Micror 02 10 Notes

8/6/2019 Micror 02 10 Notes

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2. Classical Demand and Production Theory

Klaus M. Schmidt

LMU Munich

Micro (Research), Winter 2010/11

Klaus M. Schmidt (LMU Munich) 2. Classical Demand and Production Theory Micro (Research), Winter 2010/11 1 / 92

Introduction

Microeconomics (and a large part of modern macroeconomics as well) tries toexplain economic phenomena as the outcome of individual decision making.The starting point of the analysis is always individual behavior:

What are the objectives and constraints of individual behavior?

What should a rational individual do in a given situation?

What are optimal decisions under uncertainty?

Do individuals behave fully rationally?How does the interaction between individuals in markets andorganizations shape economic outcomes?

What are the welfare properties of an economic allocation (based on thewelfare of all concerned individuals)?

The theory of individual decision making is of crucial importance for almost allsubfields in economics.

c 2010 Klaus M. Schmidt


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In this course we analyze individual decision making and the interaction ofeconomic decisions in the following contexts:

1. Classical Demand and Production Theory: individual decisions undercertainty

2. Choice under Uncertainty: individual decisions in a risky environment

3. General Equilibrium Theory: interaction of individual decisions on

competitive markets

4. Game Theory: strategic interaction of individual decision makers

5. Incentive Theory: design of optimal incentive schemes for individualdecision makers who possess private information


Literature

The chapter on classical demand and production theory is based onMas-Colell, Whinston, Green (MWG):

Chapter 1, in particular 1.A and 1.B

Chapter 2, in particular 2.A - 2.E

Chapter 3, in particular 3.A - 3.G and 3.I

Chapter 4, in particular 4.A - 4.BChapter 5, in particular 5.A - 5.D

This is required reading!

The lecture cannot cover all of this material, and it is essential that you read itin addition to the lecture notes!It may also be useful to consult one or two other text books that cover thesame material, e.g. Deaton and Muellbauer (1980), Jehle and Reny (2010),Varian (1992), Kreps (1990), Gravelle and Rees (2004).


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2.1 Classical Demand Theory: The Framework

The decision problem faced by the consumer in a market economy is howmuch to consume of the various goods and services (= commodities). Weassume that there are L commodities, indexed by l = 1, . . . L in thecommodity.

Note that two goods with identical physical characteristics that differ in thetime or location of their availability should be treated as to differentcommodities.

The consumer chooses a consumption bundle (= commodity vector)

x =

x 1x 2...

x L

∈ X ⊂ RL

+

where X is the set the consumer can choose from (his budget set). Note thatx is a column vector and that x l ≥ 0. Why?


Preference Relations

A preference relation is a binary relation with the following interpretation:

x y if and only if the consumer weakly prefers consumption bundle x to consumption bundle y, i.e. if and only if the consumer considers x to be at least as good as y.

We can use the preference relation to define

the indifference relation ∼x ∼ y ⇔ x y and y x

the strict preference relation

x y ⇔ x y but not y x


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Rationality

Definition 2.1 (Rationality)

The preference relation is called rational if and only if it possesses the following two properties:

(i) Completeness: for all x , y

∈X, we have that x

y or y

x (or both)

(ii) Transitivity: for all x , y , z ∈ X, if x y and y z, then x z.

How strong are these assumptions?

What do they imply for and ∼?


Utility Functions

It is difficult to work directly with preference relations. Therefore, it is veryuseful if can be represented by a utility function.

Definition 2.2 (Utility Function)

A function u : X → R represents the preference relation if and only if, for all x , y ∈ X ,

x y ⇔ u (x ) ≥ u (y )

Note:

A utility function is a purely ordinal concept. It only gives a ranking of thedifferent alternatives, but the numbers of the utility function have nocardinal properties.

The utility function is not unique. For any strictly increasing functionf : R→ R, v (x ) = f (u (x )) is a new utility function that represents thesame preference relation as u (x ). We say: “A utility function is unique upto a positive, monotone transformation.”

Interpersonal comparisons of utility have no meaning.


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Proof of Proposition 2.1:


Commonly Assumed Properties of Preferences

Notation:

y ≥ x means that y l ≥ x l for all l ∈ {1, . . . , L}.

y > x means that y l ≥ x l for all l ∈ {1, . . . , L} and y k > x k for at least one

k ∈ {1, . . . , L}.y x means that y l > x l for all l ∈ {1, . . . , L} y − x =

Ll =1(y l − x l )2


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Definition 2.3 (Monotonicity)

A preference relation on X is

strongly monotone iff x , y ∈ X and y > x implies y x.

monotone iff x , y ∈ X and y x implies y x.

locally non-satiated iff for every x ∈ X and every > 0, ∃ y ∈ X such that y − x ≤ and y x.

Interpretation and graphical illustration of monotonicity and non-satiation.

Show:

Strong monotonicity implies monotonicity.

Monotonicity implies local non-satiation.

Non-satiation implies that indifference curves cannot be “thick”.


Definition 2.4 (Convexity)

The preference relation is (strictly) convex iff for every x ∈ X the upper contour set {y ∈ X | y x } is (strictly) convex; i.e., if y x and z x, then αy + (1 − α)z ()x for any α ∈ (0, 1).

Interpretation and graphical illustration of convexity.

Proposition 2.2 (Quasi-Concavity)

If the preference relation is convex then every utility function representing is quasi-concave.

Proof: Follows directly from the definition of quasi-concavity.


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Remarks:

Concavity implies Quasi-concavity but the reverse is not true.

Quasi-concavity is an ordinal property that survives any positive,monotone transformation.

Example: f (x ) =√

x is quasi-concave and concave. The positive

monotone transformation g (f (x )) = (f (x ))4 = √x 4

= x 2 is convex but stillquasiconcave.


Two Examples:

1. Homothetic Preferences:A preference relation is homothetic iff x ∼ y implies αx ∼ αy for any α > 0.Note that for homothetic preferences the wealth expansion paths are straightlines starting from the origin.

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FIG. 2.1: Homothetic Preferences


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2. Quasilinear Preferences

A preference relation is quasilinear with respect to commodity 1 iff x ∼ y implies (x + αe 1) ∼ (y + αe 1) for e 1 = (1, 0, . . . , 0) and any α > 0.Note that with quasilinear preferences the indifference sets are paralleldisplacements of each other along the axis of commodity 1.

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x2

FIG. 2.2: Quasilinear Preferences


Definition 2.5

The preference relation is continuous iff for any sequence of pairs {x n , y n }∞n =0 with x n y n for all n, x = limn →∞ x n and y = limn →∞ y n , we have x y.

Remarks:

An alternative and equivalent definition of continuity requires that theupper and lower contour sets of

must be closed (contain their

boundaries).

Continuity implies that indifference curves cannot have jumps. Show thisusing the definition of continuity.

Consider the lexicographic preference relation in the two goods case, i.e.,X = R

2+ and x y if and only if x 1 > y 1 or x 1 = y 1 and x 2 ≥ y 2. Are these

preferences continuous?


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Proposition 2.3

Suppose that X = RL+. If is rational, monotone and continuous. Then there

is a utility function u (x ) that represents .

Proof sketch: We give a graphical sketch of the proof. The proof isconstructive, i.e., it shows how a utility function that represents can be

constructed if is rational, monotone and continuous.


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FIG. 2.2: Construction of a Utility Function


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Remarks:

1. Recall that a utility function is unique only up to a positive, monotonetransformation.

2. A much stronger proposition holds but is more difficult to prove: We donot have to require that X = R

L+ nor that is monotone. Furthermore, it

can be shown that there is a continuous utility function that represents .However, not all utility functions that represent have to be continuous.Why not?

3. It is often convenient to work with a differentiable utility function.

However, there are continuous preferences that cannot be representedby a differentiable utility function. Example?

Thus, in order to guarantee the existence of a differentiable utility functionwe need an additional assumption on the smoothness of preferences.

4. If preferences are (strictly) convex , then any utility function thatrepresents these preferences is (strictly) quasiconcave . This followsimmediately from the definition of quasiconcavity.

5. However, convex preferences do not imply that the utility function isconcave! Why not?


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2.2 Utility Maximization

In the following we will assume that the consumer’s preferences satisfy theconditions of Proposition 3.1 and that his utility function is continuous and atleast twice continuously differentiable.Given the consumer’s utility function his decision problem can now beexpressed as a utility maximization problem (UMP):

maxx ≥0

u (x )

s.t. p · x ≤ w

Note that p = (p 1, . . . , p L) is a row vector.

Proposition 2.4

If p 0 and u (·) is continuous, then the utility maximization problem has a solution.


Proof of Proposition 2.4


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Demand Correspondence

Let x (p , w ) be the solution to UMP. This is called the Walrasian (sometimesalso Marshallian) demand correspondence. If x (p , w ) is unique for all(p , w ), then it is called the demand function.

Proposition 2.5

The demand correspondence x (p , w ) satisfies the following properties:

(a) Homogeneity of degree 0 in (p , w )

(b) Walras’ Law

(c) If is strictly convex (so that u (·) is strictly quasiconcave), then x (p , w ) is unique for all (p , w ).

It can also be shown that x (p , w ) is continuous (upper hemicontinuous ifx (p , w ) is a correspondence). Furthermore, if x (p , w ) is a demand function,then, under a mild additional assumption, it is also differentiable.




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How to solve the UMP:

By Walras’ Law, we know that the constraint p · x ≤ w must hold with equalityin the optimal solution. Thus, we can use Lagrange’s method (withnon-negativity constraints) to find the optimal consumption bundle.

Note:

1. The Lagrange function is

L= u (x )

−λ[p

·x

−w ]

2. The Lagrange Theorem says that if x ∗ ∈ x (p , w ), then there exists aLagrange multiplier λ ≥ 0 such that for all l ∈ {1, . . . , L}

∂ L∂ x l

=∂ u (x ∗)

∂ x l − λp l ≤ 0

with equality if x ∗l > 0. This condition is necessary but not sufficient for anoptimal solution.


3. The UMP has a solution. The solution must satisfy Lagrange’s condition.Hence, if there is only one (x ∗, λ) satisfying Lagrange’s condition, then itmust be the solution.

4. Interpretation of λ?

5. Interpretation of

[u (x ∗)− λp ] · x ∗ = 0 ?

Notation: u (x ∗) = ( ∂ u ∂ x 1

, . . . , ∂ u ∂ x L

).


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The Indirect Utility Function

Let v (p , w ) = u (x ∗) for any x ∗ ∈ x (p , w ). Thus, v (p , w ) is the highest level ofutility that the consumer can achieve given (p , w ). This is called the indirectutility function.

Proposition 2.6

The indirect utility function v (p , w ) satisfies the following properties: (a) Homogeneity of degree 0 in (p , w )

(b) Strictly increasing in w

(c) Nonincreasing in p l for any l



:


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Proposition 2.7 (Roy’s Identity)

Suppose that v (p , w ) is differentiable at (p , w ). Then for every l = 1, . . . , L:

x l (p , w ) = −∂ v (p ,w )

∂ p l

∂ v (p ,w )∂ w

.




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Remarks:

1. The proof is an envelope theorem argument, but we did not use theenvelope theorem explicitly.

2. To interprete the result, it is useful to write

∂ v (p , m )

∂ p l = −x l (p , w )

∂ v (p , w )

∂ w

Suppose that price p l is reduced marginally by ∆p l . This gives additional“income” ∆p l · x l (p , w ) to the consumer. The marginal utility of a Dollar is

the same for all goods and equal to ∂ v (p ,w )∂ w

. Hence, the price reduction

increases the consumer’s utility by ∆p l x l (p , w )∂ v (p ,w )∂ w

.

3. Roys Identy it very useful. If we know the indirect utility function, then it ismuch easier to derive the demand function from indirect utility than fromthe direct utility function.


2.3 The Dual Problem: Expenditure Minimization

Instead of maximizing the utility level for a given budget constraint, theconsumer could also minimize his expenditures subject to the constraint thathe achieves at least a given level of utility u :

minx

p · x

s.t. u (x ) ≥ u

The Expenditure Minimization Problem (EMP) is the dual problem toUMP: It reverses the roles of the objective function and the constraint.However, the EMP also characterizes the efficient use of resources by theconsumer. In fact, EMP and UMP are basically equivalent:


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Proposition 2.8

Suppose that p 0.

(a) If x ∗ is optimal in UMP when wealth is w > 0, then x ∗ is optimal in EMP when the required utility level is u (x ∗).

(b) If x ∗

is optimal in EMP when the required utility level is u, then x ∗

is optimal in UMP when wealth is w = p · x ∗.




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The Hicksian Demand Correspondence

The solution to EMP is denoted by h (p , u ) and is called the Hicksian (orcompensated) demand correspondence, or function, if h (p , u ) issingle-valued.Illustrate h (p , u ) graphically.

Proposition 2.9

For any p

0 the Hicksian demand correspondence h (p , u ) has the following

properties:

(a) Homogeneity of degree 0 in p.

(b) No excess utility: For any x ∈ h (p , u ), u (x ) = u.

(c) If is strictly convex (so that u (x ) is strictly quasiconcave) then there is a unique solution to EMP for all (p , u ).

Proof: Analogous to the proof of Proposition 2.6.


An important property of the Hicksian demand function is that it satisfies thecompensated law of demand.

Proposition 2.10

Suppose that h (p , u ) is single valued for all p 0. Then the Hicksian demand function satisfies the compensated law of demand, i.e., for all p and p

(p − p ) · [h (p , u )− h (p , u )] ≤ 0 .

Remarks:

The proposition says that price changes and compensated demandchanges go in opposite directions.

In particular, if only one price goes up, the compensated demand for thisgood goes down.

Note that the proposition refers to compensated demand changes, i.e.,the consumer is compensated for the price change such that he can stillafford his old level of utility.


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:


Let e (p , u ) = p · x ∗, where x ∗ ∈ h (p , u ), be the expenditure function of theconsumer.

Proposition 2.11

The expenditure function e (p , u ) is

(a) homogenous of degree 1 in p,

(b) strictly increasing in u,(c) nondecreasing in p l for any l,

(d) concave in p.

Intuition for these results? Explain (d) graphically.


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(a) to (c) are straightforward and left as an exercise. To prove (d), considertwo price vectors p and p . Let α ∈ [0, 1] and p = αp + (1− α)p . We haveto show:

e (p , u ) ≥ αe (p , u ) + (1− α)e (p , u )

Let x , x and x be the solutions to EMP at prices p , p , and p respectively. Itmust be the case that:

Ll =1

p l x l = e (p , u )

Ll =1

p l x l = e (p , u )

Ll =1

p l x l = e (p , u )



Since x and x minimize expenditures at prices p and p , we have:

Ll =1

p l x l ≥L

l =1

p l x l

L

l =1

p

l

x l ≥

L

l =1

p

l

x

l

Multiplying the first inequality with α and the second one with (1 − α) andadding up both inequalities yields:

Ll =1

[αp l x l + (1− α)p l x l ] ≥L

l =1

αp l x l +L

l =1

(1− α)p l x l


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Hence:

e (p , u ) =L

l =1

p l x l

≥L

l =1αp l x l +

Ll =1

(1− α)p l x l

= α · e (p , u ) + (1− α)e (p , u )

Q.E.D.


The expenditure function can easily be derived from the Hicksian demandfunction by e (p , u ) = p · h (p , u ). However, the following result shows that wecan also derive the Hicksian demand function from the expenditure function:

Proposition 2.12 (Shephard’s Lemma)

Suppose that h (p , u ) 0 is single valued and differentiable. Then

h l (p , u ) =∂ e (p , u )

∂ p l

for all l = 1, . . . , L.


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Remarks:

1. The proof is an envelope theorem argument, but we did not use theenvelope theorem explicitly.

2. Interpretation: In finite approximation Shephard’s Lemma says:

∆e (p , u ) = h l (p , u ) · ∆p l

If the price for good l increases by ∆p l , then the consumer has to pay∆p l h l (p , u ) in addition, in order to buy the same consumption bundle thathe consumed before the price change. If ∆p l is very small, then this isalso the amount necessary to get back to the old utility level u . Thereason is that for small price changes and starting from an optimallychosen consumption bundle the substitution effects can be ignored.

3. Shephard’s Lemma is very useful. If we know the expenditure function, itis much simpler to derive the Hicksian demand function via e (p , u ) thanto derive it from the direct utility function.


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Shephard’s Lemma has several important implications that are summarized inthe next proposition.

Proposition 2.13

Suppose that h (p , u ) is single valued and continuously differentiable at (p , u ),and denote its L × L derivative matrix by D p h (p , u ). Then

(a) D p h (p , u ) = D 2p e (p , u ),

(b) D p h (p , u ) is a negative semidefinite matrix,

(c) D p h (p , u ) is a symmetric matrix,

(d) D p h (p , u )p = 0.




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Remarks:

1. Negative semidefiniteness of D p h (p , u ) is again the compensated law of

demand. In particular, it implies that ∂ h l (p ,u )∂ p l

≤ 0, i.e., the own substitution

effect is non-positive.

2. Symmetry of D p h (p , u ) requires that

∂ h l (p , u )

∂ p k

=∂ h k (p , u )

∂ p l

This is not obvious and it is difficult to give an intuitive explanation for thisunexpected result.

3. If∂ h l (p ,u )

∂ p k ≥ 0 then goods l and k are substitutes. If

∂ h l (p ,u )∂ p k

< 0, then they

are called complements. Property (d) implies that for each good there

exists at least one substitute. This follows immediately from∂ h l (p ,u )

∂ p l ≤ 0.


The Hicksian demand function is a very useful concept. It is particularlyimportant when it comes to the evaluation of welfare changes. But, theHicksian demand function is not directly observable. However, the followingproposition shows that the Hicksian demand function h (p , u ) can be derivedfrom the observable demand function x (p , w ).

Proposition 2.14 (Slutsky Equation)

Suppose that h (p , u ) and x (p , w ) are single valued and differentiable. Then for all (p , w ) and u = v (p , w ) we have

∂ h l (p , u )

∂ p k

=∂ x l (p , w )

∂ p k

+∂ x l (p , w )

∂ w · x k (p , w ) .


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Remarks:

1. The Slutsky Equation shows how the properties of the unobservableHicksian demand function translate to the observable demand function.

2. In particular, the Slutsky Equation implies

∂ h l (p , u )

∂ p l =

∂ x l (p , w )

∂ p l +

∂ x l (p , w )

∂ w · x l (p , w ) .

3. If good l is a normal good (∂ x l /∂ w > 0), then an increase in p l reduces

the Hicksian demand by less than the (Walrasian) demand. In the usualdemand diagram (with p on the vertical axis), the Hicksian demandfunction is steeper than the (Walrasian) demand function.

4. If good l is an inferior good (∂ x l /∂ w < 0), then an increase in p l reducesthe Hicksian demand by more than the (Walrasian) demand. In the usualdemand diagram (with p on the vertical axis), the Hicksian demandfunction is less steep than the (Walrasian) demand function. A good canbe a Giffen good only if it is inferior.


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p

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p

FIG. 2.3: Hicksian and Walrasian Demand Functionsfor Normal and Inferior Goods


2.4 Putting it all together

Because UMP and EMP are basically equivalent, we have:

1. x l (p , w ) = h l (p , v (p , w ))

The Walrasian demand at wealth w is equal to the Hicksian demand if theconsumer wants to achieve at least the utility v (p , w ).

2. h l (p , u ) = x l (p , e (p , u ))

The Hicksian demand at utility level u is equal to the Walrasian demand if

the consumer’s wealth is equal to e (p , u ).3. v (p , e (p , u )) = u

The indirect utility function is strictly increasing in w . Thus we can invertv (p , ·) which is simply the expenditure function.

4. e (p , v (p , w )) = w

The expenditure function is strictly increasing in u . Thus we can inverte (p , ·) which is simply the indirect utility function.


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UMP EMP

v (p , w )

x (p , w )

e (p , u )

h (p , u )

FIG . 2.4: Putting it All Together


We have shown that the preference based approach to consumption theoryhas the following implications for the (Walrasian) demand function x (p , w ):

1. Homogeneity of degree 0

2. Walras’ Law

3. Compensated Law of Demand (Slutsky matrix is negativesemi-definite)

4. Symmetry of the Slutsky matrix

These restrictions on Walrasian demand function are rather weak and do notimpose much structure on observable demand.


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A natural question is, whether there are any other properties of the demandfunction that are implied by the preference based approach.

The answer is no. It can be shown that for any demand function that satisfies(1) to (4) there exists utility function (representing a rational preferencerelation) such that the demand function is generated by this utility function.

This is known as the Integrability Problem (See MWH, Chapter 3.H). It alsoshows, how the preferences of the consumer can (almost, but not quite) berecovered from his observed demand behavior.


2.5 Welfare Evaluations of Economic Changes

How can we evaluate the welfare effect of an economic change on anindividual. For simplicity we will focus on a price change from p to p , butmany other economic changes could be considered as well.

An obvious measure of the welfare change involved in moving from p 0 to p 1 is just the difference in indirect utility for the consumer under consideration:

v (p 1, w )− v (p 0, w )

If this difference is positive the consumer is better off with the new prices p 1.

What is the problem of this approach?


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If we want to measure the welfare effect in monetary terms there are twopossible approaches:

First Approach: Equivalent Variation

Suppose the government considers a new policy that leads to lower prices.We could ask the consumer, how much money he has to get in the situationwith the old prices, so that he is just indifferent to the situation with the new

prices. This is called the Equivalent Variation (EV).

EV tells us, to which amount of money at current prices the policy change isequivalent to. Of course, if the policy change yields an increase in prices, EVwill be negative. In this case, EV tells us, how much the consumer is willing topay in order to avoid the price change.


To define EV more formally, let v (p 0, w ) = u 0 and v (p 1, w ) = u 1. EV is

implicitly defined by

v (p 0, w + EV ) = v (p 1, w ) = u 1 .

Note that

e (p 0, u 0) = w and

e (p 0, u 1) = w + EV .

Hence, we have:

EV = e (p 0, u 1)− w = e (p 0, u 1)− e (p 0, u 0)


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FIG . 2.5: Equivalent Variation


Second Approach: Compensating VariationWe could ask the consumer, how much money we can take away from him inthe situation with the new prices so that he is just indifferent to the situationwith the old prices. This is called the Compensating Variation (CV). Again,CV may be negative.More formally, CV is implicitly defined by

v (p 0, w ) = v (p 1, w −CV ) = u 0 .

Let v (p 1, w ) = u 1. Note that

e (p 1, u 1) = w and

e (p 1, u 0) = w −CV .

Hence,

CV = w − e (p 1, u 0) = e (p 1, u 1)− e (p 1, u 0) .


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.

.............

.

.............

.

............. .

.............

x1

x2

FIG. 2.6: Compensating Variation


If only one price (p 1) changes from p 01 to p 11 and all other prices remainconstant, then we can express EV and CV by using the Hicksian demandcurve. Recall that e (p 0, u 0) = w = e (p 1, u 1):

EV = e (p 0, u 1)− e (p 0, u 0)

= e (p 0, u 1)− e (p 1, u 1)

= p 01

p 11

∂ e (p , u 1)

∂ p 1dp 1

=

p 01

p 11

h 1(p , u 1)dp 1

The last step follows from Shephard’s Lemma.


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Similarly,

CV = e (p 1, u 1)− e (p 1, u 0)

= e (p 0, u 0)− e (p 1, u 0)

=

p 01

p 11

∂ e (p , u 0)

∂ p 1dp 1

= p 0

1

p 11

h 1(p , u 0)dp 1

Thus, EV and CV can be interpreted as the area below the Hicksian demandfunction.


. ..............................................................................................................................................................................................................................................................................................

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.......................................................

.

.............

.

.............

.

............. .

.............

x

p

. ..............................................................................................................................................................................................................................................................................................

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.......................................................

.

.............

.

.............

.

............. .

.............

x

p

FIG. 2.7: Equivalent and Compensating Variations as Areas Below theHicksian Demand Function


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The Hicksian demand function is not directly observable. This is why the areabelow the (Walrasian) demand function (consumer surplus) is often used asan approximation. Note:

∂ x (p ,w )∂ w

= 0

⇒ Hicksian and Walrasian demand are identical and EV and CV coincide.∂ x (p ,w )

∂ w > 0

⇒ change of Walrasian demand due to a price reduction is bigger than the

change of Hicksian demand.

⇒

Consumer’s surplus underestimates EV.⇒ Consumer’s surplus overestimates CV.

∂ x (p ,w )∂ w

< 0

⇒ change of Walrasian demand due to a price reduction is smaller than the

change of Hicksian demand.

⇒ Consumer’s surplus overestimates EV.

⇒ Consumer’s surplus underestimates CV.


Differences between the two approaches:

EV takes the old prices p 0 as the reference point.

CV takes the new prices p 1 as the reference point.

If there are income effects, EV and CV yield different welfare measures.Which criterion to use depends on the question under consideration.

Examples:

1. The government considers several new policies that hurt consumers. Itwants to know, how much people are “willing to pay” in order to avoid thepolicy change. It selects the policy change that is least costly forconsumers.

2. The government has introduced the new policy measure already and nowwants to decide on a compensation scheme (at the new prices) to makesure that no consumer is worse of in the new situation than he was in theold situation. In this case CV is the appropriate measure.


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If it is not entirely clear which measure to use, then there are two argumentsin favor of the equivalent variation:

1. EV measures the welfare change in monetary units at current prices. It ismuch easier to evaluate the value of a "Swiss Franken" at current pricesthan at some hypothetical prices.

2. If you are comparing more than one proposed policy change, the EV

uses the same base prices for all projects, while the CV uses differentbase prices for each project. Hence, with CV it is not possible to comparethe welfare effects across projects.


2.6 Aggregate Demand

Aggregate demand is simply the sum of all individual’s demand functions:

D (p , w 1, w 2, . . . , w I ) =I

i =1

x i (p , w i )

In general aggregate demand depends on the vector of wealth levels(w 1, . . . , w I ). However, it would be very convenient if aggregate demand couldbe written like individual demand as a function of prices and only one wealthlevel. Thus, under what conditions can aggregate demand be expressed as afunction of prices and aggregate wealth?


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Aggregate Demand and Aggregate Wealth

Aggregate demand can be written as a function of aggregate wealth

w =I

i =1 w i if and only if for any (w 1, . . . , w I ) and (w 1, . . . , w I ) withI i =1 w i =

I i =1 w i we have

D (p , w 1, . . . , w I ) = D (p , w 1, . . . , w I ) .

The following proposition offers a necessary and sufficient condition for this to

be the case:

Proposition 2.15

A necessary and sufficient condition for aggregate demand to be a function of

prices and aggregate wealth is that preferences of all individuals admit indirect utility functions of the Gorman form with the coefficients on w i the same for every consumer i, i.e.

v i (p , w i ) = a i (p ) + b (p )w i .


Remarks:

1. You are asked to prove sufficiency yourself in one of the exercises.Necessity is much more difficult.

2. These indirect utility function represent the consumers’ preferences onlylocally: If w i becomes very small, a corner solution would obtain.

3. This type of preferences is clearly quite restrictive. It assumes that thewealth expansion paths of all consumers are parallel straight lines.


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..........................................................

.

.............

.

.............

.

............. .

.............

x

p

. ..............................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................

..........................................................

.

.............

.

.............

.

............. .

.............

x

p

FIG . 2.8: Gorman Utility Functions and Wealth Expansion Paths


So far we assumed that individual wealth is exogenously given. If individualwealth is explained endogenously in the model and depends on prices andaggregate wealth, then our chances to get an aggregate demand function thatdepends only on aggregate wealth are much better. All we need is thatindividual wealth is a function of prices and aggregate wealth only.

Definition 2.6

A wealth distribution rule is a set of functions (w 1(p , w ), . . . , w I (p , w )) with I i =1 w i (p , w ) = w for all (p , w ).

If the income distribution is determined by a wealth distribution rule, thentrivially aggregate demand is a function of (p , w ) only:

D (p , w 1(p , w ), . . . , w I (p , w )) = D̃ (p , w ) .


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Structural Properties of the Aggregate Demand Function

Assuming that aggregate demand can be written as a function of aggregatewealth, we can ask what structural properties are preserved by aggregation.

Homogeneity of degree 0 is preserved. Why?

Walras law is also preserved. Why?

However, the compensated law of demand need not hold for theaggregate demand function even if it holds for all individual demandfunctions. Why not? If there is a price change we have to compensatethe group of all consumers such that the old aggregate demand vector isstill affordable at the new prices. However, we do not have to compensateeach individual consumer such that he can still afford his old consumptionbundle. Thus, depending on how consumers are compensated, there willbe additional wealth effects that may go in any direction. These wealtheffects are responsible for that the compensated law of demand may failto hold in the aggregate.


However, while aggregation may destroy structural properties of the demandfunction, it may also add new structural properties.

Examples:

If each individual demand function is discontinous, aggregate demandmay nevertheless become a continuous function, if there are sufficiently

many consumers (a continuum).Hildenbrand (1983, 1996): If the distribution of wealth satisfies somestatistical properties, then the aggregate demand function must satisfynot only the weak axiom but also the (uncompensated) law of demand.


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2.7 Production

This chapter is often called “theory of the firm”, but this is not quiteappropriate. The “neoclassical theory of the firm” sees the firm merely as aproduction function, a black box that transforms inputs into outputs.Incentives, ownership rights, boundaries of the firm, and organizational issuesare all left out of the picture.

The questions addressed here include:Description of the production possibilities

Profit maximization

Cost functions and factor demand correspondences

Aggregate Supply


Production Sets

In order to describe the production possibilities of the firm, we use thefollowing concepts:

L commodities, indexed by l ∈ {1, . . . , L}Production plan (netput vector): y = (y 1, . . . , y L) ∈ RL, y l < 0 (net input),y l > 0 (net output).

Production set: Y ⊂ RL contains all feasible production plans.Any y ∈ Y is feasible, any y ∈ Y is not.

Transformation function: F (y ), is defined by Y = {y ∈ RL | F (y ) ≤ 0}and F (y ) = 0 if and only if y is on the boundary of the production set.

Transformation frontier: {y ∈ RL | F (y ) = 0}.


Marginal rate of transformation: Suppose that F (·) is differentiable and

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that y satisfies F (y ) = 0. Then we have

∂ F (y )

∂ y k

dy k +∂ F (y )

∂ y l dy l = 0

or

MRT lk (y ) = −dy k

dy l =

∂ F (y )/∂ y l ∂ F (y )/∂ y k

Interpretation?

Production function: Suppose that each good is either an input or an

output. Let q = (q 1, . . . , q M ) be an output vector and z = (z 1, . . . , z L−M )be an input vector. Then we can describe the production set by aproduction function q = f (z ). This is usually done in the special butimportant case of a single output good.

Marginal Rate of Technical Substitution:

MRTS lk =∂ f (z )/∂ z l ∂ f (z )/∂ z k

Interpretation?


Commonly assumed properties of production sets:

Not all of the following assumptions do always make sense. Some of them aremutually exclusive.

1. Y is nonempty.

2. Y is closed, i.e., it contains its boundary.

3. No free lunch: If y

≥0 and y

∈Y , then y = 0. Put differently,

Y RL+ ⊂ {0}.

4. Possibility of inaction: 0 ∈ Y . If inaction is not possible, then there aresunk costs .

5. Free disposal: If y ∈ Y and y ≤ y , then y ∈ Y .

6. Irreversibility: if y ∈ Y and y = 0, then −y ∈ Y .


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7. Nonincreasing returns to scale: For any y ∈ Y and α ∈ [0, 1], we haveαy ∈ Y . That is, any feasible production plan can be scaled down.

8. Nondecreasing returns to scale: For any y ∈ Y and any α ≥ 1, we haveαy ∈ Y . That is, any feasible production plan can be scaled up.

9. Constant returns to scale: For any y ∈ Y and any α ≥ 0, we haveαy ∈ Y . That is, any feasible production plan can be scaled up andscaled down. Geometrically, Y is a cone (ein Kegel).

10. Additivity: If y

∈Y and y

∈Y , then y + y

∈Y . Additivity is different from

nondecreasing returns to scale, because it can also be applied to integerproblems.

11. Convexity: If y , y ∈ Y and α ∈ [0, 1], then

αy + (1− α)y ∈ Y .


2.8 Profit Maximization

Assumptions:

1. Price-taking behavior: The vector p = (p 1, . . . , p L) of prices isexogenously given and independent of the production plans of the firms.Discussion.

2. Profit maximization: The firm’s only objective is to maximize profits.Discussion.

3. Technical asumptions: The production set satisfies non-emptiness,closedness, and free disposal. Discussion.


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The profit maximization problem (PMP) is very similar to the utilitymaximization problem that we have seen already. But it has more structureand yields much more powerful empirical predictions.

maxy

p · y s.t. y ∈ Y ,

or, alternatively,

maxy p · y s.t. F (y ) ≤ 0 .

Lagrange

Geometry of profit maximization.


Let y (p ) denote the solution to this problem. We call y (p ) the supplycorrespondence.

Note:

y (p ) need not be unique.

It may be that there exists no solution to PMP. This is frequently the case

if we have non-decreasing returns to scale. Why?Recall that if y k (p ) > 0, then y k (p ) is the firm’s supply of output k . Ify l (p ) < 0, then y l (p ) is the firm’s demand for input l .


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We can now define the profit function π(p ) by

π(p ) = max{p · y | y ∈ Y } .

If y (p ) is unique, then

π(p ) = p · y (p ) .

Note that the profit function is very similar to the indirect utility function. There

is one important difference, however. The indirect utility function depends onprices and wealth , while the profit function depends only on prices! There isno budget constraint in production theory and, hence, there are no wealtheffects.


Proposition 2.16

Suppose that y (p ) is the solution of the PMP given production set Y and π(p )is the corresponding profit function. Then:

(a) y (p ) is homogeneous of degree 0.

(b) π(p ) is homogeneous of degree 1.

(c) π(p ) is convex.

(d) If Y is convex, then y (p ) is a convex set for all p. If Y is strictly convex,

then y (p ) is single-valued.

(e) Hotelling’s Lemma: If y (p ) consists of a single point, then π(·) is differentiable at p and π(p ) = y (p ).

(f) If y (·) is a function differentiable at p, then D p y (p ) = D 2p π(p ) is a symmetric and positive semidefinite matrix with Dy (p )p = 0.


Remarks:

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Remarks:

The proofs parallel those of of Demand Theory.

Hotelling’s Lemma is very useful. If we know the profit function of a firm,we can directly compute it’s supply function.

The positive semidefiniteness of the matrix Dy (p ) is the law of supply:Quantities respond in the same direction as price changes.

The law of supply can also be expressed as

(p − p ) · (y − y ) ≥ 0

It follows immediately from a revealed preference argument:

(p − p ) · (y − y ) = (p · y − p · y ) + (p · y − p · y ) ≥ 0

because y is profit maximizing at prices p and y at prices p .

Note that the law of supply holds for any price change. In contrast todemand theory, there are no budget constraints and no wealth effects.There are only substitution effects in production theory!


2.9 Cost Minimization

Cost minimization is a necessary (but not sufficient) condition for profitmaximization. It is useful to study this problem separately for several reasons:

1. When a firm is not a price taker on the output market, we can no longeruse the profit function, but the results from cost minimization continue tohold as long as input prices are given.

2. If there are nondecreasing returns to scale, PMP does not have a

solution, but the results from cost minimization can still be applied.

3. The cost minimization problem is useful to characterize the factordemand of the firm.

We only consider the single output case. In this case let z be the vector ofinputs, f (z ) the production function, q the output, and w the vector of inputprices.


Cost Minimization Problem (CMP):

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( )

minz ≥0

w · z s.t. f (z ) ≥ q .

Let z (w , q ) be a solution to this problem. It is called the conditional factordemand correspondence. For any z ∗ ∈ z (w , q ) the following first orderconditions must hold:

w l ≥ λ∂ f (z ∗)

∂ z l , with equality if z ∗l > 0,

Illustrate graphically.Show that (z l , z k ) (0, 0) implies MRTS lk = w l /w k .

Interpretation of the Lagrange multiplier λ.

We can now define the cost function as the optimized value of CMP. Ifz (p , w ) is single valued we can write:

c (w , q ) = w · z ∗(w , q )


Proposition 2.17

Suppose that c (w , q ) is the cost function of production function f (·) and that z (w , q ) is the corresponding conditional factor demand correspondence. Then

(a) z (·) is homogeneous of degree 0 in w.

(b) c (·) is homogeneous of degree 1 in w and nondecreasing in q.

(c) c (·) is a concave function of w.

(d) If the set {z ≥ 0 | f (z ) ≥ q } is convex, then z (w , q ) is a convex set. If {z ≥ 0 | f (z ) ≥ q } is strictly convex, then z (w , q ) is single-valued.

(e) Shephard’s Lemma: If z (w , q ) consists of a single point, then c (·) is differentiable and w c (w , q ) = z (w , q ).

(f) If z (·) is differentiable at w, then D w z (w , q ) = D 2w c (w , q ) is a symmetric and negative semidefinite matrix with D w z (w , q )w = 0.

All of these results follow immediately from the analysis of the expenditureminimization problem in classical demand theory. Simply replace u (·) by f (·),u by q , and x by z (i.e., interpret the utility function as a production function).


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2.10 Aggregate Supply

The aggregate supply correspondence is simply the sum of the individualsupply correspondences. Assuming that y j (p ) is single-valued for all firms j ∈ {1, . . . , J } we can write

y (p ) =J

j =1

y j (p ) .

We know that for each firm D p y j (p ) is a symmetric, positive semidefinitematrix. Hence, D p y (p ) must also be symmetric and positive semidefinitewhich implies that the law of supply holds in the aggregate.


This can also be shown directly: For all j ∈ {1, . . . , J } we have

(p − p ) · [y j (p )− y j (p )] ≥ 0 .

Summing up over j yields

(p

−p )

·[y (p )

−y (p )]

≥0 .

In contrast to demand theory there are no wealth effects, which simplifiesaggregation dramatically.

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Micror 02 10 Notes