Microeconomics, echange/trade question

[alias]

Homework Statement

There are 2 consumers, A and B with the following utility functions and endowments:
uA(xA1, xA2) = a ln xA1 + (1-a) ln xA2 , endowment for A is wA = (0, 1)

uB(xB1,xB2) = min(xB1,xB2) endowment for B is wB = (1,0)

Find the market clearing prices and the equilibrium allocation

Homework Equations

From the Solutions Manual:
x1A = ay=p1 = ap2=p1; x1B = x2B
so from budget constraint, (p1 +p2)x1B = p1,
so x1B =p1=(p1+p2). Choose p1 = 1 an numeraire and solve
ap2 +1/(1 + p2) = 1.

The Attempt at a Solution

My textbook is not helpful and I don't know how to start the question much less get to the answer in the solutions manual. Can anyone give me some help? I'm very lost.
Thanks

 

[nate808]

for reaching out for help on this problem. Let's break it down step by step to understand how to solve it.

First, let's define some variables to make the problem easier to understand:

- xA1: the amount of good 1 consumed by consumer A
- xA2: the amount of good 2 consumed by consumer A
- xB1: the amount of good 1 consumed by consumer B
- xB2: the amount of good 2 consumed by consumer B
- a: the weight given to good 1 in consumer A's utility function (with 1-a being the weight given to good 2)
- wA: consumer A's endowment (in this case, (0,1) meaning they have no good 1 and 1 unit of good 2)
- wB: consumer B's endowment (in this case, (1,0) meaning they have 1 unit of good 1 and no good 2)
- p1: the price of good 1
- p2: the price of good 2

Now, let's use the budget constraint and utility functions to solve for the equilibrium allocation and market clearing prices.

1. Budget Constraint
The budget constraint for consumer A is given by:
p1xA1 + p2xA2 = p1(0) + p2(1) = p2
This is because consumer A has no good 1 in their endowment, so they must spend all of their budget on good 2. Similarly, the budget constraint for consumer B is given by:
p1xB1 + p2xB2 = p1(1) + p2(0) = p1
This is because consumer B has no good 2 in their endowment, so they must spend all of their budget on good 1.

2. Utility Maximization
To find the equilibrium allocation, we need to maximize the utility functions of both consumers subject to their budget constraints. Since we know that p1 = 1, we can rewrite the utility functions as:
uA(xA1, xA2) = a ln xA1 + (1-a) ln xA2
uB(xB1,xB2) = min(xB1,xB2)
We can use the Lagrangian method to solve for the optimal values of xA1, xA2, xB1 and xB2