Download Microeconomics II Solutions to problem set 1

Microeconomics II
Solutions to problem set 1
Mario Tirelli
November 2010
Solution to Problem 1
1) The Edgeworth box represents all the possible distributions of the available, total resources of two commodities between two agents. We restrict
attention to non-wasteful allocations, namely those allocations which exhaust total resources.
Let the two commodities and their quantities be x and y, the two consumers
be A and B, the total resources by (ω x , ω y ). The diagram of the box can
be drawn starting from determining both its width and its height. The box
width is
Width = ω x = 5
The box height is
Height
= ω y = 10
The endowment allocation is a point in the box (hence feasible). In our
Figure 1: The Edgeworth box (Panel I) and the endowment point (Panel II)
.
case this point is:
ω A = (ωxA ; ωyA )T = (2; 1)T
ω B = (ωxB ; ωyB )T = (3; 9)T
which is shown in (Figure 1 Panel II).
2) In our economy, the set of Pareto efficient allocations (or contract curve) is
the locus of points, in the box, such that indifference curves of the two agents
are tangent (the golden line in Figure (2)). Why? because the indifference
curves are just regular hyperbolas, hence (Inada conditions apply, implying
indifference curves of i never intersect the hedges of the box which would
correspond to either x or y being equal to zero for i) efficient allocations are
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interior, (x, y) 0:
ū = xi y i
⇒
yi =
ū
, ū ∈ R, i = A, B
xi
The part of the set of Pareto efficient allocations where both consumers do
Figure 2: The set of Pareto efficient allocations (or contract curve) and the
core of the economy.
at least as well as at their initial endowment is called the core (the red line).
Let us now define the set of Pareto efficient allocation of our economy. It is
the set of allocations exhausting total resources and equalizing the marginal
rates of substitutions.
U Mxi
yi
i
M RSx,y
=
= i
i
U My
x
where U Mxi = ∂ui /∂xi = y i , U Myi = ∂ui /∂y i = xi . Hence,
yA
yB
=
xA
xB
Finally, imposing that all the resources are distributed:
yA
10 − y A
=
xA
5 − xA
2
The latest condition is equivalent to
5 − xA
10 − y A
=
xA
yA
and it is satisfied when y A = 2xA , implying SM S A = 2. Finally, the set
of Pareto efficient allocation is
A A B B
x , y , x , y : 0 ≤ xA ≤ 5, y A = 2xA , xB = 5 − xA , y B = 2xB
3a) The offer curve of the i-th agent (OC i (p)) is locus of commodity bundles
demanded by i, as relative prices change. Because preferences are strictly
monotone and satisfy Inada, every demand allocation is interior and lies on
the budget line. Therefore, offer curves can be characterized by those positive allocations which are at the tangency points between the i-th agent’s
indifference curves and the budget line, for every relative price ratio. The
OC is then a function of relative price only, and is tangent to the consumer’s
indifference curve at the endowment point.
More formally, the OC i (p) is the solution of the following maximization
problem:
(
max(xi ; y i )
ui (xi ; y i ) = xi y i
i = A, B
(1)
[Problem I]
i
i
i
subject to
w = px x + py y
where wealth w is,
(WC)
wi = p · ω i
i = A, B
Solving [Problem I] we obtain the Walrasian’s demands:
xix (p; wi ) =
wi
2p1
xiy (p; wi ) =
wi
2p2
Imposing the (WC), we get the offer curves for the agent A and B:
2px + py 2px + py T
py
;
) = (1 +
;
2px
2py
2px
3(px + 3py ) 3(px + 3py ) T
3
;
) =( +
OC B (p) = (
2px
2py
2
OC A (p) = (
px 1 T
+ )
(2)
py
2
9py 3px 9 T
;
+ ) (3)
2px 2py
2
3b) The Walrasian equilibrium is a price vector for the two commodities
which satisfies aggregate demand equal total resources in each of the two
markets. Geometrically, it is the point where the OC intersect. Because
Walras law holds here, it suffices to find the price that clears one of the two
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markets, say that of x.
total demand of good x
z
OC A for commodity x
⇒
⇒
}|
{
OC B for commodity x
z
}|
{
z }| {
2p∗x + p∗y
3(p∗x + 3p∗y )
+
2p∗x
2p∗x
p∗y
3 9p∗y
1+ ∗ + + ∗ =5
2px 2 2px
5p∗y
5
p∗x
=
⇒
=2
∗
px
2
p∗y
total supply of good x
=
z}|{
5
(4)
This is the Walrasian equilibrium price ratio. At (p∗x /p∗y ) the market for
good y clears too [you should check this]. By substituting the equilibrium
relative price into the offer curves (Equation (2) & (3)), we obtain the Walrasian equilibrium allocations:
p∗y
;
2p∗x
3 9p∗y
OC B (p∗ ) = ( + ∗ ;
2 2px
OC A (p∗ ) = (1 +
p∗x 1 T
1
1
5 5
+ ) = (1 + ; 2 + )T = ( ; )T = (1.25 ; 2.5)T
∗
py
2
4
2
4 2
3p∗x 9 T
3 9
9
15 15 T
+ ) = ( + ; 3 + )T = (
;
) = (3.75 ; 7.5)T
∗
2py
2
2 4
2
4
2
3c) According to the First Welfare Theorem (IWT), any Walsarian equilibrium allocation is Pareto efficient. Thus, in principle, if all commodities
are goods, there is nothing to be checked here, other than verifying that
individual preferences are locally non satiated. Yet, you may want to check
that the equilibrium allocation you computed is an element of the set of
Pareto efficient allocations defined above.
4) Start from the given efficient allocation. Since it is interior, and Pareto
efficient, you know that the corresponding M RSs of the two agents are
equal to 2. Because at an interior equilibrium M RSs equal relative prices,
the only candidate, relative, price is px /py = 2. Without loss of generality
(using the fact that individual demands are homogeneous of degree zero in
prices), normalize the price of y to 1, i.e. let (px , py ) = (2, 1). To find the
necessary transfers, just impose budget balance with transfers; for agent A:
2 × 2, 5 + 5 = 2 × (ωxA + tA ) + ωyA
Compute tA by solving the latest equation at the initial endowment point
(ωxA , ωyA ) = (2, 1), assuming that this is the initial situation faced by the
planner:
10 = 4 + 2 × tA + 1 =⇒ tA = 2.5, tB = −2.5
If you instead did the exercise starting from the equilibrium allocation, then
just modify the last step by setting endowments equal to that allocation:
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(ωxA , ωyA ) = (1.25, 2.5)
Solution to Problem 2
1) Walrasian equilibria can be both characterize algebrically and geometrically.
Algebrically, for any economy (i.e. given endowments) with individuals
that are locally non satiated (LNS), an interior equilibrium allocation satisfies the condition that MRSs equal the ratio of market prices. The latest holds as a consequence of individual optimality. Geometrically, in the
box-economy, a representation is in (Figure 3). To explain why an interior
equilibrium allocation equalizes MRSs, argue by contradiction. Suppose
MRSs do not all equalize; for some individual either M RS1,2 > p1 /p2 or
M RS1,2 < p1 /p2 . Then, it is easy to see that this individual is not optimizing, hence the allocation is not an equilibrium (contradiction!). Indeed,
suppose M RS1,2 > p1 /p2 ; then, it would be budget feasible for the consumer
to buy an extra unit of good 1 at the equilibrium price p1 and sell a unit of
good 2 at price p2 , therefore exchanging a unit of good 1 against p1 /p2 units
of good 2. This makes her better off, since to be as well off she should have
exchanged a unit of good 1 against M RS1,2 > p1 /p2 units of good 2; what
she saves - M RS1,2 − p1 /p2 - in terms of good 2 is what make her better off.
More formally, suppose that, in addition, preferences are represented by
differentiable utility functions, then first order individual optimality conditions of the agent with M RS1,2 6= p1 /p2 are violated. Indeed, let the
equilibrium allocation of this agent be (x1 , x2 ). Totally differentiate her
utility function,
du = Du(x1 , x2 ) · (dx1 , dx2 )
. Assuming, that commodity 2 is a good, D2 u := ∂u/∂x2 > 0,
du
D1 u
=
, 1 · (dx1 , dx2 )
D2 u
D2 u
= (M RS1,2 , 1) · (dx1 , dx2 )
= M RS1,2 dx1 + dx2
Thus, if M RS1,2 > p1 /p2 , a trade satisfying p1 dx1 = −p2 dx2 with dx1 > 0
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is budget feasible and welfare improving:
du
D2 u
p1
= M RS1,2 dx1 − dx1
p2
p1
=
M RS1,2 −
dx1
p2
p1 p1
>
−
dx1
p2 p2
= 0
Therefore, (x1 , x2 ) cannot be part of an equilibrium allocation. Notice that
the assumption that the equilibrium allocation is interior is essential; since
the above trade is budget feasible for the considered individual if her initial
allocation x2 is positive.
Figure 3: Interior equilibrium, x∗ .
2) The Walrasian equilibrium does not always require that the MRSs equal
relative prices. In fact, there are equilibria in which one or more consumers
choose not to consume some commodity, thereby demanding an allocation
that is at the boundary of the Edgeworth box (Figure 4).
3a) To draw the Walrasian equilibrium allocations, let us study consumers’
indifference curves. The consumer A has a quasilinear utility function, instead, the consumer B has a linear utility function only in the good two.
The indifference curves generated by these utility functions are plotted in
(Figure 5, Panel I).
First, observe that indifference curves are quasi-linear in commodity 1 for
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Figure 4: Boundary equilibrium, x∗ .
Figure 5: Endowment point and the existence of the Walrasian equilibrium.
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agent A.1 While agent B cares only to consume commodity 2. It is therefore
natural to conjecture that if an equilibrium exists it will be on the boundary
of the box for most economies (i.e. initial endowment distributions).
To characterize equilibria you should concentrate on consumer A; B will
always spend all his wealth on commodity 2. So, let us define the demand
of A.
Since A0 s preferences are convex, first order conditions fully characterize A0 s
consumer demand:
∂l uA ≤ λpl with equality if xl > 0, l = 1, 2
p · (x − ω) = 0
for some λ ≥ 0.
Consider a solution with x = (x1 , x2 ), with x1 > 0, x2 = 0, satisfying,
M RS(x)
=
1 + x2 |x2 =0 >
⇔
p1
<1
p2
p1
p2
(5)
A
A
A
Therefore, if pp12 < 1, at endowments ω A , A chooses (xA
1 , x2 )(p, ω ) = (ω1 +
p2 A
A
A
B
B
p1 ω2 , 0). At equilibrium, market for good 1 clears: x1 (p, ω )+x1 (p, ω ) =
B
ω 1 ; using the fact that xB
1 (p, ω ) = 0, substituting,
ω1A +
p2 A
ω = ω1
p1 2
and rearranging (with the assumption that ω1B > 0),
p1
ωA
= 2B
p2
ω1
(6)
Since condition (5) must hold, not all the economies can lead to the type
of equilibria just described; instead, only those economies with initial endowments ω2A < ω1B . Graphically, these economies are identified by the set
of points in the box lying below its main diagonal, cutting it from the left
highest corner to the right lowest one (Figure 5, Panel II).
Therefore, we can conclude that for ω2A < ω1B equilibria exist.
You can also test that if ω2A ≥ ω1B (i.e. if the economy is one on or above
ω A +1
the diagonal cutting the box) an equilibrium exists with pp12 = ω2B +1 ≥ 1.
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When the latter inequality is strict, equilibrium
allocations are such that
A
B
A∗ = ω̄ , p1 − 1 , where p1 − 1 = ω2 −ω1 . Thus,
xB∗ = (0, ω̄2 − xA∗
1 p2
2 ), x
p2
ω B +1
1
1
Drawing indifference curves of A on (x1 , x2 ). Take the exponential transformation of
ū
A0 s utility: eū = ex1 (x2 + 1), the level set ū is x2 = eex1 − 1. A0 s M RS is: ex1 dx2 +
dx2
x1
e (x2 + 1) dx1 = 0 ⇒ dx2 + (x2 + 1) dx1 = 0 ⇒ M RS := − dx
|u = x2 + 1
1
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equilibria do always exist in this box-economy.
3b) You basically have just to rap up the observations made in the last
solution set. For all the economies with 0 ≤ ω2A < ω1B the equilibrium price
is
p1
p2
=
ω2A
,
ω1B
with allocations xA∗ = (ω1A +
p2 A
B∗
p1 ω2 , 0), x
= (0, ω̄2 ).
ω A +1
For all the economies with ω2A ≥ ω1B ≥ 0 the equilibrium price is pp21 = ω2B +1 ,
1
).
with allocations xA∗ = ω̄1 , pp12 − 1 , xB∗ = (0, ω̄2 − xA∗
2
Graphically, equilibria are either along the lower hedge or along the right
vertical hedge of the box, depending whether we are considering economies
described by initial endowments lying below or above the main diagonal.
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