Pareto set and contract curve

Question

I am having some troubles in distinguishing, from a theoretical point of view, between contract curve and Pareto set.

I have looked around books and internet, and I have found that contract curve should be a subset of Pareto set, i.e. the locus of Pareto efficient allocations that can occur as a result of mutually beneficial trading between agents.

Am I right or there is something I did not understand properly? Someone can give me a formal definition of the two objects (if there is some differences)?

Answer 1

Let $N$ denote the set of consumers. Given a set of possible allocations $A$, an allocation $a$ is Pareto-optimal if no allocation $a'$ exists for which $$ \forall i \in N: a \preceq_i a' $$ and $$ \exists j \in N: a \prec_j a'. $$ So Pareto-optimality exists without any notion of property rights, you don't need to know the initial endowment is, who has what in the beginning. Given an initial endowment $\omega$ an allocation $a$ is individually rational for consumer $i$ if $$ \omega \preceq_i a. $$ An allocation is in the contract curve if and only if it is individually rational for all consumers and is also Pareto-optimal. If the allocation $a$ is not individually rational for a given consumer $i$, he would not sign the contract which would take him there from $\omega$.

Thus the contract curve is indeed a subset of all Pareto-optimal allocations. Unless the initial endowments are given, it is not possible to determine the contract curve.