Lecture Twenty Seven
How Well Do Growth Models Work?
(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
April 5, 1996
The basic growth model can be summarized in the figure below:
work with all variables in "per worker" amounts. Draw the "per unit
of labor power" production function little y = little f of little k.
Squash this production function line down toward the x-axis by, at
each point on the function, multiplying it by the savings rate s.
Draw a straight line starting at (0, 0), corresponding to the
"required" rate of investment that will keep the capital-output ratio
constant: enough additional savings and investment to boost the
per-worker capital stock by g%, to provide capital for the n% new
workers coming into the economy, and to replace the annual
depreciation of the capital stock.
And where gross savings and investment are equal to the "required"
investment to keep the capital-output ratio constant, that is the
steady state of this economy: capital per worker tends to
gravitate to this point of attraction k*; output per worker tends to
gravitate to its point of attraction y* = f(k*); it may well take
generations for this economy to get to the steady state, but that is
where it is heading.
Our production function is:
Yt = F(Kt , Et Lt
largely for the sake of convenience: this production function has (a)
diminishing returns to each factor, and (b) constant returns to
Note that we have "EL", where "E" is the efficiencyof labor is
going to be our measure of the state of technology. As time passes, E
will grow--it is as if one worker can handle two machines as well as
two workers could before, and this is going to be the source of
long-run growth in living standards in this particular model.
You could think of other ways to think about technological progress
rather than assuming that it improves the efficiency of labor
directly--that F stays the same but that each worker is, over time,
more and more valuable. But this is the simplest to work out, and it
doesn't matter much.
How well does this model work? In some cases very well indeed.
Consider Germany's recovery from the devastation of World War II:
World War II left Western German in the immediate aftermath of the
war with an output-per-worker level that, in real terms, seemed to
correspond to what output per worker had been back in the 1890s:
perhaps 1/3 of output per worke on the eve of World War II.
Yet in the immediate aftermath of World War II the rate of return on
what capital there was was very high--depreciation was very low--and
investment was a high share of national product. Almost immediately
after 1946 the German economy began to grow very rapidly, closing
perhaps 1/20 of the distance back to a pre-WWII growth path in each
And as German production began to approach pre-WWII levels, German
growth began to slow down: so that today it is, and has for a decade
been, no faster than in the United States.
- Higher level than would perhaps have been predicted before
- Present-day growth rate no faster than suggested by growth of
"efficiency" of labor elsewhere in the world.
- Return on investment now more-or-less back to normal levels.
Or consider U.S. aggregate growth in the aftermath of WWII as well:
it looks very stable: Y grows, K, grows, E and L grow:
The first column is the growth of output per worker; the second
column is the share of that growth that is due to increases in the
capital stock; the third column is the computed growth of E (with
Biggest shift is what we call the productivity slowdown. Talk
about the productivity slowdown.
But overall picture is of pretty constant growth supported by a
more-or-less stable capital-output ratio and steady growth in labor
The international distribution of productivity, income and
Log scale; output per worker on the left hand side; log of s/(n+g+d)
on the right hand side.
Recall from last time that we had:
where a, we said, was about 0.3. It seems not unreasonable to suppose
that everyone in the world has "access" to the same "technology"--in
which case a country's Y/L should be directly proportional to its
That's what the figure above seems to show. Why, then, say that this
is a failure? Because a seems to be too large. Slope = 5 which
implies an a=0.83.
And we also know that in this production function a is the "share" of
output that is attributable, in a MPK sense, to capital.
with a pre-tax real return on investment of 10% per year, and a
capital-output ratio of 3, this equation suggests that a=0.3 is the
Yet out in the real world capital appears to be vastly, vastly more
productive than the returns people get from investing suggest.
- reverse causation; demographic transition
- there are human capital people
- there are equipment investment people
- there are measurement error people
- there are public infrastructure people
The debate continues.