I attended a theory seminar at the University of Technology, Sydney (UTS) yesterday by Professor Rabah Amir from Univ. of Iowa, talking about the following theoretical IO paper:
Amir, R., Ericksonz, P., and Jin, J. "On the microeconomic foundations of linear demand for differentiated products"
This paper provides theoretical foundations of the multi-variate linear demand function for differentiated products that is widely used in industrial organization. While quasi-linear preferences with quadratic utility function (for goods) is known to yield linear demand (Singh and Vives, 1984), the authors extend the result. Namely, under quasi-linear environment, they show that the resulting demand system becomes linear and well defined if and only if the underling utility function (for goods) must be strictly concave and quadratic. The "only if" part implies that the use of linear demand that does not satisfy the law of demand ought to be regarded with some suspicion.
A theoretical punch line of the paper is that any (linear) demand system derived by consumer's utility maximization must satisfy the law of demand. I believe that we can establish a slightly more general result than this in the following way. It is known that (see, for example, Mas-Colell et.al. 1995) if a (Marshallian) demand function satisfies the weak axiom of revealed preferences and Walras' law , then its Hicksian demand satisfies the law of demand. If we further assume that there is no income effect, then the (Marshallian) demand function also satisfies the law of demand, since the two demand functions must have identical derivatives w.r.t. prices. The following is a summary the idea.
(1) weak axiom + (2) Walras's law => law of demand for Hicksian demand
(3) no income effect => Marshiallian demand = Hicksian demand
(1) weak axiom + (2) Walras's law + (3) no income effect => law of demand
The above argument does not presume any consumer's preferences or utility maximization behind the demand function. By contrast, the current paper associates consumer's demand with his/her utility function. Now, note that if a demand function is derived by utility maximization with an increasing utility function, then conditions (1) and (2) automatically hold. If utility function is quasi-linear and a consumer has enough income, then (3) is also guaranteed. Since these are exactly what the paper assumes, the sufficient conditions (1), (2), and (3) above are satisfied and thus the law of demand (for Marshallian demand) naturally follows.
Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic theory. New York: Oxford university press.
Singh, N., & Vives, X. (1984). Price and quantity competition in a differentiated duopoly. The RAND Journal of Economics, 546-554.