Classical

thermodynamics and

economic general

equilibrium theory

Eric Smith SFI

Duncan K. Foley New School for

Social Research

SFI Complex Systems Summer School SF 2007Outline

History and some conventions

•

Modern neoclassical economics

•

Structure of thermodynamics

•

The right connection

•

An example

•A (very) little historyParallel goals of

“natural” and “social”

physics circa 1900

Deﬁne and characterize equilibria

•

Points of rest “Best” resource allocations

Equations of state Discovery of price systems

Describe transformations

•

Work, heat ﬂow Trade, allocation processesThe Walrasian Analogy

Leon Walras (1909)

Equilibrium as force

•

balance in mechanics

Equilibrium as

•

balance of “marginal

utility” in exchange

“demands”Analogies from mechanics

Position (x) Holdings (x)

Potential Energy (V) Utility (U)

Force (F) Prices (p)

p =∇U

F =−∇V

(Utility is implicitly measurable)

“Ball settles in the bottom of

the bowl to minimize energy”Gibbs and thermodynamics

Distinction between particle and system

Entropy is maximized in a S(U)

•

closed system at equilibrium

S(U)−βU

For “open” subsystem, excess

•

∂S 1

env

entropy is maximized

β = ≡

∂U T

env

Helmholtz Free Energy is

•

equivalently minimized

A= U− TS

Ball settles in the bottom of the

bowl to maximize excess entropy

(by losing energy)Irving Fisher (1926)

And yet Fisher...

A particle An individual

Space (x? V?) Commodities (x)

Energy (U?, E?, V?) Utility (U)

Force (F) Marginal utility (p)

Particles and individuals are unpredictable

•

State variables are only properties of

•

thermodynamic systems at equilibrium

Fisher mixes metaphors from

•

thermodynamics and statistical mechanicsAnalogy and confusion

J. H. C. Lisman (1949)

•

A quasi-eq. system An individual

Entropy Utility (“analogon”)

pV (ideal gas) px (value)

J. Bryant (1982)

•

pV = NT px = NT (productive content) Disgust

The formal mathematical analogy between classical

thermodynamics and mathematic economic systems has now

been explored. This does not warrant the commonly met

attempt to ﬁnd more exact analogies of physical magnitudes --

such as entropy or energy -- in the economic realm. Why

should there be laws like the ﬁrst or second laws of

thermodynamics holding in the economic realm? Why should

``utility'' be literally identiﬁed with entropy, energy, or anything

else? Why should a failure to make such a successful

identiﬁcation lead anyone to overlook or deny the

mathematical isomorphism that does exist between minimum

systems that arise in different disciplines?

Samuelson 1960But duality survived

Extensive quantities

•

Energy, volume Goods

Intensive quantities

•

Temperature, pressure PricesThe marginalist

revolution and modern

“Neoclassical”

mathematical

economic theoryIndifference and utility

1 U

Suppose more than one good

•

x

2

x = (x ,x ,...,x )

0 1 n

Only try to capture the

•

notion of indifference

u(x) =U

Relative prices = marginal

x

1

•

rates of substitution of goods

∂u/∂x

i

= p /p

i j

“Absolute” price undeﬁned

•

∂u/∂x

i

Utility is now explicitly only ordinalThe separating hyperplane

(Tj. Koopmans, 1957)

“Edgeworth-Bowley” box:

•

Conserve “endowments”:

1

(allocation of resources under

x

2

conditions of scarcity)

P.S.

Prices separate agent

•

decisions from each other

(trade and production)

“Pareto Optimum” deﬁnes

•

equilibrium as no-trade

x

1

No trade any agent can propose

Trade to equilibrium must

from an equilibrium will be

•

voluntarily accepted by any other

be irreversible

agentDuality: prices and demands

U

1

x

2

x = (x ,x ,...,x )

0 1 n

u(x) =U

∂u

∝ p

“Offer prices”

i

∂x

i

x

1

Expenditure

e(p,U)≡ min[p · x | u[x]≥ U]

function

x

!

!

!

!

∂x

∂e

!

!

δe = δp· x + p· δU

= x

i

!

!

∂U

∂p

i

p

UExchange economies and

the Walrasian equilibrium

1

x = (x ,x ,...,x )

x

2

0 1 n

P.S.

p = (p ,p ,. .. ,p )

0 1 n

eq

Maximize:

0

! "

0

L = u(x)−βp· x− x

x

1

“Wealth preservation” hoped to extract

a single equilibrium from the Pareto setTrading paths to equilibrium

really aren’t determined

The equilibrium price is a

• x

2

P.S.

terminal property of real

trade

Need not restrict prior

•

paths of trading

The equilibrium price can

•

x

1

be quite unrelated to the

F. Hahn and T. Negishi (1962)

Walrasian price

“and you may ask yourself ‘how did I get here?’ ”The mathematical

structure of

thermodynamicsState relations

General statistical systems

•

S = S(V,E)

U

have E, S, not predictable

Closed-system,

Only for equilibrium

•

irreversible

systems is E also a

V

Open-system,

reversible

constraint U

S(V,U) = max(S)| deﬁnes

V,U

•

the “surface of state”

S

E

U

Equation of state is not

Reversible and irreversible

•

transformations result in the

dependent on the path by

same ﬁnal state relation

which a point is reachedDuality and Gibbs potential

!

!

∂S p

!

State: =

S(U,V )

!

V

∂V T

U

1 p

dS≡ dU + dV

S

U

T T

! " ! "

# $

1 p 1 p

δ U + V − S = U δ + V δ

T T T T

! !

! !

∂ (G/T) ∂G

G U + pV − TS

! !

= = V

=

! !

∂ (p/T) ∂p

T T

1/T TConnecting thermodynamics

to mechanics

V

F = p*area

S(U,V ) A(T,V ) = U− TS

1 p

dA=−pdV − SdT

dS≡ dU + dV

T T

!

!

!

!

∂S p

∂A

!

!

=

− = p

!

!

∂V T

∂V

U

TReversible transformations

reservoir (T)

and work

!

!

∂A

2

1

p

p

!

− = p

piston

!

∂V

T 1 2

V V

Load

!

" #

1 2 1

ΔW = p − p dV

!

" #

A

1 2

= − dA + dA

= −ΔA

Helmholtz “free energy”

Analogies suggested by duality

Surface of state Indifference surface

S(V,U) = max(S)|

u(x) =U

V,U

Increase of entropy Increase of utility

δS≥ 0 δU≥ 0

Intensive state variables Offer prices

!

!

∂S p

∂u

!

=

∝ p

i

!

∂V T

∂x

i

U

Gibbs potential Expenditure function

G = U + pV − TS

e(p,U)≡ min[p · x | u[x]≥ U]

xProblems (1): counting

Different numbers of intensive and extensive

•

state variables (incomplete duality)

(U,V ) x = (x ,x ,...,x )

0 1 n

! "

1 p

, p ˆ≡ (p ,p ,...,p )/p

0 1 n 0

T T

Entropy is measurable, utility is not

•

G(p,T) e(p,U)

Total entropy increases; individual utility does

•

δS≥ 0 δU≥ 0Problems (II): meaning

T

−pdV = dW = dU− TdS

p

x

2

P.S.

A

x

1

Essence of the mismatch

In physics, duality of state constrains transformations

•

The “price” of this power is that we must limit

ourselves to reversible transformations, and cannot

conserve all extensive state variable quantities

In economics, conservation of endowments forces

•

irreversible transformations

The result is that dual properties of state become

irrelevant to analysis of transformationsFinding the right

correspondenceThree laws in both systems

Encapsulation

•

The state of a thermodynamic Economic agents are characterized

system at equilibrium is completely by their holdings of commodity

determined by a set of pairs of bundles and dual offer price systems

dual state variables to each bundle

Constraint

•

Energy is conserved under arbitrary Commodities are neither created nor

transformations of a closed system destroyed by the process of exchange

Preference

•

A partial order on states is deﬁned A partial order on commodity

by the entropy; transformations that bundles is deﬁned by utility; agents

decrease the entropy of a closed never voluntarily accept utility-

system do not occur decreasing tradesThe construction

Relate the surface of state to indifference

•

surfaces correctly

Study economics of reversible

•

transformations

Associate quantities by homology, not by

•

analogyQuasilinear economies:

introduce an irrelevant good

x≡ (x ,x ¯)

0

Indifference surfaces are

•

translations of a single

u(x) = x +u ¯(x ¯)

0

surface in x (hence so are

0

all equilibria of an economy)

All prices on the Pareto Set

•

are equal

Differences among equilibria

•

have no consequences for

future trading behavior

Duality on equivalence classes

∂u ¯ p

i

Independent of distribution

=

∀i > 0

of x among agents

0

∂x p

i 0

Equivalence class of expenditures corresponds to Gibbs

e (p,U) = p [U− u ¯ (x ¯)] + p ¯· x ¯

p ↔ T

QL 0 0

e − p U↔ G =−TS + (U + pV )

QL 0

Resulting economic entropy gradient is normalized prices

p ¯

S = u ¯(x ¯) dS = dx ¯ ·

QL

QL

p

0Reversible trading in a closed

economy

!

" #

1 2

Ext. speculator’s proﬁt = − p dx + dx

0

0 0

!

2

" #

x

1 2 1

= p ¯ − p ¯ · dx ¯

" #

δS

1 2

P.S.

= p Δ S + S

0

QL QL

But S is a state variable!

QL

1

Same for rev. and irrev. trade

x

0

Money-metric value of trade is the amount agents

could keep an external speculator from extractingProﬁt extraction potentials in

partially open systems

e− p U p ˜· x ˜ p

0 0

= x + − u ¯(x ,x ˜)

x≡ (x ,x ,x ˜)

1 1

0 1

p p p

1 1 1

2

~

x

Economic “Helmholtz” potential

p −δA

0

P.S.

A = x − u ¯(x ,x ˜)

QL 1 1

p

1

p ˜

1

dA =− · dx ˜

QL

x +(p /p )x

1 0 1 0

p

1

reservoir (T)

!

!

∂A

! 2

1 p

p

− = p

!

piston

∂V

T

1 2

V V

LoadAggregatability and

“social welfare” functions

QL economies are the most general

•

aggregatable economies independent of

composition or endowments

(Obvious reason: dual offer prices are now

meaningful constraints on trading behavior)

For these, a “social welfare” function is the

•

sum of economic entropies

Such economies are mathematically identical

•

to classical thermodynamic systemsA small worked

exampleThe dividend-discount

model of ﬁnance

Contract Energy Conservation

1

δU =−pδV +δQ

δM =−p δN + δD

N

rδt

Constant Absolute Risk Aversion (CARA) utility model

! "

¯

Nd

2

¯

U≡ Nd 1− σ − D +φ(M)

2ν

(x ,x ,x )≡ (−D,M,N)

0 1 2

(p ,p ,p )≡ (1/rδt,1,p ) (T,1,p)

think

0 1 2 NThe state-variable description

Economic entropy and basis for the social welfare function

!

! "

¯

!

Nd dφ ∂S

2

¯ !

rδt= =

S≡U + D = Nd 1− σ +φ(M)

!

dM ∂M

2ν

N

Economic “Gibbs” part of the expenditure function

!

!

∂G

1

!

= N

G = M + p N− S

N

!

∂p

rδt

N

rδt

Economic “Helmholtz” potential for trade at ﬁxed interest

!

!

∂A

1

!

=−p

A = M− S

N

!

∂N

rδt

rδtSummary comments

Irreversible transformations are not generally

•

predictable in either physics or economics by

theories of equilibrium

They require a theory of dynamics

•

The domain in which equilibrium theory has

•

consequences is the domain of reversible

transformations

In this domain the natural interpretation of

•

neoclassical prices may be differentFurther reading

P. Mirowski, More Heat than Light, (Cambridge U. Press, Cambridge, 1989)

•

L. Walras, Economique et Mecanique, Bulletin de la Societe Vaudoise de Science Naturelle

•

45:313-325 (1909)

I. Fisher, Mathematical Investigations in the Theory of Value and Prices (doctoral thesis) Transactions

•

of the Connecticut Academy Vol.IX, July 1892

F. Hahn and T. Negishi, A Theorem on Nontatonnement Stability, Econometrica 30:463-469 (1962)

•

P. A. Samuelson, Structure of a Minimum Equilibrium System, (R.W. Pfouts ed. Essays in Economics

•

and Econometrics: A Volume in Honor of Harold Hotelling. U. North Carolina Press, 1960),

reprinted in J. E. Stiglitz ed. The Collected Scientiﬁc Papers of Paul A. Samuelson, (MIT Press,

Cambridge, Mass, 1966)

J. H. C. Lisman, Econometrics, Statistics and Thermodynamics, The Netherlands Postal and

•

Telecommunications Services, The Hague, Holland, 1949, Ch.IV.

J. A. Bryant, A thermodynamic approach to economics, 36-50, Butterworth and Co. (1982)

•

Tj. Koopmans, Three Essays on the State of Economic Science (McGraw Hill, New York, 1957)

•

G. Debrue, Theory of Value (Yale U. Press, New Haven, CT, 1987)

•

H. R. Varian, Microeconomic Analysis (Norton, New York, 1992) 3rd ed., ch.7 and ch.10

•

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