Question 1 Consider a couple, Betty and George, i = 1, 2 respectively. Each partner has private preferences over own con

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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 Betty’s and George’s
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, Betty’s total utility is:
U1 =ρu1 +(1−ρ)u2 while, similarly, George’s total
utility is:
U2 =ρu2 +(1−ρ)u1
The parameter ρ indicates the strength of the
altruistic preferences and is contained 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. In contrast, the upper
limit, ρ = 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
1
contribution g∗, made by each partner, corresponds to a
Nash equilibrium. To do this it is helpful to write each partner’s
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 Betty’s (player 1)
choice of g1? Solve this equation for g1 as a
function of g2 (this gives you Betty’s “reaction
function”). [5 marks]
c) Since the problem is symmetric George’s 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 ρ. [10 marks]
d) How does the symmetric equilibrium
contributions, g∗, to the public good G depend
on ρ? Is it increasing or decreasing in ρ? 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 ρ = 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 =μ[log(R−g1)+log(g1 +g2)]+(1−μ)[log(R−g2)+log(g1 +g2)]
for some value of μ.
What are the first order conditions the Pareto efficient levels
of for g1 and g2? What value does the
weight μ 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 ρ = 1/2 . What is the
intuition for this result? (Hint: Think about the externality that
occurs when ρ > 1/2). [10 marks]