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mdtrmsol - EEP101/ECON125 Spring 99 Prof D Zilberman TA's...

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EEP101/ECON125 Spring 99 Prof D. Zilberman TA's: Malick/Marceau Suggested Solutions Midterm Part A: Numerical Questions 1- MC(Q) = 20 + 3Q ; MB(Q) =120 – 2Q ; MSB(Q) = 120 – Q. a) Socially optimal level of output (Q*): MSB(Q) = MC (Q) => 120 – Q = 20 + 3Q => Q* = 25 b) Competitive equilibrium output (Qc) : MB(Q) = MC(Q) => 120 – 2Q = 20 + 3Q => Qc = 20 Competitive equilibrium price (Pc): MC(Qc) = > 20 +3(20) => Pc = 80 Deadweight loss (DWL): [ ( ) ( )] MSB Q MC Q dQ - 20 25 = [ ] 100 4 20 25 - Q dQ = 100 2 2 20 25 Q Q - =50 c) Consumer surplus (CS) : [ ( ) ] MB Q Pc dQ - 0 20 = [( ) )] 120 2 80 0 20 - - Q dQ = 40 2 0 20 Q Q - = 400 Producer surplus (PS) : [ ( )] Pc MC Q dQ - 0 20 = [ ( )] 80 20 3 0 20 - + Q dQ = 60 3 2 2 0 20 Q Q - = 600 d) Optimal unit subsidy ( s*): MC(Q*) - s* = MB(Q*) => s* = MC (Q*) - MB(Q*) => s* = [ 20 + 3(25) ]- [ 120 - 2 (25) ] = 25 e) Marginal external benefit: MEB(Q) =[ MSB(Q) - MB(Q) ] = (120 - Q) - (120 - 2Q) = Q => MEB(Q) = Q Change in the total external benefits: MEB Q dQ ( ) 20 25 = QdQ 20 25 = 1 2 2 20 25 Q = 112.5 Q* Qc 120 120 60 20 20 25 80 a b c d 95 MSB MB MC Q $ DWL => abc = (20)(5)(1/2) = 50 Optimal Subsidy ( s* ) => bd = 95-70 = 25 Change in total external benefit: => abdc = (1/2)(20+25)(5) = 112.5 100 70 0 MC - s* Pc Pc* Pp* s*
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EEP101/ECON125 Spring 99 Prof D. Zilberman TA's: Malick/Marceau 2- Aggregate demand of the "low income" group => AD L = 2 P L = 2 (20-q ) = 40 - 2 q. Aggregate demand of the "high income" group => AD h = 3 P h = 3 (40 -q) = 120- 3q. Aggregate demand for the public good => AD = q q if q 20 if q < 20 120 3 160 5 - - Marginal cost => MC = 5q a) Efficient output level ( q* ): AD = MC => 160 - 5q = 5q => q* = 16 b) Total cost of providing q* ( TC ): MC(q)dq 0 16 = (5q)dq 0 16 = 5 2 2 q 0 16 = 640 c) Minimum uniform fee ( F min ): The minimum uniform fee that just covers costs is Total Cost # of people = 640 5 = 128 Consumer surplus of a typical "low income" person: The consumer surplus ( CS ) of any individual is equal to the maximum total willingness to pay (WTP) for an amount of a good, minus the uniform fee ( F min ) that the person has to be pay for it. Thus, for a typical person in the "low income" group, the maximum willingness to pay for q*=16 is WTP L = P (q) dq L 0 16 = ( 20 - q )dq 0 16 = 20 0 16 q q 1 2 2 - = 192 Similarly, for a typical person in the "high income" group, the maximum willingness to pay for q*=16 is WTP h = P (q) dq h 0 16 = ( 40 - q )dq 0 16 = 40 0 16 q q 1 2 2 - = 512 Hence, the consumer surplus for a typical person in the "low income" group is CS L = WTP L - F min = 192 - 128 = 64 and the consumer surplus of a typical person in the "high income" group is CS H = WTP h - F min = 512 - 128 = 384 d) Maximum uniform fee w/o exclusion ( F ne ): The maximum uniform fee a concessionaire could charge without excluding anybody is equal to the willingness to pay of a typical person in the "low income" group. F ne = WTP L = 192 Concessionaire's profits under the uniform fee F ne : π ne = 5(192) - 640 = 320 Individual Demand Income Group # of individuals in income group P h = 40 – q High 3 P L = 20 – q Low 2
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EEP101/ECON125 Spring 99 Prof D. Zilberman TA's: Malick/Marceau e) Profit maximizing uniform fee ( F* ): Does the uniform fee of F ne = 192, obtaining π ne = 320 , maximize profits? Let's consider the case in which the concessionaire charges a uniform fee equal to the willingness to pay of a typical person in the "high income group". Thus, the uniform fee is F e = WTPh = 512.
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