Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke
Economic growth
Readings: Jones Ch 2, 3
Recall the previous equation with population growth
What happens when there is an increase in the investment rate?
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The model predicts that countries with higher investment rates will tend to have higher
levels of output per capita (and higher levels of capital per capita)
What happens when there is an increase in the rate of population growth?
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The model predicts that countries with higher population growth rates will tend to have
lower values for output per capita (and lower capital per capita)
Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke
Note: We could also examine the effects of an increase in the depreciation rate, using the
previous graph. The results are the same, both qualitatively and quantitatively. But we don’t
expect there to be great differences in depreciation rate across countries.
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There is some empirical support for the hypothesis that countries with higher population
growth rates have lower output per capita
Examining the growth rate of k:
This model maintains the key conclusions we mentioned before in a model without population
growth in terms of convergence and its transition dynamics.
Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke

If two countries have the same levels of productivity, depreciation rate, investment
rate and NOW POPULATION GROWTH, they both use the same production
function, but one country starts out at a lower level of k and y, that country will
You can also show that:
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
If two countries start at the same level of y, but one of the countries has a higher
investment rate, then that country will grow at a faster rate initially
When a country raises γ it will begin to grow at a faster rate (though not forever
because it will again reach a steady state in the long run)
This version of Solow’s model maintains a key limitation: There is no growth in k or y in the
long-run
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
That is the implication of these two variables settling down to a steady state
You might conjecture that a continual decline in the population growth rate could be a
plausible explanation for persistent increases in output per capita. But it is not.
Now we will allow for productivity growth

If productivity grows at a constant rate, g, then, we can write this as the following
differential equation

As before, we can solve this sort of differential equation for the level of A:

Productivity growth occurs whenever there is a new idea about a product innovation or a
new idea about how to produce a product more efficiently

We will see that productivity growth is the only thing that yields a plausible explanation
for growth in y, k or any other per capita variables
Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke
It would be nice if we could put the model in terms of variables that do not grow. Then we might
be able to obtain variables that achieve steady states.

We saw previously that by dividing by labor achieved steady state when that was the only
thing growing. Since we now have labor and productivity growing, we can try to divide
by both to see if that will give us a transformation of variables that achieves steady state.
Recall our newest production function
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If we divide by AL we get:

Define

Then the production function becomes
These transformed variables are defined as:

output per effective labor unit:

capital per effective labor unit:

where AL measures
Now, return to our capital accumulation equation
Define
We now have
Then dividing by K yields:
And so we obtain:
We need an expression for the growth rate of capital in terms of capital per effective labor unit.
Given
Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke
We know we can take logarithms:
Then take time derivative of this equation
And finally obtain
And the last equation becomes:
And plugging this result into:
Gives
which can be written as:
Does this model achieve a steady state?
Yes, starting from any positive level of the dynamics will push the economy in the direction of
this steady state.
Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke
Now we can examine an important question that we’ve asked before with these models: What
happens to the economy when the investment rate increases?
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An increase in the investment rate shifts the investment per effective labor unit line up
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Capital per effective labor unit begins to rise since investment exceeds depreciation, with
both of these now measured per effective labor unit
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Finally, the economy settles down to a new steady state level of capital per effective labor
unit
•
We can also analyze this model using the growth rate of capital per effective labor unit
form of the equation
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Graphing this equation yields the following which illustrates how an increase in the
investment rate causes fast growth of capital per effective labor unit initially,
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But, as the economy gets closer and closer to the steady state, the growth rate of capital
per effective labor unit slows down
Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke
•
And using the last equation it is easy to solve for the steady state value of capital per
effective labor unit:
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And steady state output per effective labor unit comes from putting the previous solution
into the production function:
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This model yields the same predictions as before:
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The farther an economy is below its steady state the faster it grows
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The farther an economy is above its steady state the slower it grows
We now examine the growth of output per capita
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We can get output per capita from the definition of output per effective labor unit. Since
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Multiplying both sides by A gives: If we divide by AL we get:
Which means
And from our production function, in per effective labor units:
We know that
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So finally,
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Using the constant growth assumption for A, we get:
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This equation tells us that:
Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke
•
Thus, when the investment rate went up, y grows faster than g for a while, but eventually
the economy returns to steady state and y growth rate returns to g
How does output per capita behave over time?
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Recall that:
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Using our expression for A, we obtain: :
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This equations tells us how y behaves over time in this model:
–
Changes in output per effective labor unit shift the level of y, but don’t affect its
long-run trend which is given by the trend in A
–
Changes in A0 will also shift the level of y, but not its trend line
•
Thus we see that an increase in the investment rate will cause the level of y to increase,
but not affect its long-run growth rate
•
The following graph illustrates precisely what would happen in this model, plotting how
the log of y changes over time:
Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke
Once again the model predicts convergence and transition dynamics
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The economy converges to a steady state for fixed values of the parameters
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The farther below/above steady state the faster/slower the economy will grow.
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Suppose there are two economies named
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InitiallyBehind (IB)
–
InitiallyAhead (IA)
Econ 701 – Survey of Macroeconomics
Spring 2015/ Manopimoke
How well does the data support the convergence hypothesis?
However, when we broaden the set of countries to include all countries in the postwar period,
there is absence of convergence
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Many of the countries that are much poorer than the developed countries of the
world are growing slower than those developed countries