Lecture Twenty Eight
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
http://www.j-bradford-delong.net
April 8, 1996
Administration
Growth Accounting: Increases in Capital Intensity
Growth Accounting: Increases in the Efficiency of Labor
Results from Growth Accounting
Growth Policy
Administration
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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