00:01
Hi, in this question given function f of x comma y equals 2x power 2 by 3 into y power 1 by 3, where x is the amount of money expanded for labor and y is the cost of capital equipment and assume that the company currently spend 15 units on equipment.
00:24
So here given that y equals 15.
00:29
So for part a, we need to find.
00:31
The x value.
00:33
Here given that the marginal productivity of labor equals to marginal productivity of capital.
00:40
So that is do f by do x equals do f by do y.
00:48
On differentiating f with respect to x then we get 2 into 2 by 3 into x power 2 by 3 minus 1 which is minus 1 by 3 into y power 1 by 3.
01:04
On differentiating with respect to y, then we get 2 into 1 by 3 into x power 2 by 3.
01:12
1 by 3 minus 1 minus 1 by 1 which is y power minus 2 by 3 which implies here 4 by 3 into y by x the whole power 1 by 3 which is equal to 2 by 3 into 2 by 3 into here x x by 1 by 1 .3 which is equal to 2 by 3 into here x x x by 1 by 1 by 1.
01:31
Y the whole power 2 by 3 so first we need to find the value of x so here on for the simplification we get 2 into y power 1 by 3 y power 2 by 3 which is equal to x power 2 by 3 into x power 1 by 3 on further simplification we get 2 y equals x here given that y is equals 15, therefore x equals 2 into 15, therefore x equals 30.
02:10
So for part a, the required answer, x equals 30.
02:19
For part b, by using the given that we can consider do f by do x is less than do f by do y.
02:28
On substituting the values, then we get 4 by 3 into y by x the whole power 1 by 3, less than 2 by 3 into x by y the whole power 2 by 3...