Optimal Output, a Specific Tax and a Proposed Price Cut
A furniture unit sells a single chair model. Weekly demand is modelled as p = 600 − 2q, where p is price in rupees and q is chairs sold per week. Total weekly cost is C(q) = 0.5q² + 100q + 10,000, which includes ₹10,000 of fixed rent and salaries. The production manager currently sets output at 150 chairs a week because that figure maximises sales revenue, and argues that maximum revenue must also mean maximum profit. The accountant's marginal cost statement shows that at 150 chairs the cost of the last chair is well above what the market will pay. Separately, the government is proposing a specific tax of ₹30 per chair, collected from the seller. The owner wants to know the profit-maximising output and price now, how the tax would change them and how much of the tax can be passed to customers, and whether a 10% price cut from the current optimum would be worthwhile.
1. Find the profit-maximising output and price, and show why 150 chairs is not optimal.
Read the worked answer
Revenue R = pq = (600 − 2q)q = 600q − 2q², so MR = 600 − 4q. Cost C = 0.5q² + 100q + 10,000, so MC = q + 100.
Set MR = MC: 600 − 4q = q + 100 → 500 = 5q → q = 100 chairs. Price: p = 600 − 2(100) = ₹400.
Profit = 400 × 100 − (0.5 × 10,000 + 100 × 100 + 10,000) = 40,000 − 25,000 = ₹15,000.
Second-order check: profit π = 500q − 2.5q² − 10,000, so d²π/dq² = −5 < 0, confirming a maximum.
At q = 150: p = 300, revenue = ₹45,000, but profit = 500(150) − 2.5(22,500) − 10,000 = ₹8,750. Revenue is indeed maximised at q = 150 (MR = 0), but profit is not, because units 101–150 each add more to cost than to revenue. The manager is confusing the revenue-maximising point with the profit-maximising point.
2. The ₹30 per chair tax is imposed. Find the new optimum and quantify how much of the tax is passed to customers.
Read the worked answer
The tax adds ₹30 to marginal cost: MC = q + 130. Set MR = MC: 600 − 4q = q + 130 → 470 = 5q → q = 94 chairs. Price: p = 600 − 2(94) = ₹412.
Profit before tax = 500(94) − 2.5(94²) − 10,000 = 47,000 − 22,090 − 10,000 = ₹14,910. Tax = 30 × 94 = ₹2,820, so after-tax profit = ₹12,090.
Effects:
- Price rises from ₹400 to ₹412, so ₹12 of the ₹30 tax (40%) is passed on; the firm absorbs ₹18 per chair.
- Output falls by 6 chairs (100 to 94).
- Profit falls by ₹2,910 (15,000 to 12,090).
The pass-through is not 100% because demand is downward-sloping: raising price loses volume. Under a specific tax on a linear-demand monopolist, pass-through is less than the full tax unless demand is perfectly inelastic.
3. The sales manager proposes cutting the ₹400 price by 10% to increase volume. Evaluate this using the point elasticity at the optimum and the profit figures.
Read the worked answer
At q = 100, p = 400, dp/dq = −2, so dq/dp = −0.5. Elasticity e = (dq/dp)(p/q) = (−0.5)(400/100) = −2.0. Demand is elastic, so a price cut raises revenue.
A 10% cut takes price to ₹360, and q = (600 − 360)/2 = 120 chairs (a 20% volume rise).
- Revenue: 360 × 120 = ₹43,200, up from ₹40,000 (8%).
- Profit: 500(120) − 2.5(14,400) − 10,000 = 60,000 − 36,000 − 10,000 = ₹14,000, down ₹1,000.
The proposal fails on profit even though it succeeds on revenue. The reason is that at q = 100, MR = 200 equals MC = 200; every chair beyond 100 adds more to cost than to revenue. Elastic demand guarantees higher revenue from a price cut only while MR is positive, not higher profit. The manager should stop using revenue as the objective; profit at 100 chairs is the highest of the three options examined (₹15,000 versus ₹14,000 and ₹8,750).