Question

P1 A marketing analyst has estimated a firm’s demand function to be ?? = 40 ? 2?? + 20??, where X is an indicator for whether the economy is in a boom (X = 1) or recession (X = 0). Marginal cost of producing the good is 10.

i) Write down the inverse demand, i.e. P as a function of Q and X.

ii) Write down the marginal revenue function during a boom as a function of Q.

iii) What is the firm’s optimal price during a boom?

iv) Write down the marginal revenue function during a recession as a function of Q.

v) What is the firm’s optimal price during a recession?

P2 A snack company’s data analysts have estimated the following elasticities for sales of the firm’s products. Price elasticity is –2.5. Income elasticity is –0.8. Cross-price elasticity with chocolate bars is +0.5. Advertising elasticity is +0.8. Marginal cost is $6.

i) What is the optimal profit margin (P – MC)/P in percent?

ii) What is the optimal price?

iii) If the price of chocolate bars increases by 10%, by how much do the sales of the company change? (Pay attention to the right sign.)

iv) If the price of chocolate bars increases by 10%, and consumer incomes decrease by 5% at the same time, by how much do the sales of the company change? (Pay attention to the right sign.)

v) If the firm wants to achieve 20% sales growth, by how much does it need to increase its advertising expenditure?

P3 Consider the cost function ?? = 40 + 3?? ? 2?? ? + ? ? ?? ? .

i) At Q = 4, what is the firm’s average fixed cost?

ii) At Q = 4, what is the firm’s marginal cost?

iii) If the firm optimally produces Q = 4, and ?? = 35 ? ????, what does a have to be?

iv) Which Q minimizes the firm’s average variable cost?

v) What is the firm’s minimum average variable cost?

P4 A firm makes two goods, A and B, with production functions ?? ? = 2?? ?(?? ?) ? and ?? ? = 16?? ?+4?? ? . The firm has a fixed total supply of capital of 1 unit, and it can hire labor at PL = $10 for the first 5 hours, and for the overtime rate of PL = $20 after that, up to 10 hours. Output prices are P A = $2 and P B = $1.

i) So that capital’s marginal revenue product is equal for A and B, what must L A be?

ii) How much labor should the firm hire to work on B?

iii) If K A = 1, what is the overall revenue from QA and QB?

iv) If K A = 0, what is the overall revenue from QA and QB?

v) What is the profit contribution (revenue minus labor cost)?

Answer 1