PS2 Q1i answer
David Ronayne
24 February 2012
I solve the problem for a more general Cobb Douglas question with I individuals and L
goods, it is up to you to do the specific question with I = L = 2 from the problem set. So
the question I am answering is:
Find the set of CE in an I-consumer, L-commodity, pure exchange economy where the
QL
αil
consumption set of each agent is RL
+ . The utility function of consumer i is ui (x) =
l=1 xil
with αil ∈ (0, 1) ∀i, l and endowments are ωi ∈ RL
++ ∀i.
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Show you can restrict attention to p 0
Before starting, realise that the CE (if a CE exists) price vector could be anywhere in RL .
We need to narrow down the options before we start if we can. It is always nice if we know
that p 0, as then several nice theorems hold, which help us solve the problem.
Let’s consider if there exists a CE with prices p : ∃ l ∈ L s.t. pl ≤ 0 (bold text means its
a vector, non-bold means a number). I will show that this is not possible in this economy.
L
A CE in this question is x ∈ RLI
such that the UMP (utility maximisa+ and p ∈ R
tion problem) of every individual is solved given p, and such that x is feasible, i.e. that
PI
PI
i=1 ωil ∀l ∈ L. We’ll show that these prices will mean no individual has a
i=1 xil =
solution to his UMP.
Consider Mr.i’s UMP:
maxxi
L
Q
l=1
s.t.
L
P
il
xα
il
pl x1l ≤
l=1
L
P
l=1
1
pl ω1l
Now consider the allocation x̃1l0 = ω1l0 ∀l0 6= l, for sure he can do this as he’s just consuming
0
his endowment of these commodities, note that because ωi ∈ RL
++ ∀i =⇒ x̃1l0 > 0 ∀ l 6= l.
The budget constraint (cancelling terms) now becomes pl xil ≤ pl ωil , only 1 commodity left
in there now.
Now notice that (compute it to check):
∂ui >0
∂xil xil0 =x̃il0 ∀l0 6=l
In other words, he always demands more of commodity l at this level of the other commodities, hence his UMP have no solution (he’d like ∞). He is not stopped by his budget constraint precisely because pl ≤ 0. Hence there cannot be a CE in this economy if
∃ l ∈ L : pl ≤ 0. Notice that we did this proof for arbitrary l ∈ L, so it does not matter
which good you consider, just that there is at least one non-positive price. Hence, we know
that if there is a CE, there cannot be any non-positive prices or all the UMP’s would have
no solution, so any CE must be where p 0.
Rest of answer to follow, but try to do it this weekend if you can. . .
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