Choosing a Mango Variety over Ten Years
A horticulture cooperative in Kolar is choosing between two tissue-culture mango varieties for a 10-year block of 400 trees. Variety A is expected to yield 80 kg per tree in year 1 and then increase by 12 kg per tree each year for years 2–5. From year 6 onward its yield increases by 5% per year over the previous year. Variety B yields 95 kg per tree in year 1 and increases by 8% each year for all 10 years. Certification rules allow only one variety in the block. A buyer fixes ₹70 per kg for years 1–5 and ₹80 per kg for years 6–10. The president wants the larger 10-year revenue. The treasurer objects that years 1–3 cash flow is tight and Variety B's early lead should decide the matter. Assume yield per tree is the same for all 400 trees and that year 6 is the first 5% increase for Variety A.
1. Write Variety A's per-tree yields for years 1–10 and find its total per-tree yield over 10 years.
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For Variety A, years 1–5 form an arithmetic progression: 80, 92, 104, 116, 128 kg. Its sum is (5/2)(80 + 128) = 520 kg.
Year 6 is 128 × 1.05 = 134.4 kg, and the next four yields form a geometric progression with first term 134.4 and ratio 1.05:
134.4, 141.12, 148.176, 155.5848, 163.36404 kg.
Their sum is 134.4 × ((1.05^5 − 1)/0.05) = 134.4 × 5.52563125 = 742.64484 kg.
Total per tree over 10 years = 520 + 742.64484 = 1262.64484 kg, about 1262.64 kg.
2. Compute total revenue per tree and for the whole block for each variety. Which variety should be planted?
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Variety A revenue per tree: Years 1–5: 520 × 70 = ₹36,400. Years 6–10: 742.64484 × 80 = ₹59,411.59. Total ≈ ₹95,811.59 per tree. For 400 trees: ₹38,324,636 approximately.
Variety B total yield per tree is a geometric sum: 95 × ((1.08^10 − 1)/0.08) = 95 × 14.48656246 = 1376.22343 kg. Years 1–5 yield = 95 × ((1.08^5 − 1)/0.08) = 95 × 5.86660096 = 557.32709 kg. Revenue years 1–5 = 557.32709 × 70 = ₹39,012.90. Years 6–10 yield = 1376.22343 − 557.32709 = 818.89634 kg. Revenue years 6–10 = 818.89634 × 80 = ₹65,511.71. Total ≈ ₹104,524.60 per tree, or ₹41,809,841 for 400 trees.
Variety B gives about ₹3.49 crore more revenue over the block, so the cooperative should plant B on revenue grounds.
3. Test the treasurer's cash-flow objection using cumulative revenue for years 1–3. Does it change the recommendation?
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Variety A yields 80 + 92 + 104 = 276 kg in years 1–3. At ₹70 per kg, revenue is 276 × 70 = ₹19,320 per tree.
Variety B yields 95 + 95(1.08) + 95(1.08)^2 = 95 + 102.6 + 110.808 = 308.408 kg. Revenue is 308.408 × 70 = ₹21,588.56 per tree.
Variety B is ahead by ₹2,268.56 per tree, or about ₹9,07,424 over 400 trees, even in the tight early period. The treasurer's concern does not overturn the recommendation: B is better both early and over ten years. The decision would change only if the cooperative faced a liquidity constraint not captured by revenue, such as an inability to finance the slightly higher initial certification or input cost.