Module:     S101 – Public Economics

Deadline Date:        12 noon GMT, 4th  Mar 2021

Critical Assessment of the Structural Model Approach in Allcott, Diamond, Dubé, Handbury, Rahkovsky and Schnell (2019)

Section 1 Introduction

This report critically assesses Allcott, Diamond, Dubé, Handbury, Rahkovsky and Schnell (2019) and focuses mainly on their structural approach. We identify three issues in their structural approach, specifically, no income effect at household level, potential endogeneity in the instrumental variable, and no consideration for corner solutions.

This report is organized as follows. Section 2 briefly summarizes the paper, highlighting their research question, findings, and the relevance of their results. Section 3 to 5 identify three    issues of their structural approach. We provide empirical evidence for the validity of these concerns and how these issues may affect their results. We also discuss possible ways to ameliorate these concerns. Section 6 concludes.

Section 2 Summary of the Paper

Allcott et al. (2019) investigate the causes of nutritional inequality and to what extent it can be explained by the supply-side and demand-side factors. Nutritional inequality is socioeconomic disparity in healthy eating where higher-income families make healthier food choices than lower-income families.

The authors first use two reduced-form event studies to examine the importance of local environment. They exploit two quasi-exogenous variation in local supply condition to control for the simultaneity between local environment and demand for healthy food.

Firstly, they investigate how the entry of supermarkets changes household food choices. Using data for 6,721 supermarket entries from Nielson’s TDLinx data, they find that improving access to healthy food has little effect on household food choices, explaining at most 1.5% of nutritional inequality. Secondly, they examine how moving to healthier counties affect household grocery purchases. Using 2,869 cross–ZIP code moves and 2,277  cross-county moves, they find that the explanatory power of healthier local environment is also negligible, at most 3%.

The authors then estimate a structural model for food demand, following Dubois et al. (2014), based on Nielsen Homescan and RMS data from 2006 to 2016. To overcome endogeneity in prices paid by households in their estimation, they use a novel instrument, the comparative price advantage of a retail chain weighted by its market presence. Their    simulation results show that nutritional inequality arises mostly from demand-side factors (e.g. preferences) which accounts for 90% of nutritional inequality, whereas supply-side factors (e.g. prices and product characteristics) can explain only 10%.

Their findings have important policy implications. Their results show that current supply-   side subsidies of healthy food are not effective to combat nutritional inequality. It is more cost-effective to provide an ad valorem subsidy of purchasing healthy food to low-income families.

Section 3 No Income Effect at Household Level

Allcott et al.(2019) follow the approach in Dubois et al. (2014) and assume that household food demand depends on prices and food characteristics. The authors assume that consumer preferences are homothetic. Preferences are said to be homothetic if the ratio of the demand for any two goods depends only on the ratio of their prices. This can be seen from the first-order-condition summed over all products in product group j (equation 6 in the paper), Σk(k)1 pkjyk(*)j  = Σc(c)= 1  Σk(k)1 akjcyk(*)j  +  . It implicitly defines the optimal consumption of product k in group j (yk(*)j ) as a function of product prices (pkj ), product characteristics (akjc ), household preferences for characteristics (βc, μj θj ), and the marginal utility of consuming the outside goods (λ). Household income does not feature in the optimality condition and for a given household, the ratio of the demand for any two goods depends only on relative prices.

One restriction imposed by their assumption of homothetic preferences is that there is no   income effect for a given household. The expenditure on each product group does not vary with household income. Though the authors allow the preference parameters to vary for different income groups and therefore account for income effect at a group level, the assumption of no heterogeneity in income effect at a household level is restrictive.

Empirical research has shown that there is income effect on food choices for a given household. Low-income families typically consumes more healthier food when their income is less constrained. McGranahan and Schanzenbach (2013) exploit the fact that EITC refund  payment is uneven throughout a year and examine how the food choice of households eligible for EITC is changed by the flow of transfer payment. Using data from Consumer Expenditure Diary Survey during the period 1982-2011, they impute the household eligibility for EITC and find that in months of EITC refund payment, the expenditure share of healthy food (e.g. fresh fruits) for a given family increases. This result shows that there is heterogenous income effect on food choice at a household level. Homothetic preferences may not be a good representation for consumer preferences and the model should be modified to allow for income effect within a household.

Moreover, Allcott et al.(2019) assume that consumers have Cobb-Douglas preferences over consumption of all food groups. One property of this functional form is the unitary income   elasticity for all product groups (Chipman, 1974). This is contradictory to empirical findings.  For example, Kumar et al. (2011) estimate the demand elasticities of food consumption for Indian households using two models, a multi-stage QUAIDS model and a FCDS model, based on data from National Sample Survey Organization from 1983 to 2005. Both models show that on average, income elasticities for all food products are below 1. There are also large variation in income elasticities of different food products. For example, staple products (e.g. wheat and rice) have low income elasticities, about 0.024 for an average consumer, while the income elasticity of meat products is much higher, at 0.669.

Without allowing for heterogenous income effect at a household level, the estimated demand parameters may absorb the omitted income effect and may be biased in Allcott et al. (2019). The authors may also overestimate the overall explanatory power of consumer preferences on the nutritional inequality phenomenon. The fact that richer households make healthier food choices than poorer households may partly arise from the income difference itself. Low-income families are more constrained and their food preferences revealed by their purchases are partly a result of their low income.

The authors may consider alternative approaches to allow for heterogenous income effects  at household level, e.g. the approach adopted by Kumar et al. (2011). They use QUAIDS with multistage budgeting to estimate the demand parameters for food choice. Multistage budgeting allows for corner solution and QUAIDS captures quadratic income effects. Their estimation procedure has two stages. The first stage models the allocation of total budget to food consumption, which captures the corner solution. In the second stage, the household decides how much to spend on different food products, featuring household income in the budget share equation. The equation also needs to be modified to incorporate preferences for food characteristics.

Section 4 Potential Endogeneity in the Instrumental Variable

Allcott et al. (2019) constructs an instrumental variable to overcome the endogeneity issue in their estimation. Their estimation equation is

lnyijt  = - ln (plJt(一) - a lJct(—) - ξ ) + δj + φm + φt + εijt

where εijt  is household i ’s idiosyncratic preferences for a product group j andplJt(一) is the average price i paid for calories from product group j. εijt  is likely to be correlated with plJt(一) . For example, local market conditions, such as higher demand for a certain product reflected by a favourable demand shock in the idiosyncratic preferences may result in retailers’ setting higher prices.

The authors instrument the average price by the comparative pricing advantage of a retailer in markets other than the local market m weighted by the presence of the retail chain across markets. Specifically, their instrument pjmt  is the weighted average of the difference  between the log price of a product k in group j of a retail chain r and that of other chains in markets other than the local market, with the weights accounting for the nationwide calorie consumption from product k, the market position of chain r in the local market, and its average sales of product group j nationwide. They argue that this instrument should be exogenous, as the comparative price advantage of a retail chain in other places should not   be correlated with the local market conditions, and hence it is uncorrelated with household idiosyncratic preferences, i.e. E (εijt pjmt ) = 0.

Yet, empirical research has found that prices in most retail chains in the US are uniform across the nation. DellaVigna and Gentzkow (2019) investigate the pricing strategy of US retail chains and its optimality. They use the same dataset as Allcott et al. (2019) (RMS and Homescan data) and their sampling period is also almost identical, from 2006 to 2014. Focusing on food stores, drug, and merchandise stores, they compare quarterly prices within a chain and find that there are almost no variation in prices set by most of retail chains in US across stores, regardless of consumer demographics and competition. Their results show that absolute differences in log quarterly prices within a chain for food stores are 0.031, approximately $1. The price across the nation is almost uniform.

The uniform pricing poses a threat to the exogeneity of the instrument, especially given the similarity of the samples used by DellaVigna and Gentzkow (2019) and Allcott et al. (2019). It suggests that the comparative price advantage of a retail chain in other markets is highly likely to be its price advantage in the local market as well, or rather it is just the price difference of a product between chain r and other chains. In response to the price difference of the chains in their local markets, households are likely to shop strategically. There is response from households to the price advantage of retailers and the instrument is likely to be correlated with household idiosyncratic preferences.

The strategic shopping of households in response to food prices has been documented by the literature. Hanson et al. (2016) investigate the coping strategies of family with children at risk of food insecurity. They select an interview sample (73 families) based on a regression analysis using data from 1998– 1999 Kindergarten cohort in Early Childhood Longitudinal Survey of Youth and then conduct face-to-face interviews with eligible families. They find that almost all subjects shop strategically as a coping strategy, e.g. shopping in multiple stores to get the best price and bulk-buying on sales.

Therefore, given the uniform pricing of retail chains and strategic shopping of consumers, the instrumental variable in Allcott et al. (2019) is likely to be endogenous. Moreover, their   instrument also weights the comparative price advantage of a chain by the chain’s presence in the local market, which may worsen the endogeneity issue. A chain may expand its presence in markets as a response to higher local demand, though still keeping its prices uniform across stores.

If the instrumental variable is endogenous, their estimation will be wrong. They adopt the method of moments estimation method in which E(εijt pjmt ) = 0 is a moment condition. If this moment condition does not hold, their system is not identified.

The authors also provide three robustness checks for the exogeneity of their instrument. Yet, their robustness tests are not convincing. Firstly, they find that price variation is retailer-specific and is also constant across areas, rather than area-specific that is constant across different retailers. They conclude that prices are not endogenous to local market conditions. Yet, this finding only reflects the uniform pricing strategy of retailers, as chains set almost the same price in all markets. Additionally, how the price variation changes across areas and retailers does not capture the response from household grocery purchasing pattern to the pricing of retailers. Secondly, they show that the instrument is not correlated with their predicted household preferences. However, the power of this test depends on how good their prediction is. Finally, they re-estimate the model after adding additional fixed effects for regional preferences for product groups, and regional preferences interacted with urban-rural status and with an indicator of whether the county median income is above national median respectively. They find that estimates are similar   and argue that exclusion restriction is likely to hold. Yet, the additional fixed effects all capture the regional characteristics. It is how heterogenous households would respond to retailers’ price differences that is more worrisome. As shown by DellaVigna and Gentzkow (2019), most retail chains do not adjust their price in response to the variation in local market characteristics. Regional fixed effects are unlikely to be correlated with the price differences across chains. As a result, their inclusion does not affect the estimation by much.

Section 5 No Consideration for Corner Solutions

In their estimation, Allcott et al. (2019) drop observations with zero purchases for some food products. The missing observations account for a relatively large proportion of the total sample, at 10.6%. Excluding corner solutions raises two concerns for their results.

Firstly, their estimation may be biased. The decision to have zero purchase of certain food   items is likely to be correlated with household unobservable. Their sample essentially has a selection bias, where households with interior solutions self-select into the sample.

Early research has shown that excluding exterior solution leads to estimation bias in demand parameters for food consumption. Heien and Wessells (1990) present a technique to account for corner solutions in consumer problem and examine whether allowing for corner solutions will lead to different results. They present a two-step Amemiya estimator.

In the first stage, they estimate a Probit regression which determines the probability that a household will consume a given product and calculate the inverse mills ratio for each food  product. This stage captures the self-selection, where household preferences and other unobservable characteristics determine whether a household will consume a positive amount of a product. In the second stage, they estimate the demand system using the inverse mills ratio as an instrument. They suggest that this technique can be applied to any demand system. Based on data for 14,930 households from USDA's household food consumption survey between 1977 and 1978, they estimate an AIDS demand system for food consumption using censored and uncensored regression respectively. They find that the two regressions differ significantly in their estimated own-price elasticities of goods with a large proportion of zero purchases. The own-price elasticities are smaller under the censored regression. The censored regression also has a higher goodness offit and improves the explanatory power of the model. Therefore, dropping observations with corner solutions may bias results in Allcott et al. (2019).

Secondly, excluding exterior solution challenges the external validity of their policy implications. Their results are only valid for consumers with interior solutions. It cannot be  generalized to the whole population. Households with corner solutions respond differently to their proposed policy. In particular, low-income families are more likely to be excluded from their estimation. In the first-stage regression of Heien and Wessells (1990), the use of   food stamps which can be viewed as a proxy for low socioeconomic status is significant. This indicates that low-income families are more likely to have corner solutions. If the real own-price elasticities for these households are also smaller than estimated by Allcott et al. (2019) , as shown by Heien and Wessells (1990), then the ad valorem subsidy proposed by Allcott et al. (2019) will have smaller effect on improving healthy eating.

The robustness check of dropping observations in Allcott et al. (2019) is also not convincing. For a given product group, they find no correlation between the differences of the share of  missing observations across income groups and the differences of average characteristics and conclude that their estimation of preference heterogeneity is not affected. Yet, this only shows that there is no correlation between corner solutions and the observable

characteristics. There can be correlation with household unobservable heterogeneity or with unobservable product characteristics.

To incorporate corner solutions, besides QUAIDS model with multistage budgeting proposed in Section 3, the authors may also consider two strategies proposed by Wales and

Woodland (1983). The first strategy is the Amemiya-Tobin model which is also used by

Heien and Wessells (1990). The second strategy is the Kuhn-Tucker approach. Wales and

Woodland (1983) use these two approaches to estimate Australian households’ preferences for meat and show that the two approaches yield similar results. Yet, the Kuhn-Tucker

approach may be more appealing for two reasons. Firstly, the Amemiya-Tobin model makes  an arbitrary assumption on the allocation of density to the feasible region. In contrast, there is no arbitrary assumption in the Kuhn-Tucker approach and there should be less concerns

on the external validity of the results. Secondly, Phaneuf et al. (2000) suggests that the

Kuhn-Tucker approach starts with a utility function and restrictions of utility theory are

automatically fulfilled. Allcott et al. (2019) also begins their structural model by specifying a utility function and the Kuhn-Tucker approach may be more suitable.

Section 6 Conclusion

This report critically assesses the structural model approach in Allcott, Diamond, Dubé,

Handbury, Rahkovsky and Schnell (2019). There are mainly three issues. Firstly, they assume homothetic preferences, which implies no income effect at a household level and unitary

income elasticity. These restrictions are contradictory to empirical findings and can bias their estimation. Secondly, their instrument may be endogenous. Thirdly, they do not

consider corner solutions in their estimation, which leads to sample selection bias and

challenges the external validity of their implications. We present QUAIDS with multistage budgeting, the Amemiya-Tobin model, and Kuhn-Tucker approach as possible ways to

ameliorate these concerns.

Despite these concerns, the paper by Allcott et al. (2019) has important policy implications and great academic value. Their results show that in contrast to the mainstream political

argument, current supply-side subsidies of healthy food are not effective in addressing

nutritional inequality. They complement reduced-form analysis with a structural approach.

Their structural model also follows a pioneer framework which allows for consumer

preferences for product characteristics. They also construct a novel instrument for food prices.