Lecture Twenty Six
Long Run Growth II
(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 3, 1996
Administration
Same Figure as Last Time--but Different Interpretation
A Model with (a) Labor Force Growth, and (b) Improvements in
Technology
Converging to the Steady State
Administration
Same Figure as Last Time--but Different Interpretation
Last time we took a look at a greatly oversimplified model--in
which the labor force was constant, in which there were no
improvements in technology, in which all that was going on was (a)
the production function (constant returns to scale); (b) capital
accumulation; and (c) depreciation.
That oversimplified model could be summarized in the figure below:
work with all variables in "per worker" amounts. Draw the "per
worker" 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 annual
depreciation of the capital stock.
And where gross savings and investment are equal to depreciation,
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.
Increases in the savings rate increase k* (and thus y*); decreases in
the savings rate decrease k* (and thus y*).
Note, also, that there comes a point after which it is not worth
raising your savings rate. After all, in this model steady-state
consumption is given by:
c* = y* - dk* = f(k*) - dk*
Where is c* at a maximum. Well, think: what happens as you move up
and to the right on the graph above. Each unit increase in per-worker
capital k adds the marginal product of capital MPK to
gross output per worker. And we recall from chapter 3
that:
MPK = a(Y/K) = af(k)/k
But each unit increase in per-worker capital k adds d to
depreciation. Diminishing returns. As long as k is relatively
low, MPK > d, and so increases in k* increase steady-state
per-worker consumption. When k is very high, MPK < d, and so
increases in k* decrease steady-state per worker production. And in
the middle MPK = d --the "golden rule" point for this particular
economy.
Today we are going to consider a much more complicated economy -- one
in which there is labor force growth, and are improvements in
technology. But the point of today is to get us back to the diagram
above: the variables on the x and y axes will have a slightly
different interpretation; and instead of having a straight "dk" line
we will have an "(n+d+g)k" line. But we will still be able to analyze
the long-run behavior of the economy using the figure above.
So let's start.
A Model with (a) Labor Force Growth, and (b) Improvements in
Technology
We start with a slightly different production function:
Y = F(K, EL)
where instead of "L" 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.
So we start with our production function. And we add to it three
equations: the first is the growth in the efficiency of labor E from
background technological progress. And let's say that E grows at a
rate g each year.
The second equation is growth of the labor force. The labor force
grows at a rate n each year.
And the third equation we have seen before: the capital stock is
equal to last year's capital stock, plus savings--the savings
rate times output--minus depreciation--the depreciation rate
times capital.
In this expanded model--with improvements in labor force efficiency,
and with population growth--the labor power of the economy measured
in efficiency units is growing at a rate of n+g a year. So let's look
for what has to be the case for the capital stock to match it--for
the capital stock to also keep growing at n+g a year. What has to
happen? Well, gross investment has to be larger than n+g times the
capital stock because of depreciation:
So given n, g, d; the capital-output ratio consistent with a capital
stock growing at n+g a year is given by taking the savings rate, and
dividing it by (n+g+d). If the capital stock is smaller as a
proportion of output than s/(n+g+d), it will be growing faster than
the labor power of the economy; if the capital stock is larger as a
multiple of output than s/(n+g+d), it will be growing slower. So over
time the capital-output ratio will head for: s/(n+g+d).
What does this mean for the long-run destiny of the economy. Well
let's take our production function, and stuff s/(n+g+d) into it where
we can:
And let's think of new definitions for y and for k; instead of
per-worker quantities, let's use:
y = Y/(EL)
k = K/(EL)
they are the per-unit-of-labor-power measures of output and of the
capital stock. So that "true" output per worker is not y but yE, and
"true" capital per worker is not k but kE.
So then:
And note that we can describe the evolution of this economy on
exactly the same figure as we used on Monday--only replacing "dk"
with "(n+g+d)k"
Golden rule changes: now MPK = n+g+d
Converging to the Steady State
So that if k is away from k*, the change in capital stock is:
(n+g+d)(k*-k) - s(MPK)(k*-k)
Recalling that:
MPK = aY/K = a(n+g+d)/s
Change in k is approximately = (1-a)(n+g+d)(k*-k)
Effects of Policies and Other Changes
Here we have to be careful. We are interested both in the
effects of shifts on the steady-state per-unit-of-labor-power
variables k* and y*; and we are also interested in k*E_{t}
and y*E_{t} -- the per-worker values.
- Economy goes to steady state, which is one in which per-worker
capital and output are growing by g each year (and total output is
growing by n+g each year), even though per-unit-of-labor power k*
and y* are constant (and the capital-output ratio k/y is constant;
which it is approximately; thus thought that this is a pretty good
model).
- Increases in s shift the investment curve up--and land you at
a steady-state with higher k* and y*; whether consumption is
higher depends on which side of the "Golden Rule" you are on: is
MPK >, =, < (n+g+d)?
- Increases in n shift the n+g+d line upward--and land you at a
capital stock per unit of labor power that is lower, a lower k*
and y*
- Increases in g have the greatest power to affect per-worker y*
and k*
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