<p>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 $\omega$.</p>

<p>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.</p>

CC BY-SA 3.0
2016-04-07T09:46:04.497Z
2016-04-07T09:46:04.497Z
Giskard
<p>I am having some troubles in distinguishing, from a theoretical point of view, between contract curve and Pareto set. </p>

<p>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.</p>

<p>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)? </p>

<microeconomics><general-equilibrium>
Pareto set and contract curve
PostDocing
# Pareto set and contract curve

(Asked by: username_0 on 2016-04-07)

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)?

**Tags:** microeconomics, general-equilibrium

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## 1 answers

### Answer by username_1 on 2016-04-07. Score: 2 | Accepted answer

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.


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