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
delong@econ.berkeley.edu

April 5, 1996

Review
Successes
Failures

Review

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 )a

largely for the sake of convenience: this production function has (a) diminishing returns to each factor, and (b) constant returns to scale.

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.

Successes

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 year.

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 WWII.
• 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 a=0.3).

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 efficiency.

Failures

The international distribution of productivity, income and wealth.

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 s/(n+g+d).

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 right number.

Yet out in the real world capital appears to be vastly, vastly more productive than the returns people get from investing suggest.

Ways out:

• reverse causation; demographic transition
• externalities
• there are human capital people
• there are equipment investment people
• there are measurement error people
• there are public infrastructure people

The debate continues.

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Econ 100b

Created 4/30/1996
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