Pareto Optimality and Walrasian Equilibrium in a 3-Consumer Exchange Economy

Questions & Problems
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Consider a two-good exchange economy with three consumers. The aggregate endowment of each good is 1. The utility functions of the three consumers are as follows:

\begin{aligned} u_1(x_1,y_1) & = x_1^2+y_1^2 \\ u_2(x_2,y_2) & = x_2^2+y_2^2 \\u_3(x_3,y_3) & = x_3^{0.5}y_3^{0.5} \end{aligned}

Determine the set of Pareto-optimal allocations for this economy when consumers 1 and 2 are treated equally (in terms of utility). What is the Walrasian equilibrium (or equilibria) if consumers 1 and 2 each have an initial endowment of (0.25,0.25) and consumer 3’s is (0.5,0.5)?

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Total Aggregate Endowments

\bar{X} = 1, \qquad \bar{Y} = 1

Utility Functions

  • Consumer 1:

    u_1(x_1, y_1) = x_1^2 + y_1^2

  • Consumer 2 :

    u_2(x_2, y_2) = x_2^2 + y_2^2

  • Consumer 3 :

    u_3(x_3, y_3) = x_3^{0.5} y_3^{0.5}

First part -Pareto-optimal allocations
As it is given that for consumer 1 and consumer 2 utility treated equally
so , u1=u2
therefore, (x_1^2+y_1^2)=(x_2^2+y_2^2)

Let their common utility be:

u_1 = u_2 = t^2

A target utility of t^2 can be achieved efficiently by specializing the bundles—giving one consumer only Good 1 and the other consumer only Good 2:

(x_1, y_1) = (t, 0) \quad \text{and} \quad (x_2, y_2) = (0, t)

Now ,consumer 3 gets:

For Good X, the total supply is 1:

x_1 = t, \quad x_2 = 0
\therefore x_3 = 1 - t

Similarly, for Good Y, the total supply is 1:

y_1 = 0, \quad y_2 = t
\therefore y_3 = 1 - t

So, Consumer 3 receives the remaining allocation bundle:

\boxed{(x_3, y_3) = (1 - t, 1 - t)}

Consumer3’s utility

u_3 = \sqrt{x_3 y_3}

substituting:

u_3 = \sqrt{(1 - t)(1 - t)}

Therefore

\boxed{u_3 = 1 - t}

so pareto set is

\boxed{(x_1, y_1, x_2, y_2, x_3, y_3) = (t, 0, 0, t, 1-t, 1-t), \quad 0 \leq t \leq 1}

Second Part - walrasian equilibrium

Endowments of each consumer:

\omega_1 = (0.25, 0.25)
\omega_2 = (0.25, 0.25)
\omega_3 = (0.5, 0.5)

setting prices i.e.
px = p , py = 1

Now ,
Consumer 1 income : p+1/4
Comsumer 2 income : p+1/4
Consumer 3 income : p+1/2

consumer 3 demand:
x_3 = \frac{m_3}{2p}
y_3 = \frac{m_3}{2}
substituting m3
x_3 = \frac{p+1}{4p}
y_3 = \frac{p+1}{2}

consumer1’s demand:

\text{Buy only } X: \quad x_1 = \frac{m_1}{p} \implies u_X = \left( \frac{m_1}{p} \right)^2
\text{Buy only } Y: \quad y_1 = m_1 \implies u_Y = m_1^2

Therefore:

  • If p<1, buying X is better
  • If p>1, buying Y is better
  • If p=1, they are indifferent

so the only possible price is p=1
​

Consumer3’s income :

m_3 = 1

Substituting :

x_3 = y_3 = \frac{1}{2}

Therefore, :

\boxed{(x_3, y_3) = \left( \frac{1}{2}, \, \frac{1}{2} \right)}

and for consumer 1 and 2 :
m1=m2=1/2

and at (1,1) each maximizes so

and as the utility is convex so the optimum are at corners:

\boxed{\left( \frac{1}{2}, 0 \right) \quad \text{or} \quad \left( 0, \frac{1}{2} \right)}

so final equilibrium is

\boxed{\left(x_1, y_1, x_2, y_2, x_3, y_3\right) = \left( \frac{1}{2}, 0, \, 0, \frac{1}{2}, \, \frac{1}{2}, \frac{1}{2} \right)}