Last updated: January
24, 2000
Note: These notes are preliminary and incomplete
and they are not guaranteed to be free of errors. Please let me know if
you find typos or other errors.
We are interested in the sources of the growth in potential output. Recall, potential
output is determined by the production function
Y = A·F(K, N)
From the production function it is clear that Y grows over time because
- Total factor productivity, A, grows over time (for reasons that are not
entirely clear)
- The capital stock, K, grows over time (due to investment)
- The labor supply, N, grows over time (due to population growth, increases in
participation rates, immigration)
In what follows let
- %DY = growth in potential output (per year)
- %DK = growth in the capital stock (per year)
- %DN = growth in the labor supply (per year)
- %DA = growth in total factor productivity (per year)
The main question is
How much does each component contribute to
the growth rate in output?
Growth Accounting answers this question.
The growth accounting equation is given by
%DY =
%DA + a_{K}·%DK
+ a_{N}·%DN
where
- a_{K} = %DY/%DK = elasticity of output with respect to capital ( holding
A and N fixed)
- a_{N} = %DY/%DN = elasticity of output with respect to labor (holding A
and K fixed)
Note: 0 < a_{K} < 1 and 0 < a_{N}
< 1 due to the production having diminishing returns to both capital and labor.
Result
- If F(K, N) exhibits constant returns to scale and market are
competitive then
- a_{K} = capital's share of output
- a_{N} = labor's share of output
- a_{K} + a_{N} = 1
For the U.S. economy, from the National Income and Product Accounts,
so the U.S. growth accounting equation is
%DY =
%DA + (0.3)·%DK + (0.7)·%DN
Implications
- 1% growth in A => 1% growth in Y holding K
& N fixed
- 1% growth in K => 0.3% growth in Y holding A
& N fixed
- 1% growth in N => 0.7% growth in Y holding A
& K fixed
Example - Growth accounting for the United States:
1950 - 1992
- Y = annual real GDP
- K = constant cost net stock of fixed private
nonresidential capital (for the Survey of Current Business) in billions of 1992 dollars
- N = Civilian employment (over 16) in millions of workers
Data: Average growth rate per year
(%) |
%DY |
%DK |
%DN |
%DA |
3.2 |
2.6 |
1.4 |
unknown |
Applying the growth equation to the above data gives
3.2% = (0.3)*(2.6%) + (0.7)*(1.4%)
+ %DA
= 0.78% + 0.98% + %DA
= 1.76% + %DA
The contribution of productivity is defined as
the residual after subtracting off the contributions from capital and labor:
%DA = 3.2% - 1.76% = 1.44%
and this is called the Solow
residual (named after the famous economist Robert Solow
who pioneered growth theory).
The relative contributions to growth are
- K: 0.78/3.2 = 24.4%
- N: 0.98/3.2 = 31.0%
- A: 1.44/3.2 = 44.6%
Productivity has the largest average contribution to postwar
growth for the United States!
It is informative to break down the contributions of each factor
to growth over sub-periods. The following table provided such a breakdown
Sources of Growth of the United
States Economy |
Years |
%DY |
(0.3)·%DK |
(0.7)·%DN |
%DA |
1950 -59 |
4.0 |
0.4 |
0.5 |
3.1 |
1960-69 |
4.1 |
0.9 |
1.2 |
2.0 |
1970-1979 |
2.9 |
1.1 |
1.5 |
0.3 |
1980-1989 |
2.5 |
0.9 |
1.3 |
0.3 |
1990-1992 |
0.6 |
0.6 |
-0.1 |
0.1 |
Notice that the slowdown in growth after 1970 corresponds generally with
a slowdown in productivity.
Statistics on Growth and Development |
Country |
GDP per capita, 1990 |
Labor Force Participation Rate, 1990 |
Average Annual Growth Rate, 1960-1990 |
Years to Double (years to half if negative) |
Rich Countries |
USA |
18,073 |
0.49 |
1.4 |
51 |
West Germany |
14,331 |
0.49 |
2.5 |
28 |
Japan |
14,317 |
0.63 |
5.0 |
14 |
France |
13,896 |
0.46 |
2.7 |
26 |
UK |
13,223 |
0.49 |
2.0 |
35 |
Poor Countries |
China |
1,324 |
0.60 |
2.4 |
29 |
India |
1,262 |
0.39 |
2.0 |
35 |
Zimbabwe |
1,181 |
0.49 |
0.2 |
281 |
Uganda |
554 |
0.49 |
-0.2 |
-281 |
Growth Miracles |
Hong Kong |
14,854 |
0.65 |
5.7 |
12 |
Singapore |
11,698 |
0.48 |
5.3 |
13 |
Taiwan |
8,067 |
0.44 |
5.7 |
12 |
South Korea |
6,665 |
0.42 |
6.0 |
12 |
Growth Disasters |
Venezuela |
6,070 |
0.35 |
-0.5 |
-136 |
Madagascar |
675 |
0.43 |
-1.3 |
-52 |
Mali |
530 |
0.48 |
-1.0 |
-70 |
Chad |
400 |
0.35 |
-1.7 |
-42 |
Source: Charles Jones, Introduction to Economic Growth, Norton, 1998.
Note: Years-to-double is based on the rule of 70.