Public Economics Field Exam
Michigan State University
Spring, 2002
Please answer question number 1 and four of the remaining questions. (Thus, the total
number of answered questions equals five.)
1. Consider an economy in which a single consumption good is produced with two
types of labor, low-skilled and high-skilled, using a constant returns to scale
technology (implying zero profits). The two types of labor enter the production
function as separate arguments, implying that the gross wage for type-i labor (i =
h, 1) is a function o f the ratio of the two types of labor, L a I : wi = wi(Lal). Let
the consumption good be the untaxed numeraire.
(a) Suppose that the two types of labor may be taxed at different constant rates.
What does the Ramsey Tax Rule tell us about the optimal taxation on these two
types of labor? Assume a representative consumer with a utility function
containing consumption and the two types of labor as arguments. No derivations
are necessary, but you should explain your rule intuitively, making sure to define
your notation.
(b) Suppose now that the economy contains two types of individuals, low-skilled and
high-skilled, where a type-i individual's utility function has consumption and
type-i labor as arguments. If the two types of labor can be taxed at different
constant rates and a uniform poll subsidy is also available (i-e., each individual
receives the same lump-sum subsidy or pays the same lump-sum tax), then how
should the two types of labor be taxed? Give another rule, and provide an
explanation. (Either derive a rule or use a special case of the rules for optimal
taxation in a many-consumer economy.)
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(c) Suppose now that the two types of labor cannot be taxed at separate constant
rates, but a nonlinear income tax is available (in which case all individuals face
the same tax schedule, but their marginal tax rates may difTer). State the optimal
nonlinear tax problem, including the objective function, constraints and control
variables. (You may assume that only the high-skilled worker's incentivecompatibility condition is binding.) Make sure to define your notation. (Hint:
Your statement must make use of the wage hctions given above.)
(d) In the standard model of optimal income taxation with a linear technology (i.e.,
fvred gross wage rates), what is the marginal tax rate on high-skilled workers?
Using the first-order conditions for the problem in (c), explain whether this result
still holds in the case of variable wage rates.
2. The behavior of voters is difficult to model, because it appears that a single voter
can rarely influence the outcome of an election. In light of this and the fact that
voting is costly, where is the incentive to vote? The following questions concern
some approaches to understanding the behavior of voters.
(a) Suppose that each voter faces the same voting cost. How can voting be
explained as an equilibrium outcome? Describe the equilibrium.
(b) Voters lack incentives needed to obtain the information needed to make
informed voting decisions. How can special interest groups take
advantage of this lack of incentives? Describe and critically evaluate two
approaches.
3. Consider the following quotation.
"Assuming that individuals possess heterogeneous preferences for a local public
good, the equilibrium for a system of communities will be Pareto efficient if and
only if individuals sort themselves across communities so that each community
contains only one type of household."
Critically evaluate this claim, including both the "if"part and the "only i f ' part.
Assume throughout the discussion that there are no restrictions on the use of head
taxes or subsidies.
4. Consider the following optimal tax problem in a one-consumer economy. Two
goods, xl and x2, are produced with one input, 1(labor), by a linear technology, so
that X I +x2 = I. Utility is a quasiconcave function u(xl, x2, I), with ul, u2 >O and us
< 0. Take 1as the numeraire and normalize the tax on it to be zero. A lump-sum
tax, T, andlor commodity taxes tl and t2 can be used to raise required revenue R.
(The consumer's budget constraint is thus (1 + t,)x, + (1 +t2)x2+T = I and the
government's is t,x, +t2x2+ T = R).
a. Set up the problem and show fiom first-order conditions that only the lumpsum tax will be used.
b. Now suppose that consumption of xl also involves a negative externality.
Utility becomes u(xl, x2, I) - b l , 6 > 0, but the consumer ignores the externality in
choosing xl, x2, and 1. How does this affect the optimal tl? Relate your answer to
Pigouvian tax theory.
c. Discuss in words how the unavailability of the lump-sum tax would complicate
the problems of choosing tl and t2 in parts a and b.
5. Do elections give good incentives to politicians? Consider this question in the
context of the following model. A continuum of citizens (mass equal to one) vote in
order to choose a leader to implement a proportional income tax and a level of public
spending. Preferences of agent i are given by
Where c denotes private consumption, g is per capita public spending and H(g) is
increasing and concave. The budget constraint for individual i is
ci = (1-t) y',
Where y' is exogenous income and it differs among citizens. Average income is
denoted y.
Elected politicians may use tax revenues for personal consumption, denoted by r.
The government budget constraint is
a. Derive the policy preferences of agent i.
b. Suppose that candidates can commit credibly to policies during the campaign (ie.
pre-election politics). Find (or explain) the equilibrium levels of public per capita
spending and rents.
c. Now suppose that there is no pre-election commitment. Find (or explain) the
equilibrium levels of public per capita spending and rents.
d. Discuss a revision to this model that can generate both positive levels of public
spending and politician rents.
6. Suppose that a local government wants to procure a computerized billing system fiom
a local firm. The billing system will be used for one period and it will generate a large
surplus equal to S. The government has limited knowledge of the costs of such a system.
Suppose that it believes that with probability p the firm is high cost with a total cost equal
to Bh- e and with probability l-p the firm is low cost with total cost equal to B1- e with
Bh> B1. It knows that the fjrms profits equal R - f(e), where R is the amount that the
government pays the firm for the system and e is the firms effort. The function q)is
strictly convex.
a. Discus the problem facing the government, the revelation principle, and the
idea of a mechanism.
b. Write down the government's problem as a maximization problem and
carefully explain the constraints.
c. One basic result is the optimal allocation can be implemented with a menu of
linear contracts. Explain this idea.
d. How does the problem change when the relationship lasts for two periods?