For a monopolistโs product, the demand function is ๐ = 72 โ 0.04๐ and the cost
function is ๐ = 500 + 30๐. At what level of output will profit be maximized? At
what price does this occur and what is the profit?
To find the level of output at which profit is maximized, we need to determine the quantity at which marginal revenue (MR) equals marginal cost (MC). Profit is maximized when MR = MC because this is the point where the additional revenue from producing one more unit (MR) is equal to the additional cost of producing one more unit (MC).
1. First, let's calculate the marginal revenue (MR) function. The monopolist's demand function is given as p = 72 - 0.04q, where p is the price and q is the quantity. MR is the change in total revenue resulting from a one-unit change in quantity (q). Since total revenue (TR) is given as p * q, we can calculate MR as the derivative of TR with respect to q.
TR = p * q = (72 - 0.04q) * q = 72q - 0.04q^2
MR = dTR/dq = 72 - 0.08q
2. Next, let's calculate the marginal cost (MC). The cost function is given as c = 500 + 30q. MC is the change in total cost resulting from a one-unit change in quantity (q). Since total cost (TC) is given as c * q, we can calculate MC as the derivative of TC with respect to q.
TC = c * q = (500 + 30q) * q = 500q + 30q^2
MC = dTC/dq = 500 + 60q
3. Now, we can set MR equal to MC to find the level of output at which profit is maximized.
72 - 0.08q = 500 + 60q
Rearranging the equation:
0.08q + 60q = 500 - 72
60.08q = 428
q โ 7.13
Therefore, the level of output at which profit is maximized is approximately 7.13 units.
4. To find the price at which this occurs, substitute the value of q back into the demand function.
p = 72 - 0.04q
p = 72 - 0.04(7.13)
p โ 71.71
Therefore, the price at which profit is maximized is approximately $71.71.
5. To calculate the profit at this output level, subtract the total cost (TC) from the total revenue (TR).
TR = p * q = 71.71 * 7.13
TR โ $511.97
TC = c * q = (500 + 30q) * q = (500 + 30 * 7.13) * 7.13
TC โ $1338.88
Profit = TR - TC = $511.97 - $1338.88
Profit โ -$826.91
Therefore, at the level of output where profit is maximized (approximately 7.13 units), the price is approximately $71.71, and the profit is approximately -$826.91 (a negative profit indicates a loss).