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Chapter 6 AppendixThe

Mathematics of Aggregate Production Functions

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© 2015 Pearson Education, Inc

Key Ideas

1. We write the aggregate production function in a specific form called the Cobb-Douglas production function.

2. 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.

6A The Mathematics of Aggregate Production Functions

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6A.1 The Mathematics of Aggregate Production FunctionsThe 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:

1/3 2/3Y = A × F(K,H)= A× K × H

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The coefficients to which K and H are raised sum to one.

This feature generates two important properties of the Cobb-Douglas production function.

6A.1 The Mathematics of Aggregate Production FunctionsThe Cobb-Douglas Production Function

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1. There are constant returns to scale in K and H

Suppose we double both K and H:

We then double Y!

1/3 2/3(2 2 ) (2 ) (2 ) A × F K, H = A× K × H1/3 2/3 1/3 2/3(2 2 ) 2 2 A × F K, H = × × A× K × H

1/3 2/3(2 2 ) 2 2 A × F K, H = × A× K × H Y

6A.1 The Mathematics of Aggregate Production FunctionsThe Cobb-Douglas Production Function

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2. In competitive markets, one-third of income is paid to physical capital and two-thirds to labor.

6A.1 The Mathematics of Aggregate Production FunctionsThe Cobb-Douglas Production Function

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1.00

Payments to U.S. Labor (% of total income)

Wages and Salaries Labor benefits

(%)

6A.1 The Mathematics of Aggregate Production FunctionsThe Cobb-Douglas Production Function

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Using the Cobb-Douglas production function, we can solve for GDP per worker, y:

1/3 2/31 1( )

Y = A × F K,H × = A× K × H ×

L L L

1/3 2/31/3 2/3

1Yy = A× K × H ×

L L × L

6A.1 The Mathematics of Aggregate Production FunctionsThe Cobb-Douglas Production Function

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We can write GDP per worker, y, in terms of capital per worker, K/L, and human capital per worker, h:

1/3 2/31/3 2/3

1/3 2/3= =

K H K Hy A× × A× ×

L L L L

1/3

2/3=

Ky A× × h

L

6A.1 The Mathematics of Aggregate Production FunctionsThe Cobb-Douglas Production Function

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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.

6A.2 The Mathematics of Aggregate Production Functions Real GDP of India with U.S. Technology

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U.S. level of technology:

1/3

2/3USUS US US

US

= K

y A × × hL

USUS 1/3

2/3USUS

US

= y

AK

× hL

6A.2 The Mathematics of Aggregate Production Functions Real GDP of India with U.S. Technology

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Indian level of technology:

1/3

2/3INDIAINDIA INDIA INDIA

INDIA

= K

y A × × hL

6A.2 The Mathematics of Aggregate Production Functions Real GDP of India with U.S. Technology

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GDP per worker of India in which India possesses U.S. technology is therefore:

1/3

2/3INDIAINDIA WITH US TECH US INDIA

INDIA

= K

y A × × hL

6A.2 The Mathematics of Aggregate Production Functions Real GDP of India with U.S. Technology

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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:

1/3 2/3

INDIA WITH US TECH = 575 19,975 1.91y × ×

INDIA WITH US TECH = $24,071y

6A.2 The Mathematics of Aggregate Production Functions Real GDP of India with U.S. Technology