Advanced Political Economics - Prof. Galasso
Problem Set 1
Solutions
1. Policy decisions in different Voting Models
(a) 1/ Bliss points of agent A over the two public goods q1 and q2 .
The preference of agent A is:
WA = ln(c) + ln(1 + q1 )
We plug in the agent budget constraint c = y(1 − τ ) as well as the
governement budget constraint τ y = q1 + q2 and we obtain:
WA = ln(y − q1 − q2 ) + ln(1 + q1 )
The bliss points over q1 and q2 are the values of q1 and q2 which
maximize the utility of the agent.
• q1 :
∂WA
=0
∂q1
−
1
1
+
=0
y − q1 − q2
1 + q1
y − q2 − 1
q1 =
2
• q2 :
The utility of agent A is decreasing in q2 . The bliss point of agent
A over q2 is therefore the lowest possible value of q2 , which is 0.
Knowing that q2∗ = 0, we get that q1∗ =
y−1
2 .
2/ Bliss points of agent B over the two public goods q1 and q2 .
The preference of agent B is:
WB = ln(c) + aln(1 + q2 )
We plug in the agent budget constraint c = y(1 − τ ) as well as the
governement budget constraint τ y = q1 + q2 and we obtain:
WB = ln(y − q1 − q2 ) + aln(1 + q2 )
The bliss points over q1 and q2 are the values of q1 and q2 which
maximize the utility of the agent.
1
• q1 :
The utility of agent B is decreasing in q1 . The bliss point of
agent B over q1 is therefore the lowest possible value of q1 , which
is 0.
• q2 :
∂WB
=0
∂q2
−
1
a
+
=0
y − q1 − q2
1 + q2
q2 =
a(y − q1 ) − 1
1+a
Knowing that q1∗ = 0, we get that q2∗ =
ay−1
1+a .
3/ Bliss points of agent C over the two public goods q1 and q2 .
The preference of agent C is:
WC = ln(c) + ln(1 + q1 ) + bln(1 + q2 )
We plug in the agent budget constraint c = y(1 − τ ) as well as the
governement budget constraint τ y = q1 + q2 and we obtain:
WC = ln(y − q1 − q2 ) + ln(1 + q1 ) + bln(1 + q2 )
The bliss points over q1 and q2 are the values of q1 and q2 which
maximize the utility of the agent.
• q1 :
∂WC
=0
∂q1
−
1
1
+
=0
y − q1 − q2
1 + q1
q1 =
y − q2 − 1
2
(1)
• q2 :
∂WC
=0
∂q2
−
b
1
+
=0
y − q1 − q2
1 + q2
q2 =
b(y − q1 ) − 1
1+b
Solving the system of equation (1) and (2), we get that q1∗ =
q2∗ = by+b−2
b+2 .
2
(2)
y−b
b+2
and
(b) Each candidate will maximize the probability of winning the election,
which is the same as to maximize the probability that the scandal δ
is small enough (below a certain threshold δ):
max P r(δ < δ) =
1 ΨX i i
α Φ (W i (q11 , q21 ) − W i (q12 , q22 ))]
+ [
2 Φ i
where Ψ is the density of the distribution
of the shock (which
P i function
is equal to 1 in this exercise), Φ =
α Φi , αi is the size of the group
of type-i agents (αi = 31 in this exercise), Φi is the group-specific
density of the distribution function of the ideology, which is the same
for each group in this exercise. Moreover, q1j and q2j , with j = 1, 2,
represents the level of q1 and q2 proposed by the candidate j. Using
the assumptions of the exercise, we can rewrite the maximization
program as:
max P r(δ < δ) =
X
1
+[
(W i (q11 , q21 ) − W i (q12 , q22 ))]
2
i
Given that 21 is fixed and that W i (q12 , q22 ) is taken as given by the
candidate 1, the maximization program for the candidate 1 simplifies
as:
X
max W =
W i (q11 , q21 )
i
max W = ln(c)+ln(1+q1 )+ln(c)+aln(1+q2 )+ln(c)+ln(1+q1 )+bln(1+q2 )
Using the agents budget constraint as well as the government budget
constraint, I can rewrite the maximization program as:
max W = 3ln(y − q1 − q2 ) + 2ln(1 + q1 ) + (a + b)ln(1 + q2 )
• q1 :
∂W
=0
∂q1
−
3
2
+
=0
y − q1 − q2
1 + q1
q1 =
2(y − q2 ) − 3
5
(3)
• q2 :
∂W
=0
∂q2
−
3
a+b
+
=0
y − q1 − q2
1 + q2
q2 =
3
(y − q1 ) − 1
2
(4)
Solving the system of equation (3) and (4), we get that q1∗ =
q2∗ = 3y−2
8 .
y−2
4
and
(c) First, all agents vote on q2 and then, all agents vote on q1 . Let’s
solve this problem by backward induction, starting by the choice of
q1 knowing q2 .
We already computed the reaction function of the 3 agents at point
a):
• q1A = y−q22 −1
• q1B = 0
• q1C = y−q22 −1
Given that agents A and C have the same reaction function, for any
q2 chosen at the first stage, the level of q1 resulting from the vote
will be such that: q1 = y−q22 −1 . This reaction function is anticipated
at the first stage of the game, it is why we should plug it in the
maximization programs of the 3 agents for the choice of q2 :
• Agent A chooses the lowest possible level of q2 , independently of
the level of q1 . Therefore q2A∗ = 0.
• When choosing the optimal level of q2 , agent B takes into account
the fact that q1 will be chosen as the following: q1 = y−q22 −1 .
Therefore, the maximization program is:
max WB = ln(y −
3
y − q2 − 1
− q2 ) + ln(1 + q2 )
2
2
∂WB
=0
∂q2
−y
2
−
1
2
q2
2
+
1
2
+
3 1
=0
2 1 + q2
3y + 1
5
• When choosing the optimal level of q2 , agent C takes into account
the fact that q1 will be chosen as the following: q1 = y−q22 −1 .
Therefore, the maximization program is:
q2B∗ =
max WC = ln(y−
y − q2 − 1 3
y − q2 − 1
−q2 )+ln(1+
)+ ln(1+q2 )
2
2
2
−y
2
∂WC
=0
∂q2
1
3 1
=0
+
2 1 + q2
− q22 + 21
q2C∗ =
4
3y − 1
7
We have that: q2A∗ < q2C∗ < q2B∗ . Therefore, the median voter is
agent C and the result of the vote is: q2∗ = q2C∗ = 3y−1
7 . Plugging
this result into the reaction function of the second stage, we get:
q1∗ = 2y−3
7 .
2. Labor and consumption taxation
Consider an economy where the preferences of individual i are quasi-linear,
namely:
wi = ci + V (xi )
where ci represents his consumption and xi his leisure. Moreover, V (.) is
increasing and concave in xi . The private budget constraint of each agent
is given by:
(1 + qC )ci ≤ (1 − qL )li + f
where qL is the income tax rate, qC the consumption tax rate and f a
fixed subsidy from government. The real wage of each agent is exogenous
and normalized to unity. Furthermore, each agent has a private productivity parameter αi so that agents have different amounts of effective time
available. More precisely, they face the following time constraint:
1 + α i ≥ xi + l i
Assume that αi is drawn from a distribution with mean α and median
αm .
(a) Compute the optimal labor supply for each individual. What are the
effects of an increase in qL (resp. qC ) on the individual labor supply?
Discuss the result.
(b) Write the government budget constraint and derive the level of the
subsidy as a function of q = (qL , qC ). Compute the policy preferences
W (q; αi ) of individual i.
(c) Does the Condorcet winner exist in that case? If yes, who is the
Condorcet winner?
(d) Compute the utilitarian welfare and determine the socially optimal
policy. What is the winning policy q when αi = α for all i? What
happens if agents are heterogeneous?
Solutions
Preferences
w = c + V (x)
Agent’s budget constraint
5
(1 + qC )ci ≤ (1 − qL )li + f,
Time constraint:
1 + α i ≥ xi + l i
(a) Labor supply
max
FOCli :
(1 − qL )li + f
+ V 1 + αi − l i .
(1 + qC )
(1 − qL )
≤ V 0 1 + αi − l i
(1 + qC )
An interior solution yields:
li (q)
=
1 + αi − V 0−1
1 − qL
1 + qC
=
= L (q) + αi − α,
1−qL
, is the average labor supply.
where L (q) = 1 + α − V 0−1 1+q
C
1
Since V is concave, LqL (·) = V1xx 1+q
< 0, and similarly for Lqc .
c
The higher the tax on labor, the lower the incentives to work, same
marginal costs, lower marginal benefits. Analogously for qC . Labor
supply is increasing in the individual productivity parameter, and
has an average form L (q) .
Equilibrium consumption is given by:
(1 − qL )li
f
+
(1 + qC )
(1 + qC )
(1 − qL )
(1 − qL ) i
f
=
L (q) +
α −α +
.
(1 + qC )
(1 + qC )
(1 + qC )
R
(b) The average labor supply is L (q) =
L (q) + αi − α dF αi . Thus,
average consumption is:
ci
=
c=
(1 − qL )
f
L (q) +
(1 + qC )
(1 + qC )
Finally, government budget constraint is
f
(1 − qL )
f = qL L (q) + qc
L (q) +
(1 + qC )
(1 + qC )
(1 + qC ) qL + qC (1 − qL )
f
= L
+
(1 + qC )
(1 + qC )
= L (q) (qc + qL ) .
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Substituting the labor supply, consumption and the subsidy f into
the indirect utility function of agent i, we get his policy preferences:
W q; αi
=
=
=
(1 − qL )
(1 − qL ) i
f
L (q) +
α −α +
+ V (1 + α − L (q))
(1 + qC )
(1 + qC )
(1 + qC )
L (q) (qc + qL )
(1 − qL )
(1 − qL ) i
L (q) +
α −α +
+ V (1 + α − L (q))
(1 + qC )
(1 + qC )
(1 + qC )
(1 − qL ) i
α − α + L (q) + V (1 + α − L (q)) .
(1 + qC )
(1−qL )
(c) Let K αi = αi −α, H (q) = (1+q
and J (q) = L (q)+V (1 + α − L (q)) .
C)
Then policy preferences can be rewritten as
(5)
W q; αi = J (q) + K αi H (q)
with K αi monotonic in αi (Kαi > 0). Therefore, agents have intermediate preferences —check Definition 4, section 2.2.2.—, a Condorcet winner exists and is given by the bliss point of the median
value αm .
(d) The utilitarian social welfare function in this economy is
Z
W q; αi dF = J (q) .
αi
Then the social optimal policy corresponds to the preferred policy of
the average citizen, α.
The average citizen has the following preference (plugging αi = α in
equation (5)):
W (q; α) = J(q)
And therefore:
∂W (q; α)
∂J(q)
∂L(q)
=
=
∂q
∂q
∂q
This last equality being due to the envelope theorem.
We have that:
∂L(q)
<0
∂q
which means that the utility of the average citizen is always decreasing in q. Her optimal value of qC and qL is therefore equal to 0.
When αi = α, we have that αm = α. Therefore, the socially optimal
policy, corresponding to the preferred policy of the average citizen,
is equal to the decentralized equilibrium, which corresponds to the
preferred policy of the median citizen. Both equilibria will be: qC =
qL = 0.
7
When agents are heterogeneous, the two equilibria might differ.
How is the utility of the median voter impacted by the level of the
taxes?
∂W (q; αm )
= LqC + K(αm )HqC
∂qc
We know that LqC and HqC are negative. If K(αm ) is positive, then
∂W (q;αm )
is negative which means that the preferred policy of the
∂qc
median citizen is the lowest possible value of qC . We get an interior
solution (qC 6= 0) only if K(αm ) is negative (i.e. if αm < α). In that
(q;αm )
= 0, which means that the optimal value
case, we can have ∂W ∂q
c
of qC is positive.
Same reasoning for qL .
Thus, f = L (q) (qc + qL ) > 0 if only if αm < α.
3. True or False
(a) False. According to the probabilistic voting model, each candidate
chooses the policy-platform that maximizes her probability of winning the election. The policy-platform which is chosen is therefore
independent from the ideology of the candidates, but aims at pleasing
the voters in the more numerous and less ideological groups.
(b) True. Strategic entry in a 3 or more candidates model: A candidate
can stay in the race, even though she knows that she will loose for
sure, in order to prevent another candidate for winning, by stealing
her votes.
(c) True. Full decentralization of spending and financing is the optimal
institutional arrangement. Policymakers’ incentives are not distorted
and the socially optimal policy emerges as an equilibrium. There
is full internalization of the costs of the local public goods, which
prevents overspending.
(d) False. cf page 1187. "‘If recognized in the first session, a member
makes a proposal to distribute δ/n to any (n − 1)/2 other members
and to keep 1 − δ(n − 1)/2n for his or her district."’ (n − 1)/2 is the
minimal number of members needed to obtain a majority vote.
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