Assume a monopolist with the following:
a. Qd = 100 – 10p
b. TC = 1 + 2Q
Find the following:
Price at profit max
Quantity at profit max
TR at profit max
TC at profit max
Profit
To find the price, quantity, total revenue, total cost, and profit at profit maximization for the given monopolist, we need to follow these steps:
1. Determine the monopolist's marginal revenue (MR) curve, which is the derivative of the total revenue (TR) curve. MR is the extra revenue generated by selling one additional unit of output. In the case of a monopolist, the MR curve is twice as steep as the demand curve (Qd), as the monopolist faces the entire market demand.
Qd = 100 - 10p
To find MR, we need to take the derivative of the demand curve with respect to quantity (Q).
dQd/dQ = -10
2. Calculate the profit-maximizing quantity by equating MR to marginal cost (MC).
MR = MC
Given that the total cost (TC) is 1 + 2Q, the marginal cost (MC) is the derivative of the total cost with respect to quantity.
MC = dTC/dQ = 2
Therefore, set MR equal to MC:
-10 = 2
Solve for the profit-maximizing quantity (Q).
3. Calculate the price (P) at the profit-maximizing quantity by substituting Q into the demand function (Qd).
Qd = 100 - 10p
Substitute Q and solve for P:
Q = 100 - 10p
4. Calculate the total revenue (TR) at the profit-maximizing quantity by multiplying the price by the quantity (P * Q).
TR = P * Q
5. Calculate the total cost (TC) at the profit-maximizing quantity by substituting Q into the total cost function (TC).
TC = 1 + 2Q
6. Calculate the profit by subtracting the total cost from the total revenue (Profit = TR - TC).
Now, let's solve these equations step by step:
Step 1: Calculate MR
dQd/dQ = -10 (since the derivative of p with respect to Q is -10)
MR = 2 * -10 = -20
Step 2: Set MR = MC
-20 = 2
-20 + 2 = 0 = -Q
Q = 10
Step 3: Calculate P
Qd = 100 - 10p
10 = 100 - 10p
10p = 100 - 10
10p = 90
p = 9
Step 4: Calculate TR
TR = P * Q
TR = 9 * 10
TR = 90
Step 5: Calculate TC
TC = 1 + 2Q
TC = 1 + 2 * 10
TC = 1 + 20
TC = 21
Step 6: Calculate Profit
Profit = TR - TC
Profit = 90 - 21
Profit = 69
Therefore, the answers to the questions are:
- Price at profit max = $9
- Quantity at profit max = 10 units
- TR at profit max = $90
- TC at profit max = $21
- Profit = $69