give the demand curve of the monopolist Q=30-.3p and function:C=2Q²+20Q+10,find the profit maximizing level of output & price
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2nd order condition
when the mp of a given variable inpute is declining but positive
total product is declining?
To find the profit-maximizing level of output and price for a monopolist, we need to determine the quantity and price at which the marginal cost (MC) is equal to the marginal revenue (MR). Let's break down the steps.
Step 1: Calculate the marginal revenue (MR).
The marginal revenue (MR) can be derived from the demand curve. The formula for calculating marginal revenue is:
MR = ΔTR / ΔQ
Given the demand curve: Q = 30 - 0.3p
To find marginal revenue (MR), we need to differentiate the total revenue function (TR) with respect to quantity (Q). Total revenue (TR) can be calculated by multiplying price (p) by quantity (Q):
TR = p * Q
Differentiating TR with respect to Q, we get:
MR = d(TR) / d(Q)
= d(p * Q) / d(Q)
= p + Q * dp/dQ
Since we are given the demand curve as Q = 30 - 0.3p, we can substitute this into the equation above to calculate MR.
Step 2: Calculate the marginal cost (MC).
Given the cost function: C = 2Q² + 20Q + 10
To find the marginal cost (MC), we need to differentiate the cost function (C) with respect to quantity (Q):
MC = d(C) / d(Q)
Step 3: Set MR = MC and solve for quantity (Q).
By setting MR equal to MC, we can find the quantity at which the monopolist maximizes profit. So, solve the equation:
MR = MC
Find the value of Q that satisfies this equation.
Step 4: Calculate the price (p).
Once we have the quantity (Q) at the profit-maximizing level, we can substitute it back into the demand curve equation to calculate the corresponding price (p).
Now, let's execute these steps to find the profit-maximizing level of output and price for the given monopolist.
Given:
Demand curve: Q = 30 - 0.3p
Cost function: C = 2Q² + 20Q + 10
Step 1: Calculate the marginal revenue (MR).
Substitute Q = 30 - 0.3p into the MR equation:
MR = p + Q * dp/dQ
Step 2: Calculate the marginal cost (MC).
Differentiate the cost function (C) with respect to Q:
MC = d(C) / d(Q)
Step 3: Set MR = MC and solve for quantity (Q).
Equate MR and MC and solve the resulting equation to find the value of Q.
Step 4: Calculate the price (p).
Substitute the value of Q obtained in the previous step back into the demand curve equation to find the corresponding price (p).