the solow model

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The Solow Model

Econ 4960: Economic Growth

Before we get to the model


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Caution:These graphs merely show some “correlation”: that the two variables are related They do not—and cannot—prove any “causation”Therefore, we cannot—yet—infer whether high income causes longevity or the other way around!We need to be very careful before claiming “causation” (This is one of the most common and serious crimes committed by news media!)


Correlates of Growth: Investment


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Correlates of Growth: Education


Kaldor facts (Balanced Growth):In the last 150 years:1 Th l f i l h d1. The real rate of return on capital shows no trend

upward or downward (which is true even in different societies)

2. Share of income accruing to capital and labor owners show no trend

3 A h f h b3. Average growth rate of output per person has been nearly constant over time

Even though Nicolas Kaldor made these observations in 1960’s, these “stylized facts” remain true today


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Now to the Solow Model…


All theory depends on assumptions which are not quite true. That is what makes it theory. The art of successful theorizing is to make the inevitable simplifying assumptions in such a way that the final results are not very sensitive.

Solow (1956, Introduction)

“Nothing is less real than realism… D t il f i It i l b l ti


Details are confusing. It is only by selection,by elimination, by emphasis, that we get at the real meaning of things.”

Georgia O’Keeffe

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1. Production TechnologyProduction function is Cobb-Douglas:

Firms are competitive and maximize profit taking prices as given:

Max (F(K,L) – rK –wL)

( ) 1,Y F K L K Lα α−= =

First order conditions:


(1 )F YwL L

F YrK K

α

α

∂= = −

∂

∂= =∂

1. Production Technology (cont’d)Two Properties:1. Zero profit!2. Labor share:

Capital share:

Transform production function into per capita terms:

wL rK Y+ =/ 1wL Y α= −

/rK Y α=

Transform production function into per-capita terms:• Define: y=Y/L and k=K/L• We have :


y kα=

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Cobb-Douglas Production:


Inada Conditions

Cobb-Douglas satisfies the Inada conditions, which rules out several types of behaviorSome of the ruled-out cases are strange, others are interestingothers are interesting.We will discuss some of these cases later


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Violations of Inada


Capital and Labor Share over Time


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2. Capital accumulation Solow makes the “behavioral assumption” that

Consumers save a constant fraction of their income:Therefore:

where

.K sY d K= −

. d KKd

=


Finally, the economy is closed to international trade.

d t

Capital accumulation per personSimple trick: /

l l lk K L

k K L≡

We have:

. . . .

log log log

further assum e:

k K L

k K L L nk K L L

= −

= − =

.. K YK sY dK s d

K K= − ⇒ = −


( ) ( )

( )

.

.

/

K K

k Ys n d sy k n dk K

k sy n d k

⇒ = − − = − +

⇒ = − +

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Steady StateSteady state is a key concept in economicsA steady state is a point (k*) such that if the system is at that point, it remains there forever.Solve for k* by setting

.0k =


( ) ( )( )

( )( )

.

1/ 1 / 1* * *

0 0

and

k sy n d k sk n d k

s sk y kn d n d

α

α α αα

− −

= − + = ⇒ − + =

⎛ ⎞ ⎛ ⎞⇒ = = =⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

(The famous!) Solow Diagram


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Comparative StaticsFrom the steady state equations:

( )/ 1* sy

α α−⎛ ⎞= ⎜ ⎟⎝ ⎠

y qA rise in investment rate increases y*A rise in population growth rate reduces y*These are consistent with empirical evidence (figs 2.6, 2.7)

yn d⎜ ⎟+⎝ ⎠


( g , )A rise in depreciation rate reduces y*

Transitional Dynamics

A striking implication of Solow’s model is that that there is no growth in the long-run!This is what a steady state means after all.There is only growth temporarily, until you converge to y*


converge to y*To see transitional dynamics, note:

( ).

1k sk n dk

α−= − +

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Transitional Dynamics


Transition after an increase in savings rate


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Transitional Dynamics in Discrete Time


Absolute Convergence or the Lack Thereof


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Absolute versus Conditional Convergence

However, the assumption that all countries have the same technology, and tastes is very strong. If one instead focuses on countries with similar characteristics, such as OECD economies you get a different picture.Barro and Sala-i Martin actually estimated that when one controls for differences in characteristics, countries (with (same human capital and life expectancy) converge to each other by about 2 percent per year.Aside: If you want to see the most colorful growth economist, check out Sala-i Martin’s website: http://www.columbia.edu/~xs23/home.html


Conditional Convergence: Seems more plausible


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Discussion questions (non-trivial)Suppose that country A has a higher y* than country B. Can you tell which one of these countries is likely to growCan you tell which one of these countries is likely to grow faster in the short-run (for example, next year)? [Think about different sources of differences in y*. Are there any conditions under which you can say something?]What effect does globalization have on convergence in the Solow model?Wh t d th b i S l M d l b t “Aid fWhat does the basic Solow Model say about “Aid for Africa”? What modification do you need to make to alter this conclusion?Suppose we relax Inada condition 1: F(0,L)>0. How does that affect the steady states?


Taking StockThe standard Solow model:

predicts that two countries will have different y* if they differ in s, n, or dThat there is no growth in output (and capital) per person in the long-run Only growth happens during transitions to the steady state. Growth rate slows down as countries become more developed.


the solow model

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