micro DSE 2005 Q.3

[nidhi]

consider a competitive exchange economy with 2 agents( 1 and 20 and 2 goods (Xand Y)
endowments of agent 1 and2 are (100,100) and (50,0) respectively.  an allocation for agent i is denoted by (xi, yi) where xi is his allocation of X and yi is his allocation of Y. agent 1's objective is to choose(x1,y1) to max is U min (x1.y1) agent2's objective is to choose (x2,y2) to maximise his u= x2+y2

Q3. an example of a pair of competitive equilib prices(p1,p2) for this economy is
a) (1,0)
b) (0,1)
c) (1/3,2/3)
d) (2/3,1/3)

u have given the answer as c)(1/3, 2/3)
 option d) i can eliminate because there is excess supply of good x at these prices
please explain how do we solve for the prices.

[vishruti]

DSE 2005 1, 2, 3, 4

U1 = min{x1,y1}

So the demand function for agent 1 is as follows

X1 = y1 = m1 / px + py

U2 = x2 + y2

So the demand function for agent 2 is

X2= m2 / px if px<py
      = 0 if px> py
      = [0, m2/px] if px=py

Now from the endowments we can calculate the income of two agents

M1= 100 px + 100 py
M2= 50 px

So we take px=1 as numeraire price then

M1 = 100 + 100py
M2 = 50

This implies (by substituting above two equations in demand function)
X1 = y1 = (100 +100py) / 1 + py = 100

And x2 = 50 and y = 0 (from initial endowment)

This will be competitive equilibrium

Clearly none of the prices could be 0 in equilibrium because it will become a free good and the markets will not clear. Therefore we have positive prices such that px < py because x and y are substitutes and y2 = 0 .at px > py  , x2 = 0 but x1 = 100 therefore markets do not clear as there is excess supply , we eliminate this case.

Now answers are

 1 – d as shown above

2 – a

because in other 3 cases if we take some amount of x ( in option b and c) from agent 1 and give to agent 2 then utility of agent 2 can be increased without any change in agent 1’s utility , hence we can make agent 2 better off without making agent 1 worse off and in option d some units if taken from 1 and given to 2 can give higher utility to agent 2 without making agent 1 worse off

3 – c

a and b has either price 0 which cannot be the case because if py=0 then it will be a free good and there will be excess demand for y and the market will not clear, If px=0 then the income of second agent will become 0 and he will demand nothing of y and infinity of x(x is a free good now) at these prices so this cannot be the case and if px > py which implies x2 = 0 which cannot be the case. Therefore answer is c

4 – b

m1 = 100 px +100 py –50
m2 = 50 px + 50

at  px = ½ and py = ½

m1 = 50
m2 = 75

Substituting in demand functions we get

X1 = y1 = 50
And
X2 =100 y2 =50 (from endowment)

[Chinni18]

Lovely explanation Vishruti, even I was confused in this question. Thanks!!!