Macroeconomics

Y

= (MPK ×

D

K )

+ (MPL ×

D

L).

The first term in parentheses is the increase in output resulting from the increase

in capital, and the second term in parentheses is the increase in output resulting

from the increase in labor. This equation shows us how to attribute growth to

each factor of production.

units of output

⎯⎯

unit of capital

units of output

⎯⎯

unit of labor

248

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P A R T   I I I

Growth Theory: The Economy in the Very Long Run

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C H A P T E R   8

Economic Growth II: Technology, Empirics, and Policy

| 249

We now want to convert this last equation into a form that is easier to inter-

pret and apply to the available data. First, with some algebraic rearrangement, the

equation becomes

17

=

(

)

+

(

)

.

This form of the equation relates the growth rate of output, 

ΔY/Y, to the growth

rate of capital, 

ΔK/K, and the growth rate of labor, ΔL/L.

Next, we need to find some way to measure the terms in parentheses in the last

equation. In Chapter 3 we showed that the marginal product of capital equals its real

rental price. Therefore, MPK

× K is the total return to capital, and (MPK × K )/Y

is capital’s share of output. Similarly, the marginal product of labor equals the real

wage. Therefore, MPL

× L is the total compensation that labor receives, and 

(MPL

× L)/Y is labor’s share of output. Under the assumption that the production

function has constant returns to scale, Euler’s theorem (which we discussed in 

Chapter 3) tells us that these two shares sum to 1. In this case, we can write

=

a

+ (1 −

a

) ,

where 

a

is capital’s share and (1 

−

a

) is labor’s share.

This last equation gives us a simple formula for showing how changes in inputs

lead to changes in output. It shows, in particular, that we must weight the growth

rates of the inputs by the factor shares. As we discussed in Chapter 3, capital’s share

in the United States is about 30 percent, that is, 

a

= 0.30. Therefore, a 10-percent

increase in the amount of capital (

ΔK/K = 0.10) leads to a 3-percent increase in

the amount of output (

ΔY/Y = 0.03). Similarly, a 10-percent increase in the

amount of labor (

ΔL/L = 0.10) leads to a 7-percent increase in the amount of

output (

ΔY/Y = 0.07).

Technological Progress

So far in our analysis of the sources of growth, we have been assuming that the

production function does not change over time. In practice, of course, techno-

logical progress improves the production function. For any given amount of

inputs, we can produce more output today than we could in the past. We now

extend the analysis to allow for technological progress.

We include the effects of the changing technology by writing the production

function as

Y

= AF(K, L),

D

L

⎯

L

MPL

× L

⎯

Y

D

K

⎯