Question

Закрытое акционерное общество является монополистом по производству спичек в Калужской области. Кривая спроса на спички описана уравнение P (Q)=100-0. 01Q. Функция общих издержек задана следующим образом:ТС(Q)=50Q+30000. Сколько Коробков спичек в день следует производить данному акционерному обществу? Какую цену за 1 коробки нужно установить? Какой будет дневная выручка?

Answer 1

Решение:

МС = dTC/dQ = TC’(Q) = (50Q + 30 000)' Q = 50.

Для определения предельной выручки нужно сначала найти функцию общей выручки:

TR(Q) = P(Q) *Q = (100 - 0,01Q) *Q = 100*Q - 0,01Q2.

Продифференцировав функцию общей выручки по Q, мы получаем функцию предельной выручки:

MR{Q) = dTR/dQ = TR'(Q) = (100 * Q - 0,01 * Q2 )’ Q = 100 - 0,02 * Q.

Используя условия максимизации прибыли, имеем:

MR (Q) = МС,

100- 0,02Q = 50,

100 - 50 = 0,02Q,

Q = 50/0,02 = 2500 коробков в день.

Определив оптимальный объем выпуска, предприятие-монополист должен установить цену на свою продукцию. Для этого нужно подставить в функцию спроса значение оптимального объема выпуска:

Р = 100 - 0,01*2500 = 100 - 25 = 75.

Следовательно, в целях максимизации прибыли необходимо назначить цену 75 коп. за один спичечный коробок.

Answer 2

Closed Joint-Stock Company as a Monopolist in Match Production in the Kaluga Region

To determine the optimal production level and price for the closed joint-stock company (CJSC) acting as a monopolist in match production in the Kaluga region, we need to analyze the demand curve for matches and the total cost function.

The demand curve for matches is described by the equation P(Q) = 100 - 0.01Q, where P represents the price per box of matches and Q represents the quantity of boxes demanded. The total cost function is given by TC(Q) = 50Q + 30000, where TC represents the total cost of production and Q represents the quantity of boxes produced.

Determining the Optimal Production Level

To determine the optimal production level for the CJSC, we need to find the quantity of boxes that maximizes the company's profit. Profit is calculated as the difference between total revenue (TR) and total cost (TC).

Total revenue is calculated by multiplying the price per box (P) by the quantity of boxes sold (Q). In this case, TR(Q) = P(Q) * Q.

To find the optimal production level, we need to find the quantity of boxes that maximizes the profit. Profit (π) is calculated as follows: π(Q) = TR(Q) - TC(Q).

To find the optimal production level, we can take the derivative of the profit function with respect to Q and set it equal to zero. This will give us the quantity of boxes that maximizes profit.

Let's calculate the optimal production level using the given equations:

Profit (π) = TR(Q) - TC(Q) TR(Q) = P(Q) * Q = (100 - 0.01Q) * Q = 100Q - 0.01Q^2 TC(Q) = 50Q + 30000

Profit (π) = 100Q - 0.01Q^2 - (50Q + 30000) Profit (π) = 50Q - 0.01Q^2 - 30000

To find the optimal production level, we take the derivative of the profit function with respect to Q and set it equal to zero:

dπ/dQ = 50 - 0.02Q = 0 50 - 0.02Q = 0 0.02Q = 50 Q = 2500

The optimal production level for the CJSC is 2500 boxes of matches per day.

Determining the Price per Box

To determine the price per box that the CJSC should set, we can substitute the optimal production level (Q = 2500) into the demand curve equation P(Q) = 100 - 0.01Q.

P(Q) = 100 - 0.01 * 2500 P(Q) = 100 - 25 P(Q) = 75

Therefore, the CJSC should set the price per box at 75 rubles.

Calculating the Daily Revenue

To calculate the daily revenue, we multiply the price per box (P) by the quantity of boxes sold (Q).

Daily Revenue = P * Q Daily Revenue = 75 * 2500 Daily Revenue = 187,500 rubles

Therefore, the CJSC will have a daily revenue of 187,500 rubles.

In summary: - The CJSC should produce 2500 boxes of matches per day. - The price per box should be set at 75 rubles. - The daily revenue will be 187,500 rubles.

Please note that the calculations provided are based on the given information and assumptions made.