Q.A firm operating under perfect competition has the short-run total cost function $TC = Q^{2} + 5Q + 50$. The prevailing market price is Rs. 45 per unit. Find the firm's equilibrium output, its total revenue, total cost, and profit at that output, and state whether the firm should continue producing in the short run.

A perfectly competitive firm produces where P=MC; derive MC from TC, set it equal to the given price, then compute TR, TC and profit at that output.

$MC=\dfrac{d(TC)}{dQ}=2Q+5$. Setting $P=MC$: $45=2Q+5 \Rightarrow Q=20$. $TR=45\times20=900$. $TC=20^{2}+5(20)+50=400+100+50=550$. Profit $=900-550=350$. $AVC=(TC-TFC)/Q=(400+100)/20=25$; since $P(45)>AVC(25)$, the firm continues producing.

Under perfect competition, a firm's equilibrium condition is $P = MC$.

From $TC=Q^{2}+5Q+50$, Marginal Cost is:

Setting $P=MC$ with $P=45$:

Total Revenue at this output:

Total Cost at this output:

Profit:

Since Total Fixed Cost here is Rs. 50 (the constant term in TC), Total Variable Cost at Q=20 is $TC-TFC=550-50=500$, so $AVC=500/20=Rs.\,25$. Because the market price (Rs. 45) is well above AVC (Rs. 25), and indeed above ATC ($550/20=Rs.\,27.5$) as well, the firm is earning a genuine super-normal profit and should certainly continue producing at this output in the short run.

A common slip is to stop at finding Q and forget to check the shutdown condition (comparing P with AVC, not just noting profit is positive) — a firm could still be making a LOSS yet correctly continue producing if AVC < P < ATC, so the continue/shutdown decision must always be checked against AVC specifically, not against profit being positive or negative.