A competitive firm faces a total cost function given by: TC = 0.5Q² + 10Q + 50 If the market price is P = 30, what is the profit-maximising output Q*, and what is the maximum profit earned by the firm?

For a profit-maximising firm under perfect competition, the equilibrium condition is: P = MC

The marginal cost is obtained by differentiating total cost with respect to output: MC = d(TC)/dQ = Q + 10

Equating marginal cost to the market price:

Q + 10 = 30

Therefore:

Q* = 20

Now calculate total revenue and total cost at Q* = 20.

TR = P × Q

= 30 × 20

= 600

TC = 0.5(20)² + 10(20) + 50

= 200 + 200 + 50

= 450

Profit = TR − TC

= 600 − 450

= ₹150

Therefore, the profit-maximising output is 20 units and the maximum profit is ₹150.