the owner of a 100hectare farm must decide how much fertiliz

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 tonner of grain sells for $50.the total production of grain (in tonnes) is given by G=38F-2(F^2) 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 $per hectare)?

Solution

(a)

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

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

Profit (Z) ($) = Total Revenue - Total Cost = (1,900F - 100F²) - 100F = 1,800F - 100F²

Profit is maximized (therefore, quantity of fertilizer used is optimized) when dZ/dF = 0

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

200F = 1,800

F = 1,800 / 200 = 9 (tonnes)

(b) When F = 9,

Total Economic rent = $100 x 9 = $900

Economic rent per hectare = $900 / 100 = $9 per hectare