Discontinuous Budget Constraints and Estimation: The Demand for Housing

Abstract

Individual choices are often characterized by economists as resulting from maximization of an implicit utility function subject to a budget constraint. Informal graphical descriptions of the outcome of this process usually presume that the budget constraint, determined by individual income and prices, is linear-that the price of a good is independent of the amount of it that is purchased, and that income is independent of the amount purchased. But governmental regulations in particular-as well as non-governmental practices-often produce non-linear, kinked, and even discontinuous budget constraints. The relationship between hours worked and income, for example, would be linear if the wage rate didn't depend on hours worked. But, because of the progressive federal income tax structure, individuals actually face a net marginal wage rate that declines with income. The budget constraint is non-linear. Negative income tax plans often prescribe one tax rate up to a so-called breakeven point, and another thereafter. There is a kink at the breakeven point.1 Social security regulations impose low tax rates on wage income up to a given level and a very high tax rate on each additional dollar of income. Most existing health insurance plans, as well as proposed national health insurance schemes, include some combination of a deductible, a coinsurance rate, and possibly a maximum health care expenditure level. The price of a dollar's worth of health care is one up to the amount of the deductible; it is the coinsurance rate between the amount of the deductible and the maximum expenditure, and is zero thereafter. Again, the implied budget constraint is non-linear; it has kinks in it. Some proposed subsidy schemes stipulate that low income families receive housing payments, but only after a minimum expenditure for housing. The implied budget constraint is discontinuous. This paper proposes a rather general method of estimation when the implicit budget constraint is non-linear. But it does so by addressing a particular problem-the analysis of data generated by treatments in the recent Housing Demand Experiment, that can be thought of as creating discontinuous individual budget constraints. It rests on the assumption that the relative value that individuals attach to purchased goods can be described by a functional relationship that assigns weights to goods, or to the dollar expenditures for these goods, a utility function. The key parameters of this function are taste parameters that are assumed to depend on individual characteristics of decision makers and to be random, given measured characteristics. That is, they depend on observed as well as unobserved attributes of individuals or of their environment. In addition, we assume that persons are not always able to match expenditures to hypothetical best, or maximizing, values. Although the approach is motivated by the idea of utility