macro 3 L1
“SOME FACTS
ABOUT PROSPERITY
AND GROWTH”
based on Chapter 2
of your Textbook
Macroeconomics III
Prof. Guido
Cozzi
February
18, 2020
1
INTRODUCTION
INTRODUCTION
•
•How do we measure prosperity? In economics, often by GDP per How do we measure prosperity? In economics, often by GDP per person!person!
•
•Higher income means higher consumption or higher saving (possibly Higher income means higher consumption or higher saving (possibly both) and saving can be used for future consumption. Consumption both) and saving can be used for future consumption. Consumption is always rooted in income and is a decent proxy for wellis always rooted in income and is a decent proxy for well--being.being.
•
•In this course we often consider In this course we often consider growth growth of income per person of income per person because the way to reach a high because the way to reach a high levellevelof income per person is of income per person is through a process of high growth in income per person.through a process of high growth in income per person.
•
•Relatively small differences in growth rates can imply large Relatively small differences in growth rates can imply large differences in levels over long periods as evidenced by next slide.differences in levels over long periods as evidenced by next slide.
•
•Conclusion: Growth is important! One focus of our classes is Conclusion: Growth is important! One focus of our classes is What What creates growth?creates growth?
2
The importance of growth
The importance of growth
3
COMPARING THE GDP LEVELS OF TWO COUNTRIES
COMPARING THE GDP LEVELS OF TWO COUNTRIES
•
•PurchasingPurchasing--power adjustment: power adjustment:
–
–When converting, say, pesos into US dollars, we want to take into When converting, say, pesos into US dollars, we want to take into account differences in the costs of living in the two countries.account differences in the costs of living in the two countries.
–
–Hence, the relevant rateHence, the relevant rate--ofof--conversion should reflect that conversion should reflect that consumer goods are usually much cheaper in less developed consumer goods are usually much cheaper in less developed countries.countries.
•
•Per worker or per capita?Per worker or per capita?
–
–GDP per capita = Official GDP divided by total population: GDP per capita = Official GDP divided by total population: Underestimates the production of less developed countries, since Underestimates the production of less developed countries, since they typically have a larger share of the population working outside they typically have a larger share of the population working outside the official market economy.the official market economy.
–
–GDP per worker = Official GDP divided by the labour force GDP per worker = Official GDP divided by the labour force (population times participation rate): corrects for differences in (population times participation rate): corrects for differences in participation rates, hence it is more of a productivity measure.participation rates, hence it is more of a productivity measure.
4
THE LORENZ CURVE
THE LORENZ CURVE
•
•… is a comprehensive way to measure the degree of inequality of a … is a comprehensive way to measure the degree of inequality of a personal income distribution, for a single country or for the world.personal income distribution, for a single country or for the world.
•
•Stylized example: Suppose there are two countries in the world, one Stylized example: Suppose there are two countries in the world, one rich and one poor and that 60% of the world population lives in the rich and one poor and that 60% of the world population lives in the poor country earning 30% of world income, while 40% lives in the poor country earning 30% of world income, while 40% lives in the rich country earning 70% of world income.rich country earning 70% of world income.
•
•The Lorenz curve is defined as the curve through a collection of The Lorenz curve is defined as the curve through a collection of points (points (xx,,yy), where ), where xxis the share of people with incomes is the share of people with incomes belowbelowa a certain level and certain level and yyis the income share of these people.is the income share of these people.
•
•The curve always starts at (0,0). In our example, the next point is The curve always starts at (0,0). In our example, the next point is (60,30), since the 60% poorest people earn 30% of world income. (60,30), since the 60% poorest people earn 30% of world income. Finally, the curve always ends in (100,100). Curve on next slide.Finally, the curve always ends in (100,100). Curve on next slide.
5
•
•Imagine that the income Imagine that the income distribution were distribution were characterized by perfect characterized by perfect equality.equality.
•
•Then the 60% poorest Then the 60% poorest people would earn 60% people would earn 60% of total income.of total income.
•
•The Lorenz curve would The Lorenz curve would be identical to the be identical to the diagonal from the lower diagonal from the lower left to the upper right left to the upper right corner indicated in the corner indicated in the figure.figure.
•
•We can use this to define We can use this to define a measure of inequality a measure of inequality called…called…
6
THE GINI COEFFICIENT
THE GINI COEFFICIENT
•
•The area between the Lorenz curve and the diagonal The area between the Lorenz curve and the diagonal divided by the full area below the diagonal. In the figure divided by the full area below the diagonal. In the figure that is GC/(GC+A+B+C). A measure between zero and that is GC/(GC+A+B+C). A measure between zero and one.one.
•
•If the Gini coefficient is zero, there is no inequality.If the Gini coefficient is zero, there is no inequality.
•
•The larger the Gini coefficient becomes, the more The larger the Gini coefficient becomes, the more inequality there is.inequality there is.
•
•Using GDP per worker and population data for 137 Using GDP per worker and population data for 137 countries and artificially associating with each person in a countries and artificially associating with each person in a country the GDP per worker of the country, we can draw country the GDP per worker of the country, we can draw Lorenz curves for the world in 1960 and 2003, Lorenz curves for the world in 1960 and 2003, respectively:respectively:
7
•
•Stylized fact 1: Stylized fact 1: Some countries are rich and some are poor; Some countries are rich and some are poor; the differences are enormous and it has pretty much stayed like the differences are enormous and it has pretty much stayed like that in relative terms over the last 40 years. However, there is that in relative terms over the last 40 years. However, there is some tendency towards a more equal world income distribution, some tendency towards a more equal world income distribution, though not much at the very bottom. though not much at the very bottom.
8
Prosperity and poverty, growth and decline
Prosperity and poverty, growth and decline
9
Prosperity and poverty, growth and decline
Prosperity and poverty, growth and decline
•
•In the ’top 20’ of the fastest In the ’top 20’ of the fastest growing countries we find growing countries we find the ’growth miracles’, e.g., the ’growth miracles’, e.g., tiger economies such as tiger economies such as Korea, Singapore, Hong Korea, Singapore, Hong Kong, China and India. In Kong, China and India. In the bottom we find the the bottom we find the ’growth disasters’, many of ’growth disasters’, many of which have had negative which have had negative average growth rates over average growth rates over the period considered.the period considered.
10
•
•Stylized fact 2:Stylized fact 2:Growth rates vary substantially between Growth rates vary substantially between countries, and by the process of growing or declining fast, countries, and by the process of growing or declining fast, a country can move from being relatively poor to being a country can move from being relatively poor to being relatively rich, or from being relatively rich to being relatively rich, or from being relatively rich to being relatively poor.relatively poor.
•
•Stylized fact 3: Stylized fact 3: Growth can break in a country, turning Growth can break in a country, turning from a high rate to a low one or vice versa.from a high rate to a low one or vice versa.
•
•In the 1960s the EastIn the 1960s the East--Asian ”growth miracles” experienced Asian ”growth miracles” experienced growth breaks, moving from low to high growth rates. growth breaks, moving from low to high growth rates. Hence, it Hence, it isispossible for poor countries to escape from possible for poor countries to escape from poverty!poverty!
11
CONVERGENCE
CONVERGENCE
Hypothesis of absolute convergence:
Hypothesis of absolute convergence:In the long run, GDP In the long run, GDP per worker (or per capita) converges to one and the same per worker (or per capita) converges to one and the same growth path in all countries, so that all countries converge on the growth path in all countries, so that all countries converge on the same level of income per worker.same level of income per worker.
12
• If two countries are to end up with the same level of GDP per worker
and one of them starts at a level much lower than the other, then that
country must grow relatively quickly compared to the other.
• That is, if we estimate across countries i a regression line:
where is the average growth rate of GDP per worker,
and is the initial level of GDP per worker, we expect the estimate
of to be significantly positive - note the minus in front of
• For a sample of 24 rich OECD countries we find:
0
0 1 0
ln ln
ln
i i
T i y y
y
T
−
= −
1
( ) 0 ln ln i i
T y − y / T
0 ln i y
1 .
13
•
•There is a tight negative correlation and the estimate of is There is a tight negative correlation and the estimate of is indeed significantly positive.indeed significantly positive.
•
•However, the countries included in the sample were all relatively However, the countries included in the sample were all relatively rich in 2003. Therefore there is a ’sample selection bias’.rich in 2003. Therefore there is a ’sample selection bias’.
1
14
•
•For a more representative sample of 65 countries we find:For a more representative sample of 65 countries we find:
•
•The negative relationship has disappeared, and is The negative relationship has disappeared, and is insignificant. We have to reject the hypothesis of absolute insignificant. We have to reject the hypothesis of absolute convergence.convergence.
1
15
Hypothesis of conditional convergence: A country’s income
per worker converges to a country-specific long-run growth
path, which depends on the basic structural characteristics of the
country. The further below its own long-run growth path a country
starts, the faster it will grow. Income per worker therefore
converges to the same level across countries conditional on the
countries being structurally alike.
• The ”correct” regression line could be an equation like:
where the last term on the right hand side captures structural
influence, where is the average investment rate of country i, and
is the average growth rate of the labour force. (The theory behind
this formulation is introduced in later chapters.)
• For the same representative sample of 65 countries we get:
( ) 0
0 1 0 2
ln ln
ln ln ln 0 075
i i
T i i i y y
y s n .
T
−
= − + − +
i s i n
16
•
•Stylized fact 4Stylized fact 4: Convergence: If one controls appropriately : Convergence: If one controls appropriately for structural differences between the countries of the world, for structural differences between the countries of the world, a lower initial value of GDP per worker tends to be a lower initial value of GDP per worker tends to be associated with a higher subsequent growth rate in GDP per associated with a higher subsequent growth rate in GDP per worker.worker.
17
THE LONG
THE LONG--RUN GROWTH PROCESSRUN GROWTH PROCESS
•
•Stylized fact 5: Stylized fact 5: Over periods of more Over periods of more than 130 years, than 130 years, many countries in many countries in Western Europe and Western Europe and North America have North America have had relatively had relatively constant annual constant annual rates of growth in rates of growth in GDP per capita in the GDP per capita in the range 1.5range 1.5--2 per cent.2 per cent.
18
• Labour’s income share turns out to be quite stable (see figures)
• and labour’s share of income can be
written AS:
where is the average real wage per
worker, and is GDP per worker.
• This implies:
• Stylized fact 6: During the long
periods of relatively constant growth
rates in GDP per worker in the typical
Western economy, labour’s share of
GDP has stayed relatively constant,
hence the average real wage of a
worker has grown by approximately the
same rate as GDP per worker.
t t w / y
t w
t y
19
• The relative constancy of the
factor income shares and the
observation that real interest
rates do not have any trend
upwards or downwards in the
long run (see figures), so that
rates of return on capital must be
relatively stable,
( ) t t t r / Y / K
t r
• and the fact that capital’s share of
income can be written as:
where is the rate of return on
capital,
• implies:
20
•
•Stylized fact 7Stylized fact 7: During the long periods of relatively : During the long periods of relatively constant growth rates in GDP per worker in the typical constant growth rates in GDP per worker in the typical Western economy, capital’s share and the rate of return Western economy, capital’s share and the rate of return on capital have shown no trend, therefore the capitalon capital have shown no trend, therefore the capital--output ratio K/Y has been relatively constant, and the output ratio K/Y has been relatively constant, and the capital intensity K/L has grown by approximately the capital intensity K/L has grown by approximately the same rate as GDP per worker.same rate as GDP per worker.
21
BALANCED GROWTH
BALANCED GROWTH
•
•summarizes stylized facts 5summarizes stylized facts 5--7 and is an important 7 and is an important concept for evaluating the growth models in the concept for evaluating the growth models in the subsequent chapters.subsequent chapters.
•
•A growth process follows balanced growth if:A growth process follows balanced growth if:
1.
1.GDP per worker, consumption per worker, the real GDP per worker, consumption per worker, the real wage and the capital intensity all grow at one and the wage and the capital intensity all grow at one and the same constant rate, g.same constant rate, g.
2.
2.The labour force (population) grows at a constant The labour force (population) grows at a constant rate, n.rate, n.
3.
3.GDP, consumption and capital grow at the common GDP, consumption and capital grow at the common rate, g+n.rate, g+n.
4.
4.The capitalThe capital--ouput ratio and the rate of return on ouput ratio and the rate of return on capital are constant.capital are constant.
22
“CAPITAL
ACCUMULATION AND
GROWTH: THE
BASIC SOLOW
MODEL”
Based on Chapter 3 of
your textbook
Macroeconomics III
Prof. Guido
Cozzi
February
19, 20 20
THE BASIC SOLOW MODEL
THE BASIC SOLOW MODEL
•
•How can a nation become rich, i.e., initiate a growth How can a nation become rich, i.e., initiate a growth process leading to higher GDP/consumption per capita process leading to higher GDP/consumption per capita in the long run?in the long run?
•
•The basic Solow model provides some first answers: The basic Solow model provides some first answers: It predicts how the evolution and the longIt predicts how the evolution and the long--run levels run levels of GDP and consumption per capita depend on of GDP and consumption per capita depend on structural parameters such as the rate of investment structural parameters such as the rate of investment and the growth rate of the labour force.and the growth rate of the labour force.
•
•Key elements of the Solow model:Key elements of the Solow model:
–
–In each period, output is determined by the supplies of In each period, output is determined by the supplies of capital and labour through the production function.capital and labour through the production function.
–
–Exogenous savings/investment rate, Exogenous savings/investment rate, ss, exogenous growth , exogenous growth rate of labour force, rate of labour force, nn, and exogenous depreciation rate, , and exogenous depreciation rate, δδ..
–
–Explicit description of capital accumulation: Explicit description of capital accumulation:
–
–Accumulation of capital is the main driving force for wealth. Accumulation of capital is the main driving force for wealth. “Basic” model: No technological progress“Basic” model: No technological progress
1ttttKKIK+=+−
2
THE ”MICRO WORLD” OF THE SOLOW MODEL
• Object: Closed economy
• Time: A sequence of periods/years
• Agents: Households and firms (and government)
• Commodities and markets: Output, capital
services and labour services (one asset = physical
capital)
• The market for output: Supply = firms’ output, .
Demand from households for consumption and
investment = . Relative price = 1.
– One-sector model: Output can be used either for
consumption or for investment.
t = 0,1,2,...
t Y
t t C + I
3
• The market for capital services: Consumers own
the capital stock, , and rent its services to firms.
Supply of capital services = . Firms’ demand =
– Relative price (in units of output) for renting one unit of
capital for one period: = real rental rate for capital
– Real interest rate: , where is the rate of
depreciation, or:
(Alternative interpretation: The firms own the capital,
borrow for the purchase of capital at an interest rate of
and bear the cost of depreciation themselves.)
– User cost
• Labour market: Households supply = = the
labour force. Demand from firms = . Relative
price: = the real wage rate.
• Competitive markets: and adjust to equate
supply and demand in all markets full (or natural)
utilization of resources.
t K
s
t K
t r
rt = t +
t t = r −
t
t t = r = +
t L
d
t L
t w
d
t K
t r t w
4
THE PRODUCTION SIDE
• is modelled as if all production (all of GDP) comes
from one profit-maximizing firm that produces value
added, , from capital services, (machine-years),
and labour services, (man-years), according to
the production function:
1. Constant returns to scale: .
The replication argument!
2. Positive marginal products:
t Y
d
t K
d
t L
( ) d d
Yt = F Kt ,Lt
( ) ( ) d d d d
t t t t F K ,L =F K ,L
( ) 0 ( ) 0 d d d d
K L F' K ,L , F' K ,L .
5
3. Marginal products are decreasing in the amount of
the factor used:
(diminishing returns) and growing in the amount of
the other factor used:
• Profit maximization: Given and , the firm
chooses and to:
The usual necessary conditions for an optimum:
(These two equations do not determine and
from given and ; they only determine .)
0 0 KK LL F , F '' ''
t r t w
d
t L d
t t Y ,K
max s.t ( ) d d d d
t t t t t t t t Y −rK −w L , . Y = F K ,L
( ) ( ) d d d d
K t t t L t t t F K ,L = r , F K ,L = w ' '
d
t K d
t L
d d
t t K / L
0 0 LK LK F , F '' ''
t r t w
6
• Competitive market clearing and ,
where and are the supplies in period t:
Since and are predetermined in any given
period, and are determined this way.
and predetermined. What does that mean?
d
t t K = K d
t t L = L
t K t L
( ) ( ) K t t t L t t t F K ,L = r , F K ,L = w ' '
t K t L
t r t w
t K t L
7
THE INCOME DISTRIBUTION
1. No pure profits. Euler’s rule
2. The functional income distribution
The income share of each factor is the elasticity of
the production function with respect to the factor in
question.
( ) ( )
( ) ( )
t K t t t t K t t t
t L t t t t L t t t
r F K ,L rK F K ,L K
w F K ,L w L F K ,L L
= =
= =
' '
' '
( ) ( ) ( ) 0 t t K t F K ,L − F K ,Lt Kt − FL Kt ,Lt Lt = ' '
( )
( )
( )
( )
and
t t K t t t t t L t t t
t t t t t t
rK F K ,L K w K F K ,L L
Y F K ,L Y F K ,L
= =
' '
8
According to empirics, labour’s share is relatively constant
According to empirics, labour’s share is relatively constant around 2/3 over long periods except for shortaround 2/3 over long periods except for short--run fluctuations:run fluctuations:Labour’s share of domestic factor incomesLabour’s share of domestic factor incomes
9
Is there a production function that fulfills all of our
assumptions and has fixed output elasticities
independently of and ? Yes, the Cobb-Douglas
production function:
where is total-factor-productivity (TFP).
Check: It follows that
The Cobb-Douglas function seems to be a realistic longrun
assumption. We even have reason to believe that
.
t K t L
( ) 1 0 0 1 t t t t t t F K ,L B K L , B , , − =
t B
( )
1
and 1
and 1
t t
t K t t L t
t t
t t t t
t t
K K
r F B w F B
L L
rK w L
Y Y
−
= = = = −
= = −
' '
1/ 3
10
HOUSEHOLDS
• The number of households in period is , which is
predetermined. Household behaviour:
1. Each supplies one unit of labour inelastically. Total supply .
2. Owns the capital stock, , which is predetermined in period .
Supply = (as long as ).
3. The representative household decides given , and hence
. The intertemporal budget constraint:
We assume that the result of the consumer’s considerations is:
4. ”Biology”:
t t L
= Lt
t K t
t K 0 t r
t C t Y
t t t S Y −C
1 0 1 t t t t K K S K , + − = −
t t S = sY .
( ) 1 1 1 t t L n L , n + = + −
11
THE COMPLETE MODEL (with book equation numbers below)
• Parameters: and . N.b. No subscript on :
”Basic” Solow model.
• Endogenous variables: and
of which and are state variables:
1
t t t Y BK L − =
1
t
t
t
K
r B
L
−
=
(1 ) t
t
t
K
w B
L
= −
t t S = sY
( ) 1 1 t t L n L , + = +
B, ,s, n t B
( ) ( ) ( ) ( ) ( ) t t t t t Y , K , L , r , w ( ) t S
( ) t K ( ) t L
t 1 t t t K K S K + − = −
(14)
(15)
(16)
(17)
(18)
(19)
12
• Given and the model determines
• Government in the model? Yes. Simply interpret
as private plus government savings.
• We viewed the capital accumulation equation:
as the household’s budget constraint.
0 K 0 L ( ) ( ) ( ) t t t Y , K , L ...
t 1 t t t K K S K + − = −
t S
13
Alternatively: By definition we have that
The condition for equilibrium in the output market, or
the national accounting identity, is: , and
by definition: . Hence
Combining and gives the capital accumulation
equation again: is the savings rate and the
investment rate.
t t t Y = C + I
t t t S Y −C
( ) t t I = S
s
1 ( ) t t t t K K I K + − = −
() ()
14
ANALYSING THE BASIC SOLOW MODEL
1. Define: and .
2. From we get the per capita production
function:
Note:
t t t y = Y / L t t t k = K / L
1
t t t Y BK L − =
0 1 t t y Bk , =
( ) 1 1
y k
t t t t t t ln y ln y lnk lnk g g − − − = − =
15
t 1 t t t K K S K + − = − t t S = sY
( ) 1 1 t t L n L + = +
( ( ) ) 1
1
1
1 t t t k sy k
n
+ = + −
+
t t y Bk =
( ( ) ) 1
1
1
1 t t t k sBk k
n
+ = + −
+
t k
( ) 1
1
1 t t t t k k sBk n k
n
+
− = − +
+
savings per
capita
Replacement investment to
compensate for depreciation
and growth of labour force t = sy
”technical term”
appearing because
of discrete time 16
The transition diagram
The transition diagram
( ( ) ) 1
1
1
1 t tt k sBkk
n
+ = +−
+
17
• About the slope, :
– It is everywhere strictly positive
– It goes to infinity as goes to zero
– It goes to as , and
. Indeed we assume that (realistic!).
• The figure shows that in the long run. Hence
, and
etc.
• The values , etc. define the ”steady state”.
(1− ) / (1+ n)
1 ddt t k / k +
(1− ) / (1+ n) 1
*
tk →k
( ) * *
t y y B k
→ = (1 ) * *
t c →c = − s y ( ) 1
* *
t r r B k
−
→ =
t k
t k →
n+ 0
* * k , y
n −
18
THE SOLOW DIAGRAM
Why does growth in and have to stop?
Diminishing returns!
( ( ) ) 1
1
1 t t t t k k sBk n k
n
+ − = − +
+
kt
sBkt
k*
(n + )kt
t k t y
19
STEADY STATE
• The long run levels , , etc. depend on
parameters. How? What makes a nation rich?
• Look at the Solow equation:
In steady state . Insert this to find
* y * k
( ( ) ) 1
1
1 t t t t k k sBk n k
n
+ − = − +
+
1 0 t t k k + − =
( ) ( )
( )
1
1
1
1
1
1
1
* *
*
* *
sB k n k
s
k B
n
s
y B k B
n
−
−
−
−
= +
=
+
= =
+
20
• Some sharp predictions of the Solow model:
The elasticity of wrt. is (since we believe
that ): an increase in of 10%, e.g., from 20 to
22%, should give an increase in of 5%!
The elasticity of wrt. and wrt. are
and , respectively. Why is the latter not one?
Capital accumulation.
• We have reached empirically testable hypotheses!
Empirics:
( )
1
ln ln ln
1 1
* y B s n .
= + − + − −
* y s
1
1 2
−
1 3
1 2
−
n+ B
s
* y
1
1 2
− −
−
1
3
* y
21
Real GDP per worker against the average investment
Real GDP per worker against the average investment share across 65 countriesshare across 65 countries
22
Real GDP per worker against the average annual labour force
Real GDP per worker against the average annual labour force growth rate across 65 countriesgrowth rate across 65 countries
23
“CAPITAL
ACCUMULATION AND
GROWTH: THE BASIC
SOLOW MODEL”
…also based on
Chapter 3 of your
textbook
Macroeconomics
III
Prof. Guido
Cozzi
February
18, 2020
1
THE COMPLETE MODEL (with book equation numbers below)
• Parameters: and . N.b. No subscript on :
”Basic” Solow model.
• Endogenous variables: and
of which and are state variables:
1
t t t Y BK L − =
1
t
t
t
K
r B
L
−
=
(1 ) t
t
t
K
w B
L
= −
t t S = sY
( ) 1 1 t t L n L , + = +
B, ,s, n t B
( ) ( ) ( ) ( ) ( ) t t t t t Y , K , L , r , w ( ) t S
( ) t K ( ) t L
t 1 t t t K K S K + − = −
(14)
(15)
(16)
(17)
(18)
(19)
2
GOVERNMENT SECTOR
• follows from the identity/definition
and the national accounting identity
• Same construction with a public sector:
• Insert into the identity to get
t 1 t t t K K S K + − = −
t 1 t t t K K I K + − = −
t t t t t t t t Y = C + I Y −C = I S = I
p g p g
t t t t t Y =C +C + I + I
p g
St St
p g p g
t t t t t t t Y −T −C +T −C = I + I
1
p g
t t t t t K K I I K + − = + −
1
St
p g
t t t t t K K S S K + − = + −
3
• We can use the model as it stands. We just have to
reinterpret as the sum of private and government
capital stock and as the sum of private and public
savings. Similarly the equation should be
reinterpreted as:
• The essential assumption underlying the Solow model
interpreted to include a government is that the sum
of private and public consumption as a fraction
of GDP is a constant, .
• This seems plausible empirically. And seems
plausible for many Western countries:
t K
t S
St = sYt
(1 ) p g p g
t t t t t t t S = S + S = sY C +C = − s Y
1−s
s 0.2
4
The consumption share of GDP in several Western countries
The consumption share of GDP in several Western countries
5
THE BASIC SOLOW MODEL, SHORT VERSION
• In the previous part we saw that the Solow model
leads to the transition equation:
• and to the Solow equation:
1
t t t Y BK L − =
( ) 1 1 t t L n L + = +
t 1 t t t K K sY K + − = −
( ( ) ) 1
1
1
1 t t t k sy k
n
+ = + −
+
( ) 1
1
1 t t t t k k sBk n k
n
+
− = − +
+
savings per
capita t = sy
Replacement investment to
compensate for depreciation
and growth of labour force
”technical term”
appearing because
of discrete time
6
The Solow diagram (repetition)
Why does growth in and have to stop?
Diminishing returns!
( ( ) ) 1
1
1 t t t t k k sBk n k
n
+ − = − +
+
kt
sBkt
k*
(n + )kt
t k t y
7
COMPARATIVE ANALYSIS IN THE SOLOW DIAGRAM
1. The economy is initially in steady state with
parameters and . What happens if the
savings rate increases permanently from to a new
and higher level, ?
B, ,s, n
s
s'
kt
sBkt
k*
(n + )kt
s’Bkt
k08
2.
2.The economy is initially in steady state. No The economy is initially in steady state. No parameters change, but an exogenous event, e.g., a parameters change, but an exogenous event, e.g., a war or natural disaster, reduces the capital stock to war or natural disaster, reduces the capital stock to half size ”overnight”. How is this analysed in the half size ”overnight”. How is this analysed in the Solow diagram?Solow diagram?
ktsBktk*(n + )ktk*/2 9
STEADY STATE
• Last time we saw that the Solow model implies
convergence to a unique steady state. From the
Solow equation
one easily computes
and then
( ( ) ) 1
1
1 t t t t k k sBk n k
n
+ − = − +
+
1
1
1
* 1 s
k B
n
−
−
=
+
( )
1
1
* * s
r B k
n
−
−
= =
+
( )
1
1
* * 1 s
y B k B
n
−
−
= =
+
( ) ( )
1
1
1 1 1 * * s
w B y
n
−
−
= − = −
10+
Some main lessons (repeated from previous note)
• The elasticity of wrt. is :
an increase in the savings rate of 10%, e.g., from 20
to 22%, gives a long-run increase in income per
worker of around 5% according to the basic Solow
model!
• The elasticity of wrt. is ! Note that the
effect is stronger than one-to-one due to capital
accumulation.
• Another steady state prediction concerns the real
interest rate…
( )
1
ln ln ln
1 1
* y B s n .
= + − + − −
* y s
1
1 2
−
1 3
1
1 2
−
B * y
11
THE ”NATURAL” INTEREST RATE
• The real rate of interest is determined by productivity
and thrift in the long run (Knut Wicksell):
– Higher capital is more productive demand of
capital per worker increases (ceteris paribus) higher
equilibrium interest rate (the price of capital).
– Higher supply of capital per worker increases
the equilibrium interest rate decreases.
• Reasonable parameter values on an annual basis,
, imply
and . This value for is very close to
empirical observations of the real interest rate!
1 1
* * s s
r
n n
− −
= = −
+ +
s / (n + )
8 3 * =1/ 3, s = 0.22, n = 0.005, = 0.05 r = . %
3 3 * = . % *
12
STRUCTURAL POLICY
1. Crowding out:
• Consider a permanent fall in caused by a permanent
increase in government consumption as a percentage of
GDP.
• What happens on impact? is unaffected and still grows
at the rate of , and is unaffected. But savings decrease
and consumption increases. There is full crowding out.
• What happens in the longer run? During a transitional
period grows more slowly than at the rate of and
falls down to a new lower steady state level. There is more
than full crowding out. And the real interest rate
increases.
• The government cannot increase GDP by raising
government expenditure in the long run. How about the
short run? (Keynes…)
s
t Y
n t y
n
t Y t y
13
2.
2.Motives for taxMotives for tax--financed public services from a longfinanced public services from a long--run perspective run perspective
•
•Public investments (that would not be made by private Public investments (that would not be made by private agents)agents)
For government consumption:
For government consumption:
•
•Public (nonPublic (non--rival and possibly nonrival and possibly non--excludable) goodsexcludable) goods
•
•Public consumption, e.g., on education and health care, Public consumption, e.g., on education and health care, replacing private consumption, which means that is not replacing private consumption, which means that is not affectedaffected
•
•Distributive reasonsDistributive reasons
•
•Externalities (education)Externalities (education)
•
•General productivity effects of public consumption, e.g., General productivity effects of public consumption, e.g., judicial system, health care, etc. judicial system, health care, etc.
s
14
3. Incentive policies:
– Policies that do not affect model parameters directly
through government expenditure/revenue, but indirectly
through the way they affect private behaviour. We cannot
analyse incentive policies explicitly because private
behaviour has not been derived from optimization.
– Golden rule:
The that maximizes , which is , is called the
golden rule savings rate.
– The model suggests structural policies that
• promote technology
• encourage savings (assuming that is considerably
below ): institutions and incentive
• reduce
( )
1
1
1
1
1
1 1
*
*
s
y B
n
s
c B s
n
−
−
−
−
=
+
= −
+
* s c ** s =
n
s
** s
15
GROWTH IN THE BASIC SOLOW MODEL
• The long run prediction of the Solow model is its
steady state. What is the growth rate of GDP per
capita in steady state?
• Zero! Not in accordance with stylized facts.
• What is the growth rate of in steady state? Since
and is constant, and must grow at
the same rate, . GDP grows, but only at the same
rate as the labour force. Why is that?
• Assume that the economy initially is below steady
state, implying that . Then ,
implying that capital per worker increases. But, once
again, because of diminishing returns the growth
in and will ultimately cease.
• However, there is transitory growth. How long-lasting
is that?
t Y
Yt / Lt = yt t y t Y t L
n
*
t k k ( ) t t sBk n k +
t y t k
16
SIMULATION
• Initially we are in steady state with the following
parameter values:
representing a developing country. This implies
that .
• With effect first time in period 1, the savings rate increases
permanently to corresponding to the savings rate
of a typical Western economy, implying
and .
• Starting with in period zero and one we now simulate
over . We also calculate ,
and etc. for and draw the
evolutions of these variables in the following figures.
B =1, =1/ 3, = 0.05, n = 0.03,
s = 0.08
1 * * k = y =
s' = 0.22
5 20 * k ' = . 1 73 * y ' = .
0 k =1
t = 2,3,...
( ( ) ) 1
1
1
1 t t t k s' Bk k
n
+ = + −
+
t t y Bk = (1 ) t t c = − s' y
1 ln ln y
t t t g y y − = − t =1,2,3,...
17
The evolution of , and after the increase
in
t y y
t g t c
s
18
The figures show that transitory growth is relatively
The figures show that transitory growth is relatively longlong--lasting.lasting.
The evolution of , and after the increase
The evolution of , and after the increase in (continued)in (continued)
t y y
t g t c
s
19
• In the Solow model, the transition towards steady
state is at least as important as the steady state
itself. And during this transition there is growth in
and . Hence, the basic Solow model is a
growth model!
• It is easy to find the growth rate of :
This is called the modified Solow equation.
• The growth rate of follows from .
t y
t k
t k
( ( ) )
( ( ))
1
1 1
1
1
1
1
t t t t
t t
t
t
k k sBk n k
n
k k
sBk n
k n
+
+ −
− = − +
+
−
= − +
+
t y y k
t t g = g
20
The modified Solow diagram
( ( )) 1 1 1
1
t t
t
t
k k
sBk n
k n
+ − −
= − +
+
21
•
•Growth in GDP per worker is higher the further below Growth in GDP per worker is higher the further below steady state the economy is. This is in accordance steady state the economy is. This is in accordance with with conditional convergenceconditional convergence..
•
•A permanent increase in gives a jump upwards in A permanent increase in gives a jump upwards in the growth rate of GDP per worker. the growth rate of GDP per worker.
s
22
CONCLUSIONS BASED ON THE BASIC SOLOW
CONCLUSIONS BASED ON THE BASIC SOLOW MODELMODEL
•
•What can a (poor) country do to create a transitory What can a (poor) country do to create a transitory growth in GDP per worker resulting in a permanently growth in GDP per worker resulting in a permanently higher level of income and consumption per worker? higher level of income and consumption per worker? The basic Solow model provides the following The basic Solow model provides the following answers:answers:
–
–Increase the savings rateIncrease the savings rate
–
–Reduce the growth rate of the labour forceReduce the growth rate of the labour force
–
–Reduce the rate of depreciation, i.e., invest betterReduce the rate of depreciation, i.e., invest better
–
–Improve the level of technologyImprove the level of technology
•
•How useful are these recommendations?How useful are these recommendations?
23
ABOUT PROSPERITY
AND GROWTH”
based on Chapter 2
of your Textbook
Macroeconomics III
Prof. Guido
Cozzi
February
18, 2020
1
INTRODUCTION
INTRODUCTION
•
•How do we measure prosperity? In economics, often by GDP per How do we measure prosperity? In economics, often by GDP per person!person!
•
•Higher income means higher consumption or higher saving (possibly Higher income means higher consumption or higher saving (possibly both) and saving can be used for future consumption. Consumption both) and saving can be used for future consumption. Consumption is always rooted in income and is a decent proxy for wellis always rooted in income and is a decent proxy for well--being.being.
•
•In this course we often consider In this course we often consider growth growth of income per person of income per person because the way to reach a high because the way to reach a high levellevelof income per person is of income per person is through a process of high growth in income per person.through a process of high growth in income per person.
•
•Relatively small differences in growth rates can imply large Relatively small differences in growth rates can imply large differences in levels over long periods as evidenced by next slide.differences in levels over long periods as evidenced by next slide.
•
•Conclusion: Growth is important! One focus of our classes is Conclusion: Growth is important! One focus of our classes is What What creates growth?creates growth?
2
The importance of growth
The importance of growth
3
COMPARING THE GDP LEVELS OF TWO COUNTRIES
COMPARING THE GDP LEVELS OF TWO COUNTRIES
•
•PurchasingPurchasing--power adjustment: power adjustment:
–
–When converting, say, pesos into US dollars, we want to take into When converting, say, pesos into US dollars, we want to take into account differences in the costs of living in the two countries.account differences in the costs of living in the two countries.
–
–Hence, the relevant rateHence, the relevant rate--ofof--conversion should reflect that conversion should reflect that consumer goods are usually much cheaper in less developed consumer goods are usually much cheaper in less developed countries.countries.
•
•Per worker or per capita?Per worker or per capita?
–
–GDP per capita = Official GDP divided by total population: GDP per capita = Official GDP divided by total population: Underestimates the production of less developed countries, since Underestimates the production of less developed countries, since they typically have a larger share of the population working outside they typically have a larger share of the population working outside the official market economy.the official market economy.
–
–GDP per worker = Official GDP divided by the labour force GDP per worker = Official GDP divided by the labour force (population times participation rate): corrects for differences in (population times participation rate): corrects for differences in participation rates, hence it is more of a productivity measure.participation rates, hence it is more of a productivity measure.
4
THE LORENZ CURVE
THE LORENZ CURVE
•
•… is a comprehensive way to measure the degree of inequality of a … is a comprehensive way to measure the degree of inequality of a personal income distribution, for a single country or for the world.personal income distribution, for a single country or for the world.
•
•Stylized example: Suppose there are two countries in the world, one Stylized example: Suppose there are two countries in the world, one rich and one poor and that 60% of the world population lives in the rich and one poor and that 60% of the world population lives in the poor country earning 30% of world income, while 40% lives in the poor country earning 30% of world income, while 40% lives in the rich country earning 70% of world income.rich country earning 70% of world income.
•
•The Lorenz curve is defined as the curve through a collection of The Lorenz curve is defined as the curve through a collection of points (points (xx,,yy), where ), where xxis the share of people with incomes is the share of people with incomes belowbelowa a certain level and certain level and yyis the income share of these people.is the income share of these people.
•
•The curve always starts at (0,0). In our example, the next point is The curve always starts at (0,0). In our example, the next point is (60,30), since the 60% poorest people earn 30% of world income. (60,30), since the 60% poorest people earn 30% of world income. Finally, the curve always ends in (100,100). Curve on next slide.Finally, the curve always ends in (100,100). Curve on next slide.
5
•
•Imagine that the income Imagine that the income distribution were distribution were characterized by perfect characterized by perfect equality.equality.
•
•Then the 60% poorest Then the 60% poorest people would earn 60% people would earn 60% of total income.of total income.
•
•The Lorenz curve would The Lorenz curve would be identical to the be identical to the diagonal from the lower diagonal from the lower left to the upper right left to the upper right corner indicated in the corner indicated in the figure.figure.
•
•We can use this to define We can use this to define a measure of inequality a measure of inequality called…called…
6
THE GINI COEFFICIENT
THE GINI COEFFICIENT
•
•The area between the Lorenz curve and the diagonal The area between the Lorenz curve and the diagonal divided by the full area below the diagonal. In the figure divided by the full area below the diagonal. In the figure that is GC/(GC+A+B+C). A measure between zero and that is GC/(GC+A+B+C). A measure between zero and one.one.
•
•If the Gini coefficient is zero, there is no inequality.If the Gini coefficient is zero, there is no inequality.
•
•The larger the Gini coefficient becomes, the more The larger the Gini coefficient becomes, the more inequality there is.inequality there is.
•
•Using GDP per worker and population data for 137 Using GDP per worker and population data for 137 countries and artificially associating with each person in a countries and artificially associating with each person in a country the GDP per worker of the country, we can draw country the GDP per worker of the country, we can draw Lorenz curves for the world in 1960 and 2003, Lorenz curves for the world in 1960 and 2003, respectively:respectively:
7
•
•Stylized fact 1: Stylized fact 1: Some countries are rich and some are poor; Some countries are rich and some are poor; the differences are enormous and it has pretty much stayed like the differences are enormous and it has pretty much stayed like that in relative terms over the last 40 years. However, there is that in relative terms over the last 40 years. However, there is some tendency towards a more equal world income distribution, some tendency towards a more equal world income distribution, though not much at the very bottom. though not much at the very bottom.
8
Prosperity and poverty, growth and decline
Prosperity and poverty, growth and decline
9
Prosperity and poverty, growth and decline
Prosperity and poverty, growth and decline
•
•In the ’top 20’ of the fastest In the ’top 20’ of the fastest growing countries we find growing countries we find the ’growth miracles’, e.g., the ’growth miracles’, e.g., tiger economies such as tiger economies such as Korea, Singapore, Hong Korea, Singapore, Hong Kong, China and India. In Kong, China and India. In the bottom we find the the bottom we find the ’growth disasters’, many of ’growth disasters’, many of which have had negative which have had negative average growth rates over average growth rates over the period considered.the period considered.
10
•
•Stylized fact 2:Stylized fact 2:Growth rates vary substantially between Growth rates vary substantially between countries, and by the process of growing or declining fast, countries, and by the process of growing or declining fast, a country can move from being relatively poor to being a country can move from being relatively poor to being relatively rich, or from being relatively rich to being relatively rich, or from being relatively rich to being relatively poor.relatively poor.
•
•Stylized fact 3: Stylized fact 3: Growth can break in a country, turning Growth can break in a country, turning from a high rate to a low one or vice versa.from a high rate to a low one or vice versa.
•
•In the 1960s the EastIn the 1960s the East--Asian ”growth miracles” experienced Asian ”growth miracles” experienced growth breaks, moving from low to high growth rates. growth breaks, moving from low to high growth rates. Hence, it Hence, it isispossible for poor countries to escape from possible for poor countries to escape from poverty!poverty!
11
CONVERGENCE
CONVERGENCE
Hypothesis of absolute convergence:
Hypothesis of absolute convergence:In the long run, GDP In the long run, GDP per worker (or per capita) converges to one and the same per worker (or per capita) converges to one and the same growth path in all countries, so that all countries converge on the growth path in all countries, so that all countries converge on the same level of income per worker.same level of income per worker.
12
• If two countries are to end up with the same level of GDP per worker
and one of them starts at a level much lower than the other, then that
country must grow relatively quickly compared to the other.
• That is, if we estimate across countries i a regression line:
where is the average growth rate of GDP per worker,
and is the initial level of GDP per worker, we expect the estimate
of to be significantly positive - note the minus in front of
• For a sample of 24 rich OECD countries we find:
0
0 1 0
ln ln
ln
i i
T i y y
y
T
−
= −
1
( ) 0 ln ln i i
T y − y / T
0 ln i y
1 .
13
•
•There is a tight negative correlation and the estimate of is There is a tight negative correlation and the estimate of is indeed significantly positive.indeed significantly positive.
•
•However, the countries included in the sample were all relatively However, the countries included in the sample were all relatively rich in 2003. Therefore there is a ’sample selection bias’.rich in 2003. Therefore there is a ’sample selection bias’.
1
14
•
•For a more representative sample of 65 countries we find:For a more representative sample of 65 countries we find:
•
•The negative relationship has disappeared, and is The negative relationship has disappeared, and is insignificant. We have to reject the hypothesis of absolute insignificant. We have to reject the hypothesis of absolute convergence.convergence.
1
15
Hypothesis of conditional convergence: A country’s income
per worker converges to a country-specific long-run growth
path, which depends on the basic structural characteristics of the
country. The further below its own long-run growth path a country
starts, the faster it will grow. Income per worker therefore
converges to the same level across countries conditional on the
countries being structurally alike.
• The ”correct” regression line could be an equation like:
where the last term on the right hand side captures structural
influence, where is the average investment rate of country i, and
is the average growth rate of the labour force. (The theory behind
this formulation is introduced in later chapters.)
• For the same representative sample of 65 countries we get:
( ) 0
0 1 0 2
ln ln
ln ln ln 0 075
i i
T i i i y y
y s n .
T
−
= − + − +
i s i n
16
•
•Stylized fact 4Stylized fact 4: Convergence: If one controls appropriately : Convergence: If one controls appropriately for structural differences between the countries of the world, for structural differences between the countries of the world, a lower initial value of GDP per worker tends to be a lower initial value of GDP per worker tends to be associated with a higher subsequent growth rate in GDP per associated with a higher subsequent growth rate in GDP per worker.worker.
17
THE LONG
THE LONG--RUN GROWTH PROCESSRUN GROWTH PROCESS
•
•Stylized fact 5: Stylized fact 5: Over periods of more Over periods of more than 130 years, than 130 years, many countries in many countries in Western Europe and Western Europe and North America have North America have had relatively had relatively constant annual constant annual rates of growth in rates of growth in GDP per capita in the GDP per capita in the range 1.5range 1.5--2 per cent.2 per cent.
18
• Labour’s income share turns out to be quite stable (see figures)
• and labour’s share of income can be
written AS:
where is the average real wage per
worker, and is GDP per worker.
• This implies:
• Stylized fact 6: During the long
periods of relatively constant growth
rates in GDP per worker in the typical
Western economy, labour’s share of
GDP has stayed relatively constant,
hence the average real wage of a
worker has grown by approximately the
same rate as GDP per worker.
t t w / y
t w
t y
19
• The relative constancy of the
factor income shares and the
observation that real interest
rates do not have any trend
upwards or downwards in the
long run (see figures), so that
rates of return on capital must be
relatively stable,
( ) t t t r / Y / K
t r
• and the fact that capital’s share of
income can be written as:
where is the rate of return on
capital,
• implies:
20
•
•Stylized fact 7Stylized fact 7: During the long periods of relatively : During the long periods of relatively constant growth rates in GDP per worker in the typical constant growth rates in GDP per worker in the typical Western economy, capital’s share and the rate of return Western economy, capital’s share and the rate of return on capital have shown no trend, therefore the capitalon capital have shown no trend, therefore the capital--output ratio K/Y has been relatively constant, and the output ratio K/Y has been relatively constant, and the capital intensity K/L has grown by approximately the capital intensity K/L has grown by approximately the same rate as GDP per worker.same rate as GDP per worker.
21
BALANCED GROWTH
BALANCED GROWTH
•
•summarizes stylized facts 5summarizes stylized facts 5--7 and is an important 7 and is an important concept for evaluating the growth models in the concept for evaluating the growth models in the subsequent chapters.subsequent chapters.
•
•A growth process follows balanced growth if:A growth process follows balanced growth if:
1.
1.GDP per worker, consumption per worker, the real GDP per worker, consumption per worker, the real wage and the capital intensity all grow at one and the wage and the capital intensity all grow at one and the same constant rate, g.same constant rate, g.
2.
2.The labour force (population) grows at a constant The labour force (population) grows at a constant rate, n.rate, n.
3.
3.GDP, consumption and capital grow at the common GDP, consumption and capital grow at the common rate, g+n.rate, g+n.
4.
4.The capitalThe capital--ouput ratio and the rate of return on ouput ratio and the rate of return on capital are constant.capital are constant.
22
“CAPITAL
ACCUMULATION AND
GROWTH: THE
BASIC SOLOW
MODEL”
Based on Chapter 3 of
your textbook
Macroeconomics III
Prof. Guido
Cozzi
February
19, 20 20
THE BASIC SOLOW MODEL
THE BASIC SOLOW MODEL
•
•How can a nation become rich, i.e., initiate a growth How can a nation become rich, i.e., initiate a growth process leading to higher GDP/consumption per capita process leading to higher GDP/consumption per capita in the long run?in the long run?
•
•The basic Solow model provides some first answers: The basic Solow model provides some first answers: It predicts how the evolution and the longIt predicts how the evolution and the long--run levels run levels of GDP and consumption per capita depend on of GDP and consumption per capita depend on structural parameters such as the rate of investment structural parameters such as the rate of investment and the growth rate of the labour force.and the growth rate of the labour force.
•
•Key elements of the Solow model:Key elements of the Solow model:
–
–In each period, output is determined by the supplies of In each period, output is determined by the supplies of capital and labour through the production function.capital and labour through the production function.
–
–Exogenous savings/investment rate, Exogenous savings/investment rate, ss, exogenous growth , exogenous growth rate of labour force, rate of labour force, nn, and exogenous depreciation rate, , and exogenous depreciation rate, δδ..
–
–Explicit description of capital accumulation: Explicit description of capital accumulation:
–
–Accumulation of capital is the main driving force for wealth. Accumulation of capital is the main driving force for wealth. “Basic” model: No technological progress“Basic” model: No technological progress
1ttttKKIK+=+−
2
THE ”MICRO WORLD” OF THE SOLOW MODEL
• Object: Closed economy
• Time: A sequence of periods/years
• Agents: Households and firms (and government)
• Commodities and markets: Output, capital
services and labour services (one asset = physical
capital)
• The market for output: Supply = firms’ output, .
Demand from households for consumption and
investment = . Relative price = 1.
– One-sector model: Output can be used either for
consumption or for investment.
t = 0,1,2,...
t Y
t t C + I
3
• The market for capital services: Consumers own
the capital stock, , and rent its services to firms.
Supply of capital services = . Firms’ demand =
– Relative price (in units of output) for renting one unit of
capital for one period: = real rental rate for capital
– Real interest rate: , where is the rate of
depreciation, or:
(Alternative interpretation: The firms own the capital,
borrow for the purchase of capital at an interest rate of
and bear the cost of depreciation themselves.)
– User cost
• Labour market: Households supply = = the
labour force. Demand from firms = . Relative
price: = the real wage rate.
• Competitive markets: and adjust to equate
supply and demand in all markets full (or natural)
utilization of resources.
t K
s
t K
t r
rt = t +
t t = r −
t
t t = r = +
t L
d
t L
t w
d
t K
t r t w
4
THE PRODUCTION SIDE
• is modelled as if all production (all of GDP) comes
from one profit-maximizing firm that produces value
added, , from capital services, (machine-years),
and labour services, (man-years), according to
the production function:
1. Constant returns to scale: .
The replication argument!
2. Positive marginal products:
t Y
d
t K
d
t L
( ) d d
Yt = F Kt ,Lt
( ) ( ) d d d d
t t t t F K ,L =F K ,L
( ) 0 ( ) 0 d d d d
K L F' K ,L , F' K ,L .
5
3. Marginal products are decreasing in the amount of
the factor used:
(diminishing returns) and growing in the amount of
the other factor used:
• Profit maximization: Given and , the firm
chooses and to:
The usual necessary conditions for an optimum:
(These two equations do not determine and
from given and ; they only determine .)
0 0 KK LL F , F '' ''
t r t w
d
t L d
t t Y ,K
max s.t ( ) d d d d
t t t t t t t t Y −rK −w L , . Y = F K ,L
( ) ( ) d d d d
K t t t L t t t F K ,L = r , F K ,L = w ' '
d
t K d
t L
d d
t t K / L
0 0 LK LK F , F '' ''
t r t w
6
• Competitive market clearing and ,
where and are the supplies in period t:
Since and are predetermined in any given
period, and are determined this way.
and predetermined. What does that mean?
d
t t K = K d
t t L = L
t K t L
( ) ( ) K t t t L t t t F K ,L = r , F K ,L = w ' '
t K t L
t r t w
t K t L
7
THE INCOME DISTRIBUTION
1. No pure profits. Euler’s rule
2. The functional income distribution
The income share of each factor is the elasticity of
the production function with respect to the factor in
question.
( ) ( )
( ) ( )
t K t t t t K t t t
t L t t t t L t t t
r F K ,L rK F K ,L K
w F K ,L w L F K ,L L
= =
= =
' '
' '
( ) ( ) ( ) 0 t t K t F K ,L − F K ,Lt Kt − FL Kt ,Lt Lt = ' '
( )
( )
( )
( )
and
t t K t t t t t L t t t
t t t t t t
rK F K ,L K w K F K ,L L
Y F K ,L Y F K ,L
= =
' '
8
According to empirics, labour’s share is relatively constant
According to empirics, labour’s share is relatively constant around 2/3 over long periods except for shortaround 2/3 over long periods except for short--run fluctuations:run fluctuations:Labour’s share of domestic factor incomesLabour’s share of domestic factor incomes
9
Is there a production function that fulfills all of our
assumptions and has fixed output elasticities
independently of and ? Yes, the Cobb-Douglas
production function:
where is total-factor-productivity (TFP).
Check: It follows that
The Cobb-Douglas function seems to be a realistic longrun
assumption. We even have reason to believe that
.
t K t L
( ) 1 0 0 1 t t t t t t F K ,L B K L , B , , − =
t B
( )
1
and 1
and 1
t t
t K t t L t
t t
t t t t
t t
K K
r F B w F B
L L
rK w L
Y Y
−
= = = = −
= = −
' '
1/ 3
10
HOUSEHOLDS
• The number of households in period is , which is
predetermined. Household behaviour:
1. Each supplies one unit of labour inelastically. Total supply .
2. Owns the capital stock, , which is predetermined in period .
Supply = (as long as ).
3. The representative household decides given , and hence
. The intertemporal budget constraint:
We assume that the result of the consumer’s considerations is:
4. ”Biology”:
t t L
= Lt
t K t
t K 0 t r
t C t Y
t t t S Y −C
1 0 1 t t t t K K S K , + − = −
t t S = sY .
( ) 1 1 1 t t L n L , n + = + −
11
THE COMPLETE MODEL (with book equation numbers below)
• Parameters: and . N.b. No subscript on :
”Basic” Solow model.
• Endogenous variables: and
of which and are state variables:
1
t t t Y BK L − =
1
t
t
t
K
r B
L
−
=
(1 ) t
t
t
K
w B
L
= −
t t S = sY
( ) 1 1 t t L n L , + = +
B, ,s, n t B
( ) ( ) ( ) ( ) ( ) t t t t t Y , K , L , r , w ( ) t S
( ) t K ( ) t L
t 1 t t t K K S K + − = −
(14)
(15)
(16)
(17)
(18)
(19)
12
• Given and the model determines
• Government in the model? Yes. Simply interpret
as private plus government savings.
• We viewed the capital accumulation equation:
as the household’s budget constraint.
0 K 0 L ( ) ( ) ( ) t t t Y , K , L ...
t 1 t t t K K S K + − = −
t S
13
Alternatively: By definition we have that
The condition for equilibrium in the output market, or
the national accounting identity, is: , and
by definition: . Hence
Combining and gives the capital accumulation
equation again: is the savings rate and the
investment rate.
t t t Y = C + I
t t t S Y −C
( ) t t I = S
s
1 ( ) t t t t K K I K + − = −
() ()
14
ANALYSING THE BASIC SOLOW MODEL
1. Define: and .
2. From we get the per capita production
function:
Note:
t t t y = Y / L t t t k = K / L
1
t t t Y BK L − =
0 1 t t y Bk , =
( ) 1 1
y k
t t t t t t ln y ln y lnk lnk g g − − − = − =
15
t 1 t t t K K S K + − = − t t S = sY
( ) 1 1 t t L n L + = +
( ( ) ) 1
1
1
1 t t t k sy k
n
+ = + −
+
t t y Bk =
( ( ) ) 1
1
1
1 t t t k sBk k
n
+ = + −
+
t k
( ) 1
1
1 t t t t k k sBk n k
n
+
− = − +
+
savings per
capita
Replacement investment to
compensate for depreciation
and growth of labour force t = sy
”technical term”
appearing because
of discrete time 16
The transition diagram
The transition diagram
( ( ) ) 1
1
1
1 t tt k sBkk
n
+ = +−
+
17
• About the slope, :
– It is everywhere strictly positive
– It goes to infinity as goes to zero
– It goes to as , and
. Indeed we assume that (realistic!).
• The figure shows that in the long run. Hence
, and
etc.
• The values , etc. define the ”steady state”.
(1− ) / (1+ n)
1 ddt t k / k +
(1− ) / (1+ n) 1
*
tk →k
( ) * *
t y y B k
→ = (1 ) * *
t c →c = − s y ( ) 1
* *
t r r B k
−
→ =
t k
t k →
n+ 0
* * k , y
n −
18
THE SOLOW DIAGRAM
Why does growth in and have to stop?
Diminishing returns!
( ( ) ) 1
1
1 t t t t k k sBk n k
n
+ − = − +
+
kt
sBkt
k*
(n + )kt
t k t y
19
STEADY STATE
• The long run levels , , etc. depend on
parameters. How? What makes a nation rich?
• Look at the Solow equation:
In steady state . Insert this to find
* y * k
( ( ) ) 1
1
1 t t t t k k sBk n k
n
+ − = − +
+
1 0 t t k k + − =
( ) ( )
( )
1
1
1
1
1
1
1
* *
*
* *
sB k n k
s
k B
n
s
y B k B
n
−
−
−
−
= +
=
+
= =
+
20
• Some sharp predictions of the Solow model:
The elasticity of wrt. is (since we believe
that ): an increase in of 10%, e.g., from 20 to
22%, should give an increase in of 5%!
The elasticity of wrt. and wrt. are
and , respectively. Why is the latter not one?
Capital accumulation.
• We have reached empirically testable hypotheses!
Empirics:
( )
1
ln ln ln
1 1
* y B s n .
= + − + − −
* y s
1
1 2
−
1 3
1 2
−
n+ B
s
* y
1
1 2
− −
−
1
3
* y
21
Real GDP per worker against the average investment
Real GDP per worker against the average investment share across 65 countriesshare across 65 countries
22
Real GDP per worker against the average annual labour force
Real GDP per worker against the average annual labour force growth rate across 65 countriesgrowth rate across 65 countries
23
“CAPITAL
ACCUMULATION AND
GROWTH: THE BASIC
SOLOW MODEL”
…also based on
Chapter 3 of your
textbook
Macroeconomics
III
Prof. Guido
Cozzi
February
18, 2020
1
THE COMPLETE MODEL (with book equation numbers below)
• Parameters: and . N.b. No subscript on :
”Basic” Solow model.
• Endogenous variables: and
of which and are state variables:
1
t t t Y BK L − =
1
t
t
t
K
r B
L
−
=
(1 ) t
t
t
K
w B
L
= −
t t S = sY
( ) 1 1 t t L n L , + = +
B, ,s, n t B
( ) ( ) ( ) ( ) ( ) t t t t t Y , K , L , r , w ( ) t S
( ) t K ( ) t L
t 1 t t t K K S K + − = −
(14)
(15)
(16)
(17)
(18)
(19)
2
GOVERNMENT SECTOR
• follows from the identity/definition
and the national accounting identity
• Same construction with a public sector:
• Insert into the identity to get
t 1 t t t K K S K + − = −
t 1 t t t K K I K + − = −
t t t t t t t t Y = C + I Y −C = I S = I
p g p g
t t t t t Y =C +C + I + I
p g
St St
p g p g
t t t t t t t Y −T −C +T −C = I + I
1
p g
t t t t t K K I I K + − = + −
1
St
p g
t t t t t K K S S K + − = + −
3
• We can use the model as it stands. We just have to
reinterpret as the sum of private and government
capital stock and as the sum of private and public
savings. Similarly the equation should be
reinterpreted as:
• The essential assumption underlying the Solow model
interpreted to include a government is that the sum
of private and public consumption as a fraction
of GDP is a constant, .
• This seems plausible empirically. And seems
plausible for many Western countries:
t K
t S
St = sYt
(1 ) p g p g
t t t t t t t S = S + S = sY C +C = − s Y
1−s
s 0.2
4
The consumption share of GDP in several Western countries
The consumption share of GDP in several Western countries
5
THE BASIC SOLOW MODEL, SHORT VERSION
• In the previous part we saw that the Solow model
leads to the transition equation:
• and to the Solow equation:
1
t t t Y BK L − =
( ) 1 1 t t L n L + = +
t 1 t t t K K sY K + − = −
( ( ) ) 1
1
1
1 t t t k sy k
n
+ = + −
+
( ) 1
1
1 t t t t k k sBk n k
n
+
− = − +
+
savings per
capita t = sy
Replacement investment to
compensate for depreciation
and growth of labour force
”technical term”
appearing because
of discrete time
6
The Solow diagram (repetition)
Why does growth in and have to stop?
Diminishing returns!
( ( ) ) 1
1
1 t t t t k k sBk n k
n
+ − = − +
+
kt
sBkt
k*
(n + )kt
t k t y
7
COMPARATIVE ANALYSIS IN THE SOLOW DIAGRAM
1. The economy is initially in steady state with
parameters and . What happens if the
savings rate increases permanently from to a new
and higher level, ?
B, ,s, n
s
s'
kt
sBkt
k*
(n + )kt
s’Bkt
k08
2.
2.The economy is initially in steady state. No The economy is initially in steady state. No parameters change, but an exogenous event, e.g., a parameters change, but an exogenous event, e.g., a war or natural disaster, reduces the capital stock to war or natural disaster, reduces the capital stock to half size ”overnight”. How is this analysed in the half size ”overnight”. How is this analysed in the Solow diagram?Solow diagram?
ktsBktk*(n + )ktk*/2 9
STEADY STATE
• Last time we saw that the Solow model implies
convergence to a unique steady state. From the
Solow equation
one easily computes
and then
( ( ) ) 1
1
1 t t t t k k sBk n k
n
+ − = − +
+
1
1
1
* 1 s
k B
n
−
−
=
+
( )
1
1
* * s
r B k
n
−
−
= =
+
( )
1
1
* * 1 s
y B k B
n
−
−
= =
+
( ) ( )
1
1
1 1 1 * * s
w B y
n
−
−
= − = −
10+
Some main lessons (repeated from previous note)
• The elasticity of wrt. is :
an increase in the savings rate of 10%, e.g., from 20
to 22%, gives a long-run increase in income per
worker of around 5% according to the basic Solow
model!
• The elasticity of wrt. is ! Note that the
effect is stronger than one-to-one due to capital
accumulation.
• Another steady state prediction concerns the real
interest rate…
( )
1
ln ln ln
1 1
* y B s n .
= + − + − −
* y s
1
1 2
−
1 3
1
1 2
−
B * y
11
THE ”NATURAL” INTEREST RATE
• The real rate of interest is determined by productivity
and thrift in the long run (Knut Wicksell):
– Higher capital is more productive demand of
capital per worker increases (ceteris paribus) higher
equilibrium interest rate (the price of capital).
– Higher supply of capital per worker increases
the equilibrium interest rate decreases.
• Reasonable parameter values on an annual basis,
, imply
and . This value for is very close to
empirical observations of the real interest rate!
1 1
* * s s
r
n n
− −
= = −
+ +
s / (n + )
8 3 * =1/ 3, s = 0.22, n = 0.005, = 0.05 r = . %
3 3 * = . % *
12
STRUCTURAL POLICY
1. Crowding out:
• Consider a permanent fall in caused by a permanent
increase in government consumption as a percentage of
GDP.
• What happens on impact? is unaffected and still grows
at the rate of , and is unaffected. But savings decrease
and consumption increases. There is full crowding out.
• What happens in the longer run? During a transitional
period grows more slowly than at the rate of and
falls down to a new lower steady state level. There is more
than full crowding out. And the real interest rate
increases.
• The government cannot increase GDP by raising
government expenditure in the long run. How about the
short run? (Keynes…)
s
t Y
n t y
n
t Y t y
13
2.
2.Motives for taxMotives for tax--financed public services from a longfinanced public services from a long--run perspective run perspective
•
•Public investments (that would not be made by private Public investments (that would not be made by private agents)agents)
For government consumption:
For government consumption:
•
•Public (nonPublic (non--rival and possibly nonrival and possibly non--excludable) goodsexcludable) goods
•
•Public consumption, e.g., on education and health care, Public consumption, e.g., on education and health care, replacing private consumption, which means that is not replacing private consumption, which means that is not affectedaffected
•
•Distributive reasonsDistributive reasons
•
•Externalities (education)Externalities (education)
•
•General productivity effects of public consumption, e.g., General productivity effects of public consumption, e.g., judicial system, health care, etc. judicial system, health care, etc.
s
14
3. Incentive policies:
– Policies that do not affect model parameters directly
through government expenditure/revenue, but indirectly
through the way they affect private behaviour. We cannot
analyse incentive policies explicitly because private
behaviour has not been derived from optimization.
– Golden rule:
The that maximizes , which is , is called the
golden rule savings rate.
– The model suggests structural policies that
• promote technology
• encourage savings (assuming that is considerably
below ): institutions and incentive
• reduce
( )
1
1
1
1
1
1 1
*
*
s
y B
n
s
c B s
n
−
−
−
−
=
+
= −
+
* s c ** s =
n
s
** s
15
GROWTH IN THE BASIC SOLOW MODEL
• The long run prediction of the Solow model is its
steady state. What is the growth rate of GDP per
capita in steady state?
• Zero! Not in accordance with stylized facts.
• What is the growth rate of in steady state? Since
and is constant, and must grow at
the same rate, . GDP grows, but only at the same
rate as the labour force. Why is that?
• Assume that the economy initially is below steady
state, implying that . Then ,
implying that capital per worker increases. But, once
again, because of diminishing returns the growth
in and will ultimately cease.
• However, there is transitory growth. How long-lasting
is that?
t Y
Yt / Lt = yt t y t Y t L
n
*
t k k ( ) t t sBk n k +
t y t k
16
SIMULATION
• Initially we are in steady state with the following
parameter values:
representing a developing country. This implies
that .
• With effect first time in period 1, the savings rate increases
permanently to corresponding to the savings rate
of a typical Western economy, implying
and .
• Starting with in period zero and one we now simulate
over . We also calculate ,
and etc. for and draw the
evolutions of these variables in the following figures.
B =1, =1/ 3, = 0.05, n = 0.03,
s = 0.08
1 * * k = y =
s' = 0.22
5 20 * k ' = . 1 73 * y ' = .
0 k =1
t = 2,3,...
( ( ) ) 1
1
1
1 t t t k s' Bk k
n
+ = + −
+
t t y Bk = (1 ) t t c = − s' y
1 ln ln y
t t t g y y − = − t =1,2,3,...
17
The evolution of , and after the increase
in
t y y
t g t c
s
18
The figures show that transitory growth is relatively
The figures show that transitory growth is relatively longlong--lasting.lasting.
The evolution of , and after the increase
The evolution of , and after the increase in (continued)in (continued)
t y y
t g t c
s
19
• In the Solow model, the transition towards steady
state is at least as important as the steady state
itself. And during this transition there is growth in
and . Hence, the basic Solow model is a
growth model!
• It is easy to find the growth rate of :
This is called the modified Solow equation.
• The growth rate of follows from .
t y
t k
t k
( ( ) )
( ( ))
1
1 1
1
1
1
1
t t t t
t t
t
t
k k sBk n k
n
k k
sBk n
k n
+
+ −
− = − +
+
−
= − +
+
t y y k
t t g = g
20
The modified Solow diagram
( ( )) 1 1 1
1
t t
t
t
k k
sBk n
k n
+ − −
= − +
+
21
•
•Growth in GDP per worker is higher the further below Growth in GDP per worker is higher the further below steady state the economy is. This is in accordance steady state the economy is. This is in accordance with with conditional convergenceconditional convergence..
•
•A permanent increase in gives a jump upwards in A permanent increase in gives a jump upwards in the growth rate of GDP per worker. the growth rate of GDP per worker.
s
22
CONCLUSIONS BASED ON THE BASIC SOLOW
CONCLUSIONS BASED ON THE BASIC SOLOW MODELMODEL
•
•What can a (poor) country do to create a transitory What can a (poor) country do to create a transitory growth in GDP per worker resulting in a permanently growth in GDP per worker resulting in a permanently higher level of income and consumption per worker? higher level of income and consumption per worker? The basic Solow model provides the following The basic Solow model provides the following answers:answers:
–
–Increase the savings rateIncrease the savings rate
–
–Reduce the growth rate of the labour forceReduce the growth rate of the labour force
–
–Reduce the rate of depreciation, i.e., invest betterReduce the rate of depreciation, i.e., invest better
–
–Improve the level of technologyImprove the level of technology
•
•How useful are these recommendations?How useful are these recommendations?
23
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