In the The Edgeworth Box and Pareto Efficiency, where do the two people end up if they trade at market prices? Each acts like the consumer of Cobb-Douglas Demand, with one twist: their income is not a fixed number of dollars.
Key idea
A competitive equilibrium is a price ratio at which each person, choosing the best bundle she can afford with the value of her endowment, demands amounts that together use up exactly what exists. With $y$ as money ($p_y = 1$) and $p$ the price of $x$: income $m_i = p\,\omega_{ix} + \omega_{iy}$, and the market for $x$ clears. By Walras's law, when one market clears the other does too.
Why it works
Each person's spending equals the value of what she brings, so total spending equals the total value of all endowments. If the $x$ market clears, the value of $x$ bought equals the value of $x$ available, so the leftover spending on $y$ must equal the value of $y$ available: the $y$ market clears too. And only relative prices matter: doubling every price doubles every income and changes nothing. That is why we can set $p_y = 1$. In the box, the price line passes through the endowment with slope $-p$, and equilibrium is where both people's best points on that line coincide.
Worked example: Identical preferences
Both have $U = x^{1/2}y^{1/2}$. A owns (20, 10); B owns (10, 20).
- Incomes: $m_A = 20p + 10$ and $m_B = 10p + 20$.
- Demands for $x$: $x_A = \dfrac{m_A}{2p}$ and $x_B = \dfrac{m_B}{2p}$. Clearing: $\dfrac{30p + 30}{2p} = 30$, so $30p + 30 = 60p$ and $p = 1$.
- $m_A = 30$, so A consumes (15, 15); B consumes (15, 15).
Answer: $p = 1$; each consumes (15, 15).
Worked example: Different tastes
$U_A = x^{1/2}y^{1/2}$ with endowment (10, 0); $U_B = x^{1/4}y^{3/4}$ with endowment (0, 30).
- $m_A = 10p$ and $m_B = 30$.
- $x$ market: $\tfrac{1}{2}\cdot\dfrac{10p}{p} + \tfrac{1}{4}\cdot\dfrac{30}{p} = 10$, so $5 + \dfrac{7.5}{p} = 10$ and $p = 1.5$.
- A: income 15, buys $x_A = 5$ and $y_A = 7.5$. B: buys $x_B = 5$ and $y_B = 22.5$. Both markets clear.
Answer: $p = 1.5$.
Common mistake
Treating income as a fixed number of dollars. Here income is the value of the endowment at the price you are solving for, so it changes as $p$ changes. Writing $x_A = \tfrac{1}{2}m_A/p$ with $m_A$ held fixed gives the wrong clearing equation. Check: write each income as $p\,\omega_x + \omega_y$ before you set up the market-clearing condition.