total Revenue) Total revenue from y is TR(y) = py or TR( C, K ) = pƒ ( C, K ). If Jack produces 10 cheesecakes and price is $4, his total revenue is $40. Watanabe Econ 23 19 PMP 9 / 90

Definitions Definition 1.5 (Profit) The economic profit generated by the production plan ( C, K, y) is π( C, K ) = If Jack raised $40 from cheesecakes sales and paid $25 for his employees and kitchen investment, his profit is $15. Watanabe Econ 23 19 PMP 10 / 90 Definitions 1.6 (Profit Structure) Jack produces cheesecakes y according to y = ƒ ( C, K ) = C K. Factor price is = ( C, K ) = (1, 1) and cheesecake sells for $4 apiece. If he hires 20 chefs and rent out 5 stand mixers, his 1 labor cost is 2 capital cost is 3 total cost is 4 sales volume is 5 total revenue is 6 profit is Watanabe Econ 23 19 PMP 11 / 90 Price-Taker Assumption Definition 1.7 (Competitive Market) Jack is in a competitive market if he is a price taker. 1 Competitive labor market: C is given. 2 Competitive capital market: K is given. 3 Competitive cheesecake market: p is given. Watanabe Econ 23 19 PMP 12 / 90

Price-Taker Assumption Jack may influence the equilibrium price in reality. 24 Monopoly 25 Monopoly behavior 26 Oligopoly Watanabe Econ 23 19 PMP 13 / 90 1 Introduction 2 Short-Run Profit Maximization Problem Fixed Cost Short-Run Profit Maximization Problem Solution to Short-Run Profit Maximization Problem Feasible Production Plans Tangency Condition 3 Comparative Statics 4 Long-Run Profit Maximization Problem Watanabe 5 Factor Econ 23 Demand 19 PMP 14 / 90 6 Returns to Scale & Profit Mazimization Problem 7 Summary Fixed Cost Long-run Jack maximizes his profit differently from short-run Jack. 1 Long-run Jack solves long-run profit maximization problem 2 Short-run Jack solves short-run profit maximization problem Watanabe Econ 23 19 PMP 15 / 90

Fixed Cost Recall Definition 5.1 from Chapter 18: Definition 2.1 (Short Run & Long Run) 1 A short run is a circumstance in which a firm is restricted in its choice of at least one input level. 2 A long run is the circumstance in which a firm is unrestricted in its choice of input levels. Watanabe Econ 23 19 PMP 16 / 90 Fixed Cost Definition 2.2 (Fixed Cost) Fixed cost is a cost that Jack has to pay for the fixed input. Jack has to pay the rent ( K K ) even when y = 0 in the short run. Suppose the size of kitchen if predetermined at K = 5. FC = Watanabe Econ 23 19 PMP 17 / 90 Fixed Cost Fixed cost may or may not be a sunk cost (cost cannot be recouped, regardless of future actions) depending on the timing: 1 It is sunk after Jack signed up the lease. 2 Not if Jack hasn t signed up the lease yet. Watanabe Econ 23 19 PMP 18 / 90

Short-Run Profit Maximization Problem In the short run, Jack solves the short-run profit maximization problem (SPMP): Problem 2.3 (Short-Run Profit Maximization Problem (SPMP)) Jack maximizes his short run profit given p, ( C, K ): max C π( C, K ) = pƒ ( C, K ) } {{ } C C K } {{ K } total revenue total cost = pƒ ( C, K ) C C FC. Watanabe Econ 23 19 PMP 19 / 90 Solution to Short-Run Profit Maximization Problem Key: separate what Jack can earn from what Jack can produce for a while and patch them together later. 1 line tells what each production plan earns him. 2 Feasible production plans tell what he can actually produce. Watanabe Econ 23 19 PMP 20 / 90 Solution to Short-Run Profit Maximization Problem Short-run profit maximization problem is rather easy. Dumb or smart, Jack s kitchen equipment is predetermined at K. All he has to do is to choose the right C, i.e., he just needs to gage the relationship between y and C. Everything else is not at his discretion ( K, p, C, K ). Watanabe Econ 23 19 PMP 21 / 90

What is the relationship among cheesecake y, employment C and profit π then? Forget the production function for a while. As an accountant, tell Jack what combination of C and y (production plan) will generate π be it feasible or not. Watanabe Econ 23 19 PMP 22 / 90 Definition 2.4 ( Line) An isoprofit line at π contains all the production plans ( C, K, y) that yield the same profit level of π. Once again, we are asking a hypothetical question here. How much profit can Jack earn if his production plan is ( C, K, y)? ( C, K, y) may not be technologically feasible. Watanabe Econ 23 19 PMP 23 / 90 ( Line) Suppose p = 4, = ( C, K ) = (1, 1), K = 10. Find the isoprofit line at π = 0 and π = 10. Watanabe Econ 23 19 PMP 24 / 90

C y py C C K K π 0 10 0 10 0 10 5 20 10 10 0 20 20 10 0 10 40 10 0 0 5 20 0 10 10 10 10 10 10 20 10 40 20 10 10 50 10 10 Watanabe Econ 23 19 PMP 25 / 90 2 80 20 70 20 60 10 50 0 40 10 10 20 60 Watanabe Econ 23 19 PMP 26 / 90 How do we read this graph? Slice off along fixed y or C. 1 Fix y = 10. The more C is, the smaller the profit will be. 2 Fix C = 10. The more y is, the larger the profit will be. Watanabe Econ 23 19 PMP 27 / 90

Proposition 2.6 (Slope of Line) The slope of isoprofit line is given by C p. Proof. Rearrange π = py C C K K. Watanabe Econ 23 19 PMP 28 / 90 Profit grows as he moves from southeast to northwest. As a profit maximizer, Jack wants to move to the northwesternmost point ( C, y) = (0, a lot)... but can he? Watanabe Econ 23 19 PMP 29 / 90 Feasible Production Plans Recall p.16 from Chapter 18: 2.7 (Feasible Production Plan) Jack s short-run production function is ƒ ( C, K ) = C K = 10 C when K = 10. His feasible production plan ( C, K, y) satisfies y ƒ ( C, K ) = 10 C. Watanabe Econ 23 19 PMP / 90

Feasible Production Plans Prod Fn 2 Watanabe Econ 23 19 PMP 31 / 90 Feasible Production Plans The slope of production function measures marginal product of a chef given K = 10. Watanabe Econ 23 19 PMP 32 / 90 Feasible Production Plans Which production plan should Jack choose? 1 On one hand, the northwestern, the better. 2 On the other, any y > ƒ ( C, K ) is off limit. Watanabe Econ 23 19 PMP 33 / 90

Tangency Condition 2 80 20 70 20 60 10 50 0 40 10 10 20 60 Watanabe Econ 23 19 PMP 34 / 90 Tangency Condition Prod Fn 2 Watanabe Econ 23 19 PMP 35 / 90 Tangency Condition 2 Prod Fn 80 20 70 20 60 10 50 0 40 10 10 20 60 Watanabe Econ 23 19 PMP 36 / 90

Tangency Condition Recall: 1 The slope of production function denotes the marginal product of C. 1 2 The slope of isoprofit line is C p ( Proposition 2.6 ). Jack earns the maximum profit at ( C, K, y) where the production function is just back to back with isoprofit. 1 given K = K. Watanabe Econ 23 19 PMP 37 / 90 Tangency Condition Condition 2.8 (Tangency Condition) At the optimal production plan ( C, K, y), C p = MP C( C, K ). Watanabe Econ 23 19 PMP 38 / 90 Tangency Condition What does tangency condition Condition 2.8 mean? C = pmp C ( C ) = C p = MP C( C, K ) C = pmp C ( C, K ) Watanabe Econ 23 19 PMP 39 / 90

Tangency Condition What if C > pmp C ( C, K )? What if C < pmp C ( C, K )? Watanabe Econ 23 19 PMP 40 / 90 2.9 (Short-Run Profit Maximization Problem) Suppose p = 4, = (1, 1), K = 10 and y = ƒ ( C, K ) = C K. Marginal product of chef is K MP C ( C, K ) = 2. How many chefs should Jack hire? C Watanabe Econ 23 19 PMP 41 / 90 2 Prod Fn 80 20 70 20 60 10 50 0 40 10 10 20 60 Watanabe Econ 23 19 PMP 42 / 90

1 Introduction 2 Short-Run Profit Maximization Problem 3 Comparative Statics Question 4 Long-Run Profit Maximization Problem 5 Factor Demand 6 Returns to Scale & Profit Mazimization Problem Watanabe Econ 23 19 PMP 43 / 90 7 Summary Question Question 3.1 (Environmental Change & Hiring Decision) How should Jack revise his hiring decision when 1 C 2 p? Watanabe Econ 23 19 PMP 44 / 90 3.2 (Environmental Change & Hiring Decision) Suppose p = 4, = (1, 1), K = 10 and y = ƒ ( C, K ) = C K. Marginal product of chef is K MP C ( C, K ) = 2. How many chefs should he hire or C lay off when 1 C = 1 2 2 p = 4 2? Watanabe Econ 23 19 PMP 45 / 90

2 Prod Fn 80 20 70 20 60 10 50 0 40 10 10 20 60 Watanabe Econ 23 19 PMP 46 / 90 2 Prod Fn 70 50 10 10 10 70 90 110 Watanabe Econ 23 19 PMP 47 / 90 2 Prod Fn 80 20 70 20 60 10 50 0 40 10 10 20 60 Watanabe Econ 23 19 PMP 48 / 90

2 20 10 Prod Fn 10 10 20 20 60 Watanabe Econ 23 19 PMP 49 / 90 0 Discussion 3.3 (Environmental Change & Hiring Decision) Why does Jack have to downscale when C or p? Watanabe Econ 23 19 PMP 50 / 90 Tangency condition Condition 2.8 : MP C ( C, K ) = C p. C or p increases the right-hand side. Marginal product is decreasing in C if Jack s technology exhibits diminishing marginal product. 2 Jack has to stop hiring early on when each chef s contribution is still high. 2 Cf. Definition 3.5 in Chapter 18. Watanabe Econ 23 19 PMP 51 / 90

What about other factors? Discussion 3.4 (Change in Fixed Cost) 1 Does K affect the optimal production plan ( C, K, y)? Does he hire more? Does he produce more? 2 Does K affect the optimal production plan ( C, K, y)? 3 Do they affect Jack s decision making process similarly or differently? Watanabe Econ 23 19 PMP 52 / 90 3.5 (Change in Fixed Cost) Suppose p = 4, = (1, 1), K = 10 and K y = ƒ ( C, K ) = C K. MP C ( C, K ) = 2. How does C the optimal production plan ( C, K, y ) change when 1 K = 1 2. 2 K = 10 5. Watanabe Econ 23 19 PMP 53 / 90 2 Prod Fn 80 20 70 20 60 10 50 0 40 10 10 20 60 Watanabe Econ 23 19 PMP 54 / 90

2 Prod Fn 70 20 10 60 10 50 0 40 10 20 20 60 70 Watanabe Econ 23 19 PMP 55 / 90 2 Prod Fn 80 20 70 20 60 10 50 0 40 10 10 20 60 Watanabe Econ 23 19 PMP 56 / 90 2 Prod Fn 85 35 25 75 25 65 15 55 5 45 5 5 15 25 35 45 55 Watanabe Econ 23 19 PMP 57 / 90

1 Introduction 2 Short-Run Profit Maximization Problem 3 Comparative Statics 4 Long-Run Profit Maximization Problem Solution to Long-Run Profit Maximization Problem Tangency Condition & MRTS 5 Factor Demand 6 Returns to Scale & Profit Mazimization Problem Watanabe Econ 23 19 PMP 58 / 90 7 Summary Solution to Long-Run Profit Maximization Problem Recall Definition 5.1 from Chapter 18: Definition 4.1 (Short Run & Long Run) 1 A short run is a circumstance in which a firm is restricted in its choice of at least one input level. 2 A long run is the circumstance in which a firm is unrestricted in its choice of input levels. Now Jack can change K along with C. Watanabe Econ 23 19 PMP 59 / 90 Solution to Long-Run Profit Maximization Problem Problem 4.2 (Long-Run Profit Maximization Problem (LPMP)) Given and p, long-run Jack solves max C, K π( C, K ) =. The same condition Condition 2.8 applies to K : K p = MP K( C, K). Watanabe Econ 23 19 PMP 60 / 90

Solution to Long-Run Profit Maximization Problem 2 Prod Fn 80 20 70 20 60 10 50 0 40 10 10 20 60 Kitchen Equipment x K Watanabe Econ 23 19 PMP 61 / 90 Tangency Condition & MRTS Tangency conditions for long-run profit maximization problem: C p = MP C( C, K ) K p = MP K( C, K ). Then, C = MP C( C, K ) K MP K ( C, K ) C = MRTS( C, K ). K (1) 3 3 Recall Definition 4.1 in Chapter 18. Watanabe Econ 23 19 PMP 62 / 90 Tangency Condition & MRTS What does (1) mean? Wait for Chapter 20: Cost Minimization Problem. Watanabe Econ 23 19 PMP 63 / 90

1 Introduction 2 Short-Run Profit Maximization Problem 3 Comparative Statics 4 Long-Run Profit Maximization Problem 5 Factor Demand 6 Returns to Scale & Profit Mazimization Problem 7 Summary Watanabe Econ 23 19 PMP 64 / 90 Question 5.1 (Jack in the Labor Market) Jack simultaneously appears in three markets: 1 Cheesecake market (as a supplier) 2 Labor market (as a buyer) 3 Capital market (as a buyer) How is Jack s hiring decision reflected in labor demand? Watanabe Econ 23 19 PMP 65 / 90 Definition 5.2 (Factor Demand Function) Factor demand function returns the optimal amount of input for each factor price given other parameters. Watanabe Econ 23 19 PMP 66 / 90

Factor demand is just another way to look at tangency condition Condition 2.8 : C p = MP C( C, K ). Tell Jack ongoing C and he will find C that satisfies Condition 2.8 above. And that is factor demand. We already know the answer ( Question 3.1 ). Watanabe Econ 23 19 PMP 67 / 90 3.2 revisited: 5.3 (Factor Demand) Suppose p = 4, = (1, 1), K = 10 and y = ƒ ( C, K ) = C K. Marginal product of chef is K MP C ( C, K ) = 2. How many chefs should he hire or C lay off when 1 C = 1 2? Watanabe Econ 23 19 PMP 68 / 90 2 Prod Fn 80 20 70 20 60 10 50 0 40 10 10 20 60 Watanabe Econ 23 19 PMP 69 / 90

2 Prod Fn 70 50 10 10 10 70 90 110 Watanabe Econ 23 19 PMP 70 / 90 C 1 2 C Watanabe Econ 23 19 PMP 71 / 90 Similarly, for other C s, Jack determines his labor demand schedule according to Condition 2.8 : C p = MP K( C, K ). In 5.3 in particular, the factor demand is K 2 = C C p C = 40. 2 C Watanabe Econ 23 19 PMP 72 / 90

5 Factor Demand x C =40/w C 2 4 Wage w ($) C 3 2 1 0 Chefs x C Watanabe Econ 23 19 PMP 73 / 90 Question 5.4 (Comparative Statics on Factor Demand) What if p plummeted in half? Watanabe Econ 23 19 PMP 74 / 90 Replace p in Condition 2.8 with p = 2, C p = MP K( C, K ). which leads to K 2 = C C p C = 10. 2 C Watanabe Econ 23 19 PMP 75 / 90

5 4 Factor Demand When p=4: x C =40/w C 2 Factor Demand When p=2: x C =10/w C 2 Wage w ($) C 3 2 1 0 Chefs x C Watanabe Econ 23 19 PMP 76 / 90 1 Introduction 2 Short-Run Profit Maximization Problem 3 Comparative Statics 4 Long-Run Profit Maximization Problem 5 Factor Demand 6 Returns to Scale & Profit Mazimization Problem IRS Will Explode 7 Summary Watanabe Econ 23 19 PMP 77 / 90 IRS Will Explode How are returns to scale related to profit maximizing behavior? Not all the technologies are compatible with profit maximization problem. Watanabe Econ 23 19 PMP 78 / 90

IRS Will Explode Fact 6.1 (Returns to Scale & PMP) If Jack is a price taker, 1 decreasing returns to scale: nice. 2 increasing returns to scale: explosion. 3 constant returns to scale: 1 explosion if π > 0 2 nice if π = 0. Watanabe Econ 23 19 PMP 79 / 90 IRS Will Explode In what follows, consider a short-run profit maximization in which fixed cost is sunk. Watanabe Econ 23 19 PMP 80 / 90 IRS Will Explode 2 Prod Fn 90 40 80 70 20 60 10 50 0 0 10 20 Watanabe Econ 23 19 PMP 81 / 90

IRS Will Explode 2 Prod Fn 90 40 80 70 20 60 10 50 0 0 10 20 Watanabe Econ 23 19 PMP 82 / 90 IRS Will Explode 2 Prod Fn 90 40 80 70 20 60 10 50 0 0 10 20 Watanabe Econ 23 19 PMP 83 / 90 IRS Will Explode 2 Prod Fn 90 40 80 70 20 60 10 50 0 0 10 20 Watanabe Econ 23 19 PMP 84 / 90

IRS Will Explode 2 Prod Fn 90 40 80 70 20 60 10 50 0 0 10 20 Watanabe Econ 23 19 PMP 85 / 90 IRS Will Explode CRS Jack is compatible with profit maximization problem only when C p = MP C( C ) and consequently raises zero profit. Watanabe Econ 23 19 PMP 86 / 90 IRS Will Explode Just because IRS Jack is not compatible with profit maximization problem does not mean he has to be a dumb Jack to find the optimal hiring and investment plan. Call for Chapter 22 Cost Minimization Problem. Watanabe Econ 23 19 PMP 87 / 90

1 Introduction 2 Short-Run Profit Maximization Problem 3 Comparative Statics 4 Long-Run Profit Maximization Problem 5 Factor Demand 6 Returns to Scale & Profit Mazimization Problem 7 Summary Watanabe Econ 23 19 PMP 88 / 90 Solving short-run and long-run profit maximization problem. Tangency condition: vs feasible production plan. Comparative statics on PMP. Factor demand. Returns to scale compatible with profit maximization problem. Watanabe Econ 23 19 PMP 89 / 90 accounting cost, 8 capital market, 65 competitive market, 12 constant returns to scale, 79 decreasing returns to scale, 79 diminishing marginal product, 51 factor demand function, 66 factor price, 7 feasible production plan, 20, financial capital, 8 fixed cost, 17 increasing returns to scale, 79 isoprofit, 20 slope of, 28 isoprofit line, 23 labor market, 65 long run, 16, 59 long-run profit maximization problem, 60 marginal product, 32 marginal rate of technical substitution, 62 opportunity cost, 8 π, see profit price taker, 12, 79 production plan, 22 profit, 5, 10 returns to scale, 78 short run, 16, 59 short-run profit maximization problem, 19 SPMP, see short-run profit maximization problem sunk cost, 18 tangency condition, 6, 38, 67 TC, see total cost total cost, 7 total revenue, 9 TR, see total revenue, see factor price