In this exercise, we consider some policy issues related to public support for schools—and the coalitions between income groups that might form to determine the political equilibrium.
A: Throughout, suppose that individuals vote on only the single dimension of the issue at hand — and consider a population that is modeled on the Hotel ling line [0,1] with income increasing on the line. (Thus, individual 0 has the lowest income and individual 1 has the highest income, with individual 0.5 being the median income individual.)
(a) Consider first the case of public school funding in the absence of the existence of private school alternatives. Do you think the usual median voter theorem might hold in this case—with the public school funding level determined by the ideal point of the median income household?
(b) Next, suppose private schools compete with public schools, with private schools charging tuition and public schools funded by taxes paid by everyone. How does this change the politics of public school funding?
(c) Some have argued that political debates on public school funding are driven by “the ends against the middle”. In terms of our model, this means that the households on the ends of the income distribution on the Hotel ling line will form a coalition with one another —with households in the middle forming the opposing coalition. What has to be true about who disproportionately demands private schooling in order for this “ends against the middle” scenario to unfold?
(d) Assume that the set of private school students comes from high income households. What would this model predict about the income level of the new median voter?
(e) Consider two factors: First, the introduction of private schools causes a change in the income level of the median voter, and second, we now have private school attending households that pay taxes but do not use public schools. In light of this, can you tell whether per pupil public school spending increases or decreases as private school markets attract less than half the population? What if they attract more than half the population?
(f) So far, we have treated public school financing without reference to the local nature of public schools. In the U.S., public schools have traditionally been funded locally—with low income households often constrained to live in public school districts that provide low quality. How might this explain an “ends against the middle” coalition in favor of private school vouchers (that provide public funds for households to pay private school tuition)?
(g) In the 1970’s, California switched from local financing of public schools to state-wide (and equalized) financing of its public schools. State-wide school spending appears to have declined as a result. Some have explained this by appealing to the fact that the income distribution is skewed to the left, with the statewide median income below the statewide mean income. Suppose that local financing implies that each public school is funded by roughly identical households (who have self-selected into different districts as our Chapter 27 Tie bout model would predict), while state financing implies that the public school spending level is determined by the state median voter. Can you explain how the skewedness of the state income distribution can then explain the decline in state-wide public school spending as the state switched from local to state financing?
B: Suppose preferences over private consumption x, a public good y and leisure ℓ can be described by the utility function u(x, y, ℓ) = xαyβℓγ. Individuals are endowed with the same leisure amount L, share the same preferences but have different wages. Until part (e), taxes are exogenous.
(a) Suppose a proportional wage tax t is used to fund the public good y and a tax rate t results in public good level y = δt. Calculate the demand function for x and the labor supply function. (Note: Since t is not under the control of individuals, neither t nor y are choice variables at this point.)
(b) Suppose instead that a per-capita tax T is used to fund the public good; i.e. everyone has to pay an equal amount T. Suppose that a per-capita tax T results in public good level y = T. Calculate the demand function for x and the labor supply function.
(c) True or False: Since the wage tax does not result in a distortion of the labor supply decision while the per-capita tax does, the former has no deadweight loss while the latter does.
(d) Calculate the indirect utility function for part (a) (as a function of L, w and t).
(e) Now suppose that a vote is held to determine the wage tax t. What tax rate will be implemented under majority rule?
(f) Suppose that y is per pupil spending on public education. What does this imply that δ is (in terms of average population income I, number of taxpayers K and number of kids N in school)?
(g) Now suppose there exists a private school market that offers spending levels demanded by those interested in opting out of public education (and assume that spending is all that matters in people’s evaluation of school quality). People attending private school no longer attend public school but still have to pay taxes. Without doing any additional math, what are the possible public school per pupil spending levels y that you think could emerge in a voting equilibrium (assuming that public education is funded through a proportional wage tax)? Who will go to what type of school?
(h) Can you think of necessary and sufficient conditions for the introduction of a private school market to result in a Pareto improvement in this model?
(i) In (e), you should have concluded that, under the proportional wage tax, everyone unanimously agrees on what the tax rate should be (when there are no private schools). Would the same be true if schools were funded by a per-capita tax T?