2) Consider a monopolist who faces the inverse demand function p = 1 y a) Find t

Question

2) Consider a monopolist who faces the inverse demand function p = 1 y a) Find the price elasticity of demand. as a function of y using the elasticity formula Price elasticity of Demand = y p p y b) Set up the profit function when the monopolist faces the cost c(y) = y/2 (marginal cost=1/2) and c(y) = y 2 (marginal cost =2y) c) Calculate the value of y for each cost function. d) What is the elasticity at the monopolists choice of output for each cost function. That is what is the value of the price elasticity of demand for the function you calculated above (All you have to do is plug the y you calculated into the function you derived above) e) What is the efficient price for each cost function? (Hint:What is an efficent price in a perfectly competitive market) gf What is the deadweight loss generated by the monopolist for each cost function? Show this graphically.

Explanation / Answer

Consider a monopolist who faces the inverse demand function p = 1 y

a) Find the price elasticity of demand. as a function of y using the elasticity formula Price elasticity of Demand

Price elasticity = slope of the demand curve x price / quantity

= -1 x p/y

Ed = -p/y

b) Set up the profit function when the monopolist faces the cost c(y) = y/2 (marginal cost=1/2) and c(y) = y 2 (marginal cost =2y)

Profit = TR – TC

i) Profit = (1 – y)y – y/2 = y/2 – y^2

ii) Profit = (1 – y)y – y^2 = y – 2y^2

c) Calculate the value of y for each cost function.

Maximize profits by keeping dPr/dy = 0

i) 0.5 – 2y = 0 or y* = 0.25

ii) 1 – 4y = 0 or y* = 0.25

d) What is the elasticity at the monopolists choice of output for each cost function.

i) Price p = 1 – 0.25 = 0.75. Elasticity = -p/y = -0.75/0.25 = -3

ii) Price p = 1 – 0.25 = 0.75, Elasticity = -p/y = -0.75/0.25 = -3

e) What is the efficient price for each cost function?

P = MC at efficient price and quantity setting

i) 1 – y = 0.5 and so y = 0.5 and thus, p = 0.5

ii) 1 – y = 2y and so y = 0.33 and thus p = 0.67

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