niehans money in static theory of optimal payment 195383

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Money in a Static Theory of Optimal Payment ArrangementsAuthor(s): Jürg NiehansSource: Journal of Money, Credit and Banking, Vol. 1, No. 4 (Nov., 1969), pp. 706-726Published by: Ohio State University PressStable URL: http://www.jstor.org/stable/1991447Accessed: 20-08-2014 15:34 UTC

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*

JURG NIEHANS

Money n a StaticTheoryof Optimal

PaymentArrangements*

INrRoDucTIoN

A MODERN CONOMYs characterizedby the virtually

universal use of media of exchange, collectively called money. This poses

certainquestions,about as old as economics tself. Why do we use such media

of exchange?What commoditiesare selected as media of exchange?What is

the economic servicewe get from money? Some of the answers,as we know,

are to be found in the fact that the economy may deviate from equilibrium,

that expectationsare subject to error and uncertainty.This includes those

motives for holding money which are usually called speculative nd pre-

cautionary. Though-or rather just because they have captured most of

the economists' attention in recent decades, they will not be the subject of

this paper. Other answers,however, are valid even in economic equilibrium.

Paragraphs n them belong to the time-honored nventoryof textbooks and

treatises.We learnthat a mediumof exchangegives more scope to the division

of labor. We learn that as media of exchangewe should choose commodities

which have a stablevalue, an appropriate alue per pound, and all those nice

propertieswith the fancy Victorian names like portability, ndestructibility,

homogeneity,divisibilityand cognizability.lWe learn that the basic service

of money consists in the convenience t oSers in facilitatingexchange.

There is the recurringmetaphor of the oil which lubricates xchange.2

* Research or this paper was supportedby a National ScienceFoundation grant.

1 Representative xamplesare offeredby J. Stuart Mill [13, pp. 5 f.], Jevons [9, pp. 30 f.],

and Menger [12, pp. 261 f.]. About the Aristotelianorigin of this tradition see Schumpeter

[16, pp. 62 f.].

2

See Marshall[11, p. 38] and Wicksell 20, Vol. 2, pp. 4 f.]. Adam Smith iked to talk about

the wheelof circulation 17, Vol. 2, pp. 18, 21].

JURGNIEHANSs professor S economics t JohnsHopkins University.

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JURG

NIEHANS

: 707

These

are

plausiblearguments,

but they do

not amount

to an

economic

anal-

ysis.

In fact,

such an analysis

s still lacking.3

This

paper s intended

o

furnish

some

components

o a

moremodern

heoryof money

n

economicequilibrium.

The problems o be foundin an equilibriumheoryof moneymay be sub-

divided

into dynamic

aspects involving

time

and static

or timeless

aspects.

The

present

analysis

will concentrate

on the

static

aspects. There

will

be no

assets

and

commodity

stocks,

but only current

lows

of production,

exchange,

and

consumption.

The

main problem

will

be the optimal

arrangement

f these

flows. This

excludes

some important

aspects

of

moneysuch as

its advantages

in bridging

he

gap between

receipts

and expenditures.

t also restricts

he

analysis

to commodity

money. In

principle,

the extension

of the present

approach

to these dynamic

problems is

straightforward,

ut

it requires

a

separate

paper.4

1. The Bilateral

Balance Requirement

Consider

an

economic system

with

a number

of agents

or traders.

Each

agent

s endowed

with

certain

resources.Each

has

a certain technical

knowl-

edge,

represented

y a

production

unction,

and

certaintastes,

represented

y

a utility

function.

Suppose

we permit

a Walrasian

equilibrium

o

be estab-

lished. This

equilibrium

will

determinea

network

of transactions,

pecifying

whatcommoditieswill flow from agentA as theirorigin to agentB as their

destination.

These

transactions

will

be called ultimate

lows. The

network

of ultimate

flows

will satisfy

the budget

constraint

or

each agent.

However,

for three

or more

traders here

s no

reasonwhy trade

between

any two

should

be

balanced. To

use

Wicksell's familiar

example: Certainly

A may

supply

wheat

to B, B

may supply

fish to

C, while C supplies

timberto A

in a tri-

angular

network[20,

Vol. II,16

f.]. Economic

efflciency

will, in

general,

mply

bilateral

mbalanceof

ultimate

lows.

An ultimateflow from origin A to destinationB may conceivablypass

through

the hands

of

intermediate

agents.

In general,

the network

of what

will

be called

the actual

lows

may be diSerent

rom the ultimate

low net-

work.

The

policing of

an exchange

economy

requires bilateral

balance

of

the actual

flows.

One way or

another,we

must

see to it that

nobody

can get

something

for

nothing.A,

when supplying

wheat

to

B, wants to

make sure

thathe

will, in

turn, get timber

rom

C, etc. For

this reason,

there has

to be

a

quid

pro quo in

every

transaction.

This

creates a problem:

While efficiency

will

generally

equire

bilateral

mbalance,

he working

of the exchange

system

requiresbilateralbalance. It may be argued that this problem s the appro-

priate

starting

point for

an equilibrium

heory

of money.

3 The discussion

of the roleof money

for

the exchange ystem

and

of the choice of

a medium

of

exchange n Samuelson's

ntroductoryext

could

well havebeen

written

more thana century

earlier

which s probably

ust

the impression

he author

wantsto convey.

[15,pp. 52

f.].

4 In addition

to

the use of a time

dimension,

he extension

nvolvesthe introduction

of hold-

ing costs for

assets.








708

: MONEY,

CREDIT, AND BANKING

Thereare two

solutions

to this problem.

In one we

force people to renego-

tiate

their ultimate

flow contracts

subject

to a bilateral balance

constraint.

The

resulting

bilateralbarter system

restricts

trade to cases

of the double

coincidenceof wants betweenpeoplelike the starving ailor andthe shivering

baker

from economic

mythology.

The loss

in welfarewhich this

entails,

while

obvious

enough, has, to

my knowledge,

never really

been analyzed.

It may

be said

to arise from the

fact that

the lack of consistent

cross-prices

between

commodities emphasized

by Walras [19,

pp. 115 ff.]

prevents the resulting

exchangesystem

from being

Pareto-optimal.

According o the

other solution,

bilateral mbalance

n the

ultimateflows is

permitted

and bilateralbalance

is

provided

by introducing

ntermediate

lows

into the transactions

network,

.e.,

flows which do not move directly from origin to destination.To use again

Wicksell'sexample,

A can

exchangehis wheat

againstfish from

B (which

he

does

not want)

and then exchange

the fish against

timber from

C. A trade

network

in which

indirectflows

provide

bilateral balance for

each pair

of

traders

will herebe called

a payments

ystemor payments

arrangement.

or a

given network

of ultimate

lows there are usually

various

ways to provide

for

bilateral

balance

and thus various

possible

paymentsarrangements.

ome

of

these

alternativesmay

be entirelynon-monetary,

while others may involve

the use

of a mediumof exchange.

The economist s thus

faced

with the familiar

twin problemof (1) determining he optimalalternative,and (2) explaining

how market orces

will

settle on one of the

alternatives.

f he happens o

be a

monetary

theorist, his first

task

will be to explain under

what circumstances

the emerging

payments

arrangementmay

be monetary n nature.

Of

course, if transactions

were

costless for all commodities,

all payments

arrangements

would be

equally

good and the market

would

probably select

one of them at

random.

This is the usual

assumption

on which general

equi-

librium

theory

is, explicitly or

implicitly, based.

Once this

assumptionis

made,

we have closed the

door to

an equilibriumheory

of money.

If, on the

otherhand, transactionscost something,we have a criterionwhich discrim-

inates

between

various alternatives

and thus

a basis for optimization.

In an

equilibrium

heory of

money, transactions

costs thus

seem to

play a crucial

role.

Oncewe have

determined

he optimal payments

arrangement,

we still

have

to compare

t with the equilibrium

rising rom bilateral

barter.

This is a case

of weighing he

welfare

osses from barter

againstthe

additional ransactions

costs, if any,

of the optimal

paymentsarrangement.

This again seems to

be a

questionwhichhas neverexplicitlybeen considered,at least not in termsof

modernanalysis.

It is easy

to conceiveof

cases where,

in view of high

trans-

actions

costs

in one system and

moderate

distortions of price

ratios

in the

other,

the comparison

goes in favor

of bilateral

barter. The

development

of

indirect

trade, it seems,

cannot be

just a matter of

progress n

knowledge; t

is also

a matterof the given

resources, astes,

and transactions

osts. The

fol-








* fl

JURG

NIEHANS

:

709

lowing

analysis

will concentrate

on the

determination

f the

optimal

pay-

ments

arrangement.

2. The

Meaning

of Transactions

Costs

The

terms transactions

osts

or

transfer

costs

shall be

used

for the

costsassociated

with the

transfer

of ownership

rom one

individual

o

another.

They

are catchall

terms

for

a rather

heterogeneous

assortment

of costs.

The

parties

have to

communicate;

nformation

will

be

exchanged;

contracts

are

drawn

up; the

goods

must

be inspected,

weighed,

and

measured;

nd

accounts

have to

be

kept. To

a certain

extent,

transactions

nvolve

additional

trans-

portation n spaceoverand abovewhatis required o move goods frompro-

ducer

to consumer.

Some

may argue

that

all transactions

costs

are really

costs

of gathering

nformation,

but it may

be better

not to

be dogmatic

about

this.5

In

part,

transactions

costs

may

vary with

the

quantity transferred:

he

transfer

of two

automobiles

may

involve

almost

twice

as much

trouble

as

thetransfer

of one. Another

part of transactions

osts

may

be fixed:

he actual

cost

of a given

stock

transaction

o the broker

s

little affected

by the number

of shares.

It

will turn

out

that this distinction

s of

far-reaching

ignificance.

It will herebe assumed hat the transactions osts also dependon the com-

modity

and on the

traders:

between

the same

persons,

cost may

be

lower for

wheat

than

for fish,and

for

the same

commodity,

hey

may

be lower

between

A and B than

between A

and

C. It will

also

be assumed

that

transactions

costs

are counted

separately

or

the

two commodities

exchanged

n

a given

transaction

nd that

the

costs for

the two

partsof

an exchange

are

ndependent.

For the

cost of

transferring

wheat

it

thus makes

no

difference

whether

t is

exchanged

for fish

or

for timber.

This

last assumption

may,

of course,

be

unrealistic,

but

it seems to

be

justified by

the

considerable

implification

t

allows.

Finally,

we shall

assume

that

the

ultimateflows

are not affected

by trans-

actions

costs. This

is

a most far-reaching

assumption.

It

means

that trans-

actions

costs

are not

paid out

of the

resources

available

for

other

purposes

but can

be charged

o

some imaginary

pecial

account.

In this

way

theproblem

of the

payments

arrangements

s

theoretically

eparated

romthe

determination

of the

ultimate

lows

in general

equilibrium

nalysis.

What

is, in fact,

a

joint

problem

s

artificially

ivided

nto separate

parts.This

procedure

will

preclude

considerationof some of the most fundamentalproblems.For example,it

will not

be

possibleto

show

how the

exchange

system

gradually

passes

from

individual

self-sufficiency

o bilateral

barter and

further

on

to multilateral

6

The

outlinesof

an optimizing

approach

to the theory

of transactions

balances

basedon

information

costs were sketched

out by Brunner

and

Meltzer

[5 pp.

258 ff.]. The

hints

they

give

seemto

suggest

that there

are manypoints

of

contactwith

the approach

ollowed

in

this

paper.








7 I O

: MONEY,CREDIT,

AND BANKING

exchange

with the gradual

lowering

of transactions

costs

relative to other

costs.

Relaxing

this assumption

will thus

be one of the important

points

on

the agenda or

further

work. At the moment,

however,this

assumption

has

the v*tue of permittingat least a beginning.

3.0ptimalPaymentsArrangementsthroughPlanning

For

the purposeof this

section

we assumethat

there is a

centralplanning

authority

rying

to arrangepayments

n

an optimal way. For

the time

being

we shall

concentrate

on the case

of variable

ransactions osts.

It will

be as-

sumed that transactions

costs are proportional

to

the amount transferred.

Theproblemsarising rom fixedtransactions osts will be dealtwithinanother

section.

All flows will be

measured

n terms of some

arbitrary nit of

account

(which is not

necessarilya medium

of exchange)

at the prices

determined

by

the general equilibrium

or ultimateflows;

they

are values, not quantities.

Supposethere

are n transactors

and n commodities.

Transactor

produces

zffi f̂

commodityh. It

will be assumed

hat each transactor

produces ust one

commodity,so

that z$^

> 0 for h = i,

Zsh = 0 for

h

#

i. On the other

hand,

ransactor

consumes

certainamounts

of the

commoditiesproduced

by others,

tg^.

The consumption

of one's

own products will

not be

considered, i.e.,

Y, = 0. For each commoditytotal productionequalstotal consumption:

y

Zxh =

SyXh

(h

= 1 ... n).

(1)

These

may be called

conditions of market

equilibrium.

The budget

Coll-

straints

require hat for

each transactor

zz*= Eysh

[i-l---(n-1)].

(2)h

There

are (n-1)

independent

budget constraints,

he nth

depending

on the

otherstogether

with (1).

(1) and (2) relate

to the

network of ultimate

flows

which,in determining

he

optimalpayments

arrangement,

must be considered

as given.

It

is conceivable

that yjz travels

straight

from producer

to consumer .

In

many cases, however,

commodities

will travel

more

or less indirectly.

Let'scall x^j he actualflow of commodityh fromtransactor to transactor

(x^j>

O). A solution

to our problemconsists

in

a specificationof

all x^,.

Under

the above

assumptionst

contains

n2(n- 1) variables.

t is a

feasible

solution

if two sets of

conditions

are satisfied.

F*st, for

transactor , the

excess

of total purchases

over total

sales of commodity

h

must be equal to

the excess of

consumption

over production:








7 I 2

:

MONEY,

CREDIT,

AND

BANKING

(3

oX

FIG. 1.

Ultimate

F10WS

.

//

A

\\

A

W

/

/

W

W

\

>

W

A

B

C

FIG. 2.

Alternative

Payments

Arrangements.

flows

corresponding

o

these

ultimateflows.

Bilateral

balanceof

actualflows

can

obviously be

provided

by

many

different

payments

arrangements,

ome

of

whichare

depicted

n

Figure2.

The

problem

consists

n

finding

he

solution

with

minimum

ransactions

osts.

Suppose the

transactions

osts per

unit of

flow

arethose

given in

Table

1,

where he

entries

consist of

arbitrarily

elected

numbersbetween0 and 9.

The

transfer

of 100

units of

commodity1

from

trader 1

to

trader 2

would

thus

cost

900,

while

the

transfer

of

commodity2

from trader

3 to

trader 2

would

be free.

Multiplying

he

various

flows,

each

of

which

amounts

to

100,

with

the

appropriate

ransactions

costs

and

adding

overall

flows,

the

total

transactions

ost

of

arrangement

above

can be

computed

as

2600.

Intuitively,

there

seems to

be no

particular

eason

to

believe

that

this

solution is

better

or

worse

than

any of

the

others.

By

solving

the

linear

programming

roblem

withthe aid of a computer8he planningauthoritywill findthat A is, in fact,

optimal.This

means that

under the

given

conditions

commodity4

should

be

8

This

problemhas

48

variables

and 21

constraints.

One

further

constraint

was

added

by the

fact

that for

computing

the

equalities

had to

be

converted nto

minima,while

the

22nd con-

straint

provided

he

maximum

or

theirsum.

Computations

or

thisand

the

following

examples

were

carried

out at

the

Johns

Hopkins

ComputingCenter

on

the basis

of a

program

written

by

Professor

George

Nemhauser.

The

author

acknowledgeshe

valuablehelp

of

Thomas

Kelly.








JURG NIEHANS : 7 I 3

TABLE 1

TRANSACTIONSOSTRATES

Buyer

1 2 3 4

Commodity 1 2 3 1 2 3 4 1 2 3 4 1 2 3 4

Seller:

1 9 5 7 8 3 8 1 1 7 2 4 4

2 1 5 8 0 5 9 1 3 8 3 5 6

3 2 6 8 8 6 0 1 1 0 3 5 4

4 3 7 8 6 7 0 1 9 1 4 5 2

assigned he role of a mediumof exchangeagainstwhichall othercommodities

are traded.It is noteworthy hat commodity4 was assigned his role though

it is not obvious from the table that it has the lowest transactionscosts.9

Since in this system there is also a non-monetaryuse for commodity 4, the

latter may be called a commoditymoney. Of course, if the number of

tradersand commodities ncreases, he size of the programwill rapidlyexceed

the capacityof even a large computer.From the presentpoint of view this is

not important,however,because he emphasis s not on the numericalaspects,

but on the generalnature of the problem.This will become clear in the next

section.

4. OptimalPaymentsArrangementshrough he Price System

There is also another way for the authorities o put the optimal solution

into effect.l°To a given commodity in the possession of a given traderthey

can assign what, in the presentcontext, may be called place values. These

are diSerentialprices at which a given commoditycan be bought and sold at

the various nodes of the network; hey have to be added to the prices deter-

mined by the equilibriumof ultimateflows. The traderswill then be free to

arbitrage, .e., to exchangecommoditiesas they please, provided hey always

pay the consequent ransactions osts. Whenever he gain in place value from

a transactionexceeds its transactionscost, profit-seeking raders will carry

it out. Whenever he reverse s true, the transactionwill not take place. By a

suitable choice of place values the authoritiescan always see to it that the

optimal arrangement merges.The correct place values are furnishedby the

solution of the dual to the above programming roblem n the form of shadow

prices.For the numerical xampleused in the preceding ectionthey are given

in Table 2.

9 It may be noted that in the above example total transactionscosts are actually lower if

commodity 4 moves from trader4 to trader 1 along an indirectpath than if it moves directly.

It thus turns out that with the particular ransactions osts of Table 1, bilateralbalance does

not impose a burdenon the economy. In a majorityof cases, this is not so.

10The aspects of duality theory used in this section are well known. No references were

thought to be necessary.








7

I

4 : MONEY,CREDIT,AND BANKING

TABLE 2

PLACEVALUES*

Trader commodity 3 4

1 0 0 2 7

2 9 0 3 7

3 0 5 0 2

4 0 1 5 0

* Shadow prices on flows in the optimal program are in italics.

(3 0 9 9 X

7 0 7 0

l 9

(i) 5 5 O (E)

FIG.

3.

OptsmalPaymentsAxtangement.

For the flows appearing n the optimal solution, we thus get the pictureof

Figure 3. The figures n the middle of each arrow indicate the trallsactions

cost, whereasthe figures at the corners express the place values or shadow

prices. Consider,say, trader2. By selling one unit of his own commodity to

trader 3, he gains 5. By buying one unit of commodity 1 from trader 1 he

gains 9. By buying one unit of commodity4 from trader3 and reselling t to

trader 1, he gains 5. But his total gain of 19 is matchedby transactions osts

of the same amount;he just breakseven. The same is true for the other trans-

actors for those flows which are actually in the optimal solution. For each

exchange n the optimalsolution, the total gain in place value is just equal to

the transactions ost. If similar onsiderations re applied o possibleexchanges

not included n the graph, t turns out that they result n a loss. With the given

shadow prices,the transactorsust breakeven if they adopt the optimal solu-

tion, whereasany deviationhas to be paid with losses. This is true not only

for transactors onsideredone at a time, but also in the aggregate: f trans-

actions costs are minimized, heir total of 2600 is equal to the aggregategain

in place value over all transactions n the example.








JURG

NIEHANS : 7 I 5

In fact, it

is not

even necessary o

considerboth

sides of a given

transaction.

It

is enough

to look at each

flow individually. t is

not

generally rue, though,

that for each flow in

the optimalsolution

the gain

in place valuejust

matches

transactions osts. This is indeed the case for the exchangesbetweentrader 1

and trader

2 and

between rader3 and

trader4. It is

not true,however,

between

trader2

and trader3.

But so far, we have

not made

use of all shadow

prices.

In

particular,we have made no

use of the shadow

prices the

dual assigns to

the bilateral balance

constraints.They can be

interpretedas

measuring he

value of

keeping he

exchangebetweenany

two

traders n balance.In the

given

example, considering

only

relations appearing n

the

optimal solution, the

computer

gives them

as:

0, for flows fromtrader2 to trader1;

4, for

flows from trader2

to trader3;

0, for flows from

trader3

to trader4.

For flows in the

reverse

direction these figures

assume

the opposite sign.

Their role

can best

be seen as follows. If

a unit of

commodity4 moves

from

trader 3 to

trader 2,

the gain in place

value is 5,

while the

transactions ost

is

only 1. But a flow

in this

directionhas to bear

an

additionalcharge of 4,

that is, it has a bilateralbalancevalue of minus 4, because t putspressureon

the

bilateralbalance

constraint.On the

other hand,

if a unit of

commodity2

movesfrom

trader2

to trader3, it gains 5

in place

value, while the

transaction

costs 9.

But there is

an additional

premiumof 4

on this transaction,

because

it relieves

the

pressure or bilateral

balance

between the two traders.

When-

ever,instead of

considering

both sides of an

exchange,we concentrate

on just

one of the

flows in the optimal

solution,

the sum of the gain in

place

value and

the value

of bilateral

balance will be just

equal-to

the transactions

ost. For

flows not

in the

optimal solution,

however, the

gain in place value

plus the

bilateralbalancevalue will be smaller han the transactions ost. If both sides

of an

exchangeare considered

ogether,

he bilateralbalance

value

drops out,

because t

enterson both sides

in equal

amounts,but with

opposite

signs.

By

assigningappropriate

place values

to commodities, he

authoritiescan

thus call

forth an optimal

payments

arrangement.f, in

addition, they

make

use of

specialpremiumsor

charges

reflectinlghe pressureon

bilateral

balance,

the tradersdo not

even have to

look at both sides of

a given

exchange,but can

consider each flow

individually. n general, the

dual assigns

place values to

commodities at given

nodes

and bilateral balance

values to

links in such a

way thatthe total gain in placevalue is maximized,subjectto the

constraint

that there

be no flow

for which he gain in

place

valueplus thevalue of

bilateral

balancebe higher

than the

transactions ost. More

formally,

the dualcan be

set out as

follows. Denote the

place

value of commodity h

at node i

by

psh

and the value of

bilateral

balance between

traders i and

j by pXj,where

pfj = -pji

. The

problem henis








7

I

6 : MANEY,CREDIT,

AND

BANKING

max E E

psh @$h _ z.h)

h

$

subject o

pjh_ p,h + bXj

,j < c,j (i

#

j)

whereb,j = O or one

pair of traders,ll

ay (n-1) and n, and

b,j- 1 other-

wise. It

may be checkedthat the dual

is relatedto

the primalproblem n the

usual way. The fact

that, say,

b(n_l)

n = O in the

dual

corresponds o the

observation hat in the

primal one bilateralbalance

constraint

ollows from

the rest.

There are thus two ways of plannillg an optimal paymentsarrangement,

one by

quantitative

planning, he other through

appropriateprices

set by the

planning

authority.The next question

is whether

central planning is really

necessary or this

purpose,or whether he market

mechanism, f

left to itself,

would spontaneously

call forth the

appropriateprices.

This

questionhas a static and a

dynamic aspect.

From the static point of

view, it is

well known that in the

optimal solution

of a linear programming

problem

he shadowpriceshave the

attributesof

equilibrium rices n perfect

competition. n the presentcase, all

transactors,

while maximizing heir indi-

vidual

profits, are just breakingeven,

and the total of all

transactionscosts

is just

coveredby the additionalvalue

of the goods.

At the sametime, there

can be no

competitive

price systemwith these

characteristics

which is not an

optimal

solution to the linear

program.Under

perfect competition,equilib-

rium

implies an optimal payments

arrangement.

f any commodities are

adopted as media of

exchange, hey will beFhe

right ones.

Under perfect

competition, t turns out that money

does indeed manage itself.

At this fun-

damental

evel, Bagehot's famous

dictum to the

contraryseems to have no

support.The adoption of money requiresneitherlaw nor convention, nor

can it be

attributed o an

invention ;t is simply he

effectof

market orces.l2

The

dynamicproblem s whether,

tarting rom a

disequilibrium rice system,

market

forces, having

propertiesanalogousto the

Simplex

algorithm,would

gradually

convergeto

equilibrium. t may be

intuitivelyplausibleto assume

that they do, forcing an

eliminationof all those

transactions

which do not

pay their way and

encouragingan

enlargeduse of those that

yield a profit.

However,

we know that this is not

certain.As a

consequence,we cannot be

ll This corresponds o the fact that in the primalproblem,bilateralbalance for one pair of

traders

ollows from the rest of the system.

1 This is

what, among others, Menger

maintainedagainst

those who emphasized he role

of the state,

custom,

convention or invention[12, pp. 253 ff.].

It may be contrastedwith the

contraryview

expressed n,

say, Samuelson'sntroductory ext

[15, p. 54]. It is

shown in section

7 that fixedtransactions osts

reverse he

conclusion. The argument n the text

is, of course,

not intendedto deny the

historicalrole of

government ntervention n monetary

matters nor

the existence

of many good

reasonsfor it.








JURG NIEHANS

: 7 I 7

TABLE

3

ULTIMATEFLOWS

II

Producer

Conser

Total

(Commodity)

1

2 3

4 Production

1

-

50

170 0

220

2

0

- 20

130

150

3

220

0

- 20

240

4

0

100

50 -

150

quite sure that monetary

arrangementswill

actually

approach he optimum.

It should

be noted, though, that

this element

of uncertaintys not

specific o

paymentsarrangements nd money, but pervades he analysisof competitive

equilibrium n general.

6.

DiffierentTypes

of PaymentsArrangements

So far,

the discussionhas been

based on

the simpleexampleof

a rectangular

networkwith arbitrary

ssumptions

about transactions

osts. Actually,

t was

not expected

beforehand hat

commodity

4 would emerge as

a medium

of

exchange n this case.The question s what paymentsystemswill result from

particularassumptions

about transactions

costs. Some

significantcases will

be discussed

n this section.

The discussion

will be based on the

somewhat

less simple

network

of ultimateflows given

in Table

3, constructedmore

or

less at random. It

may be easier

to visualize this network

with

the help of

a graphical representation

Fig.

4). In this graph,

the circles

denote the

traders,

while the

commodityflows are represented

by arrows, the width

of

which

is proportionate o the

size of the flow. The

circle of each

traderis

shaded n correspondence

o the

commodityhe produces.

It is

clear that the

ultimateflows fail to satisfythe bilateral balance requirement.The various

cases arisingout

of this network

of ultimateflows

are distinguished

by dif-

ferent

assumptions bout transactions

osts

resulting n diSerent

arrangements

for bilateralbalance.

Case

1: Medium f Exchange.

For a commodity

o emergeas

the universal

means

of payment,

t is sufficient hat its transactions

osts are sufficientlyow

relative

o the transactions osts

for other

commodities.Suppose,

or example,

that transactions

costs for commodity

1

are unity regardless

of the traders

involved, while the transactions osts for other commoditiesvarybetween4

and 9,

dependingboth on the

commodity

and on thetraders.This

assumption

will result

in an optimal payments

arrangement

f the type represented

by

Figure

5. For all

commoditiesexcept the

first, the optimal

flows are identical

with the ultimate

flows in Figure

4. Commodity 1

is then used

to provide

bilateral balance

for each pair

of traders. It thus

assumes the

function of








7 I 8 : MONEY,CR8D1T,AND BANKING

Commo ty 2

D

Cornmodity ,

* Commodity t

_ G_

FIG.4. Ultimate Flows II.

money as a medium of exchange.With some imagination, he resultingpay-

ments arrangementmay be likened to the gold standardsystem where com-

modity 1 represents old. For trader1, representing outh Africa, gold is just

an export commodity without monetary mplications.For trader 2, on the

other hand, gold appears as just an import, again without monetary signifi-

cance. For trader3, gold is commoditymoney, being used simultaneoulsy

as a medium of exchange and for non-monetarypurposes. For trader 4,

finally,gold appearsas puremoney with no final demand or it.

- It is interesting o observethat in Figure 5 none of the traderscan be said

to have a dominatingposition. In particular, he producerof the money com-

modity in no way emerges as an exchangecenter. This can immediatelybe

applied to internationalarrangements. f transactionscosts for, say, dollars

are lower than for other currencies, he dollar will emergeas an international

mediumof exchange. t does not follow, however, hat New York will become

a financial center; indeed, the low transactionscosts for dollars, wherever

they are used, may actually prevent New York from acting as a financial

center.

It should be noted that seemingly small differences n transactionscosts








i

JURG

NIEHANS

:

7

I

9

FIC}.

.

Medium

of

Exchange.

may

e

enough

to

let

some

commodity

emerge

as

medium

of

exchange.

For

example,

f

the

transactions

osts

for

commodities

1

and

2

are

zero

and

unity,

respectively,

hile

for

commodities

3

and

4

they

are

8

and

9,

the

optimal

payments

rrangement

s

unchanged.l8The smalladvantageof commodity1is noughto

safeguard

ts

position

as

medium

of

exchange

n

this

case.

In

the

context

of

the

present

model

it

is

not

possible

to

construct

a

case

where

ome

commodity

s

pure

money

for

all

traders.

To

do

this,

it

would

be

necessary

o

allow

for

commodities

which

are

only

held

in

stock

without

current

onsumption

or

production.

In

principle,

his

should

not

be

difflcult,

but

n

addition

to

4

current

outputs

this

requires

a

larger

program

han

was

available

o

the

author

at

the

time.

Implicitly

or

explicitly,

general

equilibrium

analysis

seems

to

be

based

on

thessumption, hat thereis alwayssome commoditywith zero transactions

costs

o

be

used

as

the

universal

medium

of

exchange.

n

this

case,

the

optimal

flows

an

be

determined

egardless

of

any

details

about

transactions

osts-

we

ust

look

at

the

ultimate

flows

and

use

the

medium

of

exchange

as

the

But

he

shadow

pnces

are

different.








balancing tem. Since in fully monetizedeconomies we

have indeed a closely

related group of commodities,

called money, the

transactions osts of which

are low

relativeto all others,

such a model is perfectlyappropriate or many

purposesoutsidethe field of monetaryanalysis. t is not appropriate, owever,

if the question is exactly which

of a group of rivallingcommoditiesare used

as

money, how differentmonies

are assigned heirroles,andhow the payments

mechanism s aSectedby changes n transactions osts

for different andidates

for monetary unctions.

Case2: Centerof Exchange.

The emergenceof a

medium of exchange s

by no

means necessary.Let us assume, for example,

that transactionscosts

are

differentiated ot so much

between commoditiesas between traders.To

be specific,we may assumethat trader 1 has particular bilities,having trans-

actions costs of unity for all

commodities with all

other traders, while the

transactions osts betweenother tradersvary between

S and 9. In the optimal

solution

for this case, no

commoditycan be said to be a mediumof exchange.

Instead,

trader 1, not unnaturally,emerges in the

position of a universal

broker or centerof

exchange. That is, all flows

between any of the other

traders

disappearand traders 2, 3, and 4 make all

their deals through the

intermediary f trader 1. This

resultingnetwork s

depicted n Figure 6.

.1t

O

720 : MONEY,CREDIT,AND

BANEXING

FIG.

6. Center of Exchange.








JURG

NIEHANS : 72 I

This case is

suggestive of the

payments

arrangements ound

in societies

with little

developedmonetary

systems. It also

illustratescertain

aspects of

international

payments

arrangementsn

highly

developedmonetarysystems.

In particular, t may turn out that transactions osts for variouscurrencies

are

not significantly

diSerent,

while at the same time

there are

marked dif-

ferences

between various cities,

i.e.,

between London,

Copenhagen, and

Lisbon.

Dependingon the natureof

these diSerences,

Copenhagen

nd Lisbon

may thus find it

advantageous o

make theirdeals

through

London. In this

case, it is not the

dominant

position of the pound

which makes

London a

financialcenter. It

would be

equally incorrectto say

that the

comparative

advantageof London

helps to give

the pound a

dominating

position. Indeed,

as

Figure6

shows, the exchange

enter arrangement

ends to

keep transfers

of the center'sown goods at a

relatively ow

level. For the

commoditiesof

peripheral raders,on

the other

hand, the number of

transfers s

increased.

In

the above

example, he volume

of

exchange ransactions

nvolvingDanish

and

Portuguese urrencywould

thus increase

relative o the volume

of pound

transactions.

t can thus be said

that the need

for an international

mediumof

exchange s reduced f

some financialcenter

has a

dominatingposition as an

ntermediaryor

international

xchange.

Case3: Dual

MonetarySystem.

Even if money is

used, it is not

necessarily

true that one commodityemergesas the universalmediumof exchange.It is

easily possible

that onecommodity

ulfills his

function or some

transactions,

while another

commodity s used

for others.

This is the case of a

dual more

generally,a

multiple monetary

ystem.To

illustrate, et's assume

hat trans-

actions costs

for

commodity 1 are relatively

ow

(zero) between traders 1,

2, and 3,

while transactions osts

for

commodity2 are relatively

ow between

traders2, 3, and 4. All

other

transactionscosts are

between 4

and 9. The

optimal

solution for this

case is illustratedn

Figure7. It

appears, ndeed, that

traders 1, 2,

and 3 use commodity

1 to

balance theirflows, whereastraders

2, 3, and 4 use

commodity2. Now, traders1,

2, and 4

will not be consciousof

this

duality, because

they will

regard the respective

media of

exchange ust

as ordinary

exports or

imports. Trader 3,

however,

can hardly help to be

conscious of

using two

monies, since he uses

commodities 1 and 2 not just

becausehe

has a final

demand or them but as

media of

exchange,each with a

clearlydefined

range of functions.

The most

obvious

application f this case is

bimetallism. imilarphenomena

ariseundera

gold exchange

tandard,where

gold andsome national

currencies

are simultaneously sed for internationalpayments.But the most important

cases of

multiple monetary

systems are the

individualeconomies.

Typically,

a modern

economy

uses, side by side, a

considerable

number of diSerent

media of

exchange.

They consistof diSerent

coins, bank notes,

checks, bank

deposits,gold

bars, and

the like, often

including oreign

monies.Eachof these

media has its

characteristic ange

of use,

though theseranges are

often over-

lapping.What

is used as

money to buy real

estate or

Dutch Guildersmay not








u * - - - - - --w-w-|v-s-woN-&-a - w

| ---

I

722 : MONE, CEDA, A BANNNG

FIO.

7. Dual MonetarySystem.

be used to pay a parking meter. These media of exchange can certainly be

exchangedagainsteach other, often at a fixed price, but this exchangeusually

involves transactions osts of a more or less significantamount.l4The present

analysissuggests hat a study of the divisionof labor betweenpotentialmedia

of exchange on the basis of their comparativeadvantages or payments ar-

rangements s crucial for basic monetaryanalysis and for the understanding

of monetarydevelopment. t also suggests hat an analysisof transactions osts

in the frameworkof linear programmingmay be a fruitful introduction o

such a study.

7. Fixed TransactionsCosts: Money Does not ManageItself

So far, it was assumed hat all transactions osts vary in proportion o the

respective lows. For many transactions his is not actually true, transactions

costs rising less than in proportion o the value of the flow. To approximate

14Everydayexamples: he trouble of getting a dime for a pay station on a Sundaymorning,

the time and cost of cashing a check in an out-of-townbank, the trouble of getting a certified

check for a real estate transaction,etc.








JURG NIEHANS

: 723

such cases, we

may visualize ransactions

osts as consistingof a variable

and

a fixed component.

It will thus be necessarynow

to consider the problems

created by fixed transactionscosts.

To simplify the

discussion, we will, in

turn, disregardany variablecomponent.

Again, the

problemof minimizing ransactions osts

can be represented

s

a linear programming roblem.

The resourceconstraints

3) and the bilateral

balanceconstraints

4) are not aSected

by the changednatureof transactions

costs. The objective

function, however,

becomes more complicated.

Trans-

actions costs

for commodity h

being transferred rom trader i to trader

are now

t^jA^;, where shj = 1 if x^j

> 0, and hj = O if x^j = 0. The problem

is to minimizetotal transactions

cost T = Sh Ei

Ej t^jA^; subject to the

additionalconstraintsas follows:

O < a7,j

< 1

xhJ

<

ahjU, where U is some

number at least

as large as the highest x

a^Jnteger.

The additional

constraints ee to

it that for positivex, a must be 1, while

for

x equal to zero

the minimizing

operation automatically ees to it that

a is

zero [8a, pp. 252 ff.]. This is a

problem of integer

programming. t may be

classifiedas a multi-commodity etwork-flowproblemwith fixed initialcosts.

In the context

of such problems, Operations

Research has been

mainly

concernedwith

devisingalgorithmswhichmay make

the search or the

optimal

solution more efficient han complete

enumeration

2, 3, 4]. For the present

analysis,since

it is not concernedwith numericalcomputation,

his aspect

is

of little interest.

The main question is whether there

is still a good

chance

that an optimal payments arrangement

s consistent

with competitiveequi-

librium.Whatwe have to look at,

therefore, s the interpretationf the

shadow

prices.The mainpoint is that in integerprogramming,s GomoryandBaumol

have shown [7],

the shadow prices

have generallyno straightforward

nter-

pretation in terms of competitive

prices. Specifically,

hey will usually not

represent he marginalrevenueproducts

of the inputs

and it may well be that

they are zero even though there

is no excess capacity

n the usual sense. The

conclusion is

that in integer programming

not every efficient solution

can

be achievedby

simpledecentralized ricingdecisions

or by competitive

market

pricingprocesses.There may be

no set of market

prices which leads profit-

maximizers o the optimum. Gomory

and Baumol

also note that these diHi-

culties are related o increasing eturns o scale which are well knownto lead

to troublefor

competitivepricing.

There seems to be no reason

why these general

results should not apply

to the particular

monetary problem we are confronted

with. In the

present

case, increasing eturns o scale arise

n a particularly

cuteform, becauseonce

a transfer akes place at all, its

scale can be increasedwithout limit

at zero








724 : MOMY, CEDIT,

AND BANKING

marginalcost. We

would thus expect that in the optimal solution the

shadow

pricesfor all bindingresources onstraintsbecome

zero. We have to

conclude

that wherever ixed transactions osts are at

all significant, he market

mech-

anism cannot be expected o providefor an optimal paymentsarrangement.l5

In particular,we cannot

be surethat a mediumof exchangewill emerge

when-

evereconomicefficiency

emands t, nor is there

any guarantee hat the right

commoditieswill be chosen as media of exchange.

n these cases Bagehot

was

right: money does not

manage tself, in the most fundamental ense.

Instead,

there will be scope

for conscious planning

action to improve the payments

system.This may be

the ultimaterationale or the notion, so often expressed,

that money as a medium

of exchange s an

invention, hat somebodyhad

to see its advantages, hat its adoption requires

either a command

or a

convention.The analysisalso seems to indicate that even in equilibriumwe

cannot be sure that

there is no room for further

mprovements n monetary

arrangements.A considerablepart of technological

progress n the

mone-

tary field may actuallyconsist not in shifts

in production unctions

but pre-

cisely in findinga closer

approximationo the optimalpaymentsarrangement.

At the same time, this state of aSairs may provide

a blanket ustification-or

excuse for taxes, subsidies,

and all sorts of

regulations n this area.l6

rhe question is,

of course, how important

fixed transactionscosts really

are. There seems to be little direct informationon this point, but one can

hardly help feeling that

at least for the more

advancedpayments echniques,

the fixed components

are of considerable,perhaps

often dominating mpor-

tance. Certainly, or

the handlingof a check,

the amount writtenon the face

of it is of little consequence.Also, the replacement

ost of bank notes

will

have little relation to

their face value, and

for an accountingsystem, be it

computerizedor not,

it makes hardly any diSerence

whetheran entry has 3

or S digits. If, on the

other hand, the medium

of exchange s a commodity

money dominatedby

its non-monetary unctions,

variablecosts maybe more

important. t may thusbe that in the area of paymentsarrangementsnd the

choice of media of exchange, he scope for

conscious improvement

and thus

for planning has rather

ncreased han decreased n the course of economic

history.

Concluding emark

In conclusion, a

word should be added

about the principal imitation of

this analysis, namely

the independenceof ultimate flows from payments

ar-

15 On the face of it, there

seems to be a close similarity

between ransactions nd transporta-

tion costs. KoopmansandBeckman ound that in a spatial

system n whichtransportation

osts

for each indivisibleplant

dependon the location of other

plants, the marketmechanismwould

not support an efficient

ocation pattern [10]. It might

be interesting o explore the formal

analogy of the two problems.

16 In connectionwith

the integer programming roblem

analyzedby Gomory and Baumol

Alcaly and Klevoricksuggested

a system of taxes and

subsidieswhich would see to it that the

dual pricesguide the economy

to the optimum [1].








JURG NIEHANS

: 725

rangements.J.

Stuart

Mill visualized money

as a

contrivance or sparing

time and labour,

a machine

for doing

quicklyand commodiously,

what

would

be done, though

less quickly

and commodiously,

without

it. Yet he

believedthat therelationsof commodities o one anotherremainunaltered

by money

and

thingswhich by

barterwould

exchange or one

another

will,

if sold

for money, sell for

an equal

amount of it [13,

pp. 9 S.].

It is remark-

able

how precisely his

agrees with

the implications

of the approachused

in

this

paper: Under appropriate

onditions,

money will

be used

to economize

resources pent

on transactions,

but at the

same time

the networkof ultimate

flows

and the

exchangeratio between

them

are not aSected

by the emerging

payments

arrangements.

t is all-important

o realize, however,

that

this

invariance

f ultimate

lows s not

due to theeconomizing

behavior

of economic

agents,

but solely to an

artificial

onstrainton this behavior.As soon as the

economic

agentsare permitted

o

optimizeultimate

flows and

paymentsar-

rangements

at

one stroke, it will

no longer

be true that relative

prices

(ex-

cluding

transactions

osts) are unaSected

by monetaryarrangements.

n this

case,

each set of transactions

osts

will give rise to

a different

et of ultimate

flows and exchange

ratios

each combined

with the

correspondingoptimal

paymentsarrangement.

n an integrated

ystem

one cannot have

it both

ways:

Either

moneysaves time

and labor

and, by doing

so, aSects

the allocation

of resourcesandthus the real sideof the economy,or else moneyhas no effect

on the real side

and thus

cannot legitimately

be

presentedas economizing

resources.

The classical

conception

as represented

by Mill,

therefore, s the

result

of the same

lack of a solution

to the

joint optimization

problemwhich

is the

main limitation

of the present

paper.

LITERATURE

ITED

1. ALCALY,OGER. and ALVINK. KLEVORICK. Note on the Dual Prices

of Integer

Programs, Econometrica,

34

(January,1966), pp.

206-214.

2. BALINSKY,

. L. Integer

Programming

Methods,

Uses, Computation,

ManagementScience,

12 (November,

1965),pp. 253-313.

3. BEALE,

E. L. M. Survey

of

Integer Programming,

Operations

Research

Quarterly,

16 (June,1965).

4. BELLMORE,

. and

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