Question

An industry consists of two (perfectly) firms. Firm 1 has a total cost function given by ??1(?1)=?1 +(?1)^2

while firm 2 has a total cost function given by ??2(?2)=3*?2+(1/2)*(?2)^2 .

1. (a) Let ? denote the (exogenous) price at which each firm can sell its output. Write down each firm’s profit-maximization problem and the associated first-order conditions (FOCs).

2. (b) Derive the firms’ supply functions ?∗(?) and ?∗(?) and verify that these functions are

   linearly increasing in ?.

3. (c) Derive the industry supply curve ?(?). [Hint: Draw a picture and remember the notion of horizontal summation. You should demonstrate that the industry supply curve is a piecewise function in ?]

4. Again assuming that the firms act as price takers, find the industry equilibrium when the industry demand curve is given by ??(?)=(9/2)-(1/2)p .[Hint: It may be useful to add the relevant to the graph considered in part (c)]

5. (e) Calculate the output and profit of each firm under the equilibrium characterized in part (d).

Answer 1

There are two firms: Firm 1 and Firm 2

The total cost of Firm 1 is

The total cost of Firm 2 is

(a) Let price be P_i for both firms

The profit maximisation problem for Firm 1 is

The total revenue of Firm 1 is

So, the profit maximisation problem for Firm 1 is

The FOC is

The profit maximisation of Firm 2 is

The total revenue of Firm 2 is

So, the profit maximisation problem for Firm 1 is

The FOC of Firm 2 is

(b) For deriving the supply curve for the individual firms we have to equate price and its marginal cost.

The supply curve of firm 1 is

To calculate the MC we have to differentiate the total cost of firm 1.

Hence we can say that the supply curve of Firm1 is

The supply curve of firm 2 is

To calculate the MC we have to differentiate the total cost of firm 2.

Hence we can say that the supply curve of Firm 2 is

(c) The industry supply curve is the horizontal summation of the two individual firms.

We have the supply curve as, when q₁ = q₂ = q

We have the demand function

(d) To find out the industry equilibrium we have to equate the industry demand and industry supply curve

The diagram is given below

The above diagram shows the industry supply curve and demand curve which equated together brings out the industr equilibrium ath point E.

(e) Putting p in the demand curve we get the quantity

The total revenue function is

The profit function is