A 10% price cut: right direction, wrong size
Neelam sells a kitchen appliance through a single online storefront. Her market survey for the coming season gives the demand relation p = 120 − 2x, where x is the quantity sold per month in hundreds of units and p is the price per unit in rupees; the relation is valid for 0 ≤ x ≤ 45. Her monthly cost, in hundreds of rupees, is C(x) = 20x + 300, covering procurement, packing and a fixed platform fee. At present she prices the appliance at ₹90, sells 15 hundred units a month, and believes that any price cut will 'grow the business'. A consultant proposes cutting the price by 10% from ₹90 and argues that because demand is elastic, revenue and profit must both rise. Neelam wants a marginal-analysis check of where profit is actually maximised, what the elasticity is at the present price and at that optimum, and whether the consultant's 10% cut should be accepted as it stands.
1. Form the revenue and profit functions, find the output and price that maximise monthly profit, and verify that it is a maximum.
Read the worked answer
R(x) = x·p = x(120 − 2x) = 120x − 2x² (₹ hundreds).
MR = R′(x) = 120 − 4x; MC = C′(x) = 20.
Profit P(x) = R − C = 120x − 2x² − 20x − 300 = −2x² + 100x − 300.
P′(x) = −4x + 100 = 0 ⇒ x = 25; P″(x) = −4 < 0, so it is a maximum (equivalently MR = MC: 120 − 4x = 20 gives x = 25).
Price: p = 120 − 2(25) = ₹70. Profit = −2(625) + 100(25) − 300 = −1,250 + 2,500 − 300 = 950 (₹ hundreds) = ₹95,000. Revenue at this point = 120(25) − 2(625) = ₹1,75,000.
x = 25 lies inside 0 ≤ x ≤ 45, so the interior optimum is admissible.
2. Compute the point elasticity of demand at the current price of ₹90 and at the profit-maximising price of ₹70. Explain why profit maximisation does not occur at unit elasticity.
Read the worked answer
From p = 120 − 2x we get x = 60 − p/2, so dx/dp = −1/2.
η = (p/x)(dx/dp) = −p/(2x).
- At p = 90, x = 15: η = −90/30 = −3.0, |η| = 3 → demand is highly elastic.
- At p = 70, x = 25: η = −70/50 = −1.4, |η| = 1.4 → still elastic, but less so.
Why not at |η| = 1? Unit elasticity is the condition for revenue maximisation: R is maximised where MR = 0, i.e. 120 − 4x = 0, x = 30, p = ₹60, where |η| = 60/60 = 1. Profit maximisation stops earlier, at MR = MC = 20, because every extra unit also costs ₹20 to make. Since MC > 0, the firm halts while marginal revenue is still positive - that is, while demand is still elastic. At x = 25, MR = 20 > 0 confirms this. Treating |η| = 1 as the profit target is a category error.
3. Evaluate the consultant's proposal: cut price 10% from ₹90 to ₹81. Compare revenue and profit with the present position and with the true optimum, and give Neelam a recommendation.
Read the worked answer
At p = 81: x = (120 − 81)/2 = 19.5 (i.e. 1,950 units).
- Revenue = 81 × 19.5 = 1,579.5 (₹ hundreds) = ₹1,57,950, up from 90 × 15 = ₹1,35,000.
- Cost = 20(19.5) + 300 = 690 (₹ hundreds) = ₹69,000.
- Profit = ₹88,950, up from 90(15) − 20(15) − 300 = 750 (₹ hundreds) = ₹75,000.
So the cut does raise both revenue and profit - the consultant is right about direction. It is wrong about magnitude: ₹88,950 is still ₹6,050 below the ₹95,000 available at p = ₹70.
Marginal check at the new point: MR = 120 − 4(19.5) = 42, MC = 20, so MR > MC; profit is still rising with output. The firm should keep expanding.
Recommendation. Do not stop at a 10% cut. Move to ₹70 (about a 22% cut from ₹90), where MR = MC, giving x = 25 hundred units and profit ₹95,000. Note the elasticity evidence: at ₹90 demand is so elastic (|η| = 3) that the current price is well above the profit-maximising level; the business is indeed being held back by price, but by more than 10%.