A farmer has 10 hectares of land and 800 kg of fertilizer. He can grow maize or beans. The yield per hectare depends on the amount of fertilizer applied per hectare as follows:

- For maize, total output (in tonnes) from $x$ hectares receiving $f_m$ kg of fertilizer per hectare is $Q_m = 2 f_m x$.
- For beans, total output (in tonnes) from $y$ hectares receiving $f_b$ kg of fertilizer per hectare is $Q_b = 1.5 \sqrt{f_b}\, y$.

The farmer may allocate the land between the two crops ($x + y = 10$) and must satisfy the fertilizer constraint $f_m x + f_b y = 800$. The market price is ₦1500 per tonne of maize and ₦2000 per tonne of beans. Fixed costs are ₦5000 and other variable costs are negligible.

Determine the allocation of land ($x$ and $y$) and the amount of fertilizer per hectare ($f_m$ and $f_b$) that maximizes the farmer's profit, and state the maximum profit.

Question

A farmer has 10 hectares of land and 800 kg of fertilizer. He can grow maize or beans. The yield per hectare depends on the amount of fertilizer applied per hectare as follows:

- For maize, total output (in tonnes) from $x$ hectares receiving $f_m$ kg of fertilizer per hectare is $Q_m = 2 f_m x$.
- For beans, total output (in tonnes) from $y$ hectares receiving $f_b$ kg of fertilizer per hectare is $Q_b = 1.5 \sqrt{f_b}\, y$.

The farmer may allocate the land between the two crops ($x + y = 10$) and must satisfy the fertilizer constraint $f_m x + f_b y = 800$. The market price is ₦1500 per tonne of maize and ₦2000 per tonne of beans. Fixed costs are ₦5000 and other variable costs are negligible.

Determine the allocation of land ($x$ and $y$) and the amount of fertilizer per hectare ($f_m$ and $f_b$) that maximizes the farmer's profit, and state the maximum profit.

Accepted Answer

Allocate 8 ha to maize with 100 kg/ha fertilizer and 2 ha to beans with no fertilizer; maximum profit = ₦2,395,000.

Neco Model Questions Vol1 2024 Economics Question 1

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