# Macroeconomics Problem Set: Pareto Efficiency, Walrasian equilibrium

## Problem 1:

1. Pareto Efficiency
1. Define and distinguish the following concepts in one or two sentences each: (i) Pareto optimality, (ii) Pareto improvement.

Pareto optimality refers to a situation where no individual can be made better off without making at least one other individual worse off. A Pareto improvement is a change in allocation that makes at least one individual better off without making any other individual worse off.

• Suppose that a relevant energy good is traded in a market at publicly known prices by suppliers and consumers that engage in price-taking behavior. When the market is complete, and equilibrium is achieved, is it still possible to attain Pareto improvements in this market through wealth redistribution? Explain briefly.

In a market with price-taking behavior, a Pareto efficient allocation is achieved at equilibrium prices, and there are no further [blur] Pareto improvements that can be made through trade. However, it is still possible to achieve Pareto improvements through wealth redistribution. By redistributing wealth from the richer to the poorer agents, it is possible to improve the welfare of the poorer agents without harming the welfare of the richer agents. However, the extent to which this is possible depends on the initial distribution of wealth and the preferences of [/blur]

## Problem 2:

Note that for the following question we’re looking for answers that pertain to the economic theories pertaining to efficiency and welfare within the general equilibrium framework so be sure to show a strong comprehension of these theories.

Bobby and Candace were stranded in a mountain hut and due to Candace having injured her knee, Bobby had collected and hoarded most of the food in the hut for himself.

Given the amounts each of them was able to initially obtain, it was in both of their interests to trade. After they finished trading, Bobby still had a majority of the food.

1. Assuming Bobby and Candace had been able to trade freely, would you consider this outcome efficient? Would you consider this outcome equitable? Explain why, in economics, the two may not be the same.

In the question, Candace’s boyfriend comes along and wishes to re-allocate the food to initialize another round of trading.

If Bobby and Candace were able to trade freely, the outcome would be efficient if it is a Pareto efficient allocation, i.e., [blur] there is no other allocation that would make at least one person better off without making the other person worse off. However, this outcome may not be equitable since Bobby had initially hoarded most of the food, which implies that he had more bargaining power in the trade. Equity refers to the fairness of the distribution of resources or outcomes, while efficiency refers to the optimal use of resources to maximize total welfare. In this case, it is possible for the outcome to be efficient but not equitable.[/blur]

• Bobby and Candace are already at equilibrium when Candace’s boyfriend shows up. He’s therefore concerned that by changing their initial endowments and allowing them to trade again they may not end up at an efficient outcome. Is this a valid concern? Explain why or why not.

Candace’s boyfriend’s concern is not valid if the initial endowment is reallocated in a way that preserves the efficiency of the trade. [blur] In a general equilibrium framework, the efficiency of the trade depends on the prices at which the goods are traded, which in turn depend on the initial endowments. If the initial endowments are changed, the prices will also change, but the efficiency of the trade can still be maintained if the new prices reflect the relative scarcity of the goods. Therefore, it is possible to change the initial endowments and still achieve an efficient outcome[/blur]

• Candace’s boyfriend prefers that the two friends end up with half the food each. Is it possible for him to achieve this? Justify your answer. If your answer is yes, briefly explain how he would go about this.

It is possible for Candace’s boyfriend to achieve an allocation where each person gets half of the food, but it [blur] may not be Pareto efficient. To achieve this allocation, Candace’s boyfriend could redistribute the food equally between Bobby and Candace and allow them to trade freely again. However, this allocation may not be Pareto efficient since it may not be possible to redistribute the food in a way that makes both Bobby and Candace better off without making the other person worse off. If the initial allocation was such that one person had a comparative advantage in producing one type of food, then the Pareto efficient allocation may not be an equal split of the food. Therefore, achieving an equal split of the food may not be the most efficient outcome.[/blur]

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## Problem 3: Boba and Candy

Suppose there are two agents, Rae and Anya, who have preferences over boba (b) and candy (c):

????R(b, c) = log(b) + 2log(c)

????A(b, c) = log(b) + 3c

Suppose that Rae’s endowment, ????R(b, c), is 2 units of boba and 6 candy. Anya’s endowment,

????A(b, c), is 8 units of boba and 2 candy.

• Derive the individual excess demand functions for each person

To derive the individual excess demand functions, we need to find the optimal bundle of boba and candy that maximizes each agent’s utility subject to their budget constraint.
[blur]
For Rae:
max log(b) + 2log(c)
s.t. pb + pc = peb + pec
where pb is the price of boba, pc is the price of candy, peb is Rae’s endowment of boba, and pec is Rae’s endowment of candy.
Using the Lagrangian method, we have:
L(b, c, λ) = log(b) + 2log(c) + λ(peb + pec – pb b – pc c)
Taking first-order conditions with respect to b, c, and λ, we get:
1/b = λpb (1)
2/c = λpc (2)
pb b + pc c = peb pb + pec pc (3)
Dividing (1) by (2), we get:
b/c = 1/2λ
Substituting into (3), we have:
pb b + 2pb b = peb pb + pec pc
3pb b = 2peb pb + 2pec pc
b = 2/3(peb pb + pec pc)/pb
Similarly, we can find Rae’s demand for candy:
c = 2/3(peb pb + pec pc)/2pc = 1/3(peb pb + pec pc)/pc
Rae’s excess demand for boba and candy are then:
xRb = b – peb = 2/3(peb/pb + pec/pc) – 2
xRc = c – pec = 1/3(peb/pb + pec/pc) – 6
For Anya:
max log(b) + 3c
s.t. pb b + pc c = peb pb + pec pc
Using the Lagrangian method, we have:
L(b, c, λ) = log(b) + 3c + λ(peb + pec – pb b – pc c)
Taking first-order conditions with respect to b, c, and λ, we get:
1/b = λpb (1)
3 = λpc (2)
pb b + pc c = peb pb + pec pc (3)
Dividing (1) by (2), we get:
b/c = 1/3λ
Substituting into (3), we have:
pb b + 3pb b = peb pb + pec pc
4pb b = peb pb + 3pec pc
b = 1/4(peb pb + 3pec pc)/pb
Similarly, we can find Anya’s demand for candy:
c = 1/4(peb pb + 3pec pc)/pc
Anya’s excess demand for boba and candy are then:
xA b = b – peb = 1/4(peb/pb + 3pec/pc) – 8
xA c = c – pec = 1/4(peb/pb + 3pec/pc) – 2
[/blur]

• What is the aggregate excess demand function?

The aggregate excess demand function is [blur]simply the sum of the individual excess demand functions:
x b = xR b + xA b = 1/4(peb/pb + 3pec/pc) – 6
x c = xR c + xA c = 1/3(peb/pb + pec/pc) – 8 [/blur]

• Compute a Walrasian equilibrium price and allocation

A Walrasian equilibrium occurs when [blur] the aggregate excess demand for each good is zero, that is, x b = 0 and x c = 0
Substituting the expressions for x b and x c, we get:
1/4(peb/pb + 3pec/pc) – 6 = 0
1/3(peb/pb + pec/pc) – 8 = 0
Solving for pb and pc, we have:
pb = 4(peb/3pec)^(-1)
pc = 3(peb/pec)^(-1)
Substituting these prices into the expressions for Rae’s and Anya’s demands for boba and candy, we get:
Rae’s demand: b = 2, c = 6
Anya’s demand: b = 8, c = 2
Thus, the Walrasian equilibrium allocation is (b, c) = (10, 8) with prices (pb, pc) = (2, 3).
[/blur]

• Verify that this equilibrium allocation is Pareto efficient

To verify that the equilibrium allocation is Pareto efficient, [blur]we need to check whether there exists no other allocation that would make at least one agent better off without making the other agent worse off.
Suppose we allocate 9 units of boba and 7 units of candy to Rae, and 1 unit of boba and 3 units of candy to Anya. Then, Rae’s utility is:
uR(9, 7) = log(9) + 2log(7) ≈ 4.73
Anya’s utility is:
uA(1, 3) = log(1) + 3(3) = 7
Thus, both agents are strictly better off in this allocation than in the Walrasian equilibrium allocation. Therefore, the Walrasian equilibrium allocation is not Pareto efficient. [/blur]

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