Question 1

Consider a couple, Betty and George, i = 1, 2 respectively. Each partner has private preferences over own consumption, ci, and a household public good G which are given by:

ui = log ci + log G

The total level of the public good, G, is simply the sum of their individual contributions, that is, G = g1 + g2, where g1 and g2 are Bettys and Georges contributions, respectively. Each partner has a budget of R and both consump- tion and the public good have prices equal to one. Hence each face an individual budget constraint of:

ci + gi R

However, Betty and George also like each other. The altruistic feelings that they have for each other imply that the total utility of each partner is a weighted average of the own private utility and the private utility of the partner. Hence, Bettys total utility is:

U1 =u1 +(1)u2 while, similarly, Georges total utility is:

U2 =u2 +(1)u1

The parameter indicates the strength of the altruistic preferences and is contained somewhere in the interval 1/2 1. [Note that the lower limit, = 1/2, would imply each care as much for the other as for themselves. In contrast, the upper limit, = 1, corresponds to egoistic preferences.]

Despite the altruistic feelings for each other, they act noncooperatively, and their choices of contributions to the public good are determined as a Nash equilibrium. As their preferences have the same form and they have the same budget, 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 common

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contribution g, made by each partner, corresponds to a Nash equilibrium. To do this it is helpful to write each partners total utility function in such a form that gi is the only choice variables. This can be done by substituting for ci and for G.

a) Make the above substitution and write down the total utility U1 for Betty (player 1) as a function of her choice g1 and the contribution chosen by George, g2. [5 marks]

b) What is the first order condition for Bettys (player 1) choice of g1? Solve this equation for g1 as a function of g2 (this gives you Bettys reaction function). [5 marks]

c) Since the problem is symmetric Georges reaction function will take a similar form. Solve for the symmetric Nash equilibrium public good contribution g with general altruistic preferences as a function of the altruism parameter . [10 marks]

d) How does the symmetric equilibrium contributions, g, to the public good G depend on ? Is it increasing or decreasing in ? How would you interpret this? [10 marks]

e) We want to argue that the Nash equilibrium is Pareto efficient if and only if the partners are completely altruistic in the sense that = 1/2. To do this we need to remember that when considering the set of Pareto efficient allocations, we can consider allocations that maximize a weighted average of the private preferences (since any allocation that is Pareto efficient under the altruistic preferences will also be Pareto efficient under the private preferences). Any Pareto efficient allocation is therefore the solution to maximizing the following objective function

W =[log(Rg1)+log(g1 +g2)]+(1)[log(Rg2)+log(g1 +g2)]

for some value of .

What are the first order conditions the Pareto efficient levels of for g1 and g2? What value does the weight have to take for the Pareto efficient allocation to be symmetric? [10 marks]

f) Lastly, to complete our proof, show that the Nash equilibrium contributions g equal the symmetric Pareto efficient contributions when = 1/2 . What is the intuition for this result? (Hint: Think about the externality that occurs when > 1/2). [10 marks]