1. Given demand curve of the monopolist :Q=30-0.3P,& given the cost function C=2Q2+20Q+10,
a)find the profit maximizing level of output &price
b)determine the max possible profit.
c)check for the 2nd order condition.
please I don't get the answers
To find the profit-maximizing level of output and price for a monopolist, we need to consider the demand curve and cost function. The monopolist aims to maximize profits, which occurs when marginal revenue (MR) equals marginal cost (MC). Here's how you can calculate the answers to each question:
a) Determine the profit-maximizing level of output and price:
Step 1: Start with the demand curve: Q = 30 - 0.3P
Step 2: Rewrite the demand equation in terms of price: P = (30 - Q) / 0.3
Step 3: Calculate the revenue function (R) by multiplying price (P) by quantity (Q): R = P * Q
Step 4: Rewrite the revenue equation using the demand equation: R = [(30 - Q) / 0.3] * Q
Step 5: Calculate marginal revenue (MR) by differentiating the revenue equation with respect to quantity (Q): MR = dR/dQ
Step 6: Set MR equal to marginal cost (MC) to find the profit-maximizing level of output: MR = MC
Step 7: Differentiate the cost function (C) with respect to quantity (Q) to find marginal cost (MC): MC = dC/dQ
Step 8: Solve the equation MR = MC to find the level of output (Q).
Step 9: Substitute the value of Q into the demand equation to find the corresponding price (P).
b) Determine the maximum possible profit:
Step 1: Calculate the total revenue (TR) by multiplying the price (P) by the quantity (Q): TR = P * Q
Step 2: Calculate the total cost (TC) by substituting the value of Q into the cost function (C): TC = 2Q^2 + 20Q + 10
Step 3: Calculate profit (π) by subtracting total cost (TC) from total revenue (TR): π = TR - TC
Step 4: Substitute the values of Q and P into the profit equation to find the maximum possible profit.
c) Check the second-order condition:
Step 1: Differentiate the marginal revenue (MR) equation with respect to quantity (Q) to find the second derivative: MR' = d^2R/dQ^2
Step 2: Determine the sign of the second derivative to check for concavity or convexity. If MR' < 0, it indicates concavity, which means the monopolist's profit is maximized at that output level.