Download handout Solow model

READING LIST ON ECONOMIC GROWTH
Topics and readings:The main text is Auerbach, A., and L. Kotlikoff, “Macroeconomics-An
integrated Approach”, 2nd edition, MIT Press.
Other required readings are journal articles, which are available on the course web.
Also chapters in David Weil, Economic Growth, 2nd edition, are part of the course.
Summary required readings:
Auerbach and Kotlikoff (AK): Ch. 1-3
(ch 5: review of national accounts.), ch 6, 12, Ch. 15, pp. 411-423.
David Weil; Openness, Government; Culture; Inequality; Geography.
These chapters will be presented by the students.
Journal articles.
1. Review and in-depth analysis of the Solow-model.
Required reading: This handout. Read ch. 1 in AK, which deals with the production function.
Optional readings: Relevant chapters in Weil or chapters 7-8 in Mankiw, Macroeconomics.
Extra-credit exercise: simulation exercise of the Solow-model to be handed in.
Student may collaborate, but the group should not be bigger than 3 students.
Even though extra-credit exercises are not required the return is high in terms of points per
hour of effort. Hence, I recommend you to do them.
2A. Empirics of Economic Growth
Required reading: Two journal articles available on the course-web:
J. Persson, Convergence across the Swedish Counties 1906-1993,
European Economic Review, 1997, eer.pdf
R. Barro, Human Capital and Economic Growth,
Swedish Economic Policy Review, growthbarro.pdf
2B. Introduction to econometrics
Students not previously exposed to econometrics should read Introduction to econometrics
(pdf-file).
Students should do econometric exercise and write econometric report.
For C-students this exercise is an extra credit exercise; for D-students it is required.
In addition, D-level students should do additional report related to economic growth or to the
course in general. It could e.g. be that students include an additional explanatory variable in
the regressions related to the empirical growth exercise mentioned above. For example,
include some geographic variable.
3. Endogenous growth models
Required reading: Handout below.
Optional readings: Relevant chapters in Weil or ch. 8 in Mankiw, Macroeconomics.
4. The overlapping generation model (OLG)
A. Optimal consumption, and saving by a young person: AK: chapter 2.
B. Completing the closed-economy OLG-model: AK, Ch. 3
C. Allowing for population growth in this model: AK, appendix to ch.3.
D. More in-depth analysis of the choices of the household:
Regarding saving the household may receive income in the last period of life as well, and may
choose the number of hours it want to work in both periods of life.
Required reading: Handout below (which is based on the textbook by Varian).
Extra-credit exercise: Saving and labor-supply-exercise to be handed in for extra credit.
5. Review of National Income Accounting: AK, chapter 5.
6. Factor mobility in the OLG-model: An analysis of globalization: AK, ch. 12.
7. A government sector in the OLG-model: AK, chapter 6;
8. Precautionary saving in the OLG-model: AK, ch.15 (pp. 411-423): Handout below;
During the later part of the course we study more applied topics:
These topics should primarily be presented in-class by the students.
9. Income distribution and Economic Growth.
Article on income inequality and economic growth by Robert Barro; chapter in Weil
10. Economic growth and openness. Chapter in Weil.
11. Economic growth and the government. Chapter in Weil.
12. Economic growth and Culture. Chapter in Weil.
13. Economic growth and Geography. Chapter in Weil.
14. Pollution and Economic Growth.
Persson (2008), to be added on the course web and chapter in Weil.
Students will get points for their presentations.
Not included any longer:
15. Fertility Choice and Income level.
Reading: handout below.
GROWTH ACCOUNTING, read yourselves.
Mathematics: Proportional or percentage Changes in Economics.
Expressing levels into growth rates:
Rule 1. If y(t) = x(t)*z(t), then
y
x z

 .
y
x
z
Ex.: Total Revenue (TR) = Price(P)*Quantity(Q)
 If P is raised by 10 % and sold quantity (Q) thereby decreases by 5%, then
TR increases by 5 %.
y
x z
.


y
x
z
Rule 2. If y(t) = x(t)/z(t), then
Ex.: GNP per capita (y) = GNP(Y)/Population(Pop)
 If GNP (Y) increases by 5 % and the population increases by 3 % , then GNP
per capita increases by 2 %.
Rule 3. If y(t )  x(t)a , then

Rule 4. If Yt  At  Kt  Lt
then
y
x
 a .
y
x
1
,
Y
A
K
L

 
 (1   ) 
Y
A
K
L
(1)
(2)
(3)
thus the growth rate of Y equals:
(1) The growth rate of totalfactorproductivity.
(2) The contribution of physical capital.
(3) The contribution of labor.
Question addressed by so-called growth accounting
growth accounting:
How big share of the growth rate of the GDP can be attributed to changes in
capital, to changes in the labor input and to changes in total factor productivity?
For developed countries we have good data on
We have not direct data on
Y K
L
,
and
,
Y
K
L
A
as A captures the influence on Y of many
A
different factors on Y. E.g. taxes, climate for business, educational level of work
force, infrastructure, social capital etc.
Under perfect competion,  , is the share of national income that is capital
income, and (1-  ) is the share of national income that is labor income. We have
data on labor income and national income. Thereby, we get an estimate of  .
Example:
Year
2005
2006
Y
100
103
A
?
?
K
300
306
L
1000
1010
Y
K
L
 0.03,
 0.02 and
 0.01
Y
K
L
A
 0.03 =
+ 0.3*0.02 + 0.7*0.01
A
A

= 0.03 – 0.006 – 0.007 = 0.017.
A
0.017/0.03 = 0.57 : 57 percent of the growth rate of Y can be attributed to an
increase in A. 0.006/0.03 = 0.2: 20 percent can be attributed to an increase in K.
0.007/0.03 = 0.23: 23 procent can be attributed to an increase in L.
We have not explained why K, L and A changes over time.
We have only been engaged in accounting.
The neoclassical growth model explains why K and thereby Y increase.
(A and L are exogenously given in this model; that is, they are determined
outside the model.)
ECONOMIC GROWTH, read yourselves
Aim to explain why the standard of living (GDP/GNP per capita) changes over
time. Main text: David Weil.
Math: Growth rate = Percentage Change
y y
y
, e.g. r1 = 0.02, that is, 2 % .
 1 0  r1
y
y0
where y0 = income per capita year 0, y1 = income per capita year 1.
r1 = growth rate/percentage change between year 0 and year1.
y1  y0  r1  y0  y1  r1  y0  y0  y0  (1  r1 )

Analogously:
y2  y1  (1  r2 ) , y3  y2  (1  r3 )

y3  y0  (1  r1 )  (1  r2 )  (1  r3 )
y1
y2
At a constant yearly percentage change (growth rate) income year 3 is:
y3  y0  (1  r )  (1  r )  (1  r )  y0  (1  r )3
where r = constant yearly growth rate/percentage change.
After t years and a constant growth rate income per capita equals:
yt  y0  (1  r )t ,
where t = number of years.
Exercise: If GDP per capita (in 1995 prices) in 1995 and in 2000 was 194 and
222 thousands, what is the average annual growth rate during this 5-year period?
Graphical representation of the exponential function:
yt  y0  (1  r )t . Let y0  1 and r = 0.03: yt  (1  0.03)t
4.5
4
3.5
3
2.5
y
Serie1
2
1.5
1
0.5
0
0
10
20
30
40
50
time
If r increases, steeper slope. If y0 increases, the curve shifts upwards.
Read yourselves:
t
Alternative graphical representation of the function: yt  y0  (1  r )
 ln( yt ) = ln( y0  (1  r ) )
t
 ln( yt ) = ln( y0 ) + ln( (1  r ) )
t
 ln( yt ) = ln( y0 ) + t  ln(1  r )
 ln yt  ln y0 
int ercept
ln(1  r)  t
slope coefficient
This is the equation for a straight line: y = a + b  x
If r is a small number < 0.1  ln(1 r)  r  ln yt  ln y0
 r t
The logarithm function (r=0.03)
1.6
1.4
1.2
ln y
1
0.8
Serie1
0.6
0.4
0.2
0
0
10
20
30
40
50
time
Formula: yt  y0  (1  r )
How many years does it take to double y at different growth rates?
t
 2 y0  y0  (1  r )
t
 2  (1  r )
t
 ln(2) = ln( (1  r ) )
 ln(2) = ln(1  r )  t  r  t
 ln(2)/r  t
 t  ln(2)/r
t
If r = 0.05  t  14 years.
If r = 0.015  t  46 years.
THE SOLOW GROWTH MODEL: Lecture
Aim to explain the development over time of K and Y, k=K/L, and y=Y/L.
In the model the growth rate of the technological level, A / A  g , and the
growth rate of the numbers of workers, L / L  n , are exogenous variables.
In the model everyone is a worker. Thus, the number of workers = population.
Thus, they are determined outside the model.
As A / A , and L / L are assumed to be exogenous variables, the model is
about the accumulation of physical capital, and its effects on k, Y, and y.
To simplify we start by assuming: A / A  L / L  0 .
Thus, the level of A and L are assumed to be constant over time.
Assumption (A1): The production function:
where 0 <  < 1.
Yt  A  Kt  L1 ,
Expressing production in terms of per worker (labor productivity):
Yt At  Kt  Lt1

Lt
Lt

 At  Kt  Lt   At  Kt  Lt 
K 
 At  t 
 Lt 
Labor productivity depends on:
* Totalfaktorproductivity, A. If A   Y/L 
* Physical capital per worker, K/L. If K/L   Y/L 
Note: (1-  )* Labor productivity (Y/L) = MPL (= W/P)
 y  A  k
Thus, the level of A and L are assumed to be constant over time.
As L is assumed to be constant we can assume L=1.
Small letters indicate that variables are expressed in terms of per worker.
Lecture:

y=Y/L
Labor productivity
9
8
7
6
5
4
3
2
1
0
Y/L=A(K/L)
A=2
A=2
Serie1
Serie2
A=1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
k=K/L
In figure: A=1,2, and  =0.5 .
Slope of the curve above is MPK=dY/dK.
Note: L is assumed to be a constant; for example, L=1  K=k
More complicated proof which is optional:
dy
[    A  k  1 , which is MPK:
dk
 1
dY
K
   A  K  1  L1    A  K  1  L( 1)    A    ]
MPK=
dK
L
Note that:
dY
K
A  K   L1
Y
   A  K  1  L1    A  K  1  L1  ( 1)   
 
dK
K
K
K
Thus, MPK   
Y
   ( production per unit of capital )    APK
K
Thus, the marginal and the average product of labor are linked together:
When APK decreases, the also MPK decreases.
MPK is less than APK.
Lecture:
Assumption (A2): A constant share of income is saved
(= a constant share of production is invested).
Goods market equilibrium condition: Y  C  I  G  NX
We assume a closed economy without a government sector:
 G=NX=0
 S=Y-C=I
 National saving equals gross investment.
S
(A2): S  s Y ,
where s is the share of income that is saved. s 
Y
Note1: Y=GDP=GNP
Note2: s is not saving per worker even though s is a small letter.
Y I
 s Y  I
 s 
 s y i
L L
Y CI C I

   ci
L
L
L L
 y  c  i  (1  s)  y  s  y  (1  s)  A  k   s  A  k 
Moreover,
y=Y/L,i
production and investment per worker
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
y=Ak

i=sy=sAk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Serie1
Serie2

16
17
k=K/L
In figure: s=0.3, A=1, and  =0.5. The vertical distance between the curves for
production per worker, and investment per worker is consumption per worker.
Lecture:
Assumption (A3):
K  I    K
where K is net investment, I = gross investment, and   K = depreciation of
capital per period.  is the depreciation rate, which is between 0 and 1; e.g.
0.05. (That is, 5 %).  If I    K , then K  0
If I    K , then K  0 ; If I    K , then K  0 .
Expressing (A3) in terms of per worker:  k  i    k
K I
I L
I /L
i
Derivation optional:[
     
  
K
K
K L
K/L
k
L
k K L
Using k = K/L 
, where
=0 by assumption.


L
k
K
L
k  i    k  i    k ]
k K



k
K
k k
d*k
Depreciation of capital per worker
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Serie1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
k
In figure:   0.1
The whole model:
(A1): y  A  k  ,
the production function
(A2): i  s  y
investment = saving (equilibrium condition) where
saving is constant share of income.
(A3): k  i    k ,
The time path of the capital stock per worker
The whole model can be reduced to one equation:
Inserting (A1) and (A2) into (A3):
k  i    k  s  y    k  s  A k     k
The long-run equilibrium (steady state) value for k, k * , occurs when k  0 .
That is, when gross investment equals depreciation
 s y   k
 s  A k     k
s  A  k  s  A  k 1
Solving for k in equilibrium:


k
Lecture:
1
1 1
k 

1
1
 s  A 1


  

 s  A 1
k*  

  
What is the long-run equilibrium value of y, y* ?

 s  A 1
y *  A  (k * )  A  
 If s  or A  

  
k *  and y* .
If the economy is not in its equilibrium, it converges over time towards the
equilibrium because if k< k * , then i>   k  k * , and
if k> k * , then i<   k  k * . See figure below.
Showing the equilibrium in the Solow diagram:
y, i, dk
The SOLOW MODEL
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0

y=Ak
y=A*k*exp(alfa)
Serie1
Serie2
Serie3
dk
dk
i=sy=sAk
1
2
3
4
5
6
7
8
9
k
In figure: A=1, s=0.3,   0.1 , and  =0.5.
10
11
12
13
14
15

16
17
Lecture:
The transition to equilibrium: a numerical example
Starting below the equilibrium: The initial value of k: k(year=1)=4.00.
Assume also: A=1, s=0.3,   0.1 , and  =0.5.
year k
i
 k k
 y
k
y  k 0.5 c 
(1  0.3)  y 0.3  y
k
1
2
3
4
5
…
4.00
4.2
4.395
4.584
4.768
2.00
2.049
2.096
2.141
2.184
1.4
1.435
1.467
1.499
1.529
0.6
0.615
0.629
0.642
0.655
0.4
0.420
0.440
0.458
0.477
0.2
0.195
0.189
0.184
0.178
0.049
0.047
0.045
0.043
0.05
0.046
0.043
0.040
9
3
2.1
0.9
0.9
0
0
0

The equilibrium values of k and y are calculated by using the formulas:
1
 s  A 1
k 
 ,
  
*

 s  A 1
y  A  (k )  A  

  
*
*

How to fill out the Table based on an initial value and assumed parameter
values: A=1, s=0.3,   0.1 , and  =0.5.
Start by filling out the column for k based on the formula:
k  i    k  s  y    k  s  A k     k
 k2  k1  s  A k1    k1
 k2  k1  s  A k1    k1  (1   )  k1  s  A k1
If k(year=1)=4, A=1, s=0.3,   0.1 , and  =0.5.
k2  0.9  k1  0.3  k10.5 , k3  0.9  k2  0.3  k20.5 , etc.
After the values of k has been filled out, all other values of other variables
(columns) can be calculated.
 y
y
0.0245
0.0229
0.021
0.020
0
Lecture:
Graphical description of transition to equilibrium when economy start below and
above the equilibrium ln( y*  3 )=1.1:
(1) k(t=1)=4, y (t=1)=2 , and  ln(y (t=1)=2)=0.69
(2) k(t=1)=14, y (t=1)=3.74 , and  ln(y (t=1)=2)=1.32
Transition to equilibrium
1.4
1.2
ln (Y/L)
1
0.8
Serie1
Serie2
0.6
0.4
0.2
0
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
time
According to model:
The growth rates of k and y are higher the lower k and y are. This explains why
the slope of the curves for lny becomes flatter and flatter when lny approaches
its equilibrium. Recall that the slope of lny is the growth rate of y.

If two economies share the same equilibrium; that is, have the same parameter
values on A, s (as well as on  and  ) but differ with respect to initial values,
then the economy with lower k and y experience higher growth rates of k and y
than the economy with higher k and y.

the model says that y (and k) of these two economies converge over time. In
other words, the model says that y over time converge across economies if the
economies share the same equilibrium value of y).
Main lesson of empirical work on growth:
Real per capita income tends over time to converge across economies, which are
similar with respect to “institutions”.

An economy with an initially relatively low real income per capita has on
average a higher growth rate of real income per capita than an economy with an
initially relatively high real income per capita if “institutions” are similar. Ex.:
EU-countries and regions within countries.
Read by yourselves:
Evidence from the OECD-countries (the currently rich countries)
Growth rate of GDP p.c.
Average annual growth rate of GDP p.c., 1960-2000, and GDP
p.c. in 1960
0.05
0.04
0.03
Serie1
0.02
0.01
0
0
2000
4000
6000
8000
10000
12000
14000
16000
Real GDP per capita 1960
Sample includes: Australia, Austria, Belgium, Canada, Denmark, Finland,
France, Greece, Iceland, Ireland, Italy, Japan, Netherlands, New Zealand,
Norway Portugal, Spain, Sweden, Switzerland, United Kingdom and USA.
Initially poor countries grow faster in terms of real GDP per capita during the
period 1960-2000 than initially rich countries. The correlation between the
average annual growth rate of real GDP per capita between 1960 and 2000 and
real GDP per capita in 1960 = - 0.89
Evidence from the 24 Swedish Regions, 1911-1993
Regions that were relatively poor in terms of real income per capita in 1911, on
average had a higher growth rate of real income per capita.
Higher growth rates in poor regions caused relative differences in real per capita
income to diminish across the Swedish Regions between 1911 and 1993.
The dispersion is lower for real per capita income when it is adjusted for
regional differences in cost of living as counties with high unadjusted real per
capita incomes tend to have cost of living.
Per capita Income adjusted and unadjusted for cost of living
The empirical evidence on convergence in real per capita income across the
Swedish regions is consistent with the predications of the textbook model:
Low real per capita income 
Värmland moves to top category of per capita income
when regional differences in cost of living are accounted for in 1993
Per capita income (p.c.i) is in 1980 prices
Read by yourselves:
Little capital (physical + human) per worker,
low wages, high rates of return to capital capital per worker 
 production per worker   income per capita 
Also factor mobility tends to contribute to convergence:
Low wages and high returns to capital out-migration, and foreign investment
 capital per worker   production per worker 
Evidence from the countries of the world
Growth rate of GDP per
capita
Average annual growth rate growth rate of GDP p.c., 19602000, and GDP p.c. in 1960
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01 0
-0.02
S
2000
4000
6000
8000
10000
12000
14000
Real GDP per capita in 1960
Sample includes 80 countries.
 No convergence in real GDP per capita across the countries of the world. The
correlation between the average annual growth rate of real GDP per capita
between 1960 and 2000 and real GDP per capita in 1960 = + 0.14.
Is lack of convergence in GDP per capita for the countries of the world,
evidence against the model? NO!
The model says that if countries have the same equilibrium, the poorer country
should grow faster in terms of y and k than the country that is richer in terms of
y and k.
16000
Lecture:
But if countries differ with respect to equilibrium, that is, with respect to values
on A, s (as well as on  and  ), the poorer economy need, according to model,
not grow faster than the initially richer economy.
Africa is poor because it has a low equilibrium.
Example that a rich country can grow faster than a poor country
Country A (Poor Country): A=1, s=0.2,   0.1 , and  =0.5  : k *  4 , y *  2
Assumed initial values of k and y: 4 and 2.
 Country A’s growth rate of y=0
Country B (Rich Country): A=1, s=0.3,   0.1 , and  =0.5  : k *  9 , y*  3
Assumed initial values of k and y: 5 and 2.24.
 Country B’s growth rate of y is positive.
y, i, dk
Countries with different saving rates
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
y=k**0.5
i=0.3*y
i=0.2*y
0.1*k
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
k
Time path of y of two countries
3.5
3
Y/L
2.5
2
Country B
1.5
Country A
1
0.5
0
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93
time
Read by yourselves:
The short and long run effects of an increase of L (e.g. due to immigration)
y, i, dk
The SOLOW MODEL
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0

y=A*k*exp(alfa)
y=Ak
Serie1
dk
dk
Serie2
Serie3
i=sy=sAk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
k
A one-time increase of L:L(t=0)<L(t=1)= L(t=2)= L(t=3)= L(t=4)
 At time 1: K/L and Y/L, At time 2 and onwards: K/L  and Y/L
If the economy initially is in its equilibrium, it will over time revert to the initial
equilibrium as gross investment exceeds depreciation of capital.
K/L
Time path K/L
10
9
8
7
6
5
4
3
2
1
0
Serie1
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101
time
In figure: A=1, s=0.3,   0.1 , and  =0.5, K(0)=900, L(0)=100 and L(1)=200.
The long run values of k and y are unchanged. However, adjustment takes a long
time so migration plays a role for y during a long time according to model.
What happens to the long run values of Y and K?
Y *  A  K   L1  A  k *  L  Y *  A  k *  L(1)  Y *  A  k *  L(0)
K *  k *  L(1)  K *  A  k *  L(0)
 Size of economy increases when L increases.
Example: Y * increases from 3*100= 300 to 3*200=600,
and K * increases from 9*100=900 to 9*200=1800.
In case of a pandemic, L decreases, the results are the opposite.
Lecture:
The effect of an increase in A
y, i, dk
The SOLOW MODEL
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0

y=A*k*exp(alfa)
y=Ak
Serie1
dk
dk
Serie2
Serie3
i=sy=sAk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
k
Old value: A=1, s=0.3,   0.1 , and  =0.5  k *  9 and y*  3 .
y*  6.75
New value: A=1.5  k*  20.25 ,
The transition to the new long run equilibrium
Transition to new equilibrium
8
7
6
Y/L
5
Serie1
4
Serie2
3
2
1
0
1
7
13
19
25
31
37
43
49
time
55
61
67
73
79
85
91
97
Lecture:
Long-run growth of the number of workers
Before we assumed: Lt 1  Lt
 L(0)=L(1)=L(2)=L(3)
Now we assume : Lt 1  (1  n)  Lt
where n is the constant growth rate of the number of workers; e.g. 0.01.
L
L
L
L L
L
 t 1  (1  n)
 t 1  1  n  t 1  t  n  t 1
n
Lt
Lt
Lt Lt
Lt
L
 t n
Assumption A4.
 L(0)<L(1)<L(2)<L(3)
Lt
To keep k constant gross investment (I) now needs to compensate not only for
depreciation of capital to keep k constant but also for the fact that the number of
workers increases over time:
(A3):
K  I    K
Derivation below optional:
K I
I L
I /L
i
[
     
  
K
K
K L
K /L
k
L
k K L
 n by assumption.
, where


L
k
K
L
k  n  i   ]  k  i  (n   )  k
k
K


n 
k
k
K
k
Using k = K/L 
The whole model:
(A1): y  A  k  ,
the production function
(A2): i  s  y
investment = saving (equilibrium condition) where
saving is constant share of income.
(A3)+(A4): k  i  (n   )  k , The time path of the capital stock per worker
A-level students need not know mathematical derivation below:
The whole model can be reduced to one equation:
Inserting (A1) and (A2) into (A3)+(A4):
k  i  (n   )  k  s  y  (n   )  k  s  A k   (n   )  k
The long-run equilibrium
The long-run equilibrium (steady state) value for k, k * , occurs when k  0 .
That is, when gross investment equals “depreciation”
 s  y  (n   )  k
 s  A  k   (n   )  k
Lecture:
Solving for k in equilibrium:
1
1 1
k 

s A 
n 
1
1
k
k

s  A  k 1
n 
1
 s A


 n  

 s  A 1
k*  

 n  
What is the long-run equilibrium value of y, y* ?
 s A
y *  A  (k * )  A  

 n  

1
 If n  
k *  and y* .
The transition to the equilibrium
If the economy initially is in equilibrium and n  the economy moves over time
to the new lower equilibrium because when i< (n   )  k  k:
y, i,(n+d)k
Growth rate of L increases
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
y=k**0.5
i=0.3*y
(0+0.1)*k
(0.05+0.1)*k
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17
k
In figure: A=1, s=0.3,   0.1 ,  =0.5 and n=0 and 0.05.
(1) when n=0 
k *  9 and y*  3 .
(2) when n=0.05 
k *  4 and y *  2 .
Lecture:
Transition to new equilibrium
3.5
3
Y/L
2.5
Serie1
2
Serie2
1.5
Serie3
1
0.5
0
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
Time
The growth rates of aggregate variables:
K k L k
K  k L 
   n
K
k
L
k
Y y L y
 

n

Y  yL
Y
y
L
y
In the steady-state:
k * y*
 * 0
k*
y
K * k * L
 *   0n  n
K*  k*  L 
L
K*
k
*
*
y L
Y
Y *  y*  L
 *  * 
 0n  n
L
Y
y
END OF LECTURE NUMBER 1.
Read by yourselves: Factor prices: In the model:
C + I = real GDP (Y) = real capital income + real labor income
= (W/P)*L + (r+  )K =MPL*L+MPK*K
r= real return on physical capital, for example: 0.03; that is, 3 %.
K=physical capital, which is the asset (that is, the wealth in the economy).
Note: There is only one good in the Solow model, which is consumed or
invested. If it is invested it is an asset which yields a return. K is the only asset
in the economy. There exist no bonds, stocks or money in the model.
(r+  ) is the real cost per unit of capital which under perfect competition equals
the real rental cost per unit of capital (R/P).
Profit-maximization implies that W/P=MPL, R/P=(r+  )=MPK.
Expressing the equilibrium condition above in terms of per worker:
 c + i=y=(W/P) + (r+  )k
(1) W/P=MPL= (1   )  A  k   (1   )  y
(2) r      A  k  1    y / k  If k   W/P  and r 
In poor and rich countries K/L is low and high, respectively.
If the value of A is the same in poor and rich countries, the real return on capital
is higher in poor countries. As a result, we expect capital to move from rich to
poor countries, increasing K/L in poor countries and lowering K/L in rich
countries. Thereby, mobility of capital contributes to convergence in K/L
between rich and poor economies. We expect L to move the opposite way
because W/P is higher in rich countries. Mobility of L increases K/L in poor
countries and decreases K/L in rich countries. Thereby, it also contributes to
convergence in K/L between rich and poor countries.
Why do capital not flow to Africa? In other words, why
are not large investments taking place in some African countries?
Answer: Because A is low, which means that MPK=r+d is not so high.
This can be seen in Solow-diagram. (Allow countries to differ w.r.t. A.)
Important Exercise: Derive the equilibrium expressions for the real wage and
for the real return on capital; that is, express the real wage and the real return to
capital as functions of the exogenous variables: s, A, n, the depreciation rate and
alfa.
Read by yourselves:
The golden rule level of capital:
The level of capital that maximizes consumption per worker in equilibrium
Consumption per worker is the distance between the curve for labor productivity
( y  A  k  ) and the curve for depreciation of capital per worker: (n+d)k. This
distance is maximized at the level of k where the slopes of these two curves are
dy
the same: MPK 
   A  k  1  n  
dk
Solving the equation   A  k 1  n   for k yields the answer.
A government that wants to maximize consumption per worker should choose
the saving rate (s) so that this level of capital is achieved.
An economy can save too much. That is, by decreasing the saving rate per capita
consumption can increase in the steady state.
LECTURE II:
Adding realism in the model: continuing technological progress
There is technological progress if new production techniques arise due to
innovations such as the computer, engine, electricity, etc.
A  dA / dt  g ,
A
A
gt
optional reading: A(t )  A(0)  e
Model assumption: (A5):
where g =rate of technological progress is exogenously assumed.[Optional
reading: The model is here formulated in continuous time which means that time
changes continuously. Previously the model was in discrete
lecture:
time which means that the time is in periods. If the model were in discrete time:
At  A0  (1  g )t . Note: g  g .
Only technological progress can explain long run increases in the living
standard= GDP per capita = Y/L=y
Growth rates in the long-run equilibrium:
k * y*
k * y* w*
Before: *  *  0 , Now:
 * 
g
w
k
y
k*
y
y* k *
r*
K *
Y *
*
r

d



(
y
/
k
)
,
,
,



(
 * )0

g

n

g

n
r
y*
k
K*
Y*
Technological progress is exogenous
As the rate of technological progress is unexplained by the SOLOW model (that
is, exogenous), adding g to the model does not add any more economic insights
than the version of the model with g=0.
For this reason and because it is simpler we will focus on the version of the
model where g=0. Keep however in mind that technological progress makes the
model more realistic because in the real world y typically increases over time
due to new production techniques; that is, due to innovations.
What happens if the economy is off its equilibrium growth path?
ln(Y/L)
Transition to equilibrium growth path
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Serie1
Serie2
Serie3
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
Time
When the economy approaches its equilibrium growth path, the growth rate of y
deviates from the long-run growth rate (g). If an economy starts out below
(above) the equilibrium growth path, the growth rate of y is higher (lower) than
g. Holding constant the equilibrium growth path that is holding constant A(0), s,
n, g, d and alfa, a lower y means a higher growth rate of y.
Lecture:
What happens to the growth rate and to the equilibrium growth path if the
saving rate increases (or institutions improve or population growth )?
Transition to higher equilibrium growth path
2.5
ln(Y/L)
2
Serie1
1.5
Serie2
1
Serie3
0.5
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
time
If s increases, the equilibrium shifts upwards, and the growth rate of y is higher
than the long run growth rate during to the transition to the new equilibrium
growth path.
Lecture:
A mathematical treatment:
The Solow model with continuing technological progress:
Mathematical model (no need to understand details focus on figures). We do the
model in continuous time instead of in discrete time:
(A1): Y (t )  K (t )   ( A(t )  L(t ))1 ,
A(t )  A(0)  egt
Where A(0) is a shift variable that increases if human capital of workers
increases, if business climate improves, g is the rate of technological progress.
To get nice mathematical expressions we assume that technological progress is
labor-augmenting which means that A is multiplied with L.
(In case of a Cobb-Douglas pf labor-augmenting technological progress is
equivalent to neutral technological progress that we thus far have assumed:
 Y (t )  A(t)1  K (t)   L(t)1 Let B(t )  A(t )1
 Y (t )  B(t )  K (t )   L(t )1
 Neutral technological progress.)
Model continued:
We express all variables per effective worker, AL:
(A1): y  k 
where y  Y / AL
(A2): s  y  i
(A3):

dK / dt  I    K
dK / dt I
I AL
I / AL
i
   
 
  
K
K
K AL
K / AL
k
dk / dt dK / dt dA / dt dL / dt



K
A
L
k
Using k  K / AL 
 inserting (A4) and (A5):
dk / dt dK / dt

gn
K
k

dk / dt
dK / dt
ng 
K
k
Lecture:
dk / dt
i
dk / dt i

 g  n  
  (g  n   )
k
k
k
k
 dk / dt  i  ( g  n   )  k  s  y  ( g  n   )  k  s  k   ( g  n   )  k
In equilibrium: dk / dt  0

1


1
s
k*  
 ,
g

n






1
s
y*  

 g  n  
Multiply both sides by A(t):
1


1
s
k*  
 A(t ) ,

g

n






1
s
y* (t )  
 A(t )

g

n




 The equilibrium growth paths of k(t) and y(t).
In other words, the equilibrium is no longer a point but a time path.
1


1
s
k * (t )  
 A(0)  e gt

 g  n  


1
s
 y* (t )  
 A(0)  e gt

 g  n  


1
s
 ln k * (t ) 
 ln 
  ln A(0)  g  t
 g  n  
1





s
 ln y* (t ) 
 ln 
  ln A(0)  g  t
 g  n  
1


These are the equilibrium growth paths of lnk(t) and lny(t):
If the saving rate (s) increases, or the growth rate of the labor force (n) decreases
,or if A(0) increases due to e.g. a higher educational level among the workers,
then the equilibrium growth paths of y and k shifts upwards. As the economy
moves towards its new equilibrium growth path, the growth rate of y is higher
than the long run growth rate (g).
Transition to the equilibrium growth path
If the economy is not on its equilibrium growth path, it will over time move to
its equilibrium growth path.
Assume: A(0)=1, s=0.3, and (n+g+d)=0.1, g=0.015, n=0.015, d=0.07.
 k * =9, y * =3, ln( y * =3)=1.1
We assume two different starting values:
(1) k (1) =4, y (1) =2, ln( y (1) =2)=0.69, (2) k (1) =16, y (1) =4, ln( y (1) =4)=1.39
Lecture:
To find out how K/AL develops over time, we use the transitional equation:
dk / dt  s  k   ( g  n   )  k
 k  s  k   ( g  n   )  k
 k2  k1  s  k1  ( g  n   )  k1
 k21  s  k1  (1  ( g  n   ))  k1  0.3  k1  (1  0.1)  k1
Multiplying k (t ) with A(t) gives k(t), then it is trivial to find y(t) and lny(t):
ln(Y/L)
Transition to equilibrium growth path
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Serie1
Serie2
Serie3
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
Time
When the economy approaches its equilibrium growth path, the growth rate of y
deviates from the long-run growth rate (g). If an economy starts out below
(above) the equilibrium growth path, the growth rate of y is higher (lower) than
g. Holding constant the equilibrium growth path that is holding constant A(0), s,
n, g, d and alfa, a lower y means a higher growth rate of y.
What happens to the growth rate and to the equilibrium growth path if the
saving rate increases?
Initially assume that the economy is on its equilibrium growth path and that:
A(0)=1, s=0.3, and (n+g+d)=0.1, g=0.015, n=0.015, d=0.07.
 k * =9, y * =3, ln( y * =3)=1.1
We assume that s increases to 0.4  k * =16, y * =4, ln( y * =3)=1.39
Transition to higher equilibrium growth path
2.5
ln(Y/L)
2
Serie1
1.5
Serie2
1
Serie3
0.5
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
time
If s increases, the equilibrium shifts upwards, and the growth rate of y is higher
than the long run growth rate during to the transition to the new equilibrium
growth path.
Summing up the results:
The long run growth rate of y is g; that is, the growth rate that occurs along the
equilibrium growth path.
The growth rate of y = g + growth that occurs during the adjustment to
equilibrium growth path.
y
y 
y
Holding constant the variables that determine the equilibrium growth path
constant; that is, A(0), s, and n, a lower y means a higher growth rate.
Moreover: A(0)  , s, or n 

y

y
Increasing A(0), s or decreasing n shifts the equilibrium growth path upwards,
and thereby induces transitional growth.
The growth rates of other variables in the model:
K k A L k




gn
K  k  A L 
K
A
L
k
k
Y y A L y
 


gn

Y  y  A L
Y
y
A
L
y
Lecture:
In the steady-state:
k * y*
 *  0,
y
k*
k * y* w*
 * 
g
w
k*
y
K * k * A L
 *  
 0 g n  g n
K*  k *  A L 
A
L
K*
k
Y * y* A L
Y *  y*  A  L

 *    0 g n  g n
A
L
Y*
y
y* k *
r*
r*  d    ( y / k ) ,
  ( *  * )  0
r
y
k
Testing the model empirically by regression analysis on a cross-section of
countries:
i
i
i
i
i
y2000
 y1960
/ y1960
   1  y1960
 2  (S / Y )i  3  n19602000
 etc.
where i = swe, Norway, finland, usa, etc.
According to model 1, 3 should be negative. And  2 should be positive.
The dependent variable is typically the average annual growth rate of GDP per
capita:
y2000  y1960  (1  g )40




y2000
 (1  g )40
y1960
1/40
y2000 
y1960 
1  g
Alternatively an approximate formula,
(it is approximate because in the real world data is discrete).
y(2000)  y(1960)  e g40  ln y(2000)  ln y(1960)  g  40
  ln y(2000)  ln y(1960)  / 40  g
Note: gr for small r and g=ln(1+r)>r .
Lecture:
Other variables than the standard variables in the SOLOW model
In empirical analysis often more variables than initial income per capita (or
initial income per employed), the investment rate, the population growth rate are
included. For example, educational level, variables measuring tax rates,
corruption, openness to trade, population age structure, population density.
To make the empirical analysis fully consistent with the Solow model, it is
typically assumed that these variables impact the level of technology (A) and
thereby the steady state level of production per worker.
Thus, A is assumed to depend on a host of variables.
If the model is tested in terms of per capita, often age structure variables; the
share of the population below 15 and above 65 years, are included as
explanatory variables in the regressions to account for the fact that people of
these age groups typically do not work.
HERE YOU SHOULD READ THE EMPIRICAL ARTICLES ON ECONOMIC
GROWTH.
Read by yourselves:
Factor mobility in the Solow model ((assume g=0).
K will move from richer countries with low r (due to high k) to poorer countries
where r is high due to low k.
Thereby k and y will tend to equalize across countries.
Nowadays, rich EU-countries invest capital or move production to new EUcountries or to CHINA or India.
L will move the opposite way, from low-wage countries to high wage countries,
which also contributes to equalize real wages, r and k across countries.
Workers move from new EU-countries to old EU-countries where real wages
are higher.
Specific example: assume two countries that are the same with respect to the
parameters: A, n, d, and alfa, but one country has a higher saving rate than the
other. Assume that these countries are in their respective equilibria.
Allowing for factor mobility across countries equalizes the real wage, the real
return to capital, k and y across countries.
The new equilibrium will be joint for the two countries and is determined by a
weighted average of the saving rates in the 2 countries, where the weights are
given by the size of the populations in the two countries.
Capital to labor (k) ratio is low in developing countries. As a result, one would
expect a high real rate of return on investment in those countries. Why then do
not a lot of investment (construction of new factories, etc.) take place in many of
these countries? Answer: There is a lot of corruption, which makes the actual
rate of return much lower; that is, after the investor have paid off a lot of
government official, there might not be so much money left. A is low. There
might also be a political risk. Investors might risk that some bandits take over
the factories, like in Zimbabve.
Lecture:
ENDOGENOUS GROWTH MODELS
Endogenous growth models rejects the assumption of the Solow model that
technological progess is exogenous.
Start with a simple production function: Y = AK, where Y is output,
K is the capital stock, and A is a constant measuring the amount of
output produced for each unit of capital (noticing this production
function does not have diminishing returns to capital). One extra unit
of capital produces A extra units of output regardless of how much
capital there is. This absence of diminishing returns to capital is
the key difference between this endogenous growth model and the
Solow model.
Let’s describe capital accumulation with an equation similar to those
we’ve been using: K = sY - K. This equation states that the change
in the capital stock (K) equals investment (sY) minus depreciation
(K). We combine this equation with the production function, do
some rearranging, and we get: Y/Y = K/K = sA - 
This equation shows what determines the growth rate of output Y/Y.
Notice that as long as sA > , the economy’s income grows forever,
even without the assumption of exogenous technological progress.
In the Solow model, saving leads to growth temporarily, but
diminishing
returns to capital eventually force the economy to approach a steady
state in which growth depends only on exogenous technological
progress.
By contrast, in this endogenous growth model, saving and investment
can
lead to persistent growth.
A second and more realistic class of endogenous growth models are
complementary to the Solow model builds on Paul Romer (1990).
In the model above MPK does not fall when K increases. A second and more
realistic class of endogenous growth models are complementary to the Solow
model. They try to explain the long-run growth rate, g, which is exogenous in
the Solow model, by the number of scientists, etc.
Read 2-sector model for example in Weil or in Mankiw, ch. 8
One sector, the research sector, produces new technology. The rate of
technological progress depends on the patent system, the number of scientists
and their economic incentives.
Read by yourselves (or lecture):
Factors that impact GDP per capita in the real world:
GDP
GDP
Hours worked Employment POP 19  64



POP Hours worked Employment POP 19  64
POP
POP = Population. If GDP per hour (=labor productivity) increases or the hours
worked per employed increases or the number of employed as a share of
population increases, then GDP per person increases.
In other words, get each worker to produce more or get more people in
production, then GDP per person increases.
Production per employed (= the first 1 or 2 term on the left hand side of the
equation above) is in macromodels is GDP per worker, Y/L.
GDP (or GNP) per capita as a measure of the standard of living
The income distribution
GDP per capita (=average income) can be a poor indicator of the income of the
average citizen; that is, of the median income.
The median is the person in the middle of the income distribution.
Typical income distribution
Number of income earners
60
50
40
30
Serie1
20
10
0
Income classes of equal size
The income distribution is typically assymmetric:
 median income < average income
 the more unequal income distribution the greater difference between median
and average income and a larger proportion of the population tends to have an
income below the average income.
An extreme example:
Read by yourselves (or Lecture):
Country Equal. 10 individuals each with an income of 5000.
The median and mean income is 5000.
Country Unequal. 9 individuals each has an income of 2000.
One individual has an income of 32000.
Average income is 5000. Median income is 2000.
GDP per capita as an indicator of “human development/happiness”
We have concluded that average income per capita may be a poor indicator of
the income of the average person; that is, of the median person.
What is the relationship between income per capita and other indicators of
“welfare/happiness”? We want (but cannot) measure is happiness/utility:
U = U (y, x1, x2, x3, x4,…)
Where y = income per capita, x1=literacy rate, x2=assess to clean water, x3=
infant mortality rate, x4=life expectancy, etc.
2 views:
1. The correlation between income per capita and other variables (x1,x2,x3,x4,..)
which we believe impact the welfare of people is high.
Therefore, it is sufficient to study determinants to income per capita.
2.The correlation is not necessarily high.
The UN (UNDP’s) “Human Development Index” has 3 components:
1. Life expectancy. 2. Educational level (e.g. literacy rate).
2. Income per capita.
According to this index Sweden’s is a top 5 whereas with respect to income per
capita Sweden is only top 20.
Problem of Household surveys that ask “Are you happy?” is that the meaning
of the word happy may differ across cultures.
We are rich now but are we happier?
The importance of relative position.
Harvard-students were asked what alternative they preferred:
a) USD 50000/year whereas others get half.
b) USD 100000/year whereas others get the double.
Source: The economist, Aug. 9, 2003.
Some characteristics of poor countries
Large agricultural sector.
They have a comparative advantage with respect to labor-intensive production
as they have a lot of labor but only a little capital (physical and human).
Demography: Young populations, many kids per woman.
In the overlapping generations model which is described in the textbook by
Auerbach and Kotlikoff; the individuals make an intertemporal choice; that is,
they decide how much of present income to consume today and how much to
save, which consumption when being old. This model can easily be extented to
also include a choice of how many hours to work when being young (and how
many hours to work when being old).
If we make this extention of the model the material below is relevant.
Read by yourselves (or lecture):
LABOR SUPPLY, based on Varian section 9.8 Maybe included;
In this case student should study it themselves.
LABOR SUPPLY
REAL WAGE (W/P)
20
15
Serie1
10
Serie2
5
0
1
2
3 4
5 6
7
8 9 10 11 12 13 14 15 16
Hours worked
Labor Supply: 3 possibilities:
(1) Labor supply is unrelated to the real wage.
(2) Labor supply increases when the real wage increases.
(3) Labor supply decreases when the real wage increases: when the real wage
increases the individual can afford to take more leisure, which she likes.
Factors that increase aggregate labor supply at a given real wage:
1. Labor immigration. 2. Lower unemployment benefits should increase the
labor supply of the domestic population.
The Microeconomics behind labor supply
The individual or household faces the choice between consumption and leisure.
More consumption requires more hours worked and hence less leisure.
The problem of the individual is to maximize:
U = U(C,R)
where
C= consumption during a period of time, e.g. a day.
R = hours of leisure enjoyed during a day.
Read by yourselves (or lecture):
If C U(.), and if R U(.).
The two constraints the individual faces are:
(1) The time constraint:
LR  L
where L = labor supply in hours,
L is the time endowment which is 24 hours per day.
(2) P  C  W  L  M
where P= Price of the consumption good
W = Nominal Hourly Wage
M= non-labor income, e.g. government transfers
Let M  P  C
In other words, C is the quantity of goods that the individual receives that is not
related to hours worked.
 P  C W  L  P  C  P  C W  L  P  C
 P  C W  L  W  L  P  C  W  L
 P  C W  (L  L)  P  C W  L
 P  C W  R  P  C W  L
Now we have combined the two constraints that the individual faces, and the
result is similar to the usual budget constraint: px  x  p y  y  I
Thus, the goods that the individual derives utility from (C and R) are on the lefthand-side of the equation. And in front of the quantities of these goods are the
respective prices of these goods  W is the price of leisure: it is what the
individual gives up by taking one hour of leisure. P  C W  L is called full or
potential income. If R=0, then P  C  P  C W  L .
The constraint can be rewritten in real terms:
1 C  (W / P)  R  C  (W / P)  L
where 1 = real price of consumption, W/P is the real price of leisure = the
quantity of goods the individual gives up by consuming one more unit of leisure.
Graphical illustration of the choice possibilities of the individual:
Let C  0 ,  C  (W / P)  L  (W / P)  R
Intercept
Slopecoefficient
Read by yourselves (or lecture):
An Increase of the Real Wage
Consumption
20
15
Serie1
10
Serie2
5
0
1
2
3
4
5
6
7
8
9
10
Leisure (0.0-1.0)
Note: The choice constraint cuts the x-axis where R= L .
In the figure we assume that L =1, and that W/P increases from 10 to 20.
The numbers on the x-axis are 0.0, 0.1, 0.2,…, 1.0.
Note also that labor supply (L) = L - R: When R=0, then L= L .
If W/P  the intercept increases, and the slope becomes more negative.
If W/P , the individual can afford more of both C and R. On the other hand,
when W/P , R becomes more expensive in terms of the quantity of
consumption goods the individual gives up by consuming one more unit (hour)
of leisure.
3 hypothetical possibilities on demand for leisure and on labor supply (L= L -R)
when W/P :
1.No effect on the demand for leisure and on the labor supply if the substitution
(price) effect = income effect. The substitution effect is negative for the demand
of leisure when the price of leisure (that is, the real wage) increases. The
income effect for the demand of leisure is positive as a higher real wage means
that the individual can afford and wants more leisure when income increases.
2.Negative effect on the demand for leisure (= positive effect on labor supply) if
the substitution effect > income effect.
3. Positive effect on the demand for leisure (= negative effect on labor supply) if
the substitution effect < income effect.
Read by yourselves (or lecture):
The optimal choice with positive non-labor income ( C  0 )
C  (W / P)  R  C  (W / P)  L
 C  C  (W / P)  L  (W / P)  R
Slopecoefficient
Intercept
An Increase of Non-Labor Income
Consumption
20
15
Serie1
10
Serie2
5
0
1
2
3
4
5
6
7
8
9
10
LEISURE (0.0-1.0)
In the figure we assume that L =1, W/P=10, and that C increases from 5 to 10.
The numbers on the x-axis are 0.0, 0.1, 0.2,…, 1.0.
When C increases the individual wants more of both goods as they are assumed
to be so-called normal goods. You want more of normal goods when your
income increases.
An increase of C does not change the opportunity cost of enjoying leisure, and
constitutes therefore a pure income effect.
Summary:
The effect of changes in the exogenous variables on optimal demand for C and
R, and on optimal labor supply:
If C  C * , R * , L*  L  R* 
If W/P   C * , R * ?, L*  L  R* ?
Read by yourselves (or lecture):
A mathematical note on how to derive optimal demand-functions in case of
a Cobb-Douglas (or a logarithmic) utility function:
If the individual maximizes U ( x, y)  x  y 
subject to the budget constraint: px  x  p y  y  I
where x = quantity of good x, y= quantity of good y, px = price of good x, p y =
price of good y, and I= income.
The optimal demand for x and y are such that the consumer chooses to spend a
constant fraction of its income on these goods:
p x  x*


I

 x* 

I

   px
*
py  y


I

 y* 

   py
I

Note if   1  x*  (1   ) 
I ,
px
y*   
I
py
A mathematical example on the optimal choice of leisure (optimal labor supply):
Assume that the individual has the following utility function: U  C1/ 2  R1/ 2
The constraints of the individual are: (1) L  R  L  1
(2) C  W / P  L  C
Note: W, P and C can not be affected by the individual. Thus, they are
exogenous from the point of view of the individual.
Combining the constraints yields:
1 C  (W / P)  R  C  (W / P)  L
1 C  (W / P)  R  C  (W / P)
The result is similar to the usual budget constraint: px  x  p y  y  I
Optimal demands for C and R, and optimal labor supply are:
(W / P  C)  0.5  (W / P  C)
C*  0.5  I  0.5 
pc
1
(W / P  C )  0.5  0.5  C
R*  0.5  I  0.5 
pR
W /P
W /P
L*  1  R*  0.5  0.5  C
W /P
When C  0 :If C  C * , R * , L*  L  R* 
If W/P   C * , R * , L*  L  R* : More labor is supplied when W/P .
When C  0 :
If W/P   C * , R * =0.5 and L*  L  R* =0.5. That is, labor supply and
optimal leisure are unrelated to W/P.
Thus, the substitution effect equals the income effect.
Read by yourselves (or lecture):
CONSUMPTION
According to Keynesian theory the private consumption function is:
C (Y  T )  C  MPC  (Y  T ) , Where T=net taxes=taxes – transfers
Y-T=disposable income
Current private consumption depends on current disposable income.
More elaborate theories say that current consumption depends not also on
expected future disposable income, Wealth, and on the interest rate. For
example, we expect an increase in wealth to increase current consumption at a
given level of Y-T, which would increase C in the equation above.
A Simple Model of Intertemporal Choice over the life-cycle.
Assumptions: The individual lives 2 periods.
The individual consumes in both periods and also receives incomes in both
periods. The incomes are exogenously given.
We assume a perfect capital market, which means that the individual can borrow
and lend as much as she wants at a given interest rate.
The individual receives and leaves no bequest.
Preferences are represented by the utility function: U(C1,C2)
where C1=consumption in first period of life, and C2=consumption in second
and last period of life. The individual values both goods (C1 and C2). The
marginal utility of C1 is diminishing when C1 increases (and C2 is constant),
and the marginal utility of C2 is diminishing when C2 increases (and C1 is
constant).
Diminishing marginal utilities implies that the individual wants to “smooth”
consumption rather than consuming a lot in one period and little in the other
period. The perfect capital market, which implies that the individual can lend
and borrow at a given interest rate, makes consumption “smoothing” possible.
A consequence of diminishing marginal utilities is that if income only is
received in one period of life, the individual wants to spread this income over
both periods of life. If income increases only in one period, the individual wants
to spread this increase of income over both periods.
Secondperiod
consumption
Here are the combinations of first-period and second-period consumption
the consumer can choose. If he chooses a point between A and B, he
consumes less than his income in the first period and saves the rest for
the second period. If he chooses between A and C, he consumes more that
his income in the first period and borrows to make up the difference.
Y2
B
Consumer’s
Consumer’sbudget
budgetconstraint
constraint
Saving
Vertical
Verticalintercept
interceptisis
(1+r)Y
(1+r)Y11++YY22
A
Borrowing
Horizontal
Horizontalintercept
interceptisis
YY1 ++YY2/(1+r)
1
2/(1+r)
Y1
First-period consumption
C
The constraints: (1) S=Y1-C1, (2) C2=(1+r)S+Y2
where S = Saving in first period of life (can be negative), r=interest rate, Y1 and
Y2 income net of taxes received in period 1 and in period 2.
Combining the constraints (1) and (2):
 C2=(1+r)*(Y1-C1) + Y2
The budget constraint in figure above:
 C 2  (1 r ) Y1  Y 2  (1  r )  C1
Slope of the constraint:
dC 2
 (1  r)
dC1
Giving up one unit of C1 means more than one unit of C2 can be consumed
because of positive return (interest) on saving. Thus, C1 is “more expensive”
than C2.
The budget constraint can also be written:
1
1
 1 C1
 C 2  Y1 
Y 2
1 r
1 r
Present value of life-time consumption = Present value of life-time income
What is a present value? If r=0.05, the present value, x(t), of a value next year,
x(t+1): e.g. 105 dollars, is the value you have to deposit in a bank today to
receive 105 dollars next year. Thus, x(t)*(1+r)=x(t+1). If there is no uncertainty,
and there is a perfect capital market, the individual should be indifferent
between receiving 100 dollars today and receiving 105 dollars next year if the
interest rate is 5 %.
Read by yourselves (or lecture):
The intertemporal budget constraint above corresponds to our usual budget
constraint that has prices in front of the quantities:
1
 P1 C1 P2  C 2  Y1
Y 2
1 r
where P1= price of current consumption=1, P2=1/(1+r)=the price of future
consumption. P1>P2. Because if giving up one unit of C1, positive interest on
savings means more than one unit of C2 can be consumed. Thus, C1 is more
expensive than C2.
If either Y1  or Y2 , the budget constraint shifts outwards.
 C1*  and C 2*  because of diminishing marginal utilities. The optimal
levels of C1 and C2 depends on the present value of life-time income,
1
Y1 
Y 2 :
1 r
1
Y 2   C1*  and C 2* 
1 r
Regardless of whether Y1  or Y2  increase, the consumer spread the increase
1
in Y1 
Y 2 over both periods.
1 r
If Y1 S *  (Y1 C1* )  ,
as the consumer wants to increase consumption in both periods.
If Y2 S *  (Y1 C1* )  , for the same reason.
 Thus, if Y2 C1* . This result does not happen in the Keynesian model.
Y1 
Read by yourselves (or lecture):
Secondperiod
consumption
Economists
Economistsdecompose
decomposethe
theimpact
impactof
ofan
anincrease
increasein
inthe
thereal
realinterest
interest
rate
on
consumption
into
two
effects:
an
income
effect
and
a
rate on consumption into two effects: an income effect and a
substitution
substitutioneffect.
effect.The
Theincome
incomeeffect
effectisisthe
thechange
changein
inconsumption
consumption
that
thatresults
resultsfrom
fromthe
themovement
movementto
toaahigher
higherindifference
indifferencecurve.
curve.The
The
substitution
substitutioneffect
effectisisthe
thechange
changein
inconsumption
consumptionthat
thatresults
resultsfrom
fromthe
the
change
changein
inthe
therelative
relativeprice
priceof
ofconsumption
consumptionin
inthe
thetwo
twoperiods.
periods.
Y2
New budget
constraint
B
A
Old budget
constraint
C
IC2
IC1
Y1
First-period consumption
An
Anincrease
increasein
inthe
theinterest
interestrate
rate
rotates
rotatesthe
thebudget
budgetconstraint
constraint
around
aroundthe
thepoint
pointC,
C,where
whereCCisis
(Y
(Y11,,YY22).). The
Thehigher
higherinterest
interestrate
rate
reduces
reducesfirst
firstperiod
periodconsumption
consumption
(move
(moveto
topoint
pointA)
A)and
andraises
raises
second-period
second-periodconsumption
consumption
(move
(moveto
topoint
pointB).
B).
If the interest rate increases, C1 becomes more expensive relative to C2.
The substitution effect is that you consume less of the good whose price has
increased, C1* , and more of the other good, C 2* .
For a saver: If r , a saver becomes richer: the income effect: C1*  and C 2* .
The net effect (substitution + income effect): C 2* , C1* ? , S=(Y1- C1* )?
For a borrower:
If r , a borrower becomes poorer: the income effect: C1*  and C 2* .
The net effect (substitution + income effect): C1*  , S=(Y1- C1* ), C 2* ?
In aggregate an economy typically saves: r   S=(Y1- C1* )?
It is often assumed that an increase in r has no or a positive effect on S.
Borrowing constraints: C1  Y1
Consider an individual that consumes less than she would like in period 1:
If Y1   C1* , C 2*  0 . That is, she uses all of the increase in Y1 for C1.
Borrowing constraints are facts of life: They should increase aggregate saving in
the economy, but may be an obstacle for small-business that may have profitable
Read by yourselves (or lecture):
investment projects that the banks might not want to lend money to because of
imperfect information.
The motive for saving in the intertemporal choice model is that the individual
wants to smooth consumption over the life-time. If we add uncertainty to the
model, people also save because future income may be uncertain or because the
individual might live longer than expected. This is called precautionary saving.
Modigliani’s life-cycle model, and Friedman’s permanent income hypothesis
builds on the microeconomic intertemporal choice model above.
Mathematical Example with a common utility function
The individual/household chooses C1 (and thereby S and C2) to maximize
U  C1  C 21 , where 0    1
If the individual is impatient which is a common assumption:   1/ 2
The budget constraint of the individual is:
1
1
1 C1
 C 2  Y1 
Y 2
1 r
1 r
where r, Y1 and Y2 cannot be affected by the individual (are exogenous).

Y2
  (Y1 
)
 I
(1  r )
Y2
*
C1 

   (Y1 
)
PC1
1
(1  r )
C 2* 
(1   )  I

PC 2
(1   )  (Y1 
1
(1  r )
Y2
)
(1  r )
 (1  r )  (1   )  (Y1 
Y2
)
(1  r )
S *  Y1 C1*  Y1   (Y1 Y 2 )  (1  ) Y1  Y 2
(1 r )
(1 r )
Note: The solution to the mathematical problem is such that the endogenous (the
choice) variables are expressed as functions of the exogenous variables.
1
 If Y1 
Y 2   C1* , C 2* ; S *  if Y1, S *  if Y2.
1 r
If r   C1* , S *  (Y1 C1* ) , C 2*  (1 r)  (1   ) Y1  (1   ) Y 2 
Read by yourselves (or lecture):
Borrowing constraints: C1  Y1
Consider an individual that consumes less than she would like in period 1:
If Y1   C1* , C 2*  0 . That is, she uses all of the increase in Y1 for C1.
Borrowing constraints are facts of life: They should increase aggregate saving in
the economy, but may be an obstacle for small-business that may have profitable
investment projects that the banks might not want to lend money to because of
imperfect information.
The motive for saving in the intertemporal choice model is that the individual
wants to smooth consumption over the life-time. If we add uncertainty to the
model, people also save because future income may be uncertain or because the
individual might live longer than expected. This is called precautionary saving.
Modigliani’s life-cycle model, and Friedman’s permanent income hypothesis
builds on the microeconomic intertemporal choice model above.
EXERCISE:
Combining the intertemporal choice model with endogenous labor supply
Assume that Y1 and Y2 are not exogenous from the point of view of the
individual. Assume that Y1=W1*L1, where L1=1-R1, where L1 is hours
worked in period 1, and R1 is hours of leisure in period 1. 1=L1+R1 equals time
endowment (total number of hours available) in period 1 that is normalized to 1.
Assume also that Y2=W2*L2, where L2=1-R2, where L2 is hours worked in
period 2, and R2 is hours of leisure in period 2 of life. 1=L2+R2 equals total
number of hours available in period 2 that are normalized to 1. We also assume
that W1 and W2 are exogenous from the point of view of the individual.
Assume: U  C1  C2  R1  R21  
Where the preference parameters,  ,  ,  , 1       , all are assumed to be
between zero and 1.Write up the intertemporal budget constraint of the
individual. Derive the optimal levels of C1, C2, R1, R2, L1, and L2 as functions
of the exogenous variables.What happens to the optimal levels of C1, C2,
R1,R2, L1, and L2 if W2 increases?
Do not read not included in the course:
THE FERTILITY CHOICE
Assume that a household derives utility from a consumption good and from having kids.
Assume the following utility function:
U (C , K )  C  alfa * ln K
where C is quantity of consumption goods and K is number of kids.
(You may use the greek notation for alfa.)
Alfa is assumed to have a positive value.
Assume that W is the wage income that the household receives if the household works full
time. Note that if y=lnx, then dy/dx=1/x
(Assume also that the household lack other sources of income than labor income.)
Assume that the price of the consumption good is 1.
Assume that the price of (the cost of) children is related to the wage income. This is the case
because when the household have kids, it is assumed that the household no longer can work
full time because the have to look after/raise the kids. Thus, the household give up part of the
wage income, which constitutes the price (cost) of kids. Assume that the price per kid is
W*beta, where beta is the proportion of the household’s full time that the each kid require.
For example, if beta=0.2 and the household has one kid, then 80 percent of the time of the
household is devoted to work and 20 percent is devoted to the kid. If the household has 2 kids,
60 percent of the time is devoted to work and 40 percent is devoted to raising these two kids.
a.Write up the budget restriction of the household. 2p
b. Derive the optimal levels of C and K as functions of the exogenous variables. 4p
Assume that the wage income is exogenous from the point of view of the household.
Also show the optimal choice graphically with C on the vertical axis and K on the horizontal
axis. c.What happens to the optimal choice of C and K if W increases. Show mathematically
by using the derivative.
What happens to the utility level of the household. Explain why! 2p
d. Is this theoretical effect consistent with empirical observations from the real world? 1p
e. What happens to the optimal choice of C and K if beta increases? Show mathematically.
What happens to the utility level of the household. Explain why. 1p