An industry has only two firms producing outputs y1 and y2, respectively. The first firm has a
cost function of TC(y1) = 20 + 20y1 and the second has a cost function TC(y2) = 10 + 5y2 + y22. The demand function for the product these firms make is Q = 120 - 3P, where Q is the
total output of the two firms. What are the firms' outputs in equilibrium?
In equilibrium, the firms will produce outputs such that the market price is equal to the marginal cost of production. The market price is given by the inverse demand function, P = 40 - (1/3)Q. The marginal cost of production for the first firm is MC1 = 20 and for the second firm is MC2 = 5 + 2y2.
Setting MC1 = MC2 = P, we get:
20 = 5 + 2y2
y2 = 7.5
Substituting this value of y2 into the demand function, we get:
Q = 120 - 3(40 - (1/3)Q)
Q = 90
Therefore, the output of the first firm is y1 = 45 and the output of the second firm is y2 = 7.5.
To find the equilibrium outputs of the two firms, we need to determine the point where their total cost equals the total revenue.
First, let's calculate the total cost for each firm:
Total cost for Firm 1, TC(y1) = 20 + 20y1
Total cost for Firm 2, TC(y2) = 10 + 5y2 + y2^2
The total revenue is given by the demand function:
Total revenue, TR = PQ = P(Q) = P(120 - 3P) = 120P - 3P^2
In equilibrium, TR should be equal to the total cost. Therefore, we equate TR to TC1 + TC2:
TR = TC1 + TC2
120P - 3P^2 = 20 + 20y1 + 10 + 5y2 + y2^2
Since the outputs for the firms are denoted as y1 and y2, we can rewrite the equation as:
120P - 3P^2 = 20 + 20y1 + 10 + 5y2 + y2^2
Now, let's solve for y1 and y2 by differentiating the equation with respect to y1 and y2, respectively:
∂TR/∂y1 = 20
∂TR/∂y2 = 5 + 2y2
Setting these derivatives equal to zero gives us:
20 = 0 => y1 = 0
5 + 2y2 = 0 => y2 = -5/2
Since the output cannot be negative, we discard the solution y2 = -5/2.
Therefore, in equilibrium, the first firm's output is y1 = 0, and the second firm's output is y2 = 0.
To find the firms' outputs in equilibrium, we need to solve for the optimal outputs where their costs are minimized and the total output meets the market demand.
First, let's find the marginal cost (MC) for each firm by taking the derivative of their respective cost functions with respect to their output:
For the first firm:
MC1 = d(TC1(y1))/dy1 = 20
For the second firm:
MC2 = d(TC2(y2))/dy2 = 5 + 2y2
Next, we need to find the market price. The market price (P) can be determined by setting the total market quantity (Q) equal to the sum of the firms' outputs and solving for P:
Q = y1 + y2
120 - 3P = y1 + y2
Now, substitute the cost and output functions into the market price equation and solve for P:
120 - 3P = (20 + 20y1) + (10 + 5y2 + y2^2)
120 - 3P = 30 + 20y1 + 10 + 5y2 + y2^2
110 - 3P = 20y1 + 5y2 + y2^2
Next, let's differentiate the market price equation with respect to both y1 and y2:
d(110 - 3P)/dy1 = 20
d(110 - 3P)/dy2 = 5 + 2y2
Since we are looking for an equilibrium point, the firms' outputs must satisfy the following conditions:
1. MC1 = d(110 - 3P)/dy1
2. MC2 = d(110 - 3P)/dy2
Now, substitute the expressions for MC1, MC2, and d(110 - 3P)/dy1, d(110 - 3P)/dy2 into the conditions and solve for y1 and y2:
20 = 20
5 + 2y2 = 20
From the second equation, we can solve for y2:
2y2 = 15
y2 = 7.5
Substituting the value of y2 into the first equation, we have:
20 = 20
Therefore, the firms' outputs in equilibrium are y1 = 0 and y2 = 7.5.