Exchange Economies

Exercises

Work through the prompts first, then compare against the solutions once you are ready.

Exercise 1

Walrasian Equilibrium Computation Two agents A and B have identical Cobb-Douglas preferences

u_i(x_1, x_2) = x_1^{0.5} x_2^{0.5}

. Endowments are

\omega_A = (4, 1)

and

\omega_B = (1, 4)

(a)
Find the Walrasian equilibrium price ratio $p_1/p_2$ , using the closed-form formula with $p_2 = 1$ as numeraire.
(b)
Given the equilibrium $p_1$ , compute each agent's wealth $m_i = p_1 \omega_{i1} + \omega_{i2}$ and derive the equilibrium allocation $(x_A^*, x_B^*)$ .
(c)
Compare each agent's utility at the equilibrium allocation with utility at the endowment. Are both agents better off?

Exercise 2

Asymmetric Preferences Agent A has

u_A = x_1^{0.7} x_2^{0.3}

and agent B has

u_B = x_1^{0.3} x_2^{0.7}

. Endowments are

\omega_A = (5, 5)

and

\omega_B = (5, 5)

, so total endowments are

\Omega_1 = \Omega_2 = 10

(a)
Find the Walrasian equilibrium price ratio $p_1/p_2$ .
(b)
Compute each agent's equilibrium allocation $(x_A^*, x_B^*)$ .
(c)
Which agent is a net buyer of good 1 and which is a net seller? Relate this to their preferences.

Exercise 3

Contract Curve Derivation Agents A and B have Cobb-Douglas preferences

u_A = x_1^{\alpha_A} x_2^{1-\alpha_A}

and

u_B = x_1^{\alpha_B} x_2^{1-\alpha_B}

. Total endowments are

\Omega_1

and

\Omega_2

. Derive the contract curve

x_{A2}(x_{A1})

from first principles.

(a)
Write down the MRS for each agent in terms of their own consumption bundle. Recall $MRS = -\partial u / \partial x_1 \div \partial u / \partial x_2$ .
(b)
Set $MRS_A = MRS_B$ and substitute the box constraints $x_{B1} = \Omega_1 - x_{A1}$ and $x_{B2} = \Omega_2 - x_{A2}$ . Solve for $x_{A2}$ as a function of $x_{A1}$ .
(c)
What is the contract curve when $\alpha_A = \alpha_B$ ? Describe it geometrically.

Exercise 4

Second Welfare Theorem Both agents have

\alpha_A = \alpha_B = 0.5

and total endowments

\Omega_1 = \Omega_2 = 10

. A social planner wants to implement the allocation

x_A = (8, 8)

x_B = (2, 2)

(a)
Verify that the target allocation $(x_A, x_B) = ((8,8),(2,2))$ lies on the contract curve.
(b)
Find the equilibrium price ratio $p_1/p_2$ consistent with this allocation being a Walrasian equilibrium (from the MRS condition at the allocation).
(c)
Describe a lump-sum redistribution of initial endowments such that the Walrasian equilibrium from those new endowments implements $(x_A, x_B) = ((8,8),(2,2))$ .