Question

2) For two agents, a and b, with the following utility functions over goods x and y (6)

ua=ua(xa,ya)=xaya

ub=ub(xb,yb)=xbyb

a) Determine the slope of the contract curve in the interior of an Edgeworth box that would show this two-person two-goods situation.

b) For initial endowments ωa=(12,7) and ωb=(9,10), what is the Walras allocation between the two agents a and b? (Remember that it is the relative price of the goods that matters in this consideration. Also remember that all the units of goods that exist here are will end up with one or the other agent; so, the overall 21 units of x and the 17 units of y will be fully allocated between the two. )

c) Calculate the utility level before and after trading.

3) For two agents, a and b, with the following utility functions over goods x and y (6)

ua=ua(xa,ya)=xa13ya23

ub=ub(xb,yb)=xb12yb12

a) Determine the slope of the contract curve in the interior of an Edgeworth box that would show this two-person two-goods situation.

b) For initial endowments ωa=(9,8) and ωb=(6,7), what is the Walras allocation between the two agents a and b?

c) Calculate the utility level before and after trading.

Answer 1

I have done question 2 for you. I hope it helps.

(a) The condition for the contract curve:

MRS_a = MRS_b = P_x/P_y

=> (Del u_a/Del x_a) / (Del u_a/Del y_a) = (Del u_b/Del x_b) / (Del u_b/Del y_b)

=> y_a/x_a = y_b/x_b = P_x/P_y ........(1)

=> y_a x_b = x_a y_b ..............(2)

Also, we know that the sum of the values of x and y consumed by both a and b will be equal to the supply of x and y respectively, i.e.

x_a + x_b = x .........(3)

y_a + y_b = y ..........(4)

Now, from (2),(3) and (4) :-

y_a (x-x_a) = x_a (y-y_a)

=> x y_a - y_a x_a = y x_a - y_a x_a

=> x y_a = y x_a

The above equation gives the Contract curve. Its slope is

y_a/x_a = y/x

From (1), this value is equal to P_x/P_y.

(b) u_a = x_a y_a u_b = x_b y_b

w_a = (12,7) w_b = (9,10)

m_a = 12P_x + 7P_y    m_b = 9P_x + 10P_y

Now, suppose P_x = 1. According to Walras law, if x market is in equilibrium then y market will be in equilibrium as well.

Demand for x by a + Demand for x by b = Supply for x {You can do this with market for y as well}

=> m_a/2P_x + m_b/2P_x = 12+9

=> (12P_x + 7P_y) /2P_x + (9P_x + 10P_y) /2P_x = 21

=> 6 + 7P_y/2 + 9/2 + 5P_y = 21 {Since P_x =1}

=> 17P_y/2 = 21/2

=> P_y = 21/17

Therefore, walrasian relative equilibrium price is: P_x/P_y= 17/21

Now, the walrasian allocation is:

(x_a,y_a) = (m_a/2P_x , m_a/2P_y) = ( (12P_x + 7P_y) /2P_x , (12P_x + 7P_y) /2P_y ) = (351/34 , 351/42) = (10.32 , 8.36)

(x_b,y_b) = (m_b/2P_x , m_b/2P_y) = ( (9P_x + 10P_y) /2P_x , (9P_x + 10P_y) /2P_y ) = (363/34 , 363/42) = (10.68 , 8.64)

(c) Utility before trading, u_a(x_a=12 , y_a=7) = 12*7 = 84

and u_b(x_b=9 , y_b=10) = 9*10 = 90

Utility after trading, u_a(x_a=351/34 , y_a=351/42) = (351/34)*(351/42) = 86.28

and u_b(x_b=363/34 , y_b=363/42) = (363/34)*(363/42) = 92.28

Therefore, trading has increased the utility for both a and b, thus the earlier endowment was an inefficient allocation.

NOTE: The demand for x in part (b) is calculated using lagrangean method as follows: