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HW 5 Key

# HW 5 Key - Problem Set 5 Solution Key 1 Suppose in some...

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Unformatted text preview: Problem Set 5 Solution Key 1- Suppose in some market that a typical ﬁrm’s short-run total cost curve is given by STC(Q) = 5Q2 + 10. There are 100 identical ﬁrms in the market and all ﬁxed costs are sunk. in addition, market demand is given by D(P) = 100 — 10P. a. (2 points) Derive the equation for the typical ﬁrm’s short-run supply curve. First, we need to ﬁnd the shut-down price : Total avoidable cost=5 Q2 => Average avoidable cost=5 Q. p“! = min (5 Q) = 0. Hence, ﬁrms supply for all price levels. Second, we derive the supply schedule: P 2 MC (Q) gives us the optimum quantity when the market price is P. Let’s solve it for Q to find how many units of output a firm will supply when the price is P. P : 10Q => Q : 150. Let us represent a ﬁrm’s supply function by s (P). Hence, 5 (P) = 1% for all P. b. (2 points) What is the short—run equilibrium market price? First, we need to derive the industry supply function. Let us represent a ﬁrm’s supply function by S (P). Since there are 100 ﬁrms, total quantity supplied is 10P. Hence, 5 (P) = 10P for all P. Now, we can ﬁnd the equilibrium price: S(P) = 101’ = 100 -10P — D (P) -> 20P — 100 -> P — 5. c. (1 point) At the price determined in pan b), how many units will the typical ﬁrm produce? 5 (5) = 5 d. (5 points) Suppose that the government will subsidize this market and that total subsidy expenditure will be \$375. Derive the per unit subsidy rate. Let us represent per unit subsidy by x. The supply curve after subsidy is S’ (P) = 10 (P + x). Hence, the equilibrium price after subsidy is: 10(P+x) = 100 — 101) => 20/3 = 100 — 10x _> P _ 101122;") => P _ (‘ng). The equilibrium output level after subsidyis Q: ioo—io x 907—” => Q:50+5x So, total subsidy expenditure is x (50 + 5x). So, we solve x (50 + 5x) = 375 for x: x = 5. e. (5 points) Calculate the effect ofthe government’s subsidy policy on social welfare. Before Subsidy: Consumer Surplus: (IO—3L“) = 125 Producer Surplus: 5x750 = 125 Total social welfare: 250 After Subsidy: Consumer Surplus: W : 281.25 Producer Surplus: m — W + 375 = 281‘ 25 7 Subsidy expenditure: 375 Total social welfare: 28l.25 x 2 — 375 = 187. 5 Hence, the deadweight loss due to subsidy is 250 — 187.5 : 62. 5. 2- Let a monopolist face the inverse demand of P(Q) : 120 — lOQ and the marginal cost of MC(Q) = 20Q, where Q is output. a. (2 points) Calculate the proﬁt maximizing output and price. TR (Q) = Q (120 —10Q)=> MR (Q) = 120 — 20Q. At the optimum quantity, MR (Q) = MC (Q) 120—20Q:20Q=>40Q=120=: Q: % :3 The market price is P = 120 — 10 x 3 = 90. b (5 points) Suppose now that an excise tax oft = \$40 per unit is imposed on the monopolist. Calculate the new proﬁt maximizing output and price. After tax, the marginal cost curve becomes MC’ (Q) = 20Q + 40. At the equilibrium, MR (Q) = MC' (Q) I20—20Q=20Q+40=>40Q:80=> Q22. The market price is P 2120 — 10 x 2 = 100 c. (5 points) Calculate the effect ofthe tax policy on social welfare. Before tax: Consumer Surplus: W = 45 Producer Surplus: (90 — 60) x 3 + 602” : 180 Total social welfare: 45 + 180 = 225 After tax: Consumer Surplus: w = 20 Producer Surplus: (60 — 40) x 2 + 402“ = 80 Tax Revenue: 40 x 2 = 80 Total social welfare: 20 + 80 + 80 = 180 Hence, the deadweight loss due to tax is 225 — 180 = 45 d. (3 points) What is the incidence of the tax on consumers and the monopolist if the incidence ofthe tax is deﬁned as the price rise experienced by consumers divided by the amount of the tax t? The price rise experienced by consumers : 100 — 90 = 10 So, the incidence oftax is %:%. The burden of the tax is mainly on the monopolist. ...
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