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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
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
Outline
Documentation of stylized facts
Model
Numerical analysis
Angela Redish Warren Weber Minting and Melting
Stylized facts
Silver pennies the same size until @1100
Carolingian penny 96% fine; 1.7 gms;English penny 92.5% fine; 1.5 gms.
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
Environment
Time discrete and infinite
One nonstorable, perfectly divisible consumption good
Two storable metals in fixed supply
silver – supply (ms)gold – supply (mg )
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
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)
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
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)
Angela Redish Warren Weber Minting and Melting
Trade
Information:
past trading histories private (no monitoring or commitmenttechnology)
rules out gift-giving equilibrium
agents are anonymous
rules out credit
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
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)
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
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 )
Angela Redish Warren Weber Minting and Melting
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)
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
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}
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
Results – Single Silver Coin
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
Angela Redish Warren Weber Minting and Melting
Results – Single Silver Coin
Are model’s results consistent with historical changes in single silvercoin size?
Angela Redish Warren Weber Minting and Melting
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
Angela Redish Warren Weber Minting and Melting
Single Coin: Silver or Gold?
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
Angela Redish Warren Weber Minting and Melting
Single Coin: Silver or Gold?
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
Angela Redish Warren Weber Minting and Melting