Seignorage and Growth during the
Pax Romana
John M. Hartwick
[email protected]
Abstract
We set out a stylized and simple model of the economy of Roman Italy during a period of considerable economic prosperity.
Our focus is on gold coins as currency and the seignorage being
managed by the central government. We solve numerically for a
balanced growth representation of the economy of Roman Italy, a
solution that captures the intricacies of money creation, currency
expansion and seignorage. The "use" of seignorage by the central
government is treated in two ways, leaving us with two distinct
but closely related models. We subscribe to the view that the
exhaustion of low-cost gold and silver deposits contributed signi…cantly to the ending of the economic prosperity enjoyed by
Roman 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. Central 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
1
We do not attempt to calibrate our model to …t the very few bits of data that
are available on the structure of the economy of Roman Italy. Scheidel (2012) draws
together many fragments of data on revenues and expenditures of the central government in Rome during the Pax Romana.
1
period, an activity that yielded substantial seignorage for the central government. 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 economic 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 la2
We treat gold and silver deposits as located in the the city of Rome for the most
part and thus abstract from actual transportation costs for gold and silver bullion to
mints in the Empire. We also consider minting to have been done only in the city or
Rome for the most part.
3
Nerva (96-98), Trajan (98-117), Hadrian (117-138), Antoninus Pius (138-161),
Marcus Aurelius (161-180).
4
Augustus 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 Edward
Gibbon’s …ve good or great emperors. Each of the …ve great emperors was succeeded
by his adopted son, except for Marcus Aurelius who was succeeded by his natural
son, 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
5
The slavery of the Roman Empire involved able individuals rising to senior positions in business, banking and agriculture. Most slaves would no doubt have toiled
in laborious jobs. Slaves were denied many basic rights but the life of one on the bottom 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 slavery
in order to be better fed and housed. There were clear routes for an able slave to
gain 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 slaves
succeeded in getting their freedom and as well many slaves occupied positions of
responsibility and lived comfortably. Slavery in the United States he argues was a
closed system, with very few slaves being manumitted and incentives to work were
of the "stick" variety rather than the "carrot" variety. "During the Second Century
CE as peaceful conditions sharply reduced the supply of slaves, their average cost
quadrupled." (Roberts (2011; p. 208). "At any one time, approximately 150,000
prisoners and slaves labored in Roman mines, 40,000 in the Spanish silver mines
alone." (Roberts (2011; p. 218).
6
Fractional reserves and money multiplication is not taken up by Temin (2004a).
Presumably little money multiplication was taking place since the banks were not
tightly linked in networks. The contribution of banks to the economy of ancient
Rome was presumably in intermediating between lenders and borrow-investors.
7
Publican societies, forms of joint stock companies, operated in many areas of the
economy. The stock owners were usually the large land owners in the Empire. Small
businesses were run by freedmen, often with a single slave and a "factory" joined to
a residence.
8
Archeologists point out that the numbers of wrecks of Roman ships in the
Mediterranean exhibits a peak during the early Roman Empire. This supports the
contention that the economy was doing relatively well at this time and that subsequent 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 bread
baked with grain grown in North Africa or Egypt, and …sh that had been caught
and dried near Gibralter. He cooked with oil from North Africa in pots and pans of
copper mined in Spain, ate o¤ dishware …red in French kilns, and drank wine from
Spain 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 Ethiopia
send 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 principal 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)) initiated the construction of the majestic Coliseum in the second half of
the First Century. Around 100 CE, Trajan (98-119) supervised the reconstruction 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 infrastructure: roads, ports, aqueducts, sewers and town-centers. Temples
and stadiums were erected in larger towns and cities. Good infrastructure, 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 economic activity and social relations among individuals and groups more
generally.
We focus attention here on the money supply during the Pax Romana. Trade ‡ourished and there was a general absence of in‡ation.
Our interpretation is that Augustus laid down a new system of governance and his successful successors embellished his system in various
qua¢ ng African or Aegean wine in Syrian glasses, all while Greek statues gazed down
on set tables of Moroccan lime wood."
9
Temin (2004a) argues that land was bought and sold in markets. He emphasizes
that agricultural societies that are based on mostly agricultural activity will have
less market activity because isolated farms will be relatively autarkic. Information
moved slowly in the absence of the post, phone and trains. Hence price adjustments
would 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 period was relative internal and external peace. Central to his system10
was a balancing of the interests of power-seekers at the center. In addition 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 standardized currency, one that allowed for trade and regional specialization to
‡ourish. The combination of a vast peaceful nation and a uniform system 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 social
order" (Roberts (2011; p. 170).
11
Shortly 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 in
33 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. For
instance, the total Roman coinage per capita in the early years of the Principate came
to approximately 80 percent of the current US money supply. Rome’s sources of silver
allowed it to mint enough virtually pure silver denarii to reliably maintain this money
supply. 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
gold aureus as well. This was used for international trade, the payment of taxes, and
other 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 based
on the coins previously used by Qin. These coins were round with a square hole in the
middle which was the common design for most Chinese copper coins until the 20th
century. Due to the low value of an individual coin, the Chinese have traditionally
strung a nominal thousand copper coins onto a piece of string. However government
taxes were levied in both coins and in products such as rolls of silk. Salaries were
also 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 use
lead 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 abetted 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 complete
and China experienced a period of rapid economic growth. This was re‡ected in the
growth of coining. In 1073, the peak year for minting coins in the Northern Song,
the government produced an estimated six million strings containing a thousand
copper coins each. The Northern Song is thought to have minted over two hundred
million strings of coins which were often exported to Inner Asia, Japan, and SouthEast Asia, where they often formed the dominant form of coinage. Song merchants
rapidly adopted forms of paper currency starting with promissory notes in Sichuan
called "‡ying money" (feiqian). These proved so useful the state took over production
of this form of paper money with the …rst state-backed printing in 1024. By the 12th
century, various forms of paper money had become the dominant forms of currency
in China and were known by a variety of names such as jiaozi, qianyin, kuaizi, or
guanzi." (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 sophistication
described 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 those
of 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 system
throughout the Empire. He does note however: "Rural transactions in Rome were
made with relatively uniform coins, as in eighteenth-century France, and possibly
more 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 gold
aureus as well. This was used for international trade, the payment of taxes, and
other large payments." (Roberts (2011; pp. 236-37) Between 200 BC and 200 CE
an estimated 100 million denarii were brought to Rome per year from silver mines
in 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 century
BC 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 and
silver mines per year. He is unsure whether this represents net "pro…t" to the state
or gross value of re…ned ores. "Mineral wealth alone might have covered a sizeable
share 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 government sector (a sector employing labor and capital to produce government 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
16
There were coins of lesser metals in low denominations in addition to gold and
silver 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] were
actually created for the purpose of facilitating the state’s own expenditures for goods
and services." Steve Coe on the Internet (http://ezinearticles.com/?Ancient-Silverand-GoldCoinage&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 investment 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 (debasement 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
18
This is the value of current total product net of the current value of seignorage. We experimented with savings governed by the Ramsey Rule but rejected this
formulation since imports are treated as homogeneous, leaving imports of consumer
and/or producer goods unspeci…ed. One might characterize our model as a neoclassical Solow growth model with money included and sectors disaggregated.
19
A 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, there
would 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 households, 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 government 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 accounting surprises in this second case, the case of no explicit production of
government goods.
The currency system that Augustus inherited and in a sense institutionalized 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 Romana. 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 denarius
was divided into four bronze sesterces, which were the common unit of commerce
in 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 it
was divided into six and later seven bronze obols. For calibration, one modius (6.5
kilograms) of wheat cost four to six sesterces on the private market in Roman during
the …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 Roman govenment with the necessary …anancial funds to maintain the Empire, 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 copperbased 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-lessvaluable 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=
When the gold in Spain began to run out, the Romans noticed that
21
Healy (1978) deals exhaustively with Roman mining activity but leaves one without speci…cs about volumes of high-quality metals produced.
10
there were also gold mines in Dacia (called Romania today)... The Romans 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 locations. In his Chapter 15, he documents the debasement of gold and
silver coinage during the Third Century. "A pound of silver was producing 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 supply of bullion, and in the case of Rome, reductions in the precious-metal
content of the coinage suggests strain on the bullion supply." DuncanJones 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 borders. "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
22
Gold was mined by the Romans in Wales, also, presumably in relatively small
quantities.
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 117
CE), it held it value because 25 denarii could always buy one arueus... There was
little 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 provinces
until 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 between 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 largescale 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 internal 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 during 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 government 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 production 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 highquality 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 outsiders
24
"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).)
25
Unit 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] becomes 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 presumably
had the same value in the marketplace as the equivalent quantity of metal. From very
early on, however, states from time to time attempted to establish a conventional
value for coins that, as metal, were worth less than their ’…at’or ’…duciary’value."
(Harris (2008; p. 199).
26
Hollander (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 a
demand for money that is independent of transactions needs by households.
15
as it did during the Pax Romana.
3
Government Production Funded with current seignorage: 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 producing ‡ow output per period under constant returns to scale. In our
numerical solving, we assume that each sector is operating with a CobbDouglas 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,
K g 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 pg g equal to
rK g + wN g : r is the prevailing rental rate on a capital input and w is the
local wage rate. Hence (
1)pg g is the seignorage that the government
is free to allocate. In our …rst model, this seignorage covers the cost of
local government goods produced in the current period. That is
(
1)pg g = pG G = rK G + wN G :
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)pg g:
Current seignorage is serving two functions in the current period; it pays
for the current government goods produced in the private sector and it
27
We 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 to
not 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 national 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 1
C
rK
rKI
rKg
rKG
P
NNI (entries) = rK
C
wN
wNI
wNg
wNG
P
= wN
pg (
p (
=0
g
1)g
1)g
NNP (entries)
=CpC
=IpI
=gpg (= imports)
=0
= M
In Table 1, the total value of capital in use in the current period
P
P
is
rK and for labor is
wN: 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, indicated by
M; and (b) current government product, a ‡ow good rather
than a durable good, is fully funded by current seignorage, (
1)pg g:
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 "products"), double counting in NNP is precluded by having the value of
current government product intermediate in the account. Current government 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 supports 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 Nt
Nt
(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 (K G ; K I ; K g ; N G ; 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
K g ] + w[N
NG
NI
N g ] = CpC
rK I + wN I = IpI
rK G + wN G = ((
1)pg g)
rK g + wN g = pg g ;
There are four equations de…ning current output levels28 C; I; G and
g:
[K
KG
KI
K g ] 2 [N
NG
NI
N g ]1
2
=C
[K I ]a1 [N I ]1
1
=I
[K g ]a3 [N g ]1
3
= g;
[K G ]a4 [N G ]1
4
= ((
1)pg g)=pG ;
There are four equations de…ning output prices pC ; pI ; pG ; and pg :
28
We assume that industry output came from many small …rms. Competitive
conditions are assumed to have prevailed.
19
[
2]
2
[1
2]
[1
2]
r 2 w[1
2]
= pC
[
1]
1
[1
1]
[1
1]
r 1 w[1
1]
= pI
[
3]
3
[1
3]
[1
3]
r 3 w[1
3]
= pg
[
4]
4
[1
4]
[1
4]
r 4 w[1
4]
= pG ;
The exogenous level of M de…nes the price level.29
M v = CpC + IpI + gpg + (
1)pg g 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)pg g] = 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 unchanging). In our 16th equation, g must satisfy
(
for I=K = n and
1)pg g
= n;
M
1)pg g:
M = nM = (
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;
4
= 2:1;
1
= 0:5;
2
= 0:4;
3
= 0:4;
= 0:35, we obtain the following results:
M is the money supply and v1 pQ 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 the
interest rate.
30
Savings 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 our
system and thus allows for the possibility of balanced growth.
29
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
K
K
=
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 solutions 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 gold
available 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:
31
Population growth rates in various locations around the world are inferred to
be almost zero until about 1700. Our numerical output is clearly o¤ base in this
regard. Professor Walter Scheidel in the Stanford Classics department reports an
informed guess of the population growth rate for the Empire during the Pax Romana
of 0.15%. He also points out that average heights of people in Italy during the period
of Roman hegemony were lower than earlier and later. He mentions relatively high
rates of urbanization and associated disease transmission as a possible cause. See
also Scheidel and Turchin (2009)
32
This price-vector invariance should be referred to as neutrality of the money
supply. For the money supply to be non-neutral, we must connect transactions
demand and "interest-rate" demand for money in a non-neutral fashion as in the
h
i1=2
Baumol-Tobin formulation: money demand is tpQ
; where t is the cost of going
r
to the bank. Crucial here is the exponent 1=2: If the exponent were 1:000; then
neutrality would be restored to the system. In the Appendix, we solve this model
with only a transactions demand for cash balances, inspired by the famous formula:
M v = pQ:
21
5
Balanced Growth
We distinguished between the economic assumptions "driving" balanced
growth and the technical details of solving for endogenous variables characterizing 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 = pG G:
(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
M
M
= n or (
1) =
nM
:
gpg
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 by Nt (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
N_
N
is 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
K_ t _
Kt
= kt + nkt for kt =
;
Nt
Nt
M_ t
Mt
and
=m
_ t + nmt for mt =
:
Nt
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 sector, pG [G=N ]; is equal to the "excess value" of current gold production
(
1)gpg . The 16 equations to be solved include four value balance
equations
r[kt
kG
kI
k g ] + w[1
nG
nI
ng ] = cpC
rk I + wnI = [k_ + nkt ]pI for
rk G + wnG = ((
K_
= [k_ + nkt ]
N
1)pg g)
rk g + wng = pg g for g = g=N0 ;
four production relations
[k
kG
kI
k g ] 2 [1
nG
nI
ng ]1
2
=c
[k I ]a1 [nI ]1
1
= k_ + nkt
[k g ]a3 [ng ]1
3
= g;
[k G ]a4 [nG ]1
4
= ((
1)pg g)=pG ;
four price equations
[
2]
2
[1
2]
[1
2]
r 2 w[1
2]
= pC
[
1]
1
[1
1]
[1
1]
r 1 w[1
1]
= pI
[
3]
3
[1
3]
[1
3]
r 3 w[1
3]
= pg
[
4]
4
[1
4]
[1
4]
r 4 w[1
4]
= pG ;
a savings relation
[k_ + nkt ]pI = s[ct pC + [k_ + nkt ]pI + gpg + (
a money demand equals money supply equation
23
1)pg g];
mt v = ct pC + [k_ + nkt ]pI + gpg + (
1)pg g;
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
_ by mt+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 =
1:1225
6
and g =
1:0113
;
5
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 (population) 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 balanced 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 probably 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 features 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 somewhat complicated issue of separating current gold production in funds
for current imports and funds for purchasing government goods and services. 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 m0t s 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 supporting
more 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 by G : 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 2
C
rK
rKI
rKg
rKG
P
NNI (entries) = rK
C
wN
wNI
wNg
wNG
P
= wN
pg (
pg (
=0
1)g
1)g
NNP (entries)
=pC C[1
]
I
=p I[1
] ...(I = Kt+1
g
=p g (=imports)
=pG G (= [pC C + pI I])
= M
In this formulation some excise taxes support some government production 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
Kt )
held in check during the Pax Romana. In addition the conventional narrative 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 owners. 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 Government Sector Producing Goods and Services
We alter our basic model with seignorage supporting explicit government production so that current seignorage is now transferred directly
to households as a per capita "dividend" period by period. The "reduction" 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 environment. A stock set in place many periods earlier (eg. social capital) is
producing a constant ‡ow of services such as law and order. The following Table gives an account of the Roman empire’s center (modern Italy)
with a gold mining sector and money demand for both transactions purposes 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 households
as increments to money balances. We have then an augmentation of
domestic household money balances by
33
M (=pg (
1)g) each period.
In fact part of the export of gold would go to paying the wages of troops, W T ;
on the frontier. We abstract away from this here.
27
The augmentation in household money balances is calibrated by the center 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 (population). 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
g
g
g
is costing
g
p g = rK + wN ). The "mark-up" or "surplus" in value, p (
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 = p g (
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 3
C
rK
rKI
rKg
P
NNI entries = rK
C
wN
wNI
wNg
P
= wN
pg (
=pg (
1)g
1)g
NNP entries
=pC C
=pI I ...(I = Kt+1
=pg g (=imports)
= M
The de…nition of entries in the Table 3 follows from our earlier formulation. r is the rental rate for durable capital, K: K C is then the
durable capital in use in producing consumption goods, domestically (in
Italy). w is the wage prevailing in the economy and N C 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
P
total value of capital in use in the current period is
rK and for labor
P
is
wN: 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
Kt )
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 comprises 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:
KI
C = [K
K g ] 2 [N
I = [K I ] 1 [N I ]1
1
;
g = [K g ] 3 [N g ]1
3
;
NI
N g ]1
2
;
There are three price equations, one for each of consumer goods, investment 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
K I ] + w[N
Ng
N I ] = CpC
rK I + wN I = IpI
rK g + 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
M v = 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
nM
I
with n set equal to :
g
gp
K
Hence the value of g is again endogenous under the assumption of us
(1
)=
taking up the case of balanced growth.
This 13 equation system is solved with Matlab in Appendix 3. Parameters are set at K = 6; N = 5; M = 20;
3
= 2:1;
1
= 0:5;
2
= 0:4;
= 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,
K
K
=
1:1174
6
= 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 balanced
growth 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 economy. 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 disrupted 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 Century CE, there is evidence of the banking system and long-distance trade
34
"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 active 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 Romans 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 to
an end under the pressure of a number of problems: several emperors who were
exceptionally autocratic and excessive and a series of revolts by the army which in
turn led to Rome being ruled by a series of short-term emperors. The disruption
manifested itself in many ways, including increased in‡ation in the third century
CE 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 prices
doubled after the Antonine plague of the late second century and doubled again soon
thereafter. The denarius began to be progressively debased at this time. (Temin
(2006, p. 149)). Roberts emphasizes the disruption of social stability occasioned by
the plague. We, with others, argue for a signi…cant disruption in the stability of
government caused by the exhaustion of low-cost silver and gold mines in the late
second 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 Empire 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
36
We re-solved our …rst model with becoming smaller (quality of gold deposits
deteriorating; moving from 2.0 to 1.4) and observed, in a series of equilibria, the
interest rate rising and per capita consumption declining. The rental rate for K rose
while 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 seventeenth 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 during the Pax Romana because imports were made available readily to Roman 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)
"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 in
the last decade of his life than ever before. This allowed Spain to continue its military e¤orts, but led to an increased dependency on the precious metals and jewels."
(http://en.wikipedia.org/wiki/Philip_II_of_Spain#Economy)
38
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 intelligent. 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 neighbors, 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 economic success. Adam Smith’s Wealth 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 standards 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 foreigners 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 system. 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 characteristics. 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 (K G ; K I ; K g ; N G ; 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
K g ] + w[N
NG
NI
N g ] = CpC
rK I + wN I = IpI
rK G + wN G = ((
1)pg g)
rK g + wN g = pg g
[K
KG
KI
K g ] 2 [N
NG
NI
N g ]1
2
=C
[K I ]a1 [N I ]1
1
=I
[K g ]a3 [N g ]1
3
= g;
[K G ]a4 [N G ]1
4
= ((
[
2]
2
[1
2]
[1
2]
r 2 w[1
2]
= pC
[
1]
1
[1
1]
[1
1]
r 1 w[1
1]
= pI
[
3]
3
[1
3]
[1
3]
r 3 w[1
3]
= pg
[
4]
4
[1
4]
[1
4]
r 4 w[1
4]
= pG
1)pg g)=pG ;
M v = CpC + IpI + gpg + (
with v = i0:1 ;
i = r=pI
s[CpC + IpI + gpg + (
1)pg g] = 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
1)pg g
% 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
[kt
kG
kG
kI
kI
k g ] + w[1
k g ] 2 [1
nG
nI
ng ] = cpC
K_
rk I + wnI = [k_ + nkt ]pI for
= [k_ + nkt ]
N
rk G + wnG = ((
1)pg g)
g
rk g + wng = pg g for g =
N0
I
g 1
2
n
n ]
=c
nG
[k I ]a1 [nI ]1
1
= k_ + nkt
[k g ]a3 [ng ]1
3
= g;
[k G ]a4 [nG ]1
4
= ((
[
2]
2
[1
2]
[1
2]
r 2 w[1
2]
= pC
[
1]
1
[1
1]
[1
1]
r 1 w[1
1]
= pI
[
3]
3
[1
3]
[1
3]
r 3 w[1
3]
= pg
[
4]
4
[1
4]
[1
4]
r 4 w[1
4]
= pG
ct pC + [k_ + nkt ]pI + gpg + (
1)pg g)=pG ;
1)pg g = mt v
with v = i0:1 ;
i = r=pI
s[ct pC + [k_ + nkt ]pI + gpg + (
1)pg g] = [k_ + nkt ]pI
m
_ + mt n = (
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 program.)
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 GOVERNMENT 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 program; 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; k I ; k g ; nI ; ng ; pC ; pI ; pg ; r; w;
kI
ct = [kt
k g ] 2 [1
nI
k_ + nk = [k I ] 1 [nI ]1
1
;
g = [k g ] 3 [ng ]1
3
; for g =
n g ]1
2
;
g
N0
There are three price equations, one for each of consumer goods, investment 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
k I ] + w[1
ng
nI ] = pC ct
rk I + wnI = pI [k_ + nkt ]
rk g + wng = gpg ;
The savings relation:
s[ct pC + [k_ + nkt ]pI + gpg ] = [k_ + nkt ]pI ;
and money supply equal to money demand relation:
mt v = ct pC + [k_ + nkt ]pI + 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. The Matlab 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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