Proposition 1 A competitive allocation. is PE. Proof. As above, let (an, ring) for n. = 1, 2, …,N, be a competitive allocation
and let Ill be the associated CE price. We do a proof by contradiction. That
is, we suppose that (5,“, Eng} for n. = 1, 2, .. ., N is feasible and Pareto Superior
(PS) to (timing) for n. = 1,2, …,N. Then there exists a person, say person 13, who strictly prefers (5,1, cm) to
(an, 5,2). It follows that person i could not a?ord (5,11, 51-2) at the price 15.
(Make sure you understand this claim.) But, then, 5c -|- 1555.2 1‘" we -|- 331052 (3) Moreover, because no one strictly prefers the CE allocation to the assumed
PS allocation, for all 3′ 7E i, an + 35512 3 we + We (4)

(Make sure you understand this claim.) Next, let (3’1 2 2:, Em and
02 = 2:2] Eng. Then sum (4) over all j 75 2i and add the result to (3) to get 01 + 1562 33‘ W1 + ?wa (5)
or, equivalently, _ _
(01 — W1} + £54,492 — We} 3-" D (6) Because ,6 > D, it follows from (6) that either (71 — W1 3:: D or (5’2 — W2 :- D.
But, either implies that (Em, Eng) for n = 1,2, …, N is not feasible. I Exercise 15 (i) What step in the above proof fails if people care about what
other people consume? (it) What step fails if add. numbered people and even.-
numbered people trade in separate markets and, therefore, at aft?ereht par-tees?

2.5 An important special case In what follows, we sometimes want to work with simple settings in which
we can describe equilibria. Often, the description proceeds using what is
called yeess-end—nerify. In fact, this procedure was alread}r used in one of
the exercises above. There are three steps to the procedure: (i) make a gun:
about some feature of an equilibrium; (ii) construct an entire guess based on
that feature; and (iii) verify that the step (ii) construction is an equilibrium. For the two-good pure-exchange setting, an important special case that
lends itself to gness-end—ne?fy is a setting in which all the people in the
model have the same preferences (the same utilit}r function, denoted U(c1, c2),
and indifference curve map) and in which the utility,r function U(c1,c2) is
homothetic. Here is the logic in the form of a list of claims. Consider the plane with
good 1 on the horizontal axis and good 2 on the vertical axis, the plane
in which we can represent, among other things, the common homothetic
indifference curve map. Claim (i): In a CB (with identical and homothetic
preferences), ever}r person will end up consuming along some ray,r through the
orig’n. Claim (ii): That ray has slope W2] W1. Claim (1ii): the CE price of
good 2 in units of good 1 per unit of good 2, denoted ii, satis?es l/? : U1(W15W2)/U2(W15W2)’ (7) Claim (iv): Person n’s equilibrium consumption, denoted (ewes) is the
solution to the following pair of linear equations: (51114-355712 =wn1 +15’1?ne and énQ/énl = We/Wl- (3) Exercise 16 In a few sentences, defend each of the above claims.

Exercise 18 Suppose N = 100, that the utility function of each per-eon ie
U(c1,c2) = 1/c_1 + (1/2}¢E, and that the individual endourmente are as
follouxe: (wnhiong) = (2,4) for n = 1,2,…,5D and (wnhiong) = (4,2) for
n = 51,52, …, IUD. Giue a complete numerical description of a CE for this
economy.