Question

Обратная функция спроса Р(q) = 650 - 5q, функция издержек в краткосрочном периоде ТС = 100 – 10q + q2 Определите: а) равновесные цены и объем производства монополистической отрасли в краткосрочном периоде; б) прибыль производителя.

Answer 1

Решение:
Ценовая максимизация прибыли при монополии имеет вид MC=MR.
MR=TR=Р*Q = (650 - 5q)q = 650q – q2 = 650 - 10q
MC=TC= 100 – 10q + q2 = -10+2q  
650 - 10q = -10+2q
12q=660
Q=55 ед.
Рмон. = 650-5*55 = 375 руб.
П = TR- ТС(q)= Р*Q - ТС(q)= 375*55 – (100-10*55+(55)2 = 20625-2575 = 18050 руб. Ответ: монополист будет получать максимальную прибыль 18050 руб. при выпуске объема 55 единиц и при цене 375 руб.

Answer 2

Equilibrium Prices and Production Volume in the Short Run for a Monopolistic Industry

To determine the equilibrium prices and production volume in the short run for a monopolistic industry, we need to analyze the demand and cost functions provided.

The demand function is given as: R(q) = 650 - 5q [[1]]. The cost function in the short run is given as: TC = 100 - 10q + q^2 [[1]].

a) Equilibrium Prices and Production Volume:

In a monopolistic industry, the equilibrium occurs when the marginal revenue (MR) equals the marginal cost (MC). To find the equilibrium, we need to calculate the marginal revenue and marginal cost.

The marginal revenue (MR) is the derivative of the revenue function (R(q)) with respect to quantity (q). Taking the derivative of R(q) = 650 - 5q, we get:

MR = dR(q)/dq = -5

The marginal cost (MC) is the derivative of the cost function (TC) with respect to quantity (q). Taking the derivative of TC = 100 - 10q + q^2, we get:

MC = dTC(q)/dq = -10 + 2q

Setting MR equal to MC, we have:

-5 = -10 + 2q

Simplifying the equation, we find:

2q = 5

q = 2.5

Therefore, the equilibrium production volume in the short run for the monopolistic industry is 2.5 units.

To find the equilibrium price, we substitute the equilibrium production volume (q = 2.5) into the demand function:

R(q) = 650 - 5q

R(2.5) = 650 - 5(2.5)

R(2.5) = 650 - 12.5

R(2.5) = 637.5

Therefore, the equilibrium price in the short run for the monopolistic industry is $637.50.

b) Producer's Profit:

To calculate the producer's profit, we need to subtract the total cost (TC) from the total revenue (TR).

The total revenue (TR) is the product of the equilibrium price and the equilibrium production volume:

TR = R(q) * q

TR = (650 - 5q) * q

TR = (650 - 5(2.5)) * 2.5

TR = 637.5 * 2.5

TR = 1593.75

The total cost (TC) is given by the cost function:

TC = 100 - 10q + q^2

TC = 100 - 10(2.5) + (2.5)^2

TC = 100 - 25 + 6.25

TC = 81.25

Therefore, the producer's profit in the short run for the monopolistic industry is:

Profit = TR - TC = 1593.75 - 81.25 = $1512.50.

In summary, in the short run for a monopolistic industry with the given demand and cost functions, the equilibrium price is $637.50, the equilibrium production volume is 2.5 units, and the producer's profit is $1512.50.

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