ECO 4932 Topics in Theory – Spring 2015
Homework #2 - Due Date: 03/17/15 – Answer Key
Consider a society inhabited by a continuum of citizens and normalize the size of the
population to 1. Suppose that the preferences of agent i over a publicly provided good y
and privately provided good
c i is expressed by
wi  ci   i H  g 
where H(∙) is a concave well-behaved function and  is the intrinsic parameter of
agent i that is drawn from distribution F(∙) with mean  . Again, all individuals have
initial resources only in the private good, ei  1 for all i, and one unit of private good is
required to produce one unit of public good. To finance the public good production, the
government raises a tax q on each individual so that agent i’s budget constraint is
i
ci  1  q .
a. Derive the policy preferences of each agent W  g ; i  as well as the social optimum in
this economy.
Suppose that two politician P = A, B select platforms gA and gB. Assume that each of
them maximizes the expected value of some exogenous rent R. Call  P , the vote
1

share for politician P, the P’s probability of winning the election is pP  Prob   P  
2

and his expected utility it the pP R . First, the two candidates announce their
platforms simultaneously and non-cooperatively. Then, elections are held. Last, the
elected politician implements this announced policy.
b. Assume that  i   . Determine the candidates’ probability of wining. What are the
announced platforms and which one is implemented? Discuss.
c. Suppose that agents are heterogeneous. Determine the probability of winning for
each candidate. What are the selected platforms in that case? Which one is
implemented?
d. What are the economic predictions of the model? Discuss.
Answer:
a. The policy preferences of individual i are given by:
W  g; i   1  q   i H  g 
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The social optimum is obtained by maximizing
 W  g; dF   under the
i
i
i
1
constraint  1  ci  qdF  i   0 . This results in g *  H q1   . The optimal provision
 
i
of public good is then g *  q* .
b. Votes vote for the politician whose platform provides them the highest utility. In case
of indifference, they vote for A with probability ½. Therefore, the probability of
winning for politician A is:
 0 if W  g A ;    W  g B ;  

1
p A   if W  g A ;    W  g B ;  
2
 1 if W  g A ;    W  g B ;  

Naturally, the probability that politician B win is pB  1  pA . Since  i   , all voters
prefer g * . Then each candidate increases his probability of winning by getting closer
to g * . As a consequence, there is a unique equilibrium in which politicians converge
to the same platform:
g A  g B  g*
c. If agents are heterogeneous, then only agents with type  i   prefer unambiguously
g * . Then the social optimum is expected not to be implemented. Formally, the
probability that agent i vote for politician A is:
 0 if W  g A ;  i   W  g B ;  i 

1
p A   if W  g A ;  i   W  g B ;  i 
2
 1 if W  g A ;  i   W  g B ;  i 

Recall that preferences of agents are single peaked so that, if voters make a pairwise
vote, the preferred policy of the agents with median value of  i would be selected. As
a consequence, each politician finds in his interest to select the platform
corresponding to the policy preferred by  m since it guarantees victory. As this
reasoning is the same for both politicians, the unique equilibrium is:
g A  gB  gm
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d. The social optimum is achieved only under restrictive conditions on the distribution
of the population. This implies that, absent these conditions, the majority rule
generates under-production or over-production of a public good.
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