Download Question addressed by so-called growth accounting

GROWTH ACCOUNTING, Lecture: JOAKIM PERSSON, 2011
I should use derivatives to come up with the growth accounting question,
Instead of the approximate formulas below.
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
Y  K
L
,
and
,
Y
K
L
We have not direct data on
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
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:
where t = number of years.
yt  y0  (1  r )t ,
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.
LABOR SUPPLY AND SAVING
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.
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.
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
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.
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* ?
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.
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
YY11++YY22/(1+r)
/(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:
dC2  (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 %.
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 
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
IC
IC1 2
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.
Read by yourselves:
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.
LECTURE:
Mathematical Example with a common utility function
The individual/household chooses C1 (and thereby S and C2) to maximize
U  C1  C21 , 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).

  (Y1  Y 2 )
 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, which
are the variables a single individual or single household cannot influence.
 If Y1
1
Y 2   C1* , C 2* ; S *  if Y1, S *  if Y2.
1 r
If r   C1* , S *  (Y1 C1* ) , C2*  (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 FOR Economics growth students:
Combining the intertemporal choice model with endogenous labor supply
In this model, the individual chooses labor supply in two time periods and consumption in 2
time periods.
Assume that real income in the 2 periods of life, 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 set equal to 1 to simplify.
That is, we assume that the time endowment is not 24 hours but instead equals 1. The time
period can refer to a month, a year or 30 years or whatever.
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 (=nominal wage in period 1) and
W2 (=nominal wage in period 2) are exogenous from the point of view of the individual.
Assume that the price of current consumption equals 1: P1=1.
Assume: U  C1  C2  R1  R21  
Where the preference parameters,  ,  ,  , 1       , all are assumed to be between
zero and 1.
a) Write up the intertemporal budget constraint of the individual.
b) Derive the optimal levels of C1, C2, R1, R2, L1, L2, and saving as functions of the
exogenous variables.
c) What happens to the optimal levels of C1, C2, R1, R2, L1, L2, (and Saving) if W2
increases?
d) What happens if r increases?
e) Assume now that the individual receives non-labor income in the 2 periods: N1 and
N2. Write up the intertemporal budget constraint of the individual.
f) Derive the optimal levels of C1, C2, R1, R2, L1, L2, and saving as functions of the
exogenous variables.
g) What happens to the optimal levels of C1, C2, R1, R2, L1, L2, (and Saving) if N2
increases?
EMPIRICAL EVIDENCE
One prediction of the model is that if income per capita increases the subsequent
growth rate of per capita income decreases.
For example if an economy is below its steady state and income per capita
increases, this means that the subsequent growth rate of per capita income
decreases.
This prediction can be tested by use of times series analysis:
yt 1  yt / yt    1  yt  et
Where t = 1970, 71, 72, 73,….2008.
According to this prediction 1 should be negative.
If we assume that regions in a country have the same saving rate, same
population growth rate and technology; that is, share the same steady state, then
this prediction can be tested on cross-sectional data:
i
i
i
i
y2000
 y1960
/ y1960
   1  y1960
 ei
Where i=region1, region2, region3, region4, …, region25.
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 regions and countries:
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.
This means that the model prediction is consistent with data
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 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 per capita income (and capital per worker) than
the country that is richer in terms of per capita income and capital per worker.
16000
A more complete regression model controls/takes account of the other variables
that impact growth (A, saving rate, population growth). For example, a higher
saving rate increase the growth rate of per capita income until the new steady
state has been reached.
The complete regression equation on times series data:
yt 1  yt / yt    1  yt   2  (S / Y )t  3  nt ,t 1  etc.
According to model 1, 3 should be negative. And  2 should be positive.
Number of observations= number of time periods.
The complete regression equation on cross-sectional data:
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.
Number of observations= number of countries.
Combining cross-sectional and times series data (panel or pooled data):
yti1  yti / yti    1  yti  2  (S / Y )ti  3  nti,t 1  etc.
Number of observations = number of countries * number of time periods
When variables that hold constant/control for the steady state (A, saving rate,
population growth rate) are included as explanatory variables in the regression
equation for samples of countries of the world, estimates of 1 turn from zero to
negative and statistically significant which is consistent with the model.
See Barro article.
Other variables than the standard variables in the OLG or SOLOW model
in growth regressions
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
See Barro article.
The dependent variable in growth regression 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 .
5-year or 10-year periods instead of annual data
Moreover, instead of annual data many researcher use 5-year or 10-year periods
when studying growth:



1/5
yti5 
1)  0  1  yti  ...
i 
yt 
Read also INEQUALITY ARTICLE by Barro.
Lecture:
ENDOGENOUS GROWTH MODELS
Endogenous growth models rejects the assumption of the Solow model that
technological progess is exogenous.
Converting our OLG-model to an endogenous growth model:
Exam question spring 2009.
Assume that the economy is described by the 2-period life-cycle model without a government
sector and without trade or exchange of factors of production between countries. (The
economy is thus a closed economy.) To simplify assume that the long-run growth rate of the
technology and of the population are zero; that is, A / A =0 and n=0. In this model:
Yt  A  Kt  L1t  ; Ut  cyt  c1ot1 ; kt 1  (1   )(1   ) A  kt
a) Derive the steady-state expression for k; that is, solve for k as a function of
the exogenous variables/parameters  , A , and  .
b) Derive the steady-state expressions for y, and w as functions of the exogenous
variables/parameters  , A , and  .
c)Assume now instead that the production function is: y  A  k  L1  . This is an
so-called endogenous growth model.
Rewrite the transition equation, and derive an expression for the growth rate of k;
k k
that is, derive t 1 t .
kt
Is the growth rate of k dependent on whether the economy starts out rich or poor;
That is, does the growth rate depend on the value of kt .
IN other words, does this endogenous growth model say that the lower initial k (and y) the
higher the growth rate of k (and y) when holding the steady state constant (which depends on
 ,  , A )? What is the result if we assume the standard production function: Yt  A  Kt  L1t  ,
which in per worker terms is: y  A  k 
What does the empirical evidence say about convergence in per capita income across
production among economies that are similar with respect to institutions etc; for example,
countries in Europe or regions within a country?
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.
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.
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
Read by yourselves:
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:
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.
THE ECONOMETRIC EXERCISE
INSTRUCTION for econometric exercise.
Form groups about 3 people in size, or work individually.
Use any econometrics program; eg EXCEL, SPSS and Minitab.
You can also include other variables from:
Source: http://www.worldbank.org/research/growth
Start with cross-section regressions and answer the following questions:
1a. Test whether poorer economies grow faster than richer ones?
Is there convergence in GDP per capita between OECD-countries for the period 1950-2004?
(I) In other words, do initially poor countries tend to have a higher growth rate in GDP per
capita than initially rich countries during this period?
(II) Also plot the natural logarithm of GDP per capita for the countries against time in a
diagram.
(III) Also calculate the standard deviation of GDP per capita across the OECD-countries in
1950, 1960, 1970, 1980, 1990, 2000, and 2004. The standard deviation of GDP per capita
should be expressed in percent, which is given by STD(GDP per capita)/MEAN(GDP per
capita).
The standard deviation expressed in percent is called the coefficient of variation.
Is the dispersion of GDP per capita diminishing over time between the OECD-countries when
the coefficient of variation is used? In other words, does the coefficient of variation diminish
over time?
1b. Split the period 1950-2000 in 10-year periods and run a separate regression for each 10year-period. In other words, run a regression for the 50s, one for the 60s, one for the 70s, etc.
Find out whether the result of convergence/divergence is robust over time. In other words, do
we get the same results regarding convergence/divergence in income per capita in every 10year period?
Use Real GDP per capita.
Is the result driven by outliers?
Plot the growth rate of GDP per capita against initial GDP per capita to find out.
(Do not do:1c. Sometimes the neoclassical growth model is tested in terms of GDP per
worker, where worker is defined as a person between 16-64 years. Test the hypothesis of
convergence in GDP per worker across the OECD-countries for the period 1950-2004.
In other words, test whether the average annual growth rate of GDP per worker between
1950-2004 depends on the level of GDP per worker in 1950.
You should also calculate the coefficient of variation for GDP per worker for the OECDcountries in 1950, 1960, 1970, 1980, 1990, 2000, and 2004.)
2. Multivariate (= more than one independent variable) growth regressions:
Include also other explanatory variables that according to the neoclassical growth model
impact the economic growth rate; the investment rate (Investment/GDP) and the growth rate
of the population.
Are the empirical results consistent with the model?
Test for the period 1950-2000.
The investment rate should be measured as an average for the period in question.
For the period 1950-2000 include also the variables public consumption and a measure of
openess as explanatory variables (together with initial GDP per capita, investment rate and the
growth rate in population) in the regression equation. What are their estimated impacts on the
economics growth rate? Note that the degree of openness is not explicitly included in the
neoclassical growth model; it can however be seen as one of several measures of the level of
technology, A, in this model ).
Also public consumption and openess should be measured as averages for the period 19502000.
Additional question: Is the estimated growth effect of the investment rate dependent on what
other variables are included as explanatory variables in the growth regression? Study this for
the period 1950-2000.
This question deals with the topic of multicollinearity.
Make a correlation matrix to see how the different variables correlate. Is there any reason
from this table to suspect multicollinearity.
INTERPRET AND COMMENT THE RESULTS! Are the empirical results consistent
With the neoclassical growth model?
Both the Solow model and the overlapping generations model taught by the book by
Auerbach and Kotlikoff are neoclassical growth models as the marginal product of capital
decreases when K increases, holding everything else constant.
PANELREGRESSIONER
Create a panel data set for the period 1950-2004 that is split up into 5-year periods: Thus there
are 11 observations for each country.
Answer the following questions:
1. Is there convergence in income per capita when you run a panel regression?
In other words, is it good for growth to start out poor?
1a. Create time dummy variables and include them as explanatory variables.
Does the explanatory power of the regression increase.
Interpret the estimated coefficients in front of the time-dummy variables.
What are the motives behind including time dummy variables in growth regressions?
2. Include also the investment rate and the growth rate of the population. What are the
estimated effects of these variables and of initial income per capita?
Include time dummy variables.
Next, include public consumption and the degree of openess as explanatory variables. What
are the estimated effects?
Is the estimated effect of the investment share on the economic growth dependent on what
other variables are included in the regression equation?
Also make a correlation matrix in order to see how the different explanatory variables
correlate with each other. From this matrix do you suspect the problem of multicollinearity?
Basic question: Do we get the same basic results from the cross-sectional regressions as we
get from the panel regressions? If there are any differences what can explain such differences?
3. Allow for individual-specific fixed effects (=country-specific fixed effects). Practically you
allow for country-specific fixed effects by including a dummy-variable (d1,d2,…,d23) in the
excelsheet paneldata2007. If country-specific fixed effects (=intercepts) are significant if there
are some factors we have not controlled for that impacts the growth rate of gdp per capita
which are constant over time. Could e.g. reflect the effect of geography on economic growth
rate.
In other words, run the following regression:
G(GDPper capita) = F(td1,…td10;d1,..,d23;initial gdp per capita; investment rate; population
growth rate; openness; Gov/GDP)
STRATEGY TO COMPLETE GROWTH EXCERCISE.
Start by calculating (in excel) variables that are later used in the exercise:
The average annual growth rate of GDP per capita between 1950-1990, 1950-60, 1960-70,
70-80, 80-90, 90-2000.
Also calculate the average growth rate of GDP per worker between the years 1950-2004.
For exercise 2 you also need to calculate the average investment rate between 1950-2000, the
population growth rate between 1950-2000, average government consumption (as a share of
GDP) between 1950-2000, the average value of openness between 1950-2000.
How do I perform the regression analysis in EXCEL, Swedish version.
1. Välj 1. Verktyg. 2. Dataanalys. 3. Regression.
Dependent variable: c1:c10
Independent variables: a1:b10
If the dependent variable is in column c: row1 to row10.
If two independent variables are in columns 1-2: row 1 to 10
How do I perform the regression analysis in EXCEL, English version.
2. Choose 1. Tools. 2. Dataanalysis. 3. Regression.
Dependent variable: c1:c10
Independent variables: a1:b10
If the dependent variable is in column c: row1 to row10.
If two independent variables are in columns 1-2: row 1 to 10
When you have calculated all relevant variables it is a good idea to load the data into SPSS.
How do I do that?
Open SPSS for windows.
To load data in SPSS:
File; Open; Data; select crossnewstudent.xls
To perform regression analysis in SPSS:
Analyze; regression; linear; then you choose dependent and independent(s) variables;
Then you press OK, which gives you the regression output, which you may want to print out.
In exercise 1 you are asked to provide plots. In SPSS:
Graph;Scatter;simple; choose dependent and independent variable(s); OK
ANOTHER COMMENT:
Note however that points (“.”) should be replaced by (“,”) as SPSS is a Swedish version.
By including time dummy variables one allows the intercept to vary with time.
Data set for the OECD-countries for the period 1950-2004.
Source: http://www.pwt.econ.upenn.edu
You should use Penn World Table, PWT, (Summer-Heston data set) Version 6.2
I have structured data so that the assignment can be completed.
I have created 2 files: ”cross2007.xls” and ”paneldata2007.xls”.
Data in cross2007.xls are structured for the cross-section regressions.
Data in paneldata2009b.xls are structured for panel regressions.
The variables compiled are:
POP = population in 1000s.
RGDPL = Real GDP per capita (Laspeyres index) (2000 international prices)
By intl prices we mean PPP-adjusted.
ki = Real investment share of GDP (i %) (2000 international prices)
kg = Real government share of GDP (i %)
Note that G is government consumption as a share of GDP.
RGDPW = Real GDP per worker (1985 international prices)
open = (real Exports+real Imports)/real GDP.
In panelnew.xls data are structured for the panel regressions.
The variables are:
POP = population in 1000s.
popgrow = average annual growth rate of population for each 5-year period.
The first observation is the average annual growth rate between 1950 and 1955.
The second observation is the average annual growth rate between 1955 and 1960. etc
I have used the approximative formula for the average annual growth rate.
pcinc = real GDP per capita (Laspeyres index) (2000 international prices)
growth = average annual growth rate of real pcinc for each 5-year period.
First observation is average annual growth rate between 1950 och 1955.
Second observation is the average annual growth rate between 1955 och 1960…
Openk = (Exports+Imports)/GDP.
tidsdum1 = is a time-dummy variable that has the value 1 for the first time period and the
value 0 for other periods.
tidsdum2 = = is a time-dummy variable that has the value 1 for the second time period and
the value 0 for other periods.
By including time dummy variables as explanatory variables in the panel regressions one
allows the intercept to vary with time. For information on time dummy variables borrow
an econometrics book and look up the topic.
Tidsdum1 is a dummy variable for first country. The country dummies is to be included to
allow for country-specific fixed effectsIn contrast to the cross2007.xls, paneldata.xls contains no values on I, G, P, Open for the year
2004. In regressions where these variables are included as explanatory variables use the initial
values on I , G and Open as explanatory variables.
In crossnew.xls there are values for 1990 and then one can calculate average values for I, G
and Open which should be used as explanatory variables.
Example: I = (I(1990)+I(1980))/2, G = (G(1990)+G(1980))/2, och
OPEN= (OPEN(1990)+OPEN(1980))/2 vara de förklarande variblerna.