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1
The Solow Model
Econ 4960: Economic Growth
Econ 4960: Economic Growth
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… Details are confusing. It is only by selection,by elimination, by emphasis, that we get at the real meaning of things.”
Georgia O’Keeffe
2
1. Production TechnologyProduction function is Cobb-Douglas:
Firms are competitive and maximize profit taking prices as given:
Max (F(K,L) – rK –wL)
First order conditions:
Econ 4960: Economic Growth
(1 )F YwL L
F YrK K
a
a
¶= = -¶
¶= =¶
( ) 1,Y F K L K La a-= =
1. Production Technology (cont’d)Two Properties:1. Zero profit!2. Labor share:
Capital share:
Transform production function into per-capita terms:• Define: y=Y/L and k=K/L• We have :
Econ 4960: Economic Growth
wL rK Y+ =/ 1wL Y a= -/rK Y a=
y ka=
3
Cobb-Douglas Production:
Econ 4960: Economic Growth
Inada Conditions
Capital is essential: Diminishing marginal product:
High marginal product at zero
Econ 4960: Economic Growth
y = f (0) = 0
limx→∞
f '(x) < δ
limx→0
f '(x) > δ
4
Inada Conditions (cont’d)
Cobb-Douglas satisfies the Inada conditions, which rules out several types of behaviorSome of the ruled-out cases are strange, others are interesting.We will discuss some of these cases later
Econ 4960: Economic Growth
Violations of Inada
Econ 4960: Economic Growth
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Capital and Labor Share over Time
Econ 4960: Economic Growth
2. Capital accumulation
Econ 4960: Economic Growth
Solow makes the “behavioral assumption” thatConsumers save a constant fraction of their income
Therefore:
where
Finally, the economy is closed to international trade.
K.
= sY − dK. dKK
dt=
6
Capital accumulation per person
Econ 4960: Economic Growth
Simple trick:
We have:
. . . .
/log log log
further assume:
k K Lk K L
k K L L nk K L L
º= -
= - =
( ) ( )
( )
..
.
.
/
K YK sY dK s dK K
k Ys n d sy k n dk K
k sy n d k
= - Þ = -
Þ = - - = - +
Þ = - +
Steady StateSteady state is a key concept in economicsA steady state is a point such that if the system is at that point, it remains there forever.Solve for by setting
Econ 4960: Economic Growth
.0k =
( ) ( )( )
( )( )
.
1/ 1 / 1* * *
0 0
and
k sy n d k sk n d k
s sk y kn d n d
a
a a aa
- -
= - + = Þ - + =
æ ö æ öÞ = = =ç ÷ ç ÷+ +è ø è ø
k*
k*
7
(The famous!) Solow Diagram
Econ 4960: Economic Growth
Comparative Statics
Econ 4960: Economic Growth
From the steady state equations:A 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)A rise in depreciation rate reduces y*
( )/ 1* sy
n d
a a-æ ö= ç ÷+è ø
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Transitional Dynamics
Econ 4960: Economic Growth
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*To see transitional dynamics, note:
( ).
1k sk n dk
a-= - +
Transitional Dynamics
Econ 4960: Economic Growth
9
Transition after an increase in savings rate
Econ 4960: Economic Growth
Transitional Dynamics in Discrete Time
Econ 4960: Economic Growth
10
Absolute Convergence or the Lack Thereof
Econ 4960: Economic Growth
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
Econ 4960: Economic Growth
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Conditional Convergence: Seems more plausible
Econ 4960: Economic Growth
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 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?What 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?
Econ 4960: Economic Growth
12
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.
Econ 4960: Economic Growth