Until now one consumer faced fixed prices. General equilibrium asks how a whole economy's goods end up divided. The simplest case has two people and two goods, and a picture that puts both people's indifference maps (often from the Cobb-Douglas family) in one box.
Key idea
In an Edgeworth box with $X$ units of $x$ and $Y$ units of $y$, every point is an allocation: A's bundle measured from the bottom-left corner, B's from the top-right. An allocation is Pareto efficient if no change can make one person better off without making the other worse off. At an interior efficient allocation the two indifference curves are tangent: $$MRS_A = MRS_B.$$ The set of efficient allocations is the contract curve.
Why it works
Suppose $MRS_A = 3$ and $MRS_B = 1/3$: A would give up 3 units of $y$ for one more $x$, while B would give up a unit of $x$ for just $1/3$ of a $y$. Let B give A one unit of $x$ in exchange for one unit of $y$: A pays less than her limit of 3 and B gets more than his limit of $1/3$. Both gain. In the picture, the two indifference curves through the allocation form a lens; every allocation inside it is better for both. Only where the curves touch, and the lens disappears, is there no such trade left.
Worked example: An inefficient allocation
$X = Y = 20$; both have $U = xy$. A has (5, 15), so B has (15, 5). Is this efficient?
- $MRS_A = y_A/x_A = 15/5 = 3$; $MRS_B = 5/15 = 1/3$.
- They differ, so a trade can help both: B gives A some $x$ for $y$ at any rate between $1/3$ and 3.
- For instance A gives 2 of $y$ for 2 of $x$: A moves to (7, 13), utility $91 > 75$; B moves to (13, 7), utility $91 > 75$.
Answer: Not efficient.
Worked example: Finding a point on the contract curve
$X = Y = 30$. $U_A = x^{1/2}y^{1/2}$ and $U_B = x^{1/3}y^{2/3}$. If A has 10 units of $x$ at an efficient allocation, how much $y$ does A have?
- $MRS_A = \dfrac{y_A}{x_A}$ and $MRS_B = \dfrac{1}{2}\cdot\dfrac{y_B}{x_B}$.
- With $x_A = 10$, $x_B = 20$, $y_B = 30 - y_A$: $\dfrac{y_A}{10} = \dfrac{30 - y_A}{40}$.
- $40y_A = 300 - 10y_A$, so $y_A = 6$. Check: $MRS_A = 0.6$ and $MRS_B = \tfrac{1}{2}\cdot\tfrac{24}{20} = 0.6$.
Answer: $y_A = 6$.
Common mistake
Thinking Pareto efficient means fair. The corner where A has everything is efficient: any change gives B something only by taking it from A. Efficiency rules out waste; it says nothing about how the total is shared. Check: to judge fairness you need a separate standard (equality, need, desert); the contract curve contains both very equal and very unequal allocations.