4. In a two-person, two-good exchange economy with strictly increasing utility f

Question

4. In a two-person, two-good exchange economy with strictly increasing utility functions, it is easy to see that an allocation x = (x11 , x21 , x12, x 22 ) is Pareto efficient if and only if x solves:

max xi ui (xi) subject to u j (xj ) u j (1)

x 11 + x 21 = 11 + 21 (2)

x 12 + x 22 = 12 + 22 (3)

for i = {1, 2}, i not equal to j, where ij denotes the endowment of good j for person i and x ij denotes the consumption, in the Pareto efficient allocation, of good j by person i.

(a) Prove the claim

(b) Generalize this result to the case of n goods and J consumers.

4. In a two-person, two-good exchange economy with strictly increasing utility ) is Pareto functions, it is easy to see that an allocation x = (z,z,zf efficient if and only if x solves: max uYai) subject to tr'(Z) ti for i = { 1,2), ij, where w, denotes the endowment of good j for person i and x^ denotes the consumption, in the Pareto efficient allocation, of good j by person i (a) Prove the claim (b) Generalize this result to the case of n goods and J consumers.

Explanation / Answer

Consider the given problem, here as we know that an allocation is said to be pareto efficient is we cannot make at least someone better off without worsening someone, which => we are at the position where if we try to make someone better off then some other will be adversely effected.

So, if there are 2 people and 2 goods, then to find out pareto efficient allocation, 1st we need to keep fixed the level of utility at some specific level, now we need to maximize the utility some other person, given the level of the supply of available resource.

So, mathematically the problem will look like.

Max U1=U1(X,Y) subject to U2(X,Y) >= U20 = fixed., Sum(Xi) = Wx, Sum(Yi)=Wy, here “Wx” be the total supply of X and “Wy” be the total supply of Y.

b).

let’s assume that there are “n” person and “m” goods, so the above problem will look like.

Max, “Ui=Ui(X1, X2,…Xm), subject to Uj(X1, X2, …Xm) >= Uj0 = fixed where j=1,2,…n, but not equal to “i”.

Now, the other constraint are, “Sum(X1) = Wx1”, “Sum(X2) = Wx2”,… “Sum(Xm) = Wxm”.

So, here there are (n-1+m) number of constraints in the generalized form.

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