Download 1 Introduction 2 Political-Economic Equilibrium — Direct Democracy

Politico-Economic Equilibrium
Allan Drazen
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Introduction
Policies government adopt are often quite different from a social planner’s solution.
A standard argument is because of “politics”, but how can one study in a more analytic
way the effect of politics on economic policy? The starting point is that the world is
not populated by representative agents with identical objectives, but by individuals or
groups with heterogeneous policy objectives. They may differ in how they think the world
works and desire different policies because of that. Or, for reasons of simple self-interest,
individuals with different relative endowments of capital versus labor differ in the mix of
capital and labor taxes they prefer. Hence, conflict in desired policy may reflect underlying
heterogeneity in ideology, endowments, or preferences over publicly-supplied goods. It may
a common preference for more income to less, leading to distributive conflict.
When individuals have different preferred policies, there must be a collective choice
mechanism for aggregating preferences. The societal choice of policies will then reflect the
operation of the political mechanism rather than the choice of a benign social planner, and
the results might be quite different.
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Political-Economic Equilibrium — Direct Democracy
To make these ideas more specific, let’s consider how one would find a political equilib-
rium. In a direct democracy (that is, where decisions are not delegated to policymakers),
a three-stage political-economic equilibrium could be defined as follows:
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(1) Agents optimize over their economic choice variables given policy, given their beliefs
and information;
(2) On the basis of how it affects her (that is, the individual optimization in (1)), each
individual determines his preferred, given her beliefs and information;
(The first and second steps can be summarized by each individual calculating the policy
that maximizes his expected utility.)
(3) Aggregation of the policy preferences in (2) into an economy-wide policy via some
collective choice mechanism.
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Illustration — Society’s choice of level of welfare benefits
To illustrate these ideas (and to set the stage for discussing the political economy
of growth), consider the Meltzer and Richard (1981) model of political determination of
taxes and transfer payments, using the above three-stage specification of political-economic
equilibrium.
(1) Individual optimization: Each individual chooses optimal labor supply l and consumption c for a given tax rate τ and transfers v. Individuals differ in their innate productivity ξ, which has CDF F (ξ). Before-tax income is
y (l, ξ) = ξl
Hence, we may write Ms. ξ’s maximization problem as (where leisure z = 1 − l)
M axU (cξ , 1 − lξ )
cξ , lξ
subject to a budget constraint
cξ = (1 − τ ) y (l, ξ) + v
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(1)
where τ and v are taken as given. Hence we may write the problem as
maxU ((1 − τ ) ξlξ + v, 1 − lξ )
lξ
(2)
for given τ and v. This maximization problem implies a first-order condition
(1 − τ ) ξUc (·, ·) ≤ Uz (·, ·)
(3)
with equality if l > 0. (3) implicitly defines the labor supply function l (τ , v; ξ) ≥ 0
(and optimal consumption c (τ , v; ξ) is determined by substituting l (τ , v; ξ) from (3) into
(1)). Why might (3) hold with inequality, implying l = 0? Since v > 0, individuals with
productivity ξ ≤ ξ 0 do not work, where ξ 0 is defined by
ξ0 =
Uz (v, 1)
(1 − τ ) Uc (v, 1)
This fully determines the individual’s choice problem for given economic policy. This is a
standard economic optimization problem.
(2) Most preferred policy: Given l (τ , v; ξ), we can write utility as a function of τ and
v for each individual ξ as
V (τ , v; ξ) ≡ U [(1 − τ ) ξl (τ , v; ξ) + v, 1 − l (τ , v; ξ)]
(4)
This is simply the indirect utility function for Ms. ξ as a function of exogenous taxtransfer policy. In the first-stage optimization Ms. ξ, who is “atomistic” (that is, cannot
affect aggregate values such as v by her choices) takes τ and v as given. In the second
stage, Ms. ξ asks, “What τ and v would I prefer, were it up to me, given the utility I would
get from any policy {τ , v} and the relation between τ and v in equilibrium?”
The relation between τ and v is given by the government budget constraint that any
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tax-transfer policy must satisfy:
τ ȳ = v
(5)
where average income ȳ in the economy is defined by
ȳ (τ , v) ≡
Z
ξ=∞
ξl (τ , v; ξ) dF (ξ)
(6)
ξ=ξ 0
One may show that if consumption and leisure are normal goods, then
dȳ
dτ
< 0 (details are
in the Meltzer-Richard article).
The decision problem of Ms. ξ for her most preferred tax rate may then be written as
maxU [(1 − τ ) ξl (τ , τ ȳ; ξ) + τ ȳ, 1 − l (τ , τ ȳ; ξ)]
τ
yielding a first order condition (using (3)) for Ms. ξ’s most preferred tax rate, τ̃ ξ
¶
µ
dl
dȳ
+
((1 − τ ) ξUc [·] − Uz [·]) = 0
−Uc [·] ξl − ȳ − τ
dτ
dτ
where
dl
dτ
is the total derivative of l (τ , τ ȳ; ξ) with respect to τ . Since the second term
in parentheses is the first-order condition (3) which holds with equality when l > 0, this
condition implies that for someone with positive labor supply the optimal tax rate is
τ̃ ξ =
y(l,ξ)−ȳ
dȳ/dτ .
However, since tax rates are constrained to be non-negative (relevant in the
case where income is above the mean, that is, y (l, ξ) > ȳ), we write the most preferred tax
rate as:
ξ
τ̃ = max
µ
¶
y (l, ξ) − ȳ
,0
dȳ/dτ
(7)
Note that an individual who supples no labor (l = 0) prefers the tax rate that maximizes
transfers v, that is, by (5), that maximizes τ ȳ. This is simply the tax rate for which the
elasticity of ȳ with respect to τ equals −1, which is exactly what (7) yields when y (l, ξ) = 0.
A key implication of (7) is heterogeneity over preferred tax-transfer policy, reflecting
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underlying (or “ex-ante”) heterogeneity in endowments ξ. The shape of the preferred tax
schedule as a function of ξ is easy to derive:
i) An individual whose pre-tax income is at or above the mean income will prefer a zero
tax rate and no lump-sum redistribution;
ii). Individuals with ξ ≤ ξ 0 who derive income only from transfers v will prefer the tax
rate which maximizes v, call it τ v max .
iii) Individuals who work but have income below the mean prefer positive tax rates,
which decline as y (l, ξ) rises.
Sometimes, this calculation is simple, as in our example above.
(3) Determination of the aggregate tax-transfer policy: The tax-transfer policy
actually chosen will depend on the political mechanism by which preferences are aggregated.
In a direct democracy we suppose voters vote directly on the tax rate, with the policy
gaining a majority being the optimal policy. Aggregation in the third stage need not be
via voting — an alternative would be agreement by all parties (consensus) — but it is the
leading aggregation mechanism.
A key result is that under certain conditions, the policy that would emerge under
majority voting over policies is the policy that would be favored by the median voter.
This will be the case in this model so the political equilibrium policy will be τ̃ median . We
discuss in the Appendix the Median Voter Theorem.
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APPENDIX: Median Voter Theorem in One Dimension
1. Single-peaked preferences
Single-Peaked preferences are preferences that satisfy the following two properties:
1) Each voter has a most preferred policy, say τ̃ i for Mr. i;
2) For any two other policies on same side of τ̃ i (for example, τ 1 < τ 2 < τ̃ i ), he
prefers the policy closer to τ̃ i .(spatial preference, due for example to preferences being
single-peaked).
Is this a strong restriction? If choices are over primitives in u (·), then simple concavity
of u (·) is sufficient to give single-peakedness. That is, suppose the policy choice enters
directly into the utility function. For example, suppose the policy choice is what fraction
of the budget to spend on guns versus butter, where the utility function is u (g). Consider
the following figure.
Butter
Utility
(a)
(b)
O ```
O ``
U(O*)
U(O*)
`
U(O)
Guns
O
```
O
``
*
O
O
`
O 1
Fraction
for guns
Single-Peaked Preferences
2. Single-crossing property
However, when primitives in the utility function are not policies, but they depend on
policies (for example, supply of labor as a function of the tax rate in the example in section
3 ) concavity of u (c, 1 − l) does not ensure concavity of indirect utility function V (τ )
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and hence single-peakedness. However, even when preferences are single-peaked they may
satisfy the single-crossing property which will also allow us to apply the Median Voter
Theorem.
For example, in that example, preferences are not single-peaked for a general utility
function u (c, 1 − l). However, the most preferred tax rate is inversely related to income.
Since the ordering of pre-tax incomes is not changed by the tax-transfer policy (for example,
if the tax system induced an initially higher-income individual to work so much less than
an initially lower income individual that the ranking of their pre-tax incomes switched, this
would not be the case), then the following is true. If someone with income y a prefers τ 1 to
τ 2 for all τ 1 < τ 2 , then all individuals with income y > y a will also prefer τ 1 to τ 2 for all
τ 1 < τ 2 , and vice-versa (if Mr. yb prefers τ 2 to τ 1 for all τ 1 < τ 2 , then all individuals with
income y < y b will also prefer ...). One may easily show that the preferences in the model
in section 3 satisfy the single-crossing property since pre-tax income ξl (·) is increasing in
ξ for any policy (τ , v).
3. Median Voter Theorem
When individuals have single-peaked preferences (or preferences that satisfy the singlecrossing property), one can show that under a few further assumptions,1 the Median
Voter Theorem holds, namely that no policy can defeat the policy preferred by the
median voter in a majority vote. Consider a vote between the options τ̃ med and τ 0 (or any
option on either side of τ̃ med ). It is clear that τ̃ med would defeat any “challenger”.
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What one basically requires is that participation in the vote does not depend on the two options being
voted on and that voters do not vote strategically, that is do not vote for a less preferred option given a
binary choice in order to influence the set of options in a future vote.
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Median Voter Theorem
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REFERENCES
Drazen, A. (2000), Political Economy in Macroeconomics, Princeton, N.J.: Princeton
University Press
Meltzer, A. and S. Richard (1981), “A Rational Theory of the Size of Government,”
Journal of Political Economy 89, 914-27.
Persson, T. and G. Tabellini (1994), “Is Inequality Harmful for Growth?,” American
Economic Review 84, 600-621.
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