Download Solow Model of Growth Notes

Solow Model of Growth Notes
April 15, 2011
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Introduction
What is a major de…ciency in the classical macroeconomics model? It assumes that …scal policy will only change the
composition of the economy, but not the total output or employment. Now this paints a picture of the relationship
between sectors in the economy, but what does it really tell us about long-run growth? Not a lot.
Robert Solow recognized this and improved upon the classical model, we call his work the neo-classical model.
The …rst thing he did was allow the classical model to be analyzed over time. This is reasonable given that the
classical model is the long run model, so it seems odd there is no dynamic component to it. Furthermore, the
economy seems to grow continuously, which is hard to characterize in the classical model. Many models now exist
attempting to show how and why growth occurs over time, but many (or most) of them use the Solow model as a
foundation.
So let’s take a look at what Solow did.
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2.1
Model
Assumptions
First, he made a couple of basic assumptions that aren’t too di¢ cult to grasp and only simplify the model as opposed
to change the implications. These assumptions are:
1. There is constant technology in our production function initially (A=1), and we will explore what happens
when technology changes after we develop the model.
The production function initially is Y = F (L; K)
2. There is constant population and the population level is equal to the labor force, L.
I.e., L = P opulation = LaborF orce
3. F (L; K) is a constant returns to scale production function. This means that a doubling of capital and a
doubling of labor results in a doubling in output.
I.e., F (tL; tK) = tF (L; K) = tY , where t can be any number greater than 0.
4. We are in the simplest possible economy: there is no public sector (taxes, government expenditures) and no
foreign market (net exports).
2.2
Output per Worker Production Function
We can use these assumptions to formulate our output per capita production function. In assumption 3, we said t
can be any number greater than 0. A number that satis…es this is L1 . If we replace t with L1 , then we get our output
per capita production function.
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L Y = F L L; L K
Y
K
L = F L;1
y = f (k)
Where y is output per worker and k is capital per worker. This can also be said as the output per capita is a
function of the capital that each worker is endowed.
The output per capita function can be graphed with output/capita on the y-axis and the capital per worker on
the x-axis.
Why did I draw the function like I did?
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The graph says that as k increases, the amount of output per worker increases.
It also says that as a worker gets more capital, the less they will be able to take advantage of this capital.
There is another way to say this, the slope of f(k)=change in output per worker/change in capital per worker
decreases as you increase capital. This implies that we have diminishing marginal returns to capital.
Does this contradict the assumption of constant returns to scale? Constant returns to scale (CRS) said that if
we double our factors of production, our output will double. Here, we are just changing our capital (or capital per
worker). Doubling one input will not double output in a CRS production function.
2.3
The Role of Savings in the Economy
What can people do with their money in this simple, no government, no foreign market economy? They can save or
they can consume.
How much do they save?
S = mps Y
S
Y
L = mps L
s = mps y
Where s is savings per worker and y is output per worker. We can add this savings function to our graph since
increasing k will increase y which will increase s.
But what did we learn from the classical model? We learned that if everything clears in the loanable funds
market, the amount of investment (I) in the economy must be equal to the total savings (S). Because there is no
government, private savings by workers must equal total investment.
S=I
S
I
L = L = mps y
2.4
The Evolution of Capital Stock Over Time
So our investment per worker is equal to our savings per worker. But remember that this is a dynamic economy,
meaning that the capital stock will change over time. There are two ways that the capital stock will change. The
…rst aspect to consider is that over time capital goods (machines, computers) breakdown, become outdated or are
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unusable without refurbishing. We call this breakdown of capital over time, depreciation. So the …rst thing …rms do
when the new year is up is to replace this depreciated capital.
The next step is after replacing depreciated capital, they then take the remaining investment and put it into
new capital. This will increase their capital stock. There is a possibility that they are unable to replenish their
depreciated capital, thereby lowering their overall capital stock. We call the change in the capital, net investment at
it can be de…ned as:
N et Investment( k) = gross investment(mpsf (k)) depreciation ( k)
This turns the model into a dynamic model where capital is di¤erent in each period and that change is a function
of investment (savings) and depreciation. We can characterize this mathematically in per capita terms.
k = mps f (k)
k
Let’s de…ne a few terms here. First we know that savings per worker is simply mps f (k) as we de…ned above.
The depreciation of capital is characterized by k where 0 < < 1 and simply says we lose a fraction, , of our
capital stock each period due to depreciation. If is 0.1, then we need to replace 10% of our capital each term due
to depreciation.
2.5
Finding the Steady State
So imagine that an economy is endowed with k0 units of capital. In this initial period, they will have mps f (k)
investment and spend k on their depreciated capital. This means that net investment in the initial time period is
k0 = mps f (k0 )
k0 .
The capital in the next period will be greater than the initial time period, since k0 > 0. What happens in this
next period? Now there is k1 = k0 + k0 units of capital. The net investment in this period is k1 = mps f (k1 ) k1 .
In the following time period, there is k2 = k1 + k1 units of capital. . .
The capital per worker continues to grow until mps f (k) = k. At this point, all investment is going towards
replacing depreciated capital. After depreciated capital is replaced, there is no more investment to put into new
capital. There is no more growth in k, implying the output per worker and savings per worker is constant. We call
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this point the steady state, as all the important variables of the model remain constant. The capital per worker
amount at this point is called the steady state level of capital per worker, k .
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3.1
Comparative Statics
A Change in Savings
The steady state is an important point, as it tells us that given a constant level of technological progress, a constant
level of depreciation and a constant level of savings in the economy, the maximum output that a country can reach
is the steady state level of output. This has a feel of the classical model to it, but there is a signi…cant di¤erence:
the maximum level of output can change.
What happens if the savings rate increases when an economy is in a steady state? There is no change in the
depreciation curve, but the savings/investment curve shifts up and mps become mps’where mps0 > mps. Now the
investment level at our original steady state level of k is enough to cover depreciation and have money left over.
This increases the level of k until we reach a new, higher steady state level of k .
3.2
A Change in Technology
But rarely does the savings rate permanently change. If savings increases for a certain period, then declines back to
its original level, the steady state level of output will end up remaining the same. But what can change in our model
that is almost always permanent? Technology.
Imagine that technology, A, instead of being 1 as we have assumed, doubles to 2. Now, at each level of capital per
worker, the output is doubled. If the savings rate remains unchanged, the savings/investment curve shifts up since
there is more output. This will cause the net investment at the original steady state level of capital to be positive,
increasing the level of capital in the next period, and the period after that. . . until we reach a new steady state at a
higher level of k.
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3.3
The Golden Rule
According to this model, our highest level of output would occur if everyone saved all their money. But is this what
we want, the highest level of output? No. What we want is for per capita consumption to be at its highest level.
What is this level? We can determine that level of capital by …nding where consumption per capita in the steady
state is maximized. We call this the Golden Rule Steady State Level of k.
Golden Rule steady state level of k maximizes per capita consumption. This is found on the graph where the
di¤erence between k and y = f (k) is the largest. We know this because in the steady state, the savings line will
equal the depreciation line, mps f (k) = k. The consumption per capita in the steady state is the income per
capita that is not saved, which equal depreciation. How do we maximize the di¤erence between y = f (k) and k?
We …nd the point on the production function where the slope of the production function is equal to the slope of
the depreciation line. (This can be proved mathematically.) If the steady state level of k is above the Golden Rule
steady state, the additional output per worker is less than the depreciation, cutting into consumption. The opposite
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argument holds if we are below the Golden Rule k.
So it is important to remember that this model is implying that excessive growth is not necessarily the goal.
We still want to maximize consumption, however, a balance between savings, investment, capital accumulation and
depreciation is needed.
3.4
Population Growth
So far, we have assumed that the population of the economy is …xed and everyone works. Relaxing the “everyone
works” assumption trivially changes the model. But what about assuming that the population grows over time?
Think about what is going on when one year, there are more workers than the previous year. If you’re having
trouble imagining what happens, think about what would happen if there was no depreciation, but there was a
bigger workforce every year. Your net investment in this case would be:
k = mps f (k) nk
Where n is the population growth. Why this characterization? If the labor force grew 10% per year, then a …rm
would gather their total investments, increase their capital stock by 10% for the new workers, and then put the rest
into new capital.
If we add on the fact that there is still depreciation, our net investment becomes:
k = mps f (k) nk
k
k = mps f (k) (n + ) k
What does population growth do to our steady state level of capital then? Now that we have to replenish
depreciated capital and provide capital to new workers, we have less to invest in new capital. In fact, if we are at
a steady state and population growth occurs, the net investment at our old steady state is negative, as we’re not
able to take care of depreciated capital and then provide capital to the new workers. As a result, our capital stock
decreases until we reach a new steady state at a lower level of capital.
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Implications
What are the main implications of the Solow model? First it says that if two countries have the exact same production
capabilities (capital and technology) and the same level of savings, they will reach the same steady state level of
growth. Additionally, countries grow quicker when there is less capital per worker than when there is more. These
two points imply that providing countries with the appropriate technology should inevitably get them to the same
steady state level of output. Poor countries should eventually catch up to rich one.
Is this the case? Let’s look at the GDP/capita for a couple groups of countries. The US, Singapore, UK, Italy,
South Korea. . . they have followed a similar trend since 1950. They may have varied in their rates of growth, but
not so much as to say the implications of the Solow are incorrect.
If we look at Ghana, Uganda, Niger, Congo and Sierra Leone, all these countries appear to have a similar growth
rate, which is to say, not much growth at all. The African countries almost appear to be in more of a steady state
than the OECD countries, but they are unable to catch up to the OECD countries.
So it looks as though the OECD countries have similar enough technology, culture and policies to converge with
one another. Similarly, the African countries are similar enough that they have converged with one another. But the
convergence between the two groups of countries is not seen.
The lack of convergence may be for reasons beyond technology, culture and policies. One may be the human
capital of the workforce. Human capital is the knowledge of the labor force in terms of experience, education,
etc. A labor force with more human capital will be more productive, leading to an increased steady state. Taken
together, it is likely a combination of these factors that is leading to the lack of convergence between African and
OECD countries. As a result, the convergence …nding is updated and called conditional convergence. Conditional
convergence implies that countries with the same foundations, economically and socially, tend to converge and rich
countries grow faster than poor countries.
Things to Think About
The Solow model is a nice, concise model that has some interesting implications. Speci…cally, poor countries
could be made better o¤ if they are given the same factors of production as rich countries. What can we learn from
this in the context of improving the world as whole? Can we simply provide technology to poor countries and let
them go, or is there something else that should be done? If providing technology is not the answer, what is?
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