Disequilibrium Trade in a Large Market for an Indivisible Good

Proposition A1.

For every NTA (resp. NTB), PF and ER feasible trading procedure Υ in which pt≥pCE (resp. pt≤pCE) for all t=1,…,T, we have that

Proof:

Assume Υ≡(T,{pt}t=1T,{Bt}t=1T,{St}t=1T) is a NTA feasible trading procedure satisfying PF and ER in which pt≥pCE for all t=1,…,T. Let b_∈[0,1] be the consumer such that x−1(b_)=inf{x−1(b):b∈⋃t=1TBt} and let s¯∈[0,1] be the seller such that q−1(s¯)=sup{q−1(s):s∈⋃t=1TSt}. As x(⋅)andq(⋅) are continuous and strictly monotone, x−1(⋅) and q−1(⋅) are continuous. By ER, the sets ⋃t=1T⁡Btand ⋃t=1T⁡St are intervals in [0,1] and thus so the sets {x−1(b):b∈⋃t=1T⁡Bt} and {q−1(s):s∈⋃t=1T⁡St}. Therefore, both x−1(b_) and q−1(s¯) are well defined. As pt≥pCE for all t, pT<pCE cannot hold. Suppose then that pT>pCE. By ER, we also have that b_=s¯, and therefore, x−1(b_)≥pT>pCE>q−1(s¯). But then, there exist (b,s)such thatb∈[0,1]∖⋃τ=1T⁡Bτ,s∈[0,1]∖⋃τ=1T⁡Sτandx−1(b)≥q−1(s), which violates condition (iv) of feasibility. Hence pT=pCE.

On the other hand, we have that b_=s¯≤xCE. But if b_=s¯<xCE, then we would have again that x−1(b_)>q−1(s¯), and thus a new violation of condition (iv) of feasibility. Therefore, b_=s¯=xCE. As ⋃t=1T⁡Btand ⋃t=1T⁡St are both intervals (and markets are orderly), we have that ⋃t=1TBt=[0,]=[0,s¯]=⋃t=1TSt=[0,xCE].

For a NTB feasible trading procedure satisfying PF and ER in which pt≤pCE for all t=1,…,T, the proof is analogous and thus is omitted. ∎

The next result follows directly from Proposition A1 by simply adding IC to the set of properties that are satisfied by the trading procedure.

Corollary A1.

For every NTA (resp. NTB), PF, ER and IC feasible trading procedure Υ in which pt≥pCE (resp. pt≤pCE) for all t=1,…,T, we have that

and

Proof of Proposition 2.

For any NTA, PF and OPIC trading procedure we already know that vt>vt+1 and vt=pt+pt−pt+1r, for every t.

Now, for every t such that

let

and

with ε0=0.

If, for some t,

then t is the last period in which there is trade and we let

Therefore, we can write the following algorithm that allows to compute εt at every period t in which

given the values of ε−1,εt−2,…,ε1:

We first show that, for every t, for which

there exists a unique εt>0 satisfying the above nonlinear equation. To prove this, we proceed by induction. Indeed, notice first that x−1(1−ε)<q−1(1−ε).

Then, when ε1=0,

and, when ε1=1−ε,

and so, by continuity of both x−1(⋅)and q−1(⋅) and Bolzano’s Theorem, there exist some ε1 such that, 0<ε1<1−ε, and

Now consider some period t and the t corresponding valuesεt,εt−1,…,ε1>0 satisfying the algorithm such that

and, for every t,

Then, when εt+1=0, since εt>0,

and q−1(⋅) is strictly increasing, we have that

and so

When εt+1=1−ε−∑τ=1t⁡ετ,given that

And that ∑τ=1t⁡ετ<1−ε, we have that

Therefore, again by continuity of both x−1(⋅) and q−1(⋅) and Bolzano’s Theorem, there exist some εt+1 such that, 0<εt+1<1−ε−∑τ=1t⁡ετ, and

To prove that the εt satisfying the above algorithm for every t, for which x−1(1−ε)<q−1(1−ε−∑τ=1t⁡ετ), is unique, it suffices to notice that, for every t, and given εt−1,εt−2,…,ε1, the function

is strictly decreasing in εt.

To prove that there exist some t such that

let H(⋅)=x−1(⋅)+q−1(1−ε−⋅)r Since H is continuous and strictly decreasing, then it is invertible. Therefore, we can write our algorithm as

where at=∑τ=1t⁡ετ, for each t and g(⋅)=H−1(q−1(1−ε−⋅)1+rr).^[11] Now, notice that, since at>at−1, and g′(⋅)>0, a sufficient condition for a t to exist such that

is that the fixed point for g(⋅) be higher or equal to 1−ε−q(x−1(1−ε)).

This last condition is guaranteed by assuming that

i. e. Assumption 1.

Feasibility is trivially satisfied and so the proof is omitted.

To show that this trading procedure satisfies NTA, it suffices to notice that p1=q−1(1−ε)>pPCE, there is trade in the first period, and prices are always decreasing.

Finally, to show that NWR is also satisfied, it suffices to notice that both ⋃t=1TBt and ⋃t=1TSt are connected sets. Moreover, ⋃t=1TBt=⋃t=1TSt=[0,1−ε], and the proof follows. ∎