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Article Excerpt
The federal government's Thrifty Food Plan (TFP) minimizes the difference between a proposed food plan and a current consumption bundle, subject to cost and nutrition constraints. This article adapted the TFP framework to estimate the cost of a nutritious diet, distinguishing between nutrition constraints based on food categories (meat, vegetables) or nutrients (saturated fat, calcium). The official cost target for the TFP was sufficient if one tolerated a very high difference from current consumption patterns, or if one used nutrition standards instead of MyPyramid food category standards. In other scenarios, with different constraints, the official cost target was insufficient.
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How much does a nutritious diet cost?
This question is central to debates over U.S. anti-hunger and nutrition policy. The benefit level for more than 28 million low-income participants in the Supplemental Nutrition Assistance Program (SNAP), formerly called the Food Stamp Program (FSP), is related to the federal government's official estimate of the cost of a "thrifty" but nutritious diet (Carlson et al. 2007). This question also matters for nutrition policy more broadly, because one leading explanation for the current epidemic of obesity-related chronic disease emphasizes the comparatively low cost of energy-dense foods and the high cost of healthier foods (Drewnowski and Specter 2004).
The estimated cost of a nutritious diet depends systematically on the definition of "nutritious." In Stigler's famous 1945 application of linear programming, the minimum cost required to meet narrowly defined nutrition targets was only pennies per day (Stigler 1945). He acknowledged that his cost estimate would make dietitians unhappy, and implied that they were too generous in their "cultural requirements" for palatability, variety, and prestige, which "should not be presented in the guise of being part of a scientifically-determined budget." By contrast, researchers at the Brigham and Women's Hospital in Boston estimated the monthly cost in 2003 of a "heart-healthy" and "culturally appropriate" diet for a family of four in the low-income neighborhood of Roxbury to be $692, which was $242 higher than maximum food stamp benefit at the time (Johnson et al. 2004).
The U.S. Department of Agriculture (USDA's) Thrifty Food Plan (TFP), revised most recently in 2006, offers a useful framework for studying the cost of a nutritious diet. USDA generates the TFP by solving a constrained optimization problem, choosing a diet that is as similar as possible to the current consumption pattern for low-income Americans, while simultaneously meeting a cost constraint, food group constraints drawn from the MyPyramid nutrition education materials, nutrient constraints from the Dietary Guidelines for Americans, and other miscellaneous constraints.
The maximum benefit for the FSP is related to the value of the TFP, but this policy role is commonly misunderstood. Although the TFP is described as "the basis for maximum food stamp allotments," each revision of the TFP takes an inflation-adjusted cost of the preceding plan as the cost constraint for the new food plan. The official TFP does provide the food group quantity weights that are used in USDA's annual inflation adjustments for FSP benefits, but it would make only a modest difference in the time trends if the quantity weights in the Consumer Price Index (CPI) for food at home had been used instead. The main policy role of the TFP revision is to confirm that the previous budget allotment still suffices to purchase a nutritious diet. TFP revisions after the 1970s have not sought to reopen the more fundamental question of how much a nutritious diet should cost in the first place.
In this article, we adapted the TFP framework to investigate the cost of a healthy diet. Keeping in mind the argument between Stigler and his critics, we distinguished between the impact of nutrition constraints and preference considerations on diet costs. The article used the same official data as USDA, but we made several contributions.
First, we compared and contrasted USDA's constrained optimization problem with the theory of constrained utility maximization, which is more familiar in consumer economics. In addition to the objective function from the official 2006 TFP, we explored three alternative specifications that make the objective function more similar to a utility function that has been used in the empirical economics literature. Second, by varying the cost constraint over a wide domain, instead of imposing a single fixed cost target, we measured how the difficulty of achieving a nutritious diet increased as the cost constraint became tighter. Third, to investigate how the cost depends systematically on the definition of "nutritious," we disentangled the effects of the different kinds of nutrition constraints imposed in the 2006 TFP revision. One important contrast is between dietary advice expressed in terms of foods (Pollan 2007) and nutrients (U.S. Department of Health and Human Services and U.S. Department of Agriculture 2005). We asked: (1) if one emphasizes foods by imposing the MyPyramid food category constraints, how much does the solution plan cost and how well does it satisfy specific nutrient goals? and (2) conversely, if one emphasizes nutrients by imposing specific nutrient targets from the Dietary Guidelines, how much does the solution plan cost and how different is it from the balance of broad food categories in MyPyramid?
In future work, this framework can be used to address further questions about the impact of different constraints on the cost of a nutritious diet. In addition to sharing the data and programming for this article, we have developed a Microsoft Excel-based spreadsheet program that allows one to more easily evaluate the official USDA food plans or to create a new benchmark food plan that meets one's own chosen nutrition policy goals (Wilde, Llobrera, and Campbell 2008).
METHODS
Objective Function
In the official TFP framework, the goal is to choose a food plan, composed of quantities for 59 food groups ([x.sub.1], ..., [x.sub.59]), which minimizes an objective function while simultaneously meeting a cost constraint, nutrition constraints, and other miscellaneous constraints. In the 2006 TFP report, these groups are called "categories" (Carlson et al. 2007), but to avoid confusion, we reserve the latter term for broader MyPyramid categories. The objective function (D) measures the "distance" between a proposed food plan and the current average consumption pattern for low-income Americans. This distance function is a weighted sum of the "distance contributions" ([d.sub.i]) from each food group i.
The distance contribution for each food group i gets larger as the proposed quantity ([x.sub.i]) becomes more different from the current consumption quantity ([c.sub.i]). The official TFP uses a distance contribution that is a quadratic function of the natural logarithms of ([x.sub.i]) and ([c.sub.i]):
[d.sub.i] ([x.sub.i]) = [[ln([x.sub.i]) - ln([c.sub.i])].sup.2]. (1)
For reasons discussed later, we also investigated a simpler alternative functional form for this distance contribution:
[d.sub.i]([x.sub.i]) = [[[x.sub.i] - [c.sub.i]].sup.2]. (2)
Constraints
The first constraint is the cost constraint, which ensures that the plan is affordable. The constant parameter [[beta].sub.i1] is the price for a unit of food group i in the first constraint. The cost constraint requires that the plan's total cost ([[summation].sub.i][[beta].sub.i1][x.sub.i]) cannot exceed the cost target ([y.sub.1]).
The second constraint is the lower bound on food energy, which ensures that the plan provides enough food. The parameter [[beta].sub.2] is the food energy provided by a unit of food group i in the second constraint. The lower bound on food energy requires that the plan's total food energy ([[summation].sub.i][[beta].sub.i2][x.sub.i]) must be greater or equal to the target ([y.sub.2]), which is equivalent to 95% of the Institute of Medicine's energy requirement for a person with a low active physical activity level and the median height and weight for his or her age-gender group (Carlson et al. 2007). The third constraint is the upper bound on food energy, which requires that the plan's total food energy ([[summation].sub.i][[beta].sub.i3][x.sub.i]) must be less than or equal to a higher target ([y.sub.3]), equivalent to 105% of the requirement.
The other constraints, which vary from model to model as described in the methods section later, take a similar form. Each nutrient constraint from the Dietary Guidelines has a set of parameters [[beta].sub.ij], which describe how much of the jth constraint's nutrient is supplied by one unit of the ith food, and [y.sub.j] is an upper or lower bound on consumption of the jth constraint's nutrient. For the broad MyPyramid food category targets, each parameter [[beta].sub.ij] describes how many servings of broad food category j is provided by the ith specific food, and [y.sub.j] is the target for total servings of that broad category. In this article, a "serving" is an ounce equivalent for the meat category, and a cup each for the grains, vegetable, fruit, and vegetable categories.
The Constrained Optimization Problem
Together, the objective function and constraints constitute a nonlinear programming problem:
Choose ([x.sub.1], ..., [x.sub.59]) to minimize D([x.sub.1],..., [x.sub.59]) = [summation over (i)] [w.sub.i][d.sub.i]([x.sub.i]),
subject to [summation over (i)] [[beta].sub.i1][x.sub.i][less than or equal to] [y.sub.1] (cost constraint),
[summation over (i)] [[beta].sub.i2][x.sub.i] [greater than or equal to] [y.sub.2] (lower bound on food energy),
[summation over (i)] [[beta].sub.i3][x.sub.i][less than or equal to] [y.sub.3] (upper bound on food energy),
[summation over (i)] [[beta].sub.ij][x.sub.i][greater than or equal to] [y.sub.j] (other constraintsj = 4, 5,...,K). (3)
In the official TFP, the weights in the objective function equal the expenditure shares of each food group i in current consumption:
[w.sub.i] = [[beta].sub.i1][c.sub.i]/[[summation].sub.j][[beta].sub.j1][c.sub.j]. (4)
For reasons discussed later, we also explore using weights equal to the food energy shares of each food group in current consumption.
For each model, the solution quantities and the minimum value of the objective function were found using the dual quasi-Newton algorithm for least squares minimization, implemented using the "proc nlp" procedure in SAS statistical analysis software (SAS Institute Inc. 2004).
The official functional form has an interesting feature, which apparently has not previously been noted in the literature. When the nonlinear programming problem is solved subject to the cost constraint alone, using the Lagrangian method, the prices cancel throughout, and the solution quantity can be expressed as a simple constant multiplied by the corresponding base quantity: [x.sub.i.sup.*] = [kc.sub.i], where [x.sub.i.sup.*] is the solution quantity. For example, if the TFP cost target is 80% of the cost of current average consumption, then the solution is simply to set each...
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