Q1. Eliza makes $500 a week at her summer job and spends her entire weekly income on new running shoes and designer jeans, because these are the only two items that provide utility to her. Furthermore, Eliza insists that for every pair of jeans she buys, she must also buy a pair of shoes (without the shoes the new jeans are worthless.) Therefore, she buys the same number of pairs of shoes and jeans in any given week. Please be reminded that jeans and pair of shoes are indivisible; therefore, with budget constraint, some possible adjustment to round down to the nearest integer is needed. Also, if you choose to use the graph method to answer, in your diagrams, please put jeans (J) on the vertical axis and pairs of shoes (S) on the horizontal axis.

(1) Please use a math function to express Eliza's utility function.

(2) If jeans cost $20 and shoes cost $30 how many will Eliza buy of each? What is her maximum utility?

(3) Suppose that the price of shoes reduces to $20 a pair. How many shoes and jeans will she buy? What is her maximum utility? Please explain to what effect (income, substitution or both) do you attribute the change in her utility levels between part (2) and (3).

(4) Suppose that her income reduces to $400, if jeans cost $20 and shoes cost $30, how many will Eliza buy of each? How will this affect her demand drawn from part (2)? What is her maximum utility? Please explain to what effect (income, substitution, or both) do you attribute the change in her utility levels between part (2) and (4).

Q2. No tax, no subsidy scheme

There are 1,000 identical individuals in the market for commodity X, each with a demand function given by d (X) = 16 - 2p, and 1,000 identical producers of commodity X, each with a function given by s (X) = 10p - 20.

(1) Please find the market demand function.

(2) Please find the market supply function.

(3) Please calculate the market equilibrium price and quantity mathematically.

(4) Please calculate the producer and consumer surplus.

Subsidy scheme

With the same demand and supply functions, now the government decides to grant a subsidy of $1.2 on each unit of commodity X produced to each of the 1000 identical producers of commodity X.

(5) Please calculate the market equilibrium price and quantity in the new situation.

(6) Please calculate the producer and consumer surplus in the new situation.

(7) By comparing your findings in part (4) and (6), does it make sense to you that subsidy makes the market more efficient? Please explain.

Tax scheme

With the same demand and supply functions, the government decides to collect a sales tax of $1.2 per unit sold, from each of the 1,000 identical sellers of commodity X.

(8) Please calculate the market equilibrium price and quantity with tax collection.

(9) Is the tax paid by consumers or producers? How much does each party pay?

(10) What is the total amount or taxes collected by the government?

(11) Please calculate the producer and consumer surplus after tax collection.

(12) By comparing part (4) with part (11), does it make sense to you that tax makes the market less efficient?

Q3. Suppose Vanessa has a utility function U(X, Y) = X^0.2Y^0.8, which X and Y are two goods. The prices for X and Y are $8 and $2, respectively. She has $200 in her pocket.

(1) Please explain why Vanessa's utility function is a special case of Cobb- Douglas function.

(2) How many X and Y will Vanessa buy? What is her maximum utility? Please use the Lagrangian method to answer.

(3) Please find the marginal rate of substitution when Vanessa has the maximum utility constraint with respect to her budget constraint.