Monopolist facing linear demand function p = 1000+40q and total cost function c = 500+26q.determine optimal price output and profit perfect profit
To determine the optimal price, output, and profit for a monopolist firm, we need to find the point where the firm maximizes its profits. This can be done by calculating the derivative of the profit function with respect to the quantity (q) and setting it equal to zero.
Let's start by finding the profit function. Profit is calculated as total revenue minus total cost.
Total Revenue (TR) is determined by multiplying the unit price (p) by the quantity (q).
TR = p * q
Total Cost (TC) is given by the cost function (c).
TC = 500 + 26q
Profit (Π) can be calculated as follows:
Π = TR - TC
Now, substitute the given linear demand function p = 1000 + 40q into the profit function Π to get an equation in terms of q only.
TR = (1000 + 40q) * q
TC = 500 + 26q
Π = (1000 + 40q) * q - (500 + 26q)
Expanding and simplifying, we get:
Π = 1000q + 40q^2 - 500 - 26q
Now, take the derivative of the profit function with respect to q, and set it equal to zero to find the value of q that maximizes profit.
dΠ / dq = 1000 + 80q - 26
Setting it equal to zero:
1000 + 80q - 26 = 0
Solve for q:
80q = 26 - 1000
80q = -974
q = -974 / 80
q = -12.175
Since the quantity cannot be negative, we discard the negative solution. Therefore, the optimal output quantity is 0 units. This implies that the monopolist should not produce any goods to maximize their profit.
To find the optimal price, substitute the found output quantity (q = 0) into the demand function:
p = 1000 + 40q
p = 1000 + 40(0)
p = 1000
Thus, the optimal price for the monopolist is $1000 per unit.
To calculate the perfect profit, substitute the optimal values (q = 0 and p = 1000) into the profit function Π:
Π = (1000 + 40q) * q - (500 + 26q)
Π = (1000 + 40(0)) * 0 - (500 + 26(0))
Π = 0 - 500
Π = -$500
Therefore, the perfect profit for the monopolist in this scenario is -$500.