Please show all the necessary calculations and explanations that lead to your answer. Provide your answers in the order the problems are given.

1. Suppose Niki has the following demand functions:

x_i (p₁, p₂, m) = m = 2/p_i;

for i = 1, 2. Originally she faces prices p₁ = p₂ = $2 and has an income of $100. Then the price of good 1 increases to $4. Calculate the compensating and equivalent variations, given that Niki's utility function is u (x₁, x₂) = x₁^(1/2)x₂^(1/2).

2. The demand function for a good is given by x = m/p².

a) Determine whether demand is elastic or inelastic for this good.

b) Would a price drop increase or decrease revenue from selling this good? Explain.

c) Is this a normal good? Is this a luxury or necessary good? Justify your answers.

3. Suppose the market demand function is D (P_D) = 120 P _D and the market supply function is S (P_S) = 2P_S.

a) Find the market equilibrium.

b) Suppose that the supplier is required to pay a per unit tax of t = 3. Draw a graph to show this change in policy compared to the no-tax policy. Find the equilibrium quantity and prices P_S and P_D. How much of the tax is passed on to the consumer?

c) Suppose now that the demander is required to pay a per unit tax of t = 3 instead of the supplier. Draw a graph to show this change in policy compared to the no-tax policy. Find the equilibrium quantity and prices P_S and P_D. Compare your answers in parts b. and c.

d) Calculate the change in consumer's surplus, the change in producer's surplus, the tax revenue, and the deadweight loss as a result of the new tax policy.

 A firm has a production function y = x₁ x₂.

a) What is the firm's cost function?

b) What is the firm's supply function?