Rv Adv. Micro V10

7/30/2019 Rv Adv. Micro V10

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Written Exam for M.Sc. in Economics 2010-I

Advanced Microeconomics

18. December 2009

Master course

Solution

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1.1

(UMP) is

maxx

ui(x)

s.t. p x p i and x Xi

1.2

The set Xi is closed and the set {x R

L

|p x p i} is closed. Their inter-

section is non-empty because i, p RL++ and their intersection is bounded

from below by 0 and from above by (wi/p1, . . . , wi/pL), so their intersection

is non-empty and compact. The utility function is continuous by assumption.

Therefore (UMP) has a solution as a continuous function has a maximum on

a non-empty compact set.

1.3

A preference relation X X is strictly convex if and only if (1 )x +

x x for all x, x, x X with x, x x and x = x and ]0, 1[.

Uniqueness by contradiction: Suppose that x and x are different two

solutions to (UMP), then 0.5x+0.5x X and p(0.5x+0.5x) max{px, p

x}, so 0.5x+0.5x is in the budget set. By strict convexity 0.5x+0.5x x, x.

This contradicts that x and x are two different solutions to (UMP).

1.4

Definition 1 A Walrasian equilibrium is a price vector and a list of

individual consumption bundles (p, x) such that

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xi is a solution to (UMP) given p for all i.

i xi =

i i.

Suppose that i

xi(p, p i) =i

i.

Then xi = xi(p, p i) is a solution to (UMP) given p by definition and

markets clear by assumption. Therefore (p, x), where xi = xi(p, p i) for all

i, is a Walrasian equilibrium.

1.5

Definition 2 A Pareto optimal allocation x is a feasible allocation such

that there is no other feasible allocation x with ui(xi) ui(xi) for all i and

ui(xi) > ui(xi) for at least one i.

Suppose that (p, x) is a Walrasian equilibrium. Then p RL++ and pxi =

p i because preferences are strongly monotone. If ui(xi) > ui(xi), then

p xi > p xi because xi is a solution to (UMP). If ui(xi) ui(xi), then

p xi p xi. Indeed if p xi < p xi, then ui(xi + p) > ui(xi) for > 0 by

strong monotonicity and p (xi +p) p xi for small enough contradicting

that xi is a solution to (UMP). All in all if x Pareto dominates x, theni x

f

i =

i xf

i for some becasue p

i xi > p

i xi. Therefore ifx Pareto

dominates x, then x is not feasible. Hence x is Pareto optimal.

1.6

For p R2++ the demand is x(p, p1 +p2) = ((p1 +p2)/p1, 0) and for other ps

there is no solution to (UMP) because the lexicographic preference relation

is strongly monotone. Therefore x = (1, 1) is not a solution to (UMP) for

any p.

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2.1

The problem of consumer i is

maxx

x1 ln x2

s.t. p1x1 + p2x

2 wi and x R+ R++

where wi = 6p1 + (p1, p2) and (p1, p2) is the profit of the firm.

The problem of the firm is

maxy

p1y1 + p2y

2

s.t. y {z R2|z1 0 and z2 5z1}.

2.2

A Pareto optimal allocation (x, y) can be found by solving

max(x,y)

x1 + ln x2

s.t

x R+ R++

y {z R2|z1 0 and z2 5z1}

x =

6

0

+ y.

This problem reduces to

maxz

(6 z) + ln z

The solution is z = 1. Therefore (x, y) with x = (5, 5) and y = (1, 5) is a

Pareto optimal allocation.

2.3

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There is a unique Pareto optimal allocation because the utility function rep-

resents a strictly convex preference relation and the production set is convex.Therefore according to the first welfare theorem the equilibrium allocation

has to be the Pareto optimal allocation found in 2.2. For prices the gradient

of the consumer at the consumption bundle can be used. Hence ( p, x, y),

where p = (1, 1/5), x = (5, 5) and y = (1, 5), is a Walrasian equilibrium.

3.1

Definition 3 An equilibrium with spot markets is a price system and

an allocation ((pt)tZ, (xt)tZ) such that there exist (mt)tZ and M such that

(xt, mt) is a solution to

max(x,m)

ut(x)

s.t.

ptxy + m pt

yt

pt+1xo pt+1

ot + m

x X, m R

xyt + xot1 =

yt +

ot1 and mt = M for all t.

Definition 4 A Walrasian equilibrium is a price system and an alloca-

tion ((pt)tZ, (xt)tZ) such that

(xt) is a solution to

max(x,m)

ut(x)

s.t.

p

txy + pt+1xo ptyt + pt+1ot

x X

xyt + xot1 =

yt +

ot1 for all t.

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3.2

The budget constraints in the definition of equilibrium with spot markets

can be rewritten as

ptxy + pt+1x

o ptyt + pt+1

ot

pt+1(xo ot ) m pt(

yt x

y)

Therefore if (x, m) is a solution to the consumer problem in case of spot

markets, then x is a solution to consumer problem is case of forward markets.

Hence equilibria with spots markets are also Walrasian equilibria.

3.3

Definition 5 Anordinarily Pareto optimal allocation (xt)tZ is an allo-

cation such that there is no other allocation (

t)tZ

with

xt = xt for all t TL 2 and xyt = x

yt for t = TL 1 for some TL.

ut(xt) ut(xt) for all t with at least one >.

There are two other forms of Pareto optimality: strong Pareto optimality,

where consumption can be changed at all dates, and weak Pareto optimality,

where consumption can be changed at finitely many dates. Weak optimal-

ity is the relevant notion for the welfare theorem. Ordinarily optimality is

relevant for pension-like redistributions of goods.

3.4

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The utility function is a CES function and the demand function is

x =

pbt

pbt + pbt+1

pt

yt + pt+1

ot

pt,

pbt+1pbt + p

bt+1

pt

yt + pt+1

ot

pt+1

where b = (a 1)/a.

3.5

If pt = (7/3)at for all t, then ((pt)tZ, (xt)tZ with xt = (7, 3) for all t is

an equilibrium. It is not ordinarily Pareto optimal because if xot1 = 5 and

xyt = 5 from date t = 0 and forward, then consumer 1 and forward are

better off and no consumer is worse off. Alternatively it can be used that

t=0 1/pt < .

3.6

Clearly ((pt)tZ, (xt)tZ with pt = 1 and xt = (5, 5) for all t is a monetary

equilibrium.

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Rv Adv. Micro V10

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