# Spring 2016Public EconomicsECON 4821PUBLIC ECONONOMICSECON 4821 – Spring 2016Department of Economics

Spring 2016Public EconomicsECON 4821PUBLIC ECONONOMICSECON 4821 – Spring 2016Department of EconomicsUniversity of MinnesotaProblem Set 2Due Date: Tuesday February 23th, in thebeginning class.Homework must be typed and a hard copy should be handed in. There will be a20% penalty if you do not type it.1. Consider an island with Tom Hanks and Wilson, and one good – coconuts. There is NOendowment of coconuts, and to have something to eat Tom Hanks and Wilson have to work- climb the palm trees and gather coconuts. In one hour, Tom Hanks can gather wT H andWilson wW coconuts. Both Tom Hanks and Wilson have T hours at disposal. Supposethat the utility functions for person i with i ∈ {T H, W isUi (ci , i ) = B i log ci + logwhere ci is consumption of coconuts,iiis the time spent lying on beach and surﬁng (leisure),and B T H and B W are positive constants.(a) Write down the problem of person i ∈ {T H, W, if Tom Hanks and Wilson are ontheir own (in autarky), so if no trade is possible.(b) Write the constraint for person i from part (a) in the form whereiis on the sameside as ci , keeping T on the other side of the constraint. Interpret this constraint:what is the “price” that person i has to “pay” for one hour of leisure?(c) Find the expressions for optimal choices ci∗ andi∗. How do they depend on wi andB i , explain the intuition behind this.University of Minnesota1Problem Set 2Spring 2016Public EconomicsECON 4821(d) Suppose that Tom Hanks and Wilson could trade if they wanted. Would they trade?Hint: Think about what the person who would like to buy a coconut could oﬀerin return for the coconut bought. Given your answer what can we say about thecompetitive equilibrium allocation (allocation with trade), how is it diﬀerent from theallocation cT H∗ ,T H∗, cW ∗ ,W∗from part (c)?(e) Suppose now that Tom Hanks and Wilson decide to pool the coconuts they gather.Write down the aggregate resource constraint for coconuts on this island – an equationthat shows how total consumption of coconuts by Tom Hanks and Wilson cT H + cWis limited by the amount of coconuts available that depends onTH,W, wT H , wW , T .(f) Suppose that the social welfare function of the social planner isW (U W , U T H ) = φW U W + φT H U T Hwhere U W = UW (cW ,W) and U T H = UT H (cT H ,TH). Write down the social plan-nerâ€™s problem, as a problem of maximizing this social welfare function subject to theresource constraint.(g) Write down the Lagrangean for the social plannerâ€™s problem, take ﬁrst order conditions.(h) Solve the ﬁrst order conditions to obtain the optimal allocation (cT H∗ ,SP PHint: First solve for cW , cT H ,W,THT H∗ W ∗W∗SP P , cSP P , SP P ).from their FOCs, then plug these into the FOCfor λ, after that solve the resulting equation for λ∗ and ﬁnally plug this λ∗ back toobtain (cT H∗ ,SP PT H∗ W ∗W∗SP P , cSP P , SP P ).(i) Suppose that society cares about the well being of Tom Hanks and Wilson equally sothat φT H = φW = 1, that Tom Hanks and Wilson have same preferences and timeendowment and thus B T H = B W = 1, T = 15. Finally, since it is much easier for TomHanks to climb the palm trees wT H = 6, wW = 2. Calculate cT H∗ ,from part (c). Calculate cT H∗ ,SP PT H∗ W ∗W∗SP P , cSP P , SP PT H∗, cW ∗ ,W∗from part (h). Compare them; howdoes the redistribution that the social planner would undertake look like?University of Minnesota2Problem Set 2Spring 2016Public EconomicsECON 48212. Firms A and B each currently produce 100 units of pollution. The federal governmentwants to reduce pollution levels. The marginal costs associated with pollution reductionare M CA = 100 + 2QA and M CB = 40 + 6QB for ﬁrms A and B, respectively, whereQA and QB denote the amount of reduction. Societyâ€™s marginal beneﬁt of reduction isMB = 640 − 4QT where QT is the total level of pollution reduction (QA + QB = QT ).Answer the following: Answer the following:(a) What is the socially optimal level of each ﬁrmâ€™s production?(b) How much total pollution is there in the social optimum?(c) Explain why it is ineﬃcient to give each ﬁrm an equal number of pollution permits.(d) Explain that the social optimum can be achieved if ﬁrms are given equal permits butare allowed to trade them.3. Can an activity generate both positive and negative externalities? If you believe the answeris yes, provide an example. If you believe the answer is no, give an explanation.University of Minnesota3Problem Set 2