Web-Based Technical Appendix to Measuring Aggregate Price Indexes with Demand Shocks: Theory and Evidence for CES Preferences (Not for Publication)

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1 Web-Based Technical Aendix to Measuring Aggregate Price Indexes with Demand Shocks: Theory and Evidence for CES Preferences (Not for Publication) Stehen J. Redding Princeton University and NBER David E. Weinstein Columbia University and NBER May, 08 A. Introduction This web aendix contains technical derivations, the roofs of roositions, additional information about the data, and sulementary emirical results. Section A. derives the exression for the overall CES rice index in the resence of entry and exit. We decomose the overall change of cost of living (P t /P t ) into the change in the share of exenditure on common goods (l t,t /l t,t ) and the change in the rice index for these common goods (Pt /P t ), as discussed in Section. of the aer. Section A. derives the exact CES rice index in terms of demandadjusted rices and characterizes its relationshi with the Sato-Vartia rice index, as examined in Section. of the aer. Section A. characterizes the consumer-valuation bias and shows that a ositive demand shock for a good mechanically increases the exenditure-share weight for that good and reduces the exenditure-share weight for all other goods. Section A. derives the elasticity of substitution imlied by the Sato-Vartia rice index under its assumtion of time-invariant demand for each common good. Section A.6 derives the three equivalent exressions for the CES rice index: the unied rice index in equation () in the aer, the forward dierence of the unit exenditure function in equation () in the aer, and the backward dierence of the unit exenditure function in equation () in the aer. Section A.7 characterizes the relationshi between our CES unied rice index and the major existing economic and statistical index numbers, such as the Laseyres, Paasche, Fisher and Törnqvist indexes. Each of these existing index numbers assumes no entry and exit of goods and no time-varying demand shocks for surviving goods. Section A.8 rovides further details on the Feenstra (99) estimator discussed in Section.. of the aer and used as a robustness check in Section. of the aer. Section A.9 derives the exressions for the forward and backward aggregate demand shifters in equation () in the aer. Section A.0 demonstrates that our reverse-weighting (RW) estimator generalizes to allow for a Hicks-neutral demand shifter that is common across goods. Section A. shows that the reverseweighting (RW) estimator minimizes the dierence between the change in the cost of living using (i) tastes

2 in both eriods and inverting the demand system to exress these tastes in terms of observed rices and exenditure shares, (ii) tastes in the initial eriod, and (iii) tastes in the nal eriod. Section A. roves that the RW estimator consistently estimates the elasticity of substitution as demand shocks become small (Proosition in the aer). Section A. shows that our assumtion that the forward and backward dierences of the unit exenditure function are money metric (equation (7) in the aer) is satised u to a rst-order aroximation. Section A. roves that the RW estimator consistently estimates the elasticity of substitution if the number of common goods becomes large and demand shocks are uncorrelated with rice shocks for each good and indeendently and identically distributed across goods. Section A. of the web aendix shows that the RW estimator belongs to the class of M-estimators, and uses results from Newey and McFadden (99) and Wooldridge (00) to show that the RW estimates are asymtotically normal. Section A.6 derives the GRW estimator from Section.. of the aer. Section A.7 characterizes the asymtotic bias in the RW estimator when demand and rice shocks for a given good are correlated with one another (Proosition in the aer). Section A.8 shows that the GRW estimator is consistent when demand and rice shocks are correlated with one another for each good but are indeendently and identically distributed across goods (Proosition in the aer). Section A.9 uses our inversion of the demand system to rovide bounds for the elasticity of substitution (s) regardless of the correlation between demand and rice shocks. Assuming that demand and rice shocks are correlated with one another for each good but are indeendently and identically distributed across goods, we show that the true elasticity of substitution necessarily lies within the identied set as the number of common goods becomes large. Section A.0 resents Monte Carlo evidence on the nite samle erformance of our RW, GRW and bounds estimators. Section A. develos the extension to non-homothetic CES references considered in Section. of the aer. We derive the generalizations of our common goods unied rice index and reverse-weighting estimation rocedure for non-homothetic CES. Section A. rovides further details for the extension to nested CES references considered in Section. of the aer. Section A. develos the generalization to mixed CES with heterogeneous grous of consumers considered in Section. of the aer. Section A. shows that our unied aroach to the demand system and the unit exenditure function also can be alied to the closely-related logit and mixed logit references, as widely used in alied microeconometrics. Section A. shows that our main insight that the demand system can be inverted to construct a moneymetric rice index with time-varying demand shocks is not secic to CES, but also holds for the exible functional form of translog references. Furthermore, the consumer-valuation bias is again resent for this exible functional form, because a rice index that rules out demand shocks by assumtion cannot cature the otential for consumers to increase welfare by substituting towards goods that exerience reductions in demand-adjusted rices from increases in demand. Section A.6 contains the data aendix which reorts summary statistics for each of the roduct grous in our data. Section A.7 reorts additional emirical results discussed in the aer.

3 A. Entry and Exit In this section of the web aendix, we derive the exression for the change in the cost of living in equation () in Section. of the aer in terms of the change in the share of exenditure on common goods (l t,t /l t,t ) and the change in the rice index for these common goods (P t /P t ). We start by exressing the change in the cost of living from t exenditure functions (equation () in the aer) in the two eriods: F t,t P t P t kwt ( /j ) kwt ( /j ) s s to t as the ratio between the unit # s. (A.6) Summing the exenditure share in equation () in the aer across common goods, we can exress exenditure on all common goods as a share of total exenditure in eriods t and t where l t,t resectively as: l t,t kw t,t ( /j ) s kwt ( /j ) s, l t,t kw ( s t,t /j ) kwt ( /j ) s, (A.7) is equal to the total sales of continuing goods in eriod t divided by the sales of all goods available in time t evaluated at current rices. Its maximum value is one if no goods enter in eriod t and will fall as the share of new goods rises. Similarly, l t of all goods in the ast eriod evaluated at t fall as the share of exiting goods rises.,t is equal to total sales of continuing goods as share of total sales rices. It will equal one if no goods cease being sold and will Multilying the numerator and denominator of the fraction inside the square arentheses in equation (A.6) by the summation kwt,t ( /j ) s over common goods at time t, and using the denition of l t,t in equation (A.7), the change in the cost of living can be re-written as: F t,t ( kwt,t /j ) # s s l t,t kwt ( /j ) s. (A.8) Multilying the numerator and denominator in equation (A.8) by the summation kwt,t ( /j ) s over common goods at time t, and using the denition of l t,t in equation (A.7), we obtain the decomosition of the change in the overall exact CES rice index (P t /P t ) into the change in the share of exenditure on common goods (l t,t /l t,t ) and the change in the rice index for these common goods (P t /P t ) in equation () of the aer: F t,t l t,t ( kwt,t /j ) # s s l t,t kwt,t ( /j ) s A. Derivation of Exact CES Price Index lt,t l t,t s Pt Pt. (A.9) In this section of the web aendix, we derive the exression for the exact CES rice index in terms of demandadjusted rices in equation (8) in Section. of the aer. From the common goods exenditure share (7), we can exress the change in the common goods rice index as:

4 Pt Pt ( /j ) / ( /j ) s /s s (A.60) Taking logs of both sides, and rearranging, roduces: P t Pt /j s s /j s. (A.6) If we now multily both sides of this equation by s s and sum across all common goods, we obtain: or s kw t,t s s s s s kw t,t! P t Pt P t Pt /j 0 (A.6) s s /j s s kw t,t s s! /j /j. (A.6) Re-writing this exression, we obtain the log change in our exact CES rice index in equation (8) in the aer: P # # t Pt w kw t,t w j, (A.6) j kw t,t w `W t,t s s s s s `t s `t s `t s `t, w. (A.6) kw t,t We now show that the exact CES rice index in equation (A.6) is equal to the unied rice index in equation () in the aer. Using our inversion of the demand system from equation () in the aer and our result that the demand shocks are mean zero across common goods ( ( j t / j t shocks (j /j ) in equation (A.6), we obtain: P t Pt t t + s s t s t ) 0) to substitute for the demand! s w s s, (A.66) kw t,t /Nt,t where a tilde above a variable denotes a geometric mean across common goods such that x t kwt,t x for the variable x. Using the denition of the Sato-Vartia weights (w ) from equation (A.6) above, the nal term in equation (A.66) is equal to zero, so that equation (A.66) reduces to the CES common goods unied rice index: P t Pt F CCG t,t t + s t s t s. (A.67) t

5 Finally, using equations (A.6) and (A.67) together with the denition of the Sato-Vartia rice index in equation (0) in the aer, we can exress the common goods exact CES rice index as equal to the Sato-Vartia rice index minus an additional term that we refer to as the consumer valuation bias, as in equation (6) in the aer: P t Pt F CCG t,t F SV t,t kw t,t w j, (A.68) j where the time-invariant comonent of demand (j k ) dierences out such that (j /j ) (q /q ). A. Consumer-Valuation Bias As discussed in Section. of the aer, the Sato-Vartia index is only unbiased if the demand shocks ( (q /q )) are orthogonal to the exenditure-share weights (w ); it is uward-biased if they are ositively correlated with these weights; and it is downward-biased if they are negatively correlated with these weights. In rincile, either a ositive or negative correlation between the demand shocks ( (q /q )) and the exenditureshare weights (w ) is ossible, deending on the underlying correlation between demand and rice shocks. However, there is a mechanical force for a ositive correlation, because the exenditure-share weights themselves are functions of the demand shocks. In this section of the web aendix, we show that a ositive demand shock for a good mechanically increases the exenditure-share weight for that good and reduces the exenditure-share weight for all other goods. Note that the Sato-Vartia common goods exenditure share weights (w ) can be written as: w x `Wt,t x `t, (A.69) where x s s s s, (A.70) s ( /j ) s `Wt,t (`t /j`t ) s. (A.7) Note also that demand, rices and exenditure shares at time t (j,, s ) are re-determined at time t. To evaluate the imact of a ositive demand shock for good k (q /q > and hence j /j > ), we consider the eect of an increase in demand at time t for that good (j ) given its demand at time t (j ). Using the denitions (A.69)-(A.7), we have the following two results: dw dx x w ( w ) > 0, dw `t dx x w `t w < 0. (A.7) dx s ds x s /s s s /s > 0, (A.7) where we have used the fact that ercentage changes are larger in absolute magnitude than logarithmic changes and hence: s s s s > > 0 for s > s, s

6 s s s We also have the following third result: ds dj j s s < < 0 for s < s. s (s )( s ) > 0, ds `t dj j s `t From our secication of demand in equation () in the aer, we have: (s ) s < 0. (A.7) dj dq q j. (A.7) Using this result in equation (A.7), we obtain: ds dq q s (s )( s ) > 0, ds `t dq q s `t (s ) s < 0. (A.76) Together (A.7), (A.7), (A.7) and (A.76) imly that a ositive demand shock for good k increases the Sato- Vartia exenditure share weight for that good (w ): dw dq q w dw dj j w dw dx x w dx s ds x ds j dj s and reduces the Sato-Vartia exenditure share weight for all other goods ` 6 k (w`t ): dw `t dq q w `t dw`t dj j w `t dw `t dx `t x `t w `t dx `t s `t ds `t x `t ds `t j dj s `t > 0, (A.77) < 0. (A.78) A. Elasticity of Substitution Imlied by Sato-Vartia Price Index In this section of the web aendix, we show that the Sato-Vartia rice index s assumtion of time-invariant demand for each common good imlies that the elasticity of substitution can be recovered from the observed data on rices and exenditure shares with no estimation. We rst show that under this assumtion there is an innite number of aroaches to measuring the elasticity of substitution, each of which uses dierent weights for each common good. If demand for all common goods is indeed constant (including no changes in tastes, quality, measurement error or secication error), all of these aroaches will recover the same elasticity of substitution. We next show that if demand for some common good changes over time, but a researcher falsely assumes time-invariant demand for all common goods, these alternative aroaches will return dierent values for the elasticity of substitution, deending on which weights are used. Under the Sato-Vartia assumtion of constant demand for each common good (j j k W t,t and t), the common goods exenditure share is: s ( /j k ) s j k for all `Wt,t (`t /j`) s. (A.79) Dividing the exenditure share by its geometric mean across common goods, we get: s s t /j s k, (A.80) t / j 6

7 where a tilde above a variable denotes a geometric mean across common goods. Taking logarithms in (A.80), we obtain: Taking dierences in (A.8), we have: s s ( s) t D s s t t ( s) D jk + (s ). (A.8) j t. (A.8) Multilying both sides of (A.8) by w and summing across common goods, we get: s w D s ( s) w D, (A.8) kw t,t t kw t t,t where w are the Sato-Vartia weights: w s s s s `Wt,t s `t s `t s `t s `t. Equation (A.8) yields the following closed-form solution for s: h s SV + kw t,t w s s h t kwt,t w t i s t s t i, (A.8) which establishes that the elasticity of substitution (s) is uniquely identied from observed changes in rices and exenditure shares with no estimation under the Sato-Vartia assumtion of time-invariant demand for all common goods (j j q k for all k W t,t and t). Note that we could have instead multilied both sides of (A.8) by any ositive nite share that sums to one across common goods: s x D s ( s) x D, kw t,t t kw t x, (A.8) t,t kw t,t and obtained another exression for s given observed rices and exenditure shares: h s ALT + kw t,t x s i s s t s t h i. (A.86) t kwt,t x Therefore there exists a continuum of aroaches to measuring s, each of which weights rices and exenditure shares with dierent non-negative weights that sum to one. Under the Sato-Vartia assumtion of constant demand for each good (j j j k for all k W t,t and t), each of these alternative aroaches returns the same value for s, since all are derived from equation (A.8). Now suose that some common good exeriences a demand shock (q 6 q and hence j 6 j for some k W t,t t and t), but a researcher falsely assumes that demand for all common goods is constant. Dividing the common goods exenditure share by its geometric mean, we get: s s t /q s, (A.87) t / q t 7

8 where a tilde above a variable again denotes a geometric mean across common goods. Taking logarithms in (A.87) and taking dierences, we obtain: s D s ( s) D + (s t t ) D q, (A.88) where we have used our result that q t / q t 0. Multilying both sides of (A.88) by w and summing across common goods, we get: s w D s kw t,t t Rearranging (A.89), we obtain: s q,w + ( s) kw t,t kwt,t kwt,t w w D w t t h t h s s + (s ) kw t,t w D q, (A.89) i s t s t + q q i, (A.90) Note that we could have instead multilied both sides of (A.88) by any ositive nite share that sums to one across common goods: s x D s kw t,t t where s q,x + ( s) kw t,t x D t kw t,t x, and obtained another exression for the elasticity of substitution (s): h s kwt,t s kwt,t x x t t h + (s ) kw t,t x D q, (A.9) s t s t + i q q i. (A.9) Note that both of the equations (A.90) and (A.9) return the same value for s, because both are derived from (A.88). However, suose that a researcher falsely assumes that demand for each good is constant (j j j k for all k W t,t and t) and uses equations (A.8) and (A.86) to measure s (instead of equations (A.90) and (A.9)). Under this false assumtion, equations (A.8) and (A.86) will return dierent values for s, because in general: w (q /q ) 6 x (q /q ) when w 6 x. kw t,t kw t,t Therefore, when demand for goods changes over time (q 6 q and hence j 6 j for some k W t,t and t) but a researcher falsely assumes that demand for each good is constant (j j j k for all k W t,t and t), the use of dierent weights for rices and exenditure shares (w versus x ) returns dierent elasticities of substitution in general (s SV 6 s ALT ). 8

9 A.6 Equivalent Exressions for the CES Price Index In this section of the web aendix, we derive the three equivalent exressions for the CES rice index: (i) the unied rice index in equation () in the aer, (ii) the forward dierence of the unit exenditure function in equation () in the aer, and (iii) the backward dierence of the unit exenditure function in equation () in the aer. We begin with the exression for the change in the unit exenditure function going forward in time from eriod t to t: F t,t P t P t kwt ( /j ) kwt ( /j ) s s # s. (A.9) Derivation of Ft F,t in (): Multilying the numerator and denominator of the term inside the square arentheses in (A.9) by the summation kwt,t ( /j ) s over common goods at time t, we obtain: F F t,t kwt ( /j ) s kw ( # s s t,t /j ) kwt,t ( /j ) s kwt ( /j ) s, which using the share of exenditure on common goods (A.7) can be re-written as: F F t,t ( kwt,t /j ) # s s l t,t kwt ( /j ) s. Multilying the numerator and denominator of the term inside the square arentheses by the summation kwt,t ( /j ) s over common goods at time t, we have: F F t,t ( kwt,t /j ) s kw ( s t,t /j ) l t,t kwt ( /j ) s kwt,t ( /j ) s which using the share of exenditure on common goods (A.7) can be exressed as: F F t,t l t,t ( kwt,t /j ) # s s l t,t kwt,t ( /j ) s lt,t l t,t # s, s Pt Pt, (A.9) which using the share of each common good in exenditure on common goods (7) at time t F F t,t F F t,t F F t,t l t,t l t,t kw t,t l t,t l t,t kw t,t lt,t l t,t s ( /j ) s kwt,t ( /j ) s s kw t,t s ( /j ) s ( /j ) s /j /j ## s, s, s # s. becomes: 9

10 Using our secication for demand from equation () in the aer, which imlies j /j q /q,we obtain: F F t,t lt,t l t,t which corresonds to equation () in the aer. # s s /q s s, (A.9) kw t,t /q Derivation of Ft,t B in (): From equation (A.9), using the share of each common good in exenditure on common goods at time t in equation (7) in the aer, the change in the unit exenditure function going backwards in time from eriod t to eriod t F B t,t P t P t can be re-written as follows: l t,t ( kwt,t /j ) # s s l t,t kwt,t ( /j ) s, # s s Ft,t B l t,t ( l t,t /j ) kw t,t kwt,t ( /j ) s, F B t,t F B t,t lt,t l t,t s l t,t ( l t,t /j ) kw t,t ( /j ) s s kw t,t s s /j /j s, s # s. Using our secication for demand from equation () in the aer, which imlies j /j q /q,we obtain: F B t,t lt,t l t,t s which corresonds to equation () in the aer. kw t,t s /q /q s # s, (A.96) Derivation of F CUPI t,t in (): Using the share of each common good in exenditure on common goods at times t and t in equation (7) in the aer, the change in the unit exenditure function going forward in time from eriod t to eriod t (A.9) also can be exressed as: F CUPI t,t lt,t l t,t! s /j s s /j s, (A.97) Taking logs of both sides we have: F CUPI t,t s lt,t l t,t +! j + j s s s. (A.98) Taking means of both sides across the set of common goods, we obtain: F CUPI t,t s lt,t + l t,t N t,t kw t,t N t,t kw t,t! + s s N t,t s. kw t,t j, (A.99) j 0

11 Rewriting (A.99), we obtain: F CUPI t,t lt,t l t,t lt,t l t,t s s t t!!! N t,t kw t,t s (s )N t,t N j t,t s, (A.00) kw t,t j kw t,t s t s t s j t j t. Using our secication for demand from equation () in the aer and our result that demand shocks are mean zero in logs ( N t,t N t,t k q q 0 and hence j t / j t ), we obtain: F CUPI t,t lt,t l t,t which corresonds to equation () in the aer. s t s s t, t s t A.7 Relationshi with Conventional Price Indexes In this section of the web aendix, we use the forward and backward dierences of the CES unit exenditure function in equations () and () in the aer to relate our CES unied rice index (CUPI) to existing economic and statistical rice indexes. Under our assumtion of CES references, we show that the CUPI coincides with these existing rice indexes (including Laseyres, Paasche, Fisher and Törnqvist indexes) for secic arameter values and assumtions about the entry and exit of goods and changes in demand for surviving goods. Nevertheless, there are of course other ways of rationalizing these existing rice indexes using alternative functional form assumtions on references (e.g. the Laseyres index is exact for Leontief references). According to an International Labor Organization (ILO) survey of 68 countries around the world, the Dutot (78) index is still the most rominent one for measuring rice changes (Stoevska (008)). This index is the ratio of a simle average of rices in two eriods: F D t,t N t,t kwt,t k,t N t,t kwt,t kw t,t kwt,t k,t (A.0) As the above formula shows, this index is simly a rice-weighted average change in rices, which does not have a clear rationale in terms of economic theory. A rice-weighted average of rice changes is a suciently roblematic way of measuring changes in the cost of living that most statistical agencies do not just comute unweighted averages of rices in two eriods, but select their samle of rice quotes based on the largest selling roducts in the rst eriod. If we think that the robability that a statistical agency icks a roduct for inclusion in its samle of rices is based on its urchase frequency (C`,t / kwt,t C k,t ), then the Dutot index, as it is tyically imlemented, becomes the more familiar Laseyres index, as used in U.S. imort and exort rice indexes: ercent of countries use this index although historically its oularity was much higher. For examle, all U.S. ination data rior to 999 is based on this index, and Belgian, German, and Jaanese data continues to be based on it. The ILO reort can be accessed here: htt://

12 Ft L,t kw t,t c k,t c k,t k,t kwt,t c k,t kw t,t kwt,t c k,t k,t kw t,t s. (A.0) Written this way, it is clear that the Laseyres index can be derived from the forward dierence of the CES unit exenditure function in equation () in the aer (which equals our CUPI in equation () in the aer) using the assumtions that the utility gain of new goods is exactly oset by the loss from disaearing goods (l t,t /l t,t ), the elasticity of substitution equals zero and demand for each good is constant (j /j ). The Carli index, used by 9 ercent of countries, is another oular index that can be thought of as a variant of the Laseyres index. The formula for the Carli index is F C t,t N kw t,t t,t k,t (A.0) This index is identical to the Laseyres if all goods have equal exenditure shares. However, as with the Dutot, it is imortant to remember that statistical agencies are more likely to select a good for inclusion in the samle if it has a high sales share (s k,t formula. ). In this case, the Carli index also collases back to the Laseyres Similarly, the Paasche index is closely related to the Laseyres index with the only dierence that it weights rice changes from t to t by their exenditure shares in the end eriod t: Ft P,t kw t,t c kwt,t c kw t,t s #. (A.0) We can derive the Paasche index from the backward dierence of the CES unit exenditure function in equation () in the aer (which equals our CUPI in equation () in the aer) by making the same assumtions as used to derive the Laseyres index. Finally, the Jevons index, which forms the basis of the lower level of the U.S. Consumer Price Index, is the second-most oular index currently in use, with 7 ercent of countries building their measures of changes in the cost of living based on it. The index is constructed by taking an unweighted geometric mean of rice changes from t to t: F J t,t N t,t t. (A.0) kw t,t t As we discussed earlier, this formula can be derived from our CES unied rice index (CUPI) in equation () by taking the limit in which s! and using the assumtions that the utility gain of new goods is exactly oset by the loss from disaearing goods (l t,t /l t,t ) and demand for each good is constant (j /j ). It is also related to the unied rice index through another route. Statistical agencies tyically choose roducts based on their historic sales shares. In this case the Jevons index becomes: To derive (A.0) from equation () in the aer, we use F t,t /F t,t, assume l t,t /l t,t and j /j for all k, and set s 0. The ercentages do not sum to 00 because ercent of samle resondents used other formulas.

13 s F CD t,t k,t, (A.06) kw t,t which Konyus (Konüs) and Byushgens (96) roved was exact for the Cobb-Douglas (98) functional form. This rice index is a secial case of the exact CES rice index in which the elasticity of substitution equals one, demand for each good is constant, and there are no changes in variety. Under our assumtion of CES references, our CUPI also can be related to suerlative rice indexes (Fisher and Törnqvist) that rovide an arbitrarily close local aroximation to any continuous and twicedierentiable exenditure function. Taking the geometric mean of the forward and backward dierences of the CES unit exenditure function in equations () and () in the aer, we obtain the following quadratic mean of order ( s) rice index (Diewert 976): F t,t lt,t l t,t s 6 kwt,t s kwt,t s /j s /j /j ( s) /j 7 ( s), (A.07) The Fisher index is the geometric mean of the Laseyres (A.0) and Paasche (A.0) rice indexes, and can be derived from equation (A.07) using the assumtions that s 0, the utility gain from new goods is exactly oset by the loss from disaearing goods (l t,t /l t,t ), and demand for each good is constant (j /j ): F F t,t F L t,t FP t,t /. (A.08) Closely related to the Fisher index is the Törnqvist index, which can be derived from equation (A.07) by taking the limit in which s! and using the assumtions that the utility gain from new goods is exactly oset by the loss from disaearing goods (l t,t /l t,t ), and demand for each good is constant (j /j ): Ft T (s,t kw t,t +s ). (A.09) Another way of looking at the Törnqvist index is to realize that it is just a geometric average of Cobb-Douglas rice indexes dened in equation (A.06) evaluated at times t and t. Figure A. summarizes the relationshi between our UPI and all major rice indexes. Under our assumtion of CES references, we can derive most existing rice indexes (such as the Dutot, Carli, Laseyres, Paasche, Jevons, Cobb-Douglas, Sato-Vartia-CES, Feenstra-CES) as secial cases of the UPI for articular arameter values and assumtions about the entry and exit of goods and changes in demand for surviving goods. Nonetheless, there are other otential ways of rationalizing these existing rice indexes using alternative functional form assumtions on references (e.g. the Törnqvist index is exact for translog references).

14 Figure A.: Relation Between Existing Indexes and the UPI Quadratic Mean of Order r ( ) r 6 ( ) Sato Vartia CES 6 Cobb- Douglas t/ t 6 PFW Fisher Törnqvist Feenstra CES Jevons Key : Elasticity of Substitution PFW: Purchase Frequency Weighting ' k,t /' k,t : No Demand Shifts t/ t : No Change in Variety ' k,t /' k,t t/ t 6 ' k,t /' k,t 6 Unified Price Index Aggregation 6 ' k,t /' k,t t/ t 6 Logit/Fréchet Laseyres PFW Carli PFW Paasche PFW PFW Dutot

15 Finally, we now show that the assumtion that the demand arameters for each good are constant is also central to existing continuous time index numbers, such as the Divisia index. Given a constant set of goods (W) and a unit exenditure function that deends on the vector of rices and demand arameters for each good (P ( t,j t )), and assuming that demand for each good k W remains constant (j t j), the Divisia index can be derived as follows: d P d P ( t,j) ( t,j) d d, kw dp d P ( t,j) ( t,j) d kw P d ( t,j), d P ( t,j) s d, P ( 0,,j) kw Z 0 s d. kw This derivation of the Divisia index makes exlicit the assumtion of constant demand for each good. In contrast, if demand for each good were time-varying, there would be additional terms in d j in the rst line of the derivation above. A.8 Feenstra (99) Estimator In this section of the web aendix, we rovide further details on the Feenstra (99) estimator discussed in Section.. of the aer and used as a robustness check in Section. of the aer. We estimate searate elasticities of substitution for each roduct grou in our data. We develo the estimator for a given roduct grou below, where to simlify notation we omit the subscrit g for roduct grou. We start with the double-dierenced exenditure share from the CES demand system in equation (9) in the aer: D s b 0 + b D + u, (A.0) where the rst dierence is over time and the second dierences is from the geometric mean across common goods; D denotes the time-dierence oerator such that D ( / ); a bar above a variable indicates that it is normalized by its geometric mean across common goods such that ( ) ( / t ); and the time-invariant comonent of demand (j k ) has dierenced out between the two time eriods to leave only the change in the time-varying comonent of demand (D q ); and we have used our result that demand shocks average out across common goods ( q t / q t 0). We combine this relationshi from the CES demand system in equation (A.0) above with an analogous suly-side relationshi: D s d 0 + d D + w. (A.) The identifying assumtion of the Feenstra (99) estimator is that the double-dierenced demand and suly shocks (u, w ) are orthogonal and heteroskedastic. The orthogonality assumtion denes a rectangular hyerbola for each good in the sace of the demand and suly elasticities. The heteroskedasticity

16 assumtion imlies that these rectangular hyerbolas for dierent goods do not lie on to of another. With two goods, the intersection of these rectangular hyerbolas exactly identies the elasticity of substitution. With more than two goods, the model is overidentied. In articular, following Broda and Weinstein (006), the orthogonality of the double-dierenced demand and suly shocks denes a set of moment conditions (one for each good within a roduct grou): where V b d G (V) E T [x (V)] 0, (A.) ; x u w ; and E T is the exectations oerator over time. We stack the moment conditions for all goods within a roduct grou to form the GMM objective function and obtain: n o ˆV arg min G S (V) 0 WG S (V), (A.) where G S (V) is the samle analog of G (V) stacked over all goods within a given roduct grou and W is a ositive denite weighting grou. As in Broda and Weinstein (00), we weight the data for each good by the number of raw buyers for that good to ensure that our objective function is more sensitive to goods urchased by larger numbers of consumers. A.9 Forward and Backward Aggregate Demand Shifters In this section of the web aendix, we derive the equalities between equivalent exressions for the change in the cost of living in equations () and () in the aer, as well as the exressions for the forward and backward aggregate demand shifters in equation () in the aer. We start with our three exressions for the change in the cost of living from our common goods unied rice index (equation () in the aer) and the forward and backwards dierences of the CES unit exenditure function (equations () and () in the aer): P t P t P t P t P t P t lt,t l t,t lt,t l t,t lt,t l t,t s t s s t, (A.) t s t s s /j kw t,t /j s kw t,t s /j /j s # s, (A.) s # s, (A.6) where our secication of demand in equation () in the aer imlies j /j q /q these three equations (A.), (A.) and (A.6), we obtain: Q F t,t Qt,t B s kw t,t kw t,t s s # s ( s) # s 6. Combining t s s t, (A.7) t s t t s s t, (A.8) t s t

17 where Qt F,t and QB t,t are aggregate demand shifters that are dened as: Q F t,t 6 kwt,t s s j s j 7 kwt,t s s s, Q B t,t 6 kwt,t s kwt,t s j s j 7 s s s, (A.9) where the variety correction terms for entry/exit have cancelled from both sides of the equations (A.7) and (A.8), and recall that j /j q /q. Now note the following results: s ( /j ) s (P t ) s, (A.0) which imlies: /j s s P s t /j s Pt, (A.) /j ( s) s P s t, (A.) /j s P t s s s P t P t s j j ( s), (A.) ( s) s s P t P t s j j ( s). (A.) We now use these results to rewrite the aggregate demand shifters. First, using equation (A.) in the numerator of Q F t,t in equation (A.9), we obtain: Q F t,t 6 kwt,t s kwt,t s s s P t Pt s Using equation (A.) in the denominator of equation (A.), we have: s 7 s. (A.) Q F t,t 6 kwt,t s kwt,t s s s s P t s Pt P t P t s s j j ( s) 7 s, (A.6) which simlies to equation () in the aer: Q F t,t 6 kwt,t s 7 j ( s) j s kw t,t s j j s # s, (A.7) where j /j q /q. Second, using equation (A.) in the numerator of Qt,t B in equation (A.9), we obtain: Q B t,t 6 kwt,t kwt,t s s s s P t P t s s 7 s. (A.8) 7

18 Using equation (A.) in the denominator of equation (A.8), we have: Q B t,t 6 kwt,t kwt,t s s s s s P t s Pt P t P t s s j j ( s) 7 s, (A.9) which simlies to the exression in equation () in the aer: Q B t,t 6 kwt,t s j j ( s) 7 s s kw t,t j j s # s, (A.0) where j /j q /q. We now use the denition of the forward aggregate demand shifter Q F t,t in equation (A.9) to rewrite our equality between the forward dierence of the unit exenditure function and the unied rice index in equation (A.7) as follows: Q F s kw t,t j j # s s # s s s kw t,t t s s t t s t (A.) where Q F is dened as: Q F ale ale kwt,t s kwt,t s s s j j s s ale s s j s kwt,t j. (A.) s From our inversion of the demand system in equation () in the aer, we have:! j / t + j / t s s / s t s, (A.) / s t where we have used our result that ( j t / j t demand shocks (j /j Q F kw t,t ) 0. Using equation (A.) to substitute for the unobserved ) in equation (A.), we obtain the following key result: s # ( s) s # s s s kw t,t. (A.) Now we use the denition of the backward aggregate demand shifter Qt,t B in equation (A.9) to rewrite our equality between the backward dierence of the unit exenditure function and the unied rice index in equation (A.8) as follows: Q B kw t,t s j j (s ) # s kw t,t s ( s) # s t s s t, t s t (A.) 8

19 where Q B is dened as: Q B ale kwt,t ale kwt,t s s ( s) ( s) j j (s ) s ale s s j (s ) kwt,t j. (A.6) s Using equation (A.) to substitute for the unobserved demand shocks (j /j obtain another key result: Q B s kw t,t s # s Combining equations (A.) and (A.7), we obtain: kw t,t s ( s) # s ) in equation (A.), we. (A.7) Q F Q B Q. (A.8) Additionally, equations (A.9) and (A.7) together imly: Q F t,t 6 kwt,t s s j kwt,t s s j s 7 s kw t,t s j j s # s, (A.9) and equations (A.9) and (A.0) together imly: Q B t,t 6 kwt,t s kwt,t s j s s j s 7 s s kw t,t j j s # s. (A.0) Combining the denitions of Q F and Q B in equations (A.) and (A.6) with these results for Q F, Q B, Qt F,t and QB t,t in equations (A.8), (A.9) and (A.0), we obtain the following relationshi between these variables: Q F Q B Q Q F t,t QB t,t. (A.) Using this result in equation (A.), or equivalently using this result in equation (A.7), we obtain equation (6) in the aer: s kw t,t s # s kw t,t s ( s) # s Q F t,t QB t,t Q, (A.) which must hold for any value of the elasticity of substitution and any constellation of demand and rice shocks. Finally, this relationshi in equation (A.) imlies that if the equality between the forward dierence of the unit exenditure function and the UPI in equation (A.7) is satised, the equality between the backward dierence of the unit exenditure function and the UPI in equation (A.8) must also be satised, and vice versa. To see this, suose rst that equation (A.7) is satised. Using equation (A.) to substitute for ale Qt F,t kwt,t s s s in equation (A.7), we obtain equation (A.8). Suose second that 9

20 ale equation (A.8) is satised. Using equation (A.) to substitute for Qt,t B kwt,t s ( s) in equation (A.8), we obtain equation (A.7). Therefore, there is a single value of the elasticity of substitution (s) that satises equations (A.7) and (A.8) for any constellation of demand and rice shocks. s A.0 Hicks-Neutral Shifter In this Section of the web aendix, we show that our reverse-weighting (RW) estimator generalizes to allow for a Hicks-neutral demand shifter (U t ) that is common across goods. In the resence of this Hicks-neutral shifter, the unit exenditure function in equation () in the aer becomes: P t kw t U t j and the exenditure share in equation () in the aer can be written as: s # s, (A.) s ( / (U t j )) Pt s s. (A.) Using equations (A.) and (A.), we obtain the following generalizations of the three equivalent exressions for the change in the cost of living in equations (), () and () in the aer: F F t,t F B t,t F U t,t lt,t s Pt l t,t Pt lt,t l t,t lt,t l t,t s P t P t s P t P t U t U t U t U t lt,t U t U t l t,t lt,t l t,t lt,t l t,t s t t s t s t # s, (A.) # s s /j s s, (A.6) kw t,t /j s kw t,t s /j /j s # s. (A.7) Equating (A.) and (A.6), and combining (A.) and the inverse of (A.7), the change in the Hicksneutral shifter (U t /U t ) cancels from both sides of the equation: # s /j s s kw t,t /j kw t,t s /j /j s # s t t t t s t s t s t s t # s, (A.8) # s. (A.9) Using our secication for demand from equation () in the aer, which imlies j /j q /q, and the denition of the forward and backward aggregate demand shifters in equation () in the aer, we obtain: 0

21 Q F t,t Qt,t B s kw t,t kw t,t s s # s ( s) # s t s s t, (A.0) t s t t s s t, (A.) t s t which corresonds to equations () and () in the aer. Therefore, the reverse-weighting estimator remains unchanged in the resence of a Hicks-neutral demand shifter (U t ), as in equations (8)-(9) in the aer. A. Reverse-Weighting (RW) Estimator and the Change in the Cost of Living Using Tastes in Each Period In this section of the web aendix, we show that the reverse-weighting (RW) estimator minimizes the sum of squared deviations between (i) the unied rice index evaluated using tastes in both eriods (inverting the demand system), (ii) the change in the cost of living evaluated using eriod t tastes, and (iii) the change in the cost of living using eriod t tastes. This roerty relates to the results of Fisher and Shell (97), which uses the tastes of the initial or nal eriod to bound the change in the cost of living. Here, we show that the elasticity of substitution itself can be chosen to minimize the dierence between change in the cost of living using the tastes of the initial or nal eriod. First, the change in the cost of living for common goods evaluated using eriod t 6 F t,t jt kwt,t s j kwt,t j s 7 s 6 kw t,t j s kwt,t j s 7 tastes is: s. (A.) Using the CES common goods exenditure share at time t, we have: s ( /j ) s kwt,t j s, (A.) which enables us to rewrite the change in the cost of living using eriod t F t,t jt s kw t,t tastes as: s # s. (A.) Second, the change in the cost of living for common goods evaluated using eriod t tastes is: 6 F t,t jt kwt,t j kwt,t j s s 7 s 6 kw t,t kwt,t Using the CES common goods exenditure share at time t, we have: j s j s 7 s. (A.) s ( /j ) s s, (A.6) kwt,t j

22 which enables us to rewrite the change in the cost of living using eriod t tastes as: F t,t jt # ( s) s s. (A.7) kw Third, the change in the cost of living for common goods from the unied rice index (using the demand system to substitute for tastes in eriods t and t) is given by equation () in the aer. Finally, from equations (8) and (9), the reverse-weighting estimator minimizes an objective given by the following sum of squared deviations: ( s s kw t,t + ( s # s t t kw t,t s t s s t #) s ( s) # t t s t s t #) s. From equation () in the aer and equations (A.) and (A.7) above, this objective function corresonds to the sum of squared deviations between (i) the unied rice index, (ii) the change in the cost of living evaluated using eriod t tastes, and (iii) the change in the cost of living evaluated using eriod t tastes. A. Proof of Proosition (Small Demand Shocks) Proof. If this section of the web aendix, we rove that as demand shocks become small ((q /q )! ), the reverse-weighting (RW) estimator consistently estimates the true elasticity of substitution (ŝ RW! s D ). Recall from equations (A.9), (A.7) and (A.0) in Section A.9 of this web aendix that the forward and backward aggregate demand shifters are: As (q /q As Q F t,t Qt F,t 6 kwt,t s s q kwt,t s s Q B t,t 6 kwt,t s kwt,t s q s s q s q s 7 7 s s )!, equations (A.8) and (A.9) imly that: Q F t,t! and Qt,t B!, we have the following results: # s s s kw t,t s kw t,t s kw t,t s kw t,t s q q q q s s s s, (A.8). (A.9)!, Qt,t B!. (A.60) ( s) # t t t t s t s t s t s t # s! 0, (A.6) # s! 0, (A.6)

23 which imlies that the moment condition in equation (8) in the aer is satised and the reverse-weighting estimator converges to the true elasticity of substitution: ŝ RW! s D. We now show that there exists a unique value for s that solves these two equations. We begin with equation (A.6), which can be re-written as: or equivalently s s kw t,t L F t s # L F t L D t, kw t,t s ale t + ale s t s t s t s # ale ale Lt D t s (s ) + t t s t First, we dierentiate L F t in equation (A.6) to obtain: dlt F d (s ) kw s t,t, (A.6) (A.6), (A.6). (A.66) kwt,t s s, (A.67) kw t,t s kwt,t s s s s where we have used d (a x ) /dx ( a) a x. Now note that the common goods exenditure share (7) and the CES rice index for common goods (6) imly that as j /j! : s s (P s t ) s s, k W t,t. (A.68) Using this result in (A.67), re-arranging terms and noting that kwt,t s, we obtain: dl F t d (s ) kw t,t P t s. (A.69) Note that s > 0; ( / ) < 0 for < ; and ( / ) > 0 for >. Therefore, deending on the values of the exenditure shares {s }, dlt F d(s ) can be either ositive or negative, and is indeendent of s. Second, we dierentiate Lt D in equation (A.66) to obtain: dlt D ale d (s ) t. (A.70) t Note that dl D t of rices { }, d(s ) > 0 for t > t and dld t d(s ) < 0 for t < t. Therefore, deending on the values dl D t d(s ) (A.69) and (A.70) imly that both of s. Assuming that: can be either ositive or negative, and is indeendent of s. Together equations kw t,t dlf t d(s ) and dld t d(s ) s 6 N kw t,t t,t can be either ositive or negative and are indeendent ale t, t

24 we have: dl F t d (s ) 6 dld t d (s ). (A.7) Note that the derivatives in (A.7) dier from one another and are indeendent of (s ). Therefore L F t and L D t exhibit a single-crossing roerty such that there exists a unique value of (s ) that satises (A.6), as shown in Figure A.. We next turn to equation (A.6), which can be re-written as: # s ale s kw t,t s t + ale s t s t s t, (A.7) or equivalently L B t L D t, (A.7) L B t kw t,t ale Lt D (s ) First, we dierentiate L B t in equation (A.7) to obtain: s t t + s # ale s t s t dlt B d (s ) kw s s t,t kwt,t s, (A.7). (A.7) s, (A.76) where we have again used d (a x ) /dx ( a) a x. Using the relationshi between relative rices and relative common goods exenditure shares as q /q that kwt,t s, we obtain:! from equation (A.68), re-arranging terms, and noting dlt B d (s ) s. (A.77) kw t,t Note that s > 0; ( / ) < 0 for < ; and ( / ) > 0 for >. Therefore, deending on the values of the exenditure shares {s }, dlt B can be either ositive or negative, and is d(s ) indeendent of s. Second, we dierentiate Lt D in equation (A.7) to obtain: Therefore both that: we have: dl D t d (s ale ) t. (A.78) t dlb t d(s ) and dld t can be either ositive or negative and are indeendent of s. Assuming d(s ) s 6 kw t,t N kw t,t t,t ale t, t dl B t d (s ) 6 dld t d (s ). (A.79)

25 Note that the derivatives in (A.79) dier from one another and are indeendent of (s ). Therefore Lt B and Lt D exhibit a single-crossing roerty such that there exists a unique value of (s ) that satises (A.6), as shown in Figure A.. At this unique value of s, both Lt F and Lt B equal Lt D, as also shown in Figure A.. Λ. Λ Β Λ D Λ F σ Figure A.: Single crossing between L F, L B and L D A. First-order and Second-Order Aroximations In this section of the web aendix, we show that our assumtion that the forward and backward dierences of rice index are money metric in equation (7) in the aer is satised u to a rst-order aroximation. From equation () in the aer, we have: Qt F,t s 6 kwt,t s s q kwt,t s s q s Taking a Taylor-series exansion of Q F t,t around ( / ) and (q /q ), we obtain: Qt F,t s kw t,t kw t,t s 7. s q q kw t,t + O F (s, ), Qt F,t s q + O F (s, ), (A.80) q kw t,t

26 where bold math font denotes a vector; the initial common goods exenditure shares (s at time t ; and O F (s, ) denotes the second-order and higher terms such that: ) are re-determined O F (s, ) ( s) kw t,t s ale q ( s) kw t,t s q ale q ale ( s) kw t,t `W t,t s s`t + ( s) kw t,t `W t,t s s`t (A.8) q ale q q ale q q ale ale q`t q`t ale q`t q`t + O F (s, ). where O F (s, ) denotes the third-order and higher terms. From equation () in the aer, we also have: Qt,t B # s s s q q kw t,t Taking a Taylor-series exansion of Qt,t B around ( / ) and (q /q ), we obtain: Qt,t B s q + O B (s, ), (A.8) q kw t,t where the initial common goods exenditure shares (s ) are re-determined at time t and O B (s, ) denotes the second-order and higher terms such that: O B (s, ) ( s) kw t,t s ale q q + ( s) kw t,t `W t,t s s`t ale q (A.8) q ale ale q q`t + O B (s, ). q q`t where O B (s, ) denotes the third-order and higher terms. The second-order terms in equations (A.80) and (A.8) deend on q q q`t q`t, `t `t and q`t q`t, while the third-order terms deend on higher owers of q and. For small changes in rices and q demand for each good (( / ) 0 and (q /q ) 0), these second-order and higher terms in equations (A.80) and (A.8) converge to zero (O F (s, )! 0 and O B (s, )! 0). Therefore, to a rst-order aroximation, the forward and backward aggregate demand shifters satisfy time reversibility: Qt F,t Qt,t B s q. (A.8) q kw t,t Noting that kwt,t s and (q /q ) 0 for all k W t,t, the following weighted average is also necessarily small: and hence s q 0, q kw t,t (A.8) Q F t,t Q B t,t 0. (A.86) 6

27 A. Proof of Proosition (Large Number of Common Goods) Proof. In this section of the web aendix, we rove that the reverse-weighting (RW) estimator consistently estimates the elasticity of substitution (ŝ RW! s D ) if the number of common goods becomes large and demand shocks are uncorrelated with rice shocks for each good and indeendently and identically distributed across goods. Recall from Section A.9 of this web aendix that the forward and backward aggregate demand shifters are given by: Q F t,t 6 kwt,t s s q kwt,t s s q s 7 s s kw t,t s s q, (A.87) q Q B t,t 6 kwt,t s s q s s q 7 kwt,t s s s s q q kw t,t We dene the following change of variables and weighted means and covariances: s. (A.88) s s X F, Y F q, q ( s) (s ) X B, Y B q q C C E E h i X F h i Y F h i X F YF h i X B YB h i s XF, E X B kw t,t h i s YF, E Y B kw t,t s X F kw t,t kw t,t s X B E E h i X F Y F h i X B Y B kw t,t s XB, kw t,t s YB, E E h i Y F, h i Y B, Using these denitions, the aggregate forward and backward demand shifters can be rewritten as: Q F t Q B t,t E X F,t Y F E X F E X B Y B E X B # s # s C X F Y F + E X F E Y F E X F C X B Y B + E X B E Y B E X B # s # s E Y B E Y F # s, (A.89) # s. (A.90) As N t,t!, our assumtion that the demand shocks are indeendently and identically distributed ((q /q ) i.i.d., c j for (q /q ) (0, )) imlies: C h i h! X F YF C X i B YB 0. (A.9) 7

28 Using this result in equations (A.89) and (A.90), time reversibility is satised: Q F t,t Q B t,t! Qt,t B! Qt F,t s kw t,t kw t,t s q q q q s # s, (A.9) s # s. (A.9) We now determine the asymtotic roerties of the following term from equation (A.9): s Xt F,t s q. (A.9) q kw t,t We have assumed: # q q i.i.d., c #. Using the Central Limit Theorem, we obtain: # N t,t (# ) N 0, kw t,t From the Delta method, a sequence of random variables that satises: n (Yn µ) d!n 0, c c # N t,t. imlies: h d n [g (Y n ) g (µ)]!n 0, c g 0 (µ) i. Alying this result to equation (A.9), where g ( ) ( ) s, µ, c c #, g (µ) and g 0 (µ) s, we have: Therefore, as N t,t N t,t kw t,t!, we have: # # s d!n 0, N t,t kw t,t c # (s ). N t,t # s #! 0. (A.9) We will use this result (A.9) in equation (A.9) later. First, note that equation (A.9) can be re-written as: X F t,t kw t,t s 8 < : N t,t # s +, (A.96) N t,t kw t,t s # s 9 ; +, where 8 < : N t,t ale 8 < N t,t kw t,t s : N t,t su s 9 # s ; 9 N t,t # s ;, kw t,t (A.97) 8

29 and N t,t su s N t,t su n( t /j t ) s,..., Nt,t t /j Nt,t t so N t,t k (P /j ) s, (A.98) n( su t /j t ) s,..., Nt,t t /j Nt,t t so N t,t N t,t k (P /j ) s <. Using the results (A.9), (A.97) and (A.98) in equation (A.96), we obtain the result that as N t,t! : Therefore, assuming (q /q ) i.i.d., c j Q F t,t X F t,t!. (A.99) and as N t,t!, we have: h i Xt,t F s!, (A.00) Q B t,t Q F t,t X F t,t # s!. A. Asymtotic Proerties of the RW Estimator In this section of the web aendix, we show that the reverse-weighting (RW) estimator belongs to the class of M-estimators, as characterized in Newey and McFadden (99) and Wooldridge (00). We use this result to derive the asymtotic roerties of the RW estimator. We dene X (s t, s t, t, t ) as the vector of random variables with some distribution in the oulation. We let W denote the subset of < reresenting the ossible values of X. We use s D to denote the true value of the elasticity of substitution s X < +, with X [s, s], where s > and s <. We assume that our data come as a random samle of size N from the oulation. We label this random samle {X k : k,,...}, where each X k corresonds to the vector of observed data on rices and exenditure shares for a good k in the two time eriods. Let q (X k, s) be a function of the random vector X and the arameter s. An M-estimator of s D solves the roblem: min sx N N q (X k, s), k where the true arameter s D solves the oulation roblem: (A.0) min E [q (X, s)]. (A.0) sx Our RW estimator in equation (9) solves a roblem of this form, because it can be written as: h i h i N t,t kwt,t m F (X k, s) + N t,t kwt,t m B (X k, s) min sx m F (X k, s) s `W t,t s `t `t `t s # (A.0)! s s s, (A.0) 9