WPS7566
Policy Research Working Paper 7566
Prices and Welfare
Abdelkrim Araar
Paolo Verme
Poverty and Equity Global Practice Group
February 2016
Policy Research Working Paper 7566
Abstract
What is the welfare effect of a price change? This simple welfare measures converge to approximately the same result
question is one of the most relevant and controversial ques- for price changes below 10 percent. Above this threshold,
tions in microeconomic theory and its different answers can these measures start to diverge significantly. Budget shares
lead to severe heterogeneity in empirical results. This paper play an important role in explaining such divergence,
returns to this question with the objective of providing a whereas the choice of demand system has a minor role.
general framework for the use of theoretical contributions Under standard utility assumptions, the Laspeyers and
in empirical works, with a particular focus on poor people Paasche variations are always the outer bounds of wel-
and poor countries. Welfare measures (such as Equivalent fare estimates and consumer surplus is always the median
Variation or Consumer’s Surplus) and computational meth- estimate. The paper also introduces a new simple welfare
ods (such as Taylor’s approximations or the Vartia method) approximation, clarifies the relation between Taylor’s
are compared to test how these choices result in different approximations and the income and substitution effects,
welfare measurement under different price shock scenarios. and provides an example for treating nonlinear pricing.
As a rule of thumb and irrespective of parameter choices, Stata codes for all computations are provided in annex.
This paper is a product of the Poverty and Equity Global Practice Group. It is part of a larger effort by the World Bank to
provide open access to its research and make a contribution to development policy discussions around the world. Policy
Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at Araar.
Abdelkrim@ecn.ulaval.ca and pverme@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Prices and Welfare
Abdelkrim Araar∗ and Paolo Verme †
16th February 2016
∗ ´ Pavillon de S`
CIRPEE, e Laval, Qu´
eve, Universit´ ebec, Canada, G1K 7P4; email: aabd@ecn.ulaval.ca
†
World Bank, Washington DC; email: pverme@worldbank.org. The authors wish to thank Hanan Jacoby
(World Bank) and Baoping Shang (IMF) for peer reviewing the paper, and Lidia Ceriani and participants to a
seminar held in Washington DC in January 2016 for useful comments. The work was jointly ﬁnanced by the World
Bank PSIA trust Fund and the World Bank “Fiscal and Social Policies for Poverty Reduction and Shared Prosperity
Thematic Line” (P152945).
Table of Contents
1 Introduction 4
2 Assumptions 7
3 Measures 8
3.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Computation 13
4.1 Index numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Demand functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.4 Taylor’s approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4.1 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4.2 Higher orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4.3 Convergence and the Cobb-Douglas function . . . . . . . . . . . . . . 24
4.5 Vartia’s approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.6 Breslaw and Smith’s approximation . . . . . . . . . . . . . . . . . . . . . . . 26
4.7 The ordinary differential equations methods . . . . . . . . . . . . . . . . . . . 27
4.8 Relational approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Applications 29
5.1 Individual welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.1.1 Small price variations . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.1.2 Large price variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1.3 A choice tool for practitioners . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Social welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.1 Statistical inference (Society) . . . . . . . . . . . . . . . . . . . . . . 42
5.2.2 Statistical inference (Quantiles) . . . . . . . . . . . . . . . . . . . . . 44
5.2.3 Pro-poor curves and stochastic dominance . . . . . . . . . . . . . . . . 45
6 Conclusion 47
Appendices 54
A Demand Systems 54
A.1 Linear Demand (LD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.2 Log Linear Demand (LLD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.3 The Linear Expenditure System (LES) . . . . . . . . . . . . . . . . . . . . . . 54
A.4 The Almost Ideal Demand System (AIDS) . . . . . . . . . . . . . . . . . . . . 56
A.5 The Quadratic Almost Ideal System (QUAIDS) . . . . . . . . . . . . . . . . . 57
A.6 Exact Afﬁne Stone Index (EASI) . . . . . . . . . . . . . . . . . . . . . . . . . 58
B Nonlinear price changes and wellbeing 59
2
C STATA Codes 62
Mathematical Notations
Welfare measures
CV Compensating variation
EV Equivalent variation
CS Consumer surplus variation
LV Laspeyers variation
PV Paasche variation
S Substitution effect
I Income effect
Functions
u(.) Direct utility
ν (.) Indirect utility
e(.) Expenditure
Dk (p) Demand
xk (P, m) Marshallian demand
hk (P, m) Hicksian demand
∆(.) Absolute variation
d(.) Proportional variation
λ(.) Proportion of the error term
Vectors
p Price
x Quantity
m Income
e Expenditure
sk Expenditure share of good k
Scalars
ϑ Income elasticity
η Non compensated price elasticity
Indexes
k = 1, 2, ..., n Products (subscript)
t = a, b State/Time (superscript)
o = 1, 2, ..., n Taylor degree of approximation (superscript)
3
1 Introduction
In economics, there are two established traditions for the measurement of individual utility,
well-being or welfare.1 The ﬁrst tradition, pioneered by Edgeworth (1881), argues that utility can
be measured directly with a ‘hedonimeter’ capable of capturing the physiological phenomenon
of happiness. This tradition enjoyed very few followers until the emergence and establishment of
happiness economics and prospect theory, two relatively new strands of the economics literature
that attempt, in different ways, to directly measure utility. The happiness literature tends to
measure happiness with subjective questions on happiness and life satisfaction. The prospect
theory literature has measured utility, for example, with the measurement of physiological pain.
The second tradition, pioneered by Fisher (1892), argues that utility cannot be measured
directly in any sensible way and that it is necessary to derive utility indirectly from the obser-
vation of behavioral choices.2 If we assimilate Paul Samuelson’s theory of revealed preferences
with this tradition, we can then argue that this has been the prevalent welfare theory taught in
economics over the past century. Interestingly, while Bentham himself equated happiness with
utility (as in the happiness literature), he also thought that utility was embedded in objects (as
in the revealed preferences literature): “By utility is meant that property in any object, whereby
it tends to produce beneﬁt, advantage, pleasure, good, or happiness, (all this in the present case
comes to the same thing) or (what comes again to the same thing) to prevent the happening
of mischief, pain, evil, or unhappiness to the party whose interest is considered. (p.2, Bentham
([1789]1907)).
This paper focuses on changes in welfare derived from changes in prices following the sec-
ond tradition of indirect welfare measurement. The main purpose is to estimate the difference
in welfare that derives from the choice of different welfare measures and clairify the key factors
that determine such differences. We consider ﬁve measures (henceforth called “welfare mea-
sures”) that have been proposed by the microeconomics literature to measure welfare changes
since the seminal paper by Hicks (1942): 1) Consumer’s surplus variation (CS for short); 2)
Compensating Variation (CV ); 3) Equivalent Variation (EV ); 4) Laspeyers Variation (LV ) and
5) Paasche Variation (P V ).
Building on previous contributions, we aim to: 1) Review the essential microeconomics
literature; 2) Organize and simplify this literature in a way that can be easily understood by
researchers and practitioners with different backgrounds providing algebraic, geometric, com-
putational and empirical illustrations; 3) identify and measure the essential differences across
methods and test how these differences affect empirical results; 4) Provide guidelines for the
use of alternative approaches under imperfect information on utility, demand systems, elastici-
ties and more generally incomes and quantities; 5) Provide computational codes in Stata for the
application of all welfare measures and computational methods.
While the theoretical literature regularly offers excellent review papers on the topic (see for
example, Harberger (1971); King (1983); Slesnick (1998) and Fleurbaey (2009)), we believe
that this literature remains short of providing simple guidelines for practitioners. On the other
hand, the empirical literature, which is very rich and varied, remains short of explaining clearly
the microeconomic foundations that justify the choice of one welfare measure over another. Our
1
This paper uses these three terms as synonyms and will use them interchangeably as needed.
2
See Colander (2007), for a historical comparative analysis of these two traditions.
4
main goal is to bridge these two traditions and ﬁll these gaps in an effort to serve practitioners
working with micro data, particularly those focusing on poor countries and poor people. Pre-
sumably, measuring the impact on welfare due to price changes is of interest to the policy maker
for social and distributive policies. The impact of price changes on the rich is typically small in
relative terms and less of a concern that the impact on the middle class or the poor. Hence, our
focus on the poor.
We will follow what is sometimes called the “marginal approach”. This is the estimation
of direct effects of a price change on welfare keeping the nominal budget constraint or income
constant. Price changes can eventually affect incomes of producers and other agents and these
effects can be important (see for example Ravallion (1990) and Jacoby (2015)). However, this
complicates substantially our analysis and we opted to exclude income, supply, partial or general
equilibrium effects from the paper. We will therefore follow the more common tradition of the
marginal approach as in Ahmad and Stern (1984), Ahmad and Stern (1991), Creedy (1998),
Creedy (2001), Deaton (1989), Minot and Dewina (2013) and (Ferreira, Fruttero, Leite, and
Lucchetti 2011). See also Creedy and van de Ven (1997) on the impact of marginal changes in
food subsidies on the Foster, Greer and Thorbecke (FGT) poverty indices.
The paper will cover a range of computations (henceforth called “computation methods”)
that have been proposed by the literature over the years including those based on different de-
mand systems, Taylor approximations, the Vartia method, the Breslaw and Smith method, ordi-
nary differential equation methods and a simple method based on knowledge of elasticity. There
are of course many more methods proposed by the literature and evidence on how these meth-
ods perform. Hausman and Newey (1995), for example, derive estimates of demand curves and
the consumer surplus applying nonparametric regression models. Banks, Blundell, and Lewbel
(1996) derive second order approximations of welfare effects and show how ﬁrst order approx-
imations can produce large biases by ignoring the distribution of substitution effects. In this
paper, we restrict the analysis to the most popular methods but ﬁndings likely apply to most if
not all existing methods.
With respect to computation methods, our contribution is to clarify the relation between
the ﬁve measures initially introduced by Hicks and their computation methods. Some authors
may argue that some of the computation methods we discuss such as Taylor’s approximations
of a certain degree are welfare measures themselves and different from the ﬁve measures listed
above. In this work, we will clarify the distinction between core measures and computation
methods. In addition, we clarify the decomposition of higher order Tayolor’s approximations
in substitution and income effects and propose a simple computation method based on known
elasticities.
The paper does not focus on the analysis or construction of demand systems. This literature
is rather vast and offers several alternatives. One of the critiques to simple linear expenditure
systems was that they fail to consider the Engel law, the variation of the income-expenditure
relation across the income distrbution. Muellbauer (1976), Deaton and Muellbauer (1980a),
Deaton and Muellbauer (1980b), and Jorgenson et al. (1982) contributions helped to place the
Working-Leser Engel curve speciﬁcation within integrable consumer theory thereby starting to
address this issue. Recent empirical work has shown that the popular AID demand system do
not take into consideration the full curvature of the Engel curve. Banks, Blundell, and Lewbel
(1997) showed that Working-Leser Engel types of curves may be insufﬁcient to describe con-
5
sumption behavior across income groups. They derive a demand model based on an integrable
quadratic logarithmic expenditure share system and show that this model ﬁts UK data better
than the Working-Leser Engel types of models, particulalry for selected commodities. Blundell,
Chen and Kristensen (2007) later showed that behavior changes across different types of goods
with some goods approaching a linear or quadratic shape while others having different forms.
More recently, Lewbel and Pendakur (2009) proposed the Exact Afﬁne Stone Index (EASI) im-
plicit Marshallian demand system. In the words of the authors: “In contrast to the AID system,
the EASI demand system also allows for ﬂexible interactions between prices and expenditures,
permits almost any functional form for Engel curves, and allows error terms in the model to
correspond to unobserved preference heterogeneity random utility parameters.” (p.29). Recent
empirical works that attempted to estimate demand systems directly from data in developing
countries include Attanasio, Di Maro, Lechene, and Phillips (2013) and Osei-Asare and Eghan
(2013).
With respect to demand systems, our contribution is to compare the behavior of different
welfare measures using alternative demand systems including simple Cobb-Douglas (CD), Lin-
ear Expenditure System (LES), the Almost Ideal Demand System (AIDS), the Quadratic Almost
Ideal Demand System (QUAIDS) and the Exact Afﬁne Stone Index (EASI). The paper ﬁnds that
the difference in welfare measurement is minimal as compared to changes in other parameters
such as the price change or the budget share.
Results of this paper can be relevant for a wide set of issues empirical economists are con-
fronted with. Changes in prices occur for a variety of reasons. They may be induced by global
shocks as it was the case for the global rise in commodity prices during the ﬁrst decade of the
2000s or the 2008 global ﬁnancial crisis, or they may be due to domestic shocks such as those
induced by variations in local climatic conditions. Price changes may also occur as a result of
economic policies such as changes in taxes, wages, subsidies or social transfers. In all these
cases, the policy maker may want to estimate the impact on wellbeing ex-post (for example in
the case of economic shocks) or ex-ante (for example in the case of economic policies). The
work proposed applies to both cases and provides guidelines for macro or micro economists or
for macro or micro simulation exercises of economic shocks or policy reforms.
Measuring changes in welfare due to changes in prices is also an issue very relevant for ad-
justing welfare measures (such as GDP or the poverty headcount) spatially or longitudinally and
therefore measuring changes over time correctly. As the latest round of the global Purchasing
Parity Power (PPP) surveys has shown, changes in data on prices can change welfare measure-
ments very signiﬁcantly. Changing measure or method for estimating welfare effects of price
changes can obviously amplify or reduce the effect of price changes. Practitioners as well as
international organizations engaged in measuring the impact of price changes on welfare give
surprisingly little weight to the choice of estimation method. For example, the World Bank and
the IMF use as methods of choice for spatial and longitudinal price adjustments the Laspeyers
or Paasche indexes while they almost invariably use the Laspeyers index when simulating the
impact of price changes on welfare, and this often irrespective of the magnitude of the price
change. Theoretical economists, on the other hand, tend to privilege the equivalent variation or
consumer’s surplus measures when it comes to measure changes in welfare due to price changes.
A priori, these are normative decisions and good arguments can be found to justify each of these
choices. But the outcomes of these choices can be very different in terms of welfare measure-
6
ment and this should be very clear to anyone making these choices.
The paper is organized as follows. The next section provides the underlying assumptions
used throughout the paper. Section 3 deﬁnes the welfare measures used and provides a simple
geometrical interpretation. Section 4 reviews the computational approaches provided by the
literature under speciﬁc assumptions or degree of information. Section 5 tests how the measures
and computations proposed diverge as prices and other key parameters vary. This section also
discusses statistical inference and stochastic dominance when individual welfare measures are
aggregated at the societal level. Section 6 concludes summarizing results and providing basic
recommendations for practioners.
2 Assumptions
To restrict the boundaries of the discussion that follows, we will make a number of stan-
dard assumptions. Consumers have a preference ordering R deﬁned in the commodity space X
and have well behaved utility functions (monotonic and strictly convex preferences) and single-
valued, continuously differentiable demand function where prices are strictly positive. The basic
axioms of consumer theory are observed (consumer preferences are complete, reﬂexive and tran-
sitive). Preferences are homothetic so that (x1 , x2 ) ≺ (y1 , y2 ) ⇔ (tx1 , tx2 ) ≺ (ty1 , ty2 ) for any
t > 0, although we will use at times demand models where preferences are not homothetic.
The demand function is generated by R and is not necessarily observable with data. Con-
sumers maximize utility and operate on the budget constraint with marginal utility of income
being constant throughout the space concerned by price changes. The commodity X space in-
cludes two normal goods where the ﬁrst good x1 is subject to price changes and the second good
x2 represents the bundle of all other goods available to the consumer, which may or may not
change of price.
We also assume that the budget constraint remains nominally ﬁxed under price changes so
that any price increase (reduction) results in a loss (gain) in real incomes. These assumptions
imply short-term decisions, no savings and no inter-temporal choices. Other than being stan-
dard neoclassical assumptions, we justify these choices on the ground that we are particularly
concerned with the poor and developing countries where, by deﬁnition, savings are close to zero
and consumers spend all their budget on current consumption.
Individual and household preferences are considered as one and the same. We also consider
identical behavior and utility functions across consumers and no utility inter-dependence. Social
welfare is the non-weighted sum of the outcomes of individual (household) choices implying
that we ignore any impact on the non-household sector. As discussed in the introduction, we
consider indirect utility functions on the assumption that utility cannot be observed directly and
we use money-metric utility functions as proposed by McKenzie (1957). The underlying idea is
that an indirect utility function can be represented in terms of an expenditure function.
The essential problem we are trying to solve is how to measure welfare changes when the
price of at least one of the goods considered changes and if utility, demand or both are not
known. We consider a consumer who chooses a bundle of two goods x = {x1 , x2 } subject to
prices p = {p1 , p2 }. The consumer maximizes a well behaved utility function u(x) under a
budget constraint m = p1 x1 + p2 x2 and a demand system D = d(p, m) and is subject to a price
shock (∆p1 ). What is known are current prices (p1 , p2 ), current quantities x1 , x2 , current budget
7
(m) and the price change ∆p1 . What is not necessarily known are utility u(x) and demand
functions d(p, m) and therefore the change in quantities ∆x1 and ∆x2 and the change in utility
∆u due to the price change ∆p1 . The central question is how to estimate the change in welfare
∆u in money terms and under different degrees of information on the other parameters.
Note that we will talk of partial effects when we consider variations in prices of only one
product and general effects when we can consider simultaneous variations in prices of more than
one product. We will mostly refer to the Marshallian demand function in place of Walrasian or
uncompensated demand functions and to the Hicksian demand function in place of compensated
demand function.
In real life, researchers are confronted with a general scarcity of information on consumers’
behavior and this is the more so in developing and poor countries where data are scarce. In what
follows, we will review the different ways of approximating changes in welfare under different
degrees of information on consumers’ behavior.
3 Measures
3.1 Deﬁnitions
We consider ﬁve popular measures of welfare change under price variations which were
already outlined by Hicks over 70 years ago:3 Consumer’s Surplus variation (CS), Equivalent
Variation (EV), Compensating Variation (CV), Laspeyres Variation (LV), and Paasche Variation
(PV). In this ﬁrst section, we simply outline the concepts and the basic formulations of these
measures.
The Consumer Surplus variation (CS)4 was initially introduced by Marshall and deﬁned as
“The excess of the price which he would be willing to pay rather than go without the thing, over
that which he actually does pay, is the economic measure of this surplus satisfaction. It may be
called consumer’s surplus.” (Marshall ((1890) 1961)). By deﬁnition, this measurement requires
knowledge of the Marshallian demand function (the “willingness to pay” function) and can be
represented by the area under this curve delimited by two prices. One possible formulation of
the CS is therefore as follows5
pb
CS = D(p)dp (1)
pa
where pa and pb represent initial and ﬁnal prices respectively and D(p) is a generic demand
function that applies equally to all consumers.
Perhaps the main supporter of this concept as a measure of welfare change has been Har-
berger (1971) with his letter to the profession published in the Journal of Economic Literature.
As described in this paper, the ﬁve main criticisms to the CS approach state that this approach:
1) is valid only when the marginal utility of real incomes is constant; 2) does not take into ac-
count distributional changes derived from price changes; 3) is a partial equilibrium approach;
3
See Hicks (1942).
4
Note the use of CS for consumer surplus variation rather than consumer surplus.
5
see Layard and Walters (1978).
8
4) does not apply to large price changes and 5) is made obsolete by the revealed preferences
approach.
By analogy with national accounts, Harberger (1971) responded to each of the ﬁve criticisms
but on point 1) further research has shown that the conditions for the CV approach to apply
are more restrictive than initially thought. As shown by Chipman and Moore (1977), changes
in consumer’s surplus are single valued and ordinarily equivalent to changes in utility under
the conditions of utility maximization, homogeneous utility, integrable demand functions and
constant marginal utility. In addition, with changes in prices that affect more than one product,
the CS approach is “path-dependent”, meaning that the estimation of the welfare change will
be different depending on which price changes ﬁrst. These two critiques have induced scholars
to revalue other methods and approximations of welfare change (see Slesnick (1998) for a full
critique of the CS method).
The Compensating Variation (CV) was ﬁrst named by Hicks in his “Value and Capital” but
it was Henderson (1941) who ﬁrst clariﬁed the distinction between CS and CV . Hicks (1942))
later accepted this distinction and also introduced the concept of Equivalent Variation (EV) to
distinguish Henderson’s concept of CV when welfare change is evaluated at ﬁnal rather than
initial prices. The CV is the monetary compensation required to bring the consumer back to the
original utility level after the price change. The EV is the monetary change required to obtain
the same level of utility after the price change. For changes from pa to pb of one product, these
two variations can be represented as6
CV = e(pa , ν a ) − e(pb , ν a ) (2)
pb
= h(p, ν a ).dp (3)
pa
EV = e(pa , ν b ) − e(pb , ν b ) (4)
pb
= h(p, ν b ).dp (5)
pa
where ν and e represent generic indirect utility and expenditure functions.
The Laspeyers Variation (LV) is deﬁned as the exact change in income necessary to pur-
chase, after the price variation, the same bundle of goods purchased before the price variation.
The Paasche Variation (PV) is deﬁned as the exact change in income required to purchase the
ﬁnal bundle of goods at initial prices. Hence, possible representations of the two indexes are the
following:
LV = e(pb , xa ) − e(pa , xa ) (6)
P V = e(pb , xb ) − e(pa , xb ) (7)
6
See Layard and Walters (1978) or Dixit and Weller (1979).
9
The LV and the P V derive from index number theory initially proposed by Fisher (1922).
As discussed by Fleurbaey (2009), index number theory has developed in three directions. The
ﬁrst direction aims at deﬁning desirable properties of an index and ﬁnd indexes that satisfy these
properties. For example, Diewert (1992) shows that the original indexes proposed by Fisher
satisfy a set of 21 desirable axioms which make these indexes superior to others. The second
direction is a tradition that seeks indexes that depend only on prices and quantities and that are
good approximations of welfare changes. Diewert (1992) has shown for example that it is possi-
ble to ﬁnd functional forms of the expenditure function that are both simple and ﬂexible and that
result in indexes such as the geometric mean of the Laspeyers and Paasche indexes. The third
direction initiated by Samuelson and Swamy (1974) seeks indexes that depend on individual
preferences such as the money-metric utility function. In this paper, we will not discuss further
these different developments of index number theory. What is important to stress here is that the
LV and P V indexes are routinely used to measure welfare changes under price changes and to
adjust longitudinally and spatially welfare measures such as the poverty headcount index.
As already clear from these ﬁrst formulations, while the CS , CV and EV methods require
knowledge of utility functions and demand functions while the LV and P V as deﬁned above
require information on the demand function only. The difference between measures becomes
clearer when we illustrate these methods geometrically in the next section while we will see
that it is possible to estimate these measures also in the absence of direct knowledge on the
Marshallian demand function. This is particularly important for countries where it is not possible
to measure this demand function directly because of data constraints.
3.2 Geometric interpretation
Figure 1 illustrates the ﬁve estimation methods discussed in a classic geometric setting. In
the top panel, the initial budget line is the continue blue line and the initial equilibrium is at A.
The slope of this curve is −pa . After an increase in price from pa to pb , the budget constraint
curve rotates as shown by the red line adjusting to a slope −pb and the ﬁnal state is reached in
B.
The LV measurement evaluates the change in welfare with the initial bundle of goods. Thus
LV = −xa (pb − pa ) which is (minus) the distance between A and A2 . Similarly, we can evaluate
the potential change in expenditure with the ﬁnal quantities such that P V = −xb (pb − pa ), which
is (minus) the distance between B and B2 . The CV is the required budget to offset the loss in
wellbeing with the new prices. This amount is equal to −BB 1 , which leads to D. The EV is
the price equivalent loss in wellbeing. It equals −AA1 , which leads to C .
The bottom panel of Figure 1 shows the geometric interpretation of the ﬁve methods in the
case of a price increase from pa to pb , a change in quantity from xa to xb , linear demand schedules
derived from known utility functions and changes in the price of only one product (assumptions
derived from the top panel). Following from the deﬁnitions provided in the previous section, the
CS is the area below the Marshallian demand curve and between initial and ﬁnal prices, which
is the area delimited by points pa EApb . Consequently, the EV is equal to the area pa CApb , the
CV is equal to the area pa EDpb , the LV is the rectangle pa EF pb , which is equal to −xa ∆p and
the P V is equal to the area of the rectangle pa BApb , which is equal to −xb ∆p.
From Figure 1, which is inspired from the Hicks (1942), we can derive a ﬁrst set of results:
10
Figure 1: Welfare Measures
LV = - the distance: A to A2
EV = - the distance: A to A1
CV = - the distance: B to B1
PV = - the distance: B to B2
Marshallian Money (income)
B1
D
݉
B2
A
ܫ
B
C
ܫ
A 1
A 2
0 ݔ ݔ ݔ ݔ Quantity
LV = the area: ܨܧ
CV = the area: ܦܧ
Price of good ݔ
CS = the area: ܣܧ
Hicksian demand at ݄ሺǡ ݉ሺܸ ሻሻ EV = the area: ܣܥ
PV = the area: ܣܤ
A D
F
Substitution effect
Income effect
E
C
B Marshallian demand ݔሺǡ ݉ሻ
Hicksian demand at݄ሺǡ ݉ሺܸ ሻሻ
0 ݔ ݔ ݔ ݔ Quantity
11
1. For a normal good, LV < CV < CS < EV < P V if dp = 0 and the demand schedules
are not perfectly elastic or inelastic, (see also Hicks (1942), Willig (1976) and Cory, Gum,
Martin, and Brokken (1981));
2. The welfare effect is bounded between LV = −xa dp and P V = −xb dp;
3. LV = CV = CS = EV = P V if dp = 0 and the demand schedules are not perfectly
elastic or inelastic;
4. The difference between the different measures depends on the size of the price change, on
the utility function and on the corresponding demand functions.
We can also express result (1) above in terms of changes in prices and quantities as follows:
xa dp > (xa + xc )dp/2 > (xa + xb )dp/2 > (xe + xb )dp/2 > xb dp (8)
by dividing by dp and multiplying by 2 we obtain the relation between the different methods
in terms of quantities only 7
2xa > (xa + xc ) > (xa + xb ) > (xe + xb ) > 2xb (9)
From the inequality above, we can then derive the following additional results:
5. With a perfectly inelastic (vertical) Marshallian demand schedule, xa = xb and the welfare
effect is only determined by prices and estimated at xa dp. It is also evident that LV =
CV = CS = EV = P V so that it is irrelevant which approach is used to the measurement
of welfare.
6. Vice-versa, with a perfectly elastic (horizontal) demand schedule, the consumer is not
willing to buy any quantity at the new price (xb = 0) and the welfare effects will be equal
to the loss of the original welfare xa pa . In this case too, the welfare estimates will not
depend on the approach followed and LV = CV = CS = EV = P V .
In essence, changes in welfare ∆V will be bounded between (−)xa dp and (−)xb dp. Within
these boundaries, LV = CV = CS = EV = P V if dp = 0 and the different approaches will
produce different estimates of welfare change. The difference in these estimates, in turn, will
depend on the size of the price change, on the shape of the utility function and on the derived
demand function. Choices concerning these last two parameters led to the use of different com-
putational strategies for the ﬁve indicators of welfare change illustrated. These strategies are
discussed in the next section.
7
Note that this result is based on the assumptions of moderate change in price or in the case of a straightforward
line shape of the demand function.
12
4 Computation
Given that the true utility and demand functions are mostly unknown, estimations of welfare
changes are based on approximations. The LV and P V approaches do not require utility mod-
eling and for this reason they can be estimated by simply using prices and quantities but they do
require knowledge of the demand schedule unless price changes can be considered inﬁnitesimal.
LV and P V have therefore a computational advantage when compared to the other methods but
we saw that these two approaches represent the boundaries of welfare effects estimations and
are, therefore, extreme approximations, particularly when price changes are large.
Below we ﬁrst provide a simple approach to the exact estimation of the LV and P V methods
using index number theory. We then illustrate estimations of CS , EV and CV based on known
demand functions (demand functions methods). Next, we propose a simple method based on the
own price elasticity (elasticity method). This section is followed by sections on approximation
methods including the Taylor, Vartia, Breslaw and Smith and other numerical approximations.
The section on Taylor’s approximations will also address the questions of how these approxima-
tions can be reconciled with the demand functions methods and decomposed into income and
substitution effects, two issues that we believe are not entirely clear in the existing literature.
4.1 Index numbers
The LV and P V measures are derived from the Laspeyers and Paasche price indexes. For
the case of changes in prices and quantities of two products (general effect):
LV = (pb a b a a a a a
1 x1 + p2 x2 ) − (p1 x1 + p2 x2 ) (10)
P V = (pb b b b a b a b
1 x1 + p 2 x2 ) − ( p 1 x1 + p 2 x2 ) (11)
If we consider only one product and assume no changes in prices or quantities of the other
product (partial effect), the LV and P V can be simpliﬁed to:
LV = (pb a a
1 − p1 )x = (−)∆p1 x
a
(12)
P V = (pb a b
1 − p1 )x = (−)∆p1 x
b
(13)
which correspond to the formulation provided in the geometric interpretation. The LV for-
mulation is the simplest possible computational case of the welfare effect. It only requires
knowledge of initial quantities and changes in prices, information that is known to any practi-
tioner working with micro data. In all other cases, knowledge or assumptions on the demand
schedules are required.
4.2 Demand functions
A simple shortcut is to make reasonable assumptions on the utility function and derive the
demand curve accordingly. For example, a standard approach in empirical works is to use a
utility function based on Cobb-Douglas preferences (see for example Varian, 1997):
13
1−α
u(x1 , x2 ) = xα
1 x2 (14)
with the following demand functions given an initial budget constraint m:
αm (1 − α)m
x1 = ; x2 = (15)
p1 p2
Based on these assumptions, we can estimate the CS as the change in the area under the
Marshallian demand curve over the change in price. For the change in price of only one product
(partial effect):
pb
1 αm pb
1
CS = dp1 = αm ln a (16)
pa
1
p1 p1
In the case of multiple price changes (general effect), the computation of the CS measure-
ment is more complex due to path dependency. As nicely put by Silberberg (1972): “One can
visualize the path dependence of φ (CS ) by noting that if, say, pi changes, the demand curves for
the other commodities begin to shift at the rate (∂xj /∂pi ), j = i. If, however, some other price
pj changes, the demand for commodity i shifts at the rate ∂xi /∂pj . Since these rates are not in
general equal, the way in which pi and pj are changed - for example, ﬁrst pi then pj or vice-versa
- will affect the areas under the demand curves pi dxi and hence the value of φ = pi dxi
(CS ).” (p. 944). It can be noted here that the path independency condition is always satisﬁed
if the preferences are homothetic. As an example for the Cobb-Douglas preferences, we have
that: (x1 (p, m) + x2 (p, m))dp1 dp2 = (x1 (p, m) + x2 (p, m))dp2 dp1 . Thus, in the case of
multiple price changes, the CS is the sum of the changes generated by each price change:
pb
i
CS = αi m ln (17)
i
pa
i
and this sum can be different depending on which price is changed ﬁrst.
The estimation of EV and CV based on known demand functions requires solving for these
measures an equivalence between pre and post utility functions. For example, the estimation of
EV can be done by solving for EV the following equation:
u x1 (pb , m), x2 (pb , m) = u (x1 (pa , m + EV ), x2 (pa , m + EV )) . (18)
In the case of Cobb-Douglas preferences and multiple changes in prices:
α 1−α α 1−α
αm (1 − α)m α(m + EV ) (1 − α)(m + EV )
= (19)
p1 (1 + dp1 ) p2 (1 + dp2 ) p1 p2
where pb a a
k = pk + ∆pk = pk (1 + dpk ). By solving for EV , we ﬁnd that:
1
EV = m −1 (20)
(1 + dp1 )α (1 + dp2 )1−α
Similarly, for the CV measurement, one can write:
14
u x1 (pb , m + CV ), x2 (pb , m + CV ) = u (x1 (P, m), x2 (P, m)) (21)
which, in the case of Cobb-Douglas preferences and multiple changes in prices, simpliﬁes
to:
CV = m 1 − (1 + dp1 )α (1 + dp2 )1−α (22)
A similar approach can be followed by using alternative forms of demand functions, the
most popular of which are described in Annex. Note that, by deﬁnition, EV and CV are path
independent. For the case of non linear pricing schedule, the EV and CV welfare measurements
of the CD model are reported in the appendix B.
4.3 Elasticity
It is possible to estimate CS with a parsimonious approach that makes use of information
on own price elasticity. In some cases, information about own price elasticity is available. For
example, one may not know the local demand for gasoline in a particular country but information
on own price elasticity may be largely known in the gasoline sector and roughly similar across
countries or at least similar in similar countries. If this information is available and assuming a
simple linear demand curve, one can estimate the consumer’s surplus as follows:
CS = −0.5 (x + (x + ∆x)) ∆p
= −0.5 (x(2 + η ∆p)) ∆p
= −x∆p(1 + 0.5η ∆p) (23)
where the parameter η refers to the global elasticity (∆x/x)/(∆p/p). The usual assumption
when this formula is used is that the global elasticity can be approximated to the local or initial
point elasticity and this implies a moderate change in prices. However, this is often not true.
For example, studies on the impact of subsidies reforms often use known point elasticities at
market prices of other countries to estimate the CS in a particular country using the formula
above. But subsidized prices can be very different from free market prices, sometimes several
folds different, and this implies that the local own price elasticity for free market prices cannot
be applied to the subsidized price.
There is, however, another method to estimate the global elasticity in the absence of in-
formation on the demand curve. Remember that, in the case of homothetic preferences, the
uncompensated cross-price elasticities are nil. Assume that all initial prices are normalised to 1
and that we denote the change in price of good 1 by ∆p. The ratio between the percentage of
change in quantity and that of price, or in short, the own price elasticity is deﬁned as follows:
αm αm
xb − xa p a 1+∆p
− 1 1
η1 = b ∗ = ∗ . (24)
p − p a xa ∆p αm
By performing a ﬁrst simpliﬁcation, we ﬁnd that:
15
1
1+∆p
−1
η1 = , (25)
∆p
or
−1
η1 = (26)
(1 + ∆p)
By using this formula, one can then estimate the CS with only knowledge of initial quantities,
changes in prices and own price elasticity at free market prices as follows
CS = −x∆p(1 − 0.5∆p/(1 + ∆p)) (27)
4.4 Taylor’s approximations
Taylor’s approximations are part of a larger family of expansion methods which include
Laurent’s and Fourier’s expansion methods and which can be used to derive numerous Flexible
Functional Forms (FFF) such as the normalized quadratic, generalized Leontief or translog func-
tions. We focus here on the Taylor’s expansion because of its popularity in economics for the
expansion of utility functions and because of its nice properties in relation to the Cobb-Douglas
function.
Taylor’s approximations are based on Taylor’s theorem which states that a function which
is k -times differentiable can be approximated by a n-order polynomial (with n < k ) by re-
peatedly differentiating the function around the equilibrium starting point. Applied to n-times
differentiable utility functions, this theorem allows approximating changes in utility due to price
changes with a polynomial made of n-order derivatives. The Taylor’s approximation becomes
more precise with higher order approximations but also more demanding in terms of information
required, which implies a trade-off between simplicity and data requirement.
Hicks (1942) already provided a ﬁrst quadratic expansion of utility which he used to derive
the ﬁrst general formulations of the CV and EV variations as follows:
∂xk ∂xk
EV = − xk ∆pk − 1/2 ∆pk ∆pj + 1/2 xk ∆pk ∆pk (28)
k kj
∂pj k
∂m
∂xk ∂xk
CV = − xk ∆pk − 1/2 ∆pk ∆pj − 1/2 xk ∆pk ∆pk (29)
k kj
∂pj k
∂m
and showed that the corresponding partial (one price change) variations can be written as
∂x ∂x
EV = −x∆p − 1/2 (∆p)2 + 1/2x (∆p)2 (30)
∂p ∂m
∂x ∂x
CV = −x∆p − 1/2 (∆p)2 − 1/2x (∆p)2 (31)
∂p ∂m
∂x
which can be reduced to the following if the marginal utility of money is constant ( ∂m = 0)
16
∂x
EV = CV = −x∆p − 1/2 (∆p)2 (32)
∂p
The latter expression is the Marshallian CS and this implies two important results: 1) In
the case of a price change in one product only and with marginal utility of money constant,
CS = EV = CV and 2) in other cases, CV < CS < EV always holds.
Following from the quadratic expansion described above, Hicks (1942) established the fol-
lowing relations:
CV − LV = P V − EV = −1/2 xkj ∆pk ∆pj (33)
kj
which is one half of the total substitution effect induced by a price change and
EV − CV ∂xk
=− ∆pk (34)
LV k
∂m
which purely depends on the income effect. From these relations, Hicks (1942) derived that
- with changes in prices of only one good and with the exception of the case of an inferior good
- LV < CV < CS, < EV < P V as already shown in the geometric interpretation. This result
would also normally apply to the case of multiple price changes provided that the LV is larger
when income is larger.
Moreover, Hicks (1942) shows that - in the case of changes in prices of one item - all mea-
sures can be expressed in terms of LV and substitution (S ) and income (I ) effects:
1
CV = LV + S (35)
2
1
EV = LV + S + I (36)
2
P V = LV + S + I (37)
1 1
CS = LV + S + I (38)
2 2
In essence, the difference between measures is determined by the size of the income and
substitution effects, which, in turn, is determined by the shape of the demand function. It should
also be noted that these results hold for second order approximations and quasilinear preferences
where the differences between CV and EV and between LV and P V are symmetric with respect
to CS .
For the case of homothetic preferences and a single price change, we can also generalize the
formulae above for EV and CV to higher orders of approximation o so that: 8
8
See also Dumagan and Mount (1991) for the general form of preferences and a third order of approximation.
Note that, for the rest of the paper, the higher orders refers to the case of homothetic function (i.e. ∂x
∂m = αi ).
i
17
o (o−1)
1 ∂x1
EV = −1o 1+ x1 ∆po
1 (39)
i=1
o! ∂m
o (o−1)
o1 ∂x1
CV = −1 1− x1 ∆po
1 (40)
i=1
o! ∂m
Taylor’s expansions have been used since the seminal paper by Hicks in numerous contri-
butions but the interpretation of the terms of approximations of different orders have not been
unanimous. Harberger (1971) used a Taylor’s expansion of a utility function to describe changes
in real incomes and the consumer’s surplus as:
1
∆u = pa
k ∆xk + ∆pk ∆xk + (41)
k
2 k
where the ﬁrst term on the right-hand side is the ﬁrst order change in utility, which is inter-
preted as the change in real income, the second term is the second order change in utility, which
is interpreted as the change in consumer’s surplus, and represents higher order approximations.
Weitzman (1988) provides the same formulation of the Taylor’s expansion and, as Harberger,
identiﬁes the ﬁrst term as the change in real income, but equals the second term to the substitu-
tion effect. Weitzman (1988) also proves that the sum of the two terms is an exact approximation
of the consumer’s surplus provided the correct deﬂator is used and also that the expression can
be reduced to the ﬁrst term for sufﬁciently small price changes.
This particular expression of the welfare change, even when reduced to the ﬁrst term, still
requires knowledge of the demand function as knowledge of both initial and ﬁnal quantities are
necessary to estimate ∆x. This last problem can be treated using Roy’s identity. McKenzie and
Pearce (1976), for example, showed that money-metric changes in utility can be measured using
a Taylor’s series expansion around the initial equilibrium and also noted that money metric utility
is identical to total expenditure when evaluated at the reference prices (See also Slesnick (1998)).
In this case, the marginal utility of income is one and all higher order income derivatives are zero
so that with Taylor’s approximation and Roy’s identity the change in welfare can be represented
as a function of income and price derivatives. In the case of multiple products, the change in
utility is as follows
∆ν (p, m) = − xk ∆pk
k
∂xk ∂xj
−(1/2) − xk ∆pk ∆pj
k j
∂pj ∂m
∂xk
+ 1− ∆pk ∆m
k
∂m
+ (42)
18
where represents higher order terms. Note that when the change in income is nil, the third
term disappears and the equation above is reduced to the same formulation proposed by Hicks
(1942)) for the EV approximation. 9 Moreover, in the case of a price change of only one product
and a ﬁxed nominal income m, ∆p2 = 0 and ∆m = 0. So that, the higher ﬁrst order terms are
nil and the equation above is reduced to the ﬁrst term −x∆p1 . Hence, in this particular case, the
ﬁrst order derivative of the Taylor’s expansion is equivalent to the LV measurement.
Another way to see the same result is the following. Let u(pb , m) be the level of utility after
the price change and me the equivalent income that generates the same level of utility with P a
such that:
u(pa , me ) = u(pb , m) (43)
Assuming other prices constant, we can differentiate with respect to p1 and me as follows:
∂u ∂u
dme = dp . (44)
∂me ∂p1 1
Thus:
∂u
dme = ∂p1 ∂u dp1 . (45)
∂me
By using the Roy’s identity, we ﬁnd that:
dme |me =m = −x1 dp1 (46)
This approach is sometimes referred to as the “marginal approach”, results in the same LV
formula and can be applied to approximate the impact of a price change on wellbeing regardless
of the true form of the utility function.10
The next question we wish to clarify is how the Taylor expansions of various order relate
to the income and substitution effects. We treat this question separately for the ﬁrst order and
higher orders polynomials.
4.4.1 First order
It should be obvious by now that the difference between welfare estimation methods relates
to the relative difference between the income and substitution effects, which in turn derives from
assumptions made on utility and demand functions. It is important therefore to dissect further the
role of income and substitution effects and clarify the relation with the Taylor’s approximations
which, we saw, has been the cause of some confusion in the literature.
Let us start by expressing the change in wellbeing with the ﬁrst order Taylor approximation
in the case of multiple products:
9
Note that McKenzie and Pearce (1976) derive the formulas with changes in nominal income while Hicks
2
considers only the change in prices. For the second term, we have that: ∂ e(p,u)
∂pk ∂pj =
∂hk (p,u)
∂pj . Also, note that the
∂x
k (p,u)
second term is based on the the Slutsky equation, so that: ∂h∂p j
= ∂xk (p,u)
∂pj + xk ∂mj
.
10
See among others: Ahmad and Stern (1984), Ahmad and Stern (1991), as well as the works of Newbery (1995),
Araar (1998), Creedy (1999), Creedy (2001), Yitzhaki and Lewis (1996) and Makdissi and Wodon (2002).
19
1
∆ν (p, m) = − xi ∆pi + = ν 1 (p, m) + 1
(47)
i
where ν 1 (p, m) is the ﬁrst term approximation of the welfare change and 1 the correspond-
ing residual error. In order to explore how this term can be linked to the income and substitution
components (I and S ), we start by recalling the budget constraint:
m= p1 x1 (p, m(p, u)) + p2 x2 (p, m(p, u)) (48)
Assume that all initial prices are normalized to one. By differentiating this equation with
regards to p1 , we ﬁnd that:
∆m = 0 = (∆x1 + ∆x2 ) + x1 ∆p1 + τ 1 (49)
or also:
ν 1 (p, m) = (∆x1 + ∆x2 ) + τ 1 (50)
where τ 1 is the error of ﬁrst order approximation of the budget constraint. The increase in price
of the ﬁrst good must be compensated by a decrease in the sum of consumed quantities. This
ﬁnding is important, since it shows that the welfare change depends on the change in quantities.
Given that the change in quantity implied by a price change can be decomposed into income
and substitution effects (I = ∆xI I S S
1 + ∆x2 and S = ∆x1 + ∆x2 ), we can establish the linkage
between the Taylor welfare change and the income and substitution effects. Let u denotes the
direct utility function and h(p, u) the hicksian demand function. Based on the Slutsky equation,
we have that:
∂x1 (p, m(p, u)) ∂h1 (p, u) ∂x1 (p, m(p, u))
∆p1 = ∆p1 − x1 ∆p1 (51)
∂p1 ∂p1 u=ua ∂m
The ﬁrst term on the right-hand side of equation 51 is related to the substitution effect (S ),
since it expresses the change in quantity when the level of utility is constant, while the second
term is related to the income effect. Therefore, it is not entirely correct to argue that the ﬁrst order
Taylor’s approximation is the income effect only. However, one can show that the substitution
effect of the ﬁrst order Taylor’s approximation can be assumed to be very small. Consider the
substitution and income effects for the case of two goods and a change in price of the ﬁrst good:
∂x1 (p, m(p, u)) ∂x2 (p, m(p, u))
Income eﬀect = − + x1 ∆p1 (52)
∂m ∂m
∂h1 (p, u) ∂h2 (p, U )
Substitution eﬀect = + ∆p1 (53)
∂p1 u=ua ∂p1 u=ua
The question that may arise now is about the importance of each on the two components
with a marginal change in prices. With moderate price changes, we have that: 11
11
When initial prices are equal to 1, a marginal increase in income by one unit will change the bought quantities.
However, the sum of changes in quantities is equal to one.
20
∂x1 (p, m(p, u)) ∂x2 (p, m(p, u))
+ 1 (54)
∂m p=1 ∂m p=1
Thus, the income effect can be simpliﬁed to be equal to: −x1 ∆p1 , which is simply the
Laspayers index. The latter is, in turn, an approximation of the CV measurement. This result
implies that S must converge to zero so that
∂x1 (p, m(p, u)) ∂x2 (p, m(p, u))
∆p1 − ∆p1 . (55)
∂p1 ∂p1
In other words, when the consumer re-optimize the choice after the marginal increase in price
of the ﬁrst good, it is expected that the consumer decreases x1 and increases x2 . This adjustment
is justiﬁed by the relative improvement in utility contribution of the per dollar expenditure on
good 2. However, the improvement in utility through the substitution process, even if we have
an important substitution in quantities, is practically nil. Indeed, with the marginal change in
prices, and when we are close to the consumer equilibrium, the latest consumed units of each of
the two goods will generate the same level of utility. Thus, the improvement in welfare through
the substitution process is practically nil and it can be neglected.
Let S ˜o be the approximations of the income and substitution effects according to the
˜o and I
o Taylor approximation order. Based on our discussion above, we can conclude what follow:
Proposition 1 With the ﬁrst order Taylor approximation and small price changes, the substitu-
tion effect can be neglected and the ﬁrst term is a good proxy of the income effect.
˜1 +
ν 1 (p, m) ≈ I I1
˜1 ≈ 0
S (56)
Proposition 2 With a ﬁrst order Taylor approximation and small price changes CS , CV and
EV are equal.
1 1 1
∆νCS (p, m) = ∆νCV (p, m) = ∆νEV (p, m) (57)
4.4.2 Higher orders
We start from the CS estimation by means of the second order Taylor approximation in the
case of multiple products changes (see McKenzie and Pearce (1976)):
2 ∂xi ∂xi 2
∆νCS (p, m) = ∆pi − (1/2) ∆pi ∆pj + (58)
i
∂pi i j
∂pj
For the CV and EV measurements, and as shown by Hicks (1942), we have that:
2 ∂xi ∂xi ∂xi 2
∆νCV (p, m) = ∆pi − (1/2) + xj ∆pi ∆pj + (59)
i
∂pi i j
∂pj ∂m
21
2 ∂xi ∂xi ∂xi 2
∆νEV (p, m) = ∆pi − (1/2) − xj ∆pi ∆pj + (60)
i
∂pi i j
∂pj ∂m
∂x1
For the case of a single price change and, with λ = ∂m
, we have that:
2 1 1
∆νCS (p, m) −x1 ∆p1 + λx1 ∆p2 2
i + (1 − λ)x1 ∆pi (61)
2 2
˜2 ˜ 2
ICS SCS
Similarly, starting from the decomposition of the second order, one can extend this to the
highest Taylor orders of approximation (o). For the CS measurement and a single price change,
we can write:
o o
1 1
o
∆ν (p, m)CS −1 λ(o−1) x1 ∆po
o
1+ −1o (1 − λ(o−1) )x1 ∆po
1+
o
(62)
i=1
o! i=2
o!
˜o
I ˜o
S
CS CS
This leads to the following propositions:
Proposition 3 With a marginal income change (change by one unit in income), the changes in
consumed goods converge to their expenditure shares: λ → α.
The implication of this proposition is important since, with moderate price changes, the
impact on wellbeing - regardless the initial form of consumer preferences - will converge to
those based on homothetic preferences (Cobb-Douglas for simplicity).
Proposition 4 When the expenditure share of the good is relatively small, the corrected ﬁrst
Taylor approximation term is a good proxy of the income effect, whereas the rest is a good proxy
of the substitution effect.
˜o = I
˜1 − α(o−1)
ICS CS S˜o
(63)
1 − α(o−1) CS
Figure 2 compares the sum of changes in quantities with the Taylor approximations of the
EV, CV and CS using a Cobb-Douglas model with a utility function of the form u (x1 , x2 ) =
1−α
xα1 x2 , m = 100, p1 = p2 = 1. As shown in the ﬁgure, the sum of changes in quantities
converges to the usual welfare measurements when price changes are moderate and the lower
is the expenditure share (α) the higher is the convergence. Thus, for moderate price changes,
and where initial prices are normalized, the change in quantities is close to the change in real
expenditures or welfare. This trick will help us to check for the relevance of the proposed de-
composition in equation (62). As shown in Figure 3, the three estimates of the substitution
component S converge for small price variations, and this for different values of α. With lower
values of α the convergence is faster. This conﬁrms the relevance of the new proposed interpre-
tation of the Taylor terms and their relation with the income and substitution effects. 12
12
We have that:
22
Figure 2: Price and welfare changes
0
α=0.1
α=0.2
−5
Welfare change
α=0.3
−10
Sum of change in quantities
Equivalent variation
Compensated variation
Consumer surplus
−15
0 10 20 30 40 50
The increase in price (in %)
Figure 3: Taylor approximation and the substitution effect
S: Taylor approximation (o = 100)
.1
S: Equivalent variation
α=0.4
S: Compensated variation
Substitution Effect
α=0.2
.05
α=0.1
0
0 2 4 6 8 10
The increase in price (in %)
23
4.4.3 Convergence and the Cobb-Douglas function
For the case of the Cobb-Douglas function (i.e. CS = −αmlog (1 + dp)), the Taylor ap-
proximation of EV , CV and CS measurements is a combination of a set of inﬁnite geometric
series:
o
1
EV = x1 −1o (1 + α1 )(o−1) ∆po
1 (64)
i=1
o
o
1
CV = x1 −1o (1 − α1 )(o−1) ∆po
1 (65)
i=1
o
o
1
CS = x1 −1o ∆po = −x1(ln(1 + ∆p1 ) where o −→ ∞ and p1 −→ 1 (66)
i=1
o 1
As we will discover, the Taylor approximations do not converge if the price change exceeds
the limit or the radius of convergence (R). To ﬁnd this limit, we have to apply the ratio tests as
follow:
−1(o+1) (1+α)o
1 o+1
) 1
= −1o (1+α) (o−1)
= 1 ⇒ |REV | < (67)
REV 1 +α
o o−→∞
−1(o+1) (1−α)o
1 o+1
) 1
= o (o−1)
= 1 ⇒ |RCV | < (68)
RCV −1 (1−α) 1 −α
o o−→∞
−1(o+1)
1 o+1
= −1o = 1 ⇒ |RCS | < 1 (69)
RCS o o−→∞
For instance, for the CS measurement, if the price change exceeds 100%, the Taylor approxi-
mation will give divergent results. To illustrate this, let U = x0.3 0.7
1 x2 , m = 100 and two cases of
price change, as shown in table 1. As we can observe, the additional correction terms become
large when the change in price is outside the range of convergence [-1,1]. In the divergent case
(dp1 = 2.0), we can have unexpected results such as an increase in wellbeing with the fourth
order of approximation.
4.5 Vartia’s approximation
Given that the true demand function can be modeled and estimated with only observed mar-
ket information, one question that arises is whether the marshallian demand function is sufﬁcient
• SCV = ∆xS S
1 + ∆x2 and C = m(1 + ∆p1 )
α
• ∆xS a S a
1 = (αC/(1 + ∆p1 )) − x1 and ∆x2 = (1 − α)C/(1) − x2
Also, we have that:
• SEV = ∆xS S α
1 + ∆x2 and E = m/((1 + ∆p1 ) )
• ∆ xS a S a
1 = (αE/(1)) − x1 and ∆x2 = ((1 − α)E/(1)) − x2
24
Table 1: Cobb-Douglas and the Taylor approximation
Convergent case: dp1 = 0.5 : Divergent case: dp1 = 2.0
Order (o) terms (x1 [−1o 1 o
o ∆p 1 ] ) Cumulative terms terms (x1 [−1o 1 o
o ∆p1 ]) Cumulative terms
1 -15.00 -15.00 -60.00 -60.00
2 3.75 -11.25 60.00 0.00
3 -1.25 -12.50 -80.00 -80.00
4 0.47 -12.03 120.00 40.00
True value -12.16 -32.96
to estimate the CV and the EV measures. One work that addressed this question is that of Haus-
man (1980), which, for some popular marshallian demand functions, contributed to deﬁne the
exact functional form of the CV and EV measures. However, it is the the work of Vartia (1983)
that provides a more general approach to estimate CV and EV measures.
Instead of assuming ex-ante a utility function or trying to derive welfare changes from the
Taylor’s expansion of an unknown utility function, Vartia proposes to work backward, starting
from a known demand function and deriving from this function the utility change. This approach
is based on the theory of revealed preferences and is the opposite of the approach where we ﬁrst
assume a utility function and derive from this function the demand schedules.
Vartia derives ﬁrst the conditions that relate a demand function h(.) to the indirect utility
function V (p, C ) and based on these conditions derives the ﬁrst order differential equation in
the cost (expenditure or money income) function C (.):
∂C (t) ∂pi (t)
= hk (p(t), C (t)) (70)
∂t ∂t
where t refers to the number of the iteration. With knowledge of the demand function, it is
possible to derive x iteratively and estimate ∆C .13 Hence, the CV can be expressed in terms of
sum of marginal changes over a known demand function.
t−1
CV = 1/2 (xk (pt , mt ) + xk (pt−1 , mt−1 , ))(pt
k − pk )) (71)
t k
For the CV measurement we have that pt=0 = pa is the initial price and pt=n = pb is the
ﬁnal price when the number of iterates is n and that pt − pt−1 = dp/n. The Vartia algorithm is
reversible and this enables to estimate the EV at price pt as
n−t n−(t−1)
EV = 1/2 (xk (pn−t , mt ) + xk (pn−(t−1) , mt−1 , ))(pk − pk )) (72)
t k
−t
and mt = m ˆ t−1 . Note that pn
k for t = 0 is simply the price of good k in the ﬁnal period.
Below, we use an illustration from Chen (2009) to introduce the Vartia algorithm and show
how to estimate the compensated income mc (Figure 4). We assume that the price decreases from
p0 to p3 . The compensated income at p1 is the initial income plus the area a0 . However, this latter
is simply approximated to a0 + a1 using intermediately the marshallian demand function for the
estimation. After this ﬁrst iteration, we move to the second iteration to assess the compensated
13
See Bacon (1995) for a simple spreadsheet approach to the use of the Vartia’s method.
25
income at p2 . This is based on the estimated m1 plus the CV from p1 to p2 which is approximated
to the area b0 + b1 . The same procedure is repeated to assess m3 at price p3 . At the end, the
compensated income (m3 ) at p3 is equal to: (m + (a0 + b0 + c0 ) + (a1 + b1 + c1 )). Of course, the
error term (a1 + b1 + c1 ) will converge to zero when the number of iterations is high and the price
change in each iteration becomes inﬁnitesimal. The empirical part will show a computation of
this method.
To illustrate the point of increasing precision of the Vartia algorithm with increasing itera-
tions, we report in table 2 the estimate of the CV and EV measurements for the case of Cobb
Douglas (U = x0 .3 0.7
1 x2 ) preferences and a change in the price of good 1 from 1 to 1.3. As shown
in table 2, starting from 100 iterations, the error measurement becomes very small.
Table 2: CV and EV Estimations with Vartia’s Algorithm
n CV EV
10 -8.3390394 7.5393912
100 -8.2046317 7.5653872
1000 -8.1905470 7.5687568
10000 -8.1891321 7.5691015
100000 -8.1889906 7.5691361
True value -8.1889749 -7.5691399
Figure 4: The Vartia algorithm to compute the CV measure
4.6 Breslaw and Smith’s approximation
The method of computation proposed by Breslaw and Smith (1995) is based on the second
order Taylor approximation, proposed by McKenzie and Pearce (1976). In matrix form, the
26
compensated income at step t is evaluated as follows:
Ct+1 = Ct + x(pt , Ct ) ∆pt+1 + ∆pt+1 ϕt ∆pt+1 (73)
∂x ∂x
where C0 = m, the Slutsky matrix element ϕt [i, j ] = ∂pi,t
j
j,t
− xi,t ∂C and ∆pt+1 = pt+1 − pt .
As indicated in Breslaw and Smith (1995), the estimation of the EV is based on an algorithm
which is quite similar to that of the CV such as: EV (pa → pb ) = −CV (pb → pa ).14
4.7 The ordinary differential equations methods
Welfare change measurements can be expressed as an integral of Ordinary Differential Equa-
tions (ODE). It is assumed that the welfare measurement function, like CV or EV , is continue
and ﬁrst order differentiable with respect to the price (i.e. w = f (p, m)). It follows that the
integral of the function f (p, m) between pa and pb is equal to the welfare change. Mathematics
offers numerous numerical methods mostly based on an adaptive process of estimation with suc-
cessive steps of evaluation. Economists are generally less familiar with these approaches while
they use extensively the Taylor approximation approach, even with large changes in independent
variables (price in our case). However, the Taylor approach can fail to evaluate accurately the
function when the order of approximation is low or when the price change is large.
Numerical approximations can be helpful for several reasons. First, they enable to over-
come the integrability problem, since they use an adaptive process and evaluate the impact for
successive small price changes. This also makes the substitution effect practically nil given the
small price change in each step. Second, they enable to overcome the problem of higher order
derivatives, especially in the case of complex functional forms. This simpliﬁcation derives from
using the ﬁrst derivative to approximate higher orders derivatives. Third, compared to the Taylor
approximation approach, this method is the most relevant in the case of large price changes (see
Lim (2012) for an extensive review of these methods). As an example, we introduce two popular
numerical approximation methods: The Euler method and the Fourth Runge and Kutta method.
The Euler method can be viewed as a solution of a 1st order Ordinary Differential Equations
(ODE). Formally, it is based on the following main function:
wi+1 = wi + hf (pi , wi ) (74)
where h = (pb − pa )/n; pi = pa + ih and pn = pb .
The Fourth Runge Kutta Method (RK4 in short) is based implicitly on the fourth Taylor
approximation:
wi+1 = wi + (1/6)h(ki,1 + 2ki,2 + 2ki,3 + ki,4 ) (75)
where
14
Breslaw and Smith (1995) claim that their algorithm converges faster than the Vartia’s algorithm while the work
done by Sun and Xie (2013) shows the superiority of the latter. We tested the performance of the two algorithms
both programmed in Stata and we found the Breslaw and Smith algorithm to be faster.
27
ki,1 = f (pi , wi )
ki,2 = f (pi+0.5h , wi + 0.5ki,1 h)
ki,3 = f (pi+0.5h , wi + 0.5ki,2 h)
ki,4 = f (pi+h , wi + ki,3 h)
To illustrate how these numerical methods are relevant for our purpose, we return to our sim-
ple Cobb Douglas example where w can be the compensated or equivalent income. Remember
p,m)
that ∂ν (∂p = −x(p, m). Then for the case of the CD function we have that f (p, w) = −αw/p.
More precisely, let α=0.3, pa =1, pb = 1.3, h=0.25 and m0 = w0 =100.
Table 3: Euler and RK4 Methods Simulations
EV CV
Step : i pi = pa + h ∗ i Euler RK4 True value pi = pb − h ∗ i Euler RK4 True value
0 1.00 0.000 0.000 0.000 2.00 0.000 0.000
1 1.25 -5.613 -6.475 -6.475 1.75 4.461 4.085
2 1.50 -10.041 -11.453 -11.453 1.50 9.911 9.008
3 1.75 -13.664 -15.455 -15.455 1.25 16.818 15.131
4 2.00 -16.710 -18.775 -18.775 1.00 26.048 23.091 23.114
The RK4 method is therefore helpful to estimate welfare changes under price changes. How-
ever, it requires the modeling of preferences or at least the marshallian demand functions, like
the Vartia (1983) or the Breslaw and Smith (1995) algorithms.
4.8 Relational approach
Studies on welfare measures have established a number of algebraic relations between these
measures. For example, we showed that Hicks (1942) found that CS , EV and CV could be
derived from LV , I and S so that knowledge of the latter three measures is sufﬁcient to derive
the other three measures (under certain assumptions). Chipman and Moore (1980) have also
shown that the CV and EV can be derived from the CS as follows
CV = (1 − e−CS/m )m (76)
EV = (eCS/m − 1)m (77)
which is useful in that, for example, one could derive CS using the elasticity method or
the Taylor’s approximation described and then derive CV and EV from the formulae above.
Moreover, one can also reverse the Hicks (1942) equations to ﬁnd LV or reverse the Chipman
and Moore (1980) equations to ﬁnd CS . These relations multiply the possibilities of estimating
all ﬁve welfare measures under limited information.
Other established relations between measures that help to set the relative boundaries of these
measures are Willig (1976) and Cory, Gum, Martin, and Brokken (1981). Willig (1976) has
28
developed the following rule of thumb in the case of a single price change. If |η/2m ≤ 0.05|
and CS/m ≤ 0.90, then:
η |CS | CV − CS η |CS |
≤ ≤
2m |CS | 2m
η |CS | CS − EV η |CS |
≤ ≤ (78)
2m |CS | 2m
where η and η refer to the minimum and the maximum income elasticity within the range
of price change. The relative importance between CS and m will depend on the extent of price
change and the expenditure share. Thus, one can derive that CV , EV and CS tend to converge
if the expenditure share is small or in the case of moderate price changes.
Based on the LV and P V methods, Cory, Gum, Martin, and Brokken (1981) provide inter-
esting rules under the assumption on linearity of demand curves. If we denote the error term by
λ, we have that:
CS − LV ∆x 2λ
≤ λ if ≤
|CS | x 1−λ
P V − CS ∆x 2λ
≤ λ if ≤ (79)
|CS | x 1−λ
Therefore, the lower is the price elasticity, and the change in quantity, the lower is the error
of approximation. In the extreme case where the price elasticity is nil, CS = LV = P V .
5 Applications
In empirical works, there is clearly a trade-off between the degree of approximation of true
measures of welfare and the degree of complexity in estimating these same measures. Ideally,
one would want to use the simplest of the measures proposed and the most parsimonious in terms
of data requirements. But, in some cases, this simplicity comes with a cost in terms of estimation
errors. The discussion provided thus far on the various measures proposed indicated that the
critical factor in making a choice between welfare methods and approximation strategies is the
size of the price change which has implications on the size of the income and substitution effects.
For small price changes, some of the parameters used by the various welfare measures converge
to zero and this makes simple measures accurate. For large price changes, key parameters do not
converge to zero and this makes simple measures inaccurate. Therefore, the question we want
to address now is how we can determine what is a small or large price change and what are the
best strategies to follow in empirical research under the two scenarios.
The review of the estimation methods and approximation strategies proposed revealed that
there is only one formulation of the welfare effects that does not require any modelling of pref-
erences or demand schedules and that requires the least amount of information. This is the
−x∆p formula also labelled as the “marginal approach”, which is the LV method or, equiva-
lently, the CS , EV or CV methods under certain restrictions as already discussed. It is therefore
29
instructive to discuss the potential error size of this formula before undertaking more complex
approximations of the welfare effect that invariably lead to demand modelling of some sort.
The question of the error size is, of course, non-trivial. If utility and demand functions
were known, then changes in quantities due to price changes could be calculated with precision
and one could judge which of the estimation methods come closer to reality. But if utility and
demand are latent functions largely unknown to the observer, then we don’t have a benchmark to
measure the true error size. In this case - which is the case for most analysts - the only tests we
can conduct are about the difference between estimation methods for different price increases.
This is what we do in this section treating separately individual and social welfare. Recall that
we treat social welfare as a simple non-weighted aggregation of individual welfare. Hence, the
question of small and large price changes is treated under individual welfare only while we focus
on the question of comparing distributions under the social welfare section.
5.1 Individual welfare
5.1.1 Small price variations
For small price variations, the optimal strategy is to use LV for two good reasons. One is
that this is the measure that requires the least information to be computed and the second is that
estimations produced by other measures converge to the LV measure for small price variations.
The question we want to address here is: what is “small”? What price variations are small
enough to make LV a viable tool. Or, at what point LV and other measures start to diverge
signiﬁcantly. We consider the case of a single price change or multiple price changes and the
role of the Coob-Douglas parameter “alpha”.
Single price change. We can start with an illustrative example comparing LV estimates with
EV estimates derived from Cobb-Douglas preferences in the case of a price change of only
one product. We assume that the “true” utility is deﬁned with Cobb-Douglas preferences with
1−α
U (x1 , x2 ) = xα1 x2 and α = 0.3. Assume also that initially we have income m = 100, and
prices p1 = p2 = 1. Figures 5 and 6 show respectively the impact on welfare and the error size
estimated for price increases between 0 and 50 percent. It is shown that LV provides similar
estimates to EV up to a price increase of around 10 percent while estimates diverge by about 12
percent for price increases around 20 percent (Figure 5). We can also estimate the percentage
error between the two measures -|(LV − EV )/EV |- (Figure 6) and this shows that the error
is about 6 percent for price increases of about 10 percent and doubles to around 12 percent for
price increases around 20 percent. It is also visible that the error is a linear function of the price
increase. As already discussed, the marginal approach LV tends to overestimate the negative
impact on welfare as compared to other measures. This is due to the fact that the marginal
approach essentially captures the income effect but not the substitution effect.
The bias illustrated in Figure 6 is also discussed by Banks, Blundell, and Lewbel (1996).
Using European data, they ﬁnd a bias between 5 and 10 percentage points with a price change
ranging from 10 to 20 percent. They also conﬁrm the negligible impact when price changes are
below 10 percent.
Using the same setting based on Cobb-Douglas preferences, we can now extend comparisons
to the other welfare measures considered. Note that the income effect is approximated by LV
while the substitution effect is given by:
30
Figure 5: Wellbeing and price changes Figure 6: Error size and Price change
0
30
Welfare Impact(in %)
−5
The error size (in\%)
20
−10
10
LV
EV
−15
0
0 10 20 30 40 50 0 10 20 30 40 50
Increase in p1 in (%) Increase in p1 in (%)
∂h1 (p, u) ∂h2 (p, u) α(α − 1) α(1 − α)
S= + ∆p1 = + m∆p1 (80)
∂p1 u=ua ∂p1 u=ua p2
1 p1 p 2
In Figure 7, we show the estimates of the different measurements as well as the income and
substitution effects. First, we can see that |LV | > |EV | > |CS | > |CV | > |P V | holds as
found in the theoretical part. Second, the total effect (I + S ), converges to the CS measurement.
Note that, by construction, the ﬁrst order Taylor approximation (LV ) implies nil impact of the
interaction effect. Further, for moderate price changes, the CS measurement is also a good
proxy of the CV and EV measurements given that the utility or real income earned through the
substitution effect is negligible. It is also evident that divergence across methods occurs around
price increases of 10 percent and above.
In the case discussed above, we only considered changes in prices in one of the two goods
considered. By construction, the ﬁrst order marginal approach avoids the inclusion of the it-
eration effect (i.e. the impact of cross price changes (dpi ∗ dpj )). Therefore, if one wishes
to estimate multiple price changes with the marginal approach, this would be done by sim-
ply adding up the welfare effect for each product so that the total impact would amount to
∆W = − I i=1 xi dpi . However, this is clearly an overestimation of the true effect. That is be-
cause the increase in price of a good increases demand for substitute goods and, consequently,
the price of the substitutes. Note also that, assuming homothetic preferences, the change in con-
sumer surplus takes an additive form across goods (the non compensated cross elasticities are
nil) so that ∆CS = − I i=1 xi log (1 + dpi ).
Multiple price changes. We now consider the case of simultaneous price changes across
multiple products. In addition to the increase in dp1 , assume an increase in the price of good 2:
dp2 = 0.5dp1 . As it can be observed by comparing the results of Figure (8) with those of (7), the
substitution effect is reduced signiﬁcantly after the increase in price of the second good. This is
explained by the reduction in the substitution effect due to the simultaneous increase in prices.
The role of the product’s share (α). Differences between estimation methods based on Cobb-
Douglas preferences also depend on the parameters used. The exponential “alpha” can take any
31
Figure 7: Price and welfare changes
10 0
The impact on welfare
−10
S
PV
−20
CV
CS
EV
LV
−30
0 10 20 30 40 50
The increase in price (in %)
Figure 8: Price and welfare changes
The increase in price of good 2 (in %)
0 5 10 15 20 25
0
The impact on welfare
−20 −10
S
PV
CV
CS
EV
−30
LV
0 10 20 30 40 50
The increase in price of good 1 (in %)
32
value between 0 and 1. To test how different estimation measures spread apart as prices increase
and under different alpha parameters, we compare in Figure 9 the ﬁve measures using α = 0.1
and α = 0.5. As expected, results show that for low values of alpha the different estimation
methods are very close up to 50 percent price increases while they diverge substantially for
high values of alpha already around price increases of 20 percent. Values of alpha are set by
the researcher based on notions that may come from the literature or from data. The important
lesson to keep in mind here is that the higher the alpha parameter the higher the divergence
between estimation methods so that with very low alpha one may still consider to use the LV
approach, even with price increases as large as 50 percent.
Figure 9: Price and welfare changes
0
α=0.1
−5
The impact on welfare
−15 −10
LV
α=0.5
EV
−20
CV
CS
PV
−25
0 10 20 30 40 50
The increase in price (in %)
This section has shown that the answer to the question of what is a small price change that
would sanction the use of LV as the preferred measure is not straightforward. With a given
Cobb-Douglas demand function, it depends on whether we model one or more price changes
and on the size of α in addition to the size of the price change.
5.1.2 Large price variations
For large price variations, the choice between measures becomes important as these mea-
sures diverge signiﬁcantly in estimating the welfare effect of a price change. This also means
that it is important to evaluate the role of the choice of demand function and the choice of alter-
native approximation methods such as Hicks’s or Taylor’s methods. This is what we explore in
this section.
Demand systems. So far, we used a Cobb-Douglas demand framework to assess differences
across measures but results and conclusions may have been different had we used different
forms of demand functions. To address this question, we compare EV estimations derived from
a Cobb-Douglas model with the ones derived from other popular demand systems including
the Linear Expenditure System (LES), the Almost Ideal Demand System (AIDS), the Quadratic
Almost Ideal Demand System (QUAIDS) and the Exact Afﬁne Stone Index (EASI) with a six
33
order polynomial expansion.15 As a benchmark, we also report results for LV .
For this purpose, we use data from the 1987/88 Nationwide Food Consumption Survey, which
is conducted by the United States Department of Agriculture. Demands for four categories of
food are estimated: meats, fruits and vegetables, breads and cereals, and miscellaneous. In
Figure 10, we assess the impact of a potential increase in the price of meats. Note that for
the Cobb Douglas model, the compensated own-price and cross-price elasticities for good i are
(αi − 1) and αj respectively (αl is the expenditure share of good l).16 Of course, this may
- slightly - underestimate the welfare effect because of the overestimation of the substitution
effect component.This is also conﬁrmed by the low estimated compensated cross-elasticities
with the QUAIDS model compared to those of the Cobb-Douglas model.
Figure 10 shows that the marginal approach (LV) overestimates the impact on welfare as
compared to all EV estimations irrespective of the demand model used. For instance, when
the price of meat quadruples, the impact is -80 USD, which is even higher than the maximum
reasonable impact not exceeding total expenditures (less than 50 USD on average), in the case
of perfect complement goods and non substitutable goods. Overall, the demand models show
a clear ranking with the Cobb-Douglas model providing smaller estimates as compared to the
EASI, QUAIDS, AIDS and LES models. All models provide very close estimates diverging
of a few percentage points only for very high price changes. QUAIDS, AIDS and LES are
particularly close even at price increases of 400 percent.
We shall conclude that the choice of the demand system is not a major discriminatory factor
in determining differences in welfare estimations and that the use of the simple Cobb-Douglas
model is reasonable even for very high price increases with the caveat that it may underestimate
the welfare effect as compared to other more sophisticated demand models. However, given the
simplicity of the Cobb-Douglas model and its nice properties in relation to the share of goods
consumed over total consumption, to its capacity to estimate easily all measures of welfare
change and in relation to the Taylor approximations, this model remains, in our view, the model
of choice.
Other approximation methods. The next question we want to address is how other esti-
mation methods including the elasticity method, Hicks or Taylor approximations of various
degrees compare to the Cobb-Douglas demand method. We start by comparing the Cobb-
Douglas method with the Elasticity and Taylor approximations (Figure 11). Consider the fol-
lowing example. Suppose that the true preferences are Cobb-Douglas and that we consider as
“errors” the differences from these estimates. With initial prices normalized to one, we have
CS = −mαlog (1 + dp1 ). Now assume that the price of the ﬁrst good doubles and that the
global elasticity of the linear demand curve is equal to -0.5. Based on the approximation of
equation (23), we ﬁnd that CS = −22.5 whereas the true (Cobb-Douglas) value is equal to
CS = −20.79. This error is evidently explained by the assumption of the linear demand curve
and it can become quite large as shown in Figure 11. We can also see that estimates made with
the elasticity method are rather close to the Cobb-Douglas demand method up to 100 percent
price increases. The error size can be better appreciated in Figure 12. The elasticity method
results in a rather good approximation of the Taylor’s higher order estimate and converges to
15
See Deaton and Muellbauer (1980a), Banks, Blundell, and Lewbel (1997), Lewbel and Pendakur (2009) and
Appendix A.
16
See for instance Ramskov and Munksgaard (2001).
34
Figure 10: Restricted information and welfare measurement
0 −20
The impact on welfare
−40 Marginal Approach
Cobb−Douglas Model: EV
LES Model: EV
−60
AIDS Model: EV
QUAIDS Model: EV
EASI Model: EV
Maximum expected impact
−80
0 100 200 300 400
The increase in price (in %)
this value around price increases of 100%. Hence, the elasticity method can be considered as a
viable simple alternative to Cobb-Douglas preferences up to very large price changes.
Figure 11: Cobb-Douglas Vs. Elasticity Figure 12: Cobb-Douglas Vs. Elasticity
and Taylor Methods and Taylor Methods (Error size)
0
EL (Elasticity method)
40
T1=LV (Taylor order 1)
T2 (Taylor order 2)
−2
T10 (Taylor order 10)
30
Error size (in %)
Welfare Impact
−4
20
−6
CS (Cobb−Douglas)
10
EL (Elasticity method)
−8
T1=LV (Taylor order 1)
T2 (Taylor order 2)
T10 (Taylor order 10)
−10
0
0 20 40 60 80 100 0 20 40 60 80 100
Increase in price (in %) Increase in price(in %)
We now test the performance of the Hicks approximations as compared to the Cobb-Douglas
demand function using CV and EV measures (Figure 13). The Cobb-Douglas model is u =
x0 .3 0 .7 a a
1 x2 , p1 = p2 = 1 and m = 100. For moderate price variations (say up to 20 percent),
the second order approximation proposed by Hicks (1942) is enough to approximate the Cobb-
Douglas CV and EV measurements. For large price variations, we clearly need to use higher
order approximations. As shown, in the right-hand panel of the ﬁgure, a tenth degree order of
approximation results in estimates that are very close to those provided by the Cobb-Douglas
demand function but this is true only for price variations up to about 20%. Hence, Taylor’s
35
approximations, even of high degree, diverge from Cobb-Douglas estimations early on the scale
of possible price increases.
Figure 13: Taylor approximation and welfare change
Order of approximation = 2 Order of approximation = 10
0
0
The impact on welfare
The impact on welfare
−5
−5
−10
−10
CV (Cobb−Douglas) CV (Cobb−Douglas)
CV (Hicks) CV (Hicks)
EV (Cobb−Douglas) EV (Cobb−Douglas)
EV (Hicks) EV (Hicks)
−15
−15
0 20 40 60 0 20 40 60
The increase in price (in %) The increase in price (in %)
5.1.3 A choice tool for practitioners
We are now in a position to provide more precise indications on choices of measures and
computational methods for practitioners. Table 4 summarizes welfare measures, computational
methods and minimum requirements for all measures we considered in this paper. Using these
measures, we can simulate alternative sets of price changes and compare results. For all estima-
tions that require modelling preferences, we use a Cobb-Douglas function with four parameters
described as
u(x1 , x2 ) = x0.3 0.1 0.2 0.4
1 x2 x3 x4 (81)
This represents a utility function of an individual or household with a four products con-
sumption basket and the relative weight of each product represented by the Greek letter expo-
nential. We consider six alternative scenarios for small, medium and large price increases. Sce-
narios 1-3 consider only one price change while scenarios 4-6 consider multiple price changes.
Price changes for the four products considered are reported at the top of Table 5. For all simu-
lations, we consider household expenditure m = 100. Results are shown in Table 5 and can be
summarized as follows:
• The difference between measures is small for small price variations (S1 and S4), whether
it is one or more price variations;
• The difference between measures becomes large for medium price variations (S2 and S5).
Relative to the initial expenditure m = 100, the difference between the lower and upper
bounds (PV and LV) is 5% for scenario 2 and 10% for scenario 5 with a price increase of
50% (one price or multiple prices);
• The difference between measures is extremely large and also non-sense for large price
increases (above 100%, S3 and S6). Results show that the welfare effects can be larger
36
than initial expenditure, which is a non-sense, and also varies enormously across measures
and computation methods. For some methods, even the sign changes. As a general rule,
linear approximations, as that of Taylor or CS-elasticity start to give odd results when the
change in price exceeds the unity. For the Taylor approach, we can recall the two main
sources of errors:
1. The truncation error which is related to the higher orders excluded from the estima-
tions;
2. The convergence error when the change is outside of the range of convergence of a
speciﬁc approximated function.
In the case on the Cobb-Douglas model (and for many other preference models), the Tay-
lor approximation generates an inﬁnite geometric series with a bounded range of conver-
gence, as shown in sub-section 4.4.3. This explains the odd results that we obtain with the
Taylor approximation where the price change exceeds the unity.
• The one step Taylor approximation requires higher orders when the price changes con-
verge to the unity;
• The Breslaw and Smith (1995) approach is the most efﬁcient and the closer to true values
since it uses the numerical evaluation of the Slutsky matrix - together with the second
order Taylor approximation, Note how well it performs also for very large price increases;
• In general, the measurements that requires some information related to the function form
of preferences give more precise results, as is the case for Vartia, Breslaw, Euler and RK4
methods;
• The RK4 method shows a good performance, even if the price changes are relatively large;
• When the price changes are moderate (less that 10% or even until 20%) the practitioner
can use any simple approximation method and the LV is considered to be a good proxy
of the CV ;
• When the price change exceeds unity, the error attached to the linear approximation be-
comes very large and methods based on linear approximations are inadeguate. We saw
that the particular shape of the non linear preferences function is not important and that a
simple Cobb-Douglas function is sufﬁcient;
We now construct a simple contour plot depicting the divergence (convergence) of estima-
tions resulting from different methods using the two most relevant discriminatory factors: price
change and expenditure share. As a measure of variation across methods, we use the difference
between the upper (LV) and lower (PV) bounds of the measures normalized by initial expendi-
ture (m = 100) and call this measure GAP. This provides a simple but accurate visual device
that can be used by practitioners to take decisions on when to use simple LV estimations and
when, instead, it is necessary to use more complex estimation procedures. Figure 14 shows the
contour map for one price change (Panel A) and two price changes (Panel B). The different
coloured thresholds represent the GAP measure along 5% cut points. For example, if there is
37
one price shock (Panel A) of 20% on a product that represents 10% of the consumption basket of
a household, then the GAP is largely below 5% of income. It can be noted here that, the higher
is the expenditure share, the lower is the GAP. This is mainly explained by the low substitution
effect (low compensated cross-elasticities) with the high expenditure share. It is natural that the
GAP will depend on the expenditure share and will not exceeds it. Panel B in Figure 14 shows
similar estimates but when we consider two price changes such that, α2 = 0.2 and dp2 = 0.5dp1 .
Table 4: Summary of Welfare Measure, Computation Methods and Functional Requirements
Welfare change measurement Equation Requirements
LV 12 Initial quantities/expenditures
PV 13 Marshallian demand function
CS: True value 17 Marshallian demand function
CS: Elasticity approximation 27 Initial quantities/expenditures
CS: TAYLOR (order=2) 58 Marshallian demand function
CS: TAYLOR (order=3) 62 Marshallian demand function
EV: True Value 20 Indirect utility function
EV: Taylor (order=2) 60 Marshallian demand function
EV: Taylor (order=3) 39 Marshallian demand function
EV: Vartia (10 iterations) 72 Marshallian demand function
EV: Breslaw and Smith (10 iterations) 73 Marshallian demand function/numerical derivatives
EV: Euler (10 iterations) 74 1st order derivative of the welfare function
EV: RK4 (10 iterations) 75 1st order derivative of the welfare function
CV: True Value 22 Indirect utility function
CV: Taylor (order=2) 59 Marshallian demand function
CV: Taylor (order=3) 40 Marshallian demand function
CV: Vartia (10 iterations) 71 Marshallian demand function
CV: Breslaw and Smith (10 iterations) 73 Marshallian demand function/numerical derivatives
CV: Euler (10 iterations) 74 1st order derivative of the welfare function
CV: RK4 (10 iterations) 75 1st order derivative of the welfare function
38
Table 5: Welfare Impact Simulations with Different Measures, Computation Methods and Sce-
narios
S1 S2 S3 S4 S5 S6
Price changes
dp1 0.1 0.5 5 0.1 0.5 2.5
dp2 0 0 0 0.05 0.5 5
dp3 0 0 0 0.025 0.5 10
dp4 0 0 0 0 0 0
Measures
LV -3.00 -15.00 -150.00 -4.00 -30.00 -325.00
PV -2.73 -10.00 -25.00 -3.69 -20.00 -47.94
CS True Value -2.86 -12.16 -53.75 -3.84 -24.33 -103.46
CS Elasticity -2.84 -9.38 2100.00 -3.82 -18.75 11753.13
CS TAYLOR 2 -2.85 -11.25 225.00 -3.83 -22.50 893.75
CS TAYLOR 3 -2.86 -12.50 -400.00 -3.84 -25.00 -121.88
EV True Value -2.82 -11.45 -41.58 -3.77 -21.59 -64.46
EV TAYLOR 2 -2.81 -10.13 337.50 -3.75 -18.00 1421.88
EV TAYLOR 3 -2.82 -12.24 -718.75 -3.77 -22.01 -159.11
EV Vartia -2.80 -11.16 -37.21 -3.75 -21.29 -60.46
EV-Breslaw -2.82 -11.45 -41.23 -3.77 -21.59 -64.01
EV-Euler -2.80 -11.15 -35.60 -3.75 -21.19 -55.49
EV-RK4 -2.82 -11.45 -41.58 -3.77 -21.74 -66.11
CV True Value -2.90 -12.93 -71.18 -3.92 -27.54 -181.39
CV TAYLOR 2 -2.89 -12.38 112.50 -3.91 -27.00 365.63
CV TAYLOR 3 -2.90 -12.76 -81.25 -3.92 -27.99 -84.64
CV Vartia -2.92 -13.31 -85.45 -3.93 -28.02 -214.54
CV-Breslaw -2.90 -12.93 -69.74 -3.92 -27.54 -174.95
CV-Euler -2.92 -13.29 -84.04 -3.93 -28.02 -207.57
CV-RK4 -2.90 -12.93 -71.18 -3.91 -27.31 -170.65
39
Figure 14: Contour Map of the GAP Estimations by Price Changes and Expenditure Shares
Panel A: One price change Panel B: Two price changes
500
500
0 0
−.05 −.05
−.1 −.1
−.15 −.15
400
400
−.2 −.2
−.25 −.25
−.3 −.3
−.35 −.35
−.4 −.4
300
300
−.45 −.45
−.5 −.5
−.55 −.55
−.6 −.6
(LV−PV)/m
(LV−PV)/m
−.65 −.65
200
200
−.7 −.7
Price change (in %)
Price change (in %)
−.75 −.75
−.8 −.8
−.85 −.85
−.9 −.9
100
100
−.95 −.95
−1 −1
0
0
0 .2 .4 .6 .8 1 0 .2 .4 .6 .8
40
Expenditure share (alpha) Expenditure share (alpha)
Table 6 provides a numerical example of what we showed in Figure 14 (Panel B). It is
evident that the estimates of the GAP increase quickly as we move along rows from moderate
price increases to large price increases. The GAP also increases as the share of expenditure on
the product considered (α) increases but the pace of increase is much more modest. As it is clear
by now, α and δp are the two key parameters to watch when making a choice between LV and
more complex measures with δp being the single most important factor.
Table 6: The normalised GAP Estimations by Price Changes and Expenditure Shares
The change in price dp1
α1 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
.05 -0.0011 -0.0048 -0.0106 -0.0182 -0.0273 -0.0378 -0.0495 -0.0622 -0.0758 -0.0902
.1 -0.0015 -0.0063 -0.0139 -0.0237 -0.0354 -0.0488 -0.0636 -0.0796 -0.0967 -0.1148
.15 -0.0019 -0.0078 -0.0171 -0.0291 -0.0435 -0.0597 -0.0777 -0.0970 -0.1177 -0.1394
.2 -0.0023 -0.0094 -0.0204 -0.0346 -0.0515 -0.0707 -0.0917 -0.1145 -0.1386 -0.1641
.25 -0.0026 -0.0109 -0.0236 -0.0401 -0.0596 -0.0816 -0.1058 -0.1319 -0.1596 -0.1887
.3 -0.0030 -0.0124 -0.0269 -0.0456 -0.0676 -0.0926 -0.1199 -0.1493 -0.1805 -0.2133
.35 -0.0034 -0.0139 -0.0302 -0.0510 -0.0757 -0.1035 -0.1340 -0.1668 -0.2015 -0.2379
.4 -0.0037 -0.0154 -0.0334 -0.0565 -0.0837 -0.1145 -0.1481 -0.1842 -0.2225 -0.2626
.45 -0.0041 -0.0169 -0.0367 -0.0620 -0.0918 -0.1254 -0.1622 -0.2016 -0.2434 -0.2872
.5 -0.0045 -0.0185 -0.0399 -0.0674 -0.0999 -0.1363 -0.1763 -0.2191 -0.2644 -0.3118
.55 -0.0049 -0.0200 -0.0432 -0.0729 -0.1079 -0.1473 -0.1903 -0.2365 -0.2853 -0.3364
.6 -0.0052 -0.0215 -0.0465 -0.0784 -0.1160 -0.1582 -0.2044 -0.2539 -0.3063 -0.3611
.65 -0.0056 -0.0230 -0.0497 -0.0839 -0.1240 -0.1692 -0.2185 -0.2714 -0.3272 -0.3857
.70 -0.0060 -0.0245 -0.0530 -0.0893 -0.1321 -0.1801 -0.2326 -0.2888 -0.3482 -0.4103
.75 -0.0063 -0.0260 -0.0562 -0.0948 -0.1401 -0.1911 -0.2467 -0.3062 -0.3691 -0.4349
.8 -0.0067 -0.0276 -0.0595 -0.1003 -0.1482 -0.2020 -0.2608 -0.3237 -0.3901 -0.4596
.85 -0.0071 -0.0291 -0.0628 -0.1057 -0.1563 -0.2130 -0.2749 -0.3411 -0.4111 -0.4842
.9 -0.0075 -0.0306 -0.0660 -0.1112 -0.1643 -0.2239 -0.2889 -0.3585 -0.4320 -0.5088
.95 -0.0078 -0.0321 -0.0693 -0.1167 -0.1724 -0.2349 -0.3030 -0.3760 -0.4530 -0.5334
1 -0.0082 -0.0336 -0.0725 -0.1222 -0.1804 -0.2458 -0.3171 -0.3934 -0.4739 -0.5581
5.2 Social welfare
So far, the paper has discussed differences across welfare estimation methods at the individ-
ual -household- level. When we stated the basic assumptions of this paper, we also mentioned
that we consider social welfare as the sum of individual welfare, which makes the formulation
of all social welfare measures straightforward. Individual welfare effects due to price changes
will be different across individuals whatever measure we use because, as a minimum, initial
consumption and the share of products in the consumption basket is different across individuals.
Therefore, we should expect to have a distribution of individual welfare effects for any of the
measures considered in this paper.
In this section, we maintain the initial assumptions and we will not attempt to address issues
of heterogeneity of the impact across individuals. However, working with a distribution of wel-
fare changes, as opposed to individual welfare changes, opens the possibility of using statistical
inference for testing differences across social welfare distributions. We can also test statistical
differences across measures for quantiles rather than for the entire distribution and we can start
41
discussing how different welfare estimation methods differ for different parts of the welfare
distribution. This is important because, as we stated in the introduction, this paper is mainly
concerned about poor people and poor countries. Moreover, by shifting the analysis to the dis-
tribution of welfare, we can also borrow from welfare economics and the very rich literature on
distributional analysis and use instruments such as stochastic dominance or pro-poor curves to
reﬁne our assessment of the difference between welfare estimation measures. This is what this
section is about. We will start by introducing a possible strategy to estimate standard errors at
the social and quantile levels and we will ﬁnish by introducing pro-poor curves for comparing
welfare estimation methods.
5.2.1 Statistical inference (Society)
The questions we address are the following: Are the differences between means of welfare
estimation methods statistically signiﬁcant? And can these statistical tests help us to discrimi-
nate between welfare estimation methods? Most analysts work with sample surveys and welfare
estimates such as LV or CS will contain a sampling error. For each sampled household i, one
can estimate the net impact on wellbeing: ∆νi = f (∆p1 , ..., ∆pk ; ∆m = tri ) where tri refers
simply to the transfer to the individual i. Since the required information to assess νi comes
only from the sampled households i, the variable ∆ν will be i.i.d. In general, the sample size
of national household surveys is large and, based on asymptotic theory and the case of large
sample size, the distribution of any estimator will converge to the normal distribution. We need,
therefore, to start by ﬁnding a proper approach to the estimation of the standard error of the
difference between means of welfare estimation methods.
Suppose that we want to estimate the mean difference between LV and P V , the upper and
lower bounds of our welfare estimation methods. Let ∆P V,LV denote the estimator of the mean
difference, N the simple size and I the number of goods so that
∆P V,LV = µP V − µLV (82)
N I
1
= −xb a
j,i dpi − −xj,i dpi (83)
N j =1 i=1
I
= dpi (µxa
i
− µxb
i
) (84)
i=1
where µx denotes the average of x. For the case of Cobb-Douglas preferences, we have that
xb a
i = xi /(1 + dpi ) where the initial price is assumed to be equal to 1. In this case, for multiple
price changes we have
I
1
∆P V,LV = µ xa − µ xa dpi (85)
i=1
i i
(1 + dp)
I
dp2
i
= µ xa . (86)
i=1
1 + dpi i
42
It follows that the standard error of the mean difference for a single price change can be
estimated as
dp2
1
σ∆P V,LV = σµ a (87)
1 + dp1 x1
Similarly, if we wish to estimate the standard error of the mean difference between CS with
Cobb-Douglas preferences and LV we can derive the estimation of the standard error as follows
σ∆CS,LV = (dp1 − log (1 + dp1 )) σµxa (88)
1
Since the estimator of ∆CS,LV uses two basic estimators (µCS and µLV ), the delta method
can be used to estimate the standard error (see Rao (1973)).
Single price change. We are now in a position to test statistically the means difference be-
tween welfare estimation methods at the societal level. To illustrate these tests, we use data from
the 1987/88 Nationwide Food Consumption Survey, which is conducted by the United States
Department of Agriculture. We simulate price increases between 0 and 100% of Meats and test
means differences between P V , EV , CS and LV in turn using 95% conﬁdence intervals. As
shown in Figure 15, differences across welfare estimation methods are visible starting from price
increases in between 10% and 20% while conﬁdence intervals remain rather narrow until price
increases of 100%. For example, with a price increase of 85%, the mean difference between
P V and LV is around 8 percentage points but the conﬁdence interval around that difference is
a fraction of a percentage point at the 95% conﬁdence level. In other words, conﬁdence inter-
vals do not really put into question the remarkable difference across welfare estimation methods
already very evident beyond price increases of 10-20%.
Multiple price changes. Of course, price changes may be more complex than one simple
price increase and may include increases as well as decreases in prices. Using the same set of
data, assume that the price reform concerns two goods with a decrease of 45% in the price of
Fruits and vegetables and an increase between 0% and 100% in the price of Meats. As shown
in Figure 16, two main lessons can be drawn. First, the judgment on the impact of the price
reform on welfare can depend on the selected measure of welfare change. For instance, for an
increase of 60% in the price of Meats, only the P V measurement indicates an improvement in
welfare. The second lesson concerns the gray area for which one cannot make any judgment
about the nature of the impact. For instance, for the range [83%, 90%] of increase in the price of
Meats, the impact measured by P V is not signiﬁcantly different from 0. In essence, statistical
inference can help in determining whether the impact of one welfare measure is signiﬁcantly
different from zero but beyond price increases of 10-20% conﬁdence intervals are too narrow to
overlap across measures. Hence, all the empirical results reported for individual welfare effects
and related to differences between welfare measures largely apply to mean social welfare.
43
Figure 15: The difference between welfare Figure 16: The welfare measurements and
measurements the sampling errors
10
10
DIF1: PV−LV Confidence interval of PV−LV (95 %)
DIF2: EV−LV Confidence interval of EV−LV (95 %)
DIF3: CS−LV Confidence interval of CS−LV (95 %)
5
8
0
6
−5
4
EST1: PV C.I. of PV (95 %)
−10
EST2: CS C.I. of CS (95 %)
2
EST3: LV C.I. of LV (95 %)
EST4: EV C.I. of EV (95 %)
EST5: CV C.I. of VV (95 %)
−15
0
0 20 40 60 80 100 0 20 40 60 80 100
Price change (in %) Price change (in %)
5.2.2 Statistical inference (Quantiles)
Quantiles are sub-sets of samples (populations) and, by deﬁnition, smaller sets than samples
(populations), which makes statistical inference more complex. In this section, we provide
one possible approach to estimate the standard error of the means difference between welfare
estimation methods by quantile.
Let τ1 and τ2 be two successive percentiles deﬁning a given population group (for instance,
the second quintile is the one with incomes y > Q(τ1 = 0.2) and y ≤ Q(τ2 = 0.4) and Q(τ )
is the quantile at percentile τ . The average impact of a price reform on the welfare of group
g (τ1 , τ2 ) is deﬁned as follows:
N −1 ˆ (τ2 )] − N −1
ωi yi I [yi ≤ Q ωi yi I [yi ≤ Qˆ (τ1 )] ˆ2 − α
α ˆ1
ˆ(τ1 , τ2 ) =
µ = (89)
N −1 ωi I [yi ≤ Qˆ (τ2 )] − N −1 ˆ
ωi I [yi ≤ Q(τ1 )] ˆ ˆ
β2 − β1
where I [true] = 1 and zero otherwise, ωi is the ﬁnal weight (for instance, the product of sam-
pling weights times household size).
This estimator is a function of four basic estimators (α∗ ,and β∗ ). For simplicity, let α ˆ1 =
N −1 ai and ai = ωi yi I [yi < Q ˆ (τ1 )]. As we can observe, the elements of the new generated
variable (a) are not i.i.d, since each element (ai ) is based on another estimate (Qˆ (τ1 )) and this
implicit variability will affect the variance of our basic estimator.
One way to estimate accurately the standard error is by correcting for this implicit variability
using the Rao (1973) approach. In what follows, we deﬁne the variance of the estimators α1 and
β1 (those of α2 and that of β2 are quite similar) as
ˆ1 )
V ar(α ˆ (τ1 )) I [Q
ˆ (τ1 )] − F (Q
= N −1 ST D ω yI [y ≤ Q ˆ (τ1 )
ˆ (τ1 ) ≤ y ] − τ1 /f (Q (90)
ˆ1 )
V ar(β ˆ (τ1 )) I [Q
ˆ (τ1 )] − F (Q
= N −1 ST D ω I [y ≤ Q ˆ (τ1 ) ≤ y ] − τ1 (91)
44
Table 7: Estimated statistics for the fourth quintile
Estimator Estimated value
Estimated average 8503.57
STE: Analytical approach (without considering the implicit variability) 20.26
STE: Analytical approach 67.37
STE: Bootstrap (number of replications = 400) 68.57
where ST D(V ) is the standard deviation of the variable V , and f (.) and F (.) denote respectively
the density function and its cumulative.
Is the estimated standard error of µ(τ1 , τ2 ) signiﬁcantly different form that when we
omit to take into account the implicit variability, i.e. when we assume that: V ar(α ˆ1 ) =
−1
N ST D yI [y ≤ Q ˆ (τ1 )] ? To answer this question and to validate the proposed method for
the estimation of the STE, we use a sample of 6000 randomly generated observations.17 Table
7 shows the estimates for the forth quintile. The estimated STE with the bootstrap approach is
close to that estimated with the proposed analytical approach. Instead, omitting the correction
for the implicit variability will give a wrong estimate of the STE.
5.2.3 Pro-poor curves and stochastic dominance
Having a method for statistical inference at the quantile level, we can now use it to draw
and compare pro-poor price reforms curves. There are different interpretations of what pro-poor
means and this has been the object of debate in the welfare economics literature with some
arguing that pro-poor means that the poor outperform the non-poor and other arguing that a pro-
poor reform is one that simply makes the poor better off, whether they outperform the non-poor
or not (Ravallion and Chen (2003), Kakwani and Pernia (2000) Son (2004), Duclos (2009)).
Irrespective of these different views, assessing pro-poorness implies comparing welfare impacts
across quantiles and this can be done with what is known as pro-poor curves. One can then
use a standard stochastic dominance approach to determine whether one distribution dominates
another or not. In our case, each pro-poor curve can be constructed using the pre and post
price reform for each welfare estimation method. We can then check for the pro-poorness of the
impact with the different welfare measurements by using stochastic dominance theory. Recall
that a pro-poor curve for a welfare estimation is ﬁrst order welfare dominant if and only if:
Qb (τ ) − Qa (τ )
λ1 (τ ) = − >0 ∀τ ∈ [0, τ ∗ ] (92)
Qa (τ )
where is simply the average impact of the price reform and τ ∗ is a given critical percentile.
The critical value marks the limit of the test, which is simply the highest possible proportion
of poor. Recall here that, among the Social Welfare Functions (SWFs) that obey to the ﬁrst
order dominance constraint, some do not attribute more importance to the poor. For unequivocal
comparisons, we need a second order dominance test as follows:
17
We used the Stata command gen imp wel = n*(3+0.05*uniform())
45
C b (τ ) − C a (τ )
λ2 (τ ) = − >0 ∀τ ∈ [0, τ ∗ ] (93)
C a (τ )
where C (p) is simply the Generalised Lorenz curve (see for instance Son (2004)). The
estimator of the variance of the quantile Q(τ ) and the generalised Lorenz curve can be deﬁned
as follows:
Vˆ
ar((Q(τ ))) = N −1 ˆ (τ ))
ˆ (τ )] − τ /f (Q
I [yi ≤ Q (94)
Vˆ
ar((C (τ ))) = N −1 ˆ (τ )]
τ Q(τ ) + (yi − Q(τ )) I [yi ≤ Q (95)
To illustrate these instruments, we use again data from the 1987/88 Nationwide Food Con-
sumption Survey. Assume that the price reform include an increase of 20% in the price of Meats
and a reduction of 20% in the price of Fruits and vegetables. Figures 17 and 18 show ﬁrst and
second order dominance of the pro-poor curves constructed with the LV and CS methods, the
latter with Cobb-Douglas preferences. As one can observe, the simulated price reform is pro-
poor, and this, even if the change in prices does not practically impact government revenues (the
curve averages around zero). It can also be noted that the LV and CS curves largely overlap
indicating that there is little difference between measures for this particular price reform. In this
case, the use of LV would be appropriate and also simplify computation.
As we have learned by now, the differences across measures diverge as price increase. We
therefore repeat the exercise with a large price increase, an increase in the price of meats of
100%. Figures 19 and 20 show that the LV and CS measures provide very different results and
the difference is greater for lower quantiles (the poor). The reason for the difference between
the LV and CS measures is the fact that the share of expenditure represented by the product
affected by price changes varies across the welfare distribution and this share is used by the CS
method (alpha in the Cobb-Douglas formula) but not by the LV method. This is one more reason
to be very weary of the differences across measures. As prices increase, the differences across
measures increase, they are signiﬁcant and they can be larger for lower parts of the distributions.
This makes estimates of the welfare effects on poverty even more complex than estimates for an
entire society.
46
Figure 17: Fist order pro-poor price reform Figure 18: Second order pro-poor price re-
curve form curve
.03
With LV With CS With LV With CS
.06
C.I. with LV (95%) C.I. with CS (95%) C.I. with LV (95%) C.I. with CS (95%)
.04
.02
.02
.01
0
0
−.02
−.01
−.04
0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1
Percentiles Percentiles
Figure 19: Fist order pro-poor price reform Figure 20: Second order pro-poor price re-
curve form curve
.2
With LV With CS
0
C.I. with LV (95%) C.I. with CS (95%)
.1
−.05
0
−.1
−.15
−.1
With LV With CS
−.2
C.I. with LV (95%) C.I. with CS (95%)
−.2
0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1
Percentiles Percentiles
6 Conclusion
The paper studied the theoretical and empirical properties of the ﬁve most popular measures
used in economics to capture welfare changes when prices change. We adopted a number of
standard assumptions including: monotonic and strictly convex utility functions, single-valued
and continuously differentiable demand functions, complete, reﬂexive and transitive prefer-
ences, constant marginal utility of income, utility maximizing consumers and normal goods.
The theoretical part provided deﬁnitions, geometrical illustration and various computation op-
tions for all measures and led to the following conclusions:
• The welfare effects is bounded between LV = (−)xa dp and P V = (−)xb dp;
• LV < CV < CS < EV < P V if dp = 0 (if the demand schedules are not perfectly
elastic or inelastic);
47
• LV = CV = CS = EV = P V if dp = 0 (if the demand schedules are not perfectly
elastic or inelastic);
• LV = CV = CS = EV = P V with a perfectly elastic or inelastic Marshallian demand
schedule;
• With a price change in only one product, marginal utility of money constant and a
quadratic approximation, CV , CS and EV can be approximated to the same measure;
• CV , CS , EV and P V can all be expressed as functions of LV and the income and
substitution effects;
• The ﬁrst order and higher orders Taylor’s approximations can always be split into an
income and a substitution effect;
• For small price changes, the substitution effect of the ﬁrst order Taylor’s approximation
can be approximated to zero;
• For small price changes and with a ﬁrst order Taylor’s approximation, CS , CV and EV
can be approximated to the same measure;
• With a marginal income change, changes in consumed goods converge to their expenditure
shares;
• When the expenditure share is small, the corrected ﬁrst Taylor approximation term is a
good proxy of the income effect, whereas the rest is a good proxy of the substitution
effect;
• With small price changes, the Taylor approximation converges to the CS measure;
The empirical part tested the sensitivity of the welfare measure to changes in various pa-
rameters including price shocks, demand functions, consumption bundles (shares of products in
total consumption) and elasticities. This exercise provided broad indications that can be used by
practitioners to make choices between measures. In particular:
• With a small or moderate price change, ”LV” (the marginal approach) becomes the ap-
propriate measure to assess the change in welfare;
• Price changes can be considered “small” in all cases if they are below 10%;
• Differences between the ﬁve measures considered are very sensitive to the expenditure
share of products;
• If the expenditure share on the product that is affected by the price change is small, then
price changes of up to 50% can be considered small;
• The contour plot provided by the paper can be used as a device to decide when welfare
outcomes between measures are too large to use the simplest of the measures considered
(LV ).
48
• Differences between the ﬁve measures considered are not particularly sensitive to the var-
ious demand functions considered (CD, QUAIDS, AIDS, EASI and LES);
• With changes in prices of multiple products, the welfare effect is reduced and the bounds
at which welfare measures grow apart can be relaxed;
• For large price variations, and with knowledge of the free market point elasticity, one
can use equation (27) to avoid making full assumptions about demand schedules. This
stratagem is fairly close to using Cobb-Douglas preferences up to price variations of about
100 percent and is very parsimonious in terms of information required;
• Conﬁdence intervals (95%) for the welfare measures considered do not seem to alter the
conclusions above;
• When welfare measures provide very different results, the choice of the measure is ulti-
mately a normative choice;
• In the absence of any normative argument in favor of one particular measure, the sensible
choice is CS for the simple reason that this measure is always the median measure;
• In the absence of any normative argument in favor of one demand function, the sensi-
ble choice is the Cobb-Douglas function because of its simplicity and nice properties in
relation to utility curves.
Finally, it is evident that if one considers the price shock of only one product and this product
is a small share of household expenditure, the measure of choice is invariably LV . But in the real
world, it is not very common to model situations when price increases and expenditure shares
are both small, whether we consider subsidies, taxes or changes in inﬂation. For example, it is
not uncommon in developing countries to have subsidies that are several folds the subsidized
price so that the price increase necessary to remove subsidies is measured in the hundreds of
percentage points. Similarly, imposing sales taxes on a particular product can increase prices by
20-30% and this is not uncommon in developing countries where a typical sales tax is around
20%. In these cases, the expenditure share of products may be small but the price increases are
large. In other cases, such as a longitudinal or spatial inﬂation adjustments, the price adjustment
may be relatively small but the share of expenditure concerned by the inﬂation adjustment is
very large (the whole products’ basket). And it is not atypical for developing countries to expe-
rience periods of hyperinﬂation. Hence, practitioners rarely ﬁnd situations where both the price
increase considered and the expenditure share of the products considered are small. It is also rare
for practitioners working on developing countries to be able to measure the Marshallian demand
schedule. In the case of subsidies, prices are ﬁxed by deﬁnition and surveys cannot provide the
price variations necessary to estimate demand schedules. In the case of taxes, few developing
countries have proper tax systems in place and fewer have populations that comply to the tax
systems and this makes estimations of tax elasticities impossible. This paper has provided some
initial guidelines on what to do under different circumstances and degrees of information.
49
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53
Appendices
A Demand Systems
A.1 Linear Demand (LD)
q = a − bp (96)
a 1
p= − q (97)
b b
p
η = −b (98)
q
A.2 Log Linear Demand (LLD)
q = e a p− b (99)
ln(q ) = a − ln(p)b (100)
1
p = (e−a q )− b (101)
∂ln(q )
η= = −b (102)
∂ln(p)
A.3 The Linear Expenditure System (LES)
The LES model assumes that expenditures depend linearly on income and prices. The LES
speciﬁcation derives from the maximization of a linear Stone-Geary utility function such that:
K
M ax U = βk ln (xk − γk ) ; (103)
k=1
subject to the budget constraint and the Engel aggregation condition:
K K
m= pk xk , and βk = 1 (104)
k=1 k=1
The econometric model of the demand function of household h for commodity k :
K
βk
xk = γk + m− pk γk . (105)
pk k=1
54
There are (2K − 1) parameters ( βs and γs to be estimated). Expenditure on a given good
is a linear function in income and prices. However, the expenditure system in not linear in
parameters. Note that the non-homotheticity introduced by the subsistence quantities (γk ) allows
the LES to exhibit non-unitary expenditure elasticities. The model can be estimated through
maximum likelihood using the Stata nlsur command (this ﬁts a system of nonlinear equations
by feasible generalized nonlinear least squares). The Marshallian price elasticities are:
γk (1 − βk ) pl γl
εk,k = − 1 and εk,l = −βk (106)
xk pk x
The compensated price elasticities are:
γk βk ph
ηk,k = (βk − 1) 1 − and ηk,h = − (xk − γh ) (107)
xk p k xk
The income elasticity is:
βk m
ηk = − 1. (108)
p k xk
Note that the indirect utility of the LES model takes the following form:
K
m− k=1 pk γk
V (m, p) = K βk
(109)
k=1 pk
pk
Let ςk = pk +dpk
. It can be shown that:
K K K K
βk βk
EVLES = m ς −1 + pk γk − ς (pk + dpk )γk (110)
k=1 k=1 k=1 k=1
Remember that the equivalent variation with the CD model can be written as follows:
K
EVCD = m ς αk − 1 (111)
k=1
With moderate estimates of γk , βk converges to αk , i.e, the expenditure share. Based on this,
and by comparing the two impacts, we ﬁnd that:
K K βk K
EVLES k=1 pk γk − k=1 ς k=1 (pk + dpk )γk
θ(P, m) = =1+ (112)
EVCD m K βk −1
k=1 ς
Let τ = K k=1 pk γk denote the expenditures on the initial dotations and dτ =
K
k=1 dpk γk .
We can simplify again and ﬁnd:
K βk
τ dτ k=1 ς
θ(P, m) = 1 − + K
(113)
m m 1− k=1 ς
βk
With a single price change we ﬁnd that:
55
τ dτ ς βk
θ(P, m) = 1 − + (114)
m m 1 − ς βk
Even if θ(P, m) is divergent when pk → ∞, it follows that with rational large price changes,
the EVCD is a good proxy of EVLES . Let m = 100, βk =0.2, γk =2, τ =10 and pk = 1 ∀k . With a
price change of 500%, we ﬁnd that theta ≈ 1.178. In essence, we can say that, with reasonable
large price variations, it is expected that EV estimated with the CD model can be a good proxy
of EV with the LES model.
A.4 The Almost Ideal Demand System (AIDS)
To present the AIDS model, we start by deﬁning the PIGLOG model. This model can be
represented via an expenditure function c(P, U ) that deﬁnes the minimum level of expenditure
to reach a predetermined level of utility given prevailing prices:
log (c(p, U )) = (1 − U ) log(a(p)) + U log(b(p)) (115)
where U is the utility located between 0, the level of subsistence, and 1, the level of beatitude.
The function a(p) is found through the TRANSLOG form:
K K K
∗
log (a(p)) = α0 + αi log(pi ) + γi,j log(pi ) log(pj ) (116)
i=1 i=1 j =1
The component b(p) is deﬁned as :
K
log (b(p)) = log (a(p)) + β0 pi βi (117)
i=1
Thus, we ﬁnd that:
K K K K
∗
log (c(p, u)) = α0 + αi log(pi ) + γi,j log(pi ) log(pj ) + uβ0 pi βi (118)
i=1 i=1 j =1 i=1
Using Shephard’s Lemma (∂c(p, u)/∂pk = xk ) , the expenditure share on good i becomes:
K K K
wi = αi + γi,j log(pj ) + βi uβ0 pi βi (119)
i=1 j =1 i=1
and
1 ∗ ∗
γi,j = γi,j + γj,i (120)
2
Expenditure shares also simplify as:
K
w i = αi + γi,j log(pj ) + βi log(m/a(p)) (121)
j =1
56
where a(p) can be perceived as a price index equaling:
K K K
∗
log (a(p)) = α0 + αi log(pi ) + γi,j log(pi ) log(pj ) (122)
i=1 i=1 j =1
The additional conditions of the model are:
K
I: αk = 1 Sum of expenditures shares is 1
i=1
K K
II : γi,j = 0 ∀j and βi = 0 Homogeneity of degree 0 of demand functions
i=1 i=1
III : γi,j = γj,i Symmetry of the Slutsky matrix
(123)
The indirect utility function is deﬁned as follows:
ln(m) − ln(a(p))
Ln(V ) = (124)
b(p)
A.5 The Quadratic Almost Ideal System (QUAIDS)
Banks, Blundell, and Lewbel (1997) have proposed the Quadratic Almost Ideal System
(QUAIDS) model that adds the quadratic logarithmic income term to the AIDS speciﬁcation
of Deaton and Muellbauer (1980a). This was proposed in order to take into account the po-
tential quadratic form of the Engel curve behavior for some durable and luxury goods. The
speciﬁcation is as follows:
K
λi
wi = αi + γi,j log(pj ) + βi log(m/a(p)) + log(m/a(p))2 (125)
j =1
b(p)
The price index is given by:
K K K
∗
log (a(p)) = α0 + αi log(pi ) + γi,j log(pi ) log(pj ) (126)
i=1 i=1 j =1
The price aggregator is given by:
K
b (p) = pβ
i
i
(127)
i=1
The set of constraints to obey to the usual Marshallian properties are:
K
I: αk = 1 Sum of expenditures shares is 1
i=1
K K
II : γi,j = 0 ∀j and βi = 0 Homogeneity of degree 0 of demand functions
i=1 i=1
III : γi,j = γj,i Symmetry of the Slutsky matrix
(128)
57
The income and demand elasticities are deﬁned as follow:
I: ei = µi /wi − 1 Income elasticity
nc
II : ei,j = µi,j /wi − δi,j Non compensated elasticity (129)
III : ec
i,j = µi,j /wi − ei wi Compensated elasticity
where
∂wi m
µi = ∂ log(m)
= βi + 2λi log a(p)
K 2
∂wi λi β j m
µi,j = ∂ log(pj )
= γi,j + µi αj + γk,j log(pk ) − b(p)
log a(p)
k=1
(130)
The indirect utility function is deﬁned as follows:
−1 −1
ln(m)−ln(a(p))
Ln(V ) = + λ(p)
b(p) (131)
and,
K
λ(p) = λi log(pi ) (132)
i=1
A.6 Exact Afﬁne Stone Index (EASI)
To deal with the empirical nonlinear form of the Engel curve and to propose a more ﬂexi-
ble model, Lewbel and Pendakur (2009) use the Shephard’s lemma to approximate real income.
This linear approximation implies the use of the Stone price index (SPI), as in the case of the Lin-
ear Approximate Almost Ideal Demand System (LA/AIDS). Even with this restriction, among
the advantages of the EASI model is the possibility of using a higher order of the polynomial
real income, which enables to better ﬁt the Engel function. Formally, the approximated EASI
model can be deﬁned through the implicit marshallian budget share as follow:
o K K
wi = ˜r +
br y ak log(pk ) + y+˜
bk log(pk )˜
r=1 j =1 k=1
o
= ˜r + Ap + Bpy
br y ˜+˜ (133)
r=1
˜ = log (m) −
˜ denotes here the log of the approximated real income: y ≈ y
where y
K
wk log(pk ).
k=1
The parameter o is the polynomial order of the real income and the parameter ˜ is simply the
error term of the estimation. For simplicity and compared to the Lewbel and Pendakur (2009)
58
presentation of the model, we omit the household characteristics determinants. Based on the
Shephard’s lemma and the cost function, Lewbel and Pendakur (2009) show that the exact real
income is equal to:
m − p w + p Ap
y = (134)
1 − 0.5 ∗ p Bp
Thus, the exact EASI model can be deﬁned as follows:
o r
m − p w + p Ap m − p w + p Ap
wi = br + Ap + Bp + (135)
r=1
1 − 0.5 ∗ p Bp 1 − 0.5 ∗ p Bp
As was the case for the AIDS or the QAIDS models, additional conditions are imposed:
K K
I: ai,j = 0 ∀j and bi = 0 Homogeneity of degree 0 of demand functions
i=1 i=1
II : ai,j = aj,i Symmetry of the Slutsky matrix
(136)
Among the recommended econometric methods to estimate the model is the nonlinear three
stage least squares (3SLS). Let pc denotes the vector of the log of prices after the change. The
equivalent income (EI) is equal to: 18
EI = exp [log (m) + p w − pc w + 0.5 ∗ p Bp − 0.5 ∗ pc Bpc ] (137)
B Nonlinear price changes and wellbeing
For some goods, pricing is not homogeneous across quantities consumed, a case often re-
ferred to as non-linear pricing. The generic term nonlinear pricing refers to any case in which
the tariff is not strictly proportional to the quantity purchased. This applies to regulated and non-
regulated prices for goods where, to different levels of consumption, correspond different levels
of prices. This is typically the case of utilities such as electricity or water where the price is set
according to the consumed quantity. For simplicity, assume that we have two goods, electricity,
which we call good 1, and the rest of goods which we call good 2. Further, assume that the price
of electricity is deﬁned by two blocks of consumed quantities as follows:
Table 8: The price schedule
Block Price
1 : 0 - q1 p 1 ,1
2: q1 and more p 1 ,2
59
Table 9: Non linear price schedule: an illustrative example
Block Price Consumed quantity Cost
1:[0-10] 1 10 10
2:[10-16] 2 6 12
3:[16 and more] 3 6 18
Total 22 40
Our aim is to derive the EV and CV welfare measurements in the case of nonlinear pricing
for the simple CD function. To introduce the proposed approach, ﬁrst, consider the illustrative
example in Table 9.
Also, assume that the consumer maximizes its utility by spending 40$ and consumes 22
units. It is worth noting that proposing a simple functional form of consumer preferences with
the presence of nonlinear pricing schedule is not an easy task. However, we attempt to make
this feasible based on the following proposition.
Proposition 1 : Adequacy of preferences under the veiled schedule of prices
The consumer behavior is independent of the exact nonlinear structure of the price schedule, but
it can be based on its equivalent average price schedule.
To better clarify this idea, assume that the seller will not indicate on the bill prices for each
block of consumed quantities, but simply reveals the total amount for the total quantities con-
sumed. A rational consumer should not be indifferent to how the bill is computed. However,
this consumer must select the desired quantity based on its total cost in order to maximize its
utility. The equivalent average price, for a given consumer, is simply the average price for the
bought quantity. In the example above, the equivalent average price is equal to 40/22. Remark
that different schedule prices can generate the same cost or average price. Thus, it is also irra-
tional to suppose that the marginal price is enough to model the consumer behavior. Given the
constant expenditure shares assumed by the CD model, consider the following two cases:
A- The seller adopts a linear price structure and increases the single price from 40/22=1.81
to 40/20.5=1.95;
B- The seller adopts a nonlinear price structure and increases the price in the third bracket
from 3 to 4. With a ﬁxed budget of 18, consumed quantities in the third bracket are
reduced to 4.5.
Therefore, in both cases, the consumer can buy the same quantity of 20.5 with a budget of
40$. If the veil on price schedule exists, the optimal utility in each of the two cases will be the
same. Note that with CD preferences, we have τh,k = pb a a b
h,k /ph,k = qh,k /qh,k , where ph,k refers to
the average price (linear or nonlinear cases). By assuming that initial prices are normalized to
one, the equivalent variation can be redeﬁned as follows:
1
EVh = mh K
−1 (138)
1 τh,k αh,k
18
See also Hoareau S. and Tiberti (2014) for the deﬁnition of elasticities of the EASI model.
60
K
CVh = mh 1 − τh,k αh,k (139)
1
Based on the level of expenditures on good k and the new price schedule, we can easily
estimate the component τh,k . The lesson here is that the non-linearity of prices is not translated
into different consumer preferences. It is the budget constraint that reduces the space of choices.
61
C STATA Codes
1 /************************ THE STATA CODE: A *******************************/
/* Estimating the LV, PV, EV, CV and CS welfare change measurements */
3 /* Equations: LV->(12)|PV->(13)|EV->(20)|CV->(22)|CS-> (17)|CS_ELAS->(27) */
/**************************************************************************/
5 /* =======================================================================*/
/* Input information */
7 /* =======================================================================*/
/* - List of varnames of per capita expenditures on the different goods */
9 /* - List of price changes */
/* - To estimate the expenditures after the price change the assumption is*/
11 /* that the preferences are homothetic */
/* =======================================================================*/
13 /* Outputs: Change in welfare:LV, PV, EV, PV, CS and CS_ELAS variables */
/**************************************************************************/
15 // Constructing the hypothetical data
clear
17 set obs 1000
set seed 1234
19 gen income = uniform()*_n
gen food = ( 0.3 + 0.2*uniform() )*income
21 gen clothes = ( 0.1 + 0.1*uniform() )*income
23 // Initialising the lists of items and price changes
local list_of_items food clothes // list of items
25 local price_changes 0.06 0.04 // proportions of price changes
// Estimating the welfare change with LV and PV measurements
27 gen LV = 0 // Initialising the variable LV
gen PV = 0 // Initialising the variable PV
29 gen EV = 0 // Initialising the variable EV
gen CV = 0 // Initialising the variable CV
31 gen CS = 0 // Initialising the variable CS
gen CS_ELAS = 0 // Initialising the variable CS
33
local i = 1 // number of the item.
35 foreach item of local list_of_items {
tempvar item_‘i’
37 gen ‘item_‘i’’ = ‘item’
tempvar share_‘i’
39 gen ‘share_‘i’’ = ‘item_‘i’’/income // The expenditure shares
local nitems = ‘i’
41 local i = ‘i’ + 1
}
43 local i = 1 // number of the item.
foreach dp of local price_changes {
45 local dp_‘i’ = ‘dp’
local i = ‘i’ + 1
47 }
tempvar price_index
49 gen ‘price_index’ = 1 // Initialising the Laspeyres price index
forvalues i=1/‘nitems’ {
51 tempvar item_a
gen ‘item_a’ = ‘item_‘i’’ // The expenditures in period (a)
53 replace LV=LV - ‘dp_‘i’’*‘item_a’
tempvar item_b
55 gen ‘item_b’ = (‘item_a’)/(1+‘dp_‘i’’) // The expenditures in period (b)
replace PV=PV - ‘dp_‘i’’*‘item_b’
57 replace CS=CS - ‘item_a’ *ln(1+‘dp_‘i’’)
replace CS_ELAS=CS_ELAS - ‘item_a’*‘dp_‘i’’*(1 - 0.5*‘dp_‘i’’*(1+‘dp_‘i’’))
59 replace ‘price_index’=‘price_index’*((1+‘dp_‘i’’)ˆ‘share_‘i’’)
}
61 replace EV = income * ( 1/‘price_index’ - 1 )
replace CV = income * ( 1 - ‘price_index’ )
63 /**************************************************************************/
62
1 /************************ THE STATA CODE: B *******************************/
/* Estimating the EV, PV and CS welfare change measurements */
3 /* Approach: Taylor approximation */
/* Order 1 : CV=EV=CS = LV */
5 /* Order 2 : CV-> eq_59 ||EV-> eq_60 || CS-> eq_58 */
/**************************************************************************/
7 /* =======================================================================*/
/* Input information */
9 /* =======================================================================*/
/* - List of varnames of per capita expenditures on the different goods */
11 /* - List of price changes */
/* - The assumption: the preferences are homothetic */
13 /**************************************************************************/
/* =======================================================================*/
15 /* Outputs: Change in welfare: EV, CV and CS variables */
/**************************************************************************/
17 // Constructing the hypothetical data
clear
19 set obs 1000
set seed 1234
21 gen income = uniform()*_n
gen food = ( 0.3 + 0.2*uniform() )*income
23 gen clothes = ( 0.1 + 0.1*uniform() )*income
// Initialising the lists of items and price changes
25 local list_of_items food clothes // list of items
local price_changes 0.06 0.04 // proportions of price changes
27 // Setting the order of Taylor approximation
local order = 2 // The user can set the taylor order to 1.
29
// Estimating the welfare change with LV and PV measurements
31 gen LV = 0
gen EV_TAYLOR_‘order’ = 0 // Initialising the variable EV
33 gen CV_TAYLOR_‘order’ = 0 // Initialising the variable CV
gen CS_TAYLOR_‘order’ = 0 // Initialising the variable CS
35 local i = 1 // number of the item.
37 foreach item of local list_of_items {
local item_‘i’ = "‘item’"
39 dis ‘item_‘i’’
tempvar share_‘i’
41 gen ‘share_‘i’’ = ‘item_‘i’’/income // The expenditure shares
local nitems = ‘i’
43 local i = ‘i’ + 1
}
45 local i = 1 // number of the item.
foreach dp of local price_changes {
47 local dp_‘i’ = ‘dp’
local i = ‘i’ + 1
49 }
51 forvalues i=1/‘nitems’ {
replace EV_TAYLOR_‘order’=EV_TAYLOR_‘order’ - ‘dp_‘i’’*‘item_‘i’’
53 replace CV_TAYLOR_‘order’=CV_TAYLOR_‘order’ - ‘dp_‘i’’*‘item_‘i’’
replace CS_TAYLOR_‘order’=CS_TAYLOR_‘order’ - ‘dp_‘i’’*‘item_‘i’’
55
if ‘order’ >= 2 {
57 forvalues j=1/‘nitems’ {
replace EV_TAYLOR_‘order’ = EV_TAYLOR_‘order’ - 0.5 * ( -‘share_‘i’’*‘item_‘j’’ ///
59 -‘item_‘i’’*(‘i’==‘j’))*‘dp_‘i’’*‘dp_‘j’’
replace CV_TAYLOR_‘order’ = CV_TAYLOR_‘order’ - 0.5 * ( ‘share_‘i’’*‘item_‘j’’ - ///
61 ‘item_‘i’’*(‘i’==‘j’))*‘dp_‘i’’*‘dp_‘j’’
replace CS_TAYLOR_‘order’ = CS_TAYLOR_‘order’ - 0.5 * (0 - ///
63 ‘item_‘i’’*(‘i’==‘j’))*‘dp_‘i’’*‘dp_‘j’’
}
65 }
}
63
67 if ‘order’ >= 3 {
forvalues i=1/‘nitems’ {
69 forvalues j=1/‘nitems’ {
forvalues k=1/‘nitems’ {
71 replace EV_TAYLOR_‘order’ = EV_TAYLOR_‘order’ - ///
1/3* ((‘share_‘i’’* (‘share_‘j’’*‘item_‘k’’ +‘item_‘j’’)*(‘i’==‘j’==‘k’)) + ///
73 (‘share_‘i’’*‘item_‘k’’ + ‘item_‘i’’)*(‘i’==‘j’==‘k’) )*‘dp_‘i’’*‘dp_‘j’’*‘dp_‘k’’
75 replace CV_TAYLOR_‘order’ = CV_TAYLOR_‘order’ - ///
1/3* (( -‘share_‘i’’* (‘share_‘j’’*‘item_‘k’’+ ‘item_‘j’’)*(‘i’==‘j’==‘k’)) + ///
77 ( - ‘share_‘i’’*‘item_‘k’’ + ‘item_‘i’’)*(‘i’==‘j’==‘k’) )*‘dp_‘i’’*‘dp_‘j’’*‘dp_‘k’’
79 replace CS_TAYLOR_‘order’ = CS_TAYLOR_‘order’ - ///
1/3 * ( ‘item_‘i’’*(‘i’==‘j’==‘k’))*‘dp_‘i’’*‘dp_‘j’’*‘dp_‘k’’
81 }
}
83 }
}
85 /**************************************************************************/
64
1 /************************ THE STATA CODE: C *******************************/
/* - The Vartia(1983) algorithm */
3 /* - Approach: Numerical approximation */
/* - Outputs: Change in welfare: CV (EQ. 66) // EV (EQ. 67) */
5 /* - The user can increase the number of goods or use another demand */
/* function */
7 /**************************************************************************/
9 set trace off
mata: mata clear
11 mata
num=10 /* Number of partitions or iterations */
13
//Initialising the parameters (prices and income)
15 real matrix function initialise_parameters(scalar t)
{
17 if (t==0) return (1.0 \ 1.0 \ 1.0 ) /* Initial price vector */
if (t==1) return (1.3 \ 1.4 \ 1.7 ) /* Final price vector */
19 if (t==2) return (0.3 \ 0.5 \ 0.2 ) /* income shares */
if (t==3) return (100) /* Income */
21 }
23 // Defining the demand function
real matrix function eval_quantities(real matrix x)
25 {
real matrix q
27 alpha =initialise_parameters(2)
n=cols(x)-1
29 n1=n+1
q = J(1,n,0)
31 for (r=1; r<=n; r++) {
q[r]=alpha[r]:*x[n1]:/x[r] // Evaluate quantities
33 }
return(q)
35 }
37 p0 = initialise_parameters(0)
p1 = initialise_parameters(1)
39 alpha = initialise_parameters(2)
y_cv_old = initialise_parameters(3)
41 y_ev_old = initialise_parameters(3)
y0 = initialise_parameters(3)
43
y_cv=y0
45 y_ev=y0
tcv = 0
47 tev = 0
cv=0
49 ev=0
51 del = (p1:-p0)/num
dp=p1:-p0
53 par = J(rows(alpha)+1,1,.)
par[1..rows(alpha)]=p0
55 par[rows(alpha)+1]=y0
q_cv_old=eval_quantities(par’)
57
par[1..rows(alpha)]=p1
59 par[rows(alpha)+1]=y0
q_ev_old=eval_quantities(par’)
61
for (i=1; i<=num; ++i) {
63 // initialising parameters for local derivatives (CV)
a_cv = J(rows(alpha),1,1)
65 a_cv = (((a_cv:*(i-1)):/num):*dp):+1 // prices at step i for CV
par_cv = J(rows(alpha)+1,1,.)
65
67 par_cv[1..rows(alpha)]=a_cv
par_cv[rows(alpha)+1]=y_cv
69 q_cv_new = eval_quantities(par_cv’)
71 // initialising parameters for local derivatives (EV)
a_ev = J(rows(alpha),1,1)
73 a_ev = -(((a_ev:*(i-1)):/num):*dp):+p1 // prices at step i for EV
par_ev = J(rows(alpha)+1,1,.)
75 par_ev[1..rows(alpha)]=a_ev
par_ev[rows(alpha)+1]=y_ev
77 q_ev_new = eval_quantities(par_ev’)
79 cv = (0.5*(q_cv_old:+q_cv_new))*del
ev = (0.5*(q_ev_old:+q_ev_new))*del
81
y_cv = y_cv+cv
83 y_ev = y_ev-ev
85 tcv = tcv-cv
tev = tev-ev
87
q_cv_old = q_cv_new
89 q_ev_old = q_ev_new
}
91
tcv
93 tev
95 end
97 clear all
local price_index=((1.3)ˆ0.3)*((1.4)ˆ0.5)*((1.7)ˆ0.2)
99 dis "True CV = " 100 * (1- ‘price_index’ )
dis "True EV = " 100 * (1/‘price_index’ -1)
66
/************************ THE STATA CODE: D *******************************/
2 /* - The Breslaw and Barry’s (1995) algorithm */
/* - Approach: Numerical approximation */
4 /* - Outputs: Change in welfare: EV, CV (EQ. 6 */
/* - The algorithm is programmed with Stata (exactly with mata). mata is */
6 /* the matrix language of Stat */
/* - The user can increase the number of goods or use another demand */
8 /* function */
/**************************************************************************/
10
set more off
12 mata: mata clear
mata
14 num=10 /* Number of iterations */
real matrix function initialise_parameters(scalar t)
16 {
if (t==0) return (1.0 \ 1.0 \ 1.0 ) /* Initial price vector */
18 if (t==1) return (1.3 \ 1.4 \ 1.7 ) /* Final price vector */
if (t==2) return (0.3 \ 0.5 \ 0.2 ) /* income shares */
20 if (t==3) return (100) /* Income */
}
22
// Defining the Marshallian demand functions
24 // Parameters: x_1...x_n (prices) and x_{n+1} income
real matrix function eval_quantities(real matrix x)
26 {
real matrix q
28 alpha =initialise_parameters(2)
n=cols(x)-1
30 n1=n+1
q = J(1,n,0)
32 for (r=1; r<=n; r++) {
q[r]=alpha[r]:*x[n1]:/x[r] // Evaluate quantities
34 }
return(q)
36 }
38 void eval_t2(x, v) // function used for the numerical derivative
{
40 alpha =initialise_parameters(2)
n=cols(x)-1
42 n1=n+1
v = J(1,n1,.)
44 v[1..n]=eval_quantities(x) // Evaluate the quantities
v[n1] = x[n1] // Evaluate the income
46 }
48 D = deriv_init() // function used for the num. derivative
50 p0 = initialise_parameters(0)
p1 = initialise_parameters(1)
52 alpha = initialise_parameters(2)
y_cv = initialise_parameters(3)
54 y_ev = initialise_parameters(3)
tcv = 0
56 tev = 0
del = (p1:-p0)/num
58 ndel = -del
dp=p1:-p0
60 for (i=1; i<=num; ++i) {
62 a_cv = J(rows(alpha),1,1)
a_cv = (((a_cv:*(i-1)):/num):*dp):+1 // prices at step i for CV
64
a_ev = J(rows(alpha),1,1)
66 a_ev = -(((a_ev:*(i-1)):/num):*dp):+p1 // prices at step i for EV
67
68 // initialising parameters for local derivatives (CV)
par_cv = J(rows(alpha)+1,1,.)
70 par_cv[1..rows(alpha)]=a_cv
par_cv[rows(alpha)+1]=y_cv
72 par_cv
// initialising parameters for local derivatives (EV)
74 par_ev = J(rows(alpha)+1,1,.)
par_ev[1..rows(alpha)]=a_ev
76 par_ev[rows(alpha)+1]=y_ev
78 h_cv=eval_quantities(par_cv’)’ // quantities at step i for CV
h_ev=eval_quantities(par_ev’)’ // quantities at step i for EV
80
deriv_init_evaluator(D, &eval_t2())
82 deriv_init_evaluatortype(D, "t")
84 deriv_init_params(D, par_cv’)
AA=deriv(D, 1)[1::rows(alpha),1..rows(alpha)+1]
86 A1 = AA[1::rows(alpha),1..rows(alpha)] // derivative with respect to prices
A2 = AA[1::rows(alpha),rows(alpha)+1] // derivative with respect to income
88 dhdp_cv = A1+A2*h_cv’
90 deriv_init_params(D, par_ev’)
AA=deriv(D, 1)[1::rows(alpha),1..rows(alpha)+1]
92 A1=AA[1::rows(alpha),1..rows(alpha)] // derivative with respect to prices
A2 = AA[1::rows(alpha),rows(alpha)+1] // derivative with respect to income
94 dhdp_ev = A1+A2*h_ev’
// See the equation (68)
96 cv = del’*h_cv + 0.5*del’*dhdp_cv*del
ev = ndel’*h_ev + 0.5*ndel’*dhdp_ev*ndel
98
tcv=tcv-cv
100 tev=tev+ev
102 y_cv=y_cv+cv
y_ev=y_ev+ev
104
}
106 tcv
tev
108 end
110 clear all
local price_index=((1.3)ˆ0.3)*((1.4)ˆ0.5)*((1.7)ˆ0.2)
112 dis "True CV = " 100 * (1- ‘price_index’ )
dis "True EV = " 100 * (1/‘price_index’ -1)
68
/************************ THE STATA CODE: E *******************************/
2 /* - The Euler and RK4 algorithms */
/* - Approach: Numerical approximation */
4 /* - Outputs: Change in welfare: EV/CV (EQ. 69) // EV/CV (EQ. 70) */
/* - The user can increase the number of goods or use another demand */
6 /* function */
/**************************************************************************/
8 local m = 100 // Income
local niter = 12 // Number of iterations
10 // By default, we estimate the EV. The user can set this to be "CV".
local measurement = "EV"
12
// expenditures shares (alpha’s values). The user can change the number of items
14 matrix alpha = (0.10 \ 0.05 \ 0.12 \ 0.06)
16 matrix p0 = (1.00 \ 1.00 \ 1.00 \ 1.00 ) // Initial prices
matrix p1 = (1.15 \ 1.30 \ 1.10 \ 1.25) // Final prices
18
local price_index=1
20 local nitems = rowsof(alpha) // Number of items
// loading the different values in local macros.
22 forvalues k=1/‘nitems’ {
local alpha_‘k’ = el(alpha, ‘k’,1)
24 local p‘k’_0 = el(p0, ‘k’,1)
local p‘k’_1 = el(p1, ‘k’,1)
26 local price_index=‘price_index’*(‘p‘k’_1’ˆ‘alpha_‘k’’)
28 if "‘measurement’"=="CV" {
local h‘k’ = (‘p‘k’_1’-‘p‘k’_0’)/‘niter’
30 local multi = (1/‘price_index’ -1)
}
32
if "‘measurement’"=="EV" {
34 local multi = (1 - ‘price_index’)
local h‘k’ = -(‘p‘k’_1’-‘p‘k’_0’)/‘niter’
36 forvalues k=1/‘nitems’ {
local alpha_‘k’ = el(alpha, ‘k’,1)
38 local p‘k’_1 = el(p0, ‘k’,1)
local p‘k’_0 = el(p1, ‘k’,1)
40 }
}
42 }
local m0 = 100
44 local me0 = 100
46 // A simple program to evaluate the first derivative: the case of Cobb-Douglas function.
cap program drop odf
48 program define odf, rclass
args p m alpha
50 return scalar ft = -‘alpha’*‘m’/(‘p’)
end
52
// The iterative algorithm for the Euler and RK4 methods.
54 forvalues i=1/‘niter’ {
local j=‘i’-1
56
forvalues k=1/‘nitems’ {
58 local p_‘k’_0 =‘p‘k’_0’
local m_‘k’ = ‘m‘j’’
60 local mm_‘k’ = ‘m‘j’’
local p_‘k’_‘i’ = ‘p_‘k’_0’+‘h‘k’’*‘i’
62 local p_‘k’ = ‘p_‘k’_‘j’’
odf ‘p_‘k’’ ‘m_‘k’’ ‘alpha_‘k’’
64
local k1_‘k’ = ‘r(ft)’
66 local p_‘k’ = ‘p_‘k’_‘j’’+0.5*‘h‘k’’
69
local m_‘k’=‘mm_‘k’’+0.5*‘k1_‘k’’*‘h‘k’’
68 odf ‘p_‘k’’ ‘m_‘k’’ ‘alpha_‘k’’
local k2_‘k’ = ‘r(ft)’
70 local m_‘k’=‘mm_‘k’’+0.5*‘k2_‘k’’*‘h‘k’’
odf ‘p_‘k’’ ‘m_‘k’’ ‘alpha_‘k’’
72 local k3_‘k’ = ‘r(ft)’
74 local p_‘k’ = ‘p_‘k’_‘i’’
local m_‘k’=‘mm_‘k’’+‘k3_‘k’’*‘h‘k’’
76 odf ‘p_‘k’’ ‘m_‘k’’ ‘alpha_‘k’’
local k4_‘k’ = ‘r(ft)’
78
local c‘i’ = ‘c‘i’’+1/6*(‘k1_‘k’’+2*(‘k2_‘k’’+‘k3_‘k’’)+‘k4_‘k’’)*‘h‘k’’
80 local ce‘i’= ‘ce‘i’’+ 1*(‘k4_‘k’’)*‘h‘k’’
}
82
local m‘i’ = ‘m‘j’’ + ‘c‘i’’
84 local me‘i’ = ‘me‘j’’+ ‘ce‘i’’
if "‘measurement’"=="CV" di "Iteration "‘i’ ": ‘measurement’_Euler: " ///
86 ‘me‘i’’ - ‘me0’ "| ‘measurement’_RK4: " ‘m‘i’’ - ‘m0’
if "‘measurement’"=="EV" di "Iteration "‘i’ ": ‘measurement’_Euler: " ///
88 - ‘me‘i’’ + ‘me0’ "| ‘measurement’_RK4: " - ‘m‘i’’ + ‘m0’
}
90
92 // The true value of the EV/CV measurement
dis "True value of ‘measurement’ = " 100 * ‘multi’
70