Suppose A and B live in an exchange economy with two goods x1 and x2 Answer

Suppose A and B
live in an exchange economy with two goods x1 and x2. A owns 50 of both goods,
and B own 250 of both goods.
A’s tastes are captured by the utility function
uA(x1,x2) = x2 + 50lnx1
and B’s are captured by the utility function
uB(x1,x2) = x2 + 100lnx1.
a. Let the competitive equilibrium prices be p1 and p2 respectively. Derive
both consumers’ demand functions for both goods. 

These
preferences are translog preferences.

Demand
functions are given using the equilibrium condition mRS = price ratio

MRS for A= 50/x1= p1/p2         

So
x1= 50p2/p1

x2=
( M- p1*x1)/p2= (M/p2 )- 50

 
b. Find the relationship between p1 and p2 in equilbirum. Does the relationship
depend on the endowments (assume interior solutions)? How much of each good
will each consumer choose?

General
Equilibrium requires that MRs of A = MRs of B = p1/p2

MRS=
MU of x1/ Mu of x2

MRS
of A=50/x1^A

MRs
of B = 100/x1^B

Also
x1A+x2= 300 so that

In
equilibrium 50/x1^A = 100/300- x1^A

15000
-50×1^A= 100x1A

so
x1^A= 15000/150= 100 in optimal state.

X1^B=
300-100=200

So
price ratio = 50/100=.5

Let
p2 be 1 for simplicity so that p1= 0.5

Then for A:

0 .5×1+x2= value of initial endowment = 50*.5
+50 =75

Put
x1=100 to get .5*100 +x2=75

X2=
25 this means x2^A= 25

By same logic for B

0.5×1+x2=
value of initial endowment = 250*.5 +250 =375

Put
x1=200 to get .5*200 +x2=375

X2=2
25 this means x2^B= 275

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