metadata: authors: 0: author: Amendola,Nicola
1: author: Mancini,Giulia
2: author: Redaelli,Silvia
3: author: Vecchi,Giovanni
count: Iran, Islamic Republic of
docna: 0: docna: Price Adjustments and Poverty Measurement
docty: Policy Research Working Paper
abstracts: cdata!: Measuring poverty entails making
            interpersonal welfare comparisons, that should account for
            differences in prices faced by households, both over time
            and across space. This paper investigates the impact of
            seemingly minor differences in the practical implementation
            of price adjustments, by developing an analytical framework
            that is consistent with standard consumer theory and mindful
            of the data limitations faced by practitioners. The main
            result is at odds with common sense: even when multiple
            price indexes are available, say a food and a nonfood
            Consumer Price Index, it turns out that using a single price
            index, the total Consumer Price Index, to adjust the
            consumption aggregate is recommended. The practice of
            adjusting the components of the consumption aggregate
            separately, using matching deflators&#x2014;food
            expenditure with the food index and nonfood expenditure with
            the nonfood index&#x2014;can lead to a systematic bias
            in the welfare measure, and consequently in poverty and
            inequality measures. The direction of the bias can be easily
            predicted based on the price level and household consumption
            patterns. On the interplay between spatial and temporal
            deflation, the findings show that temporal deflation should
            be carried out before implementing adjustments to spatial
            cost-of-living differences. The paper illustrates these
            findings using the Islamic Republic of Iran&#x2019;s
            2019 Household Income and Expenditure survey: the bias in
            the headcount poverty rate due to incorrect deflation is
            substantive (5&#x2013;10 percent for estimates at the
            national level, 15&#x2013;20 percent in urban and rural
            areas, and more than 30 percent for district-level headcount
            rates). Higher-order Foster-Greer-Thorbecke poverty measures
            are even more affected.
display_title: Price Adjustments and Poverty Measurement
listing_relative_url: /research/2023/04/34051216/price-adjustments-poverty-measurement
topicv3: Poverty Measurement and Analysis m1326941 707,Growth, Poverty and Inequality m1327633 1091,Deflation m1505334 3102
publishtoextweb_dt: 2023-04-27T00:00:00Z
dois: 10.1596/1813-9450-10426
author_keys: 0: author_key: 998611
1: author_key: 1317880
2: author_key: 901788
3: author_key: 458686
model_extractions: Policy Research Working Paper 10426 Price Adjustments and Poverty Measurement _Nicola Amendola Giulia Mancini Silvia Redaelli Giovanni Vecchi_ Poverty and Equity Global Practice April 2023
Policy Research Working Paper 10426

# **Abstract**

Measuring poverty entails making interpersonal welfare comparisons, that should account for differences in prices faced by households, both over time and across space. This paper investigates the impact of seemingly minor differences in the practical implementation of price adjustments, by developing an analytical framework that is consistent with standard consumer theory and mindful of the data limitations faced by practitioners. The main result is at odds with common sense: even when multiple price indexes are available, say a food and a nonfood Consumer Price Index, it turns out that using a single price index, the total Consumer Price Index, to adjust the consumption aggregate is recommended. The practice of adjusting the components of the consumption aggregate separately, using matching deflators—food expenditure with the food index and nonfood expenditure with the nonfood index—can lead to a systematic bias in the welfare measure, and consequently in poverty and inequality measures. The direction of the bias can be easily predicted based on the price level and household consumption patterns. On the interplay between spatial and temporal deflation, the findings show that temporal deflation should be carried out before implementing adjustments to spatial cost-of-living differences. The paper illustrates these findings using the Islamic Republic of Iran’s 2019 Household Income and Expenditure survey: the bias in the headcount poverty rate due to incorrect deflation is substantive (5–10 percent for estimates at the national level, 15–20 percent in urban and rural areas, and more than 30 percent for district-level headcount rates). Higher-order Foster-Greer-Thorbecke poverty measures are even more affected.

This paper is a product of the Poverty and Equity Global Practice. 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://www.worldbank.org/prwp. The authors may be contacted at sredaelli@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
# **Price Adjustments and Poverty Measurement**

Nicola Amendola nicola.amendola@uniroma2.it Giulia Mancini giulia.mancini@uniroma2.it Silvia Redaelli sredaelli@worldbank.org Giovanni Vecchi giovanni.vecchi@uniroma2.it

**JEL codes:** C43, E31, I32

**Keywords:** poverty, inequality, prices, price adjustments, price indexes, cost-of-living differences,

inflation

**Acknowledgments.** We are grateful to Giovanni D’Alessio, Carolina Diaz Bonilla, Benoît Decerf,

Dean Jolliffe, Cristoph Lakner, Laura Liliana Moreno Herrera, Zurab Sajaia, Nobuo Yoshida, for useful comments and suggestions. All remaining errors are our own.
# **1 Monetary welfare and price adjustment**

The measurement of poverty and inequality is grounded in inter-personal welfare comparisons: in order to make broad judgments on social well-being, we need to be able to clearly tell which of any two individuals is “better off”. Even when this famously difficult question is enclosed within the bounds of a welfarist approach (Ravallion, 1994, 2016), and household consumption expenditure is chosen as the basis of the welfare measure, a number of issues need solving before comparisons across individuals can be made soundly. Among the most crucial of these issues is the fact that observed household expenditures typically do not just reflect differences in consumption, and thus welfare, but differences in prices, both over time (inflation) and across areas of a country (geographical cost-ofliving differences). Nominal household expenditures need to be adjusted for these differences, to avoid mis-ranking the welfare of individuals facing different prices.

The question of how the adjustment should be performed, exactly, has many theoretical and empirical ramifications, explored by a vast literature (Chakrabarty et al. 2018; Chen et al. 2020; Gaddis 2016; Gibson and Kim 2013, 2015, 2019; Jolliffe 2006; McKelvey 2011; Ray 2018). Much has been written, for instance, on the choice of the index used to deflate the nominal welfare indicator (Deaton and Muellbauer, 1980; Diewert 1983). What concerns economists, and welfare analysts in particular, is not the search for the best measure of price differences, _per se_ . Rather, the goal is to find the best approximation of a welfare measure that is consistent with consumer theory. Ideally, this would be accomplished by using a _true cost-of-living index_ (TCLI) to adjust nominal expenditure, which maintains the interpretation of the welfare measure as a utility function derived from an expenditure function. However, implementing this approach is usually so challenging empirically, that in most applied work analysts resort to the use of consumer price indexes: deflating nominal expenditure by a Paasche index yields an approximation of money-metric utility (MMU) (Deaton and Zaidi, 2002), while using a Laspeyres index leads to an approximation of the welfare ratio (WR) (Blackorby and Donaldson, 1987).<sup>1</sup> There is a need for guidelines regarding many other practical aspects of price deflation, beyond the choice of an appropriate index, but clear indications are hard to come by. This paper contributes to the literature on price adjustments and welfare measurement by tackling some issues that have so far

> 1 Section 4 of this paper elaborates on this short summary, by defining the objects mentioned here – TCLI, Paasche, Laspeyres, MMU, WR – and going over the implications of the choice of a deflator on the welfare measure.

2
been left unaddressed. We focus on two questions in particular: first, _how many_ deflators should be used? A recurring scenario in practice is the availability of several price indexes with partial coverage – say, food and non-food – alongside an aggregate price index. The paper works out the implications of two alternative price adjustment strategies, namely adjusting the sub-components of household expenditure separately using multiple price indexes, versus using a single index to adjust the total. Second, what is the optimal _sequence_ of temporal and spatial price adjustments? When working with household survey data, the analyst typically wants to adjust for intra-survey inflation – usually with a price index that is available outside of the survey – as well as for spatial cost-of-living differences – often by computing a deflator based on survey information. We discuss the implications of temporally adjusting all expenditures before estimating spatial deflators, versus after.

The findings deliver a few clear-cut suggestions for applied work. While using multiple indexes (MI) to adjust the components of nominal household expenditure may seem like the most accurate way to make use of the available information on price variation, we find that whenever we are able to rank the two methods, the single index approach (SI) comes out on top. This seemingly counterintuitive result depends on the fact that our goal is not to adjust nominal expenditures for changes in purchasing power, but rather to derive measures of welfare that are consistently comparable across individuals. Regarding the sequencing of temporal and spatial deflation, starting with the adjustment of all elementary expenditures for intra-survey inflation ensures that any spatial price index estimated from the survey is based on accurate (non-distorted) unit values and budget shares.

The paper is organized as follows. Section 2 explains the notation used in the rest of the paper; section 3 discusses the implications of the use of multiple indexes or a single index to adjust household expenditures; section 4 constructs a benchmark to determine which approach is better; section 5 tackles the question of the sequence between temporal and price adjustments; section 6 provides an empirical illustration of the consequences of these different strategies on the estimated distribution of welfare, using the Islamic Republic of Iran’s 2019 Household Expenditures and Income Survey (HEIS); section 7 concludes, and offers a summary of the recommended deflation strategy.

# **2 Price indexes: General notation**

This section introduces the notation for dealing with price indexes, in a way convenient for the analysis carried out in the rest of the paper. We denote a generic price index by _I_ – one can think of it as a spatial or a temporal index, indifferently – and will assume that _I_ can be described by a set of 3
three parameters. Suppose that the population consists of _H_ households, with _H_ a positive integer. The first parameter that is needed to define _I_ is {r}, with r≤H: {r} is a partition of the households’ set, and defines the _aggregation level_ of the price index _._ If r= H, for instance, we say that the index is calculated at the household level; if P is the number of PSUs, and r= P, then we say that the index is calculated at the PSU level; if R is the number of regions, and r= R then the index I is defined at the regional level, and so on. If one thinks of a _monthly_ temporal CPI, the parameter r would define a partition with r= 12. All this can be summarized by denoting the price index as I(r): this makes it explicit than each price index is characterized by a certain aggregation level.

The second parameter needed is k≤K, where K is the total number of goods and services traded and consumed by the households: _k_ is useful to define the _coverage_ of the price index I. If k= K we say that the index has full coverage, that is, its elementary components capture each and every commodity consumed by the households. To account for both the aggregation level and the coverage, the generic price index will be denoted as I(r, k).

Before introducing the third parameter, it is convenient to define Li(r, k) as a Laspeyres price index, where the subscript _i_ runs from 1 to r. This is worth illustrating with a couple of examples. Consider a simple _monthly_ CPI (e.g., a monthly Laspeyres temporal index): in this case, prices are aggregated on a monthly basis, r= 12, and we denote the corresponding twelve monthly Laspeyres indexes by Li(12, k), with i= 1, 2, ... , 12. Similarly, for a spatial price index defined at the urban-rural level, r= 2 and the two corresponding indexes are denoted by Li(2, k), with i= 1, 2. In sum, the notation Li(r, k) reminds us that the Laspeyres index a) is defined with a certain aggregation level r, b) it has a certain coverage k, and c) it takes on as many values as indicated by r (i= 1, ... , r). In a similar vein, we can denote a Paasche price index by Pi(r, k). With these two definitions in mind, we introduce a third parameter α and define the following class of price indexes: Equation 1 defines a family of three price indexes, depending on the value of the parameter α. When α= 0, Ii(r, k, α) corresponds to a Paasche price index, when α= 1, to a Laspeyres price index, and when α= 1⁄2 to a Fisher price index. In addition, each index is characterized by its own aggregation level r and coverage k. In what follows, Ii(r, k, α) can always be thought of as any of the above three indexes, either spatial or temporal, with different aggregation and coverage levels.

4
A similar notation can be used to define two _sub-indexes_ restricted to _food_ and _non-food_ commodities and services. That is useful to deal with situations when prices are only available for food items, or where food- and nonfood price levels vary with different dynamics or gradients, so that the analyst may consider using two deflators instead of one. In order to define the food- and non-food price f nf indexes − Ii and Ii , respectively –, we only need to modify equation 1 slightly: where the coverage of the food price index (equation 2) is denoted by kf ≤Kf (Kf denoting the total number of _food_ items traded in the economy), and the coverage of the nonfood index (equation 3) is denoted by knf ≤Knf (Knf denoting the total number of non-food market commodities and services). Obviously, Kf + Knf = K. The parameters αf and αnf play the same role as α in equation 1: they allow notation to be encompassing of the food Paasche (αf = 0), the food Laspeyres (αf = 1) and Fisher (αf = 0.5) indexes, as well of their nonfood counterparts.

As a last step, equations 2 and 3 can be compared with equation 1: the structure is common to all definitions, and allows us to identify a price index depending on whether we need a Laspeyres, a Paasche or a Fisher index (parameters α, αf, αnf), and on the aggregation level (parameter r), and the coverage (parameters k, kf, and knf). Table 1 illustrates the use of the seven parameters.

5
**Table 1. A useful parametrization for price indexes commonly used in poverty analysis**

|**Type of index**|**Aggregation level**<br>**(**r**)**|**Coverage**<br>**(**k, kf, knf**)**|**(**α,αf,αnf**)**|**Formula**|
|---|---|---|---|---|
|Paasche|i= 1, . . , r|j= 1, ... , k|α= 0|Pi=(∑wj<br>i(<br>pj<br>0<br>pj<br>i<br>~~)~~<br>k<br>j=1<br>)<br>−1|
|Laspeyres|i= 1, . . , r|j= 1, ... , k|α= 1|Li= ∑wj<br>0 (<br>pj<br>i<br>pj<br>0<br>~~)~~<br>k<br>j=1|
|Fisher|i= 1, . . , r|j= 1, ... , k|α= 0.5|Fi= (Li× Pi)<sup>1 2</sup><br>⁄|
|Food Paasche|i= 1, . . , r|j= 1, ... , kf|αf= 0|Pi<br>f=(∑wj<br>i(<br>pj<br>0<br>pj<br>i<br>~~)~~<br>kf<br>j=1<br>)<br>−1|
|Food Laspeyres|i= 1, . . , r|j= 1, ... , kf|αf= 1|Li<br>f= ∑wj<br>0 (<br>pj<br>i<br>pj<br>0<br>~~)~~<br>kf<br>j=1|
|Food Fisher|i= 1, . . , r|j= 1, ... , kf|αf= 0.5|Fi<br>f=(Li<br>f× Pi<br>f)<br>1 2<br>⁄|
|Nonfood Paasche|i= 1, . . , r|j= 1, ... , knf|αnf= 0|Pi<br>nf=(∑wj<br>i(<br>pj<br>0<br>pj<br>i<br>~~)~~<br>knf<br>j=1<br>)<br>−1|
|Nonfood Laspeyres|i= 1, . . , r|j= 1, ... , knf|αnf= 1|Li<br>nf= ∑wj<br>0 (<br>pj<br>i<br>pj<br>0<br>~~)~~<br>knf<br>j=1|
|Nonfood Fisher|i= 1, . . , r|j= 1, ... , knf|αnf= 0.5|Fi<br>nf= (Li<br>nf× Pi<br>nf)<br>1 2<br>⁄|

Note: the table shows the formulas for the different parametrizations illustrated in equations 1, 2 and 3.

To sum up, the notation introduced in this section identifies a set of seven parameters (r, k, kf, knf, α, αf, αnf), that are used in the rest of the paper to produce theoretical propositions on how to carry out both spatial and temporal price adjustments.

# **3 Single versus multiple price indexes in welfare deflation**

An important choice faced by practitioners pertains to the use of a single price index versus the use of multiple (two or more) sub-indexes. The simplest case is the use of two different price indexes, for instance a food price index to deflate the food component of the consumption aggregate and a nonfood price index to deflate the non-food component. In what follows, we develop a theoretical 6
framework that shows how the use of these two alternative approaches leads to different welfare distributions, and has systematic implications in terms of welfare _ranking_ of households.<sup>2</sup> Following the notation introduced in section 2, consider the following two alternative price adjustment procedures. A first method of deflation uses a single index (SI): where x<sup>h</sup> is the total nominal expenditure of household _h_ belonging to area (or period) _i_ , and the index Ii(r, k, α) is defined in equation 1.

A second method consists in using a multiple index (MI) adjustment method: h where xf is the total nominal expenditure on food items of household _h_ belonging to area (or period) h _i_ , and xnf is the nominal expenditure on all the good and services included in the consumption aggregate but different from food.

The first result identifies conditions under which the use of single versus multiple price indexes leads to the same results in terms of welfare ordering and distribution.

# **Proposition 1 (Invariance).**

h h Assume (r, k, kf, knf, α, αf, αnf ) = (H, K, Kf, Knf, 0,0,0 ), then xSI = xMI.

Proof: See the appendix.

> 2 While in principle the multiple index approach can be implemented with more than two subindexes, each matched with the appropriate component of the consumption aggregate, we focus on the use of _two_ subindexes (food and nonfood) to keep the tractability of the algebra within reasonable limits. Arguably, the food-nonfood split is also most common in applied work.

7
Proposition 1 establishes that when using a _household-level_ Paasche index, one inclusive of all items (food and nonfood), then using a single or multiple index approach leads to the same real consumption h h aggregate: xSI = xMI. In other words, the choice of the method is irrelevant for the estimation of poverty and inequality indexes, as well as for any other moment of the empirical distribution function. The situation described by Proposition 1 corresponds to the conditions assumed by Deaton and Zaidi (2002).

Does such a Paasche index exist in practical work? Arguably, it is unlikely for two main reasons. First, household survey data that are typically used for welfare measurement do not include information on prices faced by each household. Rather, households within a defined geographical area are associated with same set of prices, either because expenditure data collected at household level are matched with prices from a common market in the area or because prices are proxied by unit values.<sup>3</sup> In either case, the lack of household level price data leads to r≠H, and therefore xSI ≠xMI. Second, the ideal scenario of full coverage of prices for all items included in the consumption aggregate (i.e. k= K ) is rarely realized, either because not all items are sold in the market of reference (in case household questionnaires are complemented by market level questionnaires) or because is often impossible to collect information on quantities consumed for non-food items and hence is not possible to proxy prices with unit values.

By relaxing the assumptions of Proposition 1, we obtain a new theoretical result, that clarifies how important is, in practice, the choice between the use of one deflator (SI) versus two deflators (MI).

# **Proposition 2 (Distributional bias).**

Assume (r, k, kf, knf, α, αf, αnf) = (r, k, kf, knf, α, α, α ), and k< K or r< H or both, then h h f nf h h xSI ≠xMI. Moreover, if Ii > (<)Ii , then xSI −xMI > 0 if and only if:

> 3 h As discussed in DZ, the most common approach when using unit values is to replace the individual pk by their medians over households in the same PSU or locality in order to minimize noise and address potential problems related to outliers (DZ, p. 42).

8
where wh is the food budget share of household _h_ , and ∆i is always strictly positive. Proof: See the appendix.

Proposition 2 provides a clear-cut result. In a nutshell, the proposition says that as soon the “ideal” Paasche index (one covering _all_ commodities in the economy, and calculated at for each household in the sample) is not available, and therefore the analyst commits to a second-best solution, then the choice of method SI versus MI implies a _systematic_ difference between the _real_ consumption h h aggregates computed using the two approaches: xSI ≠xMI. In particular, the difference between the two approaches depends on two factors. First, on the “price environment” in which the household h h lives: the sign of the difference between xSI and xMI depend on whether the food price index is greater or smaller than the nonfood price index. Second, the household consumption pattern: the sign h h h of the difference between xSI and xMI depends on how large the household’s food budget share wj f nf h is. When Ij > Ij and wj is above a certain threshold (∆i in proposition 2), then using multiple h h indexes implies xMI < xSI.

It is worth pausing and elaborating on the fact that MI implies a systematic “bias” with respect to SI: in any economic environment where food prices are higher than non-food prices, the use of MI will be responsible for lower levels of standard of living for households with a relatively high food budget h h share, and xMI will be lower than xSI the higher is the importance of food in the overall household budget. With respect to SI, the MI approach will tend to penalize households who spend a large share of their budget in the category of goods that is relatively more expensive. How much is “a large share”? The threshold is defined on food budget shares. So, when food is relatively more expensive f nf (Ii > Ii ), households with “high” food budget shares will be penalized by MI. Conversely, when f nf non-food is relatively more expensive (Ii < Ii ), households with “low” food budget shares will be penalized by MI. How high or how low is identified precisely by ∆i.

The last statement can be illustrated and interpreted more meaningfully for welfare analysis if we assume that the Engel law applies, i.e. that poorer households devote a higher budget share on food compared to non-food. In this case the use of multiple index deflation implies welfare measures 9
_systematically lower_ than estimates based on the single index approach, and the size of the gap tends f nf to be larger the poorer is the household. Hence, proposition 2 tells us that, if Ii > Ii , the using multiple indexes tends to make poorer households poorer with respect to using a single index.<sup>4</sup> Figure 1 makes explicit use of Engel curves, which allows to project the food budget shares (vertical axis) on the metric of _real_ expenditures (horizontal axis) and vice-versa. Depending on whether the analyst h h uses SI or MI approach, there are two possible Engel curves: wh = f(xSI) and wh = g(xMI) with f(∙) and g(∙) real continuous function such that f<sup>′</sup> < 0 and g<sup>′</sup> < 0. Assume that a value ∆i∈(0,1) exists such that f<sup>−1</sup> (∆i) = g<sup>−1</sup> (∆i) = x̃ > 0. We know, from proposition 2, that such a value is f unique, which means that the functions f(∙) and g(∙) intersect only once. In particular, if Ii > nf h h (<)Ii and wh > ∆i, then xSI −xMI > (<)0 and vice-versa for wh < ∆i. Figure 1 illustrates the f nf case for I > I . i i

> 4 This is what happens only if the threshold value ∆i is not too high: when ∆i is “very” high, then the condition wh > ∆i will not be satisfied, neither for poor nor for nonpoor households and using multiple index deflation will “reward” all households and lead to consistently lower FGT estimates compared to a single deflator approach.

10
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lying on the segment a−b, in Figure 1 would be classified as non-poor if SI were used, while will be classified as poor if MI were used.

Finally, it is interesting to note that if z= x̃ no household would change its poverty status, which implies that the headcount poverty rate would be insensitive to the deflation procedure, while higherorder FGT-poverty measures and other poverty measures that are sensitive to the distance from the poverty line would be affected by the choice between the SI and MI deflation.

The general result enunciated in proposition 2 is easier to interpret when we focus on the Laspeyres index (α= 0), as shown in Corollary 1.

# **Corollary 1 (Distributional bias with Laspeyres index)**

Assume (r, k, kf, knf, α, αf, αnf ) = (r, k, kf, knf, 1,1,1 ), and k< K. If Lfi > (<)Lnfi , then xSIh − h xMI > 0 if and only if: where wi is the average food budget share in region (or period) _i_ .

Proof: See the appendix.

When both single and multiple price index deflation approaches use a Laspeyres index, the threshold value ∆i is given by the food budget share wi, determined by the aggregation level of the index (for example, wi may be the average food budget share in region _i_ , and/or time period _i_ ), multiplied by f f nf Li /Li , that is the ratio of the food- to the general price index. If Li > Li , the ratio is always greater than 1 and the condition wh > ∆i is satisfied by households with a food budget share strictly larger than wi. If the price indexes have been computed by using the food budget shares of the poor households,<sup>6</sup> then, because of the Engel law, wi ≅max{wh} and therefore the condition wh > ∆i can be hardly binding. This implies that the use of the MI approach increases households’ consumption h h aggregates with respect to the SI approach for _almost all_ households: xMI > xSI for most _h_ . The

> 6 On this, see Deaton and Zaidi (2002).

12
wedge in the consumption aggregates is such that all poverty counts and gaps are expected to be affected, as well as inequality indexes with concave evaluation functions (GEIs, Atkinson’s indexes with a inequality aversion parameter strictly greater than 1, etc.).

# **4 Which approach is better?**

The results presented in section 3 identify the existence of systematic biases related to the choice of a single versus multiple index deflation approach, but they do not have immediate implications from the _normative_ viewpoint. Is it possible to express a preference for either approach to adjust the nominal consumption aggregate? Is the SI approach “better” than the MI approach? To answer this question, we need to identify a benchmark, a reference theoretical measure of welfare that we should approximate by using the SI or the MI adjustment procedure.

This section is technical in its content and can be skipped by readers who are not interested in the process that leads to our result. The discussion is organized into two separate blocks. Section 4.1 reviews the theory underlying the results obtained in section 4.2.

# **4.1 The theoretical framework**

The _money metric utility function_ (MMU) is a convenient monetary measure of individual welfare (Samuelson 1974). The MMU<sup>h</sup> function is defined as the _minimum expenditure_ for reaching the utility level u(q<sup>h</sup> ) associated to the consumption bundle q<sup>h</sup> , given the market price vector p<sup>h</sup> : where c(u(q<sup>h</sup> ), p<sup>h</sup> ) denotes the _expenditure_ or _cost function_ for household _h_ (Deaton and Muellbauer 1980: 47, McKenzie 1957).

Under the standard assumptions that households are rational and minimize their expenditure, the empirical counterpart of MMU<sup>h</sup> in equation 6 is given by p<sup>h</sup> q<sup>h</sup> , where q<sup>h</sup> is the consumption bundle that, given prices p<sup>h</sup> , provides the utility u(q<sup>h</sup> ) at the minimum cost. The household expenditure p<sup>h</sup> q<sup>h</sup> , however, would not be comparable across households: both equation 6 and p<sup>h</sup> q<sup>h</sup> cannot be 13
used to carry out interpersonal or intertemporal welfare comparisons.<sup>7</sup> The problem can be overcome by introducing a _reference price vector_ p<sup>0</sup> . Hence, instead of equation 6, we can consider the following money metric utility: Equation 7 would be both utility consistent and comparable across households, but it suffers from a major drawback: preferences cannot be observed and equation 7 cannot be estimated based on the consumption behavior of household _h_ available from survey data. Household _h_ minimizes its expenditure given the market price vector p<sup>h</sup> , not the reference price vector p<sup>0</sup> .

A possible solution is to introduce the _Paasche-Konüs true cost of living index_ (TCLIPKh ), defined as follows: The true-cost-of-living index (TCLI) defined in equation 8, originally due to Konüs (1939), has been used widely in applied economics (Ray 2018). Interpretation is as follows: for household _h_ , TCLIPKh compares the cost of reaching a fixed level of utility u(q<sup>h</sup> ) at two different price situations p<sup>h</sup> and p<sup>0</sup> , such as the prices in two time periods or those in different locations.

Given equation 8, equation 7 can be rewritten as follows: h where x<sup>h</sup> is the total expenditure of household _h_ . If the analyst is able to estimate TCLIPK, the expression in equation 9 would be _i)_ utility-consistent, _ii)_ observable, and _iii)_ suitable for interpersonal comparisons.

> 7 To see why, suppose that household _h_ and household _j_ face different market prices, say ph and pj, respectively; suppose that household _j_ achieves the same utility level of household _h_ , u(q<sup>h</sup> ), but does so with different expenditure level, due to different prices it faces:

When this is the case, then MMU<sup>h</sup> and MMU<sup>j</sup> differ, even if the utility level is the same for the two households: the use of MMU as a welfare measure would lead to misranking households _h_ and _j_ (when two households achieve the same utility level they should be ranked equally well off).

14
The problem with TCLIPKh is precisely its estimation, which requires the specification of a utility function, and the estimation of the preference parameters for its application. Deaton and Zaidi (2002) – henceforth DZ – put the issue as follows: “The exact calculation of money metric utility requires knowledge of preferences. Although preferences can be recovered from knowledge of demand functions, we typically prefer some shortcut method that, even if approximate, does not require the estimation of behavioral relationships with all the accompanying assumptions, including often controversial identifying assumptions, and potential loss of credibility.” For this reason it can be argued that equation (4) does not qualify as a computationally convenient welfare measure.

The shortcut method mentioned by DZ consists in approximating the MMU<sup>h</sup> in equation 9. The idea is simple and ingenious: use a first order Taylor expansion of MMU<sup>h</sup> to show that: where P<sup>h</sup> = p<sup>h</sup> q<sup>h</sup> ⁄p<sup>0</sup> q<sup>h</sup> is a household-level Paasche index. MMU<sup>h</sup> in equation 10, corresponding to equation 2.6 in DZ’s original paper: that is a monetary welfare measure compliant with a number of desirable properties: it can be calculated by the ratio of total nominal household expenditure (x<sup>h</sup> ) to a household-level Paasche index (P<sup>h</sup> ). The relation is now an approximation, whose precision ultimately depends on the gap between the prices faced by the household and the reference set of prices. If we compare equation 9 and equation 10, it is useful to note that the Paasche index P<sup>h</sup> can be interpreted as a proxy for the TCLIPKh : the closer is Ph, empirically, to TCLIPKh , the better DZ’s shortcut in equation 10 is expected to work.

Blackorby and Donaldson (1985, 1987, 1988) – BD henceforth – observed that MMU<sup>h</sup> is not necessarily a concave function of x<sup>h</sup> .<sup>8</sup> This creates problems in evaluating redistributive policies, as it may lead to results that do not satisfy the principle of transfer: the use of MMU, in other words, comes with the risk of using distributionally insensitive tools, such as poverty and inequality measures obtained by aggregating MMU (Foster, Greer and Thorbecke, 1984). To overcome this issue, BD proposed an alternative welfare measure, the _welfare ratio_ :

> 8 The reason MMU in equations 9 and 10 is not concave is that TCLIPKh , at the denominator, depends on the utility level u,  whose functional form is not necessarily known.

15
The welfare ratio is the ratio between the expenditure function of household _h_ and the minimum expenditure necessary to achieve a reference utility level uz (typically, the utility level around a poverty line _z_ ). In equation 11, the expenditure c(uz, p<sup>h</sup> ) is a sort of numeraire, so that the WR<sup>h</sup> can be interpreted as a multiplier of the minimum expenditure necessary to achieve uz. If WR<sup>h</sup> = 1.2, for instance, then the current expenditure for household _h_ is 20% higher than the expenditure needed to achieve uz. If the expenditure at the denominator is interpreted as a poverty line, then WR<sup>h</sup> tells how many poverty baskets can be purchased by the current spending of household _h_ .

To make WR<sup>h</sup> operational, it needs to be transformed into a monetary value, which can be done simply by multiplying WR<sup>h</sup> by c(uz, p<sup>0</sup> ), the monetary cost of uz at a reference price vector p<sup>0</sup> : Equation (7) can be simplified by introducing the _Laspeyres-Konüs true cost of living index_ : The TCLI in equation 13 can be compared with the Paasche-Konüs index in equation 8. Conceptually, they are both TCLIs, and they both serve the purpose of adjusting nominal expenditures for differences in purchasing power. The difference between the two indexes is the reference utility level used in their definitions: in equation 8 Paasche uses u<sup>h</sup> , while in equation 13 Laspeyres uses uz. By plugging-in equation (8) in equation (7), we obtain the following result: To interpret equation 14 it is convenient to compare it with equation 9. The key difference is that in the former TCLILKh does not depend on household expenditure xh, so that WR̃h is a linear function of x<sup>h</sup> .

Regarding equation 14, DZ observe that to the extent to which is plausible to assume that the consumption bundle around the poverty line is insensitive to market prices, then TCLILKh can be approximated by a _Laspeyres price index_ : 16
Both the welfare ratio WR defined in equation 15, and the money metric utility MMU defined in equation 10, satisfy the properties identified at the beginning of this section, and therefore represent strong candidates for proxying the living standards based on household budget survey data.

The common structure of the two measures facilitates their interpretation and practical implementation: they are both defined as the ratio between nominal total household expenditure to a true-cost-of-living index. They can both be approximated by replacing TCLI with a price index, a Paasche index in the case of MMU, a Laspeyres index in the case of the WR. The choice of the index (whether Paasche or Fisher) implies a different choice of the underlying price measure. In this sense, the question “which index is the best choice” is not a good question: No ‘best’ index exists. By choosing the price index, the analyst chooses the welfare measure: Paasche implies a MMU, Laspeyres implies a WR, Fisher, Törnquist and other indexes imply different welfare measures.

The analysis reviewed in this section hopefully helps to clarify the recommendation put forward in Deaton and Zaidi (2002) and confirmed by Mancini and Vecchi (2022), and sets the stage of the analysis carried out in the next section. The aim is not to contribute to the debate on the best welfare measure (MMU versus WR), but rather to focus on different price adjustment procedures, SI versus MI, _taking as given that the appropriate welfare concept has been chosen_ .

# **4.2 Can the SI and MI methods be ranked?**

Section 4.1 discusses two welfare concepts, _MMU_ and _WR_ , which we use here as benchmarks for assessing whether the practice of using one deflator (SI) is preferable than the practice of using two deflators (MI). Consider first the MMU approach. According to equation 9, the money-metric utility function for household _h_ , MMU<sup>h</sup> , can be written as the ratio between household _h_ expenditure and the true-cost-of-living index (TCLI) associated to the utility level _u_ achieved by household _h_ at the prevailing prices p<sup>h</sup> : where TCLIPKh is the Paasche-Konüs TCLI introduced in equation 8. The equation above, corresponding to equation 9, identifies our benchmark. While methods SI and MI can be assessed 17
against this theoretical benchmark, in practice this is difficult to achieve as the estimation of TCLIPKh requires knowledge of household preferences, which are generally unobserved: comparing the “first best” with the SI and MI approach is an almost impossible _empirical_ exercise. In this section we draw some _theoretical_ conclusions by imposing mild restrictions on household preferences.

# **Proposition 3 (SI method is preferable to MI method when MMU is the welfare measure)**

Assume that household _h_ ’s preferences (≽h) can be represented by a continuous, increasing, and f concave utility function, and that (H, k, kf, knf, α, αf, αnf ) = (H, k, kf, knf, 0,0,0 ). Then, if Ii > nf h (<)Ii and wh < (>)∆i , then SI approach provides a better estimate of MMU than the MI approach.

Proof: See the appendix.

The result illustrated in proposition 3 depends on the _ordering_ of TCLIPKh and the Paasche price index. As discussed in Section 4.1, if preferences are regular, we know that TCLIPKh > Pi(H, k) which h implies MMU<sup>h</sup> < xSI. In this case, whenever conditions determined by the price environment and household consumption patterns determine that MI approach tends to overestimate the real f nf expenditures with respect to SI approach, i.e. if Ii > Ii and wh < ∆i, using the MI approach produces an even larger deviation from MMU<sup>h</sup> compared to the SI approach. It should be observed h h that we cannot achieve a symmetric conclusion if xMI < xSI. In this last case we could have h h h xMI < MMU < xSI and the bias induced by the MI approach can be, in absolute terms, either larger or smaller than the bias implied by the SI approach. However, while – as shown by proposition 3 - we can identify conditions under which SI approach leads to better approximation of MMU<sup>h</sup> compared to the MI approach, we cannot identify any conditions under which MI should be preferred to SI.

In line with what discussed in section 4.1, a similar analysis can be carried out using the welfare ratio measure, as defined in equation 14 and here repeated for convenience: 18
## **Proposition 4 (SI method is preferable to MI method when WR is the welfare measure)**

Assume that household _h_ preferences (≽h) can be represented by a continuous, increasing, and f concave utility function and that (H, k, kf, knf, α, αf, αnf ) = (H, k, kf, knf, 1,1,1 ). Then, if Ii > (<)Iinf and wh > (<)∆i SI approach provides a better estimate of WR̃h than MI approach. Proof: See the appendix.

As for the case of MMU, proposition 4 shows that we cannot identify conditions that would lead to concluding unambiguously in favor of an MI approach.

Overall, propositions 3 and 4 show that, irrespective of choice of the welfare measure, the SI approach weakly dominates the MI approach. The result is somewhat surprising, if seen from the angle statisticians are used to: the use of multiple deflators does make sense when one seeks consistency between numerators (say, food and nonfood expenditures) and denominators (the deflators). Clearly, the use of a food deflator and a nonfood deflator does a better job in capturing the change in purchasing power of the two sub-aggregates. From the economist’s angle things look different. The main aim of welfare analyst is to approximate a well-being measure that allows to make consistent interpersonal comparisons. From this point of view a single deflator could be a preferable solution.

# **5 Sequence of temporal and spatial price adjustment**

A problem faced by practitioners when dealing with price adjustment pertains to the issue of sequencing between temporal and spatial price adjustments. This issue has been mostly overlooked by the literature which has generally focused on temporal and spatial deflation independently. Notably, for what concerns welfare measurement, the focus has been prominently on spatial adjustment, i.e. how to properly compare welfare of households living in different regions, overlooking the temporal dimension, i.e. how to compare households that are interviewed at different points in time. For instance, in Deaton and Zaidi (2002), on addressing issues related to the construction of the consumption aggregate, they observe “when we are working with a single crosssectional household survey, the price variation is less temporal than spatial; people who live in different parts of the country pay different prices for comparable goods.” Deaton and Zaidi (2002: 7) 19
This simplification does not fit well in a number of situations: for many countries within-survey inflation rates are far from being negligible especially in cases in which data collection spread through different seasons up to covering an entire year. Leaving aside countries where inflation has escalated to hyperinflation, it is not uncommon in present days to see double-digit yearly inflation rates in countries all around the world. In this case the temporal dimension of Ii(r, k, α) cannot be ignored.

National statistical offices often provide temporal consumption price indexes (CPI) that can be relied on to adjust household expenditures for variation over time of the cost of living. The strategy that we propose to deal with spatial-temporal variation takes advantage of the availability of official CPIs, take them as exogenous, and use it to adjust expenditures for within-survey inflation; in particular, we investigate if the _sequence_ of temporal and spatial price adjustment matters.

To illustrate, consider the case of a Paasche price index calculated at the household level (we use here notation introduced in Section 3): where pjh is the market price of commodity j= 1, 2, ... , k for household h, pj0 is the price of h commodity _j_ of a reference group 0, and wj is the budget share of household _h_ for commodity _j_ .

It is common practice to proxy market prices with _unit values_ (Gibson and Rozelle, 2005), so that the estimated version of the Paasche formula above becomes: h 0 where uvj and uvj denote the unit values of commodity j for the _h_ -th and reference group, h respectively. Unit values are defined as the ratio between expenditure xj on commodity j and the corresponding quantity qjh consumed.

In this situation, the key decision is whether to calculate uvjh and ŵjh at _current_ (xjht ) or _inflationadjusted_ values (x̃jht = xjht ⁄CPIt). The analyst has two strategies: 20
- 1) the “SPI first, CPI next” option consists in calculating uvjh and ŵjh based on inflation- _un_ adjusted expenditures, and then apply the CPI to the spatially-adjusted consumption aggregate.

- 2) the “CPI first, SPI next” option consists in calculating uvjh and ŵjh using inflation-adjusted expenditures, and then construct a spatial index based on CPI-adjusted “components” (unit values and budget shares).

In the rest of this section we analyze and compare the two procedures. We focus first on the impact of the two strategies on unit values (section 5.1), and next on the budget share (section 5.2). The overall conclusion is that the correct strategy is the second option, “CPI first, SPI next”. The use of nominal expenditures, that is not adjusted for inflation (first option), instead of real expenditures (second option), introduces a bias in the estimates of relative prices (unit values) and budget shares, both when using Paasche, Laspeyres and Fisher formulae as spatial price deflators.

# **5.1 The impact of spatial-temporal adjustments on unit values**

In this section we consider the impact of the two strategies “SPI first, CPI next” and “CPI first, SPI next” on the estimation of relative prices.

Let qjht denote the quantity of commodity _j_ consumed by household _h_ in month (or period) _t_ . t Similarly, let xjhdenote the nominal expenditure of household _h_ on commodity _j_ during month (or period) _t_ . Let _T_ be the number of periods covered by the survey. Unit values can be calculated by using _nominal_ expenditure (first option): Alternatively, unit values can be calculated by using _real_ (inflation-adjusted) expenditures (second option): where CPIt is the exogenous temporal price index for month/period _t_ . It can be shown that if the expenditures on item _j_ of household _h_ and of the reference group _0_ refer to the _same period_ t= t0, 21
then temporal deflation has _no impact_ on _relative_ unit values. This result holds true in the purely theoretical and unrealistic case when households concentrate their expenditures in a single month of the year, and that month coincides with the month chosen for the reference group.

The typical case is when expenditures of household _h_ on item _j_ refer to a single period th ≠t0, that is, when the temporal consumption pattern is heterogeneous across households: household purchases are spread across different months in the survey year. Even under the assumption that the same household concentrate expenditures on item _j_ in a single month during the year, we obtain h 0 h 0 uvj (CPIt0)⁄uvj (CPIt0) ≠uvj ⁄uvj . In general, adjusting unit values for inflation leads to unit value ratios (henceforth, uv-ratios) that are different from those obtained using nominal unit values, and this will lead to different estimates for the Paasche index: Equation (17) helps gauge the consequences of adjusting unit values with a temporal CPI, prior to calculating a spatial price index or vice versa. The equation shows that the difference between _real_ uv-ratios (LHS) and _nominal_ uv-ratios (RHS) is driven by CPIt0⁄CPIth, the ratio between CPIs in months t0 and th. The ratio equals 1 when prices are stable between t0 and th, but the case of interest is when CPIt0⁄CPIth ≷1. Assume a consumption pattern such that th > t0 for all items (this is likely to happen when, for example, household _h_ is observed at a later period than the reference group). Assume also that there is a positive intra-survey inflation. Under this scenario CPIt0⁄CPIth < 1, which implies that nominal uv-ratios (that is, unit values based on nominal expenditures, on the RHS of the last equation) are higher than CPI-adjusted unit values. In turn, via equation (16), this implies that using nominal uv-ratios leads to an overestimation of P̂h [and vice versa if th < t0]. This matches with intuition: the use of CPI-unadjusted unit values does not account for the loss of purchasing power due to the increase in the general price level. Ultimately, all other things being equal, headcount poverty rates, as well as other FGT-poverty measures, calculated on CPI-unadjusted budget shares are expected to be _upward_ biased with respect to poverty rates that would be obtained on consumption 22
aggregates deflated with CPI-adjusted budget shares.<sup>9</sup> The bias would be systematic and linked to the timing of the interview – this is clearly to be avoided.

# **5.2 The impact of spatial-temporal adjustments on budget shares**

In this section we consider the impact of the two strategies “SPI first, CPI next” and “CPI first, SPI next “on the estimation of household-level budget shares, the second component of the P̂h index in equation 16.

The budget share of commodity _j_ for the h-th household can be calculated using _nominal expenditure_ s (option 1): h or, alternatively, wj can be calculated using _real expenditures_ (option 2): Suppose that all expenditures of household _h_ are concentrated in one period, say th. Then, for all items _j_ we have: In this particular case, the budget shares of household _h_ do not depend on temporal deflation. The same is true even if consumption is collected with different recall periods, but the adjustment for inflation is carried on with as single deflator that refers to the interview month th. In this second case, which is the most frequent, we have:

> 9 Note that the bias involves unit values and does not depend directly on the type of index used.

23
On the other hand, if expenditures of household _h_ span over more subperiods of the survey year and h h we use the appropriate temporal deflators, then wj (CPIt) ≠wj . In particular, if item _j_ has been purchased before the other items and there is a positive intra-survey inflation rate, by using nominal expenditures (option 1) we _overestimate the budget share of item j_ with respect to the estimates obtained by using real expenditures. Such an overestimation of the budget shares translates into an underestimation of the Paasche index P̂h, which leads to underestimated poverty measures.

The same result applies when using a Laspeyres index, instead of a Paasche price index, that is Ii(r, k, α) = Ii(r, k, 1). To show this, let wjz be budget shares of “the poor” that play the role of the reference budget share for the Laspeyres price index. Hence, if we start from _nominal expenditure_ we obtain: t where xjz is the _nominal expenditure_ on item _j_ in period _t_ of the households belonging to the reference group _z_ .

By using _real expenditure_ instead, we get: It is easy to see that if all expenditures of households belonging to the reference group _z_ are concentrated in a single period for every item or the adjustment for inflation is carried on with as Z z single deflator that refers to the interview month, then wj = wj (CPIt). In all the other circumstances we obtain conclusions similar to the ones envisaged for the case of a Paasche price index.

To sum up, the use of nominal expenditures (option 1), instead of real expenditures (option 2), introduces a potential bias in the estimates of relative prices and budget shares, both when using Paasche and Laspeyres as spatial price deflators. The severity of the bias that eventually affects the real consumption aggregate depends on the extent that household expenditures are concentrated on a 24
single subperiod of the survey year: the more expenditures are concentrated, the less significant is the distortion.

At the same time, by using nominal expenditure we avoid the distortion induced by the fact that the _temporal deflator may not correctly reflect the dynamics of relative prices_ during the survey year. A correct estimate of the price of commodity _j_ should be given by: j where CPIt is a temporal price index specific to commodity _j_ . If all prices increase at a uniform j h j h growth rate, then CPIt = CPIt and uvj (CPIt ) = uvj (CPIt). However, if the price of commodity _j_ j h h j increases more than the general price level, then CPIt > CPIt ⇒uvj (CPIt) > uvj (CPIt ) and this implies that by using the real consumption aggregate to calculate unit values _we overestimate the market price_ for commodity _j_ .

The general recommendation is that if _i)_ the dynamics of the general price level during the survey year is sustained, and _ii)_ the official temporal CPI can be assumed to capture the dynamics of the most important commodities consumed by the households, the analysts should use the strategy “CPI first, SPI next”, that is, first adjust the nominal consumption aggregate for intra-survey temporal price variation and, then use the adjusted expenditures to estimate a spatial price index that can be used to further adjust households’ expenditures.

# **6 An empirical illustration**

In this section, we illustrate the empirical implications of choosing a single index (SI) versus multiple index (MI) approach in spatial (and temporal) price adjustment, using data from the Islamic Republic of Iran’s 2019 Household Expenditures and Income Survey (HEIS).

# **6.1 Construction of single and multiple price indexes**

Similar to most expenditure surveys, the Islamic Republic of Iran’s HEIS only collects information on expenditure and quantities consumed for food items. Unit values obtained from reported food expenditures and quantities allow for the estimation of a survey-based food spatial price index. On 25
the other hand, when it comes to the measurement of non-food spatial price variation, options are characteristically limited. One of these options is to focus on rent expenditures,<sup>10</sup> pin down the characteristics of a “typical” dwelling and predict what the average rent for this dwelling would be in different areas of the country, via hedonic regression. Ratios between predicted prices for each region and the country average can be interpreted as a (Paasche) price index constructed using only one commodity.

The question examined in this section is how to proceed in the construction of single and multiple indexes that are consistent with the theoretical framework presented in previous sections, _after_ the food and rent deflators have been estimated. The approach followed in this empirical application for the construction of a “single” index and corresponding food and non-food price indexes are summarized in the equations below: i where j denotes each food item (belonging to the food commodity group), wnf denotes the budget share of non-food expenditure (including rent, and all other non-food items), pr denotes predicted rents. Notice that in this application, we use the Paasche formula to estimate the single spatial price index (equivalently, α= 0). As in section 3, i= 1, ... , r, where r refers to the survey strata defined at the region – urban/rural level.

Correspondingly, the formulas used for the construction of the spatially adjusted aggregate using SI and MI approach are the following:

> 10 Rent expenditures constitute actual rents for tenants and self-reported rents for owners/non-market tenants. Hedonic regressions have been used to validate self-reported rent estimates.

26
<!-- Start of picture text --> x<br>xSI = 0 0<br>kf wi pj +wi p r (18)<br>j=1 j i nf h<br>[∑ pj pr ]<br>xf<br>xMI = 0 + xnf<br>kf wi pj (19)<br>[∑j=1 j pji ] [p p r rh0 ]<br><!-- End of picture text --> In this application, the non-food deflator is a single commodity index (based on rent only): this is tantamount to assuming that the spatial price dynamics of all non-food items is the same as the one captured by housing prices. Reflecting the inherent non-tradable nature of housing services and typical tradeable nature of food items, this deflation approach is expected to maximize the relative spatial variability of the non-food price index compared to the food price index, and it is hence well suited to showcase empirical implications of proposition 2.<sup>11</sup> Regarding the temporal dimension of price adjustments, the _aggregate_ monthly CPI index, as estimated by the Statistical Center of Iran (SCI), is used throughout the analysis.<sup>12</sup>

# **6.2 Results**

Proposition 2 delivers a prediction as to the sign of the difference between a welfare aggregate deflated using a single index and one deflated using a multiple index approach. According to proposition 2, this difference would depend on the relative level of food and non-food price indexes as well as on observed consumption patterns (implicit differences in the level of welfare), i.e. on the food share level. What makes the analysis of data for the Islamic Republic of Iran particularly useful for an empirical illustration of Proposition 2 is the significant spatial variation observed in the Islamic 11 In practice, this assumption may seem too extreme, and welfare analysists may be reluctant to adopt it. A possible alternative, feasible under typical data constraints, would be to assume that the prices of non-food/nonrent items do not vary spatially, and that only housing expenditures are associated to the price dynamics of rent. Results obtained using this alternative approach are available in Appendix 2.

12 This is to maintain a clean connection with the theory presented in section 3. Results capture entirely and exclusively the impact of the SI and MI approaches for _spatial_ deflation.

27
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particular, in Teheran the food price index is smaller than the non-food price index, and ∆ is equal to 24; therefore, proposition 2 implies that switching from an SI deflation strategy to an MI strategy will _decrease_ real expenditures of all households with a food budget share _lower_ that 24%. Vice-versa, in the rural Northeast, the food price index is larger than the non-food price index, and ∆ is equal to 43.7; accordingly, switching from an SI deflation strategy to an MI strategy will _decrease_ real expenditures for all households with a food budget share _higher_ than 43.7%. The distributional effects of the two scenarios are clarified by the bottom panel of Table 2, which shows average budget shares by decile of (nominal) per capita expenditure. According to proposition 2, we should expect the MI approach to “underestimate” welfare compared to the SI approach for the top 4 deciles of the welfare distribution in urban Tehran, and for the bottom 7 deciles in rural Northeast. In Table 2, deciles where households are “penalized” by MI have been highlighted by using the bold type.

**Table 2. Implications of Proposition 2**

||**Teheran metro urban**|**Northeast rural**|
|---|---|---|
|Food Paasche (I<sup>f</sup>)|1.06|0.96|
|Non-food Paasche (I<sup>nf</sup>)|2.17|0.41|
|∆|24.0|43.7|
|Proposition 2 implies:|xMI<br>h< xSI<br>hwhenwh< ∆|xMI<br>h< xSI<br>hwhenwh> ∆|
|**Per capita expenditure deciles**|**Average food bud**|**get shares, **wh **(%)**|
|1|40.8|**53.0**|
|2|33.0|**49.5**|
|3|28.9|**47.5**|
|4|27.8|**44.6**|
|5|25.3|**44.3**|
|6|23.3|**42.1**|
|7|**20.8**|**42.9**|
|8|**20.2**|40.9|
|9|**18.1**|42.7|
|10|**16.1**|41.6|

h h Note: Expenditure deciles where xMI < xSI are in bold type.

_Source_ : Authors calculation based on HEIS 2019 29
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**Table 3. Percent changes in poverty measures (MI-SI)**

|||$3.65<br>|||$6.85<br>||
|---|---|---|---|---|---|---|
||**H**|**PG**|**PG2**|**H**|**PG**|**PG2**|
|National|-7.02|-7.81|-11.87|-0.29|-1.21|-2.67|
|Urban|-15.24|-17.37|-23.60|-1.90|-5.04|-8.32|
|Rural|22.18|24.95|23.42|4.41|10.47|15.12|
|Tehran metro Urban|-32.93|-36.50|-47.41|-5.72|-13.12|-20.04|
|Caspian Rural|2.93|2.34|2.61|0.84|1.17|1.71|
|Caspian Urban|-5.51|-4.33|-3.47|-5.07|-3.04|-3.67|
|Northwest Rural|8.46|15.88|0.00|12.86|13.37|12.55|
|Northwest Urban|0.85|0.23|-1.34|3.15|1.57|1.61|
|Northeast Rural|84.53|87.78|118.75|14.26|31.86|45.17|
|Northeast Urban|-2.39|-3.19|-3.07|-0.03|-1.43|-1.92|
|Central Rural|21.79|23.17|19.16|1.63|8.54|12.65|
|Central Urban|-7.92|-9.87|-11.84|-1.83|-2.67|-4.36|
|Southeast Rural|24.96|30.06|29.50|0.55|10.06|17.11|
|Southeast Urban|1.10|-0.36|-0.98|0.88|0.03|-0.03|
|Persian Gulf Rural|20.24|28.48|30.32|6.51|11.66|16.36|
|Persian Gulf Urban|-0.46|0.45|-0.85|0.30|0.41|0.82|
|Zagros Rural|54.26|21.21|0.00|11.50|18.78|23.91|
|Zagros Urban|2.62|2.61|3.90|0.35|1.41|1.76|

_Source_ : Authors calculation based on HEIS 2019 Overall, differences in terms of poverty estimates between the two deflation approaches are sensitive to the value of the poverty line – in line with theoretical predictions (Figure 1). Moreover, given that spatial price indexes are defined at the region/area level and that biases in urban/rural areas would go in opposite directions, differences in poverty estimates the national level tend to be relatively smaller compared to the ones at the subnational level. Considering the $3.65 poverty line, the MI approach would result in a 7 percent underestimation of poverty at the national level, which is the composite effect of a 15 percent underestimation of poverty in urban areas and a 22 percent overestimation of poverty in rural areas, compared to the SI approach. At the higher value of the poverty line ($6.85), the bias at the national level is negligible, while poverty in urban (rural) areas still results underestimated (overestimated) when using the MI approach compared to the SI approach.

In urban Tehran, compared to single index deflation, the MI approach would result in a 33 percent (5.7 percent) underestimation of poverty at the $3.65 ($6.85) poverty line. In rural Southeast the MI 32
approach would result in a 25 percent (0.55 percent) overestimation of poverty at the $3.65 ($6.85) poverty line. Rural regions of Zagros and Northeast show an even higher poverty over-estimation f nf bias when using the MI approach, reflecting a relatively larger differential between Ii and Ii as well as relatively lower food budget shares.

The effect of different deflation methods on inequality measures is shown in Table 4. MI generally “mitigates” inequality in urban areas and “exacerbates” it in rural areas, with respect to SI, though magnitudes are very much dependent on the chosen measure. Considering these effects is relevant not only when inequality is the statistic of interest, but also when poverty dynamics is of interest (Ferreira 2012), particularly in settings where the mechanics underlying poverty changes are investigated (e.g., Datt and Ravallion 1992).

**Table 4. Percent changes in inequality measures (MI-SI)**

||**Gini**|**MLD**|**Theil**|**Atkinson**<br>ε=0.5|**Atkinson**<br>ε= 2|
|---|---|---|---|---|---|
|National|-0.95|-2.25|-2.67|-2.25|-2.02|
|Rural|2.97|6.42|5.10|5.58|5.47|
|Urban|-2.10|-4.80|-4.85|-4.50|-4.35|
|Tehran metro Urban|-5.86|-12.87|-11.95|-11.58|-11.03|
|Caspian Rural|-0.05|-0.03|-0.38|-0.19|0.21|
|Caspian Urban|-0.81|-1.69|-1.63|-1.60|-1.40|
|Northwest Rural|0.61|0.62|-2.76|-0.64|1.32|
|Northwest Urban|0.49|0.98|1.09|0.98|0.66|
|Northeast Rural|5.17|11.66|10.99|10.77|10.57|
|Northeast Urban|-0.36|-0.75|-0.68|-0.68|-0.66|
|Central Rural|1.89|4.29|3.40|3.72|3.87|
|Central Urban|-0.80|-1.84|-1.65|-1.67|-1.74|
|Southeast Rural|9.68|19.62|19.04|18.62|13.31|
|Southeast Urban|-0.03|-0.11|-0.05|-0.07|-0.15|
|Persian Gulf Rural|6.26|12.94|14.32|13.00|10.07|
|Persian Gulf Urban|0.19|0.41|0.44|0.40|0.33|
|Zagros Rural|4.71|9.60|9.94|9.39|7.64|
|Zagros Urban|0.36|0.72|0.67|0.67|0.61|

_Source_ : Authors calculation based on HEIS 2019 33
Differences between SI- and MI-based results presented so far are compatible with compensations in the re-ranking of individuals across the welfare distribution. The rearrangement of households along the distribution of expenditure can have important policy consequences. For this reason, in the final part of this section we focus on the non-anonymous distributional implications of switching from SI to MI, borrowing a few tools from the mobility literature.<sup>15</sup>

**Table 5. Probabilities of transition in and out of poverty when switching from SI to MI ($6.85 PL)**

|**Area**|**% of poor individuals**<br>**using SI**<br>**who become non-poor**<br>**using MI**|**% of non-poor**<br>**individuals using SI**<br>**who become poor**<br>**using MI**|**number of individuals**<br>**changing poverty**<br>**status**|
|---|---|---|---|
|National|3.7|1.2|1,620,300|
|Tehran metro Urban|9.3|1.7|686,288|
|Caspian Rural|0.2|0.5|12,923|
|Caspian Urban|5.4|0.1|62,248|
|Northwest Rural|6.2|3.8|133,883|
|Northwest Urban|0.0|0.9|49,406|
|Northeast Rural|5.3|3.6|102,378|
|Northeast Urban|0.7|0.2|22,455|
|Central Rural|2.0|1.1|49,644|
|Central Urban|2.2|0.1|62,538|
|Southeast Rural|4.8|8.8|197,767|
|Southeast Urban|0.5|1.1|40,399|
|Persian Gulf Rural|3.4|3.6|90,712|
|Persian Gulf Urban|0.3|0.2|14,949|
|Zagros Rural|4.2|2.9|95,337|
|Zagros Urban|0.6|0.2|18,733|

_Source_ : Authors calculation based on HEIS 2019 First, we focus on the binary outcome of whether a household is labeled as poor or not, under the two alternative deflation scenarios. Taking the SI approach as a starting point, Table 5 shows the proportion of individuals that change their poverty status when the MI approach is adopted. Nationally, 3.7% of individuals who were poor under SI are no longer poor when using MI, while

> 15 We follow, by and large, the analysis presented by Ceriani, Olivieri and Ranzani (2022) in their evaluation of the distributional impact of imputed rent estimation.

34
1.2% of those who were non-poor under SI become poor when using MI. Overall, more than 1.5 million individuals in the country see their poverty status changed (from poor to non-poor or viceversa) purely because of the choice between SI and MI. The single most relevant contributor to the size of these flows is Tehran, which is more populous to begin with, but several rural areas also display significant transitions in poverty status.

Table 6 gives a more detailed account of the re-ranking of households across the expenditure distribution: the population is sliced into decile groups, and a transition matrix is computed (the cells report the estimated probabilities of “jumping” from one expenditure group to another, again when switching from SI to MI). For example, in urban Teheran, a family that falls in the fifth decile group of real per capita expenditure when SI is used has a 57% probability of falling into the fifth decile again when using MI, a 25% chance of falling down to the fourth decile, and an 18% chance of jumping up to the sixth decile. In the rural Northeast in particular, jumps can span up to three groupings, showing that re-rankings can be considerable in magnitude.

The contents of Table 6 can be summarized using mobility indexes. The Prais-Shorrocks index (Prais 1955, Shorrocks 1978), ranges between 0 (complete immobility) and 1.1 (complete origin independence) in this application, and is 0.33 for Teheran and 0.36 for the rural Northeast. The average jump index (Bartholomew 1973) ranges from 0 (complete immobility) to 1 (complete reversal), and it is 0.30 for Teheran and 0.36 for the rural Northeast, suggesting that average crossdecile movements due to the choice of MI vs. SI are substantive.

35
**Table 6. Probabilities of transition across deciles of per capita expenditure, from SI (rows) to MI (columns)**

|||||||**Tehera**<br>Decil|**n urban**<br>es MI|||||
|---|---|---|---|---|---|---|---|---|---|---|---|
|||1|2|3|4|5|6|7|8|9|10|
||1|0.86|0.13|0|0|0|0|0|0|0|0|
||2|0.05|0.70|0.25|0|0|0|0|0|0|0|
||3|0|0.08|0.63|0.29|0|0|0|0|0|0|
||4|0|0|0.12|0.64|0.21|0|0|0|0|0|
|Deciles|5|0|0|0|0.25|0.57|0.18|0|0|0|0|
|SI|6|0|0|0|0|0.19|0.61|0.19|0|0|0|
||7|0|0|0|0|0|0.29|0.59|0.12|0|0|
||8|0|0|0|0|0|0|0.28|0.67|0.06|0|
||9|0|0|0|0|0|0|0|0.14|0.84|0.02|
||10|0|0|0|0|0|0|0|0|0.07|0.93|
|||||||**Northea**|**st rural**|||||
|||||||Decil|es MI|||||
|||1|2|3|4|5|6|7|8|9|10|
||1|0.98|0.02|0|0|0|0|0|0|0|0|
||2|0.25|0.73|0.02|0|0|0|0|0|0|0|
||3|0|0.37|0.57|0.06|0|0|0|0|0|0|
||4|0|0.02|0.27|0.57|0.14|0|0|0|0|0|
|Deciles|5|0|0|0.03|0.22|0.59|0.16|0|0|0|0|
|SI|6|0|0|0|0.01|0.22|0.54|0.22|0|0|0|
||7|0|0|0|0|0.03|0.18|0.56|0.23|0|0|
||8|0|0|0|0|0|0.04|0.13|0.6|0.23|0|
||9|0|0|0|0|0|0|0.03|0.14|0.69|0.13|
||10|0|0|0|0|0|0|0|0|0.12|0.87|

_Source_ : Authors calculation based on HEIS 2019 36
# **7 Practical guidelines**

When adjusting for price differences, welfare analysts are called to make several choices: some are the subject of long-standing debate (which index should be used?), while others have attracted far less attention (should each expenditure component be adjusted by its own price index, or should a single, aggregate index be preferred? When estimating a spatial price index from the survey, should one use data that have been adjusted for inflation, or go with nominal expenditures?). This paper contributes to the second set of issues. The main aim of the analysis has been to clarify the implications of alternative deflation strategies on the welfare aggregate and, based on those implications, to reach some conclusion of which deflation strategy should be preferred. Below is an operational summary of the conclusions.

We start from a scenario that is common in practice, and that occurs after some work has gone into estimating the components of the nominal consumption aggregate. We have _i_ ) all elementary expenditures that make up the consumption aggregate, in nominal terms; _ii_ ) an exogenous set of temporal CPIs, so that there is a question of whether one should use the aggregate or the individual indexes (food and non-food, or even a finer breakdown by categories); _iii_ ) a need to adjust for spatial price differences, and to do so by estimating a price index based on survey data.

A step-by-step deflation strategy is as follows.

- 1) Before doing anything else, adjust all elementary expenditures for inflation. The temporal price adjustment should be performed first in order to be able to estimate the spatial deflator based on inflation-adjusted expenditures. This is especially important if the price dynamics during the survey period is sustained. Omitting this step would imply that the spatial price index would be estimated using distorted unit values and budget shares (section 5).

- 2) To adjust for inflation, the analyst may choose whether to use CPIs specific to each commodity group – food CPI for food expenditures, non-food CPI for non-food expenditures – which qualifies as a “multiple index” (MI) approach, or the total CPI for each and every expenditure, or a “single index” (SI) approach. We cannot conclude that SI provides a better estimate of the welfare measure than MI in _all_ situations (section 4), but there are some conditions in which this is the case, others where the two approaches cannot be ordered, none where MI is preferable to MI. For this reason, the general recommendation is to stick with a single index (SI).

- 3) Next, the analyst estimates a spatial price index based on survey data. The options may be limited here: a typical scenario is one where the analyst may be able to compute a food SPI, and one

37
based on rent and/or energy expenditures (these are usually the only items for which survey-based unit values can be estimated). Instead of computing two different SPIs, and applying them to subcomponents of the consumption aggregate, the recommendation is to compute a single, allencompassing spatial price index (whatever its coverage). The reasoning is the same as in the previous point. In any case, the use of SI or MI has clear distributional implications, which can be assessed by simply computing the quantity ∆i in Proposition 2 (which depends on the price indexes) and comparing it with the budget shares of the households in the survey.

- 4) Finally, the analyst computes the real consumption aggregate by summing up elementary _temporally deflated_ expenditures, and dividing the result by the single _spatial_ index computed in step 3.

38
# **Appendix 1. Proofs**

## **Proof of proposition 1**

Observe that (r, k, kf, knf, α, αf, αnf) = (H, K, Kf, Knf, 0,0,0 ) implies: It follows: However ∎

## **Proof of proposition 2**

h h Under the assumption (r, k, kf, knf, α, αf, αnf ) = (r, k, kf, knf, α, α, α ), we can rewrite xB and xA as follows: 39
And f nf h h h h Assume Ih > Ih , hence xSI −xMI ≠0 and xSI −xMI > 0 if and only if the term in the squared bracket is positive which holds true if and only if: f nf It is immediate to verify that if Ih < Ih the term in the squared bracket is positive if and only if wh < ∆i.

# **Proof of the corollary**

Under the assumption (r, k, kf, knf, α, αf, αnf ) = (r, k, kf, knf, 1,1,1 ), all the indexes are of the f nf Laspeyres type and we can write Ii = wiIi + (1 −wi)Ii . Hence, the threshold value ∆i simplifies to:

# **Proof of proposition 3**

Under the assumption on preferences in the text, we know that TCLIPKh > Ph(H, k) (Konüs, 1924) which, in turn, implies MMU<sup>h</sup> = xh⁄TCLIPKh < xh⁄Ph(H, k) = xSIh . Hence (xSIh −MMUh) > 0. Suppose now that Iif > (<)Iinf and wh < (>)∆i, then xMIh > xSIh and (xMIh −MMUh) > (xSIh −MMUh) > 0. Method (MI) tends to overestimate the real expenditures with respect to method (SI).

40
# **Proof of proposition 4**

Under the assumption on preferences in the text, we know that TCLILKh < Lh(H, k), which implies WR̃<sup>h</sup> = xh⁄TCLILKh > xh⁄Lh(H, k) = xSIh . Hence (xSIh −WR̃h) < 0. Suppose now that Iif < (>)Iinf and wh > (<)∆i, then xMIh < xSIh and (xMIh −WR̃h) < (xSIh −WR̃h) < 0. Method (MI) tends to underestimate the real expenditures with respect to method (SI).

∎ 41
# **Appendix 2. Sensitivity analysis**

This appendix reports results obtained following an alternative approach for estimating the price variation of non-food items. In particular, the “single” Paasche index (SI approach) is constructed as in the equation below: where j denotes each food item (belonging to the food commodity group), wr<sup>i</sup> denotes the budget i share of housing expenditure, pr denotes predicted rents, while wnfnr denotes the budget share of _all_ non-food/non-rent items, which are assumed not to vary spatially.

Correspondingly, the food and non-food price indexes to be used in multiple index spatial deflation (MI approach) are estimated as follows: 42
<!-- Start of picture text --> .<br>a © Non-food Paasche (MI)<br>® Total Paasche (Si)<br>tater geet partes<br>iffZéfféfe;f{fjas $é$<br>é "é f Offé<br><!-- End of picture text -->
**Table A1. Percent changes in poverty measures (MI-SI)**

|||$3.65<br>|||$6.85<br>||
|---|---|---|---|---|---|---|
||**H**|**PG**|**PG2**|**H**|**PG**|**PG2**|
|National|-0.26|0.00|0.00|-0.39|-0.78|-0.72|
|Urban|-4.40|-5.76|-7.34|-0.79|-2.18|-3.15|
|Rural|3.70|5.17|5.25|0.22|0.99|1.96|
|Tehran metro Urban|-19.51|-24.36|-32.74|-4.32|-9.52|-13.35|
|Caspian Rural|0.00|0.90|0.82|0.49|0.45|0.63|
|Caspian Urban|-0.58|-0.33|0.00|0.41|-0.39|-0.69|
|Northwest Rural|1.37|1.21|0.38|0.47|0.78|1.13|
|Northwest Urban|0.17|0.38|0.00|0.79|0.53|0.61|
|Northeast Rural|8.05|9.11|10.29|0.41|1.51|3.29|
|Northeast Urban|-0.60|-1.85|-2.78|0.44|-0.61|-0.90|
|Central Rural|5.25|6.41|5.73|0.52|1.76|2.79|
|Central Urban|0.00|-5.02|-5.93|-1.20|-1.32|-2.10|
|Southeast Rural|3.75|5.84|6.25|-0.79|0.62|2.03|
|Southeast Urban|-0.20|-0.06|-0.08|0.69|0.06|0.04|
|Persian Gulf Rural|4.06|5.06|5.14|0.81|1.58|2.21|
|Persian Gulf Urban|0.82|0.85|0.81|0.00|0.43|0.48|
|Zagros Rural|2.90|3.97|2.76|0.60|1.00|1.49|
|Zagros Urban|-2.26|-0.60|-1.02|-0.14|-0.31|-0.36|

44
**Table A2. Percent changes in inequality measures (MI-SI)**

||**Gini**|**MLD**|**Theil**|**Atkinson**<br>ε=0.5|**Atkinson**<br>ε= 2|
|---|---|---|---|---|---|
|National|-0.54|-1.03|-1.46|-1.14|-0.59|
|Rural|0.61|1.35|0.87|1.08|1.20|
|Urban|-0.78|-1.67|-1.88|-1.65|-1.31|
|Tehran metro Urban|-1.92|-4.38|-3.99|-3.88|-3.73|
|Caspian Rural|0.01|0.03|-0.04|-0.01|0.07|
|Caspian Urban|-0.23|-0.49|-0.49|-0.48|-0.40|
|Northwest Rural|-0.11|-0.43|-1.46|-0.78|-0.15|
|Northwest Urban|0.16|0.32|0.38|0.33|0.22|
|Northeast Rural|1.13|2.52|2.21|2.27|2.39|
|Northeast Urban|-0.22|-0.47|-0.48|-0.45|-0.41|
|Central Rural|0.48|1.08|0.83|0.94|0.98|
|Central Urban|-0.38|-0.85|-0.75|-0.77|-0.79|
|Southeast Rural|3.46|6.64|6.55|6.39|4.38|
|Southeast Urban|-0.04|-0.10|-0.07|-0.07|-0.09|
|Persian Gulf Rural|1.60|3.21|3.59|3.26|2.50|
|Persian Gulf Urban|0.04|0.08|0.08|0.08|0.07|
|Zagros Rural|0.85|1.67|1.71|1.62|1.30|
|Zagros Urban|-0.05|-0.11|-0.10|-0.10|-0.08|

45
**Table A3. Probabilities of transition in and out of poverty when switching from SI to MI ($6.85 PL)**

|**Area**|**% of poor individuals**<br>**using SI**<br>**who become non-poor**<br>**using MI**|**% of non-poor**<br>**individuals using SI**<br>**who become poor**<br>**using MI**|**number of individuals**<br>**changing poverty**<br>**status**|
|---|---|---|---|
|National|1.2|0.3|482,400|
|Tehran metro Urban|5.4|0.2|187,498|
|Caspian Rural|0.0|0.2|4,254|
|Caspian Urban|0.8|0.3|18,189|
|Northwest Rural|1.5|1.7|51,295|
|Northwest Urban|0.0|0.3|15,823|
|Northeast Rural|1.9|1.8|48,837|
|Northeast Urban|0.0|0.2|10,411|
|Central Rural|0.5|0.5|18,982|
|Central Urban|1.5|0.1|41,593|
|Southeast Rural|1.1|1.0|33,777|
|Southeast Urban|0.0|0.5|13,622|
|Persian Gulf Rural|0.8|1.4|28,674|
|Persian Gulf Urban|0.0|0.0|0|
|Zagros Rural|1.1|1.2|35,618|
|Zagros Urban|0.3|0.1|9,783|

46