## Lecture notes for Macro I

Endogenous Growth

Prerequisites for Endogenous Growth

Take the simplest neo-classical growth model and assume away the exogenous growth rates. Can we anyway generate sustained long run growth? Let the CRS production function be

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and capital accumulation

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Now

Now , what happens to *g**y *as time goes to infinity.
The RHS of is non-negative. As long as *s *is strictly
positive, capital accumulation continues. So if *s *is kept
positive *F**K *falls forever since *F**KK *<0. In other words, if *F *shows CRS in *K *and
*AL, *then it shows DRS in *K *alone. If the Inada
condition

then the economy thus is bound to approach zero
growth due to and the fact that *s *is finite.

With depreciation or population growth we need that the limit in is large enough to account for depreciation and population growth.

What is the fix? We need to have a production
function that shows CRS *in factors of production that are
produced *(*i.e., that are accumable*). The most simple
and straightforward way is to let the productivity index *A *depend
on how much capital the economy has accumulated. We could think
of this as a *Learning by doing *mechanism.

So in the simplest endogenous growth model we
could assume that the level of knowledge, measured by *A *is
given by

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Substituting into the production function we have

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From the CRS assumption we have

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The term *F*(1,*aL*) is constant so
output is linear in *K *alone, i.e., shows CRS in *K *alone.
In some of the already classic articles by Romer and others the
constant was called *A. *Then we can write *Y=AK, *thus
these models go under the name *AK-models.*

The basic ingredient in endogenous growth models is thus to construct a social production function that has CRS in produced factors of production. There are basically to lines of models that achieve this,

Knowledge or Human Capital in the production function. Knowledge and/or human capital is surely producable as you hopefully experience during this course. If then the production function has CRS in capital and human capital

together.Increasing specialization. Since Adam Smith we know that increasing market sizes permits increasing specialization. So as production increases the economy may be more and more efficient. This may be another source of endogenous growth.

Growth rate in the simplest learning by doing (AK) model

Let the constant *F*(1,*aL*) be
called __A__*. *Then we have

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So the growth rate of the economy depends on
the growth rate of capital which in turn depends on the net
savings rate. With a given savings rate *s*

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In this simple model it is easy to endogenize the savings decision. With CRRA preferences

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the Euler equation, which we can derive using Optimal Control, states that

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where *r**t *is the net return on savings. If this equals the full
marginal return on capital (why should it not?) – *r**t *= *A* so

The only saving rate which is consistent with and the transversality conditions is that the growth rate of consumption equals the growth rate of output. What would happen otherwise? So

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Note that this model implies very simple dynamics – the economy is always in its steady state growth path. Shifts in some parameters or a shock to the capital stock implies an immediate jump to the new steady state growth path. Can you explain why?

Aggregate versus Private Knowledge

In the previous example there was no
distinction between aggregate and private knowledge. This meant
that the learning by doing effect of accumulating capital was
fully internalized in the savings decision by the individuals.
This may, reasonably, not be the case. Take the opposite view
instead. Assume that there are an infinite number of identical
small firms indexed by the rational numbers *i *on a unit
interval [0,1]. Now that the production function for the
individual firm is given by

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The level of knowledge is proportional to the aggregate stock of capital and is identical to all firms.

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Now the *aggregate* production function
can be written

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However, when a firm decides to increase its capital slightly, the effect that has on the aggregate capital stock and thus on the stock of knowledge is negligible. This since

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So the *private *return on capital is

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which is smaller than the social which equals

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Let, for example, the production function be

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The private return to capital is in this case

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Since this is the interest rate faced by consumers the Euler equation for consumers with CRRA utility gives

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So we see that growth in this model is to low compared to the welfare maximum given by

R&D

In the previous examples knowledge was produced as a by-product of capital accumulation. We can easily change that and introduce a specific sector where knowledge is produced. So let

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where *a**L *is the share of labor that is allocated to knowledge
accumulation. This sector could be thought of as an *R&D *sector
or a schooling sector. The are some decreasing returns to scale
in this sector if g is smaller than unity. The production function for
output and the capital accumulation is given by

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For now we assume that savings and the share of labor allocated to R&D is fixed and exogenous. Now lets define

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So, as in the Solow model

Now this reminds us about the Solow model and a steady state occurs when is zero. Let us look at a plot of the components in the RHS of .

We see that a large R&D sector and thus a high growth rate of knowledge implies a lower steady state capital/effective labor ratio and vice versa. What would happen if we got a shift from a low to a higher share of labor in the R&D sector?

In this model the savings rate is unimportant
for long run growth as in the Solow model. But, now growth is
(almost) endogenous. To really explain growth we would like to
let *a**L *be
determined within the model. This is the next step in the
development of this model. We would like to have some firms who
do the R&D and hire labor for this. Obviously we need to
introduce some property rights for that purpose. Note that in
such a model there is no reason to believe that growth is at its
socially optimal level.

Growth from increasing specialization

Now let us turn to the other source of sustained growth – specialization. Let us think of an economy with two sectors. Final output which is produced using labor (with inelastic supply) and a range of intermediate goods. The latter are produced with capital only. The production function for the final good is

where each *x**i,t
*represents the amount of a particular
intermediate input that is used at time *t*. *M**t *is the total number of
intermediate inputs in use at *t.* We are going to look at
symmetric equilibria where all *x**i,t
*are equal and denoted *x**t* so

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where we have normalized the labor supply to unity. The specification in implies that there is positive returns to specialization. Consider the case when we double the number of inputs but use each in half the amount. Output is then

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Now look at the intermediate goods production
sector. Here capital is transformed into intermediate goods by *M**t *firms. There is a fixed
cost of running a firm (paid in capital) so the capital
requirement of firm *i *which produces *x**i,t* can be written

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Now we assume that the final goods sector is competitive and that it buys intermediate goods so that its marginal product equals its price

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This is the inverse demand faced by the intermediate goods producing firms so their profits are

where *r**t *is the user cost of capital (interest rate plus
depreciation). From follows that the profit maximum for
intermediate goods producing firms is

where the last equality comes from the assumption of a symmetric equilibrium. Now assume that there is no barriers to entry so profits in the intermediate goods sector are zero.

Lastly, we require that the capital market is in equilibrium so that supply of capital (which is given at any point in time) equals its demand. This implies

Substituting from into we get

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Lastly, substituting this into we get

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Substituting these results into the production function for aggregate output we get

so we are back into the simple *AK *model.
Now we just have to make some additional assumption about
savings, for example that it is chosen optimally by individuals
facing some interest rate, for example the one established at the
capital market and given by if there are no capital income taxes.
That savings rate determines the growth rate of capital which
from is identical to the growth rate of output.