Starr’s ’69 paper considered Walrasian equilibria in exchange economies with non-convex preferences i.e., upper contour sets of utility functions are non-convex. Suppose ${n}$ agents and ${m}$ goods with ${n \geq m}$. Starr identified a price vector ${p^*}$ and a feasible allocation with the property that at most ${m}$ agents did not receiving a utility maximizing bundle at the price vector ${p^*}$.

A poetic interlude. Arrow and Hahn’s book has a chapter that describes Starr’s work and closes with a couple of lines of Milton:

A gulf profound as that Serbonian Bog
Betwixt Damiata and Mount Casius old,
Where Armies whole have sunk.

Milton uses the word concave a couple of times in Paradise Lost to refer to the vault of heaven. Indeed the OED lists this as one of the poetic uses of concavity.

Now, back to brass tacks. Suppose ${u_i}$ is agent ${i}$‘s utility function. Replace the upper contour sets associated with ${u_i}$ for each ${i}$ by its convex hull. Let ${u^*_i}$ be the concave utility function associated with the convex hulls. Let ${p^*}$ be the Walrasian equilibrium prices wrt ${\{u^*_i\}_{i=1}^n}$. Let ${x^*_i}$ be the allocation to agent ${i}$ in the associated Walrasian equilibrium.

For each agent ${i}$ let

$\displaystyle S^i = \arg \max \{u_i(x): p^* \cdot x \leq p^*\cdot e^i\}$

where ${e^i}$ is agent ${i}$‘s endowment. Denote by ${w}$ the vector of total endowments and let ${S^{n+1} = \{-w\}}$.

Let ${z^* = \sum_{i=1}^nx^*_i - w = 0}$ be the excess demand with respect to ${p^*}$ and ${\{u^*_i\}_{i=1}^n}$. Notice that ${z^*}$ is in the convex hull of the Minkowski sum of ${\{S^1, \ldots, S^n, S^{n+1}\}}$. By the Shapley-Folkman-Starr lemma we can find ${x_i \in conv(S^i)}$ for ${i = 1, \ldots, n}$, such that ${|\{i: x_i \in S^i\}| \geq n - m}$ and ${0 = z^* = \sum_{i=1}^nx_i - w}$.

When one recalls, that Walrasian equilibria can also be determined by maximizing a suitable weighted (the Negishi weights) sum of utilities over the set of feasible allocations, Starr’s result can be interpreted as a statement about approximating an optimization problem. I believe this was first articulated by Aubin and Elkeland (see their ’76 paper in Math of OR). As an illustration, consider the following problem :

$\displaystyle \max \sum_{j=1}^nf_j(y_j)$

subject to

$\displaystyle Ay = b$

$\displaystyle y \geq 0$

Call this problem ${P}$. Here ${A}$ is an ${m \times n}$ matrix with ${n > m}$.

For each ${j}$ let ${f^*_j(\cdot)}$ be the smallest concave function such that ${f^*_j(t) \geq f_j(t)}$ for all ${t \geq 0}$ (probably quasi-concave will do). Instead of solving problem ${P}$, solve problem ${P^*}$ instead:

$\displaystyle \max \sum_{j=1}^nf^*_j(y_j)$

subject to

$\displaystyle Ay = b$

$\displaystyle y \geq 0$

The obvious question to be answered is how good an approximation is the solution to ${P^*}$ to problem ${P}$. To answer it, let ${e_j = \sup_t [f_j^*(t) - f_j(t)]}$ (where I leave you, the reader, to fill in the blanks about the appropriate domain). Each ${e_j}$ measures how close ${f_j^*}$ is to ${f_j}$. Sort the ${e_j}$‘s in decreasing orders. If ${y^*}$ is an optimal solution to ${P^*}$, then following the idea in Starr’s ’69 paper we get:

$\displaystyle \sum_{j=1}^nf_j(y^*_j) \geq \sum_{j=1}^nf^*_j(y^*_j)- \sum_{j=1}^me_j$