A farmer has a given amount of land, denoted by cnax and can allocate it between

Question

A farmer has a given amount of land, denoted by cnax and can allocate it between two crops where & i-1,2, is the amount of land allocated to crop i. The production function for crop i is y for all i 1,2 where 1 > a > 0. The net profit from one unit of land planted with crop 1 is r while the net profit from one unit of land planted with crop 2 is s. If the farmer wishes to maximize net profit from all his land, how much of each crop needs to be planted if the total available land is to be used? Prove that you derived optimal land allocations that maximize net profits. How does an increase in s affect land planted for crop 1 and 2? Prove your answer.

Explanation / Answer

Consider the given problem here the available land is “Lmax”. Now, the production is given by “Yi = Li^a”, where “a” is between “0” to “1”. Now, since the production function for both the good is same, => it doesn’t matter whether we devote a single unit of land to “1” or “2” in both the cases the increase in production will be same that why the production decision depends on the net profit rate of each good.

Now, the net profit from “good 1” is “r” and the same for “good 2” is “s”, => if “r > s”, => it is optimum to produce only “good 1” and no “good 2”, => all the land will be devoted to the production of “good 1”. Similarly, if “r < s”, => it is optimum to produce only “good 2” and no “good 1”, => all the land will be devoted to the production of “good 2”.

So, here we can see that optimum production depends on the net profit of both the goods. So, if the net profit of “good 1” is more compare to the “good 2”, => only “good 1” will planted and if the net profit of “good 2” is more compare to the “good 1, => only “good 2” will planted.

Now, if “s > r” then increase in “s” will not have any effect on the land plantation. If initially “r > s”, then as “s” increases and become more than “r”, => the entire production plant will change, => initially only “good 1” was planted now as “s” increases such that “s>r”, => it is profitable to plant only “good 2”.

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