Prices and welfare an introduction to the measurement of well being when prices change

Prices and Welfare

An Introduction to the
Measurement of Well-being
when Prices Change

Abdelkrim Araar
Paolo Verme


Prices and Welfare


Abdelkrim Araar • Paolo Verme

Prices and Welfare
An Introduction to the Measurement of Well-being
when Prices Change


Abdelkrim Araar
Pavillon J. A. De Sève, Office 2190
Laval University
Quebec
QC, Canada

Paolo Verme
The World Bank
Washington
DC, USA

ISBN 978-3-030-17422-4
ISBN 978-3-030-17423-1 (eBook)
/>© The International Bank for Reconstruction and Development/The World Bank 2019
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FOREWORD

One of the most fundamental roles of economics is to provide policy
makers with accurate information on the impact of economic policies,
either by modeling ex-ante the effects of potential policies or by evaluating
ex-post the effects of policies that have been implemented. Among the
effects to be considered, those of price changes are among the most
relevant for household well-being. Whether price changes appear in the
financial market (interest rates), the labor market (wages), the consumer
market (commodity prices) or the government sector (taxes and subsidies),
they can have important consequences both for household income and
for the distribution of such incomes. Yet, the impact of price changes
on household well-being is one of the most sensitive topics in economic
research and possibly one of the major sources of contention in empirical
economics.
This book provides the foundations for understanding and measuring
the impact of price changes on household well-being in a unifying format
that is rarely seen in economic textbooks. It first provides a simple and
intuitive graphical representation of the problem, clarifying in the process
the normative foundations behind the different types of measures of wellbeing adopted by the economic profession. It then provides a rigorous
mathematical illustration of those measures as well as possible computation
methods. Next, it provides illustrations on how these measurement and
computational methods can be used in empirical applications under different scenarios and also offers a simple toolkit designed to help practitioners
that need to make choices between those methods. Finally, it provides
statistical instruments to increase the accuracy of estimation procedures
v


vi

FOREWORD

and offers necessary coding in Stata to estimate the measurement and
computational methods reviewed.
The authors are both experts in the field and former colleagues of
mine. During my time as Economics Professor at Université Laval, I had
the pleasure of working with Abdelkrim Araar and Paolo Verme in the
context of different projects. They are both accomplished economists
with extensive experience in the measurement of poverty and income
distribution, and they bring together a combination of skills ranging from
theory to programming, and from empirics to policy making, that is unique
and suits the scope of this book particularly well. In my view, this is one
of the most useful treatises on the subject of prices and household wellbeing and one that can be recommended to undergraduate and graduate
students, empirical economists and practitioners in economic policy.
Minister of Families, Children and Social
Development, Government of Canada
Quebec, QC, Canada

Jean-Yves Duclos


ACKNOWLEDGMENTS

This book is the byproduct of a five-year period spent by the authors
working on subsidy reforms in the North Africa and Middle East (MENA)
region. As the Arab Spring unfolded starting from 2011 and oil prices
increased, many of the countries in the region found themselves with large
budget deficits caused by energy and food subsidies inherited from the old
regimes. Confronted with these new challenges, these countries requested
support from the World Bank to reduce subsidies while managing complex

political reforms. The authors of this book would spend the next five
years working with governments in the region to reform subsidies. In
the process, they developed a subsidy simulation model (www.subsim.org)
and published a book recording the results of these simulations across
the region (The Quest for Subsidy Reforms in the Middle East and North
Africa Region, Springer, 2017). The book we present here complements
this work by providing the theory, algorithms and coding that was used for
the model and the book on the MENA region. It also expands this work by
adapting the theory and empirics to suit any kind of price reform and assist
practitioners and policy makers in taking informed decisions. The book is
dedicated to our parents.
Quebec, QC, Canada
Washington, DC, USA

Abdelkrim Araar
Paolo Verme

vii


ABSTRACT

What is the welfare effect of a price change? This simple question is one of
the most relevant and controversial questions in microeconomic theory and
one of the main sources of errors in empirical economics. This book returns
to this question with the objective of providing a general framework for the
use of theoretical contributions in empirical works. Welfare measures and
computational methods are compared to test how these choices result in
different welfare measurement under different scenarios of price changes.
As a rule of thumb and irrespective of parameter choices, welfare measures

converge to approximately the same result for price changes below 10
percent. Above this threshold, these measures start to diverge significantly.
Budget shares play an important role in explaining such divergence. Single
or multiple price changes influence results visibly, whereas the choice
of demand system has a surprisingly minor role. Under standard utility
assumptions, the Laspeyres and Paasche variations are always the outer
bounds of welfare estimates, and the consumer’s surplus is the median
estimate. The book also introduces a new simple welfare approximation,
clarifies the relation between Taylor’s approximations and the income and
substitution effects and provides an example for treating non-linear pricing.

ix


CONTENTS

1

Introduction
References

2

Assumptions and Measures
2.1 Assumptions
2.2 Measures
2.2.1 Definitions
2.2.2 Geometric Interpretation
References

9
9
11
11
14
17

3

Theory and/
/∗ − L i s t o f varnames o f p e r c a p i t a e x p e n d i t u r e s on t h e d i f f e r e n t goods
∗/
/∗ − L i s t o f p r i c e changes
∗/
/∗ − To e s t i m a t e t h e e x p e n d i t u r e s a f t e r t h e p r i c e change t h e assumpti on i s
∗/
/∗
t h a t t h e p r e f e r e n c e s a r e homotheti c
∗/
/∗ ================================================================================∗/
/∗ Outputs : Change i n w e l f a r e : LV, PV, EV , PV, CS and CS_ELAS v a r i a b l e s
∗/
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
// C o n s t r u c t i n g t h e h y p o t h e t i c a l d a t a
clear
s e t obs 1000
s e t s e e d 1234
gen income
= uni form ( ) ∗ _n
gen food

= ( 0. 3 + 0. 2∗ uni form ( ) ) ∗ income
gen c l o t h e s
= ( 0. 1 + 0. 1∗ uni form ( ) ) ∗ income
// I n i t i a l i z i n g
t h e l i s t s o f i t e m s and p r i c e changes
l o c a l l i s t _ o f _ i t e m s food c l o t h e s // l i s t o f i t e m s
l o c a l p r i c e _ c h a n g e s 0. 06 0. 04
// p r o p o r t i o n s o f p r i c e changes
// E s t i m a t i n g t h e w e l f a r e change wi th LV and PV measurements
gen LV = 0
// I n i t i a l i z i n g t h e v a r i a b l e LV
gen PV = 0
// I n i t i a l i z i n g t h e v a r i a b l e PV
gen EV = 0
// I n i t i a l i z i n g t h e v a r i a b l e EV
gen CV = 0
// I n i t i a l i z i n g t h e v a r i a b l e CV
gen CS = 0
// I n i t i a l i z i n g t h e v a r i a b l e CS
gen CS_ELAS = 0 // I n i t i a l i z i n g t h e v a r i a b l e CS
local i = 1
// number o f t h e i tem .
f o r e a c h i tem o f l o c a l l i s t _ o f _ i t e m s {
tempvar i tem_ ‘ i ’
gen ‘ i tem_ ‘ i ’ ’ = ‘ item ’
tempvar s h a r e _ ‘ i ’
gen ‘ s h a r e _ ‘ i ’ ’ = ‘ i tem_ ‘ i ’ ’ / income // The e x p e n d i t u r e s h a r e s
l o c a l nitems = ‘ i ’
local i = ‘i ’ + 1
}

local i = 1
// number o f t h e i tem .
f o r e a c h dp o f l o c a l p r i c e _ c h a n g e s {
l o c a l dp_ ‘ i ’ = ‘ dp ’
local i = ‘i ’ + 1
}
tempvar p r i c e _ i n d e x
gen
‘ price_index ’ = 1
// I n i t i a l i z i n g t h e L a s p e y r e s p r i c e i n d e x
f o r v a l u e s i =1/ ‘ ni tems ’ {
tempvar i t e m _ a
gen ‘ i tem_ a ’ = ‘ i tem_ ‘ i ’ ’
// The e x p e n d i t u r e s i n p e r i o d ( a )
r e p l a c e LV=LV − ‘ dp_ ‘ i ’ ’ ∗ ‘ i tem_ a ’
tempvar i tem_ b
gen ‘ item_b ’ = ( ‘ i tem_ a ’ ) / ( 1 + ‘ dp_ ‘ i ’ ’ ) // The e x p e n d i t u r e s i n p e r i o d ( b )
r e p l a c e PV=PV − ‘ dp_ ‘ i ’ ’ ∗ ‘ item_b ’
r e p l a c e CS=CS − ‘ i tem_ a ’ ∗ l n ( 1+ ‘ dp_ ‘ i ’ ’ )
r e p l a c e CS_ELAS=CS_ELAS − ‘ i tem_ a ’ ∗ ‘ dp_ ‘ i ’ ’ ∗ ( 1 − 0 . 5 ∗ ‘ dp_ ‘ i ’ ’ ∗ ( 1 + ‘ dp_ ‘ i ’ ’ ) )
r e p l a c e ‘ p r i c e _ i n d e x ’ = ‘ p r i c e _ i n d e x ’ ∗ ( ( 1 + ‘ dp_ ‘ i ’ ’ ) ^ ‘ s h a r e _ ‘ i ’ ’ )
}
r e p l a c e EV = income ∗ ( 1/ ‘ p r i c e _ i n d e x ’ − 1 )
r e p l a c e CV = income ∗ ( 1 − ‘ p r i c e _ i n d e x ’
)
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/


APPENDICES

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ THE STATA CODE: B ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
/∗ E s t i m a t i n g t h e EV , PV and CS w e l f a r e change measurements
∗/
/∗ Approach : T a y l o r a p p r o x i m a t i o n
∗/
/∗ Order 1 : CV=EV=CS = LV
∗/
/∗ Order 2 : CV−> ( 3 . 5 2 ) | | EV−> ( 3 . 5 3 ) | | CS−> ( 3 . 5 4 )
∗/
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
/∗ =======================================================================∗/
/∗ I nput i n f o r m a t i o n
∗/
/∗ =======================================================================∗/
/∗ − L i s t o f varnames o f p e r c a p i t a e x p e n d i t u r e s on t h e d i f f e r e n t goods
∗/
/∗ − L i s t o f p r i c e changes
∗/
/∗ − The assumpti on : t h e p r e f e r e n c e s a r e homotheti c
∗/
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
/∗ =======================================================================∗/
/∗ Outputs : Change i n w e l f a r e : EV, CV and CS v a r i a b l e s
∗/
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
// C o n s t r u c t i n g t h e h y p o t h e t i c a l d a t a
clear
s e t obs 1000
s e t s e e d 1234
gen income

= uni form ( ) ∗ _n
gen food
= ( 0. 3 + 0. 2∗ uni form ( ) ) ∗ income
gen c l o t h e s
= ( 0. 1 + 0. 1∗ uni form ( ) ) ∗ income
// I n i t i a l i z i n g t h e l i s t s o f i t e m s and p r i c e changes
l o c a l l i s t _ o f _ i t e m s food c l o t h e s // l i s t o f i t e m s
l o c a l p r i c e _ c h a n g e s 0. 06 0. 04
// p r o p o r t i o n s o f p r i c e changes
// S e t t i n g t h e o r d e r o f T a y l o r a p p r o x i m a t i o n
l o c a l o r d e r = 2 // The u s e r can s e t t h e t a y l o r o r d e r to 1 .
// E s t i m a t i n g t h e w e l f a r e change wi th LV and PV measurements
gen LV
= 0
gen EV_TAYLOR_ ‘ order ’ = 0
// I n i t i a l i z i n g t h e v a r i a b l e EV
gen CV_TAYLOR_‘ order ’ = 0
// I n i t i a l i z i n g t h e v a r i a b l e CV
gen CS_TAYLOR_‘ order ’ = 0
// I n i t i a l i z i n g t h e v a r i a b l e CS
local i = 1
// number o f t h e i tem .
f o r e a c h i tem o f l o c a l l i s t _ o f _ i t e m s {
l o c a l i tem_ ‘ i ’ = " ‘ item ’ "
d i s ‘ i tem_ ‘ i ’ ’
tempvar s h a r e _ ‘ i ’
gen ‘ s h a r e _ ‘ i ’ ’ = ‘ i tem_ ‘ i ’ ’ / income // The e x p e n d i t u r e
l o c a l nitems = ‘ i ’
local i = ‘i ’ + 1
}

local i = 1
// number o f t h e i tem .
f o r e a c h dp o f l o c a l p r i c e _ c h a n g e s {
l o c a l dp_ ‘ i ’ = ‘ dp ’
local i = ‘i ’ + 1
}
f o r v a l u e s i =1/ ‘ ni tems ’ {
r e p l a c e EV_TAYLOR_ ‘ order ’ =EV_TAYLOR_ ‘ order ’
r e p l a c e CV_TAYLOR_‘ order ’ =CV_TAYLOR_‘ order ’
r e p l a c e CS_TAYLOR_‘ order ’ =CS_TAYLOR_‘ order ’

shares

− ‘ dp_ ‘ i ’ ’ ∗ ‘ i tem_ ‘ i ’ ’
− ‘ dp_ ‘ i ’ ’ ∗ ‘ i tem_ ‘ i ’ ’
− ‘ dp_ ‘ i ’ ’ ∗ ‘ i tem_ ‘ i ’ ’

i f ‘ order ’ >= 2 {
f o r v a l u e s j =1/ ‘ ni tems ’ {
r e p l a c e EV_TAYLOR_ ‘ order ’ = EV_TAYLOR_ ‘ order ’ − 0. 5 ∗ ( −‘ s h a r e _ ‘ i ’ ’ ∗ ‘ i tem_ ‘ j ’ ’ ///
−‘i tem_ ‘ i ’ ’ ∗ ( ‘ i ’ = = ‘ j ’ ) ) ∗ ‘ dp_ ‘ i ’ ’ ∗ ‘ dp_ ‘ j ’ ’
r e p l a c e CV_TAYLOR_‘ order ’ = CV_TAYLOR_‘ order ’ − 0. 5 ∗ (
‘ s h a r e _ ‘ i ’ ’ ∗ ‘ i tem_ ‘ j ’ ’ − ///
‘ i tem_ ‘ i ’ ’ ∗ ( ‘ i ’ = = ‘ j ’ ) ) ∗ ‘ dp_ ‘ i ’ ’ ∗ ‘ dp_ ‘ j ’ ’
r e p l a c e CS_TAYLOR_‘ order ’ = CS_TAYLOR_‘ order ’ − 0. 5 ∗ ( 0
− ///
‘ i tem_ ‘ i ’ ’ ∗ ( ‘ i ’ = = ‘ j ’ ) ) ∗ ‘ dp_ ‘ i ’ ’ ∗ ‘ dp_ ‘ j ’ ’
}
}
}

i f ‘ order ’ >= 3 {
f o r v a l u e s i =1/ ‘ ni tems ’ {
f o r v a l u e s j =1/ ‘ ni tems ’ {
f o r v a l u e s k =1/ ‘ ni tems ’ {
r e p l a c e EV_TAYLOR_ ‘ order ’ = EV_TAYLOR_ ‘ order ’ − ///
1/3∗ ( ( ‘ s h a r e _ ‘ i ’ ’ ∗ ( ‘ s h a r e _ ‘ j ’ ’ ∗ ‘ i tem_ ‘ k ’ ’ + ‘ i tem_ ‘ j ’ ’ ) ∗ ( ‘ i ’ = = ‘ j ’ = = ‘ k ’ ) )
+ ///
( ‘ s h a r e _ ‘ i ’ ’ ∗ ‘ i tem_ ‘ k ’ ’
+ ‘ i tem_ ‘ i ’ ’ ) ∗ ( ‘ i ’ = = ‘ j ’ = = ‘ k ’ ) ) ∗ ‘ dp_ ‘ i ’ ’ ∗ ‘ dp_ ‘ j ’ ’ ∗ ‘ dp_ ‘ k ’ ’
r e p l a c e CV_TAYLOR_‘ order ’ = CV_TAYLOR_‘ order ’ − ///
1/3∗ ( ( −‘ s h a r e _ ‘ i ’ ’ ∗ ( ‘ s h a r e _ ‘ j ’ ’ ∗ ‘ i tem_ ‘ k ’ ’ + ‘ i tem_ ‘ j ’ ’ ) ∗ ( ‘ i ’ = = ‘ j ’ = = ‘ k ’ ) ) + ///
( − ‘ s h a r e _ ‘ i ’ ’ ∗ ‘ i tem_ ‘ k ’ ’
+ ‘ i tem_ ‘ i ’ ’ ) ∗ ( ‘ i ’ = = ‘ j ’ = = ‘ k ’ ) ) ∗ ‘ dp_ ‘ i ’ ’ ∗ ‘ dp_ ‘ j ’ ’ ∗ ‘ dp_ ‘ k ’ ’

91


92

APPENDICES

r e p l a c e CS_TAYLOR_‘ order ’ = CS_TAYLOR_‘ order ’ − ///
1/3
∗ (
‘ i tem_ ‘ i ’ ’ ∗ ( ‘ i ’ = = ‘ j ’ = = ‘ k ’ ) ) ∗ ‘ dp_ ‘ i ’ ’ ∗ ‘ dp_ ‘ j ’ ’ ∗ ‘ dp_ ‘ k ’ ’
}
}
}
}
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/

/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ THE STATA CODE: C ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
/∗ − The V a r t i a ( 1983) a l g o r i t h m
∗/
/∗ − Approach : Numeri cal a p p r o x i m a t i o n
∗/
/∗ − Outputs : Change i n w e l f a r e :
CV−> ( 3 . 6 4 ) //
EV−> ( 3 . 6 5 )
∗/
/∗ − The u s e r can i n c r e a s e t h e number o f goods or use a n o t h e r demand
∗/
/∗
function
∗/
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
set trace off
mata : mata c l e a r
mata
num=10

/∗ Number o f p a r t i t i o n s or i t e r a t i o n s ∗/

// I n i t i a l i z i n g t h e p a r a m e t e r s ( p r i c e s and income )
r e a l matrix functi on i n i t i a l i s e _ p a r a m e t e r s ( s c a l a r
{
i f ( t ==0) r e t u r n ( 1 . 0
\ 1. 0 \ 1. 0 ) /∗ I n i t i a l
i f ( t ==1) r e t u r n ( 1 . 3
\ 1. 4 \ 1. 7 ) /∗ F i n a l
i f ( t ==2) r e t u r n ( 0 . 3

\ 0. 5 \ 0. 2 ) /∗ income
i f ( t ==3) r e t u r n ( 100)
/∗ Income
}

t)
price vector
price vector
shares

∗/
∗/
∗/
∗/

// D e f i n i n g t h e demand f u n c t i o n
r e a l matrix functi on e v a l _ q u a n t i t i e s ( r e a l matrix x )
{
r e a l matrix q
alpha = i n i t i a l i s e _ p a r a m e t e r s (2)
n= c o l s ( x)−1
n1=n+1
q = J (1 ,n,0)
f o r ( r =1; r <=n ; r ++) {
q [ r ]= a l p h a [ r ] : ∗ x [ n1 ] : / x [ r ] // E v a l u a t e q u a n t i t i e s
}
return (q)
}
p0
p1

alpha
y_cv_old
y_ev_old
y0

=
=
=
=
=
=

i n i t i a l i s e _ p a r a m e t e r s (0)
i n i t i a l i s e _ p a r a m e t e r s (1)
i n i t i a l i s e _ p a r a m e t e r s (2)
i n i t i a l i s e _ p a r a m e t e r s (3)
i n i t i a l i s e _ p a r a m e t e r s (3)
i n i t i a l i s e _ p a r a m e t e r s (3)

y_ cv =y0
y_ ev =y0
tcv = 0
tev = 0
cv =0
ev =0
d e l = ( p1:−p0 )/num
dp=p1:−p0
par
= J ( rows ( a l p h a ) + 1 , 1 , . )
p a r [ 1 . . rows ( a l p h a ) ] = p0

p a r [ rows ( a l p h a ) +1] = y0
q_ cv_ ol d = e v a l _ q u a n t i t i e s ( par ’ )
p a r [ 1 . . rows ( a l p h a ) ] = p1
p a r [ rows ( a l p h a ) +1] = y0
q_ ev_ ol d = e v a l _ q u a n t i t i e s ( par ’ )
f o r ( i =1; i <=num ; ++ i ) {
// i n i t i a l i z i n g p a r a m e t e r s f o r l o c a l d e r i v a t i v e s (CV)
a_cv
= J ( rows ( a l p h a ) , 1 , 1 )
a_cv
= ( ( ( a _ c v : ∗ ( i − 1)):/num ) : ∗ dp ) : + 1 // p r i c e s a t s t e p i f o r CV
par_cv
= J ( rows ( a l p h a ) + 1 , 1 , . )
p a r _ c v [ 1 . . rows ( a l p h a ) ] = a _ c v


APPENDICES

p a r _ c v [ rows ( a l p h a ) +1] = y_ cv
q_cv_new = e v a l _ q u a n t i t i e s ( par_ cv ’ )
// i n i t i a l i z i n g p a r a m e t e r s f o r l o c a l d e r i v a t i v e s (EV)
a_ ev
= J ( rows ( a l p h a ) , 1 , 1 )
a_ ev
= − ((( a_ ev : ∗ ( i − 1)):/num ) : ∗ dp ) : + p1 // p r i c e s a t s t e p i f o r EV
par_ev
= J ( rows ( a l p h a ) + 1 , 1 , . )
p a r _ e v [ 1 . . rows ( a l p h a ) ] = a_ ev
p a r _ e v [ rows ( a l p h a ) +1] = y_ ev
q_ev_new = e v a l _ q u a n t i t i e s ( par_ ev ’ )

cv =
ev =

( 0 . 5 ∗ ( q_ cv_ ol d : + q_cv_new ) ) ∗ d e l
( 0 . 5 ∗ ( q_ ev_ ol d : + q_ev_new ) ) ∗ d e l

y_ cv = y_ cv + cv
y_ ev = y_ ev−ev
t c v = t c v −cv
t e v = tev −ev
q_ cv_ ol d = q_cv_new
q_ ev_ ol d = q_ev_new
}
tcv
tev
end
clear all
loc al price_index =((1.3)^0.3)∗((1.4)^0.5)∗((1.7)^0.2)
d i s " True CV = " 100 ∗ (1− ‘ p r i c e _ i n d e x ’ )
d i s " True EV = " 100 ∗ ( 1/ ‘ p r i c e _ i n d e x ’ −1)
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ THE STATA CODE: D ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
/∗ − The B r e s l a w and Barry ’ s ( 1995) a l g o r i t h m
∗/
/∗ − Approach : Numeri cal a p p r o x i m a t i o n
∗/
/∗ − Outputs : Change i n w e l f a r e : EV, CV−> ( 3 . 6 6 )
∗/
/∗ − The a l g o r i t h m i s programmed wi th S t a t a ( e x a c t l y wi th mata ) . mata i s ∗/
/∗
the matrix language of S tat

∗/
/∗ − The u s e r can i n c r e a s e t h e number o f goods or use a n o t h e r demand
∗/
/∗
function
∗/
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
s e t more o f f
mata : mata c l e a r
mata
num=10
/∗ Number
r e a l matrix functi on i n i t i a l i s e _ p a r a m e t e r s ( s c a l a r
{
i f ( t ==0) r e t u r n ( 1 . 0
\ 1. 0 \ 1. 0 ) /∗ I n i t i a l
i f ( t ==1) r e t u r n ( 1 . 3
\ 1. 4 \ 1. 7 ) /∗ F i n a l
i f ( t ==2) r e t u r n ( 0 . 3
\ 0. 5 \ 0. 2 ) /∗ income
i f ( t ==3) r e t u r n ( 100)
/∗ Income
}

of i t e r a t i o n s
t)

∗/

price vector

price vector
shares

∗/
∗/
∗/
∗/

// D e f i n i n g t h e M a r s h a l l i a n demand f u n c t i o n s
// P a r a m e t e r s : x_1 . . . x_n ( p r i c e s ) and x_ { n+1} income
r e a l matrix functi on e v a l _ q u a n t i t i e s ( r e a l matrix x )
{
r e a l matrix q
alpha = i n i t i a l i s e _ p a r a m e t e r s (2)
n= c o l s ( x)−1
n1=n+1
q = J (1 ,n,0)
f o r ( r =1; r <=n ; r ++) {
q [ r ]= a l p h a [ r ] : ∗ x [ n1 ] : / x [ r ] // E v a l u a t e q u a n t i t i e s
}
return (q)
}
v o i d e v a l _ t 2 ( x , v ) // f u n c t i o n used f o r t h e n u m e r i c a l d e r i v a t i v e
{
alpha = i n i t i a l i s e _ p a r a m e t e r s (2)
n= c o l s ( x)−1
n1=n+1

93

94

APPENDICES

v = J ( 1 , n1 , . )
v [ 1 . . n ]= e v a l _ q u a n t i t i e s ( x )
v [ n1 ] = x [ n1 ]
}
D = deriv_ init ()

// E v a l u a t e t h e q u a n t i t i e s
// E v a l u a t e t h e income

// f u n c t i o n used f o r t h e num . d e r i v a t i v e

p0
=
initialise_parameters
p1
=
initialise_parameters
alpha =
initialise_parameters
y_ cv
=
initialise_parameters
y_ ev
=
initialise_parameters

tcv = 0
tev = 0
d e l = ( p1:−p0 )/num
n d e l = −d e l
dp=p1:−p0
f o r ( i =1; i <=num ; ++ i ) {

(0)
(1)
(2)
(3)
(3)

a_cv
a_cv

= J ( rows ( a l p h a ) , 1 , 1 )
= ( ( ( a _ c v : ∗ ( i − 1)):/num ) : ∗ dp ) : + 1

a_ ev
a_ ev

= J ( rows ( a l p h a ) , 1 , 1 )
= − ((( a_ ev : ∗ ( i − 1)):/num ) : ∗ dp ) : + p1 // p r i c e s a t s t e p i f o r EV

// p r i c e s a t s t e p i f o r CV

// i n i t i a l i z i n g p a r a m e t e r s f o r l o c a l d e r i v a t i v e s (CV)
par_cv
= J ( rows ( a l p h a ) + 1 , 1 , . )

p a r _ c v [ 1 . . rows ( a l p h a ) ] = a _ c v
p a r _ c v [ rows ( a l p h a ) +1] = y_ cv
par_cv
// i n i t i a l i z i n g p a r a m e t e r s f o r l o c a l d e r i v a t i v e s (EV)
par_ev
= J ( rows ( a l p h a ) + 1 , 1 , . )
p a r _ e v [ 1 . . rows ( a l p h a ) ] = a_ ev
p a r _ e v [ rows ( a l p h a ) +1] = y_ ev
h_ cv= e v a l _ q u a n t i t i e s ( par_ cv ’ ) ’
h_ev = e v a l _ q u a n t i t i e s ( par_ ev ’ ) ’

// q u a n t i t i e s a t s t e p i f o r CV
// q u a n t i t i e s a t s t e p i f o r EV

d e r i v _ i n i t _ e v a l u a t o r (D, &e v a l _ t 2 ( ) )
d e r i v _ i n i t _ e v a l u a t o r t y p e (D, " t " )
d e r i v _ i n i t _ p a r a m s (D, par_ cv ’ )
AA= d e r i v (D, 1 ) [ 1 : : rows ( a l p h a ) , 1 . . rows ( a l p h a ) +1]
A1 = AA [ 1 : : rows ( a l p h a ) , 1 . . rows ( a l p h a ) ] // d e r i v a t i v e wi th r e s p e c t to p r i c e s
A2 = AA [ 1 : : rows ( a l p h a ) , rows ( a l p h a ) +1] // d e r i v a t i v e wi th r e s p e c t to income
dhdp_cv = A1+A2∗ h_cv ’
d e r i v _ i n i t _ p a r a m s (D, par_ ev ’ )
AA= d e r i v (D, 1 ) [ 1 : : rows ( a l p h a ) , 1 . . rows ( a l p h a ) +1]
A1=AA [ 1 : : rows ( a l p h a ) , 1 . . rows ( a l p h a ) ] // d e r i v a t i v e wi th r e s p e c t to p r i c e s
A2 = AA [ 1 : : rows ( a l p h a ) , rows ( a l p h a ) +1] // d e r i v a t i v e wi th r e s p e c t to income
dhdp_ev = A1+A2∗ h_ev ’
// S ee t h e e q u a t i o n ( 68)
cv = del ’ ∗ h_ cv + 0. 5∗ del ’ ∗ dhdp_cv ∗ d e l
ev = ndel ’ ∗ h_ev + 0. 5∗ ndel ’ ∗ dhdp_ev ∗ n d e l
t c v = t c v −cv

t e v = t e v + ev
y_ cv = y_ cv + cv
y_ ev = y_ ev + ev
}
tcv
tev
end
clear all
loc al price_index =((1.3)^0.3)∗((1.4)^0.5)∗((1.7)^0.2)
d i s " True CV = " 100 ∗ (1− ‘ p r i c e _ i n d e x ’ )
d i s " True EV = " 100 ∗ ( 1/ ‘ p r i c e _ i n d e x ’ −1)
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ THE STATA CODE: E ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
/∗ − The E u l e r and RK4 a l g o r i t h m s
∗/
/∗ − Approach : Numeri cal a p p r o x i m a t i o n
∗/
/∗ − Outputs : Change i n w e l f a r e : EV/CV−> ( 3 . 6 7 ) and ( 3 . 6 8 )
∗/


APPENDICES

/∗ − The u s e r can i n c r e a s e t h e number o f goods or use a n o t h e r demand
∗/
/∗
function
∗/
/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗/
local m
= 100

// Income
l o c a l n i t e r = 12
// Number o f i t e r a t i o n s
// By d e f a u l t , we e s t i m a t e t h e EV. The u s e r can s e t t h i s to be "CV" .
l o c a l measurement = "EV"
// e x p e n d i t u r e s s h a r e s ( a l p h a ’ s v a l u e s ) . The u s e r can change t h e number o f i t e m s
m a t r i x a l p h a = ( 0 . 1 0 \ 0. 05 \ 0. 12 \ 0 . 0 6 )
m a t r i x p0
m a t r i x p1

= ( 1 . 0 0 \ 1. 00 \ 1. 00 \ 1. 00 ) // I n i t i a l p r i c e s
= ( 1 . 1 5 \ 1. 30 \ 1. 10 \ 1 . 2 5 ) // F i n a l p r i c e s

l o c a l p r i c e _ i n d e x =1
l o c a l n i t e m s = rowsof ( a l p h a )
// Number o f i t e m s
// l o a d i n g t h e d i f f e r e n t v a l u e s i n l o c a l macros .
f o r v a l u e s k =1/ ‘ ni tems ’ {
l o c a l al pha_ ‘ k ’ = e l ( a l p h a , ‘ k ’ , 1 )
l o c a l p ‘ k ’ _0 = e l ( p0 , ‘ k ’ , 1 )
l o c a l p ‘ k ’ _1 = e l ( p1 , ‘ k ’ , 1 )
l o c a l p r i c e _ i n d e x = ‘ p r i c e _ i n d e x ’ ∗ ( ‘ p ‘ k ’ _1 ’ ^ ‘ al pha_ ‘ k ’ ’ )
i f " ‘ measurement ’" =="CV" {
l o c a l h ‘ k ’ = ( ‘ p ‘ k ’ _1 ’ − ‘p ‘ k ’ _0 ’ ) / ‘ n i t e r ’
l o c a l m u l t i = ( 1/ ‘ p r i c e _ i n d e x ’ −1)
}
i f " ‘ measurement ’" =="EV" {
l o c a l multi = (1 − ‘ price_index ’ )
l o c a l h ‘ k ’ = −(‘p ‘ k ’ _1 ’ − ‘p ‘ k ’ _0 ’ ) / ‘ n i t e r ’
f o r v a l u e s k =1/ ‘ ni tems ’ {

l o c a l al pha_ ‘ k ’ = e l ( a l p h a , ‘ k ’ , 1 )
l o c a l p ‘ k ’ _1 = e l ( p0 , ‘ k ’ , 1 )
l o c a l p ‘ k ’ _0 = e l ( p1 , ‘ k ’ , 1 )
}
}
}
l o c a l m0
= 100
l o c a l me0
= 100
// A s i m p l e program to e v a l u a t e t h e f i r s t
cap program drop odf
program d e f i n e odf , r c l a s s
ar gs p m alpha
r e t u r n s c a l a r f t = −‘ a l p h a ’ ∗ ‘m’ / ( ‘ p ’ )
end

d e r i v a t i v e : t h e c a s e o f Cobb−Douglas f u n c t i o n .

// The i t e r a t i v e a l g o r i t h m f o r t h e E u l e r and RK4 methods .
f o r v a l u e s i =1/ ‘ n i t e r ’ {
l o c a l j = ‘ i ’−1
f o r v a l u e s k =1/ ‘ ni tems ’ {
l o c a l p_ ‘ k ’ _0 = ‘ p ‘ k ’ _0 ’
l o c a l m_‘ k ’
= ‘m‘ j ’ ’
l o c a l mm_‘ k ’ = ‘m‘ j ’ ’
l o c a l p_ ‘ k ’ _ ‘ i ’
= ‘ p_ ‘ k ’ _0 ’ + ‘ h ‘ k ’ ’ ∗ ‘ i ’
l o c a l p_ ‘ k ’

= ‘ p_ ‘ k ’ _ ‘ j ’ ’
odf ‘ p_ ‘ k ’ ’ ‘m_‘ k ’ ’ ‘ al pha_ ‘ k ’ ’
l o c a l k1_ ‘ k ’ = ‘ r ( f t ) ’
l o c a l p_ ‘ k ’ = ‘ p_ ‘ k ’ _ ‘ j ’ ’ + 0 . 5 ∗ ‘ h ‘ k ’ ’
l o c a l m_‘ k ’ = ‘mm_‘ k ’ ’ + 0 . 5 ∗ ‘ k1_ ‘ k ’ ’ ∗ ‘ h ‘ k ’ ’
odf ‘ p_ ‘ k ’ ’ ‘m_‘ k ’ ’ ‘ al pha_ ‘ k ’ ’
l o c a l k2_ ‘ k ’ = ‘ r ( f t ) ’
l o c a l m_‘ k ’ = ‘mm_‘ k ’ ’ + 0 . 5 ∗ ‘ k2_ ‘ k ’ ’ ∗ ‘ h ‘ k ’ ’
odf ‘ p_ ‘ k ’ ’ ‘m_‘ k ’ ’ ‘ al pha_ ‘ k ’ ’
l o c a l k3_ ‘ k ’ = ‘ r ( f t ) ’
l o c a l p_ ‘ k ’ = ‘ p_ ‘ k ’ _ ‘ i ’ ’
l o c a l m_‘ k ’ = ‘mm_‘ k ’ ’ + ‘ k3_ ‘ k ’ ’ ∗ ‘ h ‘ k ’ ’
odf ‘ p_ ‘ k ’ ’ ‘m_‘ k ’ ’ ‘ al pha_ ‘ k ’ ’
l o c a l k4_ ‘ k ’ = ‘ r ( f t ) ’
local c ‘ i ’ =
l o c a l ce ‘ i ’ =
}

‘ c ‘ i ’ ’+1/ 6∗ ( ‘ k1_ ‘ k ’ ’ + 2 ∗ ( ‘ k2_ ‘ k ’ ’ + ‘ k3_ ‘ k ’ ’ ) + ‘ k4_ ‘ k ’ ’ ) ∗ ‘ h ‘ k ’ ’
‘ ce ‘ i ’ ’ + 1∗ ( ‘ k4_ ‘ k ’ ’ ) ∗ ‘ h ‘ k ’ ’

95


96

APPENDICES

l o c a l m‘ i ’ = ‘m‘ j ’ ’ + ‘ c ‘ i ’ ’
l o c a l me‘ i ’ = ‘me‘ j ’ ’ + ‘ ce ‘ i ’ ’

i f " ‘ measurement ’" =="CV" d i " I t e r a t i o n " ‘ i ’
‘me‘ i ’ ’ − ‘me0 ’ " | ‘ measurement ’ _RK4 : "
i f " ‘ measurement ’" =="EV" d i " I t e r a t i o n " ‘ i ’
− ‘me‘ i ’ ’ + ‘me0’ " | ‘ measurement ’ _RK4 : "
}

" : ‘ measurement ’ _ E u l e r :
‘m‘ i ’ ’ − ‘m0’
" : ‘ measurement ’ _ E u l e r :
− ‘m‘ i ’ ’ + ‘m0’

"

///

"

///

// The t r u e v a l u e o f t h e EV/CV measurement
d i s " True v a l u e o f ‘ measurement ’ = " 100 ∗ ‘ mul ti ’

REFERENCES
BANKS, J., R. BLUNDELL, AND A. LEWBEL (1997): “Quadratic Engel Curves
And Consumer Demand,” The Review of Economics and Statistics, 79, 527–539.
DEATON, A. AND J. M UELLBAUER (1980): “An Almost Ideal Demand System,”
American Economic Review, 70, 312–336.
H OAREAU S., LACROIX G., H. M. AND L. TIBERTI (2014): “Exact Affine Stone
Index Demand System in R: The easi Package,” Tech. rep., University Laval,
CIRPEE: />LEWBEL, A. AND K. PENDAKUR (2009): “Tricks with Hicks: The EASI Demand

System,” American Economic Review, 99, 827–63.


INDEX

B
Breslaw and Smith approximation, 40

C
Cobb–Douglas (CD) function,
25, 36–37, 60, 77
Compensating variation, 2, 11–16, 19,
22, 23, 25–27, 32–34, 36–40,
42–44, 48, 50, 56, 60, 75, 76,
87, 89
Consumer surplus, 3, 11

D
Demand functions
almost Ideal Demand System
(AIDS), 4, 53, 82–84
exact Affine Stone Index (EASI),
4, 53, 85–87
Hicksian, 10, 31
Linear Demand, 79
linear Expenditure System (LES), 4,
53, 80–82

log Linear demand, 79
Marshallian, 4, 10, 11, 14, 37, 39,

42, 58
quadratic Almost Ideal System
(QUAIDS), 4, 53, 84–85
Differential equations, 38

E
Elasticity
income elasticity, 44, 81, 85
non compensated price elasticity,
51, 85
Equivalent variation (EV), 2, 5, 11,
12, 81, 89
Euler method, 41, 42

I
Income effect, 3, 26, 31, 32, 34,
50, 76
Index numbers, 13, 19, 20
Individual welfare, 6, 48, 61, 67

© The Author(s) 2019
A. Araar, P. Verme, Prices and Welfare,
/>
97


98

INDEX

L
Laspeyres
index, 5, 13, 20, 32
variation, 2, 11, 13

Stochastic dominance, 64, 69–73
Substitution effect, 3, 19, 26–28,
30–32, 34–36, 41, 47, 50, 51,
54, 61, 75

P
Paasche
index, 5, 13
variation, 2, 11, 13
Pro-poor curves, 64, 69–73

T
Taylor’s approximations
first order, 31, 76
higher orders, 3, 76

R
Runge and Kutta method, 41

U
Utility
direct, 31
indirect, 10, 13, 38, 58, 81, 84, 85

S

Social welfare, 10, 48, 61, 64, 67
Stata codes, 90–96
Statistical inference, 6, 64–69

V
Vartia’s approximation, 38–40