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separabilitynotes - Lecture Notes on Separable Preferences...

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Lecture Notes on Separable Preferences: Ted Bergstrom UCSB Econ 210A When applied economists want to focus attention on a single commodity or on one commodity group, they often find it convenient to work with a two- commodity model, where the two commodities are the one that they plan to study and a composite commodity called ”other goods”. For example, someone interested in the economics of nutrition may wish to work with a model where one commodity is an aggregate commodity “food” and the other is “money left over for other goods.” To do so, they need to determine which goods are foods and which are not and then define the quantity of the aggregate commodity, food, as some function of the quantities of each of the food goods. In the standard economic model of intertemporal choice model of in- tertemporal choice, commodities are distinguished not only by their physical attributes, but also by the date at which they are consumed. In this model, if there are T time periods, and n undated commodities, then the total number of dated commodities is nT . Macroeconomic studies that focus on savings and investment decisions often assume that there is just one “ag- gregate good” consumed in each time period and that the only non-trivial consumer decisions concern the time path of consumption of this single good. In these examples, and in many other applications of economics, the tactic of reducing the number of commodities by aggregation can make difficult problems much more manageable. In general, such simplifications can only be purchased at the cost of realism. Here we examine the “separability conditions” that must hold if this aggregation is legitimate. To pursue the food example, suppose that the set of n commodities can be partitioned into two groups, foods and non-foods. Let there be m food commodities. We will write commodity vectors in the form x = ( x F , x F ) where x F is a vector listing quantities of each of the food goods and x F is a vector quantities of each of the non-food goods. If we are going to be able to aggregate, it must be that consumers have preferences representable by utility functions u of the form u ( x F , x F ) = U * ( f ( x F ) , x F ) f is a real-valued function of m variables and where U * is a strictly increasing function of its first argument. The function f is then a measure of the amount 1
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of the aggregate commodity food. Notice that the function u is a function of n variables, while U * is an function of only n - m + 1 variables, which denote quantities of each of the non-food variables and a quantity of the food aggregate f . Economists’ standard general model of intertemporal choice, is based on the use of dated commodities. Thus if there are n “undated commodities,” and T time periods, we define x it to be the quantity of commodity i con- sumed in period t . We define x t to be the n -vector of commodities consumed in period t and we define x = ( x 1 , . . . , x T ) to be the nT -vector listing con- sumption of each good in each period. This is sometimes called a time profile
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