Question

the production function for a country is x= x^.075y^0.25, where x stands for units of labor and y for units of capital. at present, 25 units of labor and 20 units of capital are being provided. Use differentials to estimate their change in production if an additional unit of labor is provided and if capital is decreased to 19 units.

Answer 1

1.011

Step-by-step explanation:

The two differentiation for the two parameters such as units of labor and units of capital can be represented as shown below:

If z (production) = f(x,y) = x^0.75y^0.25

Where: x represents the units of labor and y represents the units of capital.

Thus, using differential, the change in production will be:

`dz = f_(x)(x,y) dx + f_(y)(x,y)` = 0.75x^(0.75-1)y^0.25 + 0.25(x^0.75)y^(0.25-1) = 0.75x^(-0.25)y^0.25 + 0.25x^0.75y^(-0.75)

x = 25+1 = 26; y = 19

dz = 0.75*26^(-0.25)*(19^0.25) + 0.25*26^0.75*(19^(-0.75)) = 0.75*0.443*2.09 + 0.25*11.51*0.11 = 0.694+0.317 = 1.011