Question

Let’s assume that there are two goods x1= cat food, x2 = money. Consider a retirement home with two inmates, Mrs M and a grumpy old man, Mr Q. Mrs. M has the following endowment: W1=0, W2=30 , while Mr.Q has: W1=30, W2=0 . They have the following utility functions: U(x1,x2) = x1x2

(a) Illustrate the available resources and the initial allocation (=endowment) in an Edgeworth box.

(b) Set the price of money equal to one, and define p as the price of one unit of cat food. Derive Mrs M's and Mr Q's excess-demand functions for cat food. Set these two equal to each other and solve for the equilibrium price.

Answer 1

A)

Endownment of M are measured from the origin Om and endownment of W are measured from origin Ow. The size of the each axis is given by the total endowment.

The initial allocation is at point E, where M has no cat food and 30 money and W has no money and 30 cat food.

B)

U(x1,x2) = x1x2

Let the price of good x1 be p and price of good x2 be 1

Income of M = x1*p + x2 = 30

Income of W = x1*p + x2 = 30p

To get the demand function for each person we need to find the profit maximisation solution.

For M

Max U = x1x2 subject to x1*p + x2* = 30

Setting up Lagrange we get

L = x1x2 + (30 - x1*p - x2*)

First Order conditions are:

From first and second equations we get,

x2/x1 = p or x2 = x1*p

Substituting the value of x2 in third equation we get,

x1p + x1p = 30

2x1p = 30

x1 = 15/p

Now, x2 = (x1*p)

So, x2 = 15

For W

Max U = x1x2 subject to x1*p + x2* = 30p

Setting up Lagrange we get

L = x1x2 + (30p - x1*p - x2*)

First Order conditions are:

From first and second equations we get

x2/x1 = p or x2 = x1*p

Substituting the value of x2 in third equation we get,

x1p + x1p = 30p

2x1p = 30p

x1 = 15

x2 = x1*p

x2 = 15p

M's excess demand = Demand for x1 - supply = 15/p - 0

W's excess demand = Demand for x1 - supply = 15 - 30 = -15

At equilibrium M's excess demand​ = W's excess demand​

15/p = 15

p = 1

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