Advanced Microeconomics General Equilibrium Theory (GET)

Why GET? Logistics Consumption: Recap Production: Recap References

Advanced MicroeconomicsGeneral Equilibrium Theory (GET)

Giorgio [email protected]

http://www.lem.sssup.it/fagiolo/Welcome.html

LEM, Sant’Anna School of Advanced Studies, Pisa (Italy)

Introduction to the Module


How do markets work?

GET: Neoclassical theory of competitive marketsSo far we have talked about one producer, one consumer, severalconsumers (aggregation can be tricky)The properties of consumer and producer behaviors were derived fromsimple optimization problems

Main assumptionConsumers and producers take prices as givenConsumers: prices are exogenously fixedProducers: firms in competitive markets are like atoms and cannot influencemarket prices with their choices (they do not have market power)


How do markets work?

GET: Neoclassical theory of competitive marketsSo far we have talked about one producer, one consumer, severalconsumers (aggregation can be tricky)The properties of consumer and producer behaviors were derived fromsimple optimization problems

Main assumptionConsumers and producers take prices as givenConsumers: prices are exogenously fixedProducers: firms in competitive markets are like atoms and cannot influencemarket prices with their choices (they do not have market power)


How do prices emerge?

Simple story: A single marketQuantity produced as a function of price (production theory)Quantity consumed as a function of price (consumer theory)

How to solve for an equilibrium?Under suitable assumptions on the shapes of supply/demand schedules, byequating demand and supply one gets the equilibrium price/quantity pair(p∗, q∗)

Introduction Walrasian model Welfare theorems FOC characterization

Why does demand equal supply?

Simple story

Quantity produced as function of price (producer theory)

Quantity consumed as function of price (consumer theory)

Two equations in two unknowns give a solution

p

q

y(p)p

q

x(p,w)

p

q

y(p)

x(p,w)

q∗

p∗

4 / 94



Actually, the story is more complicated. . .The economy is composed of many interdependent markets (goods,services, labor, financial, . . . )Individual decisions depend on prices through profits, budget constraints,individual income and wealth, etc.Prices depend on individual choices because in all markets demand shouldequal supplyAre there prices that magically make all these decisions and constraintsmutually compatible and let them to be carried out simultaneously?

A little bit more formally:Prices are a vector pAggregate supply is a vector y(p)Aggregate demand is a vector x(p,w)General equilibrium prices satisfy y(p) = x(p,w)Potentially a very complicated system of equations to solve (given resourceconstraints)GET: Exploring solutions of this system of equations and their properties



Actually, the story is more complicated. . .The economy is composed of many interdependent markets (goods,services, labor, financial, . . . )Individual decisions depend on prices through profits, budget constraints,individual income and wealth, etc.Prices depend on individual choices because in all markets demand shouldequal supplyAre there prices that magically make all these decisions and constraintsmutually compatible and let them to be carried out simultaneously?

A little bit more formally:Prices are a vector pAggregate supply is a vector y(p)Aggregate demand is a vector x(p,w)General equilibrium prices satisfy y(p) = x(p,w)Potentially a very complicated system of equations to solve (given resourceconstraints)GET: Exploring solutions of this system of equations and their properties


GET and the “Invisible-Hand” Paradox

The homo oeconomicus “intends only his own gain, and he is in this, as inmany other cases, led by an invisible hand to promote an end which was nopart of his intention [. . . ] By pursuing his own interest he frequently promotesthat of the society more effectually than when he really intends to promoteit”. (Adam Smith, The Wealth of Nations, Book IV, chapter II, paragraph IX).

“One of the major themes of economic theory is that the behaviour of acomplex economic system can be viewed as an equilibrium arising from theinteraction of a number of economic units with different motivations.” (Scarf,1973; p. 1).

“The most intellectually exciting question in our subject remains: is it truethat the pursuit of private interest produces not chaos but coherence, and ifso how is it done?” (Hahn, 1970; p. 1)

“On the surface, at least, modern capitalist economies seem to bechaotic. . . and yet somehow the thing works. There is order — not chaos.(Katzner, 1989; pp. 1, 6–7)”


GET as a solution of the “Invisible-Hand” paradox?

1 How is it that the economy seems to function smoothly when millions ofdecision-making units operate independently and in their own selfinterest?

2 How do prices emerge?

3 Why is it that certain commodities are produced in certain quantities?

4 What determines the specific distribution of income and final goodsactually realized in the economy?


GET and the “laissez-faire” view of market functioning

The basic Walrasian conjectureThe laissez faire operation of the price mechanism, in a environment ofderegulated competitive markets where agents are motivated byself-interest, will produce not chaos but a coherence, in the sense of marketclearing, optimal outcomes (Bryant, 2010).

Pre and Post-institutional (market) conceptsSocially-optimal (Pareto efficient) allocation: an allocation (how much anyfirm produces, how much any consumer consumes) where it is impossible tomake some individuals better off without making some other individualworse off (a minimal test without any distributional or equity content)Socially desirable allocation: an allocation that can be the target of a socialplanner who wants to maximize the well-being of all individuals in theeconomyInstitutions: decentralized markets. They operate without the need of anycentral coordination device and deliver equilibrium prices and equilibriumallocations


GET and the “laissez-faire” view of market functioning

The basic Walrasian conjectureThe laissez faire operation of the price mechanism, in a environment ofderegulated competitive markets where agents are motivated byself-interest, will produce not chaos but a coherence, in the sense of marketclearing, optimal outcomes (Bryant, 2010).

Pre and Post-institutional (market) conceptsSocially-optimal (Pareto efficient) allocation: an allocation (how much anyfirm produces, how much any consumer consumes) where it is impossible tomake some individuals better off without making some other individualworse off (a minimal test without any distributional or equity content)Socially desirable allocation: an allocation that can be the target of a socialplanner who wants to maximize the well-being of all individuals in theeconomyInstitutions: decentralized markets. They operate without the need of anycentral coordination device and deliver equilibrium prices and equilibriumallocations


Positive vs Normative Qestions

1 Do equilibrium states exist?2 Are they unique? Are they stable?3 Are they socially optimal, i.e. can a socially-optimal state be reached just

by letting the market work?4 Are they socially desirable from the point of view of a social planner

wanting to maximize individual well-being?5 How do equilibrium states respond to variations in the parameters that

define the economy?


Positive vs Normative Qestions

1 Do equilibrium states exist?2 Are they unique? Are they stable?3 Are they socially optimal, i.e. can a socially-optimal state be reached just

by letting the market work?4 Are they socially desirable from the point of view of a social planner

wanting to maximize individual well-being?5 How do equilibrium states respond to variations in the parameters that

define the economy?


GET and the Real World

GET is a highly sophisticated mathematical modelGET is “logically entirely disconnected from its interpretation” (Debreu, citedin Kaldor, 1972, p. 1237)GET is the perfect example of the instrumentalist (positive) approach toeconomic theory that informs all neoclassical models“Truly important and significant hypotheses will be found to haveassumptions that are wildly inaccurate descriptive representations of reality,and, in general, the more significant the theory, the more unrealistic theassumptions” (Friedman, Essays in Positive Economics, p. 14).Models must be judged by their predictive capability, not for the realism oftheir assumptionsGoal: find a mathematical model of competitive market behavior such thatsharp implications can follow from a minimal set of assumptions onprimitives and agent behaviors

Assumptions as free goodsExample: Does an equilibrium state in the GE model exist?Yes, provided that a long series of (relatively stringent) mathematicalassumptions on preferences, technologies, etc. are satisfiedDo these assumptions have an economic meaning? Are these assumptionsrealistic? Are they verified in reality?Perhaps yes, but after all who cares...?


GET and the Real World

GET is a highly sophisticated mathematical modelGET is “logically entirely disconnected from its interpretation” (Debreu, citedin Kaldor, 1972, p. 1237)GET is the perfect example of the instrumentalist (positive) approach toeconomic theory that informs all neoclassical models“Truly important and significant hypotheses will be found to haveassumptions that are wildly inaccurate descriptive representations of reality,and, in general, the more significant the theory, the more unrealistic theassumptions” (Friedman, Essays in Positive Economics, p. 14).Models must be judged by their predictive capability, not for the realism oftheir assumptionsGoal: find a mathematical model of competitive market behavior such thatsharp implications can follow from a minimal set of assumptions onprimitives and agent behaviors

Assumptions as free goodsExample: Does an equilibrium state in the GE model exist?Yes, provided that a long series of (relatively stringent) mathematicalassumptions on preferences, technologies, etc. are satisfiedDo these assumptions have an economic meaning? Are these assumptionsrealistic? Are they verified in reality?Perhaps yes, but after all who cares...?


Oversimplifying Assumptions in GE Models

IndividualsBoth consumers and producers are fully-rational, maximizing individualswithout computational boundsBoth consumers and producers know the model of the world and act underperfect information about pricesConsumer utilities do not depend on what other consumers and producersdoFirm profits do not depend on what other firms and consumers do

Commodities and MarketsMarkets are complete (there exists a market for every commodity)Commodities are homogeneous and contingent (on time, states of nature,etc.)All the action takes place simultaneously (no explicit time in the model)

Why such oversimplifying assumptions?Model must remain analytically tractable (closed-form solutions)Look for assumptions that allow for mathematical tractability












GET: Applications, Empirics, and Policy

Economic applications of GETGET is “the” theory of competitive market behaviorGET is extremely generic and can be applied to study almost any issueswhere a GE setup is relevant (unemployment, international trade, financialmarkets, etc.)GET provides the underlying micro-structure for micro-foundedmacroeconomic models studying the business cycle, growth, etc.

Empirical validation of GETDo GE models give a satisfactorily account of actual economic data?At best very mixed evidence: empirically-testable implications rather limited(see last part of the course)

GET and policyGET tells us that all equilibria are socially optimalA very optimistic view about the ability of markets to allow for sociallydesirable states without any external interventionGET is currently used within DSGE models by central banks to decide theirpolicies












Outline

1 IntroductionConsumer and producer theory: recapThe GET model: concepts and definitionsPareto efficiency, social welfare, Walrasian equilibrium

2 Partial Equilibrium AnalysisFocus on a simple model with only one relevant commodityEquilibrium and efficiency: welfare theorems

3 Pure-Exchange ModelFocus on a more complicated (but still simple) model where there are nofirms (no production) and consumers exchange their endowmentsExistence and efficiency

4 GE with ProductionFull-fledged model with firms and consumersSimple example with 1 producer/ 1 consumerOverview of the main results about existence, efficiency and welfareUniqueness, stability, empirical testing


Outline






Outline






Outline






Logistics

Lectures and Co.14 lectures (1h30 each)Slides: http://www.lem.sssup.it/fagiolo/Teaching.htmlOffice hours: lecture days, after 16:00

TextbooksMain reference: Mas-Colell, Whinston, and Green (1995), MicroeconomicTheory, Oxford University PressRelevant chapters: Compulsory: 10,15,16; Optional: 17Other useful books:

Jehle and Reny (2000), Advanced Microeconomic Theory, Addison Wesley, 2ndedition (3rd edition also available, 2010)Hildenbrand and Kirman (1988), Equilibrium analysis: Variations on themes byEdgeworth and Walras, North-HollandIngrao and Israel (1991), The Invisible Hand, Economic Equilibrium in the Historyof Science, MIT PressKreps (1990), A Course in Microeconomic Theory, Harvester WheatsheafVarian (1984), Microeconomic Analysis, Norton

Exercises and ExamNo exercise sessions. Examples will be discussed in classGET exam (closed-books): mostly theory questions

http://www.lem.sssup.it/fagiolo/Teaching.html


Logistics







Logistics







Classical Demand Theory: Notation

n commodities

Price vector (exogenous): p ∈ Rn++

Consumer wealth level: w ∈ R++

Consumption set: X = Rn+

Consumer utility function (continuous): u : X → R



n commodities







n commodities







n commodities







n commodities






The Consumer Problem(s)

Introduction Welfare Price indices Aggregation Optimal tax

Recap: The consumer problems

Utility Maximization Problem

maxx∈Rn

+

u(x) such that p · x ≤ w .

Choice correspondence: Marshallian demand x(p,w)

Value function: indirect utility function v(p,w)

Expenditure Minimization Problem

minx∈Rn

+

p · x such that u(x) ≥ u.

Choice correspondence: Hicksian demand h(p, u)

Value function: expenditure function e(p, u)

52 / 89


Marshallian Demand: Properties

If u is continuous and represents locally non-satiated preferences, thenx(p,w)

1 is homogeneous of degree zero (no money illusion):x(λp, λw) = x(p,w) for any λ > 0

2 is a convex set if u is quasi-concave (preferences are convex)3 consists of a single element if u is strictly quasi-concave (preferences

are strictly convex), i.e. it is a function and not a correspondence4 satisfies Walras’ Law: p · x(p,w) = w , i.e. consumer spends all its

income in the optimal choice


Marshallian Demand: Correspondence or Function?

Introduction Utility maximization Expenditure minimization Wealth and substitution

Illustrating the Utility Maximization Problem

5 / 89


Marshallian Demand: Walras’ LawIntroduction Utility maximization Expenditure minimization Wealth and substitution

Implications of restrictions on preferences: non-satiation II

Proof.

Suppose that p · x < w for somex ∈ x(p,w). Then there existssome x � sufficiently close to xwith x � � x and p · x � < w ,which contradicts the fact thatx ∈ x(p,w). Thus p · x = w .

16 / 89


Utility Maximization: Solution


Solving for Marshallian demand I

Suppose the utility function is differentiable

This is an ungrounded assumption

However, differentiability can not be falsified by any finitedata set

Also, utility functions are robust to monotone transformations

We may be able to use Kuhn-Tucker to “solve” the UMP:


maxx∈Rn

+

u(x) such that p · x ≤ w

gives the Lagrangian

L(x ,λ, µ, p,w) ≡ u(x) + λ(w − p · x) + µ · x .

17 / 89FO (necessary) conditions for interior solution x∗ � 0: ∇u(x∗) = λp,λ ≥ 0

FOCs are sufficient if the problem is convex, i.e. constraint set is convex(OK) and objective function is concave (convexity of UCS, i.e.quasi-concavity of u is not sufficient)


The Expenditure Minimization Problem

Why do we need another problem?We would like to characterize ’important’ properties of Marshallian demandx(p,w)Example: Consider a price increase for one good (apples)Substitution effect: Apples are now relatively more expensive than bananas,so I buy fewer applesWealth effect: I feel poorer, so I buy (more? fewer?) applesWealth and substitution effects could go in opposite directions: we can’teasily sign the change in consumptionThis is hard to do it because in the UMP parameters enter feasible set ratherthan objective



Why do we need another problem?We would like to characterize ’important’ properties of Marshallian demandx(p,w)Example: Consider a price increase for one good (apples)Substitution effect: Apples are now relatively more expensive than bananas,so I buy fewer applesWealth effect: I feel poorer, so I buy (more? fewer?) applesWealth and substitution effects could go in opposite directions: we can’teasily sign the change in consumptionThis is hard to do it because in the UMP parameters enter feasible set ratherthan objective


Isolating the substitution effect

We can isolate the substitution effect by “compensating” theconsumer so that her maximized utility does not change

If maximized utility doesn’t change, the consumer can’t feel richeror poorer; demand changes can therefore be attributed entirely tothe substitution effect


minx∈Rn

+


i.e., find the cheapest bundle at prices p that yield utility at least u

30 / 89




Illustrating the Expenditure Minimization Problem

31 / 89


Relations Between the Two Problems

Introduction Welfare Price indices Aggregation Optimal tax

Recap: The consumer problems


maxx∈Rn

+

u(x) such that p · x ≤ w .

Choice correspondence: Marshallian demand x(p,w)

Value function: indirect utility function v(p,w)


minx∈Rn

+


Choice correspondence: Hicksian demand h(p, u)

Value function: expenditure function e(p, u)

52 / 89


Relating Hicksian and Marshallian demand I

Theorem (“Same problem” identities)

Suppose u(·) is a utility function representing a continuous andlocally non-satiated preference relation � on Rn

+. Then for anyp � 0 and w ≥ 0,

1 h�p, v(p,w)

�= x(p,w),

2 e�p, v(p,w)

�= w;

and for any u ≥ u(0),

3 x�p, e(p, u)

�= h(p, u), and

4 v�p, e(p, u)

�= u.

For proofs see notes (cumbersome but relatively straightforward)

34 / 89


Wealth and Substitution I

What about Marshallian-demand response to changes inown price?


The Slutsky equation II

Setting i = j , we can decompose the effect of an an increase in pi

∂xi (p,w)

∂pi=

∂hi

�p, u(x(p,w))

�

∂pi− ∂xi (p,w)

∂wxi (p,w)

An “own-price” increase. . .1 Encourages consumer to substitute away from good i

∂hi

∂pi≤ 0 by negative semidefiniteness of Slutsky matrix

2 Makes consumer poorer, which affects consumption of good iin some indeterminate way

Sign of ∂xi

∂w depends on preferences

43 / 89


Wealth and Substitution IIIntroduction Utility maximization Expenditure minimization Wealth and substitution

Marshallian response to changes in wealth

Definition (Normal good)

Good i is a normal good if xi (p,w) is increasing in w .

Definition (Inferior good)

Good i is an inferior good if xi (p,w) is decreasing in w .

45 / 89


Wealth and Substitution III

Engel curves show how Marshallian demand moves with wealth (locusof {x , x ′, x ′′, } in the figure below)

In the cases depicted below both goods are “normal” (both x1 and x2

increase in w)


Graphing Marshallian response to changes in wealth

Engle curves show how Marshallian demand moves withwealth (locus of {x , x �, x ��, . . . } below)

In this example, both goods are normal (xi increases in w)

✻

✲

x

x ��

x �

46 / 89


Aggregation

Aggregate demand: a suitably defined sum of all demands arising fromindividual demands of consumers in the economy

Aggregation can be trickyIndividual (Marshallian) demand can be expressed as a function ofcommodity prices and individual wealth. When can aggregate demand beexpressed as a function of commodity prices and aggregate wealth?It depends on the form of utility functions: homotetic or quasilinearpreferences do the trick

Quasilinear preferencesConsider an economy with n = 2 commodities. Quasilinear preferencesresult in utilities of the form:

u(x ,m) = φ(x) + m

where x ≥ 0 is the consumption of commodity 1 and m ∈ R is consumptionof commodity 2, φ′ > 0, φ′′ < 0, and φ(0) = 0.Commodity 1 is a consumption good, while commodity 2 can be defined as’everything else’, i.e. money left apart by the consumer for purchasing allother goods, after having made the optimal choice for commodity 1. Thelatter can be referred to as the ’consumption good’, while commodity 2 is the’numeraire’ or ’money’.


Aggregation




u(x ,m) = φ(x) + m



Aggregation




u(x ,m) = φ(x) + m



Production Theory: Notation

n commodities


Netput vector (production plan): y ∈ Rn

Production set: Y ⊆ Rn (non-empty and closed, satisfying properties ofshut-down, free-disposal and (strict) convexity)

f : Rm+ → R+ production function for 1-output technology (m = n − 1)

q = f (z) is the efficient production level given input levels z ∈ Rm+

Input prices: w ∈ Rm++

Output/input prices: p = (p,w) ∈ R++ × Rm++

Netput vector for 1-output technology: y = (q,−z) ∈ R+ × Rm+



n commodities











n commodities











n commodities











n commodities











n commodities











n commodities











n commodities











n commodities










Firm Optimal Pruduction

Introduction Production sets Profit maximization Rationalizability Differentiable case Single-output firms

The Profit Maximization Problem

The firm’s optimal production decisions are given bycorrespondence y : Rn ⇒ Rn

y(p) ≡ argmaxy∈Y

p · y

=�y ∈ Y : p · y = π(p)

�

Resulting profits are given by profit function π : Rn → R ∪ {+∞}

π(p) ≡ supy∈Y

p · y

15 / 86

Choice correspondence: Netput y(p)Value function: Profit π(p) = p · y(p)Properties:

π(p) is convex and homogeneous of degree 1y(p) is homogeneous of degree 0


Single-Output Technology


Characterizing Y : Production function I

Definition (production function)

For a firm with only a single output q (and inputs −z), defined asf (z) ≡ max q such that (q,−z) ∈ Y .

Y =�(q,−z) : q ≤ f (z)

�, assuming free disposal

53 / 86


Dividing up the problem I

With one output, free disposal, and production function f (·),

Y =�(q,−z) : z ∈ Rm

+ and f (z) ≥ q�

Given a positive output price p > 0, profit maximization requiresq = f (z), so firms solve

π(p,w) = supz∈Rm

+

pf (z)� �� revenue

−w · z��cost

z(p,w) = argmaxz∈Rm

+

pf (z) − w · z

55 / 86


Single-Output Technology


Characterizing Y : Production function I

Definition (production function)

For a firm with only a single output q (and inputs −z), defined asf (z) ≡ max q such that (q,−z) ∈ Y .

Y =�(q,−z) : q ≤ f (z)

�, assuming free disposal

53 / 86


Dividing up the problem I

With one output, free disposal, and production function f (·),

Y =�(q,−z) : z ∈ Rm

+ and f (z) ≥ q�

Given a positive output price p > 0, profit maximization requiresq = f (z), so firms solve

π(p,w) = supz∈Rm

+

pf (z)� �� revenue

−w · z��cost

z(p,w) = argmaxz∈Rm

+

pf (z) − w · z

55 / 86


Single-Output TechnologyIntroduction Production sets Profit maximization Rationalizability Differentiable case Single-output firms

Dividing up the problem II

We separate the profit maximization problem into two parts:1 Find a cost-minimizing way to produce a given output level q

Cost functionc(q,w) ≡ inf

z : f (z)≥qw · z

Conditional factor demand correspondence

Z∗(q,w) ≡ argminz : f (z)≥q

w · z

=�z : f (z) ≥ q and w · z = c(q,w)

�

2 Find an output level that maximizes difference betweenrevenue and cost

maxq≥0

pq − c(q,w)

56 / 86


Optimal-Output ProblemIntroduction Production sets Profit maximization Rationalizability Differentiable case Single-output firms

First-order conditions: Optimal Output Problem

Optimal output problem

maxq≥0

pq − c(q,w).

L(q, p,w , µ) ≡ pq − c(q,w) + µq

Applying Kuhn-Tucker here gives

p ≤ ∂c(q∗,w)

∂qwith equality if q∗ > 0

63 / 86


Supply Schedule I: Strictly Convex Technology


Supply Schedule II: CRTS Technology


Supply Schedule III: Non-Convex Technology


Supply Schedule IV: Strictly-Convex VC + Non-Sunk Setup Cost


Supply Schedule V: CRTS VC + Non-Sunk Setup Cost


Books and papers cited in these slides. . .

Arrow, K.J. and Hahn, F. H. (1972), General competitive analysis,Holden-Day.

Friedman, M. (1953). Essays in Positive Economics, Chicago Press.

Bryant, W. D. A. (2010), General Equilibrium: Theory and Evidence,World Scientific.

Kaldor , N. (1972), The Irrelevance of Equilibrium Economics, TheEconomic Journal, Vol. 82, No. 328. (Dec., 1972), pp. 1237-1255.

Katzner, D.W. (1989), The Walrasian vision of the microeconomy: anelementary exposition, University of Michigan Press.

Scarf, H. (1973), The computation of economic equilibria, Yale UniversityPress, available at http://cowles.econ.yale.edu/P/cm/m24/

Smith, A. (1904), An Inquiry into the Nature and Causes of the Wealth ofNations, London: Methuen and Co., Ltd., ed. Edwin Cannan, 1904. Fifthedition.

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Advanced Microeconomics General Equilibrium Theory (GET)