Question

2. Suppose a firm faces an inverse demand curve P = 6 − 1/2Q and has a total cost function TC = 1/4Q^2 − Q.

(a) Is this firm a price-taker or does it have market power? Explain. (2 points)

(b) Write an equation for the firm’s profit function. (1 point)

(c) Solve for the firm’s profit-maximizing level of output, Q∗ .

(2 points) (d) What price does the firm sell its product at? (1 point)

3. Draw a graph a supply and demand graph for a perfectly competitive market. Label the curves and equilibrium price and quantity. In a separate graph, do the same for a firm with market power (3 points) (a) Now assume consumers become less price sensitive (i.e. the demand curve becomes more inelastic), but this change does not affect the quantity demand at the equilibrium price (in other words, the market equilibrium does not change) (An example of this might be a prescription drug with some substitutes is taken off the market). Draw the new demand curves (label this D2) and, for the firm with market power, the marginal revenue curve (MR2). Show and explain how this affects the equilibrium price and quantity for the firm with market power. Label the new price P2 and the new quantity Q2 (3 points)

Answer 1

2. (a) The firm's inverse demand curve is P = 6 - 1/2Q
Its total revenue is P(Q)*Q = 6Q - 1/2Q2
Marginal Revenue is therefore, MR = 6 - Q

The total cost function is TC = 1/4Q^2 − Q.
Marginal Cost is, MC = 1/2Q-1

The firm will produce output where MR = MC.
=> 6-Q = 1/2Q -1
=> 3/2Q = 7
=> Q = 14/3

Now, P = 6 - 1/2(14/3) = 11/3
Also, MR = 6 - (14/3) = 4.3

We see that MR<P, therefore we conculde that this firm has some market power (it is a monopoly).

(b) The firm's profit function is given by P(Q)*Q - C(Q),
where P(Q) is the inverse demand function and C(Q) is the total cost function.

(Q) = (6-1/2Q)*Q - (1/4Q2 - Q)

(c) The profit maximising output is calculated by differentiating the profit function, (Q) w.r.t. to Q and setting it equal to zero.

(Q) = 6Q - 1/2Q2 - 1/4Q2 + Q
(Q) = 7Q - 3/4Q2
'(Q) = 7 - 3/2Q = 0
=> Q* = 14/3

(d) The price the firm sells its product at is given by, P = 6 - Q = 6 - 14/3
=> P = 4/3