Reference no: EM133016885

Q.1. Let two consUuers have preferences described by the following Utility function;

U^h(x₁^h, x₂^h) = (x₁^h)(x₂^h), h = 1, 2.

and the following endowments;

a. Calculate the consumers' demand functions by assuming that price of good 1 is p¹ and price of good 2 is p².

b. For each consUmer, write down the excess demand and supply for good 1.

Excess demand of good j for consumer i is e_i^j = (x_i^j - ω_i^j) where x_i^j is the demand of consumer i for good j and ω_i^j is the endowment of consumer i for good j. Show that the consumers' indifference curves are tangential at the equilibrium.

c. For the market of good 1, write down the aggregate excess demand for good 1. Aggregate market excess demand is z¹ = e¹₁ + e₂¹ = (x¹₁ - ω¹₁) + (x₂¹ - ω¹₂).

d. Calculate the price ratio (either p¹/p² or p²/p¹) that will clear the market for good 1.

That is to find the price ratio that sets the aggregate excess demand equal to zero, z¹ = 0. Clearing the market means there is no aggregate excess demand.

e. Show that the aggregate excess demand for good 2 is also zero at the price ratio that is calculated in d.

Q.2. In an economy with 2 people, Z and F, has the social welfare function of W = U_Z + 3U_F and the Utility possibilities frontier of UPF = Max{2U_Z + U_F , U_Z + 2U_F}
a. What type of social welfare function does the economy have?
b. At what Utility levels of Z and F, does this economy achieves social optimum?

Q.3. For each of the following policy changes, explain why the change is not likely to be a Pareto improvement:
a. Building a school, financed by an increase in the local property tax rate.
b. Expansion of certain types of cancer facilities, financed out of general revenues.
c. Replacing the system of agricultural price supports with a system on income supplements for poor farmers.
d. Protecting the domestic aluminUm industry from cheap foreign imports by imposing import taxes on the imported aluminum.

Q.4. Let there be H number of consumers all with the Utility function
Uh = log(xh) + log(G) and income 1. The private purchase of public good implies that G = gh + ∑ gh′. The equilibrium is symmetric.
a. Calculate the private purchase equilibrium level of public good, Ge.
b. Calculate the social optimal level of public good, G??, for the welfare function
W = ∑H Uh.
c. Write the ratio of G0 . Comment on the effect H getting larger on the contrast Ge between the private equilibrium and the social optimum.

Q.5. The demands for a public good, G, of three consumers are given as p1 = 10 - 1/2 G, p2 = 22 - 1/2 G, and p3 = 30 - 1/2 G where G measures the number of Units of the good and pi is the price in dollars. The marginal cost of the pUblic good is $38.

a. What is the optimal provision of the public good? Illustrate your answer with a graph.

b. Explain (in reference to the details of this problem) why the public good may not be supplied at all because of the free-rider problem.

c. If the public good is not sUpplied at all, what is the size of the deadweight loss arising from this market failure?