given the demand curve of the monopolist:Q=30-0.3p,and given the cost function:2Q2+20q+10,then find profit maximizing level of out put and price,determine the maximum possible profit,check the 2nd order condition
To find the profit-maximizing level of output and price for a monopolist, we need to follow these steps:
Step 1: Determine the monopolist's marginal revenue (MR) curve. The MR curve is derived from the demand curve by recognizing that the monopolist faces a downward-sloping demand curve. In this case, the demand curve is Q = 30 - 0.3p. To derive the MR curve, we take the derivative of the demand curve with respect to quantity (Q) and multiply it by Q. This gives us MR = 30 - 0.6Q.
Step 2: Set the monopolist's MR equal to its marginal cost (MC) to determine the profit-maximizing level of output. The MC is the derivative of the cost function. In this case, the cost function is C(Q) = 2Q^2 + 20Q + 10. Taking the derivative, we get MC = 4Q + 20.
Setting MR equal to MC: 30 - 0.6Q = 4Q + 20
Step 3: Solve the equation to find the profit-maximizing quantity (Q).
30 + 20 = 0.6Q + 4Q
50 = 4.6Q
Q = 50/4.6 ≈ 10.87
The profit-maximizing level of output is approximately 10.87.
Step 4: Use the demand curve to find the corresponding price. Substitute the value of Q into the original demand equation Q = 30 - 0.3p.
10.87 = 30 - 0.3p
Rearranging the equation to solve for p:
0.3p = 30 - 10.87
0.3p = 19.13
p = 19.13/0.3 ≈ 63.77
The profit-maximizing price is approximately 63.77.
Step 5: Calculate the maximum possible profit. Revenue (R) is the product of quantity (Q) and price (p). Cost (C) is obtained by substituting the value of Q into the cost function.
R = Q * p
R = 10.87 * 63.77 ≈ 691.26
C = 2Q^2 + 20Q + 10
C = 2(10.87)^2 + 20(10.87) + 10 ≈ 534.06
Profit (π) is given by the equation: π = R - C = 691.26 - 534.06 ≈ 157.20
The maximum possible profit is approximately 157.20.
Step 6: Check the second-order condition. The second-order condition ensures that the profit-maximizing solution is a maximum rather than a minimum. To check this condition, we take the second derivative of the profit (π) with respect to quantity (Q).
Taking the second derivative of the profit function:
d^2π/dQ^2 = d^2(R - C) / dQ^2
If the result is negative, it means the solution is a maximum. Therefore, we need to evaluate the second derivative at the profit-maximizing quantity.
Differentiating R with respect to Q:
dR/dQ = dp/dQ * Q + p
Differentiating C with respect to Q:
dC/dQ = 4Q + 20
Taking the second derivative of the profit function:
d^2π/dQ^2 = (dp/dQ + Q * d^2p/dQ^2 + dR/dQ) - (d^2C/dQ^2)
Substituting the equations:
d^2π/dQ^2 = (0 + Q * 0 + (dp/dQ * Q + p)) - (4)
At the profit-maximizing quantity (Q = 10.87) and the derived price (p ≈ 63.77), we can substitute these values into the second derivative equation and evaluate it. If the result is negative, the second-order condition is satisfied.
d^2π/dQ^2 = (0 + 10.87 * 0 + (0 * 10.87 + 63.77)) - 4
d^2π/dQ^2 ≈ (0 + 0 + 63.77) - 4
Since the result is positive, the second-order condition is not satisfied.
Therefore, the profit-maximizing level of output is approximately 10.87, the price is approximately 63.77, the maximum possible profit is approximately 157.20, and the second-order condition is not satisfied.