# **Appendix B. Is the Engel curve approach a promising alternative?**

Some studies developed and applied several Engel curve approaches to construct a spatial price index. The most common use of the Engel curve approach is motivated by temporal focus (rather than spatial), estimating and correcting a CPI bias.<sup>24</sup> For example, Dabalen, Gaddis, and Nguyen (2019) use this method and find a substantial impact of CPI bias on poverty estimates among several African countries. Other examples of the use of Engel curve methods for temporal analysis include Nakamura, Steinsson, and Liu (2016), who argue that China’s economic growth is overestimated due to CPI bias (that is, underestimation of inflation).

More relevant uses of Engel curve approaches for our study are those from spatial (or spatiotemporal) perspectives. Almas (2012) uses an Engel curve method to correct PPP bias across countries, finding that the income of poorer countries tends to be overestimated. Almas and Kjelsrud (2017) incorporate the expenditure-specific cost of living in Engel curve model to measure income inequality in India. They find that the relative price changes were pro-poor during the study period and thus growing income inequality was overestimated further. Almas and Johnsen (2018) estimate an Engel curve model to measure poverty comparable across time and space in China. They find that inflation was mild in urban/rich areas and thus poverty reduction has been more moderate than previously assumed. Most importantly, Gibson, Le, and Kim (2017) compare the performance of Engel curve methods as a tool for spatial price adjustment with other methods. Their analysis based on Vietnam’s detailed price data concludes that the Engel curve approach is not suitable for spatial adjustment.<sup>25</sup> This line of approaches assumes that households with the same expenditure share for food have the same level of utilities across space. In other words, the difference in nominal income (or expenditure) among households with the same expenditure share for food reflects spatial cost-of-living differentials. However, it is possible that many other unobserved factors affect food budget shares. As Gibson, Le, and Kim (2017) argue, however, this assumption may not hold across space.

_Engel curve method with no price data._ A potential advantage of the Engel curve method for spatial price adjustment is that price data are not necessary in some cases. Almas, Kjelsrud, and Somanathan (2019) applied such a method to estimate poverty in India. In a standard Engel curve model, the relative (aggregated) prices of food and non-food items are required. However, they argue that a restricted version of the Engel curve model below can be used by assuming that the budget share for food is not influenced by relative prices. (15), where indicates the budget share for food of household wwiijjtt = aa+ bb�lnyyiijjtt −lnPPjjtt�+ θθEEiijjtt _i_ in region + εεiijjtt _j_ (or urban/rural) at time _t_ , iijjtt � is the real income for household _i_ (nominal expenditure divided by the overall ww price level of region _j_ , ), is a vector of household-specific control variables, and is an error iijjtt jjtt iijjtt �lnyy −lnPP yy term. Given that varies only for region/time, it can be identified through region- and time-specific jjtt iijjtt iijjtt PP EE εε dummies ( ): jjtt PP DDjjtt . (16) iijjtt hsstt iijjtt jj1 jj1 jj2 jj2 iijjtt ww = aa+ bblnyy + θθEE + ∑dd DD + ∑dd DD + εε

> 24 It is noted that even in the case of temporal analysis as the primary purpose, the Engel curve method requires a regional price index to identify CPI bias (discussed later).

> 25 Other studies that construct a spatial price index and estimate poverty/inequality using an Almost Ideal Demand System (AIDS) type demand system include Chakrabarty, Majumder, and Ray (2015); Majumder, Ray, and Sinha (2015); Navamuel, Morollon, and Vazquez (2018); and Ravallion and van de Walle (1991).

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