Seignorage and Growth during thePax Romana

John M. Hartwickhartwick@econ.queensu.ca

Abstract

We set out a stylized and simple model of the economy of Ro-man Italy during a period of considerable economic prosperity.Our focus is on gold coins as currency and the seignorage beingmanaged by the central government. We solve numerically for abalanced growth representation of the economy of Roman Italy, asolution that captures the intricacies of money creation, currencyexpansion and seignorage. The "use" of seignorage by the centralgovernment is treated in two ways, leaving us with two distinctbut closely related models. We subscribe to the view that theexhaustion of low-cost gold and silver deposits contributed sig-ni�cantly to the ending of the economic prosperity enjoyed byRoman Italy and its provinces during the so-called Pax Romana(31 BC to 180 CE).

1 Introduction

We set out two related models of our stylized take1 on the economy of

Roman Italy during the Pax Romana, spanning 31 BC to 180 CE. Cen-

tral to our analysis is the minting by the central government of gold and

silver coins for much of the currency of the Roman Empire during this

1We do not attempt to calibrate our model to �t the very few bits of data thatare available on the structure of the economy of Roman Italy. Scheidel (2012) drawstogether many fragments of data on revenues and expenditures of the central gov-ernment in Rome during the Pax Romana.

1

period, an activity that yielded substantial seignorage for the central gov-

ernment. In our �rst model the seignorage funded much of government

activity in Roman Italy and in our second model the seignorage �ows

directly to households as free augmentation in their money balances. For

simplicity, we abstract from a range of coin types of di¤erent metals to

simply gold coins and gold alone. Though our attention is on an eco-

nomic model of Roman Italy during the Pax Romana, we have in mind

that the depletion of high quality gold and silver deposits2 contributed

signi�cantly to the collapse of the peace and prosperity associated with

the Pax Romana.

The early Roman Empire (the early Principate, spanning 31 BC to

180 CE, sometimes referred to as the era of Pax Romana) displayed

some remarkable economic progress, particularly in Roman Italy. The

era opens with the rule of Caesar Augustus (31 BC to 14 CE) and ends

with that of Marcus Aurelius (161 to 180) and includes, at its end, the

consecutive administrations of "the �ve great emperors",3 as designated

by historian, Edward Gibbon.4 Temin (2006) emphasizes that both la-

2We treat gold and silver deposits as located in the the city of Rome for the mostpart and thus abstract from actual transportation costs for gold and silver bullion tomints in the Empire. We also consider minting to have been done only in the city orRome for the most part.

3Nerva (96-98), Trajan (98-117), Hadrian (117-138), Antoninus Pius (138-161),Marcus Aurelius (161-180).

4Augustus was succeeded by so-called not great emperors (Tiberius (14-17),Caligula (37-41), Claudius (41-54), Nero (54-68), Galba (68-69), Otho (69), Vepasian(69-79), Titus (79-81), Domitian (81-96)). These in turn were followed by EdwardGibbon�s �ve good or great emperors. Each of the �ve great emperors was succeededby his adopted son, except for Marcus Aurelius who was succeeded by his naturalson, Commodus, considered a poor quality leader.

2

bor and capital markets functioned well5 and banking6 and long-distance

trade �ourished during the Pax Romana. Roberts (2011; Chapter 12)

reports on the organization of �rms and business more generally, among

a variety of topics on the politics and economics of Rome.7 The Empire

stretched from modern Spain to modern Syria and from north Africa

to northern England. Central were �ve large cities located around the

edge of the Mediterranean Sea. There was considerable trade between

widely separated locations8 and regional specialization. Egypt became

5The slavery of the Roman Empire involved able individuals rising to senior po-sitions in business, banking and agriculture. Most slaves would no doubt have toiledin laborious jobs. Slaves were denied many basic rights but the life of one on the bot-tom rung would not have been much di¤erent from that of an unskilled free person.Cases have been documented of a very poor free person selling himself into slaveryin order to be better fed and housed. There were clear routes for an able slave togain his freedom and many took advantage of this opportunity. See Temin (2004b).Temin argues that slavery in ancient Rome was an open system in which many slavessucceeded in getting their freedom and as well many slaves occupied positions ofresponsibility and lived comfortably. Slavery in the United States he argues was aclosed system, with very few slaves being manumitted and incentives to work wereof the "stick" variety rather than the "carrot" variety. "During the Second CenturyCE as peaceful conditions sharply reduced the supply of slaves, their average costquadrupled." (Roberts (2011; p. 208). "At any one time, approximately 150,000prisoners and slaves labored in Roman mines, 40,000 in the Spanish silver minesalone." (Roberts (2011; p. 218).

6Fractional reserves and money multiplication is not taken up by Temin (2004a).Presumably little money multiplication was taking place since the banks were nottightly linked in networks. The contribution of banks to the economy of ancientRome was presumably in intermediating between lenders and borrow-investors.

7Publican societies, forms of joint stock companies, operated in many areas of theeconomy. The stock owners were usually the large land owners in the Empire. Smallbusinesses were run by freedmen, often with a single slave and a "factory" joined toa residence.

8Archeologists point out that the numbers of wrecks of Roman ships in theMediterranean exhibits a peak during the early Roman Empire. This supports thecontention that the economy was doing relatively well at this time and that subse-quent centuries were periods of less long-distance trade. (Temin (2004a; p. 729).Roberts (2011; p. 229) cites Lionel Casson: "The Roman man in the street at breadbaked with grain grown in North Africa or Egypt, and �sh that had been caughtand dried near Gibralter. He cooked with oil from North Africa in pots and pans ofcopper mined in Spain, ate o¤ dishware �red in French kilns, and drank wine fromSpain or France."With regard to high income folk, Roberts notes: "By the Antonine period, wealthy

Romans dressed in wool from Melitus, linen from Egypt, cotton from Greece or India,and silk from China. Arabia and India supplied gens and pearls, Yeman and Ethiopiasend perfumes. Colored marble quarried in Egypt or Anatolia faced Roman houses,and Romans at food seasoned with Indonesian pepper o¤ Spanish silver dishes while

3

the principal source of grain for the city of Rome. Spain became the prin-

cipal source of gold and silver for the coinage. The standard of living

(GDP per capita) in Roman Italy, the central region, attained during

the Pax Romana has been estimated to have been at the level of the

Netherlands (or Spain or Italy) in 1700. Per capita income in Roman

Italy is thought to have been about twice that in the provinces of the

Empire. The population of the city of Rome is estimated to have risen

to one million over the �rst century CE. Nero (54 to 68 CE) supervised

the re-building to the city�s center to make it as splendid as the center

of Alexandria after a huge �re consumed large tracts of "low income"

housing in Rome, and a near-successor (Emperor Vepasian (69-79)) ini-

tiated the construction of the majestic Coliseum in the second half of

the First Century. Around 100 CE, Trajan (98-119) supervised the re-

construction of Rome�s port at Ostia on the coast of the Mediterranen

Sea and had the massive Forum built in Rome itself. His new market in

the city of Rome had three levels and comprised some one hundred and

seventy-�ve shops. The Roman Empire has become famous for its in-

frastructure: roads, ports, aqueducts, sewers and town-centers. Temples

and stadiums were erected in larger towns and cities. Good infrastruc-

ture, particularly ports and roads, obviously helped to promote trade.9

Showy stone temples and stadiums signalled a permenance and stability

to the social system, which in turn probably encouraged orderly eco-

nomic activity and social relations among individuals and groups more

generally.

We focus attention here on the money supply during the Pax Ro-

mana. Trade �ourished and there was a general absence of in�ation.

Our interpretation is that Augustus laid down a new system of gover-

nance and his successful successors embellished his system in various

qua¢ ng African or Aegean wine in Syrian glasses, all while Greek statues gazed downon set tables of Moroccan lime wood."

9Temin (2004a) argues that land was bought and sold in markets. He emphasizesthat agricultural societies that are based on mostly agricultural activity will haveless market activity because isolated farms will be relatively autarkic. Informationmoved slowly in the absence of the post, phone and trains. Hence price adjustmentswould necessarily be sticky in economies before the nineteenth century.

4

ways and his unsuccessful successors were unable to destroy the system

by their various forms of mismanagement until the latter part of the

Second Century. Since Augustus presided over the opening of a two

hundred year period of Pax Romana, a distinguishing feature of the pe-

riod was relative internal and external peace. Central to his system10

was a balancing of the interests of power-seekers at the center. In ad-

dition he institutionalized payments for troops, at reasonable cost, both

inside the Empire and at its frontiers. Another component of his system

was a stable currency11 that contributed stability to prices as well as

fostering local and long-distance trade.12 The large dimensions of the

Roman Empire as an economic entity were supported by the standard-

ized currency, one that allowed for trade and regional specialization to

�ourish. The combination of a vast peaceful nation and a uniform sys-

tem of currency and laws was not seen elsewhere in the world, with the

possible exception of China,13 until the early twentieth century. Banking

10"the forty-�ve years of his rule allowed him to painstakingly contruct a new socialorder" (Roberts (2011; p. 170).11Shortly after the death of Augustus, there was a monetary crisis that failed to

in�ict lasting damage on the economy of Rome. "There even was a liquidity crisis in33 CE in which interest rates rose, loans were called in, and land prices collapsed."(Temin (2001), p. 176. See also Roberts (2011; p.168)).12"From the time Rome subjugated the Greek city-states of southern Italy to the

beginning of the third century CE, its money supply was large and fairly stable. Forinstance, the total Roman coinage per capita in the early years of the Principate cameto approximately 80 percent of the current US money supply. Rome�s sources of silverallowed it to mint enough virtually pure silver denarii to reliably maintain this moneysupply. After Augustus conquered northern Spain, twenty thousand pounds a yearof gold from its mines, joined later by those of Romania, furnished the eight-gramgold aureus as well. This was used for international trade, the payment of taxes, andother large payments." (Roberts (2011; p. 236-37)13"As part of the Uni�cation of China, Qin Shi Huang (260 BC �210 BC) abolished

all other forms of local currency and introduced a national uniform copper coin basedon the coins previously used by Qin. These coins were round with a square hole in themiddle which was the common design for most Chinese copper coins until the 20thcentury. Due to the low value of an individual coin, the Chinese have traditionallystrung a nominal thousand copper coins onto a piece of string. However governmenttaxes were levied in both coins and in products such as rolls of silk. Salaries werealso paid in both the Qin Dynasty and Han dynasties in "stones" of grain.During the early Song dynasty ( 960�1279), China again reunited the currency

system displacing coinages from ten or so independent states. Among pre-Song coins,the northern states tended to prefer copper coins. The southern states tended to uselead or iron coins with Sichuan using its own heavy iron coins which continued to

5

in the private sector was much in evidence during the Pax Romana.14 A

system of laws and courts provided a social infrastructure which abet-

ted business as well as civic order. There were no large-scale revolts by

slaves, in contrast with say the period 100 to 31 BC. Augustus is known,

among his other accomplishments, for conquering northern Spain and in

so doing acquiring rich silver and gold mines.15 The bullion from Spain

�owed into the mints of the government of the Empire.

circulate for a short period into the Song dynasty. By 1000, uni�cation was completeand China experienced a period of rapid economic growth. This was re�ected in thegrowth of coining. In 1073, the peak year for minting coins in the Northern Song,the government produced an estimated six million strings containing a thousandcopper coins each. The Northern Song is thought to have minted over two hundredmillion strings of coins which were often exported to Inner Asia, Japan, and South-East Asia, where they often formed the dominant form of coinage. Song merchantsrapidly adopted forms of paper currency starting with promissory notes in Sichuancalled "�ying money" (feiqian). These proved so useful the state took over productionof this form of paper money with the �rst state-backed printing in 1024. By the 12thcentury, various forms of paper money had become the dominant forms of currencyin China and were known by a variety of names such as jiaozi, qianyin, kuaizi, orguanzi." (http://en.wikipedia.org/wiki/Chinese_currency#Imperial_China.... July 4, 2012.)14"A good �nanical system promotes growth, and indeed there appears to have

been growth during the Roman Republic and early Roman Empire.... the existence[of growth] is consistent with the development of the Roman �nancial sophisticationdescribed here." (Temin 2004a; p. 729)"The surprising result is that �nancial institutions in the early Roman Empire

were better than those of eighteenth-century France, albeit not as developed as thoseof eighteeth-century England and Holland." (p. 729)Temin never makes a clear claim that a source of great economic e¢ ciency of the

early Roman Empire was the presence of a smoothly functioning currency systemthroughout the Empire. He does note however: "Rural transactions in Rome weremade with relatively uniform coins, as in eighteenth-century France, and possiblymore easily than in eighteenth-century rural England." (p. 728).15"After Augustus conquered northern Spain, twenty thousand pounds a year of

gold from its mines, joined later by those of Romania, furnished the eight-gram goldaureus as well. This was used for international trade, the payment of taxes, andother large payments." (Roberts (2011; pp. 236-37) Between 200 BC and 200 CEan estimated 100 million denarii were brought to Rome per year from silver minesin Spain. (Rorberts (2011; Table 15, citing Badian (1972; p. 34). Scheidel (2012)reports that gold mines in Spain were yielding 6.5 tons per year (88 million sesteres)and in Bosnia 5.9 tons per year (80 million sesteres) An estimate for the �rst centuryBC has 35.4 tons of silver (44 million sesteres) being extracted in Spain. For the�rst century CE, Scheidel estimates some 200 million sesteres of value from gold andsilver mines per year. He is unsure whether this represents net "pro�t" to the stateor gross value of re�ned ores. "Mineral wealth alone might have covered a sizeableshare of total expenditure, perhaps anywhere from a tenth to a quarter."

6

Our interest is in Augustus�s monetary system, a system centered

on silver and gold coins,16 minted by the government in Rome as well

as in satellite mints in other centers in the Empire. Of special interest

is the system of government �nance by seignorage. If a silver coin can

be produced, including minting, at a cost of say $Y in terms of capital

and labor (ignoring Hotelling rent or user cost) and can then buy goods

worth $X in terms of capital and labor (X>Y), then the producer of

the coin has some "free" purchasing power of $X-$Y. This surplus is the

seignorage which the government can use to purchase goods and services;

alternatively it may choose to simply mail out "cheques" to households

with a value of the total annual surplus (drop money from helicopters

on households, as some have phrased it). We set out a simple abstract

growth model of Roman Italy during Pax Romana that has seignorage

at its center. In our �rst model, seignorage supports an explicit gov-

ernment sector (a sector employing labor and capital to produce gov-

ernment goods, �ows per period) in Roman Italy. In our second model,

seignorage is transferred directly to households as in a reverse head tax;

there is no explicit production of government goods. The government is

viewed as part of the social environment. In each model, money (gold

coinage we refer to it) is produced with capital and labor and households

have explicit demands for money balances over each period. The cost

of production of the gold coins currently is less than the value of goods

and services that can be traded for the gold coins. This mark-up or

surplus represents the seignorage that is available to the government to

allocate.17 More on the level of the mark-up below.

We distinguish between Roman Italy during our period of interest

16There were coins of lesser metals in low denominations in addition to gold andsilver coins.17"Contrary to the widely held belief that ancient coinage was invented simply

to promote trade, most scholars now believe this contention has not foundation.The more likely reason was that coins [rather than gold and/or silver bullion] wereactually created for the purpose of facilitating the state�s own expenditures for goodsand services." Steve Coe on the Internet (http://ezinearticles.com/?Ancient-Silver-and-Gold-Coinage&id=6225050), March 26, 2012.

7

(Pax Romana) and the provinces comprising the rest of the Empire. We

obtain a balanced growth con�guration for the economy of Roman Italy

by assuming that gold produced by the center, net of seignorage, funds

imports to Roman Italy from the provinces and abroad. In addition to

gold production, Roman Italy produces consumption goods and invest-

ment goods under constant returns to scale. Population (the labor force)

in Roman Italy is assumed to be growing at a constant rate, presumably

very small. We assume that the center organizes gold production (gold

supply) to match the increment in population over each period. Current

savings is taken as a constant fraction of the value of current output.18

Balanced growth, our reference case, has the prices unchanging while the

real side of the economy is expanding at the rate of population growth.

An interval of balanced growth is taken in our view to represent the era

of Pax Romana. Of interest to us, and many others, is that currency

problems in the Empire appear to co-incide with the un-ravelling of the

prosperity of Pax Romana. Our view is that the currency problems (de-

basement of the coinage and later in�ation in prices) were likely triggered

by the exhaustion of low-cost gold and silver deposits in the Empire.

Though our analysis seems to be a novel investigation of seignorage19

in a stylized model of Roman Italy during the Pax Romana, we delve

brie�y into a curiosity of national accounting and seignorage. In our �rst

model with our explicit production of government goods, the government

goods must be treated in the national accounts as entirely intermediate

to the economy. This occurs because the current seignorage has joint

(two) payo¤s: the seignorage is funding current government output and

it is augmenting current money balances of households. Only the value

of the increment to household money balances ends up being counted

18This is the value of current total product net of the current value of seignor-age. We experimented with savings governed by the Ramsey Rule but rejected thisformulation since imports are treated as homogeneous, leaving imports of consumerand/or producer goods unspeci�ed. One might characterize our model as a neo-classical Solow growth model with money included and sectors disaggregated.19A special case of our money supply would be costless production of paper money

rather than costly production of gold coin money. With such paper money, therewould be no residual value of currency that could be used to fund importing activity.

8

in the value of current national product. The account avoids double

counting and ends up with the value of current government product as

"intermediate". Money balance increments, which are goods to house-

holds, are "�nal" entries in net national product on the production side

of the national account. In our second model with current seignorage

mailed out to households each period (and an absence of explicit govern-

ment production) money balance increments of households again become

a "�nal" entry in the product side of the national accounts and there

are no valuations that are intermediate. In a sense there are no account-

ing surprises in this second case, the case of no explicit production of

government goods.

The currency system that Augustus inherited and in a sense institu-

tionalized was based largely on silver coins. Higher "denominations" of

currency were gold coins and lower were bronze or copper coins.20 Our

analysis here was motivated directly by the question of the role of money

during Pax Romana. However our model of Roman Italy is abstract and

stripped-down and has not been calibrated to �t the facts of Pax Ro-

mana. This is in part because the facts, particularly the sizes of �ows of

gold and silver into Rome, are not well established. We remain convinced

that the fairly well-documented economic growth of Roman Italy during

the Pax Romana was directly tied to the orderly management of money

by "the government" and we share the opinion of many historians: the

"collapse" of the Augustan system at the end of the Pax Romana was

directly linked to the unravelling of the money control system that the

center was able to maintain during the Pax Romana.

We present some observations, culled from the internet in March and

20"The monetary system of Rome was based on the silver denarius... The denariuswas divided into four bronze sesterces, which were the common unit of commercein the early Roman Empire. Sesterces were divided in turn into four copper asses,and the European, Latin set of coins was linked to a Middle Eastern, Greek set by a�xed exchange rate. The silver drachma was the equivalent of the sestertius, and itwas divided into six and later seven bronze obols. For calibration, one modius (6.5kilograms) of wheat cost four to six sesterces on the private market in Roman duringthe �rst century CE, and the daily wage was between three and four sesterces."(Temin (2006; p. 138).

9

April, 2012.

"Ancient Rome was heavily dependant on trade, and golden coins

were the main currency used....

The Romans maintained an Imperial Treasury consisting of gold

coins, the treasury was supervised by the senate, and provided the Ro-

man govenment with the necessary �anancial funds to maintain the Em-

pire, pay salaries and import goods from other parts of the world.

Contrary to popular beliefs, the amount of gold produced and used in

ancient Egypt was small, accounting for its limited Royal and religious

use. The annual production of gold during pharaonic times is thought

not to have exceeded one ton per year....

Spain alone, a major producing center in Roman times shipped 1400

tons of gold to Rome every year."21 ("Gold" the Internet (www.aldokkan.com/art/gold.htm),

March 27, 2012)

The scale of mining at Riotinto (in Spain) fundamentally altered the

Roman economy. "Basically, it ensured Rome a constant supply of fresh

metal for increased minting of silver and lower-denomination copper-

based coins," says Jonathan Edmondson (York University). Rome used

silver denarii to pay and feed its army, fund public building programs

in its capital city, and subsidize the price of (and eventually allow free

distribution of) grain to the city�s residents. But following the invasion

of Spain by the North African Mauri in the late second century, mining

activity dropped o¤ and the denarius plummeted from 97 percent silver

to 40 percent, leading to outsized in�ation as Roman minted ever-less-

valuable coinage. "The Roman state experienced major problems, since

taxes were paid in coin," Edmondson notes. "People started handing

over these debased coins in payment of taxes, while hoarding the [older]

higher-percentage silver coins." By the fourth century, he says, gold re-

placed silver as Rome�s main currency. (http://factsanddetails.com/world.php?itemid=2050&catid=56&subcatid=369)

When the gold in Spain began to run out, the Romans noticed that

21Healy (1978) deals exhaustively with Roman mining activity but leaves one with-out speci�cs about volumes of high-quality metals produced.

10

there were also gold mines in Dacia (called Romania today)... The Ro-

mans used the Dacian gold to pay their army... When the Dacian mines

ran out of gold,22 about 275 AD, the Romans abandoned Dacia and went

home. (http://www.historyforkids.org/learn/science/mining/gold.htm)

�Duncan-Jones (1994) has analyzed hundreds of hoards of Roman

coins, unearthed by archeologists over centuries in many di¤erent loca-

tions. In his Chapter 15, he documents the debasement of gold and

silver coinage during the Third Century. "A pound of silver was produc-

ing twice as many denarii under Severus as it had under Augustus two

centuries earlier. By the late Severan period the �gure was more than

twice what it had been under Antoninus Pius, only seventy years earlier."

(p. 228) "The �rst decisive fall [in precious metal content] took place in

AD 64, when �neness was reduced to about 93.5%." (p. 224)23 "That

second-century armies received cash payment in precious-metal currency

in not in serious doubt... No government has access to an unlimited sup-

ply of bullion, and in the case of Rome, reductions in the precious-metal

content of the coinage suggests strain on the bullion supply." Duncan-

Jones moves on directly to attempt to infer how the �ow of coins was

related to trade in the Empire and to payments to soldiers along the bor-

ders. "Since army pay was the biggest running expense of the Empire,

the government�s dependence on recycled cash to meet the expense was

inevitable, and recycling must have taken place. (p. 176). Recycling

involved "taxes paid in coin �nding their way back from the provinces

to Rome... and being sent out again to pay the troops in the provinces."

This recycling "should have had a smoothing e¤ect, which would tend to

22Gold was mined by the Romans in Wales, also, presumably in relatively smallquantities.23"Caesar�s aureus had 8.2 grams of gold content. Augustus�s 7.8 and Nero�s 7.25.

Although the silver content of the denarius was halved by Trajan�s reign (98 to 117CE), it held it value because 25 denarii could always buy one arueus... There waslittle in�ation and military pay was increased only slightly throughout this period...Interest rates remained at 4 to 6 percent in Rome and 8 to 12 precent in the provincesuntil the third century CE." (Roberts (2011, p. 237).

11

obliterate local characteristics in provincial coin-populations. Yet as we

have seen, local characteristics remain clearly visible." (p. 176) "Trade

did not "make the coin-population homogeneous throughout the empire.

It is worth recalling Gaius�(110 to 179 CE) remark that in his world

not only did the prices of grain, wine and oil, but also interest rates

varied widely from place to place... that suggests an economy divided

into small local cells, rather than something large and uni�ed." (p. 176)

We would expect that where transportation costs between places were

relatively low, local prices would diverge little, as say between cities on

the Mediterranean coast. Once goods had to be moved over land be-

tween places, prices of similar products would be expected to remain

quite distinct. The fairly popular notion that considerable inter-regional

trade took place during the Pax Romana should be taken as correct,

until substantial contradictory evidence emerges. The work of Kessler

and Temin (2007) on grain prices was pursued in order to shed light on

market integration across regions in the Roman Empire during the Pax

Romana. Temin is one of a number of observers who emphasize that

long-distance trade was a distinguishing feature of the good economic

performance of the Roman Empire during the Pax Romana. And these

same observers note that the volume of this trade shrank considerably

during the Third Century CE. Banking activity which has been found

to be widespread during the Pax Romana also shrank hugely during the

Third Century.

Duncan-Jones�s re�ections about the recycling of payments to troops

by the center are well worth taking up. During the Pax Romana large-

scale warfare did not occur. Border security was maintained without

large numbers of soldiers. There is a principle here. When order is

maintained over a period, people become accustomed to orderliness and

order can be maintained with small-scale enforcers. The British during

their colonial period were able to maintain order over large areas and

over millions of "subjects" with relatively few troops and police. Border

security during the Pax Romana worked in a similar fashion. Our view

12

is that local tax revenues would have been adequate to maintain these

border forces. Hodge (2002) contends that much infrastructure in the

Roman Empire was organized by the military and they in turn could

draw on slave workers for much of the physical laboring. Hence though

large infrastructure projects in the Empire, such as sewer and aqueduct

construction, may have appeared to be costly, they may in fact have

required modest boosts in revenues to be successfully undertaken. The

military were available to organize construction, given the absence of in-

ternal and external violence, and the military could control a large slave

workforce on the various projects. The center may well have supported

certain projects in the provinces such as port construction but in our

view, there was not a huge �ow of revenue from taxation at the center

which went out to the provinces. As the Third Century unfolded large

raises were given to soldiers and large tax increases were imposed from

Rome as well. Thus the �ow of funds in the public sector was no doubt

quite di¤erent from that during the Pax Romana as the Third Century

unfolded. Our view is that indeed large-scale long-distance inter-regional

trade took place during the Pax Romana and large �ows of funds did

not �ow in the public sector from Rome to the provinces. We believe

that relatively large-scale importing was being done by Roman Italy dur-

ing the Pax Romana and a principal source of funds for this importing

was the direct exporting of gold coins. This form of trade, gold out-

�ow supporting the import of food and other consumption goods plus

some investment goods, is built into our formal model of the economy

of Roman Italy during the Pax Romana.

2 The Level of seignorage

The level of mark-up over cost for the gold coins (the seignuerage gap)

is tricky to quantify. Under barter, goods are presumed to be trading at

their cost of production. When a commodity emerges from barter as the

currency, such as cows, a cow can take on a value in trade di¤erent from

the cost of production of a cow. A socially-sanctioned "mark-up" over

unit cost for a cow can develop under long-term sustained "usage". The

13

level of the mark-up can only be sustained if every person goes along with

the valuation. Long-term social practice is the force sustaining the level

of the mark-up. If the mark-up becomes large, some people will breed

cows to add to the money supply and the breeder can reap the mark-up

personally. A currency with a large and sustained mark-up requires that

"outsiders" cannot easily enter the business to providing currency to the

economy. With gold coins for the currency, similar issues are involved.

However, once the mark-up over cost emerges, a strong central govern-

ment can add strength to sustaining the level of the mark-up by taking

over control of the supply of gold and preventing outsiders from mining

gold, minting coins and enjoying the seignorage themselves. Hence for

a mark-up to be sustained it is almost essential for a strong center to

monopolize the production of currency. In fact the Empire had mints

in many places.24 However one presumes that much more activity took

place in the city of Rome itself since the central government would have

been controlling mining activities in Spain. It is worth distinguishing

between the minting of recently mined bullion and the re-minting of

melted-down coins. A new Emperor would typically spread word of his

arrival with the minting-up of new coins with his image on them. This

could induce citizens to trade-in their older coins so that they were not

stuck with currency that was hard to pay with.

This still leaves the level of the mark-up somewhat of a loose end.

One can think of the level of the mark-up as a gap in the cost of produc-

tion of a coin between the government and an outsider. If outsiders have

access to low-quality gold deposits that are associated with $X per unit

for a minted coin (X is the cost of mining the gold, processing the ore

and minting the coin) and the government monopolizes access to high-

quality deposits with $Y per unit for minted coin, then the government is

in a reasonable position to maintain the mark-up $[X-Y]. There is then a

natural mark-up for gold coins.25 Of course the cost of keeping outsiders24"Roman coins found at Oxyrhynchus in Egypt came from mints at Antioch,

Alexandra, Nicomedia, Cyzicus, Constantinople, Rome, Aquilaea, Arles, Treves,Taraco, and even London." (Roberts (2011, p. 220).)25Unit cost plus mark-up is referred to as the �duciary value of a coin by Harris

14

away from high quality deposits is not trivial. The central government

must back up control of all high-quality sources of supply to the system

with force. Any outsider that is found with a high-quality deposit must

have her deposit taken over by the government so that central control

of the supply of new coins is maintained. We neglect quantifying this

control cost and adding it to our analysis at this point and. $[X-Y] be-

comes the approximate natural mark-up that the government maintains

on gold coins. Below we have the cost to the government of producing

a gold coin as pg and the cost to the outsider of (� � 1)pg; with � > 1:More on this below.

One can argue that Augustus inherited a good currency system and

his support for the system led to it functioning smoothly over the long

run of Pax Romana.26 We share the view of some obervers: the depletion

of the high quality gold and silver deposits toward the end of the Second

Century CE led to a severe shrinkage in the size of the gap, X-Y, and

the severe erosion of seignorage as a source of revenue for the center in

the Roman Empire.

One naturally re�ects on the possibility of an adjustment in the value

of a unit of currency in order to restore the previous level of seignorage.

Debasement is the form of the typical adjustment and this shrinks the

value of seignorage. In addition, any �ddling by the center with the metal

content of the silver and gold coins would induce a strategic response by

the public, typically the hoarding of older high-quality coins. Hoarding

would a¤ect the volume of money and would result in some seizing up

of trade in goods and services. The details of such a process are beyond

the scope of our analysis here. It seems apparent that the money system

in the Roman Empire never functioned as smoothly after Pax Romana

(2008). "When the earliest Greek and Roman coins were minted, they presumablyhad the same value in the marketplace as the equivalent quantity of metal. From veryearly on, however, states from time to time attempted to establish a conventionalvalue for coins that, as metal, were worth less than their ��at�or ��duciary�value."(Harris (2008; p. 199).26Hollander (2008) confronts the rapid growth in the currency stock in the decades

just before Augustus and the absence of in�ation in that period. He introduces ademand for money that is independent of transactions needs by households.

15

as it did during the Pax Romana.

3 Government Production Funded with current seignor-age: Model 1

We turn to our formal model of the economy of Roman Italy during

the Pax Romana. There are four sectors in our economy, each pro-

ducing �ow output per period under constant returns to scale. In our

numerical solving, we assume that each sector is operating with a Cobb-

Douglas production function. There are then consumer goods, denoted

with a superscript C; investment goods denoted by I; government goods

denoted by G; and gold coin output indicated by g: Output �ows per

period are also indicated by C; I;G and g: The gold sector uses capital,

Kg and labor, N g to produce �g gold coins per period (� > 1) and yet

the value of capital and labor used up in production equals pgg equal to

rKg+wN g: r is the prevailing rental rate on a capital input and w is the

local wage rate. Hence (�� 1)pgg is the seignorage that the governmentis free to allocate. In our �rst model, this seignorage covers the cost of

local government goods produced in the current period. That is

(�� 1)pgg = pGG = rKG + wNG:

The remaining gold, g is exported from Roman Italy to fund imports

to Roman Italy from the provinces and from further a�eld. The other

key assumption is that labor is growing at exogenous rate n and the

government organizes gold production to increase across periods at this

same rate. The payment for government goods by the center with the

current seignorage simultaneously increases the current money supply,27

Mt: Hence current money supply increase is

�M = (�� 1)pgg:

Current seignorage is serving two functions in the current period; it pays

for the current government goods produced in the private sector and it27We take the money supply to be the value of the stock of coinage in the economy

of Roman Italy. We consider money supply multiplication via fractional reserves tonot be operating to a signi�cant extent in the economy.

16

ends up augmenting the current supply of money in the economy. This

double-duty results in the current value of government goods produced

ending up as intermediate in the national accounts. We turn to the na-

tional account for our economy of Roman Italy during the Pax Romana.

In our "static" national accounts Table 1, a row sum is represented

by the right hand entry; a column sum by the corresponding entry in

the bottom line. Net national income (NNI) is the sum of entries in the

bottom row and national product (NNP) is the sum of entries in the

right hand column.

17

Table 1NNP (entries)

rKC wNC =CpC

rKI wNI =IpI

rKg wNg =gpg (= imports)rKG wNG �pg(�� 1)g =0

pg(�� 1)g =�MNNI (entries) =

PrK =

PwN =0

In Table 1, the total value of capital in use in the current period

isPrK and for labor is

PwN: Observe that the column sum is the

current value of total product for Italy over the accounting period and

the row sum is the total value of inputs over the accounting period. The

accounts re�ect the fundamental principle that the value of inputs equals

the value of product over the accounting period. We are assuming as

we noted above that goods are produced with constant returns to scale

production functions and that input pricing conforms with values of

marginal products (a competitive, undistorted equilibrium).

The two features of this representation of our economy are (a) new

money balances (new gold coins) are a component of current NNP, indi-

cated by �M; and (b) current government product, a �ow good rather

than a durable good, is fully funded by current seignorage, (� � 1)pgg:Hence the so-called government budget constraint is satis�ed. (There is

no taxation in this simple economy. The absence of taxation simpli�es

our analysis. In fact there was taxation in Roman Italy at this time and

it partly funded the purchases of government goods by the center. We

take up the introduction of taxation explicitly below.) Our treatment

of money demand, seignorage and government production implies that

�ow G is purely intermediate in the economy. Since current seignorage

provides for both funding for current government product and current

increments to household money balances (seignorage has joint "prod-

ucts"), double counting in NNP is precluded by having the value of

current government product intermediate in the account. Current gov-

ernment product, G is produced using capital and labor and is paid for

with current seignorage AND current seignorage also ends up "funding"

18

current additional money balances of households.

4 Solving for Balanced Growth (seignorage sup-ports G)

We proceed to solve for a numerical example of a steady state. First

we solve for direct values, treating investment, It (=Kt+1-Kt) and Mt as

�nal goods. We use these numerical results of our �rst calculation to

calibrate a value for n = Nt+1�NtNt

(current population growth rate) and

g that yield a steady state in per capita "quantities". The value for � is

exogenous, given by nature.

Solution I: We solve for 16 unknowns (KG; KI ; Kg; NG; N I ; N g; I;

C; r; w; i; pI ; pC ; pg; pG; g) in the following 16 equations of our model.

There are four equations de�ning the match for each sector of the

value of inputs with the value of current product.

r[K �KG �KI �Kg] + w[N �NG �N I �N g] =CpC

rKI + wN I = IpI

rKG + wNG=((�� 1)pgg)

rKg + wN g = pgg ;

There are four equations de�ning current output levels28 C; I;G and

g:

[K �KG �KI �Kg]�2 [N �NG �N I �N g]1��2 =C

[KI ]a1 [N I ]1��1 = I

[Kg]a3 [N g]1��3 = g;

[KG]a4 [NG]1��4 =((�� 1)pgg)=pG;

There are four equations de�ning output prices pC ; pI ; pG; and pg:

28We assume that industry output came from many small �rms. Competitiveconditions are assumed to have prevailed.

19

[�2]��2 [1� �2]�[1��2]r�2w[1��2]= pC

[�1]��1 [1� �1]�[1��1]r�1w[1��1]= pI

[�3]��3 [1� �3]�[1��3]r�3w[1��3]= pg

[�4]��4 [1� �4]�[1��4]r�4w[1��4]= pG;

The exogenous level of M de�nes the price level.29

Mv = CpC + IpI + gpg + (�� 1)pgg with v = i0:1;

The interest rate in a steady state satis�es

i =r

pI:

The savings rate, s is taken as exogenous.30 Hence:

s[CpC + IpI + gpg + (�� 1)pgg] = IpI :

It turns out that balanced growth requires a speci�c value of g; the

initial value of gold output (in balanced growth, the ratio g/N is un-

changing). In our 16th equation, g must satisfy

(�� 1)pggM

= n;

for I=K = n and �M = nM = (�� 1)pgg: �This model is essentially static. Given initial values for K;N;M and

�; we solve this model (16 simultaneous equations in 16 unknowns) with

Matlab. See Appendix 1.

For K = 6; N = 5; M = 20; � = 2:1; �1 = 0:5; �2 = 0:4; �3 = 0:4;

�4 = 0:35, we obtain the following results:

29M is the money supply and 1vpQ is money demand, with p a vector of current

prices and Q a vector of current products. Our velocity v is interest rate sensitive.Hence our money demand depends on transactions volume and the value of theinterest rate.30Savings as a constant fraction of output involves the value of output net of the

current value of seignorage. This formulation preserves the scale neutrality of oursystem and thus allows for the possibility of balanced growth.

20

I=1.1225, C=2.1787, KI=1.5348, NI=0.8210, pI=3.2223, pC=3.3621,

pg=3.3637, Kg=0.8664, Ng=1.0806, r=1.1781, w=2.2034, i= 0.3656, g=1.0113,

pG=3.3815, KG=1.1122, NG=1.1035.

Observe that the interest rate is 36.58% and the population growth

rate is now set at �KK= 1:1224

6= 18:71%:31 We experimented with dif-

ferent parameters (K larger than N and the savings rate lower) and ran

into solutions emerging with imaginary values. The existence of solu-

tions positive in all outputs is clearly not an issue that is trivial. We

proceeded to verify that the price system did not vary when K;N; and

M were perturbed proportionately.32 This property (homogeneity in

"quantity" variables) is necessary if a balanced growth solution is to be

found. The model exhibits price-independence: when M is varied alone,

"quantities" remain unchanged while prices change.

Note that � � 1 = 1:1; indicating a 110 percent above-cost for goldavailable as seignorage. � is exogenous (given by nature) and (�� 1)pg

is functioning as a price. Clearly this latter price is endogenous since pg

is endogenous. Given values for n (= I=K) and g; we turn to calculating

a balanced growth solution. Central to the balanced growth solution is

an exgenous population growth rate, n:

31Population growth rates in various locations around the world are inferred tobe almost zero until about 1700. Our numerical output is clearly o¤ base in thisregard. Professor Walter Scheidel in the Stanford Classics department reports aninformed guess of the population growth rate for the Empire during the Pax Romanaof 0.15%. He also points out that average heights of people in Italy during the periodof Roman hegemony were lower than earlier and later. He mentions relatively highrates of urbanization and associated disease transmission as a possible cause. Seealso Scheidel and Turchin (2009)32This price-vector invariance should be referred to as neutrality of the money

supply. For the money supply to be non-neutral, we must connect transactionsdemand and "interest-rate" demand for money in a non-neutral fashion as in the

Baumol-Tobin formulation: money demand ishtpQr

i1=2; where t is the cost of going

to the bank. Crucial here is the exponent 1=2: If the exponent were 1:000; thenneutrality would be restored to the system. In the Appendix, we solve this modelwith only a transactions demand for cash balances, inspired by the famous formula:Mv = pQ:

21

5 Balanced Growth

We distinguished between the economic assumptions "driving" balanced

growth and the technical details of solving for endogenous variables char-

acterizing a balanced growth. The economics assumptions are

(1) population is growing at a constant exogenous rate n:

(2) the money stock is growing as Mt+1 �Mt = (�� 1)gpg:(3) government goods produced currently are funded by seignorage

in (�� 1)gpg = pGG:(4) the government extracts gold for minting each period in quantities

that match the current population size; g=N is unchanging.

(5) the value of gold currently produced at cost of production, gpg

funds imports to Roman Italy from the provinces and also further abroad.

For balanced growth, the value of g=N , unchanging, must allow for�MM

= n or (� � 1) = nMgpg: Hence given �; essentially from nature, the

government must get the value of g=N correct for balanced growth to

obtain. The somewhat tricky policy issue is for the center to �nd the

correct value of �; thrown up by nature, and to select g=N correctly.

Steering the economy between in�ation and de�ation involves getting

and using the correct quantity of gold each period, namely the quantity

that tracks the current population increase, this latter assumed to be

based on a constant rate. More on the correct value of g=N below.

To actually solve for the balanced growth realization, we divide all

quantities byNt (expressing the per capita variables in lower case letters)

and solve for kt+1 and mt+1 (as well as other variables) given current

values for kt and mt: Recall that_NNis set at the constant n: The value

of n is de�ned by I=K in our preceding problem. The only subtlety in

dividing though by Nt is that

_Kt

Nt= _kt + nkt for kt =

Kt

Nt;

and_Mt

Nt= _mt + nmt for mt =

Mt

Nt:

Our sixteen unknowns to solve for are kt+1, mt+1, pI , pC ,pg, pG, r, w, i,

22

C/N=(c), KI/N(=kI), NI/N(=nI), Kg=N(=kg), Ng/N(=ng), KG/N(=kG),

NG/N(=nG): Recall that we assume that output g is growing at the same

rate as N:We indicate g=N by g: And the output of the government sec-

tor, pG[G=N ]; is equal to the "excess value" of current gold production

(� � 1)gpg. The 16 equations to be solved include four value balanceequations

r[kt � kG � kI � kg] + w[1� nG � nI � ng] = cpC

rkI + wnI = [ _k + nkt]pI for

_K

N= [ _k + nkt]

rkG + wnG=((�� 1)pgg)

rkg + wng = pgg for g = g=N0;

four production relations

[k � kG � kI � kg]�2 [1� nG � nI � ng]1��2 = c

[kI ]a1 [nI ]1��1 = _k + nkt

[kg]a3 [ng]1��3 = g;

[kG]a4 [nG]1��4 =((�� 1)pgg)=pG;

four price equations

[�2]��2 [1� �2]�[1��2]r�2w[1��2]= pC

[�1]��1 [1� �1]�[1��1]r�1w[1��1]= pI

[�3]��3 [1� �3]�[1��3]r�3w[1��3]= pg

[�4]��4 [1� �4]�[1��4]r�4w[1��4]= pG;

a savings relation

[ _k + nkt]pI = s[ctp

C + [ _k + nkt]pI + gpg + (�� 1)pgg];

a money demand equals money supply equation

23

mtv = ctpC + [ _k + nkt]p

I + gpg + (�� 1)pgg; with v = i0:1;

a de�nition of the interest rate i in

i =r

pI;

and a money expansion equation

_m+mn = (�� 1)gpG:

With _k represented by kt+1�kt; and _m bymt+1�mt we solve numerically

an example based on parameters and outputs from Solution 1 above. The

solving, with Matlab, is reported in Appendix 2.

The outputs of our run with our second Matlab program, given n =1:12256

and g = 1:01135; are:

kt+1 =1.2000, C/N=0.4358, KI/N=0.3068, NI/N=0.1642, pI=3.2226,

pC=3.3622, pg=3.3636, Kg/N=0.1733, Ng/N=0.2161, r=1.1786, w=2.2029

, i= 0.3657, mt+1=4.0000, pG=3.3815, KG/N=0.2223, NG/N=0.2208.

These outputs constitute a balanced growth trajectory with prices

unchanging and kt+1=kt; ct+1=ct; and mt+1=mt: The labor force (pop-

ulation) is growing at the constant rate n: Key "quantity" variables are

all growing at rate n: Prices are unchanging.

We have here a fairly complicated dynamic system in k and m:

We would of course be interested in the dynamics of the model in

the neighborhood of the balanced growth solution. Unfortunately the

system seems too complicated for a detailed technical analysis. We

did experiment with numerical runs in the neighborhood of the bal-

anced growth solution and observed that starting with k0 and m0 away

from the balanced growth values kt converged to the balanced growth

value while mt failed to converge. Recall that our base model exhibits

price-independence. Thus this convergence property is consistent with

that price-independence. The price-independence property involves the

24

"quantity" or real side of the model independent of the money-supply

side. However, dynamics near the balanced growth solution are proba-

bly not interesting since our model has no explicit reaction behaviors for

agents to price changes that are occuring o¤ the balanced growth path.

The balanced growth solution does appear to capture essential fea-

tures of the economy of Roman Italy during the Pax Romana. Capital

was being accumulated and in�ation was not an issue. Imports �owed

in, funded with gold coins in value equal to the cost of production.

From the standpoint of policy at the center, there would be the some-

what complicated issue of separating current gold production in funds

for current imports and funds for purchasing government goods and ser-

vices. Presumably those at the center would be aware of the value of

�; nature�s coe¢ cient of "mark-up" for the value of gold. Most delicate

however is settling on a value of current gold production, a �ow we take

as tracking current population increase. But not only should the volume

of current gold use be correct, relative to current population increase,

there is in addition the initial condition on gold production for balanced

growth. This seems like the trickiest management issue, namely getting

on a balanced growth path "at the beginning", by bringing current gold

production in line with the current population size, getting the value of

g = g=N0 correct. We carried out a policy experiment with the model

on this question.

We simply perturbed the value of the balanced growth value for g and

solved for the resulting sequence of outputs, particularly kt+1 and mt+1:

For g a small amount too large, we observed the sequence of m0ts to rise,

along with prices, or in�ation to be apparent. This is not surprising since

gold is being injected into the economy somewhat faster than population

is growing. However, surprising to us was the unchanging values of

the "quantity" variables as the price variables increased. In�ation was

neutral in the model, an in�ation based on the wrong initial condition

for the pace of gold infusion. The model�s real and monetary sides cleave

cleanly. This in fact suggests that money supply management may not

25

have been complicated. The center has the "freedom" to steer the pace

of gold infusion into the economy in order to achieve zero in�ation at no

cost in terms of perturbations to the current "quantity" variables. We

have not dwelt on how agents react to price changes. The reactions of

agents are implicit and are, by default, supportive of the "neutrality" of

the money supply process. It remains of course a large leap to contend

that the behavior of variables in our model is precisely tracking behavior

in the Roman economy, particularly since we have not devoted much

attention to behavior of the model o¤ the balanced growth path.

6 The Model above with some taxation supportingmore Government Production

In fact Roman Italy would have had taxation that was supporting some

portion of current government activity. We indicate this level of extra

government production byG� : In Table 2, this extra government product

is supported by excise taxes on current consumer goods production and

current investment goods production. This is a simple extension to our

basic model. We still have the curious accounting "result" that much of

current government product is intermediate to the economy.

Table 2NNP (entries)

rKC wNC =pCC[1� � ]rKI wNI =pII[1� � ] ...(I = Kt+1 �Kt)rKg wNg =pgg (=imports)rKG wNG �pg(�� 1)g =pGG� (=� [pCC + pII])

pg(�� 1)g =�MNNI (entries) =

PrK =

PwN =0

In this formulation some excise taxes support some government pro-

duction at level G� . This government production is in addition to that

supported by current seignorage, pg(�� 1)g:Historians suggest that a principal source of government revenue was

from taxing large estates, essentially a land tax, rather than excise taxes.

This conventional narrative has levels of taxation being su¢ ciently high

so that in some sense the owners of such estates saw their potential power

26

held in check during the Pax Romana. In addition the conventional nar-

rative suggests that the break-down of the Augustan system involved the

emergence of new higher levels of taxation on estates. Roberts (2011)

suggests that this induced considerable tax avoidance by estate own-

ers. A key course of action involved an ower of a large estate becoming

somewhat autonomous and autarkic. Regional specialization became

attenuated as well as inter-regional trade.

7 Model 2: Seignorage without an Explicit Gov-ernment Sector Producing Goods and Services

We alter our basic model with seignorage supporting explicit govern-

ment production so that current seignorage is now transferred directly

to households as a per capita "dividend" period by period. The "re-

duction" in our basic model implies that there are now no government

goods, G in the national accounts and money-balance increments now

enter �nal demand (NNP) as a standard item each period. That is, new

money balances held by households are an item in current net national

product. There still can be a production of government services in this

economy but the activity becomes part of the background economic envi-

ronment. A stock set in place many periods earlier (eg. social capital) is

producing a constant �ow of services such as law and order. The follow-

ing Table gives an account of the Roman empire�s center (modern Italy)

with a gold mining sector and money demand for both transactions pur-

poses and purposes related to the current interest rate. Domestic gold

production in the current period, �g tons per year, comprises a por-

tion g which is fully paid for or balanced with wage and capital rental

disbursements and this gpg is used purely for consumer goods imports

to Italy.33 � exceeds unity. Once the g �ow is produced and paid for,

there is remaining gold (� � 1)g tons that is transferred to householdsas increments to money balances. We have then an augmentation of

domestic household money balances by �M (=pg(�� 1)g) each period.33In fact part of the export of gold would go to paying the wages of troops, WT ;

on the frontier. We abstract away from this here.

27

The augmentation in household money balances is calibrated by the cen-

ter to keep the local or domestic price level unchanging (no in�ation or

de�ation). To this end, we assume that gold is extracted, processed and

released so that g is increasing at the same rate as the labor force (pop-

ulation). In balanced growth, the labor force will again be assumed to

be growing at the constant rate n:

Central here is the fact that the cost gpg of producing the current

gold �ow �g is less than the market value of pg�g (�ow g� is costing

pgg = rKg + wN g). The "mark-up" or "surplus" in value, pg(� � 1)g;becomes an incremental �ow into the money balances of households in

the current period. Mt the current stock of money balances, has price,

unity. Hence, Mt+1�Mt = pg(��1)g: In our "static" national accounts

Table 3, a row sum is represented by the right hand entry; a column

sum by the corresponding entry in the bottom line. Net national income

(NNI) is the sum of entries in the bottom row and national product

(NNP) is the sum of entries in the right hand column.

Table 3NNP entries

rKC wNC =pCCrKI wNI =pII ...(I = Kt+1 �Kt)rKg wNg =pgg (=imports)

pg(�� 1)g =�MNNI entries =

PrK =

PwN =pg(�� 1)g

The de�nition of entries in the Table 3 follows from our earlier for-

mulation. r is the rental rate for durable capital, K: KC is then the

durable capital in use in producing consumption goods, domestically (in

Italy). w is the wage prevailing in the economy and NC is the amount

of labor involved in producing consumption goods in the current period.

Superscript I indicates the investment goods sector, one producing new

durable capital K: The third sector is the gold producing sector. The

total value of capital in use in the current period isPrK and for labor

isPwN: Observe that the column sum is the current value of total

product for Italy over the accounting period and the row sum is the

28

total value of inputs over the accounting period. The accounts re�ect

the fundamental principle that the value of inputs equals the value of

product over the accounting period. We assume that goods are produced

with constant returns to scale production functions (each industry com-

prises many small �rms) and that input pricing conforms with values of

marginal products (a competitive, undistorted equilibrium).

Our new "reduced" model is thirteen equations in thirteen unknowns

(C, I, KI , NI , Kg, Ng, pC , pI , pg, r, w, i, g). We have 3 production

relations for consumer goods, investment goods and gold:

C = [K �KI �Kg]�2 [N �N I �N g]1��2 ;

I = [KI ]�1[N I ]1��1 ;

g= [Kg]�3 [N g]1��3 ;

There are three price equations, one for each of consumer goods, invest-

ment goods and gold:

pC = [�2]��2 [1� �2]�(1��2)[r]�2 [w]1��2

pI = [�1]��1 [1� �1]�(1��1)[r]�1 [w]1��1

pg = [�3]��3 [1� �3]�(1��3)[r]�3 [w]1��3 ;

There are three value-balance equations, one for each of our three sectors:

r[K �Kg �KI ] + w[N �N g �N I ] =CpC

rKI + wN I = IpI

rKg + wN g = gpg:

The savings relation is (exogenous constant savings rate):

s[CpC + IpI + gpg] = IpI ;

and money supply equal to money demand relation is:

29

Mv = CpC + IpI + gpg with v = i0:1;

and our interest rate de�ned in

i =r

pI:

In anticipation of solving for a balanced growth outcome, we solve for g

in

(1� �) = nM

gpgwith n set equal to

I

K:

Hence the value of g is again endogenous under the assumption of us

taking up the case of balanced growth.

This 13 equation system is solved with Matlab in Appendix 3. Pa-

rameters are set at K = 6; N = 5;M = 20; � = 2:1; �1 = 0:5; �2 = 0:4;

�3 = 0:3: These parameters are the same as those employed in solving

the earlier model. The solution is:

I=1.1174, C=3.2979, KI=1.4946, NI=0.8354, pI=3.2441, pC=3.3701,

pg=3.3569, Kg=0.8382, Ng=1.0920, r=1.2123, w=2.1703, i=0.3737, g=1.0087.

We observe the interest rate at 37.37% and our value for population

growth, �KK

= 1:11746

= 18:62%: These rates are similar to those we

obtained for our model above without a sector producing government

goods. We veri�ed that proportionate changes in K,N, and M leave

prices unchanged but "quantities" altered. As we observed earlier, this

model possesses a price-independence property. When we alter M alone,

"quantities" remain unchanged and prices change.

Under the assumptions: (1) labor force (population) is growing a the

constant rate n; (2) gold supply increase matches the current population

size increase (g=N unchanging) and (3) current seignorage "funds" the

current increase in Mt (�M = (1 � �)gpg), we solve for a balancedgrowth solution in a numerical example. We re-de�ne our "quantity"

variables in the equations above in per capita terms, and input values

for kt and mt in anticipation that solution values kt+1 and mt+1 satisfy

kt+1 = kt and mt+1 = mt:We set n = I=K; kt = K=N; mt =M=N; with

30

I solved for above at 1.1174 and g =1.0087. See Appendix 4. We obtain

there a balanced growth solution (numerical example). The results are:

kt+1=1.2000, C/N=0.6596, KI/N=0.2990, NI/N=0.1670, pI=3.2444,

pC=3.3702, pg=3.3568, Kg=0.1675, Ng=0.2185, r=1.2128, w=2.1698, i=

0.3738, mt+1 =4.0000.

Observe that kt+1 = kt and mt+1 = mt: Hence we have a balanced

growth solution exhibited in our numerical run. Prices are unchanging

and quantity variables are all increasing at rate n:

This second model (helicopter injection of new money) involves a

seemingly implausible route for currency increments to get into the econ-

omy. It is thus of less interest in capturing large forces of history. The

model does provide an alternate route to balanced growth and does solve

in some sense the accounting problem that we observed with Model

I. The center�s management problem is not signi�cantly di¤erent from

the one we discussed earlier with the more complicated model. For

this model we carried out the same "policy experiment" we did earlier,

namely perturbing the initial value, g=N0:We observed the same result.

Steady in�ation resulted with an initial value above the value for the

balanced growth path. As well, the model exhibited in�ation neutrality

in the sense that as the prices rose, the "quantity" variables remained

unchanging.

8 End of the Pax Romana

An epidemic (a plague of some sort, brought in from Persia by soldiers

returning from the frontier) took hold of the Empire in 165 CE and dis-

rupted society.34 Roberts (2011) considers the arrival of this plague to

have triggered the end of the Pax Romana. 180 CE marked the death

of Emperor Marcus Aurelius. His successor, his natural son, Emperor

Commodus launched an expenditure binge that emptied the treasury at

the center. He was in due course assassinated. During the Third Cen-

tury CE, there is evidence of the banking system and long-distance trade34"Egyptian wages doubled after the major Antonine plague of 165-175 CE." Temin

(2004b; p. 519).

31

in the Empire shrinking severely. The Coliseum in Rome remained ac-

tive for contests into the sixth century but, the economy of Roman Italy

essentially contracted after about 180 CE until the Renaissance.35 A

civil war of sorts occurred near the end of the Pax Romana, followed

by the Severan Dynasty (193-235). Stability returned during the Fourth

Century but, relative to the situation during the Pax Romana, a less

productive political and economic order prevailed. Political power in

Roman Italy was slowly ceded after the Third Century to leaders in

Constantinople. As early as 285 Diocletian established a co-emperor for

the Eastern part of the empire. These co-leaders held di¤erent amounts

of power until the government in Rome simply disintegrated. The Ro-

mans retreated from governing in England in 410 CE. The Visigoths

brie�y held the city of Rome in 410 but the loss of Africa triggered the

�nal fall. (Roberts (2011); p. 252).

The remarkable aspect of the Roman Empire is that it was able to

function as a fairly uni�ed entity over so much territory for such a long

spell. The eastern part never gave up using the Greek language and the

subjects could not be said to have been assimilated by Roman Italy. One

might argue, with hindsight, that the government in Rome simply bit o¤

more than it could chew and would inevitably be faced with a splitting

o¤ of the eastern provinces from those in the west. Roberts (p. 245)

suggests that "Business survived [the collapse of the Roman Empire] in

Africa and the eastern half of the Roman Empire; Byzantium and the

Arab forces that triumphed in the seventh century inherited an urban

35"Around the start of the third century CE, the early Roman Empire came toan end under the pressure of a number of problems: several emperors who wereexceptionally autocratic and excessive and a series of revolts by the army which inturn led to Rome being ruled by a series of short-term emperors. The disruptionmanifested itself in many ways, including increased in�ation in the third centuryCE that is visible to us through debased coinage and occasional price quotations.In�ation was less than 1 percent in the �rst and second centuries CE, but pricesdoubled after the Antonine plague of the late second century and doubled again soonthereafter. The denarius began to be progressively debased at this time. (Temin(2006, p. 149)). Roberts emphasizes the disruption of social stability occasioned bythe plague. We, with others, argue for a signi�cant disruption in the stability ofgovernment caused by the exhaustion of low-cost silver and gold mines in the latesecond century CE.

32

system of money, markets and entrepreneurial business that it would

return to Europe during the early Renaissance."

There is considerable documentation of currency problems in the Em-

pire during the Third Century CE. Historians have not however traced

a path from the later dates of the well-functioning of the money system

during the Pax Romana to the relative instability of the money system

during the Third Century.36 What is clear is that the economy of the

Empire was prosperous during the period of a well-functioning money

system and was becoming fragmented during the Third Century. A

popular interpretation is that stable government at the center fell apart;

many emperors held o¢ ce for short periods and many failed to impose

order on the Empire at a reasonable cost. Tax rates became high and

prices were not stable. Another interpretation is that the high-quality

gold and silver mines in Spain and Dacia were depleted and those in

power at the center were unable to develop a new well-functioning money

system, similar to that which was in e¤ect during the Pax Romana. The

well-functioning money system, sound �scal management and law and

order were in some sense su¢ cient for markets to open up and trade to

�ourish. Clearly a good system of law and order as well a border security

are necessary for markets for goods, capital and labor to function well.

There is reasonable evidence of vast quantities of gold and silver being

mined in Spain, Dacia and elsewhere during the Pax Romana. This

is indirect evidence for the view that depletion of high-quality deposits

around 180 CE became an issue for those in power. We do not have good

evidence for a serious scarcity of gold and silver bullion around 180 CE

and a crisis of management of the money system after that date. There

is no evidence to contradict this view, either.

It makes sense to distinguish leadership problems from management

problems. Frequent turnover of leaders often accompanied by coups and

36We re-solved our �rst model with � becoming smaller (quality of gold depositsdeteriorating; � moving from 2.0 to 1.4) and observed, in a series of equilibria, theinterest rate rising and per capita consumption declining. The rental rate for K rosewhile the wage rate rose and then declined.

33

assassinations are leadership issues. It is possible for managers at the

center to keep the nation going in an economic and public security sense,

even with leadership problems. One usually links the two, however. A

power vaccuum (leadership crisis) aften co-incides with a breakdown in

management. The reverse can occur. A management crisis can trigger

and leadership crisis. Historians emphasize the leadership problems that

developed in the center after the Pax Romana. We are putting forth

the idea that management problems may well have triggered leadership

crises. Our argument is that without a steady �ow of gold and silver

bullion to the center, managers fumbled their remit. They turned to

new taxes and tax rates in order to get revenue and the high rates met

with various forms of resistence. In addition the managers turned to

in�ationary �nance in an attempt to maintain �scal order. The orderly

currency system of the Pax Romana became a fairly disorderly system.

Leadership problems co-incided with the management problems and the

economy of Roman Italy as well as that of the Empire as a whole never

returned to a properous long-run trajectory.

9 Seignorage and the Economy

Our Model 1 involves (1) the center (Roman Italy) getting goods and

services (government goods) without imposing high tax rates and (2)

the economy seeing the money supply grow in an orderly fashion. Large

contemporary economies enjoy seignorage but one presumes that the

magnitudes, relative to aggregate government expenditures, are small.

"Printing money" for a modern economy is centrally about "setting" the

price level or altering aggregate economic activity; not so much about

getting "free" government output. It is interesting to re�ect on the

case of Spain after 1530 when huge quantities of gold and silver �owed

into the treasury from mines as well as from direct plunder in central

and south America. This has been viewed as the case of a government

�oating on a sea of "free" revenue. The conventional wisdom is that the

"free" currency that was spent led directly to "excessive" importation of

goods to Spain and contributed to a stunting of the economy of Spain.

34

Local producers could not establish themselves to compete with low

cost imports. This is an early form, or a variant, of what today is

referred to as the resource curse. Abundant exports such as oil in a

nation today fund a large volume of imports and local producers are

often unable to get rooted and to compete successfully with exporters

located abroad. Local producers get shouldered out of supplying local

markets and as well are generally unable to sell to foreign markets. A

central mechanism here becomes the relatively high cost of local products

to buyers abroad because the exchange rate has made local products

expensive in international markets (so-called Dutch Disease).

Conventional wisdom, in addition, accepts that the large volumes of

gold and silver �owing to the center in Spain in the sixteenth and seven-

teenth centuries contributed directly to a signi�cant in�ation in Europe.

Under the reign of Philip II, Spain saw a �vefold increase in prices.37 The

low-cost �ows of gold and silver moving into Spain during the sixteenth

and seventeenth centuries are thus considered to have been bad for the

economic development of Spain.38 We would be remiss to fail to note

that Spain managed to build up a huge colonial empire and to conquer

large areas of Europe during its lengthy period of massive "importing"

of gold and silver bullion. Nevertheless it is not unreasonable to say that

Spain fell victim to a version of "resource curse" whereas Roman Italy

managed to avoid such a trap. Some might argue that "the Augustan

money system" stunted the economic development of Roman Italy dur-

ing the Pax Romana because imports were made available readily to Ro-

man Italy via payment with the low-cost gold and silver. This negative

judgement might include a sti�ing of technical progress because �rms

could not get rooted locally expand and innovate. Roman Italy during

the Pax Romana was essentially an importing nation in this view. This

37(http://en.wikipedia.org/wiki/Philip_II_of_Spain#Economy)38"Even though Philip II was bankrupt by 1596 (for the fourth time, after France

had declared war on Spain), more silver and gold were shipped safely to Spain inthe last decade of his life than ever before. This allowed Spain to continue its mili-tary e¤orts, but led to an increased dependency on the precious metals and jewels."(http://en.wikipedia.org/wiki/Philip_II_of_Spain#Economy)

35

view is worth re�ecing on. Roman Italy has, however, been accepted as

the high income region of the early Empire. The local economy may not

have been import-oriented but was apparently doing well. In contrast

with Sixteenth century Spain, there was no signi�cant in�ation in the

Empire during the Pax Romana. The center may have been lucky in

its management of the currency or it may have been prudent and in-

telligent. A by-product of having abundant gold and silver for funding

imports, was the expansion in the stock of currency each year. The rate

of expansion in the currency appears to have been right since in�ation

was avoided.

It is not di¢ cult to see how early observers would de�ne a nation as

an economic successs if the nation was coping militarily with its neigh-

bors, possessing large cities with bustling activity, and engaging in the

importing of goods in large volumes. A nation with large in�ows of

low-cost gold and silver could with a little luck pass this test of eco-

nomic success. Adam Smith�sWealth of Nations emphasizes the folly of

achieving economic success via gold and silver accumulation, the folly of

mercantilism. Smith emphasized that successful economic development

was about high standards of productivity of local workers and high stan-

dards of living for the citizens of a nation, not about the accumulation

of large reserves of gold via unbalanced exporting activity. For Smith

balanced international trade involved both e¢ cient local production and

access to foreign and often exotic goods. Access to imports was to be

achieved via successful low-cost local production of goods that foreign-

ers were keen on, not by the corralling of a stream of low-cost gold and

silver that was to fund importing. Successful economies would be active

exporters of locally-produced goods, not of low-cost gold and silver coins

and bullion.

Our view of the economy of Roman Italy during the Pax Romana is

largely mercantilist. Seignorage was at the center of the economic sys-

tem. However as gold and silver �owed out to fund imports to Roman

Italy, the outer provinces were bene�tting twice-fold. They were getting

36

infusions of new currency to lubricate their own economies and they were

being subjected to signi�cant demand for their products from "abroad".

Though Roman Italy may have been operating as a mercantilist nation,

the drawbacks of such a system were o¤set to a considerable extent by

bene�ts accruing to the economies of the provinces. The economy of the

empire was a special center-periphery model with mercantilist character-

istics. The managers in Rome appear to have found the Goldilocks point

�the expansion in the coin stock in circulation was not too hot and not

too cold �in contrast with what managers achieved in Spain. There was

a balance of sorts in injecting new coinage into the Empire at large. In

addition, the demand for imports by Roman Italy meant revenues and

encouragement for producers in the provinces. An interesting balance in

economic development between the center and the periphery took place.

Good management by government led to relative tranquility within the

Roman Empire. Borders were kept secure. Internal security allowed for

the development of a business culture. A uni�ed and stable currency led

to large-scale inter-regional trade.

37

APPENDIX 1: 16 EQUATIONS IN (KG; KI ; Kg; NG; N I ; N g; I;

C; r; w; pI ; pC ; pg; pG; g)

Our 16 equations in our 16 unknowns are below.

r[K �KG �KI �Kg] + w[N �NG �N I �N g] =CpC

rKI + wN I = IpI

rKG + wNG=((�� 1)pgg)

rKg + wN g = pgg

[K �KG �KI �Kg]�2 [N �NG �N I �N g]1��2 =C

[KI ]a1 [N I ]1��1 = I

[Kg]a3 [N g]1��3 = g;

[KG]a4 [NG]1��4 =((�� 1)pgg)=pG;

[�2]��2 [1� �2]�[1��2]r�2w[1��2]= pC

[�1]��1 [1� �1]�[1��1]r�1w[1��1]= pI

[�3]��3 [1� �3]�[1��3]r�3w[1��3]= pg

[�4]��4 [1� �4]�[1��4]r�4w[1��4]= pG

Mv=CpC + IpI + gpg + (�� 1)pgg

with v= i0:1;

i= r=pI

s[CpC + IpI + gpg + (�� 1)pgg] = IpI ;

(�� 1)= (I=K)M=gpg:

We solve our 16 equation system numerically, employing Matlab.

The Matlab program is as follows:

function f=nle(x)

a1=0.5;a2=0.4;a3=0.3;a4=0.35;IVK=0.0;s=0.2;

% Aug 13...

K=6*(1+IVK);N=5*(1+IVK);M=20*(1+IVK);

mu=2.1;

38

% quantities solved

f(1)=x(1)-((x(3)^a1)*x(4)^(1-a1));

ZK=K-x(8)-x(3)-x(15);

ZN=N-x(9)-x(4)-x(16);

f(2)=x(2)-(ZK)^(a2)*(ZN)^(1-a2);

f(3)=x(13)-((x(8))^(a3)*(x(9))^(1-a3));

% values matched

f(4)=x(10)*x(3)+x(11)*x(4)-(x(5)*x(1));

f(5)=x(10)*(ZK)+x(11)*(ZN)-(x(6)*x(2));

f(6)=x(10)*x(8)+x(11)*x(9)-(x(7)*x(13));

% prices of outputs

f(7)=(a1^(-a1)*(1-a1)^(-(1-a1))*(x(10))^(a1)*(x(11))^(1-a1))-x(5);

f(8)=(a2^(-a2)*(1-a2)^(-(1-a2))*(x(10))^(a2)*(x(11))^(1-a2))-x(6);

f(9)=(a3^(-a3)*(1-a3)^(-(1-a3))*(x(10))^(a3)*(x(11))^(1-a3))-x(7);

G=((mu-1)*x(13)*x(7))/x(14);

% money supply equals money demanded

f(10)=(M*(x(12)^(0.1)))-(x(5)*x(1)+x(6)*x(2)+x(7)*x(13)+G*x(14));

% exog savings rate s

f(11)=s*(x(5)*x(1)+x(6)*x(2)+x(7)*x(13)+G*x(14))-(x(1)*x(5));

% interest rate in x(12)

f(12)=x(12)-(x(10)/x(5));

% x(13) is g, here endogenized;

f(13)=mu-1-((M*x(1)/K)/(x(13)*x(7)));

% Govt sector equations

f(14)=(a4^(-a4)*(1-a4)^(-(1-a4))*(x(10))^(a4)*(x(11))^(1-a4))-x(14);

f(15)=G-(x(15)^a4)*(x(16)^(1-a4));

f(16)=G*x(14)-(x(10)*x(15)+x(11)*x(16));

Results are:

I=1.1225, C=2.1787, KI=1.5348, NI=0.8210, pI=3.2223, pC=3.3621,

pg=3.3637, Kg=0.8664, Ng=1.0806, r=1.1781, w=2.2034, i= 0.3656, g=1.0113,

pG=3.3815, KG=1.1122, NG=1.1035. (These values correspond to the

39

order of the x(i)�s in the Matlab program; x(1)=I=1.1224, x(2)=C=2.1903.

etc.)

�

40

APPENDIX 2: THE BALANCED GROWTH SOLUTION

Our 16 equation system to solve for kt+1, c, kI , nI , pI , pC , pg, kg, ng,

r, w, kG, nG, pG, mt+1; i, is:

r[kt � kG � kI � kg] + w[1� nG � nI � ng] = cpC

rkI + wnI = [ _k + nkt]pI for

_K

N= [ _k + nkt]

rkG + wnG=((�� 1)pgg)

rkg + wng = pgg for g =g

N0[kt � kG � kI � kg]�2 [1� nG � nI � ng]1��2 = c

[kI ]a1 [nI ]1��1 = _k + nkt

[kg]a3 [ng]1��3 = g;

[kG]a4 [nG]1��4 =((�� 1)pgg)=pG;

[�2]��2 [1� �2]�[1��2]r�2w[1��2]= pC

[�1]��1 [1� �1]�[1��1]r�1w[1��1]= pI

[�3]��3 [1� �3]�[1��3]r�3w[1��3]= pg

[�4]��4 [1� �4]�[1��4]r�4w[1��4]= pG

ctpC + [ _k + nkt]p

I + gpg + (�� 1)pgg=mtv

with v= i0:1;

i= r=pI

s[ctpC + [ _k + nkt]p

I + gpg + (�� 1)pgg] = [ _k + nkt]pI

_m+mtn=(�� 1)gpG:

The Matlab program yielding the steady state (solution II) is:

function f=nleB(x)

a1=0.5;a2=0.4;a3=0.3;a4=0.35;

% augment parameters proportionately with IVK

N=5;K=6/N;M=20.0/N;s=0.2;g=1.0113/N;

% g=1/N......

mu=2.1;n=1.1225/6;

41

% quantities

f(1)=((x(1)-K)+n*K)-((x(3)^a1)*x(4)^(1-a1));

ZK=K-x(8)-x(3)-x(15);

ZN=1-x(9)-x(4)-x(16);

f(2)=x(2)-(ZK)^(a2)*(ZN)^(1-a2);

f(3)=g-((x(8))^(a3)*(x(9))^(1-a3));

% value balances

f(4)=x(10)*x(3)+x(11)*x(4)-(x(5)*((x(1)-K)+n*K));

f(5)=x(10)*(ZK)+x(11)*(ZN)-(x(6)*x(2));

f(6)=x(10)*x(8)+x(11)*x(9)-(x(7)*g);

% price de�ning

f(7)=(a1^(-a1)*(1-a1)^(-(1-a1))*(x(10))^(a1)*(x(11))^(1-a1))-x(5);

f(8)=(a2^(-a2)*(1-a2)^(-(1-a2))*(x(10))^(a2)*(x(11))^(1-a2))-x(6);

f(9)=(a3^(-a3)*(1-a3)^(-(1-a3))*(x(10))^(a3)*(x(11))^(1-a3))-x(7);

G=((mu-1)*g*x(7))/x(14);

% money supply equals demand

f(10)=(M*(x(12)^(0.1)))-(x(5)*((x(1)-K)+n*K)+x(6)*x(2)+x(7)*g+G*x(14));

% exog savings rate

f(11)=(x(5)*((x(1)-K)+n*K))-s*(x(5)*((x(1)-K)+n*K)+x(6)*x(2)+x(7)*g+G*x(14));

% interest rate is x(12)

f(12)=x(12)-(x(10)/x(5));

% g solve

f(13)=((x(13)-M)+M*n)-(mu-1)*g*x(7);

% 4 SECTOR STUFF

f(14)=(a4^(-a4)*(1-a4)^(-(1-a4))*(x(10))^(a4)*(x(11))^(1-a4))-x(14);

f(15)=G-(x(15)^a4)*(x(16)^(1-a4));

f(16)=G*x(14)-(x(10)*x(15)+x(11)*x(16));

Output from our computer run are:

kt+1 =1.2000, C/N=0.4358, KI/N=0.3068, NI/N=0.1642, pI=3.2226,

pC=3.3622, pg=3.3636, Kg/N=0.1733, Ng/N=0.2161, r=1.1786, w=2.2029

, i= 0.3657, mt+1=4.0000, pG=3.3815, KG/N=0.2223, NG/N=0.2208.

42

(These values correspond to the order of the x(i)�s in the Matlab pro-

gram.)

These outputs constitute a balanced growth trajectory with prices

unchanging and kt+1=kt and mt+1=mt: The labor force (population) is

growing at the constant rate n: Key "quantity" variables are all growing

at rate n: Prices are unchanging.

Appendix 3: SOLUTION OF THE MODEL WITHOUT GOVERN-

MENT GOODS (HELICOPTER MONEY-DROP TO HOUSEHOLDS)

The parameter choices are K = 6; N = 5; M = 20; g = 1; s = 0:2;

�1 = 0:5; �2 = 0:4; �3 = 0:3: The Matlab program is

function f=nle(x)

a1=0.5;a2=0.4;a3=0.3;a4=0.35;IVK=0.0;s=0.2;

% Aug 13...

K=6*(1+IVK);N=5*(1+IVK);M=20*(1+IVK);

% g=1.0*(1+IVK);

mu=2.1;

% quantities solved

f(1)=x(1)-((x(3)^a1)*x(4)^(1-a1));

ZK=K-x(8)-x(3);

ZN=N-x(9)-x(4);

f(2)=x(2)-(ZK)^(a2)*(ZN)^(1-a2);

f(3)=x(13)-((x(8))^(a3)*(x(9))^(1-a3));

% values matched

f(4)=x(10)*x(3)+x(11)*x(4)-(x(5)*x(1));

f(5)=x(10)*(ZK)+x(11)*(ZN)-(x(6)*x(2));

f(6)=x(10)*x(8)+x(11)*x(9)-(x(7)*x(13));

% prices of outputs

f(7)=(a1^(-a1)*(1-a1)^(-(1-a1))*(x(10))^(a1)*(x(11))^(1-a1))-x(5);

f(8)=(a2^(-a2)*(1-a2)^(-(1-a2))*(x(10))^(a2)*(x(11))^(1-a2))-x(6);

f(9)=(a3^(-a3)*(1-a3)^(-(1-a3))*(x(10))^(a3)*(x(11))^(1-a3))-x(7);

% money supply equals money demanded

43

f(10)=(M*(x(12)^(0.1)))-(x(5)*x(1)+x(6)*x(2)+x(7)*x(13));

% exog savings rate s

f(11)=s*(x(5)*x(1)+x(6)*x(2)+x(7)*x(13))-(x(1)*x(5));

% interest rate in x(12)

f(12)=x(12)-(x(10)/x(5));

% x(13) is g

f(13)=mu-1-((M*x(1)/K)/(x(13)*x(7)));

The results of the numerical run are:

I=1.1174, C=3.2979, KI=1.4946, NI=0.8354, pI=3.2441, pC=3.3701,

pg=3.3569, Kg=0.8382, Ng=1.0920, r=1.2123, w=2.1703, i=0.3737, g=1.0087.

(These values correspond to the order of the x(i)�s in the Matlab pro-

gram; x(1)=I=1.1173, x(2)=C=3.3066, etc.)

Appendix 4: Balanced Growth (no explicit government production)

Our 13 equation system to solve follows ( 13 unknowns: kt+1;mt+1; c; kI ; kg; nI ; ng; pC ; pI ; pg; r; w; i):

ct= [kt � kI � kg]�2 [1� nI � ng]1��2 ;_k + nk= [kI ]�1[nI ]1��1 ;

g= [kg]�3 [ng]1��3 ; for g =g

N0

There are three price equations, one for each of consumer goods, invest-

ment goods and gold:

pC = [�2]��2 [1� �2]�(1��2)[r]�2 [w]1��2

pI = [�1]��1 [1� �1]�(1��1)[r]�1 [w]1��1

pg = [�3]��3 [1� �3]�(1��3)[r]�3 [w]1��3 ;

There are three value-balance equations, one for each of our three sectors:

44

r[kt � kg � kI ] + w[1� ng � nI ] = pCctrkI + wnI = pI [ _k + nkt]

rkg + wng = gpg;

The savings relation:

s[ctpC + [ _k + nkt]p

I + gpg] = [ _k + nkt]pI ;

and money supply equal to money demand relation:

mtv = ctpC + [ _k + nkt]p

I + gpg with v = i0:1;

and our interest rate de�ned in

i =r

pI:

Money growth is de�ned in

_m+ nm = (1� �)gpg: �

We turn to a numerical solution of the above system. The parameter

choices are K = kt = 6=N;N = 5;M = mt = 20=N; g = 1:0087=N;

�1 = 0:5; �2 = 0:4; �3 = 0:3; n = 1.1174/6, � =2.1. TheMatlab program

for solving for a balanced growth outcome is:The Matlab program is

function f=nleB(x)

a1=0.5;a2=0.4;a3=0.3;a4=0.35;

% augment parameters proportionately with IVK

N=5;K=6/N;M=20.0/N;s=0.2;g=1.0087/N;

% g=1/N......

mu=2.1;n=1.1174/6;

% quantities

f(1)=((x(1)-K)+n*K)-((x(3)^a1)*x(4)^(1-a1));

ZK=K-x(8)-x(3);

45

ZN=1-x(9)-x(4);

f(2)=x(2)-(ZK)^(a2)*(ZN)^(1-a2);

f(3)=g-((x(8))^(a3)*(x(9))^(1-a3));

% value balances

f(4)=x(10)*x(3)+x(11)*x(4)-(x(5)*((x(1)-K)+n*K));

f(5)=x(10)*(ZK)+x(11)*(ZN)-(x(6)*x(2));

f(6)=x(10)*x(8)+x(11)*x(9)-(x(7)*g);

% price de�ning

f(7)=(a1^(-a1)*(1-a1)^(-(1-a1))*(x(10))^(a1)*(x(11))^(1-a1))-x(5);

f(8)=(a2^(-a2)*(1-a2)^(-(1-a2))*(x(10))^(a2)*(x(11))^(1-a2))-x(6);

f(9)=(a3^(-a3)*(1-a3)^(-(1-a3))*(x(10))^(a3)*(x(11))^(1-a3))-x(7);

% money supply equals demand

f(10)=(M*(x(12)^(0.1)))-(x(5)*((x(1)-K)+n*K)+x(6)*x(2)+x(7)*g);

% exog savings rate

f(11)=(x(5)*((x(1)-K)+n*K))-s*(x(5)*((x(1)-K)+n*K)+x(6)*x(2)+x(7)*g);

% interest rate is x(12)

f(12)=x(12)-(x(10)/x(5));

% g solve

f(13)=((x(13)-M)+M*n)-(mu-1)*g*x(7);

Our numerical output is:

kt+1=1.2000, C/N=0.6596, KI/N=0.2990, NI/N=0.1670, pI=3.2444,

pC=3.3702, pg=3.3568, Kg=0.1675, Ng=0.2185, r=1.2128, w=2.1698, i=

0.3738, mt+1 =4.0000. (These values correspond to the order of the x(i)�s

in the Matlab program.)

Observe that kt+1 = kt and mt+1 = mt: Hence we have a balanced

growth solution exhibited in our numerical run.

46

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