(1) There is a society consisting of n voters, and voters' utility function is given as u i (B) log B (1 t)I i where B is government spending on a public good, t is the income tax rate (proportional tax), and I i is voter i's income The government's budget constraint is given as i t I i B Find voter i's utility function excluding t by substituting the government's budget constraint and using n and the average income I As an illustrative example, draw a rough graph where the horizontal axis is B, and the vertical axis is a voter's utility Find the amount of B to maximize the sum of n people's utilities Suppose that n 3 and the amount of B is determined by the equilibrium of the Downs model of the two parties Compare the amount of B between the cases when all three are I i 100 and when I 1 40, I 2 60, and I 3 200 (2) Consider a monopoly with the marginal cost c and the inverse demand function p a bx Confirm that the production maximizing the monopoly's profit is ((a bx)x cx)' 0 x (a c) 2b and the price at that time is p (a c) 2 Find the price at which the consumer surplus is maximized in the range in which the firm does not shut down its operation Find the production at that time Illustrate the three changes of regions due to the price change from Q(1) to Q(2) corresponding to (a)the social surplus increase, (b)the consumer surplus increase, and (c)the firm profit decreases, on the graph below f

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(1) There is a society consisting of n voters, and voters' utility function is given as: u i (B) = log B + (1 -

(1) There is a society consisting of n voters, and voters' utility function is given as:

u_i(B) = log B + (1 - t)I_i

where B is government spending on a public good, t is the income tax rate (proportional tax), and I_i is voter i's income. The government's budget constraint is given as ?_i t I_i= B

Find voter i's utility function excluding t by substituting the government's budget constraint and using n and the average income I.
As an illustrative example, draw a rough graph where the horizontal axis is B, and the vertical axis is a voter's utility.
Find the amount of B to maximize the sum of n people's utilities.
Suppose that n = 3 and the amount of B is determined by the equilibrium of the Downs model of the two parties. Compare the amount of B between the cases when all three are I_i = 100 and when I₁ = 40, I₂ = 60, and I₃ = 200.

(2) Consider a monopoly with the marginal cost c and the inverse demand function p = a - bx.

Confirm that the production maximizing the monopoly's profit is ((a - bx)x - cx)' = 0 x = (a - c)/2b and the price at that time is p = (a+c)/2.
Find the price at which the consumer surplus is maximized in the range in which the firm does not shut down its operation. Find the production at that time.
Illustrate the three changes of regions due to the price change from Q(1) to Q(2) corresponding to (a)the social surplus increase, (b)the consumer surplus increase, and (c)the firm profit decreases, on the graph below.