004: Macroeconomic Theory

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004: Macroeconomic Theory

Lecture 22

Mausumi Das

Lecture Notes, DSE

November 11, 2014

Das (Lecture Notes, DSE) Macro November 11, 2014 1 / 12

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AK Production Technology: First Step Towards Endogenous Growth?

In our earlier discussion we have seen that we can have long run growth of per capita income in the Solow model without technical progress if we relax some of the properties of the neoclassical production function.

In particular, steady growth along a balanced growth path is possible if the production function exhibits non-diminishing returns.

A speci…c example of such non-diminishing returns technology is the AK technology where output is a linear function of capital:

Yt =AKt.

The AK model is the precursor to the latter-day ‘endogenous growth theory’which (unlike Solow) explains technological progress in terms endogenous factors within the economy.

When technology interacts with the factor accumulation, there is no reason why the production function would exhibit diminishing returns.

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Justi…cations for AK Technology:

There are many justi…cations as to why the ‘aggregate’production technology might be linear. Here we look at three distinct examples - each providing a di¤erent justi…cation as to why diminishing returns might not work:

1 Fixed Coe¢ cient Technogy - exhibiting complemetarity between factors of production;

2 Production Technology with Learning-by-Doing and Knowledge Spillover (Frankel-Romer);

3 Production Technology with Producible Inputs (Barro).

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Justi…cation for AK Technology: Leontief Production Function

The …rst example of an AK technology is of course the Fixed Coe¢ cient (Leontief) Production Function:

Y_t =min[aK_t,bN_t],

wherea andb are the constant coe¢ cients representing the productivity of capital and labour respectively.

Suppose the economy starts with a (historically) given stock of capital and a given amount of labour force at timet.

Then …xed coe¢ cient technology implies:

Yt = aKt ifaKt <bNt (capital constrained economy);

Y_t = bN_t ifbN_t <aK_t (labour constrained economy).

In the …rst case, the production technology would in e¤ect be AK - although notice that labour is still used in the production process.

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Justi…cation for AK Technology: Learning by Doing &

Knowledge Spillover

A more nuanced justi…cation was provided by Frankel (1962), which was later exploited by Romer (1986) in developing the …rst model of endogenous growth.

Consider an economy with S identical …rms - each having access to an identical …rm-speci…c technology:

Yi =A^¯tF(Kit,Nit) A^¯t(Kit)^α(Nit)¹ ^α; 0<α<1.

Note that the …rm-speci…c production function exhibits all the neoclassical properties.

The term A¯_t represents the current state of the technology in the economy, which is treated as exogenous by each …rm.

Frankel relates the A¯t term to the aggregate capital labour-ratio in the economy - due to ‘learning by doing’:

A¯_t =g K_t

N_t ; g⁰ >0; whereK_t =SK_it; N=SN_it.

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AK Technology: Frankel-Romer Explanation

Without any loss of generality, let us assume:

g K_t Nt

= ^K^t Nt

β

; β>0, Corresponding Aggregate Production Function:

Yt =

∑

^Yît ⁼^S^hÂ^¯^t⁽^Kît⁾^α⁽^Nît⁾¹ ^αⁱ

= A^¯_t(SK_it)^α(SN_it)¹ ^α

= A^¯t(Kt)^α(Nt)¹ ^α.

Notice thatA¯_t is the total factor productivity term - which is given for each …rm, but not so for the aggregate economy.

Replacing the value of A¯_t in the aggregate production technology:

Yt =A^¯t(Kt)^α(Nt)¹ ^α = (Kt)^α⁺^β(Nt)¹ ^α ^β. In the special case where α+β=1, the aggregate production technology is AK.

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Justi…cation for AK Technology: Producible Inputs

Another interesting justi…cation for the AK Technology comes from Barro (2004).

Barro considers a production technology which uses capital and human capital (skilled labour) as the two factors of production.

The important di¤erence between this production function and the Solovian production function is that now both the factors are

‘producible’inputs.

Just as one can invest in physical capital formation to augment capital stock, similarly, one can invest in skill formation (education) to augment the stock of human capital.

In fact Barro makes the extreme assumption that physical capital and human capital can be instantaneously converted into each other - one to one.

However we do not need such a strong assumption. Let us assume instead that physical capital and human capital areex ante convertible, butnot ex post.

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AK Technology: Barro Explanation (Contd.)

Suppose the economy starts with some historically given stocks of physical capital and human capital at time 0.

Then these stocks cannot be converted into one another immediately.

But in deciding about tommorrow’s stock, a household can invest in either form of capital. One unit of investment in terms of …nal commodity can be converted either into one unit of physical tomorrow or into one unit of human capital tomorrow.

Let us assume that there is 100% deprciation of both types of capital so that the entire stock of tomorrow’s physical as well as human capital must come from today’s investment.

Let us also endow the households with perfect foresight - such that in making their portfolio decision toady they can perfectly guess the returns to the two factors that will prevail tomorrow.

What would be these returns? For that we have to turn to the production side story.

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AK Technology: Barro Explanation (Contd.)

As before, consider an economy with S identical …rms - each having access to an identical …rm-speci…c technology:

Yit =F(Kit,Hit) (Kit)^α(Hit)¹ ^α; 0<α<1.

(One can think of H_it asA_tN¯ whereA_t is the skill level of per unit of labour and N¯ is the …xed labourforce.)

Corresponding aggregate production technology:

Yt =

∑

^Yît ⁼^S⁽^Kît⁾^α⁽^Hît⁾¹ ^α ^{= (}^SKît⁾^α⁽^SHît⁾¹ ^α

) ^Y^t = (Kt)^α(Ht)¹ ^α

Perfect Competition would imply that the two factors are paid their respective marginal products:

(r_K)_t = α(K_t)^α ¹(H_t)¹ ^α; (r_H)_t = (1 α) (K_t)^α(H_t) ^α.

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AK Technology: Barro Explanation (Contd.)

Since physical and human capital are convertible to each other ex ante, and since households have perfect foresight so that they can foresee these returns when they make their portfolio choice, at any point of time after the initial point, returns from the two forms of capitals must be the same.

Thus,

(r_K)_t = (r_H)_t for all t >0 i.e., α(K_t)^α ¹(H_t)¹ ^α = (1 α) (K_t)^α(H_t) ^α

i.e., H_t

K_t = ¹ ^α

α for all t >0.

Thus for all t >0 the reduced form aggregate technology would look as follows:

Y_t = (K_t)^α(H_t)¹ ^α = ^H^t Kt

1 α

K_t == ¹ ^α α

1 α

K_t. Once again the aggregate production technology is de-facto AK.

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AK Technology: General Insight

Notice that the AK explanations do notnecessarily rule out role of labour in the production process. But the role of labour becomes implicit - hidden within a broader concept of capital.

Even though the AK models hint at some explanation for the

technology parameter (learning by doing, skill formation), they do not model technological progress explicitly.

The subsequent endogenous growth models (developed in 1980s) explicitly specify a process of technological change in terms of R&D, education etc. They also move away from the CRS technology to allow for IRS - which necessitates a movement away from competitive market structure.

Even more recent developments in growth theory go beyond technology and explain growth in terms of institutions, political economy, income distribution etc. These are however beyond the purview of the current course.

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Reference for AK Technology:

1 P. Aghion & P. Howitt (1998): Endogenous Growth Theory, pages 24-26.

2 R.Barro & X. Sala-i-Martin (2004): Economic Growth, pages 211-213.

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