@deepaush,
deepaush wrote:
Question:
A farmer produces paddy (Y1) and potato (Y2) with labour(X1) and fertilizer (X2) on his lands. Competitive markets exist for output and inputs. The production relationships are as follows:
Y_1=2+8.05X_1-2X_1^2+10.3X_2-0.5X_2^2
Y_2=4+5.04X_1-0.5X_1^2+15.24X_2+2.5X_2^2
Indicate the optimum quantity of paddy and potato to be produced if the price of paddy is `200/quintal, the price of potato is `250/qui, while the wages of labour is `10 and cost of fertilizer is `60 per unit.
I'm going to assume X1 can take a different value in each of the two equations.
Is my first assumption correct?
I'm also going to assume the last plus in the second equation should be a minus.
(Meaning, the equation should read Y_2=4+5.04X_1-0.5X_1^2+15.24X_2-2.5X_2^2.)
Is my second assumption correct?
If so...
The rate at which Y_1 increases as a function of X_1 is d(Y_1)/d(X_1) = 8.05 - 4X_1.
That means for every `10 you spend increasing X_1 by 1 unit, you increase the money from Y_1 by `200 * (8.05 - 4X_1).
You want to keep putting labour into paddy until `200 * (8.05 - 4X_1) = `10, at which point the extra paddy exactly pays for the extra labour.
So, 8.05 - 4X_1 = .05.
So, - 4X_1 = -8.
So, X_1 = 2. (You put 2 units of labour into paddy.)
By similar calculation, you can show:
--> that you should put 10 units of fertilizer into paddy.
--> that you should put 5 units of labour into potato.
--> that you should put 3 units of fertilizer into potato.
In the first equation, X_1 = 2 and X_2 = 10.
So Y_1=2+8.05*2-2*2^2+10.3*10-0.5*10^2 = 2 + 16.1 - 8 + 103 - 50 = 63.1
In the second equation, X_1 = 5 and X_2 = 3.
So Y_2=4+5.04*5-0.5*5^2+15.24*3-2.5*3^2 = 4 + 25.2 - 12.5 + 45.72 - 22.5 = 39.92
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