# Classical thermodynamics and economic general equilibrium theory

Mechanics

Oct 27, 2013 (4 years and 6 months ago)

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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
“Pareto Optimum” deﬁnes

x
1
No trade any agent can propose
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
Need not restrict prior

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
!
" #

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

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
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
“social welfare” functions
QL economies are the most general

aggregatable economies independent of
composition or endowments
(Obvious reason: dual offer prices are now
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)

(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

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
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