Economics 58 Solutions to PS # 8

Economics 58 Fall 2009 Solutions to PS # 8 1. C = q2 + wq MC = 2q + w a. C = q2 + 10q w = 10 MC = 2q + 10 = P q = 0.5P − 5 1000 Industry Q = ∑ q = 500P − 5000 1 at 20, Q = 5000; at 21, Q = 5500 b. Here, MC = 2q + .002Q for profit maximum, set = P q = 0.5P − 0.001Q 1000 Total Q = ∑ q = 500P − Q Q = 250P 1 P = 20, Q = 5000 where Supply is more steeply sloped in this case Expanded output bids up wages. P = 21, Q = 5250 2. a. The long-run equilibrium price is 10 + r = 10 + .002Q. So, Q = 1050 -50(10 + .002Q) = 550 - .1Q so Q = 500, P = 11, r = 1. b. Now Q = 1600 -50(10 + .002Q) = 1100 - .1Q Q = 1000, P = 12, r = 2. c. Change in PS = 1(500) + .5(1)(500) = 750. d. Change in rents = 1(500) + .5(1)(500) = 750. The areas are equal. e. With tax PD = PS + 5.5 PS = 10 + .002Q PD = 15.5 + .002Q Q = 1050 -50(15.5 + .002Q) = 275- .1Q 1.1Q = 275 Q = 250 PD = 16 r = 0.5 Total tax = 5.5(250) = 1,375 Demanders pay 250(16 - 11) = 1,250 Producers pay 250(11 -10.5) = 125 f. CS originally = .5(500)(21 -11) = 2,500 CS now = .5(250)(21 -16) = 625 PS originally = .5(500)(11 -10) = 250 PS now = .5(250)(10.5 -10) = 62.5 g. Loss of rents = .5(250) + .5(250)(.5) = 187.5 This is the total loss of PS in part b. Occurs because the only reason for upward sloping supply is upward slope of film royalties supply. 3. The Ramsey formula for optimal taxation a. Use the deadweight loss formula from Problem 12.9: n n ⎛ ⎞ L = ∑ DW (ti ) + λ ⎜ T − ∑ ti pi xi ⎟ i =1 i =1 ⎝ ⎠ ∂L / ∂ti = .5[eD eS /(eS − eD )]2 ti pi xi − λpi xi = 0 n ∂L / ∂λ = T − ∑ ti pi xi = 0 i =1 Thus ti = −λ ( eS − eD ) / eS eD = λ (1 / eS − 1 / eD ) b. The above formula suggests that higher taxes should be applied to goods with more inelastic supply and demand. A tax on a good discourages the consumption and production of that good. Thus, taxing a good with more inelastic supply and demand would result in less change in the consumption of the good. Therefore, the tax would produce smaller distortions: the DWL would be smaller. c. 4. This result was obtained under a set of very restrictive assumptions. First, it was obtained under partial equilibrium (the welfare analysis is undertaken in each market separately), ignoring the general equilibrium interactions between markets. Also, income effects and cross-price elasticities are not taken into account. a. Long-run equilibrium requires P = AC = MC. k k AC = + a + bq = MC = a + 2bq Hence q = P = a + 2 kb q b b. Want supply = demand nq = n Hence, n = k = A − BP = A − B( a + 2 kb ) b A − B( a + kb ) . k b c. Apparently A has a definitely positive effect on n. That makes sense since A reflects the “size” of the market. If a > 0, the effect of B on n is clearly negative. d. If appears that fixed costs (k) have a negative effect on n. The effect of a is also negative (assuming it is a positive constant). The effect of b also seems to be negative. These results make sense. The first shows how greater fixed costs increase the typical firm’s optimal size. The second and third show how higher marginal costs raise price and therefore reduce the number of firms.