# Solow Growth Accounting

The Solow growth model presents a theoretical framework for understanding the sources of economic growth, and the consequences for long-run growth of changes in the economic environment and in economic policy. But suppose that we wish to examine economic growth in a freer framework, without necessarily being bound to adopt in advance the conclusions of our economic theories. In order to conduct such a less theory-bound analysis, economists have built up an alternative framework called growth accounting to obtain a different perspective on the sources of economic growth. We start with a production function that tells us what output Yt will be at some particular time t as a function of the economy’s stock of capital Kt, its labor force Lt, and the economy’s total factor productivity At. The Cobb-Douglas form of the production function is: Yt = At × ( Kt ) (Lt ) α 1− α

If output changes, it can only be because the economy’s capital stock, its labor force, or its level of total factor productivity changes.

Changes in Capital

Consider, first, the effect on output of a change in the capital stock from its current value Kt to a value Kt + ∆K—an increase in the capital stock by a proportional amount ∆K/Kt. In this production function Kt is raised to a power, α, so we can apply our rule-of-thumb for the proportional growth rate of a quantity raised to a power to discover

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that the proportional increase in output from this change in the capital stock is: ∆Y ∆K =α Yt Kt Thus if the diminishing-returns-to-scale parameter α were equal to 0.5, and if the proportional change in the capital stock were 3%, then the proportional change in output would be: ∆Y = 0.5 × 3% = 1.5% Yt

Changes in Labor

Now consider, second, the effect on output of a change in the labor force from its current value Lt to a value Lt + ∆L—an increase in the capital stock by a proportional amount ∆L/Lt. In this production function Lt is raised to a power, 1-α, so we can apply our rule-ofthumb for the proportional growth rate of a quantity raised to a power to discover that the proportional increase in output from this change in the labor force is: ∆Y ∆L = (1− α ) Yt Lt Thus if the diminishing-returns-to-scale parameter α were equal to 0.5, and if the proportional change in the labor force were 1%, then the proportional change in output would be:

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∆Y = (1− 0.5) × 1% = 0.5% Yt

Changes in Total Factor Productivity

Last consider, third, the effect on output a change in total factor productivity. A proportional increase in total factor productivity produces the same proportional increase in output: ∆Y ∆A = Yt At Thus if the proportional change in total factor productivity were 2%, then the proportional change in output would be: ∆Y = 2% Yt

Putting It All Together

So if we consider a real-world situation in which all three—the capital stock, the labor force, and total factor productivity are changing—then the proportional growth rate of output is: ∆Y ∆K ∆L ∆A =α + (1 − α ) + Yt Kt Lt At with the first term α(∆K/K) giving the contribution of capital to the growth of output, the second term (1-α)(∆L/L) giving the contribution of labor to the growth of output, and the third term (∆A/A) giving the contribution of total factor productivity to the growth of output.

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Thus this equation is the key. If we know the proportional growth rates of output, the capital stock, and the labor force, and if we know the diminishing-returns-to-scale parameter a in the production function, then we can use this growth-accounting equation to calculate the (not directly observed) rate of growth of total factor productivity A, and to decompose the growth of total output Y into (i) the contribution from the increasing capital stock K, (ii) the contribution from the increasing labor force L, and (iii) the contribution from higher total factor productivity A. One way to view this growth-accounting equation is that it allows one to break down...

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