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have only one limit price. Therefore, the partitioning of E should work in exactly the same way. 

<img src='content_image/300013.jpg'>

There is no outstanding order or limit price associated with the fifth Arrow-Debreu security. Thus, we can define only one subset of the set E : E 1 $_{5}$. 

<img src='content_image/300012.jpg'>

We introduce new notation as (17). The idea is as follows. Having n$_{1}$ distinct limit prices associated with the first security yields 2 n$_{1}$ + 1 distinct subsets of E . Choose one subset out of these 2 n$_{1}$ + 1 subsets. Similarly, choose one of 2 n$_{2}$ + 1 subsets of E that we generate from the limit prices associated with the second security. Repeat this process for the remaining Arrow-Debreu securities. Once we have one subset for each type of Arrow-Debreu security, we can obtain the intersection of those K subsets, which is shown in (17). There are a total of Π K k $_{=1}$(2 n$_{k}$ + 1) ways to choose a combination of subsets. 

<img src='content_image/300011.jpg'>

Now consider the optimization problem (9). Imagine replacing the constraint N ∑ i =1 ξ$_{i}$ = 1 , ξ ≥ 0 with a more restrictive one (17). The part that causes problem (9) to be non-convex is (18). However, once the feasible set of the state price vector ξ is restricted to a smaller set E ( ℓ$_{1}$, ℓ$_{2}$, ..., ℓ$_{K}$ ), (18) can be replaced with x$_{j}$ = 0 or x$_{j}$ ∈ [0 , Q$_{j}$ ] or x$_{j}$ = Q$_{j}$ for ∀ j . 

<img src='content_image/300009.jpg'>

