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Incorporating subsistence into hedonic price and nutrient demand equations.

The characteristics model of consumer choice explicitly introduces food nutrients into the decision-making process (Eastwood, Brooker, and Terry 1986; Hager 1985; Ladd and Suvannunt 1976; LaFrance 1985). Solution of the model leads to hedonic price and nutrient demand equations. The former can be thought of as relating the market price of a food to its nutritional composition. The latter refers to the amount of a nutrient consumed as a function of socioeconomic variables.

Habit persistence was introduced into traditional demand analyses via the Klein-Rubin utility function, which led to its use in the linear expenditure system, the extended linear expenditure system, and most recently the almost ideal demand system. The interpretation, given originally by Samuelson (1947-1948), is that a consumer needs minimum amounts of goods, so utility is derived from the consumption of goods over and above these subsistence levels. To date none of the characteristics models, and especially those for food, has incorporated the concept of subsistence. This omission appears to be a serious limitation. Consumers need minimum amounts of nutrients in order to survive, let alone have healthier lives, ceteris paribus. These subsistence levels are primarily attained through the regular consumption of food. Utility arises from the consumption of nutrients in excess of the subsistence levels.

A direct consequence is that the arguments of the utility function ought to accommodate subsistence levels of nutrients. To the extent that this formulation is a better approximation of consumer decision-making, three implications follow directly. First, the characteristics model needs to be revised. Second, the hedonic price and nutrient demand equations should be estimated with adjusted measures of nutrient consumption. Third, public policies directed toward enhancing dietary status may need to be reevaluated in light of the revised model. These considerations are particularly important given current interest in new standards for nutritional labeling of foods; current food policy emphases on understanding relationships among consumers' dietary attitudes, knowledge, behaviors, and food consumption; and the introduction of many new foods in response to dietary awareness.(1)

This paper introduces subsistence into the characteristics framework. The utility function is written in terms of nutrients consumed above the subsistence level. Solving the model leads to empirical hedonic price equations and nutrient demand equations that reflect nutrition levels associated with minimum dietary standards. The approach yields hedonic price equations that can be consistent with declining marginal utilities of nutrients. Estimates of the equations are presented, and the results are discussed.


Assume the utility derived from food nutrients is separable from the other arguments of the utility function. Let |x.sub.i,j~ be the amount of the jth nutrient in the ith food, n is the number of food items, and m equals the number of nutrients. The amount of a nutrient that is consumed, |X.sub.j~, is derived from the quantities of foods (|q.sub.i~) and their exogenous nutrient compositions, shown as equation (1).

|X.sub.j~ = |X.sub.j~(|X.sub.1,j~, . . . , |X.sub.n,j~, |q.sub.1~, . . . , |q.sub.n~)

Let |S.sub.j~ denote the subsistence amount of the jth nutrient for the consumer and |D.sub.j~ be the excess amount. |S.sub.j~ is considered to be based on dietary needs exogenous to the consumer. |X.sub.j~ = |D.sub.j~ + |S.sub.j~. The utility function is equation (2).

U = u|(|X.sub.1~ - |S.sub.1~), . . . , (|X.sub.m~ - |S.sub.m~)~ = u(|D.sub.1~, . . . , |D.sub.m~)

Because the consumer must purchase market goods in order to obtain the characteristics, an important relationship is the marginal utility of each good. Recalling equations (1) and (2), a change in |q.sub.i~ causes a change in the vector of attributes received, which generates a change in utility. That is,

|Mathematical Expression Omitted~

The budget constraint is

|Mathematical Expression Omitted~

where |P.sub.i~ is the unit price of the ith food.

Maximizing equation (2) subject to the budget constraint (4) and the consumption technology equation (1) yields the first order conditions which can be rearranged to yield the hedonic price equation (5).

|Mathematical Expression Omitted~

The sufficient condition for a unique solution to the utility maximization problem is that the Hessian of equation (2) be negative definite. This is equivalent to assuming diminishing marginal utility for the |q.sub.i~. However, in the characteristics model this requirement entails the evaluation of equation (5). As long as goods generate characteristics that yield positive utility and there is declining marginal utility of attributes, the Hessian is negative definite.

Hendler (1975) was the first to point out that a unique consumption bundle could still arise even if some of the characteristics generate negative utility. (For example, a particular nutrient may yield a bitter taste.) In these cases the modified condition is that each good must yield a positive net utility to the consumer. That is, some marginal utility expressions on the right-hand side of equation (5) may be negative, but the sum must be positive. This condition, coupled with declining marginal utility, means the Hessian of the utility function is negative definite.

Two assumptions are generally made to estimate equation (5) for food (e.g., Eastwood, Brooker, and Terry 1986; Ladd and Suvannunt 1976). One is a linear consumption technology (|delta~|D.sub.j~/|delta~|q.sub.i~ = |x.sub.i,j~. The other is a constant marginal implicit price of each nutrient |(|delta~U/|delta~|D.sub.j~)(|delta~I/|delta~U = |b.sub.j~. These conditions yield a linear hedonic price equation (6).

|Mathematical Expression Omitted~

The linear consumption technology appears to be reasonable for foods. For example, the amount of protein in an ounce of steak is constant whether one ounce or eight ounces are eaten. However, the constant marginal price assumption is less defensible. There are two ways in which it can hold. One is that neither the marginal utility of income nor the marginal utility of the jth nutrient changes. The other possibility is that the change in marginal utility could be exactly offset by the reciprocal of the marginal utility of income, which is unlikely.

Equation (6) should allow for nutrition levels in excess of subsistence as the expressions for marginal utilities. In addition, the equation should be consistent with declining marginal utility. In order for declining marginal utilities to occur, Morse (1988) has shown that for goods that have positively valued attributes the following conditions must hold

|delta~|P.sub.i~/|delta~|D.sub.j~ |is greater than~ 0 and (7)

|Mathematical Expression Omitted~

The form used in the present analysis is equation (9). This is consistent with Morse (1988). It assumes a household's marginal implicit price (|b.sub.j~) changes with the level of |D.sub.j~.

|Mathematical Expression Omitted~

where g denotes a relationship based on being above or below subsistence. Households that are below the defined subsistence level of all nutrients are placed in the first group (g = 1). Intermediate households, those under the subsistence level for at least one nutrient, but not all nutrients, are placed in the second group (g = 2). Households that are over the defined subsistence levels for all nutrients are placed in the third group (g = 3).

Economic theory does not impose restrictions on |Mathematical Expression Omitted~. The marginal utility of income is considered to be positive and declining with income. Because the marginal utility of an attribute can be positive or negative, |b.sub.j~ can be positive or negative. Aside from |b.sub.j~ taking on positive or negative values, |Mathematical Expression Omitted~ or |Mathematical Expression Omitted~. Some additional insight is possible through comparisons of the |Mathematical Expression Omitted~. If |Mathematical Expression Omitted~ declines more (less) than |Mathematical Expression Omitted~ as a household crosses from g = 1 to g = 3, then |Mathematical Expression Omitted~ would become greater (less) than |Mathematical Expression Omitted~. Thus, it is possible to infer which marginal utility has the greater relative change as a household changes from under subsistence to over subsistence nutrition levels.

Solution of the characteristics model also yields nutrient demand relationships. Levels of nutrient consumption are functions of the marginal implicit prices of nutrients and socioeconomic variables (V), shown as equation (10). The important difference between equation (10) and previous work is that the relationship is estimated by subgroups defined by nutritional adequacy.

|Mathematical Expression Omitted~


For 50 years the USDA has prepared guides for selecting nutritious diets at different levels of cost (Peterkin and Kerr 1982). The Thrifty Food Plan (TFP) specifies nutritious diets at specified fixed-cost levels for adults and children by age and sex (Cleveland et al. 1983). The plan is derived using a quadratic programming formulation that finds the combinations of food groups that represent as little change from the consumer's food consumption as needed to meet the RDA at a specified cost from a subsample of the Nationwide Food Consumption Survey. The TFP plan is the least costly of the four food plans developed. Consequently, TFP guidelines reflect the importance of eating patterns, income levels, and nutritional constraints associated with an adequate diet.


Research has found that typical consumers are more aware of nutritional aggregates than individual nutrients (e.g., Parato and Bagali 1976; Weimar 1980). Other investigations (Federal Trade Commission 1976; Morgan 1987) have found that individuals evaluate foods in terms of key dietary factors or lump nutrients into aggregates. The level of knowledge necessary to understand the function of each nutrient and evaluate its impact and necessary level would require high information access cost. Thus, consumers are optimizing by lumping nutrients rather than spending the time necessary to refine nutrient distinctions and/or reduce the information overload potential.

Dietary guidelines for 11 nutrients by age-sex classification were fixed in the TFP. These were used to develop the set of household level measures of subsistence. For example, consumers were more likely to know about minerals as opposed to the separate nutrients. The aggregate nutrients were minerals (calcium, iron, and magnesium), B vitamins (niacin, riboflavin, thiamin, B6, and B12), and food energy (a linear combination of carbohydrates, fats, and protein). B vitamins were converted to the same units (mg) and combined. Enough information was believed to be available to consumers about vitamin C to include it separately. At the time the data were collected (see below), vitamin A was measured in international units, so it was included separately, as well. Subsistence levels for each nutrient were defined by the 1975 TFP. This definition was selected because it was developed just prior to the time of the food consumption survey described.

These groups were based on previous studies and the TFP as the most logical for combining. Terry (1985) and Morgan, Metzen, and Johnson (1979) have shown that estimations of nutrient demand and/or hedonic equations explained more of the variations in the dependent variables when dietary components were combined in ways more in keeping with the way consumers perceive nutrients.

Food consumption data are from the 1977-1978 Nationwide Food Consumption Survey (NFCS) because these are the most recent. These data pertain to at home food consumption. The spring quarter of the sample was used to limit the availability of food purchases across seasons and to keep the sample size manageable. Households that purchased at least 20 food items were included. This was done for two reasons. One was to delete households that were consuming primarily out of unmeasured stocks. The other was to have enough foods (i.e., observations) for the household level estimations of equation (6). Fewer than 200 households were deleted in this step, so the effect of sample selection bias was likely to be quite small. However, the inferences (described below) pertain to households purchasing at least 20 food items per week. Missing, incorrect, or inappropriate data further reduced the sample to 2,133 households. These were observations that fell outside the permissible values according to the 1977-1978 NFCS documentation.

Data included the kind, form, quantity, and cost of each food item used during a seven-day period. Home food production, the number of meals eaten at home and away, and guest meals data were also gathered. Nutritional contents of foods were part of the data. Use of the household as the unit of analysis was based on the assumption the dependent members of a consuming unit did not have the resources to make extensive independent purchase decisions. Food prices were calculated by dividing the expenditures on foods by their respective quantities. Table 1 lists the variables used.


A household's actual consumption of each of the aggregated nutrient groups was compared to the respective TFP subsistence level of the corresponding age-sex composition. If the household met all its requirements, it was placed in "over" group (g = 3). |D.sub.j~ = |X.sub.j~ - |S.sub.j~ was positive for the five nutrient groups in this case. If for all household members, the requirements as established by the TFP were not met, the household was placed in the "under" group. |D.sub.j~ was negative for the five nutrient groups. Households that met some but not all of their nutrient requirements were placed in an "intermediate" group. At least one |D.sub.j~ was negative. There were 310 households TABULAR DATA OMITTED in the over group, 412 households in the under group, and 1,411 in the intermediate group.


Equation (9) was estimated for each subgroup using OLS. Then, as OLS regression was calculated for the entire sample. Because three groups were involved, an F-test as explained by Intriligator (1978) and adapted by Eastwood (1990) was used. Let G equal the number of subgroups and m equal the number of estimated coefficients within a subgroup. There are k = Gm estimated coefficients altogether and h = (G - 1)m differences among the sets of estimates TABULAR DATA OMITTED for the subgroups. The h differences among the k estimated coefficients can be expressed as a set of linear constraints. Let A denote this constraint matrix. It has dimensions h by k. Let B equal a vector of the k estimated coefficients. H equals the matrix of independent variables for the entire sample. The null hypothesis is that there are no significant differences among the subgroups. The computed F-value with h and n - k degrees of freedom is

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~ is a block diagonal matrix composed of the variance-covariance matrices of each separate subgroup regression. As the data are cross-sectional and the observations are independent from one another, there would not be mutual correlation or autocorrelation among the error terms for each observation. Therefore, the off-diagonals in |Mathematical Expression Omitted~ are zero.

Table 2 presents the estimated equations. It also contains the no-intercept adjusted |R.sup.2~s (Eastwood, Gray, and Brooker 1989), the mean squared error, the overall F-value, and the number of usable observations (foods consumed by the households in the respective groups). The constructed F-test using equation (11) led to the inference that the three sets of coefficients were significantly different from one another. The |s.sup.2~ used was the pooled value because a test of the differences of the variance for each subgroup equation showed no significant differences from the poooled value. Comparing the estimated coefficients between the over and under groups, all parameters were significantly different at the .01 level. An inference is that the changes in marginal utilities did not offset each other. There were significant changes in the marginal utilities of the nutrients and/or significant changes in the marginal utilities of income between the subgroups.

All of the estimated implicit prices were significant, and except for Vitamin, A, all were positive and less than one. Aside from vitamin A, households within the subgroups and overall had positive valuations of food energy, minerals, B vitamins, and vitamin C. Comparing the two most distinct groups (ie., under, over), the under group had smaller marginal implicit prices for food energy, B vitamins, and vitamin A. This suggests that, as a household that purchased at least 20 food items moved from the under to over subgroups, the marginal utilities of these nutrients fell more than the marginal utility of income, thereby causing the increase in |b.sub.j~. Minerals and vitamin C marginal implicit prices declined between the under and over subgroups suggesting that the marginal utilities of these nutrients fell less than the marginal utilities of income. An implication is that consumers' valuations of nutrients change in nonuniform ways as dietary status changes.

Only limited comparisons with other studies can be made due to differences in definitions of variables and the present groupings of households based upon dietary status. With respect to the hedonic price equation, the |R.sup.2~s in Table 2 are all more than twice as high as those reported by Eastwood, Brooker, and Terry (EBT) (1986). Most of the estimated marginal implicit prices in EBT fall within the range of those shown here.

Table 3 gives the percentage change in the price for a percentage change in nutrient use based on sample averages of households purchasing at least 20 food items. B vitamins, followed closely by food energy, have the largest impacts. Those for minerals are less than half of those for food energy. The low values of vitamin C may reflect consumers' belief that it is found in so many foods, or easily supplemented, that less concern was given to this nutrient. One should also remember that the time period of the survey was well before the recent attention give to vitamin C.
Price Flexibilities for Subsistence Groups
Variable Under Middle Over
Food Energy .2624 .3166 .3293
Minerals .1243 .1126 .1015
B Vitamins .3181 .3521 .3490
Vitamin C .0872 .0500 .0467
Vitamin A -.0496 -.0374 -.0315

The data also permit point estimates of each household's valuations of nutrients. These equations were estimated in order to obtain household specific marginal implicit prices for use as independent variables in the nutrient demand equations. It is not feasible to analyze the results of 2,133 separate regressions for each of the five nutrients, but Table 4 summarizes the results of the household level hedonic price equation estimates. They suggest that there is enough variation in the marginal implicit prices to include them as explanatory variables in the nutrient demand equations. The table shows that, although the means are similar, the distributions, as reflected in TABULAR DATA OMITTED the minimums and maximums, are different. Negative minimum values suggest that at least one household had a negative valuation of the respective nutrient, and as the food was consumed, positive valuations of the other nutrients contained in the food prompted the household to consume it. The mean values are fairly close to those of EBT, but less variability is found in the sets of household level estimated prices, as reflected in the coefficients of variation versus EBT.

Table 5 presents a summary of the nutrient demand equation estimates for households purchasing 20 or more food items. Four equations were estimated for each nutrient--one for each subgroup and the total sample. The signs of significant coefficients are noted along with the equations in which the significance occurred. The range of the |R.sup.2~s overall F-values (all significant) are reported as well.

All of the own-price coefficients, except vitamin C, are significant, and only vitamin A has one that is positive. Significant cross-price elasticities are negative with the exception of minerals in the B vitamins equation. Complementarity exists among the nutrients with food energy and B vitamins having the broadest cross-group impact, followed by minerals. Vitamin C's coefficient is not significant in any equation, perhaps indicating a unique position for this nutrient during the sample period.

With respect to the nutrient demand equations, EBT have higher |R.sup.2~s. However, both studies have significant own-price elasticities for nutrients which are negative except for vitamin A which is positive in each. This is similar to other studies (e.g., Adrian and Daniel 1976; Basiotis et al. 1983; Chavas and Kepplinger 1983; Davis and Neenan 1979; Eastwood, Brooker, and Terry 1986; Pitt 1983; Scearce and Jensen 1979). Income and the bonus value of food stamps have positive effects on nutrient consumption. With respect to income, the effect is across all groups. The food stamp impact appears to be centered on the middle group.(2) This suggests that the program did not have a significant effect on households that were consistently under the TFP's guidelines, but it did have positive impacts on the middle group households that were not always below or above subsistence. TABULAR DATA OMITTED Because the middle group has many more households, this result indicates the program did have positive effects on households buying at least 20 food items per week but did not induce them to achieve balanced diets across the five nutrients.

The presence of a working female decreased the levels of nutrient consumption in the middle and over groups. This suggests that the income generated by the working female is not necessarily allocated to food at home consumption of households with marginal or adequate diets. It is also consistent with households with working females eating more meals away from home. Adrian and Daniel (1976) and Capps and Love (1982) found that families with employed housewives had higher consumption of carbohydrates and fats using TABULAR DATA OMITTED the 1965-1966 NFCS. The absence of a significant coefficient in the food energy equation here may reflect the shift from part-time to full-time employment of housewives during this period (Eastwood 1985).

Most location variables were not significant in any equation. This is consistent with Basiotis et al. (1983). The exceptions were primarily households in suburban areas. Within the over group, households in suburban areas had higher levels of nutrient consumption. This is similar to Akin, Guilkey, and Popkin (1983) who found urban households had higher caloric, iron, and vitamin C intakes. On the other hand Devaney and Moffitt (1991) found negative effects for suburban residents in their sample of low income households in 1979-1980.

Where significant, the level of education of the meal planner was negative. This occurred primarily in the under group. The pattern of significant coefficients suggest that the level of education has little to do with the dietary status of a household purchasing 20 or more food items per week. A similar result was also found by Adrian and Daniel (1976). Akin, Guilkey, and Popkin (1983) found that education had a positive impact on iron and vitamin C consumption, but not on the other nutrients they considered.

Life cycle stages have broad impacts on levels of nutrient consumption across all groups. D5 (female head present and average age of children 19 or older) is significant and negative in only one instance, whereas households with children of various ages have significantly higher levels of consumption. Other studies have also found household composition to be an important determinant (e.g., Adrian and Daniel 1976); Blanciforti, Green, and Lane 1981; Davis and Neenan 1979; LaFrance 1985; Scearce and Jensen 1979; Schafer and Keith 1981; Windham et al. 1983).

The pattern of significant race coefficients is similar to other studies (e.g., Adrian and Daniel 1976; Basiotis et al. 1983; Devaney and Moffitt 1991; LaFrance 1985; Scearce and Jensen 1979). Whites had lower food energy, B vitamins, and vitamin A intakes in the over groups. Blacks had lower intakes in the under groups for food energy, minerals, and B vitamins, whereas their intakes of vitamin A were higher in the middle group.


The characteristics model of consumer choice was modified to accommodate subsistence levels of nutrients. Hedonic price equations were shown to be consistent with declining marginal utility, and nutrient demand equations contained implicit prices that were adjusted for subsistence. The TFP was selected as the empirical measure of subsistence for individuals based on their age and sex. Using the most recent data available, the 1977-1978 NFCS, a household's weekly food at home consumption was converted to nutrient levels which were compared to TFP subsistence to determine whether the household was above or below subsistence for each nutrient.

Significant differences among the sets of estimated marginal implicit prices led to the inference that valuations of nutrients by households buying 20 or more food items per week changed relative to subsistence. Because American diets were oriented toward consumption of foods that provide minerals and B vitamins--especially meat products as opposed to fruits and vegetables--during the survey period, the pattern of estimated implicit prices is not surprising. This also resulted in low levels of vitamin A. Income and household size were not significantly different by subgroup, which suggests that the significant differences among subgroup nutrient valuations did not reflect differences in budget constraints. An implication is that public programs directed toward improving American diets need to be sensitive to different valuations of nutrients. Public policies aimed at improving diets could stress food energy and B vitamins because all groups have the highest estimated flexibilities for these two aggregates.

The estimated nutrient demand equations related levels of nutrients consumed to socioeconomic characteristics of households that purchased at least 20 food items. Negative own-prices, except for vitamin A, were found. Most significant cross-price effects were negative. Food energy and minerals had the dominant cross-price effects. Income had consistently positive effects on nutrient consumption for all groups. The bonus value of food stamps was positive and significant only for the middle (and total) group. This suggests that the program had a partial impact on improving diets. The negative impact of working females may reflect lower incomes of female-headed households and, therefore, poorer diets. The level of education had negative impacts which suggests that nutrition education at all levels is needed.

Christanna M. Cook is Assistant Professor, Marketing Department, Husson College, Bangor, ME; and David B. Eastwood is Professor, Department of Agricultural Economics and Rural Sociology, University of Tennessee, Knoxville.

1 Gallo (1991) reported that over 75,000 new food items were introduced between 1982 and 1990.

2 Because food stamps decrease with income and increase with family size, ceteris paribus, the coefficients of income and life-cycle variables may contain some bias. The magnitudes of the biases are felt to be small due to the low proportions of households using food stamps in all three categories, the insignificance of the coefficient of BN in two of the three subgroups, and the presence of households that qualified for food stamps but failed to acquire them.


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Author:Cook, Christanna M.; Eastwood, David B.
Publication:Journal of Consumer Affairs
Date:Dec 22, 1992
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