Kayak Rental Price and Profit Maximization
A kayak-rental operator on a busy coastal strip currently charges €25 per rental. Daily demand is modelled by q = 200 − 5p, where p is the price in euros and q is the number of rentals per day. Fixed daily costs are €300, and each rental incurs €12 in variable costs (staff, cleaning, insurance). The operator is considering raising the price to €30 for the peak season. The model is valid only for prices between €10 and €40. Last season the average was 70 rentals per day at €25, but the operator believes demand is more price-sensitive. The accountant warns that fixed costs are unavoidable in the short run. The operator asks whether the price increase is justified and what price maximizes daily profit. The operator has no capacity constraint for up to 100 rentals per day.
1. Construct the daily profit function P(p) and state its domain.
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Revenue R = p(200 − 5p) = 200p − 5p². Variable cost = 12(200 − 5p) = 2400 − 60p. Total cost C = 300 + 2400 − 60p = 2700 − 60p. Profit P = R − C = 200p − 5p² − 2700 + 60p = −5p² + 260p − 2700. Domain: 10 ≤ p ≤ 40 and q ≥ 0; q ≥ 0 gives p ≤ 40, already satisfied.
2. Find the price that maximizes daily profit. Compare €25, €26, and €30.
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The vertex of P = −5p² + 260p − 2700 is p = −260 / (2 × −5) = 26. Then P(26) = −5(676) + 260(26) − 2700 = −3380 + 6760 − 2700 = 680. P(25) = −3125 + 6500 − 2700 = 675. P(30) = −4500 + 7800 − 2700 = 600. Thus €26 maximizes daily profit at €680; €30 reduces profit by €80/day relative to €26 and by €75/day relative to €25. The proposed increase to €30 is not justified by this model.
3. Interpret the fixed cost and discuss the main limitation of the model when deciding the peak-season price.
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The €300 fixed cost shifts profit down but does not change the optimal price, because it is constant in p. It does matter for the shut-down decision: at p = 26 profit is €680, so the operator should operate. The model is a local linear approximation; if the peak season changes demand or capacity, the coefficients must be re-estimated. The model also assumes all rentals are served; with capacity 100/day, at p = 26 q = 70, so capacity is not binding.