1. UNIT II:Firms & Markets Theory of the Firm Profit Maximization Perfect Competition/Review 7/15 MIDTERM 7/1

2. Profit Maximization Last Time The Short-Run and the Long-Run Firm and Market Supply Perfect Competition (Part 1)

3. We saw last time that we can solve the firm’s cost minimization problem analogously to the consumer’s utility maximization problem. Cost minimization requires that the firm produce using a combination of inputs for which the ratios of the marginal products, or the marginal rate of technical substitution, equals the ratio of the input prices: MRTS = w/r Last Time 2 Provisos: Only in the Long-Run Only part of the firm’s problem

4. Profit Maximization The firm wants to maximize this difference: Profit (  = Total Revenue(TR) – Total Cost(TC) TR(Q) = PQ TC(Q) = rK + wL PPrice LLabor QQuantity KCapital w Wage Rate rRate on Capital Q = f(K,L) Revenue Cost

5. Last time: we considered a firm that produces output according to the following production function. Q = 4K ½ L ½ and w = $18 and r = $36. How much will it cost this firm to produce 10 units of output in the long-run? Q units? Profit Maximization

6. Last time: we considered a firm that produces output according to the following production function. Q = 4K ½ L ½ and w = $18 and r = $36. How much will it cost this firm to produce 10 units of output in the long-run? Q units? The long-run total cost curve (TC(Q)) represents the minimum cost to produce Q units of output. Profit Maximization

7. How much will it cost this firm to produce 10 units of output in the long-run? Q = 4K 1/2 L 1/2 w = 18; r = 36 MRTS = MP L /MP K MP L = 2K 1/2 L -1/2 MP K = 2K -1/2 L 1/2 MRTS = K/L. = w/r = 18/36 = L = 2K. Cost Minimization in the Long-Run The firm’s optimal factor proportion (given technology and factor prices).

8. Cost Minimization in the Long-Run L* L K K* Q = 10 How much will it cost this firm to produce 10 units of output in the long-run? L = 2K Tangency between the isoquant and an isocost curve shows the economically efficient combination K*, L*. Hence, the condition for optimal factor proportion is: MRTS = w/r

9. Cost Minimization in the Long-Run L* L K K* Q = 10 How much will it cost this firm to produce 10 units of output in the long-run? L = 2K The condition for optimal factor proportion is: MRTS = w/r. This is LR condition! Why? Because some factors (K) are fixed in the SR.

10. Cost Minimization in the Long-Run L* L K K* Q = 10 How much will it cost this firm to produce 10 units of output in the long-run? L = 2K Another way to think about this: TC is a projection of the firm’s long-run output expansion path: the locus of optimal factor bundles (K,L) for different levels of Q.

11. Cost Minimization in the Long-Run K K* Q = 10 How much will it cost this firm to produce 10 units of output in the long-run? TC= 18L + 36K = 9Q/(2) 1/2 + 9Q/(2) 1/2 = 18/(2) 1/2 (Q) = 12.73Q MC = 12.73 = AC We can solve for the firm’s long run total cost function for any level of output.

12. Cost Minimization Graphically: Q = 4K 1/2 L 1/2 w = 18; r = 36 Q $ In the SR, K = 16 TC sr = 576 + 18(Q/16) 2 Fixed Costs = rK = $576 TC lr = 12.73Q MC = 12.73 At this point, 16 units of capital is optimal.

13. Profit Maximization Profit (  = Total Revenue(TR) – Total Cost(TC) Q1 Q2Q3 Q $ Consider a firm with long-run total costs TC. TC

14. Profit Maximization Profit (  = Total Revenue(TR) – Total Cost(TC) Q1 Q2Q3 Q $ To maximize profits, the firms finds Q where distance between TC and TR is greatest. the same slope. TC TR = PQ

15. Profit Maximization Profit (  = Total Revenue(TR) – Total Cost(TC) Q1 Q2Q3 Q $ To maximize profits, the firms finds Q where distance between TC and TR is greatest. same slope. TC TR = PQ

16. Profit Maximization Profit (  = Total Revenue(TR) – Total Cost(TC) Q1 Q2Q3 Q $ To maximize profits, the firms finds Q where distance between TC and TR is greatest. same slope. TC TR = PQ

17. Profit Maximization Profit (  = Total Revenue(TR) – Total Cost(TC) $ To maximize profits, the firm finds Q where distance between TC and TR is greatest. This will be where they have the same slope.. slope. TC TR = PQ  max Q1 Q2Q3 Q* Q

18. Profit Maximization Profit (  = Total Revenue(TR) – Total Cost(TC) $ Marginal Analysis: If TC is rising faster than TR, reduce Q. If TR is rising faster than TC, increase Q. TC TR = PQ  max Q1 Q2Q3 Q* Q

19. Profit Maximization Profit (  = Total Revenue(TR) – Total Cost(TC) $ Marginal Analysis: Recall: slope TR = MR slope TC = MC Hence, to maximize profits: MR = MC TC TR = PQ  max Q1 Q2Q3 Q* Q

20. Profit Maximization The firm wants to maximize this difference: Profit (  = Total Revenue(TR) – Total Cost(TC) TR(Q) = PQ TC(Q) = rK + wL PPrice LLabor QQuantity KCapital w Wage Rate rRate on Capital Q = f(K,L)

21. Profit Maximization The firm wants to maximize this difference: Profit (  = Total Revenue(TR) – Total Cost(TC) TR(Q) = PQ TC(Q) = rK + wL PPrice LLabor QQuantity KCapital w Wage Rate rRate on Capital Q = f(K,L) Revenue Cost

22. Profit Maximization Demand for the firm’s output is given by Q = 100 – 2P. Find the firm’s profit maximizing level of output. Q = 4K 1/2 L 1/2 w = 18; r = 36 Q = 100 – 2P => P = 50 – 1/2Q TR = PQ = (50 – 1/2Q)Q = 50Q – 1/2 Q 2 MR = 50 – Q = MC = 12.73 => Q* = 37.27; P* = 31.37

23. Profit Maximization Demand for the firm’s output is given by Q = 100 – 2P. Find the firm’s profit maximizing level of output. Q = 4K 1/2 L 1/2 w = 18; r = 36 Q = 100 – 2P => P = 50 – 1/2Q TR = PQ = (50 – 1/2Q)Q  = TR – TC = 50Q – 1/2 Q 2 – 12.73Q FOC:d  /dQ = 50 – Q – 12.73 = 0 => Q* = 37.27; P* = 31.37

24. Profit Maximization in the Long-Run We solved the firm’s optimization problem focusing on the profit output level, Q*, but it is important to emphasize that the optimization principle also tell us about input choices. When the firm chooses an output level Q* that maximizes  for given factor prices (w, r), the firm has simultaneously solved for L* and K*. To produce Q* = 37.27 (given the production function, Q = 4K 1/2 L 1/2, and optimal factor proportion, L = 2K), we find: L* = 6.58; K* = 3.29. Finally,  = TR–TC = PQ–12.73Q = (31.37–12.73)37.27 = $694.71.

25. Profit Maximization in the Short-Run We saw that the firm maximizes profit by choosing a level of output such that marginal revenue equals marginal cost MR = MC. In the long-run, this implies that the firm will be utilizing its optimal factor proportion, such that MRTS = w/r. In the short-run, however, K is fixed, so the firm will not be able to optimally adjust factor proportions. Does the firm maximize profits by setting MR = MC in the short-run? How?

26. Profit Maximization in the Short-Run We solved the firm’s optimization problem focusing on the output level, but it is important to emphasize that the optimization principle also tell us about input choices. When the firm chooses an output level (Q*) that maximizes profit for given factor prices (w, r), the firm has simultaneously solved for L* and K*. In the short-run, the firm also maximizes  by setting MR = MC. But because K is fixed, the problem is simply one of optimal utilization of labor. In other words, the firm asks, “On the margin, can I increase my profit by adding (or subtracting) another unit of labor from the production process?”

27. Profit Maximization in the Short-Run Q = 4K 1/2 L 1/2 w = 18; r = 36 K = 16P = 10 MR = MC TR = PQ = 10QTC = rK + wL = 36(16) + 18L [ Q = 16L ½ => L = (Q/16) 2 ] TC = 36(16) + 18(Q/16) 2 = 576 + 18(Q/16) 2 MR = 10 = MC = (36/256)Q => Q = 71.1; L = 19.8 Revenue Cost

28. Profit Maximization in the Short-Run In the short-run, the firm’s profit maximizing calculus weighs the benefit of hiring an additional unit of labor versus its cost. If the firm can add (subtract) another unit of labor and increase revenue by more than it increases cost, it should add (subtract) it, and it should keep on adding (subtracting) until MR = MC.

29. Profit Maximization in the Short-Run What does an additional unit of L contribute to  ? But hiring an additional unit of L also increases the firm’s total costs. MFC L = w Marginal Factor Cost (MFC) The dollar amount added to total cost from an additional unit of labor (capital). NOTE (for fixed K): MC = dTC/dQ = w (dL/dQ) MC = w (1/MP l )

30. Profit Maximization in the Short-Run So the firm’s short-run optimality condition can be rewritten as: MRP L = MFC L If the firm can add (subtract) another unit of labor and increase revenue by more than it increases cost, it should add (subtract) it, and it should keep on adding (subtracting) until MRP L = MFC L. This, in turn, is what determines the firm’s (short-run) demand for labor: P (MP L ) = w

31. Profit Maximization in the Short-Run Consider a price-taking firm that is currently producing 450 units of output at a price of $2.50 per unit. In the short run, the firm’s capital stock is fixed at 16 machine- hours. The firm is currently employing 100 hours of labor at a wage of $10/hour, and the at a rental rate (r) is $10/hour. With this mix of inputs MP K =2 and MP L = 4. Is the firm maximizing profits in the short run?

32. Profit Maximization in the Short-Run Is the firm maximizing profits in the short run? Yes. MRP L = P(MP L ) = 2.50(4) = 10 = w = MFC L MR = P = $2.50; MC = w/MP L = $10/4 = $2.50 Does it matter that MP L /MP K = w/r (4/2 = 10/10) ? Does it matter that  = TR-TC = 1125 – 1160 = -35 ? Shut down rule: If P < AVC, Q = 0. The firm would lose its fixed costs ($160) if it shut down, or $35 if it produces Q = 450. What matters is variable costs, because fixed costs are spent (“sunk”) in the short run. Profit max = Loss min.

33. $$$$  = TR – TC = 0 PRICE TAKER P = MR TC MC Q AC Supply in the Short-Run P = MR TR Q* Q At price P o, the firm earns 0 profit.

34. Q Q $$$$  = TR – TC > 0 PRICE TAKER P = MR TC MC Q AC Supply in the Short-Run If the price rises … Output rises … TR P = MR

35. Q Q $$$$  = TR – TC > 0 PRICE TAKER P = MR TC MC Q AC Supply in the Short-Run If the price rises … And the firm earns profit TR P = MR

36. $$$$  = TR – TC < 0 PRICE TAKER P = MR TC MC Q AC Supply in the Short-Run P = MR TR If the price decreases… Output falls Q Q

37. $$$$  = TR – TC < 0 PRICE TAKER P = MR TC MC Q AC Supply in the Short-Run P = MR TR If the price decreases… And the firm has losses Q Q

38. Q $$$$  = TR – TC < 0 PRICE TAKER P = MR TC MC Q AC Supply in the Short-Run P = MR TR The firm will continue to produce (< Q*) iff P > AVC … Losses are less than fixed costs. AVC

39. Q $$$$  = TR – TC < 0 PRICE TAKER P = MR TC MC Q AC Supply in the Short-Run TR The firm’s short-run supply curve is in orange Where MR = MC and P > AVC. AVC

40. Supply in the Long-Run In the short run, firms adjust to price signals by varying their utilization of labor (variable factors). In the long-run, firms adjust to profit signals by – varying plant size (fixed factors); and – entering or exiting the market. What determines the number of firms in the long-run?

41. Short-run equilibrium with three firms. Firm A is making positive profits, Firm B is making zero profits, and Firm C is making negative profits (losses). Firm A Firm B Firm C q q q $P$P AC MC AC q: firm Q: market Supply in the Long-Run

42. Short-run equilibrium with three firms. Firm A is making positive profits, Firm B is making zero profits, and Firm C is making negative profits (losses). Firm A Firm B Firm C q q q $P$P MC AC AVC AC

43. Supply in the Long-Run Short-run equilibrium with three firms. Firm A is making positive profits, Firm B is making zero profits, and Firm C is making negative profits (losses). Firm A Firm B Firm C q q q $P$P In the long run, Firm C will exit the market. MC AC MC AC

44. Supply in the Long-Run In the long-run, inefficient firms will exit, and new firms will enter, as long as some firms are making positive economic profits. Firm A Firm B Firm D q q q $P$P MC AC MC AC MC AC

45. Supply in the Long-Run In the long-run, if there are no barriers to entry, then new firms have access to the most efficient production technology. We call this the efficient scale. Firm A Firm D Firm E q* q q* q q* q $ P* MC AC MC AC

46. Supply in the Long-Run The long-run supply curve. P* is the lowest possible price associated with non-negative profits, P* = AC min. $ P* Q s 1 =(mc 1 ) s2s2 s3s3 s4s4 S n, where n is the number of firms in the market.

47. Supply in the Long-Run The long-run supply curve. We can eliminate portions of the individual firms’ supply curves below P* (firms will exit). $ P* Q s1s1 s2s2 s3s3 s4s4

48. Supply in the Long-Run The long-run supply curve. We can also eliminate portions of the individual firms’ supply curves above P* (demand will be met by another firm’s supply curve). $ P* Q s3s3 s4s4 s1s1 s2s2

49. Supply in the Long-Run The long-run supply curve. As the number of firms increases, the long run supply curve approximates a straight line at P* = AC min. $ AC min = P* Q LRS

50. Supply in the Long-Run Long-run equilibrium. Firms are producing at the efficient scale. P* = AC min ;  = 0. $ P* q* q Q* Q $ LRS MC AC D

51. Perfect Competition Equilibrium and Efficiency General Equilibrium Welfare Analysis

52. Perfect Competition Profit (  = Total Revenue(TR) – Total Cost(TC) $ To maximize profits, the firm finds Q where distance between TC and TR is greatest. This will be where they have the same slope: MR = MC. TC TR = PQ  max Q1 Q2Q3 Q* Q Price-Taker P is given; can sell any amount at P. Later we will look at other possible assumptions.

53. Perfect Competition Profit (  = Total Revenue(TR) – Total Cost(TC) $ To maximize profits, the firm finds Q where distance between TC and TR is greatest. This will be where they have the same slope: MR = MC. TC TR = PQ  max Q1 Q2Q3 Q* Q Price-Taker P is given; can sell any amount at P. Later we will look at other possible assumptions, e.g., monopoly.

54. Perfect Competition Assumptions Firms are price-takers: can sell all the output they want at P*; can sell nothing at any price > P*. Homogenous product: e.g., wheat, t-shirts, long- distance phone minutes. Perfect factor mobility: in the long run, factors can move costlessly to where they are most productive (highest w, r). Perfect information: firms know everything about costs, consumer demand, other profitable opportunities, etc.

55. Perfect Competition Implications 1)Firms earn zero (economic) profits. 2)Price is equal to marginal cost. 3)Firms produce at minimum average cost, i.e., “efficient scale.” 4)Market outcome is Pareto-efficient. Now we want to consider the welfare implications

56. Perfect Competition The Short-Run & the Long-Run In the short-run, firms adjust to price signals by varying their utilization of labor (variable factors). In the long-run, firms adjust to profit signals by – varying plant size (fixed factors); and – entering or exiting the market. We can use this info to solve for the long-run competitive equilibrium. 

57. Perfect Competition Consider a perfectly competitive industry characterized by the following total cost and demand functions: TC = 100 + q 2 Q D = 1000 – 20P Find the market equilibrium in the long-run. How many firms are in the market?

58. Perfect Competition TC = 100 + q 2 Q D = 1000 – 20P $ q* = 10 q 600 Q $ LRS MC = 2q AVC = q AC = 100/q + q  1)  = TR – TC = 0 = Pq – (100 + q 2 ) = 0  = (2q)q – (100 + q 2 ) = 0 2q 2 – 100 - q 2 = 0 q 2 = 100; q* = 10

59. Perfect Competition TC = 100 + q 2 Q D = 1000 – 20P $ q* = 10 q 600 Q $ LRS MC = 2q AVC = q AC = 100/q + q  1)  = TR – TC = 0 = Pq – (100 + q 2 ) = 0  = (2q)q – (100 + q 2 ) = 0 2q 2 – 100 - q 2 = 0 q 2 = 100; q* = 10

60. Perfect Competition TC = 100 + q 2 Q D = 1000 – 20P $ q* = 10 q 600 Q $ LRS MC = 2q AVC = q AC = 100/q + q  1)  = TR – TC = 0 = Pq – (100 + q 2 ) = 0  = (2q)q – (100 + q 2 ) = 0 2q 2 – 100 - q 2 = 0 q 2 = 100; q* = 10 2) P = MC = 2q

61. Perfect Competition TC = 100 + q 2 Q D = 1000 – 20P q* = 10 q 600 Q $ LRS MC = 2q AVC = q AC = 100/q + q  1)  = TR – TC = 0 = Pq – (100 + q 2 ) = 0  = (2q)q – (100 + q 2 ) = 0 2q 2 – 100 - q 2 = 0 q 2 = 100; q* = 10 $ P* = 20

62. Perfect Competition TC = 100 + q 2 Q D = 1000 – 20P q* = 10 q 600 Q $ LRS MC = 2q AVC = q AC = 100/q + q  3)  Firms produce at AC min i) AC = MC 100/q + q = 2q 100 + q 2 = 2q 2 q 2 = 100; q* = 10 ii) AC’ = 0 -100/q 2 + 1 = 0 q 2 = 100; q* = 10 $ P* = 20 q* is the efficient scale

63. Perfect Competition TC = 100 + q 2 Q D = 1000 – 20P $ P* = 20 q* = 10 q Q* = 600 Q $ LRS MC = 2q AVC = q AC = 100/q + q D n = 60

64. Perfect Competition TC = 100 + q 2 Q D = 1000 – 20P $ P* = 20 q* = 10 q Q* = 600 Q $ LRS MC = 2q AVC = q AC = 100/q + q D n = 60

65. Perfect Competition TC = 100 + q 2 Q D = 1500 – 20P $ P* = 20 q* = 10 q Q* = 600 Q $ LRS MC = 2q AVC = q AC = 100/q + q D n = 60 D’

66. Perfect Competition TC = 100 + q 2 Q D ’ = 1500 – 20P q. n = (MC/2)n = (P/2)n => Q S = ??? SRS $ P* = 20 q’ = 15 q Q’ = 900 Q $ LRS MC = 2q AVC = q AC = 100/q + q n = 60 SRS D D’ P’= MR

67. Perfect Competition TC = 100 + q 2 Q D ’ = 1500 – 20P q. n = (MC/2)n = (P/2)n => Q S = 30P SRS $ P* = 20 q’ = 15 q Q’ = 900 Q $ LRS MC = 2q AVC = q AC = 100/q + q n = 60 SRS D D’ P’ = 30

68. Perfect Competition TC = 100 + q 2 Q D ’ = 1500 – 20P $ P* = 20 q* = 10 q Q** = 1100 Q $ LRS MC = 2q AVC = q AC = 100/q + q n = 110 D D’

69. Perfect Competition The Short-Run & the Long-Run In the short-run, firms adjust to price signals by varying their utilization of labor (variable factors). In the long-run, firms adjust to profit signals by – varying plant size (fixed factors); and – entering or exiting the market. What determines the number of firms in long run equilibrium?

70. Perfect Competition In the Long-Run … 1)Firms earn zero (economic) profits. 2)Price is equal to marginal cost. 3)Firms produce at minimum average cost, i.e., “efficient scale.” 4)Market outcome is Pareto-efficient.

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