the monopolist face the denand function and cost function as Q=10p^3 and Tc=2Q Determin Short Run Obtimal Outpt Price and profit
To find the short-run optimal output and price, as well as the profit for a monopolist, we need to determine the level of output where the marginal cost equals the marginal revenue.
First, let's find the marginal cost (MC) and marginal revenue (MR) functions. The total cost function (TC) given is Tc = 2Q, so the marginal cost (MC) is the derivative of the total cost function with respect to quantity (Q). Taking the derivative of Tc = 2Q gives us MC = 2.
To find the marginal revenue (MR), we need to express the demand function in quantity terms. The demand function given is Q = 10p^3. To express it in quantity terms, we can solve for p.
Rearranging the equation, we get p = Q^(1/3) / 10.
The marginal revenue (MR) is the derivative of the revenue function with respect to quantity (Q). The revenue (R) function is given by multiplying the price (p) by the quantity (Q).
R = p * Q
R = (Q^(1/3) / 10) * Q
R = Q^(4/3) / 10
To find the marginal revenue (MR), we take the derivative of the revenue function with respect to quantity (Q).
MR = dR/dQ
MR = (4/3) Q^(1/3) / 10
Now that we have the marginal cost (MC) and marginal revenue (MR) functions, we can find the short-run optimal output level where MC equals MR.
Setting MC equal to MR, we get:
2 = (4/3) Q^(1/3) / 10
To solve for Q, we can multiply both sides by 30:
60 = (4/3) Q^(1/3)
Now, we can solve for Q by raising both sides to the power of 3/4:
(Q^(1/3))^3/4 = (60)^3/4
Simplifying further:
Q^(3/4) = (60)^3/4
Taking the fourth power of both sides:
Q = 60^3/4
Now, substitute the value of Q back into the demand function to find the corresponding price (p):
p = (Q)^(1/3) / 10
p = (60^3/4)^(1/3) / 10
Finally, to calculate the short-run profit, we need to calculate the total revenue (TR) and total cost (TC) at the optimal quantity and price values.
TR = p * Q
Substituting the values of p and Q:
TR = [(60^3/4)^(1/3) / 10] * (60^3/4)
TC = 2Q
Substituting the value of Q:
TC = 2 * (60^3/4)
The profit (ฯ) is given by:
ฯ = TR - TC
Substituting the values of TR and TC:
ฯ = [(60^3/4)^(1/3) / 10] * (60^3/4) - 2 * (60^3/4)
Calculating these expressions will give you the short-run optimal output, price, and profit for the given monopolist.