Question

A monopolist produces two commodities 1 and 2 in quantities \(x_1\) and \(x_2\) at a constant average cost of Rs.2.50 and Rs.3.00 per item, respectively. If \(p_1\) and \(p_2\) stand for the prices charged and the market demands are: \[ x_1 = 5(p_2 - p_1) \quad \text{and} \quad x_2 = 32 + 5p_1 - 10p_2 \] The price \(p_1\) for commodity 1 at which the monopolist's total profit is maximised is (rounded off to two decimal places).

Hint:

To maximise the monopolist's profit, derive the profit function with respect to the price of the product and solve for the optimal price.

Answer 1

The monopolist's profit is given by: \[ \pi = \text{Total Revenue} - \text{Total Cost} \] Where:
- Total Revenue is \(TR_1 = p_1 \times x_1 + p_2 \times x_2\),
- Total Cost is \(TC = 2.5 \times x_1 + 3 \times x_2\).
Step 1: Substitute the demand functions into the Total Revenue and Total Cost equations: \[ TR_1 = p_1 \times (5(p_2 - p_1)) + p_2 \times (32 + 5p_1 - 10p_2) \] \[ TC = 2.5 \times 5(p_2 - p_1) + 3 \times (32 + 5p_1 - 10p_2) \] Step 2: Simplify the expressions for \(TR_1\) and \(TC\): \[ TR_1 = 5p_1(p_2 - p_1) + p_2(32 + 5p_1 - 10p_2) \] \[ TC = 12.5(p_2 - p_1) + 3(32 + 5p_1 - 10p_2) \] Step 3: Differentiate the profit function with respect to \(p_1\) to find the price that maximises profit. Set the derivative equal to zero and solve for \(p_1\).
Final Answer: \[ \boxed{p_1 = 5.10} \]