Olivier Bochet
Advanced Micro (Micro II)
Answer-key Midterm Exam Fall 2011
Exercise 1:
1) False. First, yes it is true that there are no Walrasian equilibria in
this economy. Since preferences are monotonic, we know by the first welfare
theorem that any Walrasian equilibrium will be efficient. There is a unique
efficient allocation x = ((1, 0), (0, 1)). Hence this is the only candidate allocation for a Walrasian equilibrium. Given the endowments’ position, the
only price vector p candidate to support x as Walrasian equilibrium allocation is p = (0, λ) where λ > 0. But at this price, agent 1 will ask for an
unbounded amount of good 1 –since good 1 is free. Going now to the second
welfare theorem, it would however be wrong to claim that the second welfare
theorem fails. Indeed, by performing the appropriate transfer, e.g. by setting
ω 0 = ((1, 0), (0, 1)), any price vector p0 = (λ, α) 0 will support ω 0 as a price
equilibrium with transfers.
2) False. Consider an economy with two agents in which preferences are
given by ui (x1i , x2i ) = x1i + x2i . Let ω̄ = (1, 1). Notice that the set of efficient
allocations is the whole box. Let ω = ((0, 1), (1, 0)) and consider the price
vector p∗ = (1, 1). Then any allocation x∗ = ((x∗11 , x∗21 ), (1 − x∗11 , 1 − x∗21 )) such
that x∗11 + x∗21 = 1 is supportable by p∗ as a Walrasian equilibrium.
Exercise 2: A demand system
1) Homogeneity of degree 0 implies that a = b = c = 1. Walras’ law implies
that θ = β = λ = 1 as well: since the equality θp1 + βp2 + λp3 = p1 + p2 + p3
must hold for all price vector, this is possible only if θ = β = λ = 1
1
2) Given the reduction in the demand system, it is easy to see that all
substitution effects are 0 and S(p, w) is just the 0 matrix. Goods are perfect complements –that is, these demands functions are easily identified as
coming from the maximization under budget constraint of the utility function
u(x1 , x2 , x3 ) = min{x1 , x2 , x3 }. Given S(p, w), it has all the required properties to be generated by a process of utility maximization!
Exercise 3: Exchange Economies
1) I consider in turn each of the possible values of α
Case 1: α = 0
Since agent 2 likes only good 2, it is clear that, at any Pareto efficient allocation,
he should get none of good 1. A quick inspection reveals that the set of efficient
allocation is the right boundary of the Edgeworth box. Formally,
P E(E) = {x ∈ R4+ : x11 = 2, 0 ≤ x21 ≤ 2}
Case 2: α = 1
When α = 1, preferences of agent 2 are of the perfect substitute type. His
indifference curves are just parallel straight lines of slope −1. At any interior
efficient allocation, marginal rates of substitution must be equalized:
1
2
= M RS1,2
⇐⇒
M RS1,2
1
1 −1/2
x11 = 1 ⇐⇒ x11 =
2
4
Note that we have just found the set of interior efficient allocations. We now
have to check whether there are efficient allocations on the boundary of the
Edgeworth box. A quick inspection reveals that for any allocation x such that
1
2
|x11 < 1 ,x21 =0 > M RS1,2
|x12 > 7 ,x22 =2 . This
x11 < 14 and x21 = 0, then M RS1,2
4
4
indicates that the intersection of the strict upper contour sets is empty for
allocations at which x11 < 14 and x2 1 = 0. Likewise, for any x such that
1
2
|x11 > 1 ,x21 =2 < M RS1,2
x11 > 14 and x21 = 2, then M RS1,2
|x12 < 7 ,x22 =0 . This
4
4
indicates that the intersection of the strict upper contour sets is empty for
allocations at which x11 > 14 and x21 = 2.
The set of efficient allocations is thus
1
1
1
P E(E) = {x ∈ R4+ : (i) x11 < , x21 = 0, or x11 = , 0 ≤ x21 ≤ 1, or x11 > , x21 = 2}
4
4
4
Case 3: α = 21
When α = 21 , agents have the same quasi-linear preferences. At any interior
efficient allocation, marginal rates of substitution must be equalized:
1
2
M RS1,2
= M RS1,2
⇐⇒
2
1 −1/2 1 −1/2
x
= x12
2 11
2
Using the resource constraint for good 1 –i.e. x11 + x12 = 2, we get that at any
interior and efficient allocation, x11 = 1. Note that we have just found the set
of interior efficient allocations. We now have to check wether there are efficient
allocations on the boundary of the Edgeworth box. A quick inspection reveals
that
1
|x11 <1,x21 =0 >
(i) For any allocation x such that x11 < 1 and x21 = 0, then M RS1,2
2
M RS1,2 |x12 >1,x22 =2
1
(ii) For any allocation x such that x11 > 1 and x21 = 2, then M RS1,2
|x11 >1,x21 =2 >
2
M RS1,2 |x12 <1,x22 =0
The set of efficient allocations is thus
P E(E) = {x ∈ R4+ : (i) x11 < 1, x21 = 0, or (ii) x11 = 1, 0 ≤ x21 ≤ 1, or (iii) x11 > 1, x21 = 2}
2) I consider in turn each of the possible values of α
Case 1: α = 0
I first compute the demand functions of agent 1 since these will be used for
each case. His demand functions are found by maximizing his utility subject to
his budget constraint. At an interior solution, his marginal rate of substitution
must equal the price ratio, i.e.
1 −1/2 p1
x
=
2 11
p2
Solving we obtain that
x11 (p, p · ω1 ) = (
p2 2
p2
) and x21 (p, p · ω1 ) = 2 −
2p1
4p1
By the first welfare theorem, every Walrasian equilibrium allocation is efficient. Given the shape of the set of efficient allocations, we then know that
at each Walrasian equilibrium, agent 1 must buy all of good 1. Thus, at each
Walrasian equilibrium (p∗ , x∗ ), x∗11 = 2. Since only relative prices matter, let
us normalize p∗1 = 1 and use the previous information to find p∗2 . That is,
(
√
p∗2 2
∗
)
=
2
⇐⇒
p
=
8
2
2p∗1
We know that agent 2 spends all his money on good 2, i.e.
x∗22 (p, p · ω2 ) =
3
p · ω2
2
=√
p2
8
Therefore there is a unique Walrasian equilibrium given by,
√
2
2
(p∗ , x∗ ) = ((1, 8); ((2, 2 − √ ), (0, √ ))
8
8
Case 2: α = 1
Let us normalize again the price of good 1 to 1. By the first welfare theorem, any Walrasian equilibrium is efficient. Hence, at an interior Walrasian
equilibrium, agent 1’s demand for good 1 must equal 1/4. That is,
p2 2 1
) =
⇐⇒ p2 = 1
2p1
4
Hence prices must be the same at equilibrium. Given these prices, we
simply compute the remaining demands. We find that
x11 (p, p · ω1 ) = (
1 7 7 1
(p∗ , x∗ ) = ((1, 1); (( , ); ( , ))
4 4 4 4
Are there Walrasian equilibria on the boundaries? One quickly sees that
this is not possible in this economy.
Case 3: α = 21
Let us normalize the price of good 1 to 1. Again, by the first welfare theorem,
every Walrasian equilibrium allocation is efficient. At any interior Walrasian
equilibrium, agent 1 must get a quantity of good 1 x11 such that x11 = 1. That
is,
p2 2
) = 1 ⇐⇒ p2 = 2
2p1
Computing the remaining demands, we get
x11 (p, p · ω1 ) = (
3
1
(p∗ , x∗ ) = ((1, 2); ((1, ); (1, ))
2
2
Are there equilibria on the boundaries? One quickly sees that this is not
possible in this economy.
3) I consider in turn each of the possible values of α
Case 1: α = 0
Since local non-satiation holds, the first welfare theorem also holds. It is easy
to see that any efficient allocation√x can be obtained as a price equilibrium
with transfers by choosing p = (1, 8) and setting ω = x.
For the remaining two cases preferences fulfill all the conditions laid down
for the second welfare theorem to hold.
4