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1. Suppose demand for tennis shoes is estimated to be [tex]Q = 1000 - 4p + 10pX - 2pZ + 0.1Y[/tex]. If [tex]p = 60[/tex], [tex]pX = 40[/tex], [tex]pZ = 100[/tex], and [tex]Y = 1500[/tex], answer the following questions:

a. Find the price elasticity of demand.

b. Find the cross-price elasticity of demand with respect to the price of commodity X ([tex]pX[/tex]). Is commodity X a substitute or a complement?

c. Find the income elasticity of demand. Are tennis shoes a normal good or an inferior good for the consumers in this market?

2. Suppose a tax on canned beans of $0.20 per can is levied on the consumers. As a result of the tax, the price consumers pay increases from $1 to $1.08 per can. Answer the following questions:

a. What is the tax incidence on producers? (Hint: pay attention to the sign!)

b. Suppose the price elasticity of supply is 0.6. Find the price elasticity demand given the $0.20 specific tax on canned beans.

3. Suppose a market has the following supply and demand functions: [tex]QD = 500 - 10P[/tex] and [tex]QS = 50 + 5P[/tex]. If the government imposes a specific tax [tex]\tau[/tex] on the producers, find the value of tax [tex]\tau[/tex] that maximizes the tax revenue.

4. Write out and solve the partial derivative that shows the comparative static effect of a change in the price elasticity of demand on the incidence of a specific tax on consumers. Is the value of the partial derivative positive or negative?

Answer :

The sign of the partial derivative will determine whether the value is positive or negative, indicating the direction of the relationship between the price elasticity of demand and the tax incidence on consumers.

a. To find the price elasticity of demand, we can use the formula: Price Elasticity of Demand = (% Change in Quantity Demanded) / (% Change in Price) Since we are given specific values for p, pX, pZ, and Y, we need to calculate the corresponding quantity demanded. Let's calculate the initial quantity demanded (Q1) and the quantity demanded after a small change in price (Q2) using the given equation:

Q1 = 1000 - 4p + 10pX - 2pZ + 0.1Y

Q2 = 1000 - 4(p + Δp) + 10pX - 2pZ + 0.1Y

Now we can calculate the percentage change in quantity demanded:

% Change in Quantity Demanded = [(Q2 - Q1) / Q1] * 100

Similarly, we can calculate the percentage change in price:

% Change in Price = [(p + Δp - p) / p] * 100

By substituting the given values into the formulas, we can find the price elasticity of demand.

b. The cross-price elasticity of demand with respect to the price of commodity X (pX) can be calculated using the formula:

Cross-Price Elasticity of Demand = (% Change in Quantity Demanded) / (% Change in Price of X)

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