Chapter Three

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Chapter Three

• Topics in Consumer
• Behavior

2

• This chapter contains four sessions
• Session one
• Session two
• Session three
• Session four

Finish
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Session One

• General goal
• To be familiar with a variety of specific
• types of utility functions duality theorem
• Detailed goals
• 1. Stone-Geary utility function
• 2. Separable utility functions
• 3. Additive utility functions
• 4. Homogeneous homothetic utility functions
• 5. Duality in consumptions
• 6. Evaluation
• Back

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1. Stone-Geary utility function Linear
Expenditure system

• a. Description
• b. Demand functions optimization
• c. Expenditure functions
• Back to the main menu

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2. Separable utility functions

• a. Strongly separable utility functions
• 1. Description
• 2. Properties
• 3. Example
• b. Weakly separable utility functions
• 1.Description
• Back to the main menu

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3. Additive utility functions

• a. Strongly additive utility functions
• 1. Description
• 2. Properties
• 3. Example
• b. Weakly additive utility functions
• 1. Description
• 
  Back to the main menu

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4. Homogeneous homothetic utility functions

• a. Homogeneous utility functions
• 1. Description
• 2. Properties
• b. Homothetic utility functions
• 1. Description
• 2. Properties
• 3. Example
• Back to the main menu

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5. Duality in consumption

• a. Dual problem Shepherd lemma
• 1. Description
• 2. Dual problem shepherd lemma
• 3. Generalization
• b. Roys identity
• 1. Description
• 2. Generalization
• c. Slutsky equation
• d. Example
• Back to the main menu

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Evaluation

• 1. Questions 3.1, 3.2, 3.3, 3.4
• 2. Problems 5.6, 5.7, 5.8 Nicholson

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Session Two

• General goal
• 1. The theory of revealed preferences
• 2. Revealed preferences indifference curves
• Detailed goals
• 3. Weak axioms of revealed preferences
• 4. Strong axioms of revealed preferences
• Evaluation
• Evaluation
• Back

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2. Revealed preferences indifference curves

• a. From revealed preferences to preference
• 1. The principle of revealed preference
• 2. Indirect revealed preference
• (fig.7.2 Varian)
• b. From revealed preferences to indifference
  curves
• 1. Description
• 2. Graph (fig.7.3 Varian)
• Back

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3.The weak axiom of reveal preferences

• a. Definition
• b. Graph (fig.7.4, 7.5 Varian)
• (fig. 3.2a , 3.2b)
• c. Mathematics
• d. Generalization
• d. Checking WARP
• Back

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4. The Strong axiom of revealed
preferences (SARP)

• a. Definition
• b. Graph (fig.7.2 Varian)
• c. Checking SARP
• d. Revealed preferences substitution effect

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Evaluationses.2
ch.3

• 1. Question 3.5
• 2. Problems 7.1, 7.2, 7.3 Nicholson

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Session Three

• General goal
• Consumer surplus welfare changes
• Detailed goals
• 1. Compensating variation
• 2. Equivalent variation
• 3. Consumer surplus
• 4. Welfare changes Continue

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Session Three

• Introduction
• Constant real income and the compensating
  variation in money income (C.V)
• Constant real income and the equivalent variation
  in money income (E.V)
• Consumers surplus
• Evaluation
• Example

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I. Introduction

• The income substitution effects (Review)
• (fig.3.3 Laidler)
• b. Money income real income
• (fig.3.4 Laidler)
• Back

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II. Constant real income the compensating
variation money income

• a. Description
• 1. Constant real income
• 2. Compensating variation
• b. Graph
• ooo(fig. 3.4, 3.11 Laidler) (fig. 14.4
  Varian)
• c. Mathematics
• 1. Expenditure function
• 2. Shepherds lemma
• 3. C.V (fig. 5.9 Nicholson)
  Continue
• 
  

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III. Constant real income the equivalent
variation in Money income (E.V)

• a. Description
• b. Graph (fig.3.8, 3.11 Laidler)
• (fig. 14.4 Varian)
• c. Mathematics
• d. Example
• Back

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IV. Consumer Surplus

• a. Introduction
• b. Description
• 1. Definition
• 2. Measurement
• c. Mathematics a comparative approach
• (fig. 4.8Laidler) (fig. 5.10 Nich)
• d. Example

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Evaluation

• 1. Problem 5.10 Nicholson (1992)
• 2. Question 3.7
• 3. Question 4.17 Laidler
• Back

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Session Four

• General goal
• Consumer behavior under risk uncertainty
• Detailed goals
• 1. Expected utility theory
• 2. Attitudes towards risk
• 3. Risk insurance

Continue
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Session Four

• Introduction
• Expected utility theory
• Risk aversion Attitudes towards risk
• Risk insurance
• Evaluation

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I. Introduction

• a. Background
• b. Axioms for consumer behavior in uncertain
  situations
• 1.Complete ordering axiom
• 2.Continuity axiom
• 3.Independence axiom
• 4.Unequal probability axiom
• 5.Compound-lottery axiom
• c. Expected value

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II. Expected utility theory

• a. Introduction Utility functions probability
• 1. Description
• 2. Examples
• b. Description
• c. Properties (theorems)
• d. Why expected utility is reasonable?

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III. Risk aversion Attitudes towards
risk(Behavior under Uncertainty) (Preference
towards risk)

• a. Introduction
• 1. Specified expected utility function
• 2. Risk Uncertainty
• b. Risk averse
• 1. Definition
• 2. Graph (fig.3.5) , (fig.3.1, 3.3
  Griffiths)
• (fig. 12.2 Varian)
• 3. Mathematics
• 4. Example

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III. Risk aversion Attitudes towards
risk(Behavior under Uncertainty) (Preference
towards risk)

• c. Risk neutral
• Definition
• Math
• Graph (fig. 3.1 Griffiths)
• d. Risk lover
• Definition
• Math
• Graph (fig.3.1 Griffiths) (fig. 12.2
  Varian)
• e. Measuring the risk premium (degree of risk)
• Definition
• Math
• Graph (fig. 3.3 Griffiths)

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IV. Risk Insurance

• a. Description
• b. Graph (fig.3.5, 3.6 Griffiths) Explain
• c. Mathematics
• 1. Introducing the model
• 2. Insurance premium

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Evaluation

• 1. Example 8.4 Nicholson
• 2. Problem 8.5 Nicholson
• 3. Problem 3.2 Griffiths
• 4. Questions 3.8, 3.9, 3.10
• 5. Questions 12.2, 12.3 Varian

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Fig.7.1 Varian, ch.3
Explanation
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fig.7.2Varian Ch. 3
Back to11 Back to 13
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fig.7.3Varian Ch. 3
Back Explanation
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fig.7.4Varian Ch. 3
Back Explain
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fig.7.5Varian Ch. 3
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fig.3.2aQuandt, Ch. 3
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fig.3.2bQuandt Ch. 3
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fig.3.3Laidler, Ch. 3
Back Explain
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Explain 3.3Laidler Ch. 3
Back to fig Back to text
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Fig.3.4Laidler, Ch. 3
Back to 17 Explain
Back to 18
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Explain 3.4Laidler, Ch. 3
Px? vary money income A?C respective
C.D.C
Back to 18
Back to fig Back to 17
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Fig.3.11Laidler, Ch. 3
Back Explain
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Explain 3.11 Ladler, Ch. 3
CV the amount of income needed to buy y0-y1
units of Y since BC lie on the indifference
curves corresponding to y0 , y1
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Fig.14.4Varian, Ch. 3
Back to 18
Back to 19
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Fig.5.9Nicholson, Ch. 3
Back Explain
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Explain 5.9Nicholson, Ch. 3
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Fig.3.8Laidler, Ch. 3
Back Explain
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Explain 3.8Laidler, Ch. 3

• A initial point
• B 2nd equilibrium
• To find a point (C) that lies on the I.C.C
  that passes through A and brings about the same
  satisfaction as B and is the tangency point of U
  to a budget line with the slope as the first B.C
  (3ll 1)

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Fig.3.11 Laidler, Ch. 3
Back Explain
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Explain 3.11Laidler, Ch. 3
EVY2-Y0 the amount of income needed to buy Y2-Y0
of Y gives the consumer a gain in utility
equivalent to the one obtained moving from A to B.
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Q.4.17Laidler, Ch. 3
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fig.3.5 Quant, Ch. 3
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fig.12.2 Varian, Ch. 3
W15 W215 P .5 gamble
Explain
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fig.3.5 Griffiths Ch.3
Back Explain
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fig.3.6 Griffiths, Ch.3
Back Explain
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Example .8.4nicholson Ch. 3
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problem 8.5Nicholson, Ch. 3
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Problem 3.2Griffiths, Ch.3
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Q.12.2Varian, Ch. 3
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Q.12.3Varian, Ch. 3
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fig.3.1 Griffiths, Ch.3
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fig.3.3 Griffiths, Ch.3
Explain
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Problem .5.10 Nicholson, Ch. 3
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a. Description

• It shows the relation between consumer demand
  expenditure relations.
• It clarifies the minimum subsistence quantity.
• It is useful for empirical works
• minimum subsistence quantity of Q1 ,
  Q2.
• It shows the share of every commodity in the U.F
  the D.F.
• Monotonic transformation
• Back

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b. Demand functions optimization

• F.O.C

if
if
continue
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S.O.C
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Expenditure functions

• Multiply by p1 p2 gives
• They are linear income prices suitable for
  linear regression system.

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Strongly separable utility functions

• Description
• U.F are assumed to be strictly quasi-concave,
  differentiable, increasing .
• A U.F is strongly separable in all of its
  arguments if it can be written as
  
• F fi are increasing
• Back

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Example
back
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Properties

• - Rcs depends only upon the quantities qi qj

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Weakly separable U.F

• If the variables can be partitioned in two (or
  more) groups (q1 ,,qk) and (qk1 ,,qn ) such
  that
• U F f1(q1 ,,qk) f2 (qk1 ,,qn )
• RCS for pairs of variables within the same group
  are unaffected by quantities for variables
  outside the group.
• e.g. U Ln (q1 q2 )2 v q3 q4
• Back

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Description

• A U.F is strongly addictive if it can be written
  as
• U S fi (qi) fi is increasing
• It is a special case of seperability.
• Back

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Example

• U q1a q2ß q3?
• q2ß f2 (q2) , f2 gt 0
• Back

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Properties

• Any U.F that has a monotonic transformation
  which is additive may be treated as being
  additive for all theorems applicable to additive
  function. e.g. Uq1a q2 is separable but not
  additive but F(U)alnq1lnq2 (log-transformation)
  is additive. Also the antilog of U ln(q1a q2ß
  q3? ) is strongly additive.
• U colog lnq1a q2 colog (Ln q1a
  Ln q2 ) F(Sf i(qi)).
• Continue

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• (ii) RCS q1q2 is only a function of q1 , q2.
• all cross partials are zero i.e.
• Regular strict quasi-concavity condition is
  f11f12 f22f12lt 0

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Weakly additive U.F

• The variables can be partitioned in two
• (or more) groups (q1 ,,qk) and (qk1 ,,qn )
  such that
• U F f1(q1,,qk) f2 (qk1 ,,qn )
• Cross partials for pairs of commodities in
  different groups are zero.
• e.g. U ( q1q2 )2 vq3q4

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Description

• A U.F is HDk if
• f (tq1 ,,tqn) tkf (q1,,qn)
• tgt0 , kcte
• Back

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Properties

• The partial derivatives of a function of HD(k)
  are HD(k-1)
• RCS is invariant with respect to proportionate
  change in consumption levels.
• proof
• (iii) If a consumer is indifferent between two
  consumption bundles, he will be indifferent
  between any other two bundles that use the same
  multiple of the first pair.
• back

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Description

• These properties can be exhibited for monotonic
  transformation of homogeneous functions.
• The U.F within this broad class, which includes
  homogeneous functions are called homothetic.

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Properties

• RCS will depend upon relative rather than
  absolute commodity quantities.
• Examination of RCS can indicate whether or not a
  U.F is homothetic.

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Example

• is not homogeneous but is homothetic
• Since

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Description

• It is an indirect approach to calculate
• O.D.C , C.D.C , expenditure function and
  Slutsky equation.
• It is more applicable than direct approach.

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Dual problem shepherd lemma

• According to the envelope theorem
• Applying the envelope theorem to the expenditure
  minimization problem provides the compensated
  demand functions, i.e. it expresses how
  compensated D.F can be derived from Exp.F
• continue

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This result sometimes called Shepherds lemma
after the economist who discovered it in the
context of input demand by firms (1953). back
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Generalization

• E (p1,, pn,U0 ) is the minimum expenditure
  necessary to achieve a given level of utility
  (U0). The partial derivation of expenditure
  function with respect to the ith price gives the
  ith compensated demand function
• Shepherd's lemma
• 
  Continue

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• Since the compensated demand function is
• obtained by minimizing expenditures for a
  given level of U0 , change in total expenditures
  that is due to a small change in a price is zero.
  then
• 
  
  
  back

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Description

• It shows how Marshallian demand functions can be
  derived from the indirect utility function
• Continue

(I.U.F)
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- Applying envelope theorem to the Laggrangian
form
This result is called Roys identity after its
discoverer (Roy 1942) BACK
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Generalization

• Normalizing prices
• (HDO supports this
  modifications)
• Continue
• 1. Ordinary demand function

89
-Applying composite-function rule for
(3)Partial differentiation of (1) with
respect to
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Optimal commodity demand are related to the
derivatives of the I.U.F and the optimal value of
Lagrange multiplier. Or
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Slutsky equation
At the optimal point original and dual solution
are equal. Back
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Example

• Considering the following utility function
• Calculate
• a. O.D.F, I.U.F Roys identity
• b. C.D.F, Expenditure F. Shepherd lemma
• c. Substitution effects
• d. Slutsky equation

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Evaluation

• questions 3.1 , 3.2 , 3.3 , 3.4
• Problems 5.6 , 5.7 , 5.8 Nicholson
• Back

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The principles of revealed preferences

• If (x1 ,x2) is the chosen bundle when prices are
  (p1, p2) , and if (y1, y2) be some other bundle
  such that p1 x1 p2 x2 p1 y1 p2 y2 , then
  if the consumer is choosing the most preferred
  bundle she can afford, we must have
• (x1 ,x2) gt (y1, y2)
• This principle is not circular, since
• i revealed preferences means that X was
  chosen to Y when Y was affordable
• ii Preference means that the consumer ranks
  X ahead of Y.
• iii Preference is the consequence of revealed
  preferred
• ( consumer behavior)
• iv If bundle X is chosen over Y bundle Y, then
  X must be
  preferred to Y.

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Indirect revealed preference

• Definition if (y1, y2 ) is demanded at (q1 ,
  q2) and it is revealed preferred to (z1, z2 ),
  i.e. q1 y1 q2 y2 q1z1 q2 z2 , then we know
  that (x1 , x2 ) gt (y1, y2 ) and (y1, y2 ) gt (z1,
  z2 ), from transitivity assumption (x1 , x2 )
  gt(z1, z2 ) so we say (x1 , x2 ) is indirectly
  preferred to (z1, z2 ) .
• If a bundle is either directly or indirectly
  revealed preferred to another bundle we say that
  the first bundle is revealed preferred to the
  second.
• (x1 , x2 ) is revealed preferred, either directly
  or indirectly to all of the bundle in the shaded
  area it is in fact preferred to those bundle.
• The true indifference curve through (x1 , x2 )
  must lie above the shaded area.
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Description

• As we observe more and more choices, we can get
  a better better estimate of what the consumers
  preferences are like.
• If we add more assumptions about consumer
  preferences, we can get more precise estimate
  about the shape of indifference curves.
• If Y Z are revealed preferred to X, then all
  of the weighted averages of Y Z are preferred
  to X. and if preferences are monotonic, then all
  the bundles that have more of both goods than X,
  Y, Z or any of their weighted averages are also
  preferred to X. Back

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Graph (fig.7.3 Varian)

• All of the bundles in the upper shaded area
  better than (x1,x2) and that all the bundles in
  the lower shaded area are worse than (x1,x2) .
• The true indifference curve through
• must lie somewhere between the two shaded
  areas.

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(No Transcript)
99
1.Revealed preferencea. Introductionb.
Definitionc. Graph (fig .7.1 Varian )d.
Mathematicse. Generalization
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100
Introduction

• 1- The idea of revealed preferences was
  introduced into consumer theory by Paul Samuelson
  in 1938.
• 2- The revealed preferences hypothesis has made
  it possible to establish the law of demand by
  direct observation of consumer behavior without
  having to depend on the rather restrictive
  assumption that we have noted as being necessary
  for the use of indifference curve analysis. Since
  in real life we notice the behavior.
• 3- We will adopt a maintained hypothesis that the
  consumers preferences are stable over the time
  period for which we observe his choice behavior.
• Back

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Definition

• If a consumer chooses bundle of goods A, in
  preference to other bundle B,C and D, which also
  available, then if none of the latter bundles is
  more expensive than A, we can say that A has been
  revealed preferred to the other bundles.
• Back

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Explanation Fig.7.1 Varian

• (x1 , x2) a consumers demanded bundle the
  optimal bundle.
• (y1 , y2) an affordable bundle at the given
  budget.
• (x1 , x2) is better than (y1 , y2 ). This holds
  for any bundle on
• B.C or beneath it ( x1 , x2 ).
• - We assume unique demanded bundle by adopting
  the convention that the preferences are strictly
  convex.
• If not , there will be more than one demanded
  bundle since indifference curves have flat spots.
• The shaded area is revealed worse than the
  demanded bundle.
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Mathematics

• P1y1 p2y2 m
• P1x1 p2x2 m
• P1x1 p2x2 P1y1 p2y2 (x1 , x2 ) is
  chosen over (y1 , y2 )
• (x1, x2) is directly revealed preferred to (y1,
  y2)
• Revealed preference is a relation that holds
  between the bundle that is actually demanded at
  some budget the bundles that could have been
  demanded at that budget.
• It is better to say X is chosen over Y
  instead of saying X is revealed preferred to Y
  since inherently we have not anything to do
  with preferences.
• Back

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Generalization

• Commodities prices
• - If p0 q1 p0 q0 and q0 is chosen then q0 is
  revealed to be
  preferred to q1
• Back

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Definition

• If (x1 , x2 ) is directly revealed preferred to
  (y1 , y2), and the two bundles are not the same,
  then it cannot happen that (y1 , y2) is directly
  preferred to (x1 , x2 ).
• Back

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Explanation Figs.7.5, 7.4 Varian

• Fig.7.5 consumer choices satisfy WARP. It is
  possible to find indifference curves for which
  his behavior is optimal behavior.
• Fig.7.4 violation to WARP if both of (x1 , x2
  ),(y 1,y2)
• are revealed preferences. we know that
• 1- the consumer is not choosing the best bundle
  he can afford.
• 2- there is some other aspect of the choice
  problem that has changed that we have not
  observed e.g. the consumers taste or some other
  aspects of his economic environment have changed.
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Mathematics

• Simple case
• If (x1 , x2 ) is purchased of (f1 , f2) and a
  different bundle (y1 , y2 ) is purchased ot
  prices (q1 , q2 ), then if p1 x1 p2 x2 p1 y1
  p2 y2
• it must not be the case that
• q1 y1 q2 y2 q1 x1 q2 x2
• If y-bundle is affordable when the x-bundle is
  purchased, then when the y-bundle is purchased,
  the x-bundle must not be affordable.
• Back

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Generalization

• If q0 is revealed preferred to q1 , the latter
  must never be revealed to be preferred to q0.
• The only way in which q1 can be revealed to be
  preferred to q0 is to have the consumer purchase
  the combination q1 in some price situation in
  which he could also afford to buy q0.
• i.e. q1 is R.P to q0 if p1 q0 p1 q1 ()
• WARP states that if p0 q1 p0 q0 then () can
  never hold so, if p0 q1 p0 q0 then p1 q0
  p1 q1
• Back

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Checking WARP

• WARP is a condition that must be satisfied by a
  consumer who is always choosing the things he can
  afford.
• Fig(7.1) (p1t , p2t ) the tot observation of
  prices
• and
• (x1t , x2t ) the tth
  observation of choices.
• Fig(7.2) The diagonal terms measures how much
  money the consumer is spending at each choice
  (aij ij )
• a31 How much the consumer would have to spend
  at the third set of prices to purchase the first
  bundle.
• a31 lt a33 bundle one was affordable when
  bundle 3 is R.P to one.
  Continue

110

• We put a star in the entry in row s, column t ,
  if the number is that entry is less than in the
  aij ij .
• a22 How much the consumer actually spent at the
  2nd set of prices to purchase the 2nd bundle.
• A violation to WARP consists of two observations
  t s such that row t , column s , contains a
  star and row s , column t , contains a star.
  Since this would wear that the bundle purchased
  at s is revealed preferred to the bundle
  purchased at t and vice versa.
• a12 a21 contains a star observation 2 could
  have been chosen when the consumer actually chose
  observation one vise versa.
• No stable preferences. Back

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Definition

• If (x1,x2) is revealed preferred to (y1,y2)
  (either directly or indirectly) and (y1,y2) is
  different from (x1, x2) , then (y1, y2) can not
  be directly or indirectly revealed preferred to
  (x1, x2).
• If qo is revealed preferred to q1 which is
  revealed preferred to q2 , , which is revealed
  preferred to qk , qk must never be revealed
  preferred to qo .
• This axioms answer the transitivity of revealed
  preferred.
• Continue

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• SARP is a necessary sufficient condition for
  optimizing behavior. If the observed choices
  satisfy SARP , we can always find nice,
  well-behaved preferences that could have
  generated the observed choices.
• back

113

• F.O.C If choosing the best we can do?SARP
• S.O.C If SARP?we can find well-behaved
  preferred I.C
• Back Graph

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Checking SARP

• Direct revealed preferred
• Indirect revealed preferred
• If there is a situation where there is a star in
  row t , column s , and also a star in row s ,
  column t, then observation t is revealed
  preferred to observation s , either directly or
  indirectly and at the same time , observation s
  is revealed preferred to t violation to SARP
• Continue

115

• If not , observation s are consistent with , eco.
  Theory of consumer.
• a12 10 1 is revealed preferred to 2
  directly
• a23 15 2 is revealed preferred to 3
  directly
• 1 is indirectly revealed preferred to 3
• a1322()
• Back

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Revealed preferences substitution effect

• The consumer is preferred to move along a given
  indifference hyper face in n directions.
• qo,q1 lie on the same indifference hyper face.
• If PPo he purchase qo
• If PP1 he purchase q1
• ? poqo poq1 ? po(qo-q1) 0 or -po(q1-qo)
  0
• p1q1 p1qo ? p1(q1-qo) 0
• Sum (p1-po)(q1-qo) 0 ?
• back

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Evaluation

• Question 3.5
• Problem 5.9 Nicholson
• Questions 7.1 , 7.2 , 7.3 Varian
• back

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A initial equilibriumB 1nd equilibrium if px?
total effect AB a movement along P.C.CB is
also a point or an I.C.CC is another point along
I.C.C where B.C (y2x2) has the same slope and is
tangent to I.ABACCB S.EI.E
Fig.3.3 Laidler Back
The income substitution effects (Review)
119
Money income real income

• O.D.C is derived holding money income Py
  constant constant money income D.C
• Real income is changed through income effect
• If we neglect , we will find a constant real
  income curve compensated D.C
• Back

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The ability to gain a particular constant level
of satisfaction from consumption compensated D.C
is derived.
Back
Constant real income
121
Compensating variation (CV)

• The required change in money income just
  compensates the consumer for income effects and
  keep it on the initial satisfaction level, is
  called compensating variation. i.e. we compensate
  the welfare cost (gain) of the price change.
• Back

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Expenditure function
Minimum expenditure necessary to achieve a
desired level of utility given the prices. Back
123
Shepherds lemma
Back
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C.V
Nich shaded area Back
125
Description

• An alternative way of analyzing the income the
  substitution effect of a price change
• The required change in money income just
  equivalent to the income effect that keep the 2nd
  utility level is called equivalent variation. It
  is equal to the AC.
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Mathematics
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Example

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Introduction

• I.E measures the change in satisfaction or
  utility as a result of a change in the price of a
  good.
• If ?U the gained utility because of an
  equivalent variation in money income E.V
• An alternative measure of the same gain is the
  amount by which money income may be diminished in
  order to leave the consumer just as well off as
  initially (C.V).
• The changes in utility we are discussing is
  changes in C.S
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Definition

• A consumer normally pays less for a commodity
  than the maximum amount that he would pay rather
  than forgo its consumption.
• The difference between the actual payment of the
  consumer and the potential willingness of him to
  pay is called C.S.
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Measurement

• (fig 4.1Laidler) (fig 3.3 , 3.4) , (fig 14.2 ,
  14.3 Varian )
• C.V method
• E.V method
• Using marshallian D.
• Comparison

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I. C.V method

• If we prohibit the consumption of X , how much
  would we have to increase the consumers income
  in order to compensate for this prohibition ?
• 3.3 a 1st E D , I2 , OA M
• If Q0 ? D?A , I2?I1 , CVABcC.S
• 4.1 1st E A , I2 , oy1M
• If X0 ?A?B , I2?I1
• C.V BC y1-y0 the amount of income needed to
  buy y1-y0 of y CS
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II. E.V method

• Offering the consumer a choice between being
  forbidden to consume X or accepting a reduction
  in income. How large a reduction in income would
  leave the consumer just as badly off as the
  prohibition on the consumption of X.
• The amount necessary to buy Y0-Y2 is
  equivalent to this reduction in utility.
  
  
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III. Using marshallian D
IV. Comparison (fig 3.3.b)
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Mathematics a comparative approach
The change in C.S is a rough average of the
equivalent compensating variation.
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Example
5.5 Nicholson H.Q (back) D(p) 20 2p
?CS? P12 , P23
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Example
Additional money needed to keep the consumer on
the same u

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• Continue

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a. Background

• The additional theory of consumer behavior does
  not include an analysis of uncertain situations.
• Von Neumann Morgenstain showed that under some
  circumstances it is possible to construct a set
  of numbers for a particular consumer that can be
  used to predict his choices in uncertain
  situations.
• Great controversy has centered around the
  question of whether the resulting utility index
  is ordinal or cardinal

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• It will be shown that Von Neumann Morgenstain
  utilities possess at least some cardinal
  properties.
• The consumer faces with a choice between two
  alternatives
• A choice whit certain outcome , (p1)
• He can obtain a lottery ticket with a chance of
  winning a satisfactory situation (A) or an
  unsatisfactory one (B)
• L(P,A,B) A?B B?1-p

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b. Axioms for consumer behavior in uncertainty
situations

• It is possible to construct a utility index can
  be used to predict choice in uncertain situations
  if the consumer conforms to the following five
  axioms.

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1. Complete ordering axiom

• APB or BPA or AIT completeness
• If APB BPC ?APC transitivity

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2.continuity axiom

• If APB BPC , then there exists 0ltplt1 , such
  that the consumer is indifferent between B with
  certainty L(P,A,C)

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3. Independence axiom

• If AIB , C any outcome , L1(P,A,C) , L2(P,B,C)
  ?L1 I L2
• If APB , C any outcome , L1(P,A,C) , L2(P,B,C)
  ?L1 P L2

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4. Unequal probability axiom
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5. Compound-lottery axiom

• If L1(P1,A,B) , L2(P2,L3,L4) L3(P3,A,B) ,
  L4(P4,A,B) L2 is a compound lottery in which the
  prices are lottery tickets then L2 is equivalent
  to L1 if P1 P2P3(1-P2)P4
• P2P3 the probability of obtaining A through L3
• (1-P2)P4 the probability of obtaining A through
  L4
• P2P3(1-P2)P4the probability of obtaining A
  through L2

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c. Expected value

• E(x)P1X1 P2 X2
• Expected value in uncertain situation is a
  weighted average of the values associated with
  each possible outcome , Pi are the weights.

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Description

• If the consumer has reasonable preferences about
  consumption in difference circumstances we will
  be able to use a utility function to describe
  these preferences.
• How a person values consumption in one state as
  compared to another will depend on the
  probability that the state in question will
  actually occur. The preferences for

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• consumption in different states of nature will
  depend on the beliefs of the individual about how
  likely those states are.
• In the states are mutually exclusive (rain
  shine) u(C1, C2, ?1,1- ?1) if not u(C1, C2, ?1,
  ?2). This is a function that represent the
  individuals preferences over consumption in each
  state.

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Examples

• Perfect substitutes u(C1, C2, ?1, ?2) ?1C1 ?2
  C2
• Expected value of C1 , C2 -weighted average.
• Cobb-Douglas U.F
• Monotonic transformation
• Ln u(C1, C2, ?1, ?2) ?1Ln C1 ?2LnC2 represents
• the same preference.

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Description

• If there is a utility index which conforms to the
  five axioms, the expected utility for the
  tow-outcome lottery L(P,A,B) is
  Eu(L)Pu(A)(1-P)U(B)
• Varian One convenient form of U.F might be

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• Utility is a weighted sum of some function of
  consumption in each state
• Perfect substitutes or expected value U.F V(c)
  C
• The Cobb-Douglas in logarithm V(c) LnC
• It shows the average utility or the expected
  utility of the pattern of consumption (C1, C2).
• It is called expected utility of the pattern or
  Von-Neumann-Morgenstain U.F
• The U.F has the additional form

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Properties (theorems)

• Expected utility Ranking corresponds to lottery
  rankings.
• Expected utility monotonic transformation.
• Expected utility Increasing linear
  transformation (positive affine transformation)
• Expected utility and utility numbers
• Cardinal nature of expected utility

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Expected utility ranking corresponds to lottery
rankings.

• If L1(P1,A1,A2) is preferred to L2(P2,A3,A4)
  then EU(L1)gtEU(L2)
• Proof
• We select outcomes B best ,W worst under
  consideration
• Continuality axioms there exist a probability
  Qi such that Ai is indifferent to (Qi, B, W)
  (i1, , 4)

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Compound-lottery axiom
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Unequal probability axiom
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Expected utility monotonic transformation.

• For certain outcomes any positive monotonic
  transformation of the utility functions leaves
  the ranking of outcomes unchanged.
• H.Q - for uncertain outcomes this result doesnt
  hold.
• U(A1)25 U(A2)64 U(A3)36 U(A4)49
• L1(0.5,A1,A2) gt L2(0.4,A3,A4) since
  EU(L1)44.5gt43.8EU(L2)
• But if VU.5 EV(L1)6.5lt6.6EV(L2)

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• Varian
• Any monotonic transformation of an expected
  utility function is a utility function that
  describes the same preferences but the additive
  form representation turns out to be especially
  convenient.
• E.g. If the consumers preferences are
  described by . But the latter doesnt have the
  expected utility property, while the former dose.

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Expected utility Increasing linear
transformation (positive affine transformation)

• The expected utility function can be subjected to
  same kinds of monotonic transformation and still
  have the expected U. property.
• V(U)a(U)b , agt0
• It is not only represents the same preferences
  (this is obvious since and affine transformation
  is just a special kind of monotonic
  transformation) but it also still has the
  expected utility property.

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• Proof

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Expected utility and utility numbers

• The E.u formula may be used to construct utility
  numbers for a person who conforms to the Von
  Neumann Morgenstern axioms.
• A2 gtA1 , U(A1)20 , U(A2)1000 A3 is between A1 ,
  A2 Arbitrary assign utility numbers to two
  certain incomes.
• According to continuity axiom there is a
  probability P such that (P, A1, A2) A3

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• If P .8 ? U (A3) .8 U (A1) 0.2U (A2)
  216
• If A4 gtA2gtA3gtA1 then ask the consumer for a value
  of P such that he is indifferent between A2 and
  (P,A1,A4). If P .6 then
• U(A2) .6 U(A1) .4 U(A4)
• 1000 .6 (20) .4 U(A4) ? U(A4) 2470
• The process can be continued indefinitely, and
  will not lead to contradictory results as long as
  the five axioms are obeyed.

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5. Cardinal nature of expected utility

• The utilities in the Von-Neumann-Morgenstern
  analysis are cardinal in a restricted sense.
  Since
• (i) They are derived by presenting him with
  mutually exclusive choices it is meaningless to
  infer from the utility of event A and the utility
  of event B, the utility of the joint extent A and
  B.
• (ii) If U (A) KU (B) it is not meaningful to
  assert that the consumer prefers A, k times as
  much as B.

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• (iii) Utility ratios are not invariant under
  linear transformation.
• But the utility numbers provide an interval
  scale, and differences between them are
  meaningful (since the relative magnitudes of
  differences between utility numbers are invariant
  with respect to linear transformation

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d. Why expected utility is reasonable?

• Why would we think that preferences over
  uncertain choices would have the particular
  structure implied by the expected utility
  function?
• The fact that outcomes of the random choice are
  consumption goods that will be consumed in
  different circumstances means that ultimately
  only one of these outcomes is actually going to
  occur.
• E.g. either it will be a rainy day or a sunny
  day.

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• Thus in choice under uncertainty there is a
  natural kind of independence between the
  different outcomes because they must be consumed
  separately in different states of nature.
• This assumption is known as independence
  assumption. It implies that utility function for
  contingent consumption will take a very special
  structure it has to be additive across the
  different contingent consumption bundles. i.e.

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• If C1, C2, C3 consumptions in different
  states of nature and p1,p2,p3 are probabilities
  that these 3 different states of nature
  materialize, then U(C1, C2, C3) p1U(C1)
  p2U(C2) p3U(C3) (expected U.F)
• (independent of other goods)

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1. Specified expected utility function

• We claimed that the expected utility function in
  general had some very convenient properties for
  analyzing choice under uncertainty.
• It is now assumed that the utility function
• 1. has the single argument wealth
  measured in monetary units
• 2. is strictly increasing
• 3. is continuous with continuous first
  second- order derivatives.

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2. Risk uncertainty

• Although we shall often use the ideas of
  uncertainty risk interchangeably, they strictly
  refer to different situations.
• Uncertainty refers to situations in which many
  outcomes of a particular choice are possible but
  the likelihood (probability) of each outcome is
  unknown.
• Risk It can only be measured accurately on the
  assumption that we know all the possible outcomes
  and the probability of each outcome occurring.

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1. Definition

• He prefers a situation in which a given income
  which is certain to a situation yielding the same
  expected value but which involves uncertainty.

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2. Mathematics

• L (P, W1, W2) ? EWPW1 (1-P)W2
• The utility of its expected value is greater than
  the expected value of its utility (expected
  utility of the gamble)
• For all 0ltPlt1
• U PW1 (1-P)W2gtPU(W1) (1-P)U(W2)
• or UE(W)gtEU(W)
• The utility function is strictly concave over its
  domain since (1) is identical to strict
  concavity.
• If (d2U/dw2)lt0 U.F is strictly concave he is
  risk averter (Diminishing marginal utility
  exists).

(1)
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3. Graph (fig.3.5), (fig.3.1 Griffiths)


(fig.12.2 Varian)

• UE(W) UPW1(1-P)W2 U2.5 7.5 U(10) gt
  .5 U(5) .5 U(15) PU(W1) (1-P) U(W2)
  EU(W)
• He currently has 10 of wealth certainly.
• The expected value of the gamble 10
  E(W) 2.5 7.5 10

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173
1. Definition

• He is indifferent between certain and uncertain
  outcomes with the same expected value.
• Constant MU along the relevant segment of the
  total Uncured.

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2. Mathematics

• The utility of the expected value of the game
  equals the expected utility of the game.

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1. Definition

• He would prefer a random distribution of wealth
  to its expected value in the certain situation.

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2. Mathematics
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1. Definition

• It measures the amount of income that an
  individual would give up to leave him indifferent
  between a risky choice and a certain one.

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2. Mathematics

• The sign of provides and indication of the
  consumers attitude, but since it is invariant
  under a linear transformation gt Not suitable.
• Absolute risk aversion
• Linear transform

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2. Mathematics

• Relative risk aversion
• The willingness to pay to avoid a given gamble
  depends on the individuals level of wealth.
• It might be approximately constant.

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3. Graph (fig.3.3 Griffiths)

• D reflects the expected value of the risky
  situation (A of B P -5)
• DE risk premium in that a certain income of 15
  gives the same
• Expected utility (U4) as the uncertain income of
  2o.
• The consumer is willing to give up 5 of expected
  income from the
• Risky choice to be indifferent between the
  certain uncertain outcome.
• The more risk averse, the greater R.P

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181
4. Examples

• (i) quadratic U.F

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4. Examples

• (ii) Constant risk aversion U.F

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a. Description

• The essence of insurance is that in return for a
  premium paid at the start of the period, some
  agent guarantees to reimburse the decision maker
  for any loss incurred during the period. i.e. by
  paying a fee the decision maker can put him in a
  position of certainty.
• The trade off is as follows by reducing the
  level of income (wealth) with which he or she
  begins the period by the around of this fee
  (insurance premium) he can guarantee that he will
  end the period with the initial level of income
  mines the premium.

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b. Graph (fig.3.5 3.6 Griffiths)

• No insurance U(0.5, A, B) U(0.5,W1,W2)
  0.5U(W1)0.5U(W2)
• U(0.5W10.5W2)U3gt 0.5U(W1)0.5U(W2)U4
• U(W3)U3gtU(D)U4
• If insurance U(W3)U3
• CD V3 V4 Consumer surplus that arises from
  certainty from buying insurance that would make
  the wealth outcome W3 certain at the end of the
  period.
• W4 Certainty equivalent level of wealth to W3,
  i.e. that wealth which if received with certainty
  , yields the same utility (U4) as the fifty-fifty
  gamble.

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• As long as the insurance premium is no more than
  the difference W3 W4 (the risk premium) then
  the decision maker will achieve higher utility by
  insurance then by non-insurance, i.e. will be
  above E on the total utility curve.
• ED Maximum insurance premium that would be paid
  by the decision maker given the degree of risk
  aversion implied by the shape of the total
  utility curve.
• So long as max I.P gt minimum insurance premium
  required by the insuring agent, there is
  insurance market.
• Risk Neautral (3.6.a) Risk Premium 0 gt No
  insurance market
• Risk lover (3.6.b) Risk Premium lt 0 gt No
  insurance market

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c. Mathematics

• 1.Introducing the model
• Consumer faces a risk that will suffer A
  dollars with P likelihood.
• Equivalent to a gamble of (P, W0 A, W0).
  Where W0 denotes initial wealth.
• Insurance payment (Premium) R they will give
  him a dollars if fire takes place.
• The maximum insurance premium can be obtained by
  solving
• U(W0-R) U (P, W0 A, W0)
• U(W0-R) PU(W0 A) (1-P)U(W0)

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c. Mathematics

• Risk averse
• E(w0-A)P(W0-A)(1-P)W0W0- PAE(A)
• PA(1-P)(0)
• The expected value of loss PA
• If R gt PA, he buys insurance if its price is no
  greater than R if the price is greater than R
  gt No insurance buying despite his risk aversion
• Profitability of insurance companies
  gt price gt PA
• In a perfect market all risk lovers, all risk
  neutrals some risk averters will not buy
  insurance.

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c. Mathematics

• 2.Insurance premium
• Fair bet expected value of the prize is zero
• ( E(h) 0)
• P Size of insurance premium fair bet (h)
  paying P with certainty to avoid gamble
• EU(wh) U(w-p)
• Expanding by Taylors series

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2.Insurance premium
The amount that a risk averse is willing to pay
to avoid a fair bet is proportional to Pratts
risk aversion measure.
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Fig.4.8 Laidler, ch.3
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Fig.5.10 Nicholson, ch.3
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The End Of Chapter III