Question

The inverse demand function for a monopolist is given by \( P = 100 - kQ \), where \( P \) is the unit price of the good, \( Q \) is the quantity and \( k \) is a constant. The cost function facing the monopolist is given as \( C(Q) = 50 + 2Q(1 + Q) \). If the profit maximizing output is 7, the maximum profit is ________________ (in integer).

Hint:

In monopoly problems, first express profit in terms of output \( Q \), then apply the first-order condition \( \frac{d\pi}{dQ} = 0 \) to find unknown constants, if required.

Answer 1

Step 1: Compute Total Revenue (TR).
The price function is given as:
\[ P = 100 - kQ \] So, Total Revenue (TR) is:
\[ TR = P \cdot Q = (100 - kQ)Q = 100Q - kQ^2 \] Step 2: Compute Total Cost (TC).
Given:
\[ C(Q) = 50 + 2Q(1 + Q) = 50 + 2Q + 2Q^2 \] Step 3: Profit function.
\[ \pi(Q) = TR - TC = (100Q - kQ^2) - (50 + 2Q + 2Q^2) \] Step 4: Use given output to find \( k \).
Given: profit-maximizing output is \( Q = 7 \)
Now plug \( Q = 7 \) into \( \frac{d\pi}{dQ} = 0 \) to find \( k \).
First, find \( \frac{d\pi}{dQ} \):
\[ \frac{d\pi}{dQ} = \frac{d}{dQ}(100Q - kQ^2 - 50 - 2Q - 2Q^2) = 100 - 2kQ - 2 - 4Q \] Set this to zero for \( Q = 7 \):
\[ 100 - 2k(7) - 2 - 4(7) = 0 \] \[ 100 - 14k - 2 - 28 = 0 \Rightarrow 70 = 14k \Rightarrow k = 5 \] Step 5: Now calculate profit.
Substitute \( k = 5 \) and \( Q = 7 \):
Price:
\[ P = 100 - 5 \cdot 7 = 65 \Rightarrow TR = P \cdot Q = 65 \cdot 7 = 455 \] Total Cost:
\[ TC = 50 + 2(7)(1 + 7) = 50 + 2(7)(8) = 50 + 112 = 162 \] Step 6: Compute profit.
\[ \pi = TR - TC = 455 - 162 = \boxed{293} \]