# Profit Maximisation for a Monopolist

Suppose a monopolist faces the following inverse demand function : $$p = exp(-Q)$$ The monopolist can produce any positive level of output with zero variable cost. Its fixed cost is $$F$$. Find the profit-maximising output for two cases (i) when $$F = 0.1$$ and (ii) when $$F= 0.5$$. I tried solving it using, $$MR=MC$$, the latter in this case is zero, so in both the cases I am getting 1 as the answer, and I am not sure if it is correct!

• Your answer is correct. The profit-maximizing price and quantity are independent of fixed cost as fixed cost does not appear in the first-order condition. In the former case, profit is positve, while the latter corresponds to negative profit. – dlnB Mar 19 '19 at 19:40
• Can you describe the cost function? Is it $C(q) = F$ OR $C(q) = \begin{cases} F & \text{if } Q > 0 \\ 0 & \text{if } Q = 0 \end{cases}$ ? – Amit Mar 19 '19 at 22:25
• @Amit they haven't mentioned the cost function specifically, but since the variable cost is zero I think the former is the right interpretation! – Dr. Dre Mar 20 '19 at 6:34
• @dlnB how did you arrive at the conclusion that the profit is negative in the second case without plotting the graph? – Dr. Dre Mar 20 '19 at 6:36