THE SOLOW MODEL Ghulam Samad

1
THE SOLOW MODEL (Ghulam Samad) Introduction What
are the sources of output growth in capitalist
economies? The now-standard model of economic
growth was devised by the Nobel prizewinner
Robert Solow in the 1950s. It forms the subject
of the next two lectures. First, note some
stylized facts about growth in developed
economies. A good model should account for these.
(3) and (4) are, perhaps, the puzzling ones. 1.
Output (Y) per worker (N) grows through time. 2.
Capital (K) per worker grows through time. 3. Y/K
has no long-run trend and is typically around
1/3. 4. The rate of profit on capital, p, has no
long-run trend. 5. The share of labour income in
national income has no long-run trend and is
typically around 2/3. N.B. (5) follows from (3)
and (4), assuming that all income is wages or
profits, because they imply that the profit
share, pK/Y, is trendless and therefore so is the
wages share, wN/Y1- pK/Y. A simple implication
of wN/Y being roughly constant is that w, the
wage, grows at about the same rate as
output/head, Y/N.
2
Building blocks of the Solow model The first is
the aggregate production function, which we
write Y F(K,N) (1) This involves a drastic
simplification of reality, by suppressing
heterogeneity of (a) (Value-added) output. There
is only one type of output. (b) Machines. Not
only are all machines the same, but all are made
of output. (c) Labour. No skill differences,
for example. Given this, eq.(1) states that the
amount of output produced depends in a definite
way on the amount of capital input and labour
input utilized. F is the function that tells us
what Y will be for given K and N. We can think of
F as depending on the state of technology, which
you might think of as a set of blueprints. Are
you going to swallow the story so far? If you are
worrying that advanced technology is partly
embodied in machines, so that the separation of
capital and technology is not really
acceptable, then I agree. Put this worry to one
side for the moment. 
3
Returns to factors and returns to scale We assume
factors have positive but diminishing marginal
products. Think of adding computer terminals,
one-by-one, to, say, the SOCCUL School Office,
while keeping labour input constant. We also
assume constant returns to scale (CRS) doubling
K and N results in a doubling of Y. This seems
reasonable - surely we could clone the economy
- but lumpy capital is a complication (you
cant do much with half a bridge!). Algebraically,
CRS implies that 2Y F(2K, 2N) (2) and indeed
for any number x xY F(xK, xN) (3) Now for a
fancy trick. Let x 1/N in eq(3), then Y/N
F(K/N, 1) (4) This says that, given CRS, output
per head is determined just by capital per head,
K/N. Well, that is the whole point of CRS -
scaling up K and N in equal proportion raises Y
in the same proportion, so it leaves Y/N
unchanged.
4
We can now save a bit of ink by writing output
per head as y and capital per head as k.
Therefore (4) becomes y F(k, 1) (4) And to
save even more ink, we can define a new function,
f, linking capital per head to output per head,
thus (eq.5 just defines f) y f(k) F(k, 1)
(5) The following picture from BLA summarizes
where we have got to
5
Here we see constant returns to scale (which is
what allows us to plot Y/N against K/N otherwise
we coulddnt). We also see diminishing returns
to capital, embodied in the decreasing slope of
the curve. To see this, it is easiest for a
moment to think of N as fixed. Then AB and CD,
equal to one another, correspond to equal
additions of K but to quite different additions
of output - CD is much smaller than AB.
6
The sources of growth It is clear from the
preceding picture that, in this model, there are
just two sources of growth in output/head.
Capital accumulation, which is clearly going to
be linked to the amount of saving, will move us
along the curve, provided that it raises capital
per head. Technological progress will shift the
curve upwards, raising output per head for given
capital per head. This is illustrated by another
BLA picture.
7
Will positive savings necessarily raise capital
per head and therefore output per head? This
breaks down into several questions. (a) Are
savings translated into gross investment. The
Solow model assumes the answer is yes. (b) Does
gross investment translate into more capital?
The answer is not entirely, as we must subtract
the depreciation of old machines. And the more
capital we have, the higher is depreciation. (c)
Does more capital mean more capital per head?
That depends on how fast population is growing.
If population growth is fast enough, capital per
head might not be growing at all. In that case
all savings would be going into equipping new
workers with the same tools as the old ones have.
This is called capital widening. Where capital
per head is rising, we refer to capital
deepening. So net investment goes into a
combination of capital widening and capital
deepening. What the Solow model is going to tell
us is that there is a natural limit to capital
deepening - a point will be reached where all
savings are soaked up by depreciation or
population growth.
8
From output to capital accumulation The
production function gives us the link from
capital (per head) to (output per head). We now
need the inverse link from output to capital
growth. We follow BLA ch11 and temporarily assume
zero population (and labour force) growth. We
also make three simplifying assumptions 1. The
economy is closed. 2. There is no government, so
G and T are zero. 3. Savings are proportional to
income. Simplifying assumptions are supposed
not to matter much, in the sense that wed get
similar (but more complicated) results if we
dropped them. We should ask later on whether
these assumptions are really of this kind. (2)
and (3) are, but maybe not (1).
9
Using assumptions (1) and (2) we have two
identities Y C I (6) (whatever part of
output that is not consumed is invested) Y C
S (7) (whatever part of output/income that is not
consumed is saved) Using (3) we have S
sY (8) Since (6) and (7) give us SI, we can
plug that into (8), and put in some time
subscripts, to get It sYt (9) How are we
going to interpret equation (9)? In the simple
(Keynesian) income-expenditure model that you are
familiar with, the causation in (9) runs left to
right. Investment, assumed exogenous, causes
income. So a fall in investment can cause
recession and unemployment, via the implied fall
in aggregate demand.
10
In the Solow model, we assume that causation in
(9) runs right to left. A mechanism is assumed to
exist which converts full-employment savings into
investment (so investment is certainly not
exogenous). What is the mechanism? In essence,
the idea is that - in the medium-run - the real
rate of interest adjusts to achieve this (see BLA
7 and especially BLA 14.4, p.302). One empirical
justification for this assumption, for
medium-term analysis, is the fact that recessions
do not last for ever To put this key assumption
more simply, the Solow model assumes full
employment of the factors of production output
and income are determined from the supply side by
the production function, and savings cause
investment. We now derive from (9) an equation
for capital accumulation. We allow for
depreciation by supposing that a fraction d of
the capital stock decays each year - think of d
equal to, say, 0.08 (it varies across types of
capital good). So we can write
(10)
And substituting in from (9), we get
(11)
11
We can write this in per-worker terms by dividing
through by N, so
(12)
Remember that we have assumed N constant here, so
I have left off the time subscript. Finally,
since we are going to be interested in the rate
of capital deepening, I can move capital per head
at t to the left hand side
(13)
This equation says that net investment during
year t equals savings minus depreciation (all
expressed in per capita terms). Or capital
deepening equals savings per head minus
depreciation per head. So (13) gives us the
dynamic behaviour of capital per head, but -
given the production function - we also get from
this the dynamics of output per head. Before
seeing this in a diagram, we can write (13) more
compactly as
(14)
12
What we now have can be shown in a series of
diagrams in which output per head is plotted
against capital per head. So we can plot (gross)
investment per worker - equal to savings per
worker - by scaling down the output per worker
curve. Then we can superimpose depreciation per
worker as the red straight line. These are the
two parts of the right hand side of equation (12).
13
Now we consider an economy that starts at point
B, with capital per head as indicated by point A.
There is going to be positive capital deepening,
since point C lies above point D.
14
So, over time, the economy is moving to the right
in the diagram. Capital per head and output per
head are both rising. But there is a natural
limit, at capital per head K/N. Why is that?
Diminishing returns are critical. More capital
means both more output (and saving) and more
depreciation. But as K rises, its marginal
product falls, so that eventually the extra
saving is all absorbed by extra depreciation. And
that is where output/head and capital/head reach
steady state.
15
What happens if an economy that has reached its
steady state, K0/N, increases its rate of saving?
We can see below that (a) the investment per
worker line shifts up, so (b) we get a period of
capital deepening and rising output per head, but
(c) eventually diminishing returns bring the
advance to an end, so we finish at a new steady
state with higher capital per head (K1/N) and
higher output per head.
16
The time path of output per head after a rise, at
time t, of the savings rate from s0 to s1 can be
seen below.
17
Assessment This model may seem far-fetched after
all, its prediction that output per head reaches
a steady state is contrary to the facts. What has
been left out is technological change. We will
soon see that this can be added to the model,
with the effect that the economy converges to a
steady-state growth rate of output per head,
rather than a steady-state level of output per
head. We can illustrate this with another BLA
picture, showing the dynamic effects of a rise in
the savings rate in the model with technological
change.
18
But, aside from this fix, are any of the Solow
models predictions borne out by the facts? In
other words, is the model of any use?! The answer
is yes. Blanchard has a box about French economic
growth after World War II. The model predicts
that destruction of part of the capital stock of
an economy that starts in steady state will be
followed by a period of faster-than-normal growth
of output per head. For France we
have Population loss in WWII 550,000, or about
1.3 of the pre-war total of 42 mn. Capital loss
in WWII roughly 30 of pre-war total. This
constitutes a substantial negative shock to
capital per head. Between 1946 and 1950, French
GDP per head grew at 9.6 per year, well above
the growth rate before or since. So, as the Solow
model predicts, post-war capital accumulation in
France yields particularly rapid growth, because
(a) the marginal product of capital is high, and
(b) the amount of savings absorbed in
depreciation is low. Is this the whole story?
No. Technological advances embodied in the new
capital goods were partly responsible for the
rapid growth in this period.
19
Too much of a good thing? Optimal saving and the
golden rule. Go back to the Solow model without
technological change. Is there an optimal rate of
saving? Since, we assume, utility depends on
consumption, how does consumption per head, c,
depend on s. Plainly, in steady state, s0
means c0, but also s1 means c0! In fact,
steady state c as a function of s looks like this
sG is the so-called golden rule rate of
savings. If sltsG, there is a trade-off involved
in raising s - future versus present
consumption. If sgtsG, a rise in s would be bad
for consumption now and in the future. In
practice s is usually far below estimates of sG.
Why is that? Does it matter? See BLA box on page
231.