A structural econometric model of joint consumption of goods and recreational time: an application to pick-your-own fruit.

Traditional economic models of consumer behavior assume that the demand for goods originates from an optimization problem where consumers maximize utility from the consumption of goods subject to a monetary budget constraint. The joint effects of time in the utility function and as a resource constraint (time constraint) have not received much attention in the context of demand for goods. Simultaneous consideration of these aspects has mainly been restricted to the areas of environmental and transportation economics.

Two constraint models of consumer behavior without time included in the utility function have received some attention in the economics literature (e.g., Larson and Shaikh 2001; Hanemann 2006). In these studies, only quantities of the commodities consumed determine utility and time spent acquiring (consuming) the goods does not provide utility. In an earlier theoretical article, DeSerpa (1971) included time in both the utility function and as a constraint on consumer choice and derived some comparative statics of the problem. DeSerpa's work has been mainly used to analyze travel consumer behavior (e.g., Gonzalez 1997).

In this study, we develop a fully structural econometric consumer demand model for situations in which the goods consumed have time and monetary costs, and where time spent consuming/obtaining the goods also enters into the utility function. This framework allows modeling the choice between different types of a good and the quantity of the good to buy. This model of consumer behavior is an extension of Hanemann's (1984) work on discrete/continuous choice modeling. Hanemann's (1984) model is extended in two ways. First a time constraint is added to the model, and second, time is also included into the utility function.

In the empirical part of the article, the structural econometric model is used to estimate demand functions for customers visiting pick-your-own (PYO) fruit operations, where customers choose between harvesting fruit and buying preharvested fruit from the PYO operations. This application seems quite appropriate for the model because of the joint nature of consumption of goods and time, where time spent picking fruit provides utility to the consumer. In the context of the PYO fruit application, this model is able to explain the type of fruit chosen by the household, and explain the quantity of fruit purchased. The choice margin we focus on is the decision to buy PYO fruit versus preharvested fruit. (1) We also derive a procedure to calculate the price elasticities of the total amount of fruit sold at the farm using the estimates of the price elasticities for PYO and preharvested fruit.

There are other situations where the model can be used. For example, there are several types of community-supported agriculture arrangements where consumers commit both time and money to obtain products from the farm being supported instead of buying products from the grocery store. In this context, the time spent on the farm is not only an input in the production process but can also be seen as an experience attribute. From a larger perspective, this type of model allows us to explore the value that consumers place on the farm experience. Therefore, farmers can add value to their products not only through processing but also if they are able to attach to their agricultural products some recreational/experiential value.

Theoretical Framework

The model of choice used for the theoretical framework assumes that utility is random (Hanemann 1984). Random utility can be motivated by the fact that although the utility function is deterministic for the consumer, it contains elements that are unobservable to the investigator. Utility is defined over the quantity of the goods, time spent obtaining the goods, and the consumer's perceived quality for each of the goods. The inclusion of time into preferences is the first novel characteristic of this model. The utility function is defined over two goods. The first good is available in R alternative forms, which can represent different varieties of a product. The second is a numeraire. The utility function has the following form:

(1) u(x, z, o, Tv, [psi], b, s, [epsilon])

where [psi] = [[psi].sub.1], [[psi].sub.2], ... [[psi].sub.R] is a R-dimensional vector and [[psi].sub.i] represents the consumer's evaluation of quality for the ith alternative, x = [[x.sub.1], [x.sub.2], ..., [x.sub.R]] is a R-dimensional vector and [x.sub.i] represents the quantity of the ith variety of the first good, z represents the quantity of a good numeraire, o represents the quantity of a time numeraire, Tv = [[T.sub.1],[T.sub.2], ..., [T.sub.R]] is an R-dimensional vector and [T.sub.i] represents the time spent obtaining the ith variety. The vector [b.sub.i] = [[bi.sub.1], [bi.sub.2], ..., [bi.sub.H]] is an H-dimensional vector defining H different dimensions of quality, where [b.sub.i1] is the amount of the hth characteristic associated with a unit of consumption of variety i. The R-dimensional vector [epsilon] = [[[epsilon].sub.1], [[epsilon].sub.2], ..., [[epsilon].sub.R]] is a random vector representing the unobservable characteristics of the consumer and/or attributes of the commodities. Finally, s = [[sub.1], [sub.2], ..., s[sub.L]] is an L-dimensional vector with observed characteristics of the consumer.

The consumer's problem is to choose x, z, and o to maximize utility subject to a budget constraint and a time constraint:

(2) [R.summation over (i = 1)] [p.sub.i][x.sub.i] + z = y

(3) [R.summation over (i = 1)] [T.sub.i]([x.sub.i]) + 0 + [T.sub.w] = T

where y is income, [T.sub.w] represents the number of hours worked, and o the time numeraire. If [[mu].sub.M] is the Lagrangian multiplier corresponding to the monetary budget constraint (2), [[mu].sub.T] is the Lagrangian multiplier corresponding to the time constraint (3), and the total amount of time required to obtain each variety [T.sub.i] is a linear function of the amount obtained ([T.sub.i]([x.sub.i]) = [t.sub.i][x.sub.i]) then the parameter [theta] = [[mu].sub.T]/[[mu].sub.M] can be defined so that the two constraints can be merged into a single constraint as follows: (2)

(4) [R.summation over (i = 1)] [[pi.sub.i][x.sub.i] + k = I

where [[pi].sub.i] = [p.sub.i] + [theta][t.sub.i], k = z + [theta]0 and I = y + [theta] T. Notice that [theta] corresponds to an endogenously determined value of time since both [[mu].sub.M] and [[mu].sub.T] are functions of the endogenous variables.

In order to develop a structural econometric model of variety choice, specific assumptions regarding the functional form of the direct utility function and the distribution of the errors are necessary. The following utility model is used:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [u.sup.*] is a bivariate utility function. Maximization of (5) subject to (4) leads to a corner solution where only one of the varieties is selected. (3) Equation (5) extends Hanemann's model to include the time spent obtaining the ith variety as a variable in the utility function. In this model [xi] is a parameter that measures the effect of time in the utility function.

Given that the consumer has selected variety j, the conditional direct utility function is [[bar.u].sub.j]([x.sub.j], [[psi].sub.j], [t.sub.j], k) = [[bar.u].sub.j]([[psi].sub.j], k + ([[psi].sub.j] + [xi][t.sub.j])). Conditional ordinary demand functions and the indirect utility function associated with [[bar.u].sub.j] can be obtained by solving the following conditional maximization problem:

(6) Max [bar.u].sub.j]([x.sub.j], k + ([[psi].sub.j] + [xi][t.sub.j])[x.sub.j]) st x [x.sub.j][pi].sub.j] + k = I

Defining Z = k + ([[psi].sub.j] + ([phi].sub.j]) + [xi][t.sub.j])[x.sub.j], substituting back into the utility function, and adding and subtracting ([[psi].sub.j] + [xi][t.sub.j])[x.sub.j] in the budget constraint, problem (6) can be rewritten as: Max [[bar.u].sub.j](x.sub.j], Z)st.[x.sub.j] [[pi].sub.j] + k + ([[psi].sub.j] + [xi][t.sub.j])[x.sub.j] - ([[psi].sub.j] + [xi][t.sub.j])[x.sub.j] = I or Max [[bar.u].sub.j]([x.sub.j], y) st. [x.sub.j][[pi].sub.j] - ([[psi].sub.j] + [xi][t.sub.j])] + Z = I. The solutions to this maximization problem have the form: [x.sup.*.sub.j] = [x.sub.j]([[pi].sub.j] - [[psi].sub.j] - [xi] [t.sub.j], I) and [Z.sup.*] = ([[pi].sub.j] - [[psi].sub.j] - [xi][t.sub.j], I) and therefore the conditional ordinary demand functions and indirect utility functions associated with [[bar.u].sub.j] are:

(7) [[bar.x].sub.j]([[pi].sub.j], [[psi].sub.j], [t.sub.j], I) = [[bar.x].sub.j]([[pi].sub.j] - [[psi].sub.j] - [xi][t.sub.j], I),

(8) [[bar.v].sub.j]([[pi].sub.j], [[psi].sub.j], [t.sub.j], I) = [[bar.v].sub.j]([[pi].sub.j] - [[psi].sub.j] - [xi][t.sub.j], I).

Because [[bar.v].sub.j] is decreasing in [[pi].sub.j], it follows from (8) that the single variety selected is the one for which [[pi].sub.j] - [[psi].sub.j] - [xi][t.sub.j] (i.e., quality-time effect adjusted full price) is lowest. In equation form, alternative j is preferred to alternative i if

(9) [[pi].sub.j] - [[psi].sub.j] - [xi][t.sub.j] < [[pi].sub.i] - [[psi].sub.i] - [xi][t.sub.i], [for all] = 1, ..., R, and i [not equal to] j.

The function [[psi].sub.j] can be viewed as an index of the overall quality of the jth variety. This index depends on the quality characteristics of the variety [b.sub.j], the characteristics of the individual s, and the error term [[epsilon].sub.j]. Assume the function [[psi].sub.j] has the form

(10) [[psi].sub.j][(b.sub.j], s, [[epsilon].sub.j]) = [[alpha].sub.j] + [gamma]'[b.sub.j], + [[phi].sub.l]'s + [[epsilon].sub.j],