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# Growth Accounting

Winter 2000

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.

## Sources of Growth

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 Equation

The growth accounting equation is given by

%DY = %DA + aK·%DK + aN·%DN

where

• aK  =  %DY/%DK = elasticity of output with respect to capital ( holding A and N fixed)
• aN  =  %DY/%DN = elasticity of output with respect to labor (holding A and K fixed)

Note: 0 < aK < 1 and 0 < aN < 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
• aK = capital's share of output
• aN  = labor's share of output
• aK  + aN  = 1

For the U.S. economy, from the National Income and Product Accounts,

• aK = 0.3
• aN  = 0.7

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.

## Some International Comparisons of Growth

 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.