I suppose this is really an economics question, but I'm posting here for want of a more appropriate forum. My question concerns an aggregate demand system in which we have $n$ variants of a product, and the demand function of variant $i$ is given by $$D_i(p_1,\dots,p_n)$$ where $p_i$ is the price of variant $i$. We assume that $D_i$ is strictly positive for all nonnegative prices, and satisfies the gross substitutes property which says that $$\frac{\partial D_i}{\partial p_j}\geq0 ~~ \forall j\neq i$$ i.e. that if I increase the price of one variant, then the demand for all other variants can only increase. My question is: let's suppose I have some desired demand vector $\mathbf{d} = (d_1,\dots,d_n)$ and I would like to determine a price vector $\mathbf{p}=(p_1,\dots,p_n)$ such that $D_i (p_1,\dots,p_n) = d_i$ for all $i$. What is known about the existence of such a price vector, and how difficult is it to find one? I would imagine that the primary useful tools would be fixed-point type theorems, but I'm also particularly curious if there are known algorithms for finding this vector. We can assume, for example, that for any price vector, the $D_i$'s must sum to one, so that overall demand for the $n$ variants remains constant.
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$\begingroup$ Are there additional assumptions? It seems to me if each $D_i$ is constant, equal to say $1/n$, then all your conditions are satisfied, but you cannot obtain any demand vector by varying the prices. $\endgroup$– Kevin VentulloCommented Mar 1, 2013 at 8:35
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$\begingroup$ Good point -- there are actually two more technical assumptions although I omitted them here since I was trying to get a general sense of the difficulty of this problem rather than address the specific case I'm working with; the paper that I'm referring to is "Demand for Differentiated Products, Discrete Choice Models, and the Characteristics Approach" by Anderson, De Palma, and Thisse, from Vol. 56 of The Review of Economic Studies. $\endgroup$– Eric AnderssonCommented Mar 1, 2013 at 17:40
1 Answer
You can find this in an standard graduate-level microeconomics textbook, such as Mas Colell-Whinston-Green. A solution exists with or without the assumption of gross substitutes, and does indeed use a fixed-point theorem, such as the Brouwer or Kakutani fixed point theorem. Abraham Wald has a proof for a special case that's simpler -- I think his case includes gross substitutes, but I don't know for sure. In the gross substitutes case, the solution is unique.
In terms of algorithms, there is an algorithm based on Sperner's lemma due to Scharf, but practically speaking any general-purpose nonlinear solver will do. Again, there might be something special about the gross substitutes case that allows a simpler algorithm, but I don't know for sure.
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$\begingroup$ Many thanks -- with regard to your statement "practically speaking any general-purpose nonlinear solver will do" -- is the resulting problem, say, a convex optimization problem and therefore known to be easy? The use of Sperner's lemma would suggest to me that it's somewhat more combinatorial and therefore not as nicely behaved. $\endgroup$ Commented Feb 28, 2013 at 21:23
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$\begingroup$ The problem is essentially equivalent to the Brouwer fixed point theorem (at least without gross substitutes). $\endgroup$ Commented Mar 1, 2013 at 1:42
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$\begingroup$ It's not a convex optimization problem, so you can't plausibly solve a problem with a million variables, but for problems with 20 variables, non-convex solvers will work. Google "Computational General Equilibrium" for more information. $\endgroup$– arsmathCommented Mar 1, 2013 at 10:35