Demand for food-away-from-home: a multiple-discrete–continuous extreme value model

1. Introduction

Despite the apparent lack of consensus on whether food-away-from-home (FAFH) is responsible for the rise in obesity, several jurisdictions remain convinced that taxes are an effective means of changing consumers' behaviour (Colville, 2009; Vogel, 2011). Indeed, easy access to relatively inexpensive, well-advertised and calorie-dense restaurant meals is frequently cited as a critical factor in declining diet quality among Americans (Binkley and Eales, 2000; Chou, Grossman and Saffer, 2004). Although connecting FAFH to obesity seems reasonable, the empirical evidence is mixed. While French, Harnack and Jeffery (2000), Niemeier et al. (2006), Davis and Carpenter (2009) and Currie et al. (2010) find some evidence of a small effect of FAFH, specifically access to fast food, on obesity, Anderson and Matsa (2011) find no evidence at all. Even if FAFH is responsible, empirical research documents the likely failure of taxes in regulating the consumption of fast food (Schroeter, Lusk and Tyner, 2008). Taxing FAFH is based on (at least) four assumptions that may not be supported by the data: (i) consumers do not offset high-calorie consumption occasions by eating less at other times (Mancino, Todd and Lin, 2009; Anderson and Matsa, 2011), (ii) FAFH is nutritionally inferior to food-at-home (FAH), (iii) the own-price elasticity of demand for FAFH is relatively high and (iv) that the cross-price elasticity of substitution between types of FAFH and FAH is low. In fact, the logic behind taxing FAFH may be predicated on public policy officials' lack of understanding of the structure of FAFH demand. In this study, we investigate consumers' response to changing FAFH prices with an econometric framework that is able to capture many of the unique features of FAFH demand.

There are many empirical studies that document different aspects of the demand for FAFH (Kinsey, 1983; Lee and Brown, 1986; McCracken and Brandt, 1987; Yen, 1993; Byrne, Capps and Saha, 1996; Jekanowski, Binkley and Eales, 2001; Stewart et al., 2005). None, however, consider the fundamental question of the structure of demand, namely, how prices affect the demand for FAFH, and how consumers substitute among different types of FAFH and between FAFH and FAH. While these studies isolate several important drivers that underlie the rise in FAFH consumption, they do not address the essential fact that meals away from home are differentiated goods that are consumed in discrete increments, often more than once by some consumers and not at all by others. Estimating the structure of FAFH demand is, therefore, a complex problem in that it is not only discrete–continuous, but also multiple-discrete–continuous.

Multiple-discrete–continuous choice problems are not unique to FAFH. Hendel (1999) and Dube (2004) develop a model of multiple-discrete brand choices in personal computers and carbonated soft-drinks, respectively. Hanemann (1984) develops a model of discrete–continuous demand that has subsequently been extended to model the demand for variety (Kim, Allenby and Rossi, 2002), transportation services (Bhat, 2005, 2008; Pinjari and Bhat, 2010) and recreational amenities (Phaneuf, 1999; von Haefen and Phaneuf, 2005). Each of these extensions involves an application of the general Kuhn–Tucker approach of Wales and Woodland (1983). Assuming that corner solutions result naturally from diminishing marginal utility and satiation, Bhat (2005, 2008) develops a model of multiple-discrete transportation choices, and a continuous amount of travel demand, which he calls the multiple-discrete continuous extreme value (MDCEV) model. In our data, households are surveyed over a 2-week period over which they can visit multiple types of restaurant, visit each type a number of times and take their remaining meals as FAH. Modelling the food-choice decision process in this way is not only more flexible than existing approaches, but also is more realistic and, therefore, likely to generate more policy-relevant elasticity estimates.

We contribute to the empirical literature on the demand for FAFH in four ways. First, we use a unique data set on FAFH expenditure and intake that has not yet been exploited for the purpose of analysing public policy. Second, we use a more appropriate empirical framework, the MDCEV model, to address the multiple-discrete–continuous nature of FAFH demand in a single, utility-maximising framework. Third, we account for the endogeneity of individual attributes in a model of FAFH demand. Fourth, we provide estimates of the structure of FAFH demand, including a FAH option, that can help to gauge the potential efficacy of price-based strategies for regulating the consumption of FAFH.

The remainder of the paper is structured as follows. The next section provides a brief description of the FAFH data set. The following section presents the two-stage empirical model used to estimate the demand for FAFH. Estimation results for the food-demand stage are then discussed, while a final section concludes and offers some policy implications that follow from the research results.

2. Data description

The data for this study consist of two survey data sets collected by NPD Group, Inc. – the National Eating Trends (NET) and the Consumer Reports for Eating Share Trends (CREST). The NET sample used in this research consists of a survey of 4,792 US households. Respondents report all FAH and FAFH consumption occasions over a 2-week period, including for FAFH meals the restaurant group (casual dining, fine dining etc.), restaurant segment (full service or quickservice), restaurant category (Asian, bagel, hamburger etc.) and restaurant channel (independent, major chain, local chain etc.). For all meals, respondents report the meal occasion (breakfast, lunch, dinner), and the day and month in which they took place. The respondent file includes demographic and socioeconomic data as well as measures of physical activity, several indicators of health status, and the body mass index (BMI) of all household members. All surveys were conducted between 24 February 2003 and 29 February 2004.

Physical activity is measured by nine separate fields in the NET data, consisting of self-reported exercise frequency (occasions per week), occasions of seven different types of activity (walking, running/jogging, swimming, bicycling, aerobics, weightlifting and other) and a measure of exercise history (Likert scale defined as 1 = never to 5 = frequently). Because there is no way of measuring the length or intensity of each session in the data, we create an index of physical activity by summing exercise frequency and history. For example, if a respondent does aerobics three times per week, swims twice and has exercised frequently in the past, he or she will have a physical activity value of 10. This is admittedly a rough measure of actual physical activity, but developing a more complex measure would be imputing false precision into the data.

The data also include information on several health conditions, such as diabetes, food allergies, heart disease and high blood pressure, as well as 10 different binary variables indicating whether the respondent is on a diet and, if so, what type. However, because many of the health conditions are likely to be highly correlated with each other, and others are due entirely to genetic and not behavioural causes, we only include whether a household member has heart disease, high blood pressure or high cholesterol to form a health status index that ranges in value from 0 (no health problems) to 3 (several health problems). Again, this is also only an approximate measure of actual health status, but is likely to be highly correlated with actual health status.

One weakness of the NET data set is that it does not contain food prices or meal expenditures. Data describing firm pricing and meal expenditure is critical to understanding the economic incentives consumers face in their purchase decisions. Therefore, we first develop an estimated price data set using the meal-expenditure data reported in CREST that includes all foods reported in NET. We use a novel statistical estimation procedure to do so. CREST respondents report purchases of the same foods that are reported in NET, but unlike NET, also report the amount paid at each meal. Meal expenditures from CREST (EXP_ht) are used to impute prices for similar items purchased in the NET data using the hedonic estimation procedure employed by Richards and Padilla (2009).¹

FAH is modelled as the numeraire good as it is consumed by all households in the data set. We include the demand for FAH in the demand model described below by calculating the number of at-home meals as a residual to the total number of meals taken less the number of FAFH meals,. Specifically, we assume that each respondent household faces M total ‘meal occasions’ where M = 3 × 14 × N_h, where N_h is the number of household members, each facing three meals per day for 14 days. M less the total number of FAFH meals taken over each 2-week period is defined as the number of FAH meals. Further, we use a FAH price index from the Bureau of Labor Statistics (USDOL-BLS) matched to each household's region of residence as the numeraire price. The BLS maintains a price index for FAH that is calculated by sampling foods in a representative shopping basket monthly in a large number of markets throughout the USA (http://www.bls.gov/cpi/cpi_methods.htm).

3. Empirical model of FAFH demand

In this section, we derive a demand system that reflects multiple-discrete–continuous choices among restaurant choices. During each 2-week diary period, a household is assumed to consider visiting several different restaurant types – fast food, casual, mid-range and fine dining – over the 2-week period, so the system is defined over restaurant visits. Wales and Woodland (1983) describe two ways of estimating econometric models of demand in which there are many corner solutions: (i) the Amemiya–Tobin approach, and (ii) the Kuhn–Tucker approach. In the former approach, econometric error terms are interpreted as ‘errors in measurement’ or ‘errors in optimization’, are assumed to be truncated normal and are added to the share equations ex post in an ad hoc way. All consumers are assumed to possess the same utility function. In the Kuhn–Tucker approach, utility funtctions are instead assumed to distributed randomly throughout the population and stochasticity derived directly from consumer heterogeneity. The result is an estimable demand system in which corner solutions arise from a single utility-maximisation problem.

We separate the demographic and physiological variables, because it is likely that the elements of Z^h are endogenous. Despite their likely endogeneity (we instrument for them as explained below), it is necessary to include physical attributes in the quality index as measures of physical heterogeneity are expected to be as important in identifying variation in unobserved quality as are measures of demographic heterogeneity. Based on empirical evidence that FAFH tends to be more calorically dense and higher in fat than FAH (French et al., 2001; Bowman and Vinyard, 2004), we expect the demand for all types of FAFH to rise in BMI. Although Drewnowski (1997) suggests that physical activity should have a negative effect on FAFH demand, the fact that physically active individuals demand higher energy meals leaves this effect uncertain (Nestle et al., 1998; Manore, 2004). While we would hope that health status has a negative effect on all types of FAFH demand, once the endogeneity of health status is properly accounted for, it is possible that individuals who have little concern for their health, in fact, consume more FAFH. Within FAFH, we expect to find a stronger positive effect of BMI on fast food demand.

The parameters α_i and γ_i allow the utility function to represent both corner and interior solutions. In mathematical terms, γ_i is a translation parameter that determines where the indifference curve between q₁ and q₂ becomes asymptotic to the q₁ or q₂ axis, and thereby where the indifference curve intersects the axes. For example, if γ₁ = 2, then the indifference curve becomes asymptotic to the q₁ axis at q₂ = −2. Because the value of q₂ is less than zero, the indifference curve necessarily defines a corner solution at some positive value of q₁. In more intuitive economic terms, γ_i is a satiation parameter in that higher values of γ_i imply a stronger preference for q_i. Thus, γ_i governs the slope of the indifference curve. In the context of this study, higher values of γ₁ imply a higher marginal rate of substitution of restaurant-type 2 for restaurant-type 1. The parameter α_i, on the other hand, is also interpreted as a satiation parameter in that it determines how the marginal utility of restaurant-type i changes as q_i rises. If α_i = 1, then the marginal utility of i is constant, indifference curves are linear, and the consumer allocates all income to the restaurant with the lowest quality-adjusted price (Deaton and Muellbauer, 1980). As the value of α_i falls, satiation rises, the utility function in restaurant i becomes more concave and satiation occurs at a lower value of q_i. Figures 1 and 2 demonstrate numerically how γ_i ≠ 0 leads to corner solutions in which at least one of the restaurants is not visited, and how different values of α_i affect the shape of the utility function. Importantly, if the values of are approximately equal across all types, and if the individual has relatively low values of α_i then he or she can be described as ‘variety seeking’ and visit some of all choices, while the opposite will be the case if α_i are relatively high (close to 1.0) and the perceived qualities differ (Bhat, 2005).⁴

In this estimating equation, σ is the logit scale parameter. When M = 1, or only one alternative is purchased, the MDCEV model becomes a simple logit.⁶ Therefore, (10) is appropriately described as a multiple-choice version of a simple logit model that also allows for continuous purchase decisions. This expression is convenient as it represents a closed form that is easily estimated using maximum likelihood methods. In the section ‘Results’, we compare the results obtained from a non-nested testing procedure with which we compare the fit of the MDCEV model relative to a simple logit alternative.⁷

4. Estimation method and identification strategy

Many of the same unobservable factors that lead NET respondents to be obese, sedentary or suffer obesity-related health problems are likely some of the same factors that cause them to consume high levels of FAFH. Prices are also likely to be endogenous. Because we impute household-level prices using the CREST data, the residuals in the demand model are likely to be correlated with these imputed prices. Unfortunately, in cross-section data sets such as ours, demand theory does not suggest a set of available and valid instruments. We follow the framework in Park and Davis (2001) and use a variant of the method of moments approach developed by Lewbel (1997), which uses the second and third moments of the exogenous variables as instruments for the potentially endogenous variables. We estimate with these instruments, but use maximum likelihood. We summarise the NET data in Table 1.

We test the exogeneity and relevancy of our chosen instruments using the testing procedure suggested by Godfrey and Hutton (1994) and Shea (1997). Godfrey and Hutton's (1994) two stage test proceeds as follows: in the first stage, a J-test is developed to determine whether endogeneity or errors-in-variables are the source of misspecification. If the J statistic is large, then the validity of the chosen set of instruments is in question, and alternatives should be considered. If the J statistic is small, then the second stage test is carried out. In the second stage, the null hypothesis is that the variables thought to be endogenous are, in fact, exogenous, consistent with the Hausman (1978) general specification test. Further, we follow Shea (1997) and evaluate the validity of our instruments through their partial, rather than total, explanatory power. Because Shea (1997) does not suggest a value for the partial R² that would form a threshold for being ‘too low’, we follow Staiger and Stock (1997) and interpret an F-statistic in the partial regressions <10 as indicating weak instruments.

As a final model-validation test, we conduct a non-nested test comparing the MDCEV model to the most logical discrete-choice alternative: a multinomial logit (MNL) model of FAFH-type choice.⁸ However, the dependent variable in a MNL model is fundamentally different from the MDCEV model. Although the MDCEV collapses to the simple logit in the case of M = 1, that is not a feature of our data. Therefore, we treat multiple-discrete observations as truly discrete in defining the alternative model. Rather than simply exclude multiple-purchase observations, we choose the restaurant-type with the highest implied utility for each observation and deem that to be the discrete choice. We then apply a simple logit model to the resulting data set and conduct a Vuong (1989) test for non-nested alternatives, which compares two models f(θ) and g(γ); if the Vuong test statistic, V, is greater than the critical standard-normal test value, V > c, then we reject the null hypothesis that the two models are equivalent in favour of the hypothesis that f(θ) is preferred. If V<−c, then we conclude the opposite, and if V lies between −c and c then we cannot reject the null that the models are, in fact, the same.⁹ In the NET data, the Vuong test statistic value is 34.299, easily rejecting the null hypothesis that the two models are the same, and supporting the MDCEV model.

5. Results and discussion

In Table 2, we see that the Godfrey–Hutton J-statistic for the fast food equation is 0.887, for casual dining is 0.992, for mid-range restaurants is 1.381 and for fine dining establishments is 1.034. Therefore, we fail to reject the null hypothesis in each case that our instruments are endogeneous and conclude that our set of IV are likely to be valid. Second, the Hausman H-statistic obtained with this set of instruments is 5.782, whereas the critical Chi-square value is 14.067, so we fail to reject the null hypothesis of exogeneity for the chosen set of instruments. Third, we find that the total explanatory power of the chosen instrument set is very good in each case (for cross-sectional data), ranging from a F-statistic of 19.923 for the physical activity index to 80.156 for health status. Applying the partial-regression procedure of Shea (1997); however, the F-statistics vary from 533.541 in the BMI regression to 2,082.804 for the price of mid-range restaurants. Clearly, the chosen instruments have a high degree of partial explanatory power, and cannot be described as ‘weak instruments’ in the sense of Staiger and Stock (1997).

The results obtained from estimating the MDCEV model are found in Table 3 below, along with non-IV estimates to demonstrate the extent of bias present. Interpreting the results in the IV panel, we find that all four curvature parameters are significantly different from zero. Comparing these values to the range of γ_i shown in Figure 1 illustrates a notion that some public health experts regard as being at the core of the obesity problem – that the satiation level for fast food is much higher than for other types of FAFH.¹⁰ Further, while not a formal statistical test of the specification (we conduct non-nested model selection tests below) the fact that all four parameters are statistically significant suggests that the fundamental assumption of the MDCEV model – that corner solutions are a feature of the data and consumer decision process – cannot be rejected. Third, the τ_i estimates, which are interpreted as restaurant-type-specific preference parameters, suggest a rank-ordering of preferences from fast food at the top, to fine dining restaurants, and then mid-range and casual restaurants the least preferred. Because these estimates are driven as much by volume and visitation as price, they are consistent with our prior expectations and lend further support to the validity of the MDCEV model.

Table 4 presents the elasticity estimates.¹¹ Focusing first on own-price elasticities, the estimates in Table 4 show that casual restaurants are the most price elastic, followed by mid-range restaurants. This suggests that these types of restaurants are likely to be regarded as luxuries by a segment of consumers who are particularly price sensitive. Fine dining restaurants are the least price elastic as consumers of high-end dining may be less price sensitive as those who frequent lower-priced chain restaurants. Fast food restaurants are also relatively inelastic. The elasticity estimates in Table 4 also show FAH to be inelastic in demand. Whether some of this reduction in FAH consumption represents substitution among FAH and FAFH is revealed by the cross-price elasticities. Perhaps not surprisingly, the cross-price elasticities of FAH with respect to all types of FAFH are quite low, ranging from a low of 0.005 for fine dining to 0.062 for fast food restaurants. Among different types of FAFH, the cross-price elasticities are all very low, but fast food substitutes relatively strongly with mid-range restaurants (0.019) and casual restaurants (0.016), but only weakly with fine dining establishments (0.003). In fact, the cross-price elasticity between fine dining and all restaurant formats is uniformly low. Considering all FAFH types in the NPD data, the demand for fine dining is likely to be driven by attributes of the experience not captured in our data: ambience, service quality, food quality and the other aesthetic factors. Because cross-price elasticities within FAFH are low, taxing one will likely reduce consumption at that FAFH type according to the own-price elasticity without causing consumers to substitute among different types of venues.

While previous research finds that consumer attributes, such as BMI, physical activity and health, are important determinants of a consumer's demand for FAFH (Stewart et al., 2005), none have not estimated their effects on FAFH demand using comparable elasticity measures. Therefore, we use the MDCEV model to calculate elasticities of each type of FAFH demand with respect to BMI, physical activity and health (see Table 5). These estimates are potentially important for policy purposes because, after appropriately controlling for the endogeneity of each measure, they provide more specific information on the type of individual that frequents each restaurant-type. For example, the ‘BMI’ row in Table 5 shows the elasticity of demand for each FAFH type with respect to variation in obesity. Notice also that all BMI elasticities are positive and, as expected, higher BMI levels correspond with more frequent visits to fast food restaurants. Obesity is also estimated to be strongly related to the demand for fine dining. Relative to FAH, therefore, the demand for any type of FAFH rises in the level of obesity.

Our results are similar for physical activity. Consumers who tend to exercise more frequently tend to consume more of each type of FAFH. Among the different types of FAFH, the demand for fast food appears to be most closely related to physical activity – consumers who tend to exercise more also visit fast food restaurants more frequently – perhaps because they are the least likely to worry about the caloric-density of their food. Finally, recall that the health status index is calculated such that higher values imply more health problems. With respect to the fast food elasticity, our estimate implies that, ceteris paribus, less-healthy people are significantly more likely to frequent fast food restaurants. Clearly, the implication here is that recognition of their own health condition does not stop many consumers from eating fast food.

The policy implications of our findings are readily apparent. If local jurisdictions were to place a tax on fast food with an objective of reducing consumption, the policy would be only moderately successful as demand is inelastic (−0.743). However, taxing fast food specifically would cause a limited amount of substitution into mid-range and casual restaurants as the ‘spillover effects’ of such a tax are likely to be small. Because the cross-price elasticities are so low, targeted taxation may indeed reduce FAFH consumption as a whole. Although the cross-price elasticity with respect to FAH is also small, a small percentage change in FAH occasions easily absorbs all of the lost fast food visits due to an increase in the tax.

We demonstrate this by simulating an increase in the price of fast food by 10, 25 and 50 per cent, and measuring the resulting changes in FAH, fast food, casual dining, fine dining and mid-range restaurant demand.¹² Table 6 shows these results. This simulation shows that the own-price effect of a specific tax on fast food is significant, while the cross-effects on other FAFH types and FAH is very small indeed. In interpreting these results note that the FAH measure is an index, so a small change represents a larger number of FAH meals, enough to account for the larger absolute decline in fast food visits.

While the tax propositions considered in Schroeter, Lusk and Tyner (2008) and Richards, Patterson and Tegene (2007) show that substitution opportunities thwart the intent of targeted taxes so that calories consumed actually increase, our results show the opposite. If consumers are not willing to substitute away from one type of FAFH, then targeted taxes may indeed be effective. However, we also find that tax effectiveness depends on both the type of restaurant, and the type of individual paying the tax.

6. Conclusion

Nutritionists, public health officials and economists typically place blame for the obesity epidemic on excessive consumption of restaurant meals, or FAFH more generally. Even casual observation of the data shows that FAFH consumption and obesity have both been moving upward over time so the apparent statistical association between the two cannot be denied. Uncovering the true structural factors underlying FAFH demand, however, is a much more complicated problem. In this study, therefore, we use a detailed, household-level data set to estimate the structure of FAFH demand, and how physiological attributes – obesity, physical activity and BMI – are associated with the demand for different types of FAFH. Our data consist of two survey data sets collected by NPD, Inc. that are commonly used by firms in the foodservice industry to track restaurant demand and to better understand their key market segments. For the purposes of this study, however, we use one data set – CREST – to impute prices for foods consumed away from home by respondents to a second NPD survey –NET.

In the NET data, we observe consumers visiting many different types of restaurants during each 2-week period, and consuming various amounts of food each time. For that reason, we estimate a MDCEV model of demand that accounts for satiation effects and multiple corner solutions in a structural way. Our model is structural in the sense that all decisions, including the demand for FAH as an outside option or numeraire, are derived from a single utility-maximisation model. Validation tests show that the MDCEV model performs well in an absolute sense, and in comparison with the most plausible, discrete-choice alternative.

We find that all types of FAFH are price elastic in demand, particularly fine dining, while FAH is still elastic, but less so. FAH is relatively inelastic, as consumers have a limited ability to substitute away from home meals. In terms of the cross-price elasticities of demand, we find little willingness to substitute FAFH for FAH or other types of FAFH. In that regard, our cross-price elasticity estimates show that consumers will not readily substitute between fast food, casual and mid-range restaurants, and that fine dining establishments are even more independent in demand. This result is likely due to the fact that many non-price variables enter into the decision to visit fine dining establishments. We also find that the demand for different types of FAFH varies according to the physiological profile of the consumer, measured by their BMI, the level of physical activity and health status. While all FAFH response elasticities are positive with respect to BMI, fast food and fine dining establishments appear to be the primary beneficiaries of the obesity epidemic.

Acknowledgements

Support from the Economic Research Service of the USDA and excellent comments from the reviewers and editor are gratefully acknowledged.

References