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.