ECOS2001 Chap.5 Exchange Economy & Welfare
Exchange Economy & Welfare
This chapter moves from one consumer to two, trading an endowment in an Edgeworth box. An allocation is Pareto efficient if no one can be made better off without making another worse off; for well-behaved preferences that means the two consumers' marginal rates of substitution are equal, MRSᴬ = MRSᴮ, and the set of all such points is the contract curve.
A Walrasian (competitive) equilibrium is a common price ratio at which both consumers optimise and markets clear (excess demand is zero), so MRSᴬ = MRSᴮ = p₁/p₂. The two welfare theorems link the two ideas: every competitive equilibrium is Pareto efficient (First), and any Pareto-efficient allocation can be supported as a competitive equilibrium after suitable endowment transfers (Second).
This is the last topic on the midterm.
What this chapter covers
- 01
The Edgeworth box, feasibility and allocations
- 02
Pareto efficiency and the condition MRSᴬ = MRSᴮ
- 03
The contract curve as the locus of Pareto-efficient allocations
- 04
Walrasian equilibrium: common price ratio, MRSᴬ = MRSᴮ = p₁/p₂, markets clear
- 05
First Welfare Theorem: competitive equilibrium ⇒ Pareto efficient
- 06
Second Welfare Theorem: any efficient allocation is decentralisable with transfers
Edgeworth box: contract curve & competitive equilibrium
- 2 marksB has perfect complements, so B always chooses x₁^B = x₂^B. Feasibility then forces A onto the 45° line too: x₁^A = x₂^A. The contract curve is this diagonal.
- 2 marksOn the diagonal, A's MRS = x₂^A/x₁^A = 1, so the equilibrium price ratio is p₁/p₂ = 1, i.e. p₁ = p₂ (set both = 1).
- 1 markA's income from the endowment at these prices: m^A = p₁·24 + p₂·6 = 24 + 6 = 30.
- 2 marksA is Cobb-Douglas with equal exponents, so spends half of income on each good: x₁^A = x₂^A = (1/2)·30/1 = 15.
- 1 markBy feasibility B receives the rest: x₁^B = 30 − 15 = 15 and x₂^B = 30 − 15 = 15.
Key terms
- Edgeworth box
- A diagram whose dimensions are the total amounts of two goods; each point is an allocation between two consumers measured from opposite corners.
- Contract curve
- The set of all Pareto-efficient allocations in the box, where MRSᴬ = MRSᴮ so no mutually beneficial trade remains.
- Walrasian (competitive) equilibrium
- A price ratio at which both consumers optimise and total demand equals the total endowment (markets clear), implying MRSᴬ = MRSᴮ = p₁/p₂.
- First Welfare Theorem
- Any competitive equilibrium allocation is Pareto efficient — the formal sense in which competitive markets do not waste gains from trade.
Exchange Economy & Welfare FAQ
What is the difference between Pareto efficiency and a competitive equilibrium?
Pareto efficiency is a property of an allocation (no costless improvement is possible), defined by MRSᴬ = MRSᴮ. A competitive equilibrium is a market outcome at specific prices where both optimise and markets clear. The First Welfare Theorem says every competitive equilibrium is one of the Pareto-efficient allocations, but efficiency alone says nothing about which one or about fairness.
Why does this topic appear on the midterm, not the final?
The exchange economy and welfare theorems are Topic 5, the last consumer-side topic, and the mid-semester exam covers Topics 1–5. The final exam starts at Topic 6 (uncertainty) and contains no Topic 1–5 questions, so revise the Edgeworth box for the midterm only.
Exam move
Practise finding the contract curve from each consumer's MRS, then reading the equilibrium price ratio off it, before computing incomes from the endowment. Note that a perfect-complements consumer collapses the problem onto a ray, which is a frequent exam shortcut.
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