A firm has an inverse demand function P = 30 – 2Q. It has a fixed cost of 50, and a per unit variable cost of 5. This firm breaks even at the output level(s):
a.
Q = 0
b.
Q = 15 and Q =50
c.
Q = 15 and Q = 30
d.
Q = 2.5 and Q = 10
e.
Q = 55
To determine the output level(s) at which the firm breaks even, we need to find the point(s) where the firm's total revenue equals its total cost.
The firm's total revenue can be calculated by multiplying the price (P) by the quantity (Q). The price is determined by the inverse demand function given as P = 30 - 2Q.
The firm's total cost consists of the sum of the fixed cost and the variable cost per unit multiplied by the quantity. The fixed cost is given as 50, and the variable cost per unit is 5.
Setting the total revenue equal to the total cost, we can form the equation:
Total Revenue = Total Cost
(P * Q) = (Fixed Cost + Variable Cost per unit * Q)
Substituting the values, we get:
(30 - 2Q) * Q = 50 + 5Q
Expanding and rearranging the equation, we have:
30Q - 2Q^2 = 50 + 5Q
2Q^2 + 25Q - 50 = 0
Now we need to solve this quadratic equation to find the output level(s) where the firm breaks even. We can do this using factoring or the quadratic formula.
Factoring the quadratic equation, we get:
2(Q + 5)(Q - 5) = 0
Setting each factor equal to zero and solving for Q, we find:
Q + 5 = 0 => Q = -5
Q - 5 = 0 => Q = 5
Since negative quantities do not make sense in this context, we ignore the solution Q = -5.
Therefore, the firm breaks even at the output level(s) Q = 5.
Therefore, the correct answer is (e) Q = 5.