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?
To find the firms' outputs in equilibrium, we need to determine the profit-maximizing output levels for both firms. In equilibrium, both firms will produce the quantity that maximizes their profits.
First, let's find the profit function for each firm. Profit is calculated as total revenue minus total cost.
For the first firm:
Total revenue (TR1) = y1 * P
Total cost (TC1) = 20 + 20y1
Profit (π1) = TR1 - TC1
For the second firm:
Total revenue (TR2) = y2 * P
Total cost (TC2) = 10 + 5y2 + y2^2
Profit (π2) = TR2 - TC2
Now, we need to find the price at which the firms will operate. To do this, we can equate the quantity demanded (Q) to the sum of the outputs produced by both firms (y1 + y2) and solve for price (P).
Q = y1 + y2
120 - 3P = y1 + y2
Now, let's substitute the profit functions into the price equation to express profit purely in terms of outputs and price:
π1 = (y1 * P) - (20 + 20y1)
π2 = (y2 * P) - (10 + 5y2 + y2^2)
Substituting π1 and π2 into the price equation:
120 - 3P = (y1 * P) - (20 + 20y1) + (y2 * P) - (10 + 5y2 + y2^2)
Simplifying the equation:
120 - 3P = y1P - 20 - 20y1 + y2P - 10 - 5y2 - y2^2
Since we want to find the equilibrium outputs, we need to maximize the combined profits of both firms. To do this, we maximize the sum of their profit functions:
Maximize: π1 + π2 = (y1 * P) - (20 + 20y1) + (y2 * P) - (10 + 5y2 + y2^2)
To find the equilibrium outputs, we need to take the first-order condition by differentiating the profit function with respect to each firm's output and setting it equal to zero.
For Firm 1:
d(π1 + π2)/dy1 = P - 20 = 0
For Firm 2:
d(π1 + π2)/dy2 = P - 5 - 2y2 = 0
Solving these equations simultaneously:
P - 20 = 0 => P = 20
P - 5 - 2y2 = 0 => P = 5 + 2y2
Setting both P equal to each other:
20 = 5 + 2y2
Solving for y2:
2y2 = 15 => y2 = 7.5
Substituting y2 back into the equation P = 5 + 2y2:
P = 5 + 2(7.5) = 20
Therefore, in equilibrium, the firms' outputs are y1 = 20 and y2 = 7.5.
To find the firms' outputs in equilibrium, we need to determine the point at which their total cost is equal to the market demand.
Let's start by determining the profit maximization condition for each firm. Each firm aims to maximize its profit by setting its output level where marginal cost (MC) equals marginal revenue (MR). In a perfectly competitive market, the marginal revenue is equal to the market price.
The price in this market can be obtained by solving the demand function for P. The demand function is given as Q = 120 - 3P, where Q is the combined output of the two firms.
First, let's find the total cost function by summing up the costs of the two firms:
TC_total = TC(y1) + TC(y2)
Substituting the cost functions, we have:
TC_total = 20 + 20y1 + 10 + 5y2 + y2^2
Simplifying, we get:
TC_total = 30 + 20y1 + 5y2 + y2^2
Next, let's determine the marginal cost (MC) for each firm by differentiating their cost functions with respect to their respective output levels.
MC1 = d(TC(y1))/dy1 = 20
MC2 = d(TC(y2))/dy2 = 5 + 2y2
Now, let's find the market price (P) by substituting the total output (Q) into the demand function:
Q = y1 + y2
120 - 3P = y1 + y2
Rearranging the equation, we have:
3P = 120 - y1 - y2
P = (120 - y1 - y2)/3
Now, we can set up the profit maximization condition for each firm:
MC1 = MR and MC2 = MR
Substituting the market price (P) as the marginal revenue (MR), we have:
20 = P and 5 + 2y2 = P
Substituting the value of P into the second equation, we get:
5 + 2y2 = 20
Simplifying, we find:
2y2 = 15
y2 = 7.5
Substituting the value of y2 into the first MC equation, we get:
MC1 = 20 = P
20 = (120 - y1 - 7.5)/3
Simplifying and rearranging the equation, we have:
60 = 120 - y1 - 7.5
y1 = 52.5
Therefore, in equilibrium, the first firm's output (y1) is 52.5, and the second firm's output (y2) is 7.5.