Umm i'm getting the set of pareto efficient outcomes as all allocations (x11,x12)=(12,m) and (x21,x22)=(0,n) where m>=0,n>=0, m+n=8. And the competitive equilibrium is p1/p2=1. Demands are (12,0) and (0,8) for 1 and 2 respectively.
Well done Vasudha. You correctly found the set of efficient allocations. But the competitive equilibrium in this problem does not exist. At price ratio 1, consumer 1 will demand (0, 12) and not (12, 0).
Sir for pareto efficient outcomes, I am only getting (12,2) for consumer 1 and (0,6) for consumer 2.
I went according to consumer 1's preferences - either the total consumption has to increase from (10+2 = 12) to 12+2 = 14 or the total consumption can stay constant at 12 units but > or = of good 2 needs to be present in the bundle but definitely not < 2.
So his PO outcome must be (12,2) and thus for the other consumer its (0,6). How did m+n=8 situation arise? Maybe I am interpreting the question wrong?
And for competitive equilibrium, yes indeed it doesn't exist because consumer 1's demand would be (0,12)...
hey aditi u don't hv to take the endowment as the starting point. all allocations from which nobody can be made better off w/o making anyone worse r PO. for eg (12,6) and (0,2) is PO bcoz u can't make either of them better off w/o making the other one worse off starting from this point..
When the price ratio = 1 consumer 2's income becomes m2 = 2p1+6p2
so demand for good 2 = m2/p2 = 2p1+6p2 /p2 = 2p1/p2 + 6 = 8 when p1/p2 = 1
Also, consumer 1 - income = (10p1+2p2)
he needs to have minimum 12 in his consumption bundle (total of 1+2) , so if we give him 12 of good 1 and 0 of good 2, consumer 2 gets 8 of good 2 , isn;t that a competitive equilibrium?
(10p1+2p2) / p1 = 10 + 2 p2/p1 = 12 and both markets clear... what's wrong with my approach?
Hello MR. For finding the PO allocations we can see immediately that the second person must be getting 0 of the first good. And the first good can be divided any way between the two people. Try to see that starting from any such allocation (and no other allocation), nobody's utility can be increased without making the other person's utility decrease.
As for the competitive equilibrium, we need two things-that agents observe prices and choose the utility maximizing bundles given their incomes at those prices, and that the market clears. Assume price of good 2 is 1 and price of first good is p. See that agent 2's income is 2p+6, and his demand is (0,2p+6). Now you know that for markets to clear, the first agent must demand the entire social endowment of good 1, i.e 12 units. Also, his income is 10p+2. Given his preferences, he would not demand any units of good 1 unless p=<1. So we know that p=<1 in any equilibrium. If p<1, he would buy only good 1. His demand would be 10p+2 units. Equating this to 12, we get p=1. Since this isn't <1, we rule out p<1. Next consider p=1. Now we need him to demand 12 units of good 1, i.e spend all of his income on it. But note that he wouldn't do that. Given his preferences, he would instead spend all his income on good 2. Hence there is no competitive equilibrium.