Growth Accounting (Economics 100b; Spring 1996)

Professor of Economics J. Bradford DeLong
601 Evans, University of California at Berkeley
Berkeley, CA 94720
(510) 643-4027 phone (510) 642-6615 fax
delong@econ.berkeley.edu

April 8, 1996

Growth Accounting: Increases in Capital Intensity
Growth Accounting: Increases in the Efficiency of Labor
Results from Growth Accounting
Growth Policy

Growth Accounting: Increases in Capital Intensity

One of the key questions to be answered in any study of economic growth is as follows: what share of increases in per-capita (or in total) income and productivity is due to capital accumulation--is due to increased capital-per-worker? And what share is due to technological progress--that is, the increased "efficiency" of labor in the production function:

(1) Y = F(K, EL) or Y/L = F(K/L, E) ?

Consider, first, just changes in the capital-labor ratio--increases in the amount of capital that the average worker has at his or her disposal. Suppose we (because the internet still does not make it easy to use Greek letters) use "D" for the Greek "Delta" to mean "difference", and investigate the effect of boosting the K/L ratio by some amount D(K/L) on Y/L.

In other words:

(2) Y/L + D(Y/L) = F (K/L + D(K/L), E)

We can rewrite this as:

(3) Y/L + D(Y/L) = F(K/L, E) + (MPK) x D(K/L) + [small higher order terms], using either the definition of the marginal product of capital (as the change in output associated with a small change in capital input) or the definition of a derivative as the best linear--the is, best line--approximation to a function at a particular point.

For any change in capital-per-worker that is not infinitesimally small, of course, the marginal product of capital MPK will not be constant: it will diminish as capital intensity increases. But let's ignore that. Now simply subtract off equation (1) and see:

(4) D(Y/L) = MPK x D(K/L)

Now let's multiply the first term on the right hand side by the capital-labor ratio K/L, and divide the second term by the capital/labor ratio--that doesn't change anything. And let's divide both sides of equation (4) by the initial output per worker Y/L:

(5)

This is a very nice formula for two reasons: first, it gives the proportional growth in output per worker as a function of the proportional growth of capital per worker: if the variable on the left hand side is something like a two percent per year rate of growth of output-per-worker, the variable on the right hand side will be something like a five percent per year rate of growth of capital per worker.

Second, we have seen the expression (MPK x K)/Y before: it is the share of income received by capital back in chapter 3, and it is just the parameter a in the production function:

(6) Y/L = (K/L)a(E)1-a

And in the U.S. today, a is roughly equal to 30%.

Growth Accounting: Increases in the Efficiency of Labor

Now suppose we ran through the same exercise for our same production function:

(1) Y = F(K, EL) or Y/L = F(K/L, E)

but instead of considering increases in the capital-labor ratio, we consider, instead, increases in the efficiency of labor E:

(7) Y/L + D(Y/L) = F (K/L, E + D(E))

(8) D(Y/L) = MPL x D(E), where "MPL" is the marginal product of labor-power, which shows up here because boosting "E" is increasing the efficiency of labor: making a given worker and a given amount of the work day worth more in terms of labor-power.

So once again divide both sides of the equation by the initial "Y/L", and multiply the first and divide the second term on the right hand side by the total amount of labor power--E x L--to obtain:

(9)

And once again (MPL x EL)/Y should look familiar: it is the share of income going to labor, and is equal to 1-a in our familiar production function from chapter 3 (and will be equal to about 70% in the U.S. today).

So we combine equations (5) and (9) to get equation (10);

(10)

This breaks down proportional growth in output per worker--in percent per year, say--into a component that is attributable to increases in the capital-labor ratio, and the component that is due to improvements in labor efficiency--in better "technology" is the word economists use, but this is perhaps not the best possible word. And as long as the capital and labor shares of income do not wander around too much, equation (10) will provide you with a very accurate decomposition of output-per-worker growth.

Results from Growth Accounting

In the post-WWII U.S.: only some 30% of growth in output per worker is attributable to growth in capital per worker; the rest is attributable to increased labor efficiency. (Not unexpected: steady-state growth).

In post-WWII Germany, say: in the 1980s some 30% of growth in output per worker is attributable to capital deepening; but in the 1950s some 70% of growth in output per worker was attributable to capital deepening.

In East Asia over the past generation, a high proportion of output per worker growth is due to capital deepening: in Singapore, some estimates suggest that as much as 100% of output per worker growth is due to capital deepening...

Growth Policy

The Golden Rule again: MPK = r = n + g + d

How can we see that this is the golden rule? Well, suppose we boost the steady-state capital/effective labor ratio k* by a small amount--and keep it boosted forever. our gain to production is MPK (=r); but we have to boost investment by n+g+d in order to maintain the higher k*.

So consumption changes by Dc = r - (n+g+d). Hence the "Golden Rule"

Where is the U.S. today? r is approximately 10% (real, pretax, risky); n=1%; g=1.5% (if we are lucky); d=4%. We would have to boost our capital-output ratio (or our steady-state savings rate) by 50% to get to the "golden rule".

Should we do so? How much do we care about future generations? (They are, after all, going to be richer [we think]).

The savings side:
deficit reduction
IR
As lower capital gains taxes

The investment side:
R&D
infrastructure
equipment/embodiment

Lecture Twenty Eight Handout

(1) Y = F(K, EL) or Y/L = F(K/L, E) ?

(2) Y/L + D(Y/L) = F (K/L + D(K/L), E)

(3) Y/L + D(Y/L) = F(K/L, E) + (MPK) x D(K/L) + [small higher order terms]

(4) D(Y/L) = MPK x D(K/L)

(5)

(6) Y/L = (K/L)a(E)1-a

(7) Y/L + D(Y/L) = F (K/L, E + D(E))

(8) D(Y/L) = MPL x D(E)

(9)

(10)

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Econ 100b

Created 4/30/1996
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