What is the firms optimal price during a boom write down

Question

Assignment: Managerial Economics & Business Strategy

P1 A marketing analyst has estimated a firm's demand function to be Q = 40 - 2P + 20X, 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 C = 40 + 3Q - 2Q² + ½Q³.

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 P = 35 - aQ, 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 Q^A = 2K^A (L^A)² and Q^B  = 16K^B + 4L^B. The firm has a fixed total supply of capital of 1 unit, and it can hire labor at P_L = $10 for the first 5 hours, and for the overtime rate of P_L = $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 Q^A and Q^B?
iv) If K^A = 0, what is the overall revenue from Q^A and Q^B?
v) What is the profit contribution (revenue minus labor cost)?