September | 2012 | The Leisure of the Theory Class

A paper by Azevdo, Weyl and White in a recent issue of Theoretical Economics caught my eye. It establishes existence of Walrasian prices in an economy with indivisible goods, a continuum of agents and quasilinear utility. The proof uses Kakutani’s theorem. Here is an argument based on an observation about extreme points of linear programs. It shows that there is a way to scale up the number of agents and goods, so that in the scaled up economy a Walrasian equilibrium exists.
First, the observation. Consider ${\max \{cx: Ax = b, x \geq 0\}}$ . The matrix ${A}$ and the RHS vector ${b}$ are all rational. Let ${x^*}$ be an optimal extreme point solution and ${\Delta}$ the absolute value of the determinant of the optimal basis. Then, ${\Delta x^*}$ must be an integral vector. Equivalently, if in our original linear program we scale the constraints by ${\Delta}$ , the new linear program has an optimal solution that is integral.

Now, apply this to the existence question. Let ${N}$ be a set of agents, ${G}$ a set of distinct goods and ${u_i(S)}$ the utility that agent ${i}$ enjoys from consuming the bundle ${S \subseteq G}$ . Note, no restrictions on ${u}$ beyond non-negativity and quasi-linearity.

As utilities are quasi-linear we can formulate the problem of finding the efficient allocation of goods to agents as an integer program. Let ${x_i(S) = 1}$ if the bundle ${S}$ is assigned to agent ${i}$ and 0 otherwise. The program is

$\displaystyle \max \sum_{i \in N}\sum_{S \subseteq G}u_i(S)x_i(S)$

subject to
$\displaystyle \sum_{S \subseteq G}x_i(S) \leq 1\,\, \forall i \in N$

$\displaystyle \sum_{i \in N}\sum_{S \ni g} x_i(S) \leq 1 \forall g \in G$

$\displaystyle x_i(S) \in \{0,1\}\,\, \forall i \in N, S \subseteq G$

If we drop the integer constraints we have an LP. Let ${x^*}$ be an optimal solution to that LP. Complementary slackness allows us to interpret the dual variables associated with the second constraint as Walrasian prices for the goods. Also, any bundle ${S}$ such that ${x_i^*(S) > 0}$ must be in agent ${i}$ ‘s demand correspondence.
Let ${\Delta}$ be the absolute value of the determinant of the optimal basis. We can write ${x_i^*(S) = \frac{z_i^*(S)}{\Delta}}$ for all ${i \in N}$ and ${S \subseteq G}$ where ${z_i^*(S)}$ is an integer. Now construct an enlarged economy as follows.

Scale up the supply of each ${g \in G}$ by a factor of ${\Delta}$ . Replace each agent ${i \in N}$ by ${N_i = \sum_{S \subseteq G}z_i^*(S)}$ clones. It should be clear now where this is going, but lets dot the i’s. To formulate the problem of finding an efficient allocation in this enlarged economy let ${y_{ij}(S) = 1}$ if bundle ${S}$ is allocated the ${j^{th}}$ clone of agent ${i}$ and zero otherwise. Let ${u_{ij}(S)}$ be the utility function of the ${j^{th}}$ clone of agent ${i}$ . Here is the corresponding integer program.

$\displaystyle \max \sum_{i \in N}\sum_{j \in N_i}\sum_{S \subseteq G}u_{ij}(S)y_{ij}(S)$

subject to
$\displaystyle \sum_{S \subseteq G}y_{ij}(S) \leq 1\,\, \forall i \in N, j \in N_i$

$\displaystyle \sum_{i \in N}\sum_{j \in N_i}\sum_{S \ni g} y_{ij}(S) \leq \Delta \forall g \in G$

$\displaystyle y_{ij}(S) \in \{0,1\}\,\, \forall i \in N, j \in N_i, S \subseteq G$

Its easy to see a feasible solution is to give for each ${i}$ and ${S}$ such that ${z_i^*(S) > 0}$ , the ${z_i^*(S)}$ clones in ${N_i}$ a bundle ${S}$ . The optimal dual variables from the relaxation of the first program complements this solution which verifies optimality. Thus, Walrasian prices that support the efficient allocation in the augmented economy exist.

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