# Production Functions and Marginal Products: A Comprehensive Guide

## What are the marginal product of capital and the marginal product of labor in production functions?

How can we calculate the marginal product of capital and labor for different production functions?

## Answer:

The marginal product of capital (MPK) for the production function Y = 5K^0.3 * L^0.7 is 1.5K^-0.7 * L^0.7, and the marginal product of labor (MPL) is 3.5K^0.3 * L^-0.3. For the production function Y = a ln K + b ln L, the MPK is a/K and the MPL is b/L.

To find the marginal product of capital (MPK) and the marginal product of labor (MPL) for the given production functions, we need to take partial derivatives with respect to the respective factors.

## Calculating MPK and MPL:

**a) Y = 5K^0.3 * L^0.7:**

To find MPK, we take the partial derivative of Y with respect to K:

MPK = ∂Y/∂K = 0.3 * 5K^0.3-1 * L^0.7 = 1.5K^-0.7 * L^0.7

To find MPL, we take the partial derivative of Y with respect to L:

MPL = ∂Y/∂L = 0.7 * 5K^0.3 * L^0.7-1 = 3.5K^0.3 * L^-0.3

**b) Y = a ln K + b ln L:**

To find MPK, we take the partial derivative of Y with respect to K:

MPK = ∂Y/∂K = a/K

To find MPL, we take the partial derivative of Y with respect to L:

MPL = ∂Y/∂L = b/L

Understanding and calculating the marginal product of capital and labor in production functions is crucial for analyzing production efficiency and resource allocation.