Question

Фирма с функцией общих затрат TC = 80 + 6Q + Q2 может продать любое количество своей продукции по цене Р = 22. Рассчитайте максимальную величину прибыли производителя.

Answer 1

Решение:

МС =(TC)'= 2Q + 6

Цена по условию равна Р = 22.

Р = МС

22 = 2Q + 6

2Q = 22-6

Q = 8

Answer 2

Calculation of Maximum Profit for the Producer

To calculate the maximum profit for the producer, we need to consider the total cost (TC) function, the selling price (P), and the quantity of production (Q).

The given total cost function is: TC = 80 + 6Q + Q^2.

The selling price is given as: P = 22.

To calculate the maximum profit, we need to find the quantity of production (Q) that maximizes the difference between the selling price and the total cost.

To find this quantity, we can take the derivative of the total cost function with respect to Q and set it equal to zero. This will give us the critical point where the maximum profit occurs.

Let's calculate the derivative of the total cost function:

d(TC)/dQ = 6 + 2Q.

Setting this derivative equal to zero and solving for Q:

6 + 2Q = 0, 2Q = -6, Q = -3.

Since the quantity of production cannot be negative, we discard the negative solution.

Therefore, the critical point is Q = -3.

To determine if this critical point is a maximum or minimum, we can take the second derivative of the total cost function and evaluate it at the critical point.

Let's calculate the second derivative:

d^2(TC)/dQ^2 = 2.

Since the second derivative is positive (2), the critical point Q = -3 corresponds to a minimum.

Since we are looking for the maximum profit, we need to consider the endpoints of the feasible range. In this case, the producer can sell any quantity of their product.

Therefore, there are no constraints on the quantity of production, and the maximum profit occurs at the critical point Q = -3.

To calculate the maximum profit, we can substitute the critical point into the total cost function and subtract it from the total revenue.

Total revenue = P * Q = 22 * (-3) = -66.

Total cost = TC = 80 + 6Q + Q^2 = 80 + 6(-3) + (-3)^2 = 80 - 18 + 9 = 71.

Maximum profit = Total revenue - Total cost = -66 - 71 = -137.

The maximum profit for the producer is -137.

Please note that the negative value indicates a loss rather than a profit. This means that the producer will incur a loss of 137 units of currency.