Imagine that citizens get utility from private market consumption, c, and consumption of publicly provided goods, g. A representative citizen i has the following utility function:
U(c, g) = ci + α ln(g),
where ln is the natural logarithm function and α is a parameter which is strictly positive.
a. If individuals consume all of their after tax income, then ci = (1 − t)yi , where 0 < t < 1 is the proportional income tax rate and yi is the income of individual i. Re-write the utility
function for individual i in terms of t, yi and g.
b. When forming policy preferences for g, individual i chooses g to maximize the function you wrote in part (2). Utility maximization for individual i equates the marginal cost of g with the marginal benefit of g. For this functional form, the marginal cost of g is yi y¯ and the marginal benefit is α g. Solve for the per-capita expenditure on public goods preferred by individual i, g∗i , as a function of yi, ¯y and α.
c. Solve the per-capita expenditure on public goods preferred by individual i, g∗i, as a function of yi , and α.
d. Imagine that α = 4 and ¯y = 45. Draw a graph of the marginal cost and marginal benefit curves for an individual i with income yi = 30. Solve for the level of government spending preferred by this individual.
e. Keep α = 4 and ¯y = 45. Solve for the most preferred level of spending for individuals with the following income levels: y1 = 20, and y5 = 75. Explain how realtive income affects most preferred policy choices.
f. How do the policy preference change if:
1. the society over all gets richer (¯y), ceteris paribus?
2. the social preferences for public goods get weaker (α ↓), ceteris paribus?