Olivier Bochet
Advanced Micro (Micro II)
Answer-key Homework 2
Fall 2011
Exercise 1: Neutral goods and efficient allocations
Agent 1 likes only good 1, whereas agent 2 likes only good 2. Observe that
each indifference curve of agent 1 is a vertical line –agent 1 doesn’t care about
good 2– whereas each indifference curve of agent 2 is a horizontal line –agent
2 doesn’t care about good 1. Regardless of the distribution of the aggregate
endowment, there is a unique efficient allocation x = ((1, 0), (0, 1)). That is
agent 1 gets all of good 1 while agent 2 gets all of good 2.
Exercise 2: Efficient allocations II
Agent 1’s indifference curve are concentric circles centered at the bundle
(0, 1). Agent 1 reaches the lowest utility level exactly at this bundle. Agent
2’s preferences are quasi-linear and indifference curves have infinite slope at
points along the good 2-axis –like in Figure 18 in Lecture Note 2. Notice that
at the corners, agent 1 reaches utility levels,
u1 (0, 0) = u1 (0, 2) = 1, and
u1 (2, 0) = u1 (2, 2) = 5
Hence it is clear that both allocations ((0, 2); (2, 0)) and ((2, 2); (0, 0)) cannot be efficient. Indeed at the former allocation, agent 1 is indifferent between
getting the bundle (0, 2) and the 0 bundle. At the latter allocation, agent 1 is
indifferent between getting the bundle (2, 0) and the entire stock of resources.
AS it turns out, no boundary allocation at the top of the box can be efficient.
Also one quickly notices that the same reasoning applies for any allocation
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that is not on the bottom boundary of the box –indifference curves of agent 1
in the box are arc-shaped. The set of efficient allocations is thus
P E = {x ∈ F̄E : 0 ≤ x11 ≤ 2 and x21 = 0}
P.S. All my apologies for the typo in the hint –the hint said that there was
a finite number of efficient allocations. Indeed there is a continuum of such
allocations. I hope this did not create problems for anyone. Otherwise, I will
adjust the grading accordingly.
Exercise 3: Shall we trade?
1) Agent 1’s preferences are quasi-linear while agent 2’s preferences are
Cobb-Douglas. If there are interior allocations which are efficient, then for sure
at these we will have equality between agents’ marginal rates of substitution,
1
x22
(x11 )− 2
=
M RS1 = M RS2 ⇐⇒
2
x12
Given that feasibility holds, we must have that x11 + x12 = 1 and x21 =
x22 = 1. That is,
1
(1 − x21 )
(x11 )− 2
=
2
(1 − x11 )
We want to solve this equation for x21 as a function of x11 . We first get
that
1
(1 − x11 )
(x11 ) 2 =
2(1 − x21 )
So that,
x21 = 1 −
1 − x11
1
2(x11 ) 2
Notice that when x11 = 1, then x21 = 1 but when x11 = 0, then x21 tends
to −∞. Also when x21 = 0, then we obtain that
−(x11 )2 + 6x11 − 1 = 0
This second degree polynomial has two roots, one at 5.82 and one at 0.17.
So the curve we just found cuts the lower boundary of the Edgeworth box at allocation x = ((0.171573, 0); (0.828427, 1)). To conclude, observe now that each
allocation starting from x and moving towards the origin along the boundary
of the box is also efficient: to see this, observe that at each such allocation, the
MRS of agent 1 is bigger than the one of agent 2 –agent 2’s indifference curve
is flatter than agent 1’s indifference curve– implying that there is an empty
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intersection of the strict upper contour sets at these allocations. The set of
efficient allocations is thus composed of two parts written as the union of two
sets,
P E = x ∈ F̄E : x11 ≤ 0.171573, x21 = 0 ∪ x ∈ F̄E : 0.171573 ≤ x11 , x21 = 1 −
2) We want to compute the Walrasian equilibria of this economy. Let
us first compute the demand functions of both agents. These are found by
maximizing utility subject to the respective budget constraint of the consumer.
We obtain that
if 41 ( pp21 )2 ≤ 1
1 otherwise
p1
p2
x11 (p, p · ω1 ) =
x21 (p, p · ω1 ) =
1 p2 2
( )
4 p1
− 14 pp21 if 14 ( pp21 )2 ≤ 1
0 otherwise
p2
2p1
1
x22 (p, p · ω2 ) =
2
x12 (p, p · ω2 ) =
At equilibrium, supply must be equal to demand. Let us look at the market
for good 1 and let us normalize the price of good 1 to be equal to 1. Then we
have that
p2
1
(p2 )2 +
=1
4
2
We obtain that
(p2 )2 + 2p2 − 4 = 0
This second degree polynomial has two roots but only the second one
is positive with s2 = 1.236068. The equilibrium price vector is thus p∗ =
(1, 1.236068). The Walrasian allocation is
1
1
∗
x ≈ (0.381966, ), (0.618033, )
2
2
This is the unique Walrasian equilibrium of this economy.
Exercise 4: Lexicographic Preferences and the Welfare Theorems
1) Recall that for lexicographic preferences, indifference sets are just singletons. Based on this observation, and since agent 2 likes only good 1, there
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1−x11
1
2(x11 ) 2
can be no efficient allocation at which agent 2 gets a positive quantity of good
2. It is then easy to see that the set of efficient allocations is the top boundary
of the box,
P E = {x ∈ F̄E : 0 ≤ x11 ≤ 1 and x21 = 1}
2) The condition that we used to prove the first welfare theorem –i.e. local
non-satiation– is satisfied so the first welfare theorem holds: agent 1’s preferences are strictly monotonic, and agent 2’s preferences are monotonic.
As for the second welfare theorem, let me recall one important fact: the
conditions used to prove respectively the first and second welfare theorems are
sufficient conditions only. For instance, when strict monotonicity fails, it
is possible to construct economies for which the second welfare theorems does
not hold. But this does mean that the second welfare theorem fails
for any economy in which strict monotonicity is violated.
Therefore, to answer this question, it is not sufficient to simply
state that one of the key assumption in the second welfare theorem
is violated. You actually have to check at all efficient allocations
whether the second welfare theorem fails or not.
Suppose the conditions used to prove a given theorem are necessary conditions. Then the theorem does not hold for all economies in which one (or
more) of these necessary conditions is violated.
Suppose the conditions used to prove a given theorem are sufficient conditions –as is the case for the first and second welfare theorem. Then if one of
these sufficient condition is violated, one can find an economy for which the
theorem does not hold. But it is not possible to say that the failure of the
theorem applies to all economies in which this condition is violated.
Do you see now the difference between necessary and sufficient conditions?
When conditions used to prove a given theorem are both necessary and sufficient, then the theorem uses an if and only if statement. When conditions are
necessary only, then the theorem uses an if statement. When the conditions
are sufficient only, then the theorem uses an only if statement.
Ok, going back to the question. Let’s pick an efficient allocation. If p
is such that p 0, then both agents demand only good 1. Hence supply
cannot be equal to demand. If p = (1, 0) then agent 1 demands an unbounded
amount of good 2, which is free. Again supply is not equal to demand. Finally,
if p = (0, 1), both agents demand an unbounded amount of good 1, which is
free. This reasoning is valid at any efficient allocation. The second welfare
theorem fails.
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Notice that there is no Walrasian equilibrium in this economy, and this
whatever the initial endowments are. One can notice here the tight connection between the second welfare theorem and the existence of a Walrasian
equilibrium.
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