Multimarket Equilibrium

1
Chapter 9

• Multimarket Equilibrium

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2
Session One

• General goal
• General Equilibrium in pure exchange
• a graphical approach
• Detailed goals
• 1. Edgworth box
• 2. Competitive Equilibrium
• 3. Tatonnement process
• 4. Walras law

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3
1.Introduction ses.1
ch.9

• a. Introduction to chapter 9
• 1. Levels of price determination
• 2. Requirements for a theoretical
• analysis of Equilibrium
• b. The Edgworth box
• 1. Definition
• 2. Graph (fig.28.1, 2 Varian)
• (fig.9.2Varian(a) )

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1.Introduction ses.1
ch.9

• 3. Basic concepts
• - Allocation
• - Feasible allocation
• - Initial endowment allocation
• - Final allocation
• - Pareto efficient allocation
• - Contract curve
• - Trade region (core)

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1.Introduction ses.1
ch.9

• c. Offer curve
• 1. Definition
• 2. Graph (fig.9.2b),
• (fig. 8.7 Nic.92)
• d. Market trade
• 1. Description Tatonnement process
• 2. Graph (fig.9.2a), (fig.24.3Laidler)
• (fig.28.3 Varian)

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1.Introduction ses.1
ch.9

• e. Market Equilibrium
• 1. Description
• 2. Graph (fig.9.2b), (fig.28.4 Varian),
• (fig.2.6 Layard), (fig. 8.8
  Nic.92)
• 3. Analysis (fig.24.3Laidler)
• - First round
• - Second round
• - Properties of the equilibrium

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1.Introduction ses.1
ch.9

• 4. Special cases
• - Nonexistence of well-defined
• equilibrium (fig.9.3a)
• - Multiple Equilibria (fig.9.3b),
• (fig.2.7
  Layard)

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2. Mathematical Approach General Equilibrium
ses.1 ch.9

• a. The algebra of Equilibrium
• -Varian approach
• -Nicholson approach
• b. Walras law
• c. Examples
• 1. 16.4 Nicholson(92)
• 2. Varian

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9
Session Two

• General goal
• General Equilibrium in pure exchange
• a Mathematical Approach
• Detailed goals
• 1. Equilibrium for one consumer 2. Market
  Equilibrium
• 3. Multimarket Equilibrium

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10
1. Equilibrium for one consumer ses.2
ch.9

• a. Assumptions
• b. Introducing the model
• 1. The budget constraint (excess
• demand function)
• 2. The consumer utility index
• c. Solution of the model
• (first second order conditions)

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1. Equilibrium for one consumer ses.2
ch.9

• d. Properties
• (homogeneous of degree zero)
• e. Graph (fig.9.1)
• 1. First round
• 2. Second round

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1. Market Equilibrium for one commodity n
consumer ses.2 ch.9

• a. Aggregate excess demand
• function
• b. Equilibrium conditions

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3. Multimarket Equilibrium ses.2
ch.9

• a. assumptions
• b. Aggregate budget constraint
• c. Equilibrium conditions
• d. Solution of the model
• e. An alternative Approach
• f. Example

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Evaluationses.2 ch.9

• 1. Questions one to four Ch.28 Varian
• 2. Questions 9.1 , 9.2 , 10.1 , 10.2
• 3. Problem 16.7 Nicholson
• 4. Questions 27.1, 27.3, 27.4 Laidler

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a. assumptions 1- n individuals and m commodities
with fixed quantions. 2- each individual
possesses an initial endowment of one or move of
the commodities and is free to buy or sell at the
prevailing market prices. 3- A Consumer will sell
a portion of his inifial endownent of some
commodities and add to his stock of others to
increase his utility. 4- Purchases f sales may be
interpreted as barter transactions.
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b. Introducing the Model -Goal to Maximize his
utility function subject to his budget to that
there is no excess demand or Es. - the budget
constraint (excess demand function) Eij excess
demand of i th consumer for j th commodity qij
the quantity he consumer of j th commodity
his initial endowment
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Eij gt 0 consumptions of Qj excoeds his initial
endowment he purchase Qj in the MKT
Eijlt0
He sells Qj (excess supply)
- Consumers in come
(value of his initial endowment)
-the amount of purchasing power he would obtoin
if he sold his entire endoument.
(value of commodities he purchases)
- consumers expenditive
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- assuming he sells his entire endowment uses
it to purchase commoditions.
- the net value of the consumers ED must equal
zero. - B.L the value of commodition he buys
equals the value of commodition he
sells.
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2- the consumers utility index - it can be
stated as
so we have
but we have
(u.f in terms of E.D)
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c. solution of the Model
j1,,m (familiar F.O.C)
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S.O.C it is satisfied by the assumption of
regular strict quasi-concavity of U.R
(ED for i th consumer is a function of prices)


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d. Properties 1- homogeneous of degree zero -
it was proved that D.F is HDO . a similar theorem
can be proved for he pure exchange barter
economy. - the excess demands on HDO in prices
e.g. doubling all prices will double both the
value of the consumers initial endowment and the
cost of the commodities she purchases.
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then divide the first (m-1) equation by the m th
to eliminate
k , and factor k out of the (m1) th.
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e. Graph (fig . 9-1) - initial endowment R -
income line
the locos of all quantity combinations with
the same market value as his initial endowment.
- U. Max T - RS will be sold of Q2
- ST will be purchased of Q1 -
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-this process will be continued up to the point

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III Market Equilibrium for One Commodity n
consumers a. Aggregate excess demand function for
one commodity. - it is the summation of
individual excess demand functions of the n
consumers for Qj
- Aggregate ED is also a function of the m
commodity prices.
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b. Equilibrium condition - Partial equilibrium is
attained in the j th MKT if the
when the remaining (m-1) prices are assigned
fixed values.
- it is equivalent to the condition that
S(P)D(P) - Pj is obtained by solving
for Pj and depends upon
the prices assigned to the other (m-1)
commodition.
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- the purchases and sales of the individual
consumers are determined by substituting the
equilibrium price into the individual excess
demand function.
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a. assumptions 1- n consumers m commodities
cuith fixed quautities 2- all prices are
threateal as variables 3- all prices are positive
(non negative)
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b. aggregate budget constraint - one cons
- n cons

- it is not equilibrium condition but are
identities satisfied for any set of prices. It is
coled walras law.
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c. Equilibrium conditon - every aggregate excess
demand should equal zero if all prices are
positive.
(j1,,m)
- if Ej0 , the value of
must also equal zero.
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- if the first (m-1) markets are in eqnilibrium,
it is automatically attained in m th market.
Subtrating
from gives
If fouows that
- Multimarket equililrium is completely described
by (m-1) equations of Adition of m th equation
which is dependent upon the other (m-1) adds no
new in formation.
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d. solution of the Model since them equations (2)
are functionally dependent, their jacobian is
idenfically zero, and a locally unique solution
doesnot exist for the Pj.
(linearly independent)
Reminding e.g 2 simultaneous equations which are
dependent


(have no unique solutions)
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- inability to determine absute price levels
should not be a surprising result if it is
remembered that consumers one interested only in
exchange ratios in a barter-type economy. - he
can omit the m th equation (variable) solve the
model us-ing relative priees and get rid of the
problem of linearly dependente of equation.
Using HDO if excess demand functions in prices
are the exchange
- variables of
ratios relative to Q1
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- this system of diffentiable equations has a
unique math solution for the (m-1) prive ratios
if its jacobion does not vanish in a small
neighberhood. - the math. Solotion is a
multimarket equilibrium if it contains real,
nonnegatine price ratios quantities.
by substituting into Eij
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e. An alternative Approach finding multimarket
equilibriom directly without recourse to
Ag.ED HDO
(mn equations)
(mo equations) Clearing of every MKT
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f. Example



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S.O.S is satisfied
(walras law) (result of B.L)


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HDO of ED Walras Law
(B.Constraint is satisfied for every price
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problem sII

(HDO)

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(B. constraint always funik walras low) Invoking
the condition that each MKT must be cleared.
either equation is sufficient for deter mination
of the equilibrium exchange ratio.
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In equilibriom 1 unit of Q1 can be exchanged for
2 units of Q2. Substitutins
into indiuidual ED (Eij)
1 gives 41 units of Q1 to II in exchange for 82
units of Q2

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Session Three

• General goal
• General Equilibrium in production
• exchange
• Detailed goals
• 1. Equilibrium for one consumer
• 2. Equilibrium for one firm
• 3. Market Equilibrium
• 4. Walras law
• 5. Multimarket Equilibrium

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1. Introduction ses.3
ch.9

• a. Assumptions of the model
• b. Introducing the model

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2.Equilibrium for one consumerses.3
ch.9

• a. Excess demand functions
• b. The income budget constraint
• 1. Consumers income
• 2. Budget constraint
• c. The optimization
• 1. The first second order conditions
• 2. Results

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3. Equilibrium for one firm ses.3
ch.9

• a. Production function
• b. Profit function
• c. The optimization
• (the first second order conditions)
• d. Excess demand functions for inputs
• e. Excess demand functions for outputs
• f. Properties of ED functions

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4. Market Equilibrium ses.3
ch.9

• a. The Aggregate excess demand
• functions for a factor
• b. Aggregate excess demand
• functions for a commodity
• c. Equilibrium conditions in the
• short run long run

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4. Walras law ses.3
ch.9

• a. Profit as a function of Excess demands
• b. The result

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5. Multimarket Equilibrium ses.3
ch.9

• a. Description
• b. Equilibrium conditions
• relative prices
• c. Properties

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Evaluationses.3 ch.9

• 1. Questions 9.3

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• assumptions of the mode
• 1- goods are both produced exchanged
• 2- the consumers initial endowments consists of
  primary factor, land , labor
• 3- all profits earned by firms are distributed to
  consumers as wage, rant
• 4- A consumer generally sells factors and uses
  the proceeds together with his profit income to
  purchase commoditioes.
• 5- He may withold a portion of his factor
  endowment for direct consumpt
• 6- Entrepreneurs use both factor produced goods
  for the production of commodities.
• 7- the produced commodities are useful both as
  inputs and final consumper goods.

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b. Introducing the Model - n consomers in the
economy - m goods in the economy s primary
goods / m-s produced commodities (s1 to
m) - initial endowments of i th consumer
- market prices of endowments i th consumer

• utility is a function of both primary factors he
  retains
• the quantife q(m-s) produced goods

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a. Excess demand functions 1- excess demand for
factors
- it may be bwt will most often be negative,.
Since the consumer generally sells factors in
order to buy commodities. 2- excess demand for
commodiny (no initial stock) - it must be
positive or zero - it equals the quantity he
consumes.
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b. the income and budget constraint 1- consumers
Income - it equab the value of her factor
endowments plus his profit earnings.
the value of factor endowments for i th
consumer
the profit of h th firm which produces the k th
commodity
the i th consumers proportionate share of
these profits
the umber of firms prodocing the k th commodity
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2. Budget constraint - the value of the factors
commodities that an individual consumes should be
equal to his income.
(Budget constraint) (the net value of his excess
dom equals his profit earnings)
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c. the Optimization Max
S.t
1. F.O.C
- the consumer equate the RCS for every pair of
goods to their price ratio. 2. S.O.C - the
assumaption of regular strict quasi-loncauly of
u.function over a region ensures satisfaction of
the S.O.C
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3. results -the sonsumers excess D.F can he
othained by solving for the m excess demands, as
functions of the profit levels, in which he has
an in terest, and the m prices. - we will show
that profits may he expressed as functions of
commodity and factor prices. So.
- Profits are HD in prices. - ED for consumer is
HDO in prices of all commodities f factors? Since
it we multiply Pj by k, the resolts of F.O.C
should be the same as.
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a. production function -each firm commbines
inputs for the production of a single commodity
accrding to the rechnical rules specified in its
prod. F.
the output level of the h th firm in the j the
industry
the quantity of k th good which the
entrepreneur uses as an input Both the s
factors and (m-s) commodities serve as inputs.
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b. profit function - the entrepreneurs profit is
his competitive revenue less the cost of his
inputs.
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c. the opthimization excess demand
functions 1. F.O.C Profit Partial derivativey
with r.t. inputs equal to zero.

- the entrepreneur will utilize each input up to
a point at which the value of its marginal
physical product equals its price .
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2. S.O.C If the P.F is strictly concave over a
region, the S.O.C is satisfied over that
region. - imply that
imply that - If he utilizos his own output ar
an input, as a wheat farmer utilizes wheat seed,
he will utilize up to a point at which its MPP1

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d. the result Excess demand functions for
inputs - E.D for his inputs. For a strictly
concave region of his P.F are obtained by solving
the m equations of for
E.D for One input
Of one firm
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The quantity of each input he purchases is a
function of all prices Since the eatraprencur
never supplies (sells) inputs, his E.D is always
nonnegative. - If the j th industry contains Nj
identical firms, its aggregate excess D. for the
k th input equals the E.D of a representative
firm multiplied by the number of firms within the
industry. Aggregate ED for input
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In an industry
An industrys E.D for an input is a function of
all prices the number of firms within industry.
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e. Excess demands for output - we know that
(since No initial endwment) - so S.E for the
output for the entreprenevr is realy the excess
supply of outpunt so
putting in P.F
(E.D for output for h th firm in j th industry)
- Indrstrys E.D for output
An industrys E.D for output is a function of all
prices the number of firms of firms within the
industry.
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f. properties of E.D functions the entrepreneurs
E.D for his output and inputs are HDO in all
prices. Proof If all prices are changed by the
factor tgt0 , profit be come.
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F.O.C
since
The same sesult is obtained as the F.O.C before
the event. - S.O.C also remain unchanged
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a. the aggregate E.D for a factor - it is the sum
of the E.D of the n consumer (1) and the (m-1)
industries on input account
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b. Aggregate E.D for a commodity - it is the sum
of the E.D by the n consumers (1) , the (m-s)
industries on input account (2), and its
products(3)
- the aggregate E.D given by (4) (5) can be
stated simply as
the E.D for each good (factor lomnndity) is a
function of the m prices and the numbers of firms
within the (m-s) producing industries.
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1- SR equilibrium - a SR equilibrium price can
be determined for any of the m th markete
considered in isolation from the other (m-1)
markets by setting the aggregate E.D for the good
onder consideration equal to zero. - the number
of firms in the industry as well as the prices of
the other (m-1) goods and the number of firms
within the other (m-s-1) industries are treated
as parameters.
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2- LR equilibrium - Utility production, and the
E.D functions are defined for a longer penlod of
time in the LR analysis. - the number of firns
within the industry is a variable in the
determination of a LR equ. For a commodity MKT. -
E.D and profit are both set equal to zero, and
the resultant equations are solved for profit
the number of firms.
-SR LR equilibrium prices are nonnesative
generate consumpt. prod quantities within the
region for which the excess. D. functions are
defined.
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V. Walras Law
a. profit as a function of excess demands
we know E.D input
so
the net value of a firms E.D equals the negative
of its profit.
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b. the Result - summing the budget constraint of
the i th consumer over all consumers
(a ) (budset constraint for all consumers)
- summing the profit function (b) for all
produser
(b) (profit function for all producers)
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comparing (a) (b) we have
walras Law
- walras Law holds as an identity for any
set of prices in the production and exchange
system. - Total profits appear as a negative term
in the aggresation of B.L as positive term in
the aggnegation of profit fun.
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a. Descriptiun -A LR multimarket equilibrium
requires 1. every market be cleared. 2. Profit
equal zero in every industry.
the profit of a representative firm in the j
th industry? - Again walras law results in a
functional dependence amony E.D and it is not
possible to solve (1) for absolute price levels.
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b. Equilibrium conditions relative prices -
since the E.D of every consumer producer are
HDO in prices, the aggregate E.D are HDO in
prices. - the Profits of each entrepre never are
HDI in prices.
- Doubling all prices will not affect E.D
functions profit levels will remain equal to
zero and no new firms will be induced to enter
any industry. - if Q1 is chosen as an arbitrary
commodity, all the prices divided by P1
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1(2m-s-1) independent equations can be solved for
the equilibrium values of the (m-1) exchange
ratios relative to Q1 and the (m-s) firm
numbers. - all the uanables in equilibrium are
non negative.
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c. Properties - the equilibrism exchange ratios
and the firm unmbers are determined, the E.D of
every consumer entrepreneur can be computed by
substituting their values into the individ E.D
functions. - A LR equilibrium solutions
sarisfies 1. every consumer max, utility 2.
every entrepreneur max profit
3. every MKT is cleared 4. every entrepreneur
earns a zero profit - the equilibrium values of
the individual consumption and prod. Levels are
within the regions for which the individual E.D
fun are defined.
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Evaluation
1. Wuesion 9.3
(consumer)
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(ED for all commodities factors)
(P.F)
(Producer)
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market equilibrium
(Agg. ED for factor)
(Agg E.D for comm)
(4) (5)
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Walras law
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nulti-MKT
(every MKT)
(2m-s-1 equ.)
m-1 exch. Ratio . m-s firm
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Session Four

• General goal
• Money in General Equilibrium
• models
• Detailed goals
• 1. Money as a numeraire
• 2. Monetary Equilibrium
• 3. Money in utility functions

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1. Introduction ses.4
ch.9

• a. Introduction
• b. Nature functions of money
• c. Commodity Money
• d. Fiat Money the classical
• dichotomy

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2. Money as a Numeraire ses.4
ch.9

• a. Independent exchange ratios
• b. Nonuniqueness of numeraire
• c. Numeraire as a standard of
• value
• d. Money as a numeraire

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3. Monetary Equilibrium in an exchange
economy ses.4 ch.9

• a. Assumptions
• b. Excess demand for money
• c. Equilibrium conditions
• d. Results
• d. Properties

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4.Money in utility functions ses.4
ch.9

• a. The optimization
• b. Results
• c. Example

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Evaluationses.4 ch.9

• 1. Questions 9.4, 9.5

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90
Fig.28-1 Varian, Ch9
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29
Back
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Fig.28-2 Varian, Ch9
30
Back
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92
Fig.9.2aQuant, Ch9
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Back
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Fig.9.2aQuant, Ch9
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Back to 5
Back to 6
94
Fig.8.7Nicholson(92), Ch9
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Back explain
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Explain 8.7Nicholson(92), Ch9
Back to fig back to text
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96
Fig.24.3 Laidler, Ch9
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Back to 5 explain
Back to 6
97
Explain 24.3 Laidler, Ch9
Back to fig
Back to 6
Back to 5
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Fig.28-3 Varian, Ch9
Back
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99
Fig.28-4 Varian, Ch9
Back
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100
Fig.2.6 Layard, Ch9
Back
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101
Fig.8.8Nicholson(92), Ch9
Back explain
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102
Explain 8.8Nicholson(92), Ch9
Back to fig back to text
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103
Fig.9.3aQuant, Ch9
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Back
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Fig.2.7 Layard, Ch9
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Back
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Fig.9.1Quant, Ch9
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Back
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106
Q.16.7Nicholson, Ch9
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107
Q.27.1 Laidler, Ch9
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108
Q.27.3 Laidler, Ch9
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109
Q.27.4 Laidler, Ch9
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110
Fig.9.3bQuant, Ch9
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Review Questions (Varian Ch. 28)

• 1. Is it possible to have a Pareto efficient
  allocation where someone is worse off than he is
  at an allocation that is not Pareto efficient ?
• 2. Is it possible to have a Pareto efficient
  allocation where everyone is worse off than they
  are at an allocation that is not Pareto efficient
  ?

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Review Questions (Varian Ch. 28)

• 3. True or false ? If we know the contract curve
  , then we know the outcome of any trading .
• 4. Can some individual be made better off if we
  are at a Pareto efficient allocation ?

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