EC3346 Family EconomicsAssignment Problem SetAutumn 2020Dan AnderbergThe assignment has two questions, each worth 50 marks in total. Completeboth questions.Question 1Consider a couple, Betty and George,i= 1,2 respectively. Each partner hasprivate preferences over own consumption,ci, and a household public goodGwhich are given by:ui= logci+ logGThe total level of the public good,G, is simply the sum of their individual“contributions”, that is,G=g1+g2, whereg1andg2are Betty’s and George’scontributions, respectively. Each partner has a budget ofRand both consump-tion and the public good have prices equal to one. Hence each face an individualbudget constraint of:ci+gi≤RHowever, Betty and George also like each other. The altruistic feelings thatthey have for each other imply that thetotalutility of each partner is a weightedaverage of the own private utility and the private utility of the partner. Hence,Betty’s total utility is:U1=ρu1+ (1−ρ)u2while, similarly, George’s total utility is:U2=ρu2+ (1−ρ)u1The parameterρindicates the strength of the altruistic preferences and iscontained somewhere in the interval 1/2≤ρ≤1.[Note that the lower limit,ρ= 1/2, would imply each care as much for the other as for themselves. Incontrast, the upper limit,ρ= 1, corresponds to “egoistic” preferences.]Despite the altruistic feelings for each other, they act noncooperatively, andtheir choices of contributions to the public good are determined as a Nashequilibrium. As their preferences have the same form and they have the samebudget, the Nash equilibrium will naturally be symmetric.We would like to solve for the symmetric Nash equilibrium public good con-tributions with general altruistic preferences, i.e. we want to find what common1
contributiong∗, made by each partner, corresponds to a Nash equilibrium. Todo this it is helpful to write each partner’s total utility function in such a formthatgiis the only choice variables. This can be done by substituting forciandforG.a)Make the above substitution and write down the total utilityU1for Betty(player 1) as a function of her choiceg1and the contribution chosen by George,g2.[5 marks]b)What is the first order condition for Betty’s (player 1) choice ofg1? Solve thisequation forg1as a function ofg2(this gives you Betty’s “reaction function”).[5 marks]c)Since the problem is symmetric George’s reaction function will take a similarform. Solve for the symmetric Nash equilibrium public good contributiong∗with general altruistic preferences as a function of the altruism parameterρ.[10 marks]d)How does the symmetric equilibrium contributions,g∗, to the public goodGdepend onρ? Is it increasing or decreasing inρ? How would you interpretthis?[10 marks]e)We want to argue that the Nash equilibrium is Pareto efficient if and only ifthe partners are completely altruistic in the sense thatρ= 1/2. To do this weneed to remember that when considering the set of Pareto efficient allocations,we can consider allocations that maximize a weighted average of the privatepreferences (since any allocation that is Pareto efficient under the altruisticpreferences will also be Pareto efficient under the private preferences). AnyPareto efficient allocation is therefore the solution to maximizing the followingobjective functionW=μ[log(R−g1) + log(g1+g2)] + (1−μ)[log(R−g2) + log(g1+g2)]for some value ofμ.What are the first order conditions the Pareto efficient levels of forg1andg2?What value does the weightμhave to take for the Pareto efficient allocation tobe symmetric?[10 marks]f)Lastly, to complete our proof, show that the Nash equilibrium contributionsg∗equal the symmetric Pareto efficient contributions whenρ= 1/2 . What isthe intuition for this result? (Hint: Think about the externality that occurswhenρ >1/2).[10 marks]2
Question 2Consider a couple consisting of spouseaand a spouseb. Spouseafaces anuncertain income. With probabilitypa “loss-state” occurs and she has zeroincomeya= 0. But with probability 1−pthe loss-state does not occur and shehas incomeya= 2. In contrast spousebhas a certain income ofyb= 1. Bothspouses obtain utility from consumption with utility-of-consumption functionu(c) =√c.a)Write down the expected utility for each spouse in the absence of any risk-sharing arrangement.[5 marks]Given that spouseafaces an uncertain income, there is scope for a Paretoimprovement via risk-sharing. In such an arrangement, there will be transfersoccurring between them. Specifically, in the loss-statebwill make a transferτb>0 toa. However, in returnawill have to make a transferτa>0 tobif the loss-state does not occur. Hence let{τa,τb}describe the risk-sharingarrangement.b)Write down the expected utility for each spouse under the risk-sharing ar-rangement{τa,τb}.[5 marks]c)Suppose that spouseacan suggest an arrangement{τa,τb}. In doing so, shehas to ensure that spousebis not made worse off compared to the case withno arrangement (since otherwise spousebwould reject). Show that the transferarrangement thatawould suggest would have the following form for loss-statetransferτb,τb= 1−1(p+ (1−p)√3)2whereas she would in return offer to transferτa= 2−3τbin the no-loss state.[20 marks]d)Determine the risk-sharing arrangement (i) in the limiting case wherep→1(that is, as the loss-state becomes effectively a certainty) and (ii) in the limitingcase wherep→0 (that is, where the loss-state becomes vanishingly unlikely).