EC3346 Family Economics

EC3346 Family Economics
Assignment Problem Set
Autumn 2020
Dan Anderberg
The assignment has two questions, each worth 50 marks in total. Complete
both questions.
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 consumption 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 contributions 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]
2
Question 2
Consider a couple consisting of spouse a and a spouse b. Spouse a faces an
uncertain income. With probability p a “loss-state” occurs and she has zero
income y
a = 0. But with probability 1 −p the loss-state does not occur and she
has income y
a = 2. In contrast spouse b has a certain income of y
b = 1. Both
spouses obtain utility from consumption with utility-of-consumption function
u (c) = √
c.
a) Write down the expected utility for each spouse in the absence of any risksharing arrangement. [5 marks]
Given that spouse a faces an uncertain income, there is scope for a Pareto
improvement via risk-sharing. In such an arrangement, there will be transfers
occurring between them. Specifically, in the loss-state b will make a transfer
τ b > 0 to a. However, in return a will have to make a transfer τ a > 0 to b
if the loss-state does not occur. Hence let {τ a, τ b} describe the risk-sharing
arrangement.
b) Write down the expected utility for each spouse under the risk-sharing arrangement {τ a, τ b}. [5 marks]
c) Suppose that spouse a can suggest an arrangement {τ a, τ b}. In doing so, she
has to ensure that spouse b is not made worse off compared to the case with
no arrangement (since otherwise spouse b would reject). Show that the transfer
arrangement that a would suggest would have the following form for loss-state
transfer τ b,
τ b = 1 −
1

p + (1 − p)

3
2
whereas she would in return offer to transfer τ a = 2 − 3τ b in the no-loss state.
[20 marks]
d) Determine the risk-sharing arrangement (i) in the limiting case where p → 1
(that is, as the loss-state becomes effectively a certainty) and (ii) in the limiting
case where p → 0 (that is, where the loss-state becomes vanishingly unlikely).
Provide an intuition for each case. [20 marks

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