1 The owner of a 100 hectare farm must decide how much ferti

1. The owner of a 100 hectare farm must decide how much fertilizer to apply to produce grain. A tonne of fertilizer costs $100 and a tonne of grain sells for S50. The total production of grain (in tonnes) is given by G-38 F -2 F2 where F indicates tonnes of fertilizer applied a) How much fertilizer is this farmer expected to use? b) What is the rent for this land (in S per hectare)?

Solution

(a)

Total revenue (TR) ($) = Grain price x Quantity of grain = 50G = 50 x (38F - 2F²) = 1,900F - 100F²

Total cost (TC) ($) = Fertilizer price x Quantity of Fertilizer = 100 x F

Profit (Z) ($) = TR - TC = 1,900F - 100F² - 100F = 1,800F - 100F²

Profit is maximized (and fertilizer usage is optimized) when dZ/dF = 0

dZ/dF = 1,800 - 200F = 0

200F = 1,800

F = 9 tonnes

(b) When F = 9, G = (38 x 9) - (2 x 9 x 9) = 342 - 162 = 180

Economic rent = $50 x G = $50 x 180 = $9,000