1. The Mathematics of Aggregate Production Functions
Chapter 6 Appendix
The Mathematics of Aggregate Production Functions

2. 6A The Mathematics of Aggregate Production Functions
Key Ideas
We write the aggregate production function in a specific form called the Cobb-Douglas production function.
We then use the Cobb-Douglas production function to solve for the hypothetical value of GDP per worker of India, in which India possesses U.S. technology.

3. 6A.1 The Mathematics of Aggregate Production Functions
The Cobb-Douglas Production Function
The aggregate production function can be written in many different functional forms.
Economists like to use a specific form called the Cobb-Douglas production form:
Instructor: You may want to elaborate on a general function form like the aggregate production function and a specific function form like the Cobb-Douglas production function.

4. The coefficients to which K and H are raised sum to one.
6A.1 The Mathematics of Aggregate Production Functions
The Cobb-Douglas Production Function
The coefficients to which K and H are raised sum to one.
This feature generates two important properties of the Cobb-Douglas production function.

5. There are constant returns to scale in K and H
6A.1 The Mathematics of Aggregate Production Functions
The Cobb-Douglas Production Function
There are constant returns to scale in K and H
Suppose we double both K and H:
We then double Y!

6. 6A.1 The Mathematics of Aggregate Production Functions
The Cobb-Douglas Production Function
In competitive markets, one-third of income is paid to physical capital and two-thirds to labor.

7. 6A.1 The Mathematics of Aggregate Production Functions
The Cobb-Douglas Production Function
Instructor: The dark green space is wage and salary payments, and the light green space is labor benefits (e.g., retirement, healthcare) payments.

8. 6A.1 The Mathematics of Aggregate Production Functions
The Cobb-Douglas Production Function
Using the Cobb-Douglas production function, we can solve for GDP per worker, y:
Instructor: Remind students that the law of exponents is used to go from first equation to second equation.

9. 6A.1 The Mathematics of Aggregate Production Functions
The Cobb-Douglas Production Function
We can write GDP per worker, y, in terms of capital per worker, K/L, and human capital per worker, h:
Instructor: You may want to state clearly that the equation predicts that GDP per worker is equal to technology times capital per worker to the one-third power times human capita per worker to the two-thirds power.

10. 6A.2 The Mathematics of Aggregate Production Functions
Real GDP of India with U.S. Technology
We can use the Cobb-Douglas GDP per worker formula to calculate the hypothetical GDP per worker of India in which India possesses U.S. technology.

11. U.S. level of technology:
6A.2 The Mathematics of Aggregate Production Functions
Real GDP of India with U.S. Technology
U.S. level of technology:
Instructor: We rearrange terms from our Cobb-Douglas production function to solve for technology in terms of capital per worker and human capital per worker.

12. Indian level of technology:
6A.2 The Mathematics of Aggregate Production Functions
Real GDP of India with U.S. Technology
Indian level of technology:
Instructor: We rearrange terms from our Cobb-Douglas production function to solve for technology in terms of capital per worker and human capital per worker.

13. 6A.2 The Mathematics of Aggregate Production Functions
Real GDP of India with U.S. Technology
GDP per worker of India in which India possesses U.S. technology is therefore:

14. 6A.2 The Mathematics of Aggregate Production Functions
Real GDP of India with U.S. Technology
Using data on U.S. technology and Indian capital per worker and human capital, we can now estimate the hypothetical value of GDP per worker of India in which India possesses U.S. technology:
Instructor: The human capital estimate is obtained by applying years of schooling to a Mincerian wage equation.
The same methodology can be used for each country to derive column 5 in Exhibit 6.12.