HW1,Sp2000sols

HW1,Sp2000sols - EEP101/ECON125 Spring 00 Prof.: D....

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1 EEP101/ECON125 Spring 00 Prof.: D. Zilberman GSIs: Malick/McGregor/St-Pierre Key to PROBLEM SET 1 1. The graph below illustrates the case where a monopolist supplies the market. (Note that the graph is not exactly to scale, so coordinates should be calculated as the intersection of curves, or by substituting values in the relevant functions. Ex: at point “g” MB = MSC; at point “k”, MC = 40 +2*100 = 240) 40 50 300 200 100 100 200 300 400 k MB MSC g MC MC (with subsidy) d b h 90 220 240 310 l e c j i 275 MR m a f 0 a) Social Optimum is attained at a point (g on the graph) where the marginal social cost (sum of private and external marginal costs) is just equal to the marginal social benefit (simply the inverse demand in this case). MSC = MC + MEC = (40 + 2Q) + (10 + 0.5Q) = 50 + 2.5Q Social Equilibrium => MSC = MB => Point g => 50 + 2.5Q = 400 – Q => => Q* = 100 Total External Cost , TEC, at Q * => Area ekgl => 100 (10 + 60)(1/2) = 3,500 => TEC* = 3,500 CS* (varies, but assuming quota) => area agh => (1/2)(400-300)(100) = 5,000 => CS* = 5,000 PS* (varies, but assuming quota) => area ekgh => (1/2)(260+60)(100) = 16,000 => PS* = 16,000 Total Welfare, W* = CS* + PS* - TEC* = 5,000 + 16,000 – 3,500 = 17,500
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2 but you should also see that W* => area lga => (1/2)(350)(100) = 17,500 => W* = 17,500 A common mistake was “PS* = 12,500” (perhaps representing the area ghl). Recall that producer surplus is the difference between price received and marginal cost , over the actual quantity sold . It is often a triangle, but not always. b) The monopolist produces at a level where the private marginal cost equals private marginal revenue: Total Revenues TR = P(Q)Q = (400 - Q)Q = 400Q - Q 2 . Marginal Revenue MR = ƒ ƒ TR Q = 400 - 2Q Equilibrium Quantity => MR = MC => Point d => 400 - 2Q = 40 + 2Q => Q m = 90. Equilibrium Price => 400 - Q m => 400 - 90 => 310 => P m = 310 c) Remember that the dead-weight loss is the welfare deficit, relative to the socially optimal level. Consumer's Surplus, CS m => Area abc => (400 - 310)(90)(1/2) = 4,050 => CS m = 4,050 Producer's Surplus, PS m => Area bcde => (90)(270 + 90)(1/2) = 16,200 => PS m = 16,200 Total external cost, TEC m => Area elfd => (90) (10+55)(1/2) = 2,925 => TEC m = 2,925 Total Welfare, W m = CS m + PS m - TEC m = 4,050 + 16,200 – 2,925 = 17,325 also see that W m => area acde minus elfd is alcf => (1/2)(350+35)(90) = 17,325 => W m = 17,325 Deadweight Loss => DWL m = W*- W m = 17,500 – 17,325 = 175 also see that DWL m => area agl minus alcf is cgf => (1/2)(35*10) = 175 => DWL m = 175 d) Correction of the externality The government plans to intervene with a tax or a subsidy to shift the private MC curve so that it intersects the MR at the desired optimal level of output Q*. Let's call "x" the amount by which the MC curve should change. The way we have written the equation, a negative value for x implies a subsidy (reducing MC), while a positive value implies a tax (increasing MC). MC(Q*) + x = MR(Q*) (40 + 2 Q*) + x = (400 - 2 Q*) We know from a) that Q* = 100, so x is the only unknown. Therefore 40 + 2(100) + x = 400 - 2 (100) 240 + x = 200 x* = - 40. The government should thus grant the monopolist a unit subsidy of $40 per unit produced. This seems strange--- subsidizing a polluter—but the intuition is simple. Relative to the social optimum, the market power of a monopolist make her produce “not enough” (and charge too high a price), but the unregulated externality makes her produce “too much”, so the two effects work in opposite directions. If the externality is relatively severe, the government should encourage the monopolist to restrict even further its production by charging a tax. But if the
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HW1,Sp2000sols - EEP101/ECON125 Spring 00 Prof.: D....

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