Department of Agricultural and Resource EconomicsUniversity of CaliforniaEconomics EEP101/ECON125, Spring 2004Professor ZilbermanMidterm Solutions1 Question 1 (30 points)The inverse demand facing a farm sector is denoted by p =20− x where x is output and p is output price. Thesector’s activities also generate positive externalities and the marginal externality benefits is MEB =10− 0.5x.The marginal cost of production is MC =2+x.a: How much output will be produced and at what consumer price if the industry operates as amonopoly in the output market? (4 points)Solution: Monopoly will achieve its optimal quantity when the marginal revenue (MR) equals the marginal cost(MC).p =20− x ⇒ MR =20− 2xMC =2+xMR = MC ⇒ 20 − 2x =2+x ⇒ Xm= 6 and Pm=20− Xm=14b: What will be the welfare loss in this case, relative to the socially optimal outcome? (8 points)At the socially optimality, the marginal social benefit (MSB) equals the marginal social cost (MSC). In thiscase, the marginal social benefit includes the marginal private benefit (MPB) and the marginal externality benefit(MEB).MSB = MPB + MEB =[20− x]+[10− 0.5] = 30 − 1.5xMSC =2+xMSB = MSC ⇒ 30 − 1.5x =2+x ⇒ Xs=11.2 and Ps=20− 11.2=8.8OI20 = ABCE30 = FGHMRMCMPBMSB$XXmPmXsPs1020Figure 1:Welfare Comparison between Monopolistic and Socially Optimal OutcomesAs shown by Figure 1, we can identify the area for the deadweight loss in a comparison between the monopolisticand socially optimal outcomes.1• The net social welfare under the social optimality is given bySWs= total benefits − total costs =∆FHXsO − ∆EHxsO =∆FHE• The net social welfare under monopolistic outcome is given bySWm= consumer surplus + producer surplus + externality benefits=∆ABPm+∆PmBCE +∆FGBA =∆FGCETherefore, the welfare loss relative tot he socially optimal outcome is given bySWm− SWs=∆FGCE − ∆FHE = −∆GHC= −12[MSB(Xm) − MC(Xm)] (Xs− Xm)= −12[(30 − 1.5Xm) − (2 + Xm)] (Xs− Xm)= −12[(30 − 1.5 × 6) − (2 + 6)] (11.2 − 6)= −33.8c: What policies can be used to attain the social optimum? How will the welfare of various groupsaffected by these policies? (10 points)There are several policies can be used to attain the social optimum such as subsidies and quota.• Subsidies: The optimal subsidies are given bySoptimal= MC(Xs) − MR(Xs)=(2+Xs) − (20 − 2Xs)=(2+11.2) − (20 − 2 ∗ 11.2) = 15.6.As shown by Figure 1, Table 1 demonstrates the welfare analysis.Table 1:Welfare analysis for monopoly with and without subsidiesWithout any regulation With subsidies ChangeConsumer surplus ABPm=18 ARPs=62.72 PmBRPs=34.72Producer surplus PmBCE =54 PsNE − NHR + JHST =94.56 50.56Externality benefit AF GB =51 GHRA =80.64 GHRB =29.64government net gain 0 −JHST = −81.12 −81.12social welfare FGCE = 123 156.80 GHC =33.8Therefore, consumers, producers, and the environment all gain from subsidies, while government loses.Notes:1. Producer surplus has three elements including:– PsNE is the surplus of producing units at which the marginal benefit is at least as great as themarginal cost.2– NHR is the loss due to the fact that this monopoly produces additional xs− xmunits where themarginal cost exceeds the marginal benefit if selling at ps.– JHST is the gain from subsidies.2. The net welfare gain from subsidies is exactly represented by the area GHC if you carefully workout the area for producer surplus.OQ20 = ABCE30 = FGHIJKLMNRSTMRMCMPBMSB$XXmPmXsPs1020Figure 2:Welfare Comparison between Monopolistic and Socially Optimal Outcomes• quota: The government sets up production quota to the monopolistic producers such that they shouldproduce the socially optimal level (11.2). If the government has no supplementary policy, producer surplusis represented by PsNE − NHR =3.52 > 0, which implies that this monopolistic firm will still stay inbusiness. Following the same principle, we obtain the following welfare analysis in Table 2. Therefore,consumers and the environment gain from quota, and producers lose. This direct control has no effect ongovernmental welfare.d: How much will be produced under competition and at what price? (4 points)At the competitive outcome without any regulation, the marginal private benefit (MPB) equals the marginal cost(MPC).MPB = MPC ⇒ 20 − x =2+x ⇒ Xc=9and Pc=20− 9=11 (1)3Table 2:Welfare analysis for monopoly with and without quotaWithout any regulation With quota ChangeConsumer surplus ABPm=18 ARPs=62.72 PmBRPs=34.72Producer surplus PmBCE =54 PsNE − NHR =13.44 −50.48Externality benefit AF GB =51 GHRA =80.64 GHRB =29.64government net gain 0 0 0social welfare FGCE = 123 156.80 GHC =33.8e. What policies are needed to move from the competitive outcome to the optimal outcome? Whowill be the gainers and losers? (4 points)Solution: Both subsidies and quota can move the competitive to the socially optimal outcome. If subsidies arechosen, consumers, producers, and the environment are gainers, and government incurs cost of subsidies. If quotais chosen, consumers and the environment gain but producers lose.2 Question 2 (36 points)A community has n members. The marginal benefit of each person from clean air is MBx=10− 0.4x.a: Is it a public good? Why? (4 points)Solution: Clean air is a public good. An individual cannot prevent others from using clean air (non-excludibility),and the use of clean air by one person will not reduce the benefit to others (non-rivalry).b: What is the socially optimal quality for a community of n =5individuals if the marginal costof improving x is MC =10+2x (2 points)? What is the net social welfare per member of thecommunity if they share the cost of clean up equally? (4 points)Solution: To get the aggregate marginal benefit of public goods, we need vertically sum all individual marginalbenefits together.MB5=5MBx=5[10− 0.4x]=50− 2xMC =10+2x⇒ MB5= MC ⇒ 50 − 2x =10+2x ⇒ x5=10 (2)Thus, the net welfare for each member of the community is given by100[10 − 0.4x]dx  individual benefit−100[10 + 2x]dx5  shared cost of an invidual=40 (3)4c: What is the socially optimal quality for n =10and n = ∞? (6 points)Solution:n =10⇒MB10= 10[10 − 0.4x] = 100 − 4xMC =10+2x⇒ MB10= MC ⇒ 100 − 4x =10+2x ⇒ x10= 15n = ∞⇒MB10= n[10 − 0.4x]=10n − 0.4nxMC =10+2xMBn=∞= MC ⇒ 10n − 0.4nx =10+2x ⇒ xn=∞=10n − 100.4n + 2=25.d: Suppose that the community has two groups of citizens. Group 1 of 5