A Chair-Making Unit: Profit Maximisation, a Carpenter Shortage and Cost Recovery
A furniture unit near Guwahati makes wooden chairs and sells them in a market where each additional chair forces a lower price. Market study gives the demand relation p = 1,500 − x/2, where x is chairs sold per month and p is the price in rupees. The accountant's cost sheet shows fixed cost of ₹90,000 per month (rent, supervisor, machine hire) and variable cost of 300x + 0.5x², where 300 is material and casual labour per chair and the squared term captures overtime and wastage as output rises. The workshop can currently make only 450 chairs a month because of a shortage of skilled carpenters, although the market would absorb up to 1,000 chairs at the ruling prices. The accountant argues that because fixed costs are ₹90,000 every month, the owner should raise the price above ₹1,200 to 'recover the fixed cost faster'.
1. Write the monthly profit function and find the output that maximises profit if the carpenter limit did not exist. State the price and the maximum profit.
Read the worked answer
Revenue. R = x·p = x(1,500 − x/2) = 1,500x − 0.5x². Cost. C = 90,000 + 300x + 0.5x². Profit. π = R − C = 1,500x − 0.5x² − 90,000 − 300x − 0.5x² = −x² + 1,200x − 90,000.
dπ/dx = −2x + 1,200 = 0 → x = 600 chairs. d²π/dx² = −2 < 0, so this is a maximum.
Price: p = 1,500 − 600/2 = ₹1,200. Profit: π = −(600)² + 1,200(600) − 90,000 = −3,60,000 + 7,20,000 − 90,000 = ₹2,70,000 per month.
Check by marginal reasoning. MR = 1,500 − x = 900 at x = 600; MC = 300 + x = 900. MR = MC as required. Producing 601 chairs would cost ₹901 and earn ₹899, so the unit stops at 600. Note that the profit-maximising price is exactly the ₹1,200 the accountant wants to exceed.
2. At the present capacity of 450 chairs, should the owner expand? Also evaluate the accountant's proposal to raise the price above ₹1,200, using numbers.
Read the worked answer
Current position. π(450) = −(450)² + 1,200(450) − 90,000 = −2,02,500 + 5,40,000 − 90,000 = ₹2,47,500. Marginal profit at 450 = dπ/dx = 1,200 − 2(450) = ₹300 per chair.
The 300th rupee of marginal profit is still positive, so every extra chair up to 600 adds profit; the whole 150-chair expansion is worth π(600) − π(450) = 2,70,000 − 2,47,500 = ₹22,500 a month. Equivalently, the owner can pay up to about ₹300 per chair of extra carpenter cost for the first additional chair and slightly less thereafter, so a wage rise that costs ₹25,000 a month for 150 chairs (about ₹167 each) is comfortably worth it. Expand, but negotiate.
The accountant's price rise. Raising price means selling fewer chairs. At p = ₹1,250, demand gives x = 500, and π = −2,50,000 + 6,00,000 − 90,000 = ₹2,60,000 - less than ₹2,70,000. At p = ₹1,300, x = 400, π = −1,60,000 + 4,80,000 − 90,000 = ₹2,30,000.
The error is treating ₹90,000 as something the price decision must 'recover'. Fixed cost does not change with x, so it shifts the profit level but not the profit-maximising quantity; it matters only for the decision to stay in business at all, and a monthly profit of ₹2.7 lakh answers that decisively.
3. Given marginal cost MC = 300 + x and fixed cost ₹90,000, obtain total cost by integration. Compute average cost at 450 and 600 chairs and explain the pattern using marginal cost.
Read the worked answer
Integration of marginal cost. Variable cost is the area under MC from 0 to x: VC = ∫₀ˣ (300 + t) dt = [300t + t²/2]₀ˣ = 300x + x²/2. Total cost: TC = 90,000 + 300x + x²/2, which matches the cost sheet.
At x = 450. VC = 300(450) + 450²/2 = 1,35,000 + 1,01,250 = ₹2,36,250; TC = ₹3,26,250; AC = 3,26,250/450 = ₹725. At x = 600. VC = 1,80,000 + 1,80,000 = ₹3,60,000; TC = ₹4,50,000; AC = ₹750.
Explanation. AC falls while MC is below AC and rises once MC is above AC. Here MC at 450 is 300 + 450 = ₹750, already above AC of ₹725, so AC is rising at that output. Average cost is minimised where MC = AC: 90,000/x + 300 + x/2, differentiated: −90,000/x² + 0.5 = 0 → x² = 1,80,000 → x ≈ 424 chairs, AC ≈ ₹724.
So the 450-chair workshop is already just past its least-cost scale, and at 600 chairs the rising overtime and wastage component pushes average cost to ₹750. That is not a reason to stop at 450 - profit is maximised where MC = MR = ₹900, not where AC is lowest - but it is a reason to resist expanding much beyond 600, and to look at reducing wastage rather than only hiring carpenters.