A firm has two variable factors and a production function f(x 1 , x 2 ) = x^1/2 x_2^1/4 . Please find the marginal product using formulas in the lecture notes by yourself. The price of its output is 4. Factor 1 receives a wage of ω1 and factor 2 receives a wage of ω2 . Required:
Write an equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1.

The equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1 is as follows:

1/2( x_2^1/4 / x_1^1/2) = ω1

Explanation:

Given:

f(x_1 , x_2 ) = x^1/2 x_2^1/4

Since x = x_1, we have:

f(x_1 , x_2 ) = x_1^1/2 x_2^1/4 ……………………………….. (1)

To obtain the marginal product of factor 1 (MP_1), the partial derivative of equation with respect to x_1 is taken as follows:

MP_1 = df(x_1 , x_2 )/dx_1

MP_1 = (1/2)x_1^((1/2)-1) x_2^1/4

MP_1 = (1/2)x_1^-(1/2) x_2^1/4, or 1/2( x_2^1/4 / x_1^1/2)

The equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1 can be obtained as follows:

MP_1 = ω1 …………………….. (2)

Substituting MP_1 = 1/2( x_2^1/4 / x_1^1/2) in equation (2), we have:

1/2( x_2^1/4 / x_1^1/2) = ω1 …………………… (3)

Equation (3) is therefore the equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1.

answer; i believe a ( newsletter ) should be sent;

1   all nations face the problem of scarcity because there are limited resources and unlimited wants. nations do not have enough factors of production to produce everything that their population wants. entrepreneurial resources initiate businesses to make new goods and services; labor resources produce goods and services.

2 there are four factors of production: natural resources, labor resources, capital resources, and entrepreneurial resources. natural resources are raw materials from nature that are used to produce goods. trees, water, and grains are natural resources. some natural resources, such as wheat and cattle, are renewable

3 wants

things that you do not need to have in order to survive, but would like to have

needs

things that you must have in order to survive

goods

physical products

services

tasks that a business perform for customers

resources

items that people can use to make or obtain what they need or want

determine

to settle or decide by choice of alternatives or possiblities

4 i really dont know