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Unformatted text preview: Macroeconomics II Lecture 4
Growth theory:
the Solow Model Macroeconomics
Instructor: Dang Vu
University of Economics and Business  VNU In this chapter, you will learn… the closed economy Solow model how a country’s standard of living depends on its
saving and population growth rates how to use the “Golden Rule” to find the optimal
saving rate and capital stock LECTURE 4 Economic Growth: Solow model slide 1 Why growth matters Data on infant mortality rates: 20% in the poorest 1/5 of all countries 0.4% in the richest 1/5 In Pakistan, 85% of people live on less than $2/day. Onefourth of the poorest countries have had
famines during the past 3 decades. Poverty is associated with oppression of women
and minorities.
Economic growth raises living standards and
reduces poverty….
LECTURE 4 Economic Growth: Solow model slide 2 1 Macroeconomics II Income and poverty in the world
selected countries, 2000
100 Madagascar % of popu
ulation
living on $2 per day or less 90 India
Nepal
Bangladesh 80
70
60 Botswana Kenya 50 China 40 Peru 30 Mexico Thailand 20
Brazil 10
0
$0 Russian Chile
Federation $5,000 S. Korea $10,000 $15,000 $20,000 Income per capita in dollars Why growth matters Anything that effects the longrun rate of economic
growth – even by a tiny amount – will have huge
effects on living standards in the long run.
annual
growth rate of
income per
capita …25 years …50 years …100 years 2.0% 64.0% 169.2% 624.5% 2.5% 85.4% 243.7% 1,081.4% LECTURE 4 percentage increase i
t
i
in
standard of living after… Economic Growth: Solow model slide 4 Why growth matters If the annual growth rate of U.S. real GDP per
capita had been just onetenth of one percent
higher during the 1990s, the U.S. would have
generated an additional $496 billion of income
during that decade. LECTURE 4 Economic Growth: Solow model slide 5 2 Macroeconomics II The lessons of growth theory
…can make a positive difference in the lives of
hundreds of millions of people.
These lessons help us understand why poor
countries are poor design policies that
can help them grow learn how our own
growth rate is affected
by shocks and our
government’s policies
LECTURE 4 Economic Growth: Solow model slide 6 The Solow model due to Robert Solow,
won Nobel Prize for contributions to
the study of economic growth a major paradigm: widely used in policy making benchmark against which most
recent growth theories are compared looks at the determinants of economic growth
and the standard of living in the long run
LECTURE 4 Economic Growth: Solow model slide 7 How Solow model is different
from Lecture 3’s model
1. K is no longer fixed: investment causes it to grow,
depreciation causes it to shrink
2.
2 L is no longer fixed: population growth causes it to grow
3. the consumption function is simpler LECTURE 4 Economic Growth: Solow model slide 8 3 Macroeconomics II How Solow model is different
from Lecture 3’s model
4. no G or T (only to simplify presentation;
we can still do fiscal policy experiments)
5.
5 cosmetic differences LECTURE 4 Economic Growth: Solow model slide 9 The production function In aggregate terms: Y = F (K, L) Define: y = Y/L = output per worker
k = K/L = capital per worker Assume constant returns to scale:
zY = F (zK, zL ) for any z > 0 Pick z = 1/L. Then
Y/L = F (K/L, 1)
y = F (k, 1)
y = f(k)
LECTURE 4 where f(k) = F(k, 1) Economic Growth: Solow model slide 10 The production function
Output per
worker, y
f(k)
MPK = f(k +1) – f(k)
1
Note: this production function
exhibits diminishing MPK. Capital per
worker, k
LECTURE 4 Economic Growth: Solow model slide 11 4 Macroeconomics II The national income identity Y=C+I (remember, no G ) In “per worker” terms:
y=c+i
where c = C/L and i = I /L LECTURE 4 Economic Growth: Solow model slide 12 The consumption function s = the saving rate,
the fraction of income that is saved
(s is an exogenous parameter)
Note: s is the only lowercase variable
y
that is not equal to
its uppercase version divided by L Consumption function: c = (1–s)y
(per worker) LECTURE 4 Economic Growth: Solow model slide 13 Saving and investment saving (per worker) = y – c
= y – (1–s)y
= sy National income identity is y = c + i
Rearrange to get: i = y – c = sy
(investment = saving, like in lecture 3!) Using the results above,
i = sy = sf(k)
LECTURE 4 Economic Growth: Solow model slide 14 5 Macroeconomics II Output, consumption, and investment
Output per
worker, y f(k) c1
sf(k) y1
i1 Capital per
worker, k k1
LECTURE 4 Economic Growth: Solow model slide 15 Depreciation
Depreciation
per worker, k = the rate of depreciation
= the fraction of the capital stock
that wears out each period k 1 Capital per
worker, k
LECTURE 4 Economic Growth: Solow model slide 16 Capital accumulation
The basic idea: Investment increases the capital
stock, depreciation reduces it.
Change in capital stock
k = investment – depreciation
i
–
k
= Since i = sf(k) , this becomes: k = s f(k) – k
LECTURE 4 Economic Growth: Solow model slide 17 6 Macroeconomics II The equation of motion for k k = s f(k) – k The Solow model’s central equation Determines behavior of capital over time… …which, in turn, determines behavior of
all of the other endogenous variables
because they all depend on k. E.g., income per person: y = f(k)
consumption per person: c = (1–s) f(k)
LECTURE 4 Economic Growth: Solow model slide 18 The steady state k = s f(k) – k
If investment is just enough to cover depreciation
[sf(k) = k ],
then capital per worker will remain constant:
k = 0.
This occurs at one value of k, denoted k*,
called the steady state capital stock. LECTURE 4 Economic Growth: Solow model slide 19 The steady state
Investment
and
depreciation k
sf(k) k*
LECTURE 4 Economic Growth: Solow model Capital per
worker, k
slide 20 7 Macroeconomics II Moving toward the steady state
k = sf(k) k Investment
and
depreciation k
sf(k) k investment depreciation k1
LECTURE 4 k* Capital per
worker, k Economic Growth: Solow model slide 21 Moving toward the steady state
Investment
and
depreciation k = sf(k) k
k
sf(k) k k1
LECTURE 4 k* Capital per
worker, k Economic Growth: Solow model slide 22 Moving toward the steady state
Investment
and
depreciation k = sf(k) k
k
sf(k) k
k1 k2
LECTURE 4 k* Economic Growth: Solow model Capital per
worker, k
slide 23 8 Macroeconomics II Moving toward the steady state
k = sf(k) k Investment
and
depreciation k
sf(k) k investment depreciation k2
LECTURE 4 k* Capital per
worker, k Economic Growth: Solow model slide 24 Moving toward the steady state
Investment
and
depreciation k = sf(k) k
k
sf(k) k k2
LECTURE 4 k* Capital per
worker, k Economic Growth: Solow model slide 25 Moving toward the steady state
Investment
and
depreciation k = sf(k) k
k
sf(k) k
k2 k3 k*
LECTURE 4 Economic Growth: Solow model Capital per
worker, k
slide 26 9 Macroeconomics II Moving toward the steady state
Investment
and
depreciation k = sf(k) k
k
sf(k) Summary:
As long as k < k*,
investment will exceed
depreciation,
and k will continue to
grow toward k*. k3 k*
LECTURE 4 Capital per
worker, k Economic Growth: Solow model slide 27 Now you try:
Draw the Solow model diagram,
labeling the steady state k*.
On the horizontal axis, pick a value greater than k*
for the economy s initial capital stock. Label it k1.
economy’s
stock
Show what happens to k over time.
Does k move toward the steady state or
away from it? LECTURE 4 Economic Growth: Solow model slide 28 A numerical example
Production function (aggregate): Y F (K , L ) K L K 1 / 2L1 / 2
To derive the perworker production function,
divide through by L: LECTURE 4 Economic Growth: Solow model slide 29 10 Macroeconomics II A numerical example, cont.
Assume: s = 0.3 = 0.1 initial value of k = 4.0 LECTURE 4 Economic Growth: Solow model slide 30 Approaching the steady state:
A numerical example Assumptions: y k ; s 0.3; 0.1; initial k 4.0
k Year k y c i dk 1 4.000 2.000 1.400 0.600 0.400 0.200 2 4.200 2.049 1.435 0.615 0.420 0.195 3 4.395
4 395 2.096
2 096 1.467
1 467 0.629
0 629 0.440
0 440 0.189
0 189 4
4.584
2.141 1.499
…
10
5.602
2.367 1.657
…
25
7.351
2.706 1.894
…
100
8.962
2.994 2.096
…
¥
3.000 2.100
CHAPTER 9.000
7 Economic Growth I 0.642 0.458 0.184 0.710 0.560 0.150 0.812 0.732 0.080 0.898 0.896 0.002 0.900 0.900 0.000 31
slide Exercise: Solve for the steady state
Continue to assume
s = 0.3, = 0.1, and y = k 1/2
Use the equation of motion
q
k = s f(k) k
to solve for the steadystate values of k, y, and c. LECTURE 4 Economic Growth: Solow model slide 32 11 Macroeconomics II An increase in the saving rate
An increase in the saving rate raises investment…
…causing k to grow toward a new steady state:
Investment
and
depreciation k
s2 f(k)
s1 f(k) LECTURE 4 k* k 2* 1
Economic Growth: Solow model k
slide 34 Prediction: Higher s higher k*. And since y = f(k) ,
higher k* higher y* . Thus, the Solow model predicts that countries
with higher rates of saving and investment
will have higher levels of capital and income per
worker in the long run. LECTURE 4 Economic Growth: Solow model slide 35 International evidence on investment
rates and income per person
Income per 100,000
person in
2000
(log scale) 10,000 1,000 100
0 5 10 15 20 25 30 35 Investment as percentage of output
(average 19602000) LECTURE 4 Economic Growth: Solow model slide 36 12 Macroeconomics II The Golden Rule: Introduction Different values of s lead to different steady states.
How do we know which is the “best” steady state? The “best” steady state has the highest possible
consumption per person: c* = (1–s) f(k*). An increase in s leads to higher k* and y*, which raises c* reduces consumption’s share of income (1–s),
which lowers c*. So, how do we find the s and k* that maximize c*?
LECTURE 4 Economic Growth: Solow model slide 37 The Golden Rule capital stock
*
k gold the Golden Rule level of capital, the steady state value of k
that maximizes consumption.
To find it, first express c* in terms of k*:
c* = y* i* = f (k*)
= f (k*) LECTURE 4 i* k* In the steady state:
i* = k*
because k = 0. Economic Growth: Solow model slide 38 The Golden Rule capital stock
steady state
output and
depreciation Then, graph
f(k*) and k*,
look for the
point where
the gap between
them is biggest.
*
*
y gold f (k gold ) LECTURE 4 k*
f(k*) *
c gold
*
*
i gold k gold
*
k gold Economic Growth: Solow model steadystate
capital per
worker, k*
slide 39 13 Macroeconomics II The Golden Rule capital stock
c* = f(k*) k*
is biggest where the
slope of the
production function
equals
l
the slope of the
depreciation line: k*
f(k*) *
c gold MPK = *
k gold LECTURE 4 steadystate
capital per
worker, k* Economic Growth: Solow model slide 40 The transition to the
Golden Rule steady state The economy does NOT have a tendency to
move toward the Golden Rule steady state. Achieving the Golden Rule requires that
policymakers adjust s
s. This adjustment leads to a new steady state with
higher consumption. But what happens to consumption
during the transition to the Golden Rule?
LECTURE 4 Economic Growth: Solow model slide 41 Starting with too much capital
*
If k * k gold then increasing c*
requires a fall in s.
In the transition to
the Golden Rule,
consumption is
higher at all points
in time. y
c
i t0
LECTURE 4 Economic Growth: Solow model time slide 42 14 Macroeconomics II Starting with too little capital
*
If k * k gold
then increasing c*
requires an
increase in s. Future generations
enjoy higher
consumption,
but the current
one experiences
an initial drop
in consumption.
LECTURE 4 y
c i
t0 Economic Growth: Solow model time slide 43 Population growth Assume that the population (and labor force)
grow at rate n. (n is exogenous.) L n
L EX: Suppose L = 1,000 in year 1 and the
population is growing at 2% per year (n = 0.02). Then L = n L = 0.02 1,000 = 20,
so L = 1,020 in year 2.
LECTURE 4 Economic Growth: Solow model slide 44 Breakeven investment ( + n)k = breakeven investment,
the amount of investment necessary
to keep k constant. Breakeven investment includes: k to replace capital as it wears out nk to equip new workers with capital
(Otherwise, k would fall as the existing capital stock
would be spread more thinly over a larger
population of workers.)
LECTURE 4 Economic Growth: Solow model slide 45 15 Macroeconomics II The equation of motion for k With population growth,
the equation of motion for k is k = s f(k) ( + n) k actual
investment LECTURE 4 breakeven
investment Economic Growth: Solow model slide 46 The Solow model diagram
Investment,
breakeven
investment k = s f(k) ( +n)k
( + n ) k sf(k) k*
LECTURE 4 Capital per
worker, k Economic Growth: Solow model slide 47 The impact of population growth
Investment,
breakeven
investment ( +n2) k
( +n1) k An increase in n
causes an
increase in breakeven investment,
leading to a lower
steadystate level
of k. sf(k) k2*
LECTURE 4 Economic Growth: Solow model k1* Capital per
worker, k
slide 48 16 Macroeconomics II Prediction: Higher n lower k*. And since y = f(k) ,
lower k* lower y*. Thus, the Solow model predicts that countries
with higher population growth rates will have
lower levels of capital and income per worker in
the long run. LECTURE 4 Economic Growth: Solow model slide 49 International evidence on population
growth and income per person
Income 100,000
per Person
in 2000
(log scale) 10,000 1,000 100
0 1 2 3 4 5 Population Growth
(percent per year; average 19602000)
LECTURE 4 Economic Growth: Solow model slide 50 The Golden Rule with population
growth
To find the Golden Rule capital stock,
express c* in terms of k*:
c* = y* = f (k* ) i* ( + n) k* c* is maximized when
MPK = + n
or equivalently,
MPK = n
LECTURE 4 In the Golden
Rule steady state,
the marginal product
of capital net of
depreciation equals
the population
growth rate. Economic Growth: Solow model slide 51 17 Macroeconomics II Alternative perspectives on
population growth
The Malthusian Model (1798) Predicts population growth will outstrip the Earth’s
ability to produce food, leading to the
impoverishment of humanity. Since Malthus, world population has increased
sixfold, yet living standards are higher than ever. Malthus omitted the effects of technological
progress. LECTURE 4 Economic Growth: Solow model slide 52 Alternative perspectives on
population growth
The Kremerian Model (1993) Posits that population growth contributes to
economic growth. More people = more geniuses, scientists &
engineers, so faster technological progress. Evidence, from very long historical periods: As world pop. growth rate increased, so did rate
of growth in living standards Historically, regions with larger populations have
enjoyed faster growth.
LECTURE 4 Economic Growth: Solow model slide 53 Chapter Summary
1. The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate negatively on its p p
g
y
population g
growth rate
2. An increase in the saving rate leads to higher output in the long run faster growth temporarily but not faster steady state growth.
LECTURE 4 Economic Growth: Solow model slide 54 18 Macroeconomics II Chapter Summary
3. If the economy has more capital than the Golden Rule level, then reducing saving will
increase consumption at all points in time,
making all generations better off.
If the economy has less capital than the Golden
Rule level, then increasing saving will increase
consumption for future generations, but reduce
consumption for the present generation. LECTURE 4 Economic Growth: Solow model slide 55 19 ...
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