Análisis comparativo por preguntas de la encuesta

In order to derive divergent tax rate policy in equilibrium, we must look for non-UCR preferences. In our model, given any positive tax ratet, a higher voter type always prefers to allocate more tax revenue to good 0, a lower voter type prefers more to good 1. Then what we can expect if we fix a voter type, and ask her a question like “if the government increases the tax rate, how should the extra revenue be spent?” For homotheticv(x0, x1, θ), we have known that the allocation rule that a voter prefers is unchanged with the tax rate. However, Jacoby (1994) performs an analysis of mass government spending preferences across a set of policy areas using a national survey data, and finds that mass attitudes toward spending on social programs are significantly positively related to overall attitudes on government spending, but other programs including defense are either not related or negatively related.

Free and Cantril (1967) argues that the U.S. public want more expenditure on social programs, but repeatedly demands for smaller government. Our understanding of these

is strictly smaller than the case where t⁰= t¹, so there is no equilibrium with t⁰6= t¹.

findings is that as tax rate (and the implied government revenue) increases, voters expect increasingly more share of tax revenue to be allocated to social program. Security is like a kind of necessity, when the tax rate is low, voters may prefer the candidate who allocates more on it, but as tax rate increases to some high level and beyond, voters may become prefer the candidate who allocates more on social program. As tax rate increases, the benefit from extra spending on security decreases dramatically beyond some level, while social program still has higher substitutability with private consumption compared with security program like defense, because it is more directly related to a voter’s well being.

This leads to a particular structure on voters’ utility functions. Analogous to the UCR property defined above, we present the following strict single-crossing property (SSCP) for voter preferences over two public goods:

Assumption 1.1 (Strict Single-Crossing Property). All voters’ utility functions over two public goods: v(α, t), satisfy the following strict single-crossing property: ⁹

For all α > α⁰ and t > t⁰,v(α, t⁰) −v(α⁰, t⁰) ≥ 0 implies v(α, t) − v(α⁰, t) > 0.

The property simply says that the utility difference,v(α, t) − v(α⁰, t), as a function of t, crosses the horizontal line only once at only one point, the usual (non-strict) single-crossing property allows that the difference crosses the horizontal line for a continuum value of t (see Figure1.6for the difference between single-crossing property and strict single-crossing property). Actually, it’s not difficult to see that the UCR property implies single-crossing property, but not strict single-crossing property.

In our model, we assume α⁰ > α¹. Loosely speaking, this assumption implies that, at lower tax rates, a voterθ may strictly prefer candidate 1 who allocates more tax revenue to good 1; as tax rate increases, the voter will be indifferent between the two candidates

9see Milgrom and Shannon (1994), Edlin and Shannon(1998), for use of single-crossing property in modeling politics, seeAshworth and Bueno de Mesquita(2006).

t

(a) Strict single-crossing property

t

(b) Single-crossing property

Figure 1.6: Strict and non-strict single-crossing property

at certain tax rate; then as the tax rate increases further, the voter will strictly prefer candidate 0 to candidate 1 at any higher tax rate. In this sense, the property is so called strict single-crossing. Here we can understand good 1 as a kind of necessity (e.g., security), good 0 is some social program more directly related to voter welfare, such as social security and/or health care.

For the utility functionsv(x₀, x₁, θ), if v(x₀, x₁, θ) is “more concave” in x₁ than inx₀,¹⁰ then the strict single-crossing property is satisfied. Because if a voter’s utility function is “more concave” in x1 than in x0, the marginal benefit from extra spending on good 1 decreases faster than that from good 0. This captures the idea that as tax rate increases, voters prefer an increasingly higher fraction of tax revenue to be spent on good 0 (e.g., social programs) instead of good 1 (e.g., security-related programs). And, since v(x0, x1, θ) is strictly concave, a voter’s preference is single-peaked with respect to tax rate, so given that the two candidates propose the same tax rate, a voter likes more and more the candidate

10Here “more concave” means that the Arrow-Pratt coefficient of relative risk aversion of v1(·) is greater than that of v0(·), i.e., −x1v1⁰⁰/v⁰1> −x0v⁰⁰0/v0⁰.

who allocates more on good 0.¹¹

Graphically, the expansion line of a voter θ is curved toward the expansion line of candidate 0 (see the right panel of Figure 1.5). It’s not difficult to see that under this condition, candidate 0 has comparative advantage when the tax rate is high, i.e., running a big government, while candidate 1 has comparative advantage when the tax rate is low, i.e., running a small government. This is different from the homothetic cases, where both candidates and all voters have straight expansion lines from the origin, so there are no comparative advantage or disadvantage for both candidates at either high or low tax rate. We thus expect that in equilibrium, a candidate should choose the tax rate which corresponds to his advantage, that is, candidate 0 should choose a higher tax rate to run a big government, candidate 1 should choose a lower tax rate to run a small government.

This is indeed what the equilibrium looks like.

Proposition 1.3 (Policy Divergence). Suppose the utility function of two public goods satisfy the strict single-crossing property in Assumption1.1, andθm has a strictly positive density, then:

1. In the Nash equilibrium defined by the system of equations in Proposition 1.1, we have policy divergence: ¯t⁰ 6= ¯t¹. That is, two candidates propose different tax rates as their platform in Nash equilibrium, actually, we have: ¯t⁰ > ¯t¹, candidate 0 proposes a higher tax rate.

2. Conversely, if (¯t⁰, ¯t¹, ¯θ) satisfy Equation 1.3, 1.4and 1.5, then it is a local equilibrium.

The equilibrium like this is as in the right panel of Figure 1.5. The idea for proving divergent tax rates is simple. Consider the system of equations in Proposition 1.1, the

11Actually, it is easy to check that if −x1v⁰⁰1/v1⁰ = −x0v⁰⁰0/v0⁰, i.e., v(x0, x1, θ) has the same concavity between good 0 and good 1, then it is homothetic (the vice versa is also true), we go back to the first case where we have policy convergence in equilibrium.

summation of two partial derivatives,^∂v(x_∂x^0,x1,¯θ)

0 α+^∂v(x_∂x^0,x1,¯θ)

1 (1−α) = ^∂vθ¯_∂t^(α,t) = 1. Suppose on the contrary there exists a solution (¯t, ¯t, ¯θ) solving the system, then the system implies that vθ¯(α⁰, t) and vθ¯(α¹, t) are equal at ¯t (by Equation 1.5), and their partial derivatives with respect tot are also equal at ¯t (by Equation 1.3and 1.4), which contradicts with the fact thatvθ¯(α, t) has the strict single-crossing property in Assumption1.1. By Assumption 1.1,vθ¯(α⁰, t) and vθ¯(α¹, t) can cross only once at only one point and from below, so if at ¯t the two are equal, then for anyt > ¯t, vθ¯(α⁰, t) > vθ¯(α¹, t), so their derivatives with respect tot cannot be equal at ¯t.

Remember that candidate 0 has comparative advantage of running a big government, candidate 1 has comparative advantage of running a small government, this gives the centrifugal pressure for tax policy divergence. However, there is also centripetal pressure for convergence because voters have trade-offs between private consumption and public goods. In equilibrium, candidate 0 would not offer more good 1 than candidate 1, i.e., we have x⁰₀ > x¹₀, x⁰₁ < x¹₁. So Democrats will not impose a terribly high tax rate to have a very big government, similarly, Republicans will not impose an ignorable tax rate to have a very small government. Both Democrats and Republics choose not to trespass on the issues (public goods) owned by their opponents. Candidate 0 offers more public good 0 than 1, while candidate 1 offers more public good 1 than 0. In this respect, the result connects to the issue ownership literature which argues that candidates do not converge but rather choose to emphasize the issues they have strength in, because they cannot benefit from simply replicating their opponent platform (Petrocik (1996),Petrocik et al. (2003)).

The result also has its connection withEgan(2008), which demonstrates that Republicans enjoy a long-run public opinion advantage over Democrats on issues of taxation, anti-crime and national security; while Democrats are mostly trusted in large expenditure issues such as education, health care and social security.

In both cases (policy divergence and policy convergence), we have a cut-off voter type θ who is indifferent between the two candidates, voters with θ > ¯¯ θ vote for candidate 0, voters with θ < ¯θ vote for candidate 1. A candidate wins an election if and only if the realized median voterθ_m strictly prefers him over his opponent, so if θ_m > ¯θ, candidate 0 wins, if θ_m < ¯θ, candidate 1 wins. We denote the cumulative distribution of θ_m asF , so the winning probability of candidate 0 is 1 −F (¯θ), the winning probability of candidate 1 is F (¯θ). So if a candidate is strictly preferred by the median median voter, i.e., θ^∗_m such that F (θ^∗_m) = 0.5, then he wins with probability greater than a half. However, we should notice both the distribution ofθm and the distribution ofθ only affect the probability of winning or who wins an election ex-post, the equilibrium results, including the cut-off voter type and proposed tax rates are all rigid.