Perloff (2014, 3e, GE), Section

Transcription

1 3. Part 3C. Profit Maximization & Supply Short-Run Supply & Competitive Equilibrium 短期供給與均衡 Short-Run Output Decision Short-Run Shutdown Decision Short-Run Firm Supply Curve Short-Run Market Supply Curve Short-Run Competitive Equilibrium Short-Run PMP: Analytical Results Perloff (214, 3e, GE), Section

2 Short-Run Output Decision SR PMP choose q to maximize the economic profits, Max π( q) R( q) C( q) pq C( q) { q } F.O.C. d ( q ) dc ( q ) p pmc( q) p MC( q) dq dq Because a competitive firm s MR equals the market price, a profit-maximizing competitive firm produces the amount of output at which its marginal cost equals the market price. 2

3 S.O.C. 2 d d '( q) dmc( q ) dmc ( q ) dq 2 dq dq qq* qq* dq The S.O.C. holds as long as the lope of MC is positive. 3

4 Solution * q q S ( p) the output supply p p S ( q) the inverse supply MC ( q ) Profit: π pq C( q ) [ pac( q )] q * * * * * 4

5 Comparative Statics F.O.C.: p MC( q) Totally differentiating the F.O.C. w.r.t. p and q gives dmc( q) 1 dp dq dq dq dp dmc ( q )/ dq The slope of output supply is positive. 5

6 Figure: How a Competitive Firm Maximizes Profit Perloff (214, 3e, GE), Figure 8.3, p

7 Solved Problem 8.1: The Effect of a Unit Tax If a specific tax of is collected from only one competitive firm, how should that firm change its output level to maximize its profit, and how does its maximum profit change? Perloff (214, 3e, GE), Solved Problem 8.1, pp

8 Answer: Step 1: Find the firm s profit-maximizing output before the tax is imposed. The Before-Tax PMP Max π( q) pq C( q) { q } FOC F.O.C. d ( q ) dc ( q ) p dq dq p MC( q) q * q 1 8

9 Step 2: Find the firm s profit-maximizing output after the tax is imposed. The After-Tax PMP Max π( q) pq C( q) q { q } F.O.C. d ( q) dc( q) p dq dq p MC ( q ) q * q q 2 9

10 Step p3 3: Calculate the effect of tax on output. d ( q) dc( q) F.O.C. p dq dq Differentiating the F.O.C. w.r.t.,, we obtain 2 dcq ( ) dq 1 2 dq d dq 1 d dmc/ dq q q 2 1 1

11 Step 4: Calculate the effect of tax on profit. PMP: Max π( q) pq C( q) q { q} By the Envelope Theorem, * d * q q d 2 Tax will reduce the profit. Alternative Way: Show that π( q ) π( q ) 2 1 π( q ) pq C ( q ) q pq C ( q ) ( q ) ( q ) π( q ) profit at q 2 before the tax ( q ) π( q ) Because the profit is maximized at q 1 before the tax. 11

12 Figure: The Effect of a Specific Tax, MC + MC AC + AC p AC(q 2 ) + AC(q 1 ) A e 2 B e 1 p = MR q 2 q 1 q, Units per year Perloff (214, 3e, GE), Solved Problem 8.1, p

13 Short-Run Shutdown Decision Shutdown Condition: pq < VC(q) ) The firm can gain by shutting down only if its revenue is less than its short-run variable cost. VC( q) p AVC q A competitive firm shuts down if the market price is less than the minimum of its short-run run average variable cost curve. 13

14 = pq VC(q) F pq VC(q) = + F pq VC(q) < + F < The firm can gain by shutting down only if the sum of its profit and the fixed cost is negative. or < F The firm can gain by shutting down only if its loss is greater than its fixed cost. 14

15 Producer Surplus * q S P. S. pq p ( q ) dq * pq * q * MC ( q ) dq pq VC ( q ) * * * F Quasi Economic Rent ( 準經濟租 ) As long as P.S. (i.e., + F ), firms will not shut down. 15

16 Figure: The Short-Run Shutdown Decision Production: p 62 62, min AVC Shutdown : sd F 1, Perloff (214, 3e, GE), Figure 8.4, p

17 Short-Run Firm Supply Curve Tracing Out the Short-Run Supply Curve If the price falls below the firm s minimum average variable cost (p < SAVC min ), the firm shuts down. The competitive firm s short-run supply curve is its marginal cost curve above its minimum average variable ibl cost. 17

18 Figure: How the Profit-Maximizing gq Quantity Varies with Price Perloff (214, 3e, GE), Figure 8.5, p

19 Factor Prices and the Short-Run Firm Supply Curve An increase in factor prices causes the production costs of a firm to rise, shifting the firm s supply curve to the left. q eg e.g., w 19

20 Figure: Effect of an Increase in the Cost of Materials on the Vegetable Oil Supply Curve Perloff (214, 3e, GE), Figure 8.6, p

21 Solved Problem 8.2: SR Supply Curve Given that a competitive firm s short-run cost function is C(q) = 1q 4q 2 +.2q , what is the firm s short-run supply curve? If the price is p = 115, how much output does the firm supply? Perloff (211, 2e), Solved Problem 8.2, pp

22 Answer: Step1: Determine the firm s output levels for which MC >minavc. dc( q) 2 MC( q) 1 8q.6q dq VC( q) AVC ( q ) 1 4 q.2 q q The MC cuts AVC at its minimum: MC(q) = AVC (q) 1 8q.6q 1 4q.2q 2 4q.4q 2 2 q 1 2 davc( q) or 4.4q dq The SR supply curve is the MC curve for q 1. 22

23 Step 2: Determine the quantity where p = MC = 115. p MC q q q 2 ( ) q 8q 15 2 ( 8) ( 8) 4.6 ( 15) q q (115)

24 Figure: Short-Run Cost Curves Perloff (214, 3e, GE), Figure 7.1 (b), p

25 Example: SR Supply Curve: Cobb SR Cost Curve C(w, v, q, K ) = vk + wq 1/a K -b/a Cobb-Douglas Case where K is the level of capital held constant in the short run. SR Marginal Cost Curve 1a C w a SMC ( w, v, q, K ) q K q a Profit Maximization: p = SMC b a w p q K b 1a b a a Nicholson and Snyder (212, 11e), Example 11.3, p

26 SR Supply Curve 1 a a b w 1a 1 q K p a Shutdown Decision: p = SMC < SAVC SVC a a SVC q 1 b 1a b a a a a wq K SAVC wq K 1 a b C C w a a SMC( w, v, q, K) q K q a Since SAVC < SMC for all values of a < 1. There e is no price low enough that the firm will want to shut down. 26

27 Figure: The Short-Run Supply Curve: Cobb-Douglas Case SMC p e SAVC AC p = MR AC(q * ) q * q, Units per year 27

28 Short-Run Market Supply Curve The Market Supply Curve the horizontal sum of the supply curves of all the individual firms in the market. n S S S( p) Q ( p) q ( p) i1 1 where n # of firms in the market i 28

29 Notes: The number of firms, n, is fixed in the SR. Because new firms need time to enter the market. Cost structures of firms may be identical (perfect info. about tech.) or different from each other (e.g., different accessibility of materials). 29

30 SR Market Supply with Identical Firms The market supply curve flattens as the number of firms in the market increases. Because the market supply curve is the horizontal sum of more and more upwardsloping firm supply curves. The more identical firms producing at a given price, the flatter (more elastic) the SR market supply curve at tthat tprice. S n i S i1 Q ( p) q ( p) nq ( p) 3

31 Figure: Short-Run Market Supply with 5 Identical Firms Perloff (214, 3e, GE), Figure 8.7, p

32 SR Market Supply with Firms That Differ Stairlike Market Supply Curve Where firms differ, only the low-cost firm supplies goods at relatively low prices. As the price rises, the other, higher-cost firm starts supplying, creating a stairlike market supply curve. The more suppliers there are with differing costs, the more steps there are in the market supply curve. Differences in cost structures are one explanation for why some market supply curves are upward sloping. As price rises and more firms are supplying goods, the market supply curve flattens. 32

33 Figure: Short-Run Market Supply with Two Different Lime Firms Perloff (214, 3e, GE), Figure 8.8, p

34 Short-Run Competitive Equilibrium SR Market EQM By combining the SR market supply curve and the market demand curve, we can determine the SR competitive EQM. S D S( p) Q ( p ) Q ( p ) D ( p ) 34

35 Figure: Short-Run Competitive Equilibrium Perloff (211, 2e), Figure 8.9, p

36 Solved Problem 8.3: The Effect of a Unit Tax What is the effect on the short-run equilibrium of a specific tax of τ per unit that is collected from all n firms in a market? What is the incidence of the tax? Answer: Step 1: Show how the tax shifts the MC & AC curves and hence the individual firm s supply curve. MC, AVC & AC shift up by. The SR supply curve of a firm also shifts up by. Step 2: Show how the market supply curve shift. The SR supply curve is the sum of all the individual firm s supply curves, so it too shifts up by. Perloff (214, 3e, GE), Solved Problem 8.3, p

37 Step p3 3: Determine how the SR market EQM change. The Pretax SR Market EQM: E 1 EQM Price = p 1 EQM Quantity = Q 1 = nq 1 The After-Tax SR Market EQM: E 2 EQM Price = p 2 EQM Quantity = Q 2 = nq 2 Comparison of E 1 & E 2 : p 2 > p 1 q 2 < q 1 Q 2 < Q 1 37

38 Step p4 4: Discuss the incidence of the tax. p 2 < p 1 + The EQM price increases, but less than the full amount of the tax. The incidence of the tax is shared between consumers & producers. 38

39 Figure: The Effect of a Unit Tax on the Short-Run Competitive Equilibrium Perloff (214, 3e, GE), Solved Problem 8.3, p

40 Short-Run PMP: Analytical Results SR PMP To maximize profit given prices of output and inputs, p, w, v, and the amount of capital, K. Max pq( wlvk) { qlk,, } s.t. q f ( LK, ) and K K Step 1: Simplifying i the Problem by Substitution Max pf ( L, K ) ( wl vk ) { L} 4

41 Step 2: F.O.C. (for an Interior Minimum) f ( L ): L p L w w p MP VMP L L Value of the Marginal Productivity MP L w p p w MP L 41

42 Implications of F.O.C. w pmp VMP L L SR profit is maximized where the wage rate equals its value of marginal productivity. w MPL p SR profit is maximized where the marginal productivity of labor equals the real wage rates. w dl dc p w MP dq dq p SMC L SR profit is maximized where the market price equals the marginal cost. 42

43 Step p3 3: S.O.C. f MP L L L L 2 2 p p 2 2 (It holds if PF is DMP L, i.e. the marginal productivity of labor is diminishing.) 43

44 Step p4 4: Solution to the PMP Choice Functions * L L p w v K SR Demand for Labor D (,,, ) Output Supply Function * * q f L K q S p w v K (, ) (,,, ) Maximum Value Functions pq ( wl vk ) ( pwvk,,, ) * * * (Indirect) Profit Function 44

45 Step p5 5: Comparative Statics SR Labor Demand Function L L D ( p, w, v, K ) + SR Output Supply Function q q S p w v K + (,,, ) + Profit Functions ( pwvk,,, ) + 45

46 Comparative Statics of L F.O.C.: pf ( L, K ) w L Totally differentiating the F.O.C. w.r.t. L and p, w, v, K, respectively. dl fl pflldl fldp dp pf LL pf dl dw LL pf dl LL pf dl pf dk LL LK dl 1 dw pf LL dl dv dl f LK > if f LK > dk f LL 46

47 Comparative Statics of q Output Supply: q p L L P [ (,,, ), ] * q f L p w v K K Differentiating the q w.r.t. p, w, v, K, respectively. f L q w f L L w q L f L v v q K L fl fk K 47

48 Comparative Statics of Profit Function: * * * By the Envelope Theorem, * q p L w K v KK * v pq ( wl vk ) 48

49 Step 6: Short-Run Competitive Equilibrium SR EQM Conditions: p = SMC dsmc/dq > p SAVC (or + F ) Given the market price, p * SAVC, the SR market EQM is: SR EQM Labor: SR EQM Quantity: * * L L p w v K D (,,, ) * * q q S p w v K (,,, ) Firm s Profit: * * (,,, K) ( p w v K ) F 49