1. Members of a primitive tribe may own bundles of various goods, which anthropologists have numbered {1,….,m}. The tribe has various ritual exchange activities, numbered {1,….,n}. In each activity j, there is a “host” and a “guest,” and the host gives the guest some net quantity θij of each good i (where a negative θij denotes the guest giving -θij units of i to the host). Any tribesman may do each activity any number of times, as guest or host. Prove a theorem of the following form: “Given any such matrix of parameters θij , exactly one of the following two conditions is true: (1) There is a way to use some combination of these exchange activities to increase one’s holdings of every good by at least one unit. (2) …”
[You may assume that people can also do a activity j at a fractional level xj , which would then yield a net transfer θijxj of each good i. But this assumption is not actually necessary.]If you cannot do the proof here, at least try to formulate a conjecture as to what condition (2)might be.