A Model of Indivisible Commodity Money with …...Stylized facts By the 12th century, the silver content of the penny varied across Europe Year Weight Fineness Fine weight England

A Model of Indivisible Commodity Moneywith Minting and Melting

Angela RedishWarren Weber

February 12, 2010

Angela Redish Warren Weber Minting and Melting

Introduction

Focus: Early (800 - 1500) commodity money systems (coins)

Commodity money: Money that is

Intrinsically useful, orRedeemable on demand into something that is


Introduction

Stylized facts about evolution of commodity money systems:

1 Started with a single silver coin

Initially same size across countriesOver time, debased in some countries, not debased in others

2 Next added a second silver coin

If original silver coin small, then new silver coin largerIf original silver coin large, then new silver coin smaller

3 Much larger gold coin added even later


Introduction

Purpose: Build model of commodity money that provides a rationalefor such an evolution

Features we incorporate in the model:

1 Coins have limited divisibility (coins cannot be too small)

2 Limited number of types of coins

3 No fixed exchange rate between coins of different metals


Outline

Documentation of stylized facts

Model

Numerical analysis


Stylized facts

Silver pennies the same size until @1100

Carolingian penny 96% fine; 1.7 gms;English penny 92.5% fine; 1.5 gms.


Stylized facts

By the 12th century, the silver content of the penny varied acrossEurope

Year Weight Fineness Fine weight

England 1160 1.46 gms .925 1.35 gmsCologne 1160 1.4 gms .925 1.30 gms

Paris 1160 1.28-.85 gms < .5 < .60 gmsMelgueil 1125 1.1 gms .42 0.46 gmsLucca 1160 0.6 gms 0.6 0.35 gmsBarcelona 1160 0.66 gms 0.2 0.13 gmsVenice 1160 0.05 gms


Stylized facts

Gold coins introduced later

Coin Metal Fine weight Value in Silverunit of account equivalent

Venice1170 Denarius Silver .10 gms 1d1194 Grossus Silver 2.1 gms d1284 Ducat Gold 3.5 gms 50gmsFrance1200 Denier Silver 0.36 gms 1d1266 Gros Silver 4.2 gms 12d

tournois1290 Royale Gold 6.97gms 300d 100 gmsEngland1200 Penny Silver 1.4 gms 1d1279 Farthing Silver .34 gms .25d1344 Noble Gold 8.8 gms 80d 103 gms1351 Groat Silver 5.18 gms 4d


Environment

Time discrete and infinite

One nonstorable, perfectly divisible consumption good

Two storable metals in fixed supply

silver – supply (ms)gold – supply (mg )


Environment

Silver and gold can be held as coins or jewelry (bullion)

Silver and gold coins are indivisible, but can be minted or melted

Possibly two silver coins

first (original) silver coin contains bs ounces of silversecond silver coin (if exists) contains ηbs ounces of silver

η ∈ {2, 3, . . .}restriction on η consistent with actual coinage

Only a single gold coin

contains bg ounces of gold


Environment

Agents hold

s1 silver coins of first types2 silver coins of second typej s units silver jewelryg gold coinsjg units gold jewelry

V Only coins can be used in trade

V Only jewelry yields utility (similar to Velde-Weber)


Agents

[0, 1] continuum, infinitely-lived

Preferences:

u(qn+1)− qn + µ(bs js , bg jg )− γ(s1 + s2 + g)

c consumption; q production

u(0) = 0, u′ > 0, u′(0) = ∞, u′′ < 0

γ utility cost of holding a coin

Maximize expected discounted (β) lifetime utility

Prob 12 can either consume or produce, but not both, in each period

θ prob of a single coincidence match


Trade

Each period has two subperiods

1 First subperiod: decentralized trade in bilateral matches

Preference assumption rules out double coincidence matches

2 Second subperiod: agents can alter

coin/coin mixcoin/jewelry mix (minting or melting)


Trade

Information:

past trading histories private (no monitoring or commitmenttechnology)

rules out gift-giving equilibrium

agents are anonymous

rules out credit


Trade

Single coincidence matches: potential consumer makes TIOLI offer(q, ps

1, ps2, p

g )

q ∈ <+ quantity of production demanded

psi ∈ Z quantity of silver coins of type i offered

pg ∈ Z quantity of gold coins offered

psi , p

g < 0 interpreted as making change

Let Λ be set of all feasible TIOLI offers

Lotteries not permitted


Portfolio adjustment

After trade agents make portfolio adjustment (z , zs1 , zs

2 , zg )

z ∈ Z quantity of silver coins of type 2 exchanged for coins of type 1

z si ∈ Z quantity of silver coins of type i minted (> 0) or melted (< 0)

zg ∈ Z quantity of gold coins minted (> 0) or melted (< 0)


Model: Value functions

Expected value of holding yt = (s1,t , s2,t , gt , jst , jgt ) beginning second

subperiod

vt(yt) = maxzt ,zs

1,t ,zs2,t ,z

gt

βwt+1(s1,t + ηzt + zs1,t , s2,t − zt + zs

2,t ,

j st − zs1,t − ηzs

2,t , gt + zgt , jgt − zg

t )

−S(j st , zs1,t , z

s2,t , j

gt , zg

t )− φzt

S(j st , z s1,t , z

s2,t , j

gt , zg

t ) seigniorage tax on mintingφ cost of converting a silver coin of type 2 into η coins of type 1

Note: Coins minted must be of same metal as brought to mint


Model: Value functions

Expected value of holding yt beginning of first subperiod

wt(yt) =θ

2

∑yt

πt(yt) maxΛ

[u(qt)

+vt(s1,t − ps1,t , s2,t − ps

2,t , jst , gt − pg

t , j st , jgt )]

+(1− θ

2)vt(yt) + µ(bs j

st , bg jg )− γ(s1,t + s2,t + gt)

πt(yt) = fraction of agents with yt beginning first subperiod

y denotes seller


Model: Asset distributions

Define λb(ks1 , ks

2 , kg ; yt , yt) probability buyer with yt meeting sellerwith yt leaves with s1 = ks

1 , s2 = ks2 , g = kg

λs(ks1 , ks

2 , kg ; yt , yt) similar for seller

Post-trade (pre-mint/melt) asset distribution is

πt(ks1 , ks

2 , kg , j s , jg ) =θ

2

∑yt ,yt

πt(yt)πt(yt)[λb(·) + λs(·)]

+(1− θ

2)πt(k

s1 , ks

2 , kg , j s , jg )


Model: Asset distributions

Define δ(ks1 , ks

2 , kg , hs , hg ; yt) to be probability agent with yt hass1 = ks

1 , s2 = ks2 , g = kg , j s = hs , jg = hg after second subperiod

Pre-trade next period asset distribution is

πt+1(ks1 , ks

2 , kg , hs , hg ) =∑yt

πt(yt)δ(ks1 , ks

2 , kg , hs , hg ; yt)


Model: Equilibrium

Steady state symmetric equilibrium:

Value functions w , v ; asset holdings π and π; quantitiesps1, p

s2, p

g , z , zs1 , zs

2 , zg , q that satisfy

1 Bellman equations2 asset transitions3 market clearing


Results

Numerical – analytic results not possible

Assume:

θ = 23

β = 0.9ms = 0.1mg = 0.01σs = 0.04σg = 0.02γ = 0.001φ = 0.0001u(q) = q1/4

µ(bs js , bg jg ) = 0.05(bs js)1/2 + 0.1581(bg jg )1/2

S(j st , z s1,t , z

s2,t , j

gt , zg

t ) =

max{µ(bs jst , bg jgt )− µ[bs j

st − bsσs(z

s1,t + ηz s

2,t), bg jgt − σgbgzgt ], 0}


Results – Single Silver Coin

For large coins, welfare increases as ms increasesOpposite true for small coins

1.50

1.55

1.60

1.65

1.70

1.75

1.80

0 0.003 0.006 0.009 0.012 0.015 0.018

Welfare

sillver coin size

ms = .1

ms = .05

ms = .15



Welfare increases as trading opportunities increase

0.60

1.00

1.40

1.80

0 0.003 0.006 0.009 0.012 0.015 0.018

Welfare

silver coin size

theta = 1/3

theta = 1/10



Are model’s results consistent with historical changes in single silvercoin size?


Single Coin: Silver or Gold?

Ex ante welfare by silver coin size

1.5

1.55

1.6

1.65

1.7

1.75

0 0.003 0.006 0.009 0.012 0.015 0.018 0.021

Welfare

sillver coin size

max



Ex ante welfare by gold coin size

1.50

1.55

1.60

1.65

1.70

1.75

0.0000 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018 0.0021

welfare

gold coin size

max



Aside Interesting finding:ms

bs=

mg

bgwhereˆdenotes optimal

Intuition: Given preferences for output and jewelry and γ, there is someoptimal amount of jewelry that (on average) agents want to trade foroutputAside: we might use the q from different ms results as evidence for thisintuition


Documents

A Model of Indivisible Commodity Money with …...Stylized facts By the 12th century, the silver content of the penny varied across Europe Year Weight Fineness Fine weight England