Exchange Economies

Solutions

Exercise 1

(a) The closed-form equilibrium price ratio is $

p_1 = \frac{\alpha_A \omega_{A2} + \alpha_B \omega_{B2}}{(1-\alpha_A)\omega_{A1} + (1-\alpha_B)\omega_{B1}} = \frac{0.5 \times 1 + 0.5 \times 4}{0.5 \times 4 + 0.5 \times 1} = \frac{2.5}{2.5} = 1.

(b) With

p_1 = 1

:

m_A = 1 \times 4 + 1 = 5

.

m_B = 1 \times 1 + 4 = 5

. Cobb-Douglas demands:

x_{i1}^* = \alpha_i m_i / p_1

and

x_{i2}^* = (1-\alpha_i) m_i

. So

x_{A1}^* = 0.5 \times 5 / 1 = 2.5

,

x_{A2}^* = 0.5 \times 5 = 2.5

. Similarly

x_{B1}^* = 2.5

,

x_{B2}^* = 2.5

. Market clearing check:

x_{A1}^* + x_{B1}^* = 5 = \Omega_1

and

x_{A2}^* + x_{B2}^* = 5 = \Omega_2

. Both markets clear. (c) At the endowment:

u_A(\omega_A) = 4^{0.5} \times 1^{0.5} = 2.0

and

u_B(\omega_B) = 1^{0.5} \times 4^{0.5} = 2.0

. At the equilibrium:

u_A(x_A^*) = 2.5^{0.5} \times 2.5^{0.5} = 2.5

and

u_B(x_B^*) = 2.5$. Both agents achieve utility 2.5 at the equilibrium versus 2.0 at the endowment. The gains from trade are symmetric and strictly positive.

Exercise 2

(a) Applying the closed-form formula: $

p_1 = \frac{0.7 \times 5 + 0.3 \times 5}{0.3 \times 5 + 0.7 \times 5} = \frac{3.5 + 1.5}{1.5 + 3.5} = \frac{5}{5} = 1.

The symmetric endowments give

p_1 = 1

regardless of the asymmetry in

\alpha

. This follows from the formula: numerator

= (\alpha_A + \alpha_B) \times 5

, denominator

= (2 - \alpha_A - \alpha_B) \times 5

; since

\alpha_A + \alpha_B = 1

, we get

p_1 = 1/1 = 1

. (b) With

p_1 = 1

:

m_A = 1 \times 5 + 5 = 10

.

m_B = 10

.

x_{A1}^* = 0.7 \times 10 / 1 = 7

,

x_{A2}^* = 0.3 \times 10 = 3

.

x_{B1}^* = 0.3 \times 10 / 1 = 3

,

x_{B2}^* = 0.7 \times 10 = 7

. Market clearing:

7 + 3 = 10 = \Omega_1

and

3 + 7 = 10 = \Omega_2

. (c) Agent A starts with 5 units of good 1 and ends with 7, so A is a net buyer of good 1 (and a net seller of good 2). Agent B does the reverse. This reflects the preference asymmetry: A places more weight on good 1 (

\alpha_A = 0.7

) and B places more weight on good 2 (

\alpha_B = 0.7$). Each agent trades away the good they value less for the good they value more, which is precisely what the Walrasian mechanism achieves.

Exercise 3

(a) For Cobb-Douglas

u = x_1^\alpha x_2^{1-\alpha}

: $

MRS = \frac{\partial u / \partial x_1}{\partial u / \partial x_2} = \frac{\alpha x_1^{\alpha-1} x_2^{1-\alpha}}{(1-\alpha) x_1^\alpha x_2^{-\alpha}} = \frac{\alpha}{1-\alpha} \cdot \frac{x_2}{x_1}.

So

MRS_A = \frac{\alpha_A}{1-\alpha_A} \cdot \frac{x_{A2}}{x_{A1}}

and

MRS_B = \frac{\alpha_B}{1-\alpha_B} \cdot \frac{x_{B2}}{x_{B1}}

. (b) Setting

MRS_A = MRS_B

and substituting

x_{B1} = \Omega_1 - x_{A1}

,

x_{B2} = \Omega_2 - x_{A2}

:

\frac{\alpha_A}{1-\alpha_A} \cdot \frac{x_{A2}}{x_{A1}} = \frac{\alpha_B}{1-\alpha_B} \cdot \frac{\Omega_2 - x_{A2}}{\Omega_1 - x_{A1}}.

Let

k_A = \alpha_A/(1-\alpha_A)

and

k_B = \alpha_B/(1-\alpha_B)

. Cross-multiplying:

k_A \, x_{A2} (\Omega_1 - x_{A1}) = k_B (\Omega_2 - x_{A2}) x_{A1}.

Expanding and collecting terms in

x_{A2}

:

x_{A2} \bigl[ k_A (\Omega_1 - x_{A1}) + k_B x_{A1} \bigr] = k_B \Omega_2 x_{A1}.

Reverting to

\alpha

notation,

k_A = \alpha_A/(1-\alpha_A)

and

k_B = \alpha_B/(1-\alpha_B)

, the solution simplifies to:

x_{A2}(x_{A1}) = \frac{\alpha_A(1-\alpha_B)\,\Omega_2\, x_{A1}}{(1-\alpha_A)\alpha_B (\Omega_1 - x_{A1}) + \alpha_A(1-\alpha_B)\,x_{A1}}.

(c) When

\alpha_A = \alpha_B = \alpha

, the contract curve simplifies to

x_{A2} = (\Omega_2 / \Omega_1) x_{A1}$, which is the main diagonal of the Edgeworth box. With identical preferences, any allocation on the diagonal is Pareto optimal — the agents agree on relative valuations at every point, so there are no further gains from reallocation.

Exercise 4

(a) With

\alpha_A = \alpha_B = 0.5

and

\Omega_1 = \Omega_2 = 10

, the contract curve is the main diagonal:

x_{A2} = (\Omega_2/\Omega_1) x_{A1} = x_{A1}

. At the target,

x_{A1} = x_{A2} = 8

, so the point satisfies

x_{A2} = x_{A1}

. Also

x_{B1} = 2 = x_{B2}

, consistent with

x_{B2} = x_{B1}

. The allocation is on the contract curve. (b) At any interior Pareto-optimal allocation, all agents face the same price ratio equal to their common MRS. For agent A at

(8,8)

: $

MRS_A = \frac{\alpha_A}{1-\alpha_A} \cdot \frac{x_{A2}}{x_{A1}} = \frac{0.5}{0.5} \cdot \frac{8}{8} = 1.

So the supporting price ratio is

p_1/p_2 = 1

. One can verify

MRS_B = (0.5/0.5)(2/2) = 1

as well. (c) By the second welfare theorem, any Pareto-optimal allocation can be supported as a Walrasian equilibrium after an appropriate redistribution of endowments. With

p_1 = p_2 = 1

, agent A's budget must equal

p_1 x_{A1}^* + p_2 x_{A2}^* = 8 + 8 = 16

. Set new endowments

\hat{\omega}A = (\hat{\omega}{A1}, \hat{\omega}_{A2})

satisfying

\hat{\omega}{A1} + \hat{\omega}{A2} = 16

and

\hat{\omega}{B1} + \hat{\omega}{B2} = 4

, with

\hat{\omega}{A1} + \hat{\omega}{B1} = 10

and

\hat{\omega}{A2} + \hat{\omega}{B2} = 10

. One natural choice:

\hat{\omega}_A = (8, 8)

and

\hat{\omega}_B = (2, 2)

. With these endowments the agents are already at the target allocation, so the equilibrium price

p_1 = 1

supports it with zero trade. More generally, any split satisfying the budget constraints at

p_1 = 1$ works; the planner has a one-dimensional family of redistributions available.