General Equilibrium

[bhavookb21]

2 agents, 2 commodities. Utility function is min(x,y) for both the agents.

initial endowment for 1st agent (0,5)  and for 2nd agent (10,0).

1. Is the allocation (3,3) and (7,2) competitive equilibrium and pareto efficient?

2. Is the allocation (10,5) and (0,0) competitive equilibrium and pareto efficient?

3. Assuming the sum of prices is 1, then the price vector is (0,1) , (1,0) , (1/2,1/2) or (1/3,2/3)

[Dr. Strange]

Demand of 1 for x= 5py/px+py
Demand of 1 for y= 5py/px+py
Demand of 2 for x= 10px/px+py
Demand of 2 for y= 10px/px+py

Total demand for x= (5py+10px)/px+py    Total supply=10
Total demand for y= (5py+10px)/px+py     Total supply=5
This gives px=0
Hence (10,5) for 1 and (0,0) for 2 is competitive as well as pareto efficient
and px=0 py=1 can be price vectors

[bhavookb21]

what is the reason for getting px=0?

On Thu, May 5, 2016 at 12:09 PM, Dr. Strange [via Discussion forum] <[hidden email]> wrote:

Demand of 1 for x= 5py/px+py
Demand of 1 for y= 5py/px+py
Demand of 2 for x= 10px/px+py
Demand of 2 for y= 10px/px+py

Total demand for x= (5py+10px)/px+py    Total supply=10
Total demand for y= (5py+10px)/px+py     Total supply=5
This gives px=0
Hence 10,5 is competitive as well as pareto efficient
and px=0 py=1 can be price vectors

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[duck]

Hi bhavook :)

For Competitive equilibrium, the following two conditions should be satisfied:
1) Consumers are maximizing their utility ie. finding the demand.
2) Market clearing for all goods i.e demand = supply for both the goods.  

In your question:
Both the consumers have "min" utility function so they will demand equal amount of good x and good y in equilibrium. But the quantity for good x (10 units) is more than the quantity of good y (5 units) available in the economy. Therefore, to clear the market, price of good which is available in abundance (here, it is good x ) should be zero.