The dynamics of individuals' fat consumption.

In the past few decades, consumers have become increasingly aware of the link between their lifestyle choices and the risk of noncommunicable diseases such as heart ailments and cancer (Chern, Loehman, and Yen 1995: Cutler, Glaeser, and Shapiro 2003; Ippolito and Mathios 1995; Variyam et al. 1998). Several scientific studies have found an association between diet, physical activity, and health risks (Stoeckli and Keller 2004: van Dam et al. 2002; Giovannucci et al. 1993). These studies view the diet-health risk association as a result of consumption decisions of the past, present, and the future. For instance, van Dam et al. (2002) tracked over 40,000 people between 1986 and 1994 and found that frequent consumption of meat, a proxy for total and saturated fat intake, increases risk of type 2 diabetes. Others have identified significant correlations between total fat intake and incidences of several types of cancer (Stoeckli and Keller 2004). The diet-health risk association has prompted the U.S. Departments of Agriculture and Health and Human Services, American Cancer Society and others to issue dietary guidelines, which urge a change in the composition of fat intake from meat products to dairy and fish products.

Several factors make it difficult to incorporate health management into an optimizing framework in order to derive empirical specifications of health-based food demand functions. For instance, assuming consumers have access to some health information, it is not clear whether this information should be represented as a preference or constraint in an optimizing decision model. Moreover, recent health-based food consumption models are static, while a more realistic representation of a health optimization would require a dynamic model (Park and Davis 2001; Nayga and Capps 1999).

This article introduces a dynamic approach to incorporate health management in consumer's demand for meat, fish, and dairy (MFD) products. We derive MFD consumption from a two-step optimization problem. The first step consists of a utility maximization problem with two constraints: an expenditure constraint and a fat intake constraint. We assume that consumers earn positive utility from the consumption of foods (goods), even those with high fat content, but earn negative utility from their cumulative fat level (bads) in the body. (1) That is, consumers enjoy consuming fatty meats (steak, pizza) but are limited by self-imposed (or doctor-imposed) health requirements. Therefore, consumers face not only expenditure constraints but also constraints on the amount of fat they are willing to absorb each time period. The solution to the first-step problem is an indirect utility function (IUF) with properties not unlike those of the standard IUF. However, fat intake and the cumulative fat level are arguments of the IUF, in addition to prices and expenditures. In the second step, consumers maximize utility over time by regulating their fat intake in order to control the cumulative fat level. The choice of fat intake is presented as a dynamic optimization problem, whose solution is a dynamic IUF (DIUF). Using the duality properties of the DIUF (equivalents of Roy's Identity), we derive dynamic consumer demand functions with a law of motion incorporating health decisions.

A second-order approximation of the DIUF allows us to represent the dynamic demand functions as expenditure shares and to derive an explicit equation of motion for cumulative fat levels. They are estimated as a censored system using data on 250 U.S. households on a monthly basis between December 1997 and January 2001, which are obtained from ACNielsen's Homescan Panel (ACNH) database. The products included in our estimation are beef, pork, chicken, fish, milk, and cheese. The fat content of the MFD products is calculated using a recent report of the Agricultural Research Service, U.S. Department of Agriculture (Gebhardt and Thomas 2002). The price and expenditure elasticities of demand for various MFD products are computed using the estimates of the dynamic demand functions. To the best of our knowledge, this is the first study to identify how consumers adjust fat intake over time and to derive the effects of accumulated fat on demand for individual MFD products (elasticities).

A Dynamic Demand Model with Fat Intake

The first step in deriving dynamic consumer demand functions with health attributes is to set up an IUF, which represents the solution to a static two-constraint utility maximization problem. This problem can be specified as

(1) [psi] (p, f, E, F, C) = max U([x.sub.1], [x.sub.2], ..., [x.sub.n], C)

s.t. [n.summation over (i=1)] [p.sub.i][x.sub.i] [less than or equal to] e; [n.summation over (i=1)] [f.sub.i][x.sub.i] = F

where [psi] is the IUF, p is a nx1 vector of prices of consumption goods x, E is expenditures, [f.sub.i] [member of] f is the fat content of consumption good [x.sub.i], F and C are respectively fat intake and cumulative body fat level and U is a direct utility function.

The first constraint in equation (1) represents the typical expenditure constraint and is assumed to be binding. The second constraint represents a health or fat-intake constraint. Each food product contains a certain amount of fat ([f.sub.i]), which is written as a proportion of the product's consumption in equation (1). For example, if [f.sub.1] were 0.01, it would mean that 1% of the amount of [x.sub.1] consumed is fat. The specification in equation (1) assumes that consumers care about the total fat intake rather than fat intake from individual foods, which allows for substitutability among "bads." (2) A special case of equation (1) is one where F and C are scalars implying that people care only about total fat and do not manage fat from each MFD product.

The utility function, U, is increasing and concave in the n goods, [x.sub.i], but decreasing in the amount of cumulative fat, C. (3) The properties of the static IUF are similar to those of a standard utility function. That is, [psi] is decreasing and convex in prices p, increasing and concave in E, and homogenous of degree zero in p and E. Additionally, [psi] is increasing and concave in F. (4) This reflects the assumption that consumers enjoy and would consume more fatty meats, if self--(or doctor-) imposed fat constraints did not limit such consumption. However, consumers earn negative utility from their total cumulative fat levels C. We assume that + is decreasing in C. This view is analogous (but opposite in sign) to the roles of investment and capital in production theory with adjustment costs (Epstein 1981: Vasavada and Chambers 1986). (5)

Similar to dynamic production models, the solution to the first (allocation) stage is then used to optimize over time the level of fat intake (F) and cumulative fat (C). To determine the level of "C" we introduce dynamics to the consumer problem through an equation of motion describing fat absorption in each period,

C = F - [gamma]C (2)

where [gamma] is an average rate of fat decay in the body for all MFD products. The decay rate is determined by biological factors and assumed to be exogenous for our purposes. (6) We focus on fat absorption because of the scientific link between meat fat consumption and higher risks of noncommunicable diseases. In addition, the law of motion represents expected behavior of an average consumer based on dietary guidelines from health organizations. Evidence suggests that consumers choose food products based on fat content (Cowley 1998; Food Marketing Institute 2005). (7) Moreover, equation (2) represents a health management rule not unlike a body mass index in related studies (Cutler, Glaeser, and Shapiro 2003).

Note that health-conscious consumers face a dynamic problem analogous to a standard dynamic investment problem in production economics (Epstein 1981; Sargent 1978; Vasavada and Chambers 1986). While fat intake raises consumer utility each period, cumulative fat lowers utility. Consumers control C by managing F each period. That is, in each time period, consumers vary the amount and types of meat they consume to meet their optimal choice of F. So, cumulative fat follows an equation of motion and decays at the rate of [gamma]. This choice problem can be represented as a standard dynamic optimization problem, where consumers maximize their indirect utility across time. The control variable is F, representing each period's fat intake, which is used to manage the state variable C. To focus our attention on fat intake, we do not specify an equation of motion for expenditures. This makes our dynamic demand functions dependent on expenditures rather than income in the following sections.

The dynamic problem to represent this consumer can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where V is a DIUF and a is a consumer's subjective time discount rate. In the dynamic problem in equation (3), a consumer manages his/her cumulative fat level, C, by choosing the level of fat intake each period, F, so as to maximize discounted utility over time. Since consumers enjoy fatty foods, a rise in F increases consumer's static utility, but the cumulative build up of fat, C, lowers long-run utility. The DIUF has prices, expenditures, and the fat content of each product, and total cumulative fat as its arguments.

As in any dynamic problem, there exists a Bellman equation, which is the static equivalent of the dynamic problem. The Bellman equation is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)