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In Question #2, “p”should be the probability of NO market failure.Midterm Questions (Due 23:59PM Wednesday, April 5, 2023.)

1

Question #1 (50 Marks)

Consider our two-period OLG model of a monetary economy. Assume that young

individuals are born and receive an endowment of e1 units of the consumption good

while old individuals receive no endowments. A stock of fiat money of size M0 is divided

equally across the initial N old individuals at time t = 0. Each period, a cohort of young

individuals of size N is born and the previous period’s old leaves the economy. The size

of the youth population is constant over time. Let the size of the nominal money stock

in period t be denoted by Mt and the gross growth rate of the money stock in period t is

zt = MMt t 1 . Assume that the growth rate of money, ẑt is constant over time so ẑt = ẑ with

the relationship that zt = 1 + ẑt . Newly printed money in period t is equally divided up

amongst the current old population. Let A2,t denote the amount of money transferred

to an old individual of period t.

Individuals live for two periods. Their objective of an individual born in period t is

to maximize lifetime utility u(c1,t ) + u(c2,t+1 ) by choice of consumption during youth,

c1,t , and consumption in old age, c2,t+1 . The utility function u(c) has the properties that

the marginal utility of consumption is strictly positive, u0 (c) > 0, there is diminishing

marginal utility of consumption u00 (c) < 0 and the marginal utility of consumption goes
to infinity as consumption approaches zero, limc!0 u0 (c) = 1.
Each period, the young of period t can exchange some of their endowment goods for
money and use this money to finance purchases of the consumption good in their old
age. The money held by a young individual in period t, M1,t , serves as the only means
to transfer wealth over time and Pt is the number of money units required to purchase a
single unit of the consumption good in period t.
1. Write down the period budget constraints of the individual. (5 Marks)
2. Derive the individual’s intertemporal consumption trade-o↵ and interpret it in
words explaining how the individual views trading o↵ a unit of youth consumption and the return when old. (5 Marks)
3. Define a competitive monetary equilibrium. (5 Marks)
4. Consider a stationary monetary equilibrium in which consumption of the young
and the inflation rate are constant over
and old, real money balances, m1,t = MP1,t
t
1 ↵
time. Assume that the utility function takes the form u(c) = c1 ↵ , 0 < ↵ < 1 so
that u0 (c) = c ↵ .
(a) Show that in a stationary equilibrium, money market clearing requires a2 =
1 ⇡1 m1 . (5 Marks)
1
(b) Derive an expression for the inflation rate in a stationary equilibrium. (5
Marks)
(c) Using the above result, show that in a stationary equilibrium, m1 = e1 1 . (5
1+z ↵
Marks)
(d) Suppose ↵ > 0 and that money growth is pushed towards infinity, that is,

z ! 1. What happens to m1 , c1 and c2 in this limiting case? Provide some

intuition with reference to the individual’s optimal intertemporal consumption

trade-o↵ condition. (5 Marks)

5. Read the attached article from The Economist magazine. How would this model

explain the (virtual) abandonment of the domestic currency in Zimbabwe. Make

reference to specific behaviour of the individual behaviour in this model and the

equilibrium determination of the return to money. (15 Marks)

2

Question #2 (50 Marks)

Consider our two-period OLG model of a monetary economy. Assume that young

individuals are born and receive an endowment of e1 units of the consumption good

while old individuals receive e2 units of endowments, e1 > e2 . A stock of money of size

M is divided equally across the initial N old individuals at time t = 0. Each period,

a cohort of young individuals of size N is born and the previous period’s old leaves the

economy. The size of the youth population is constant over time as is the stock of nominal

money balances, Mt = M .

Individuals live for two periods. Their objective of an individual born in period t is to

maximize lifetime utility u(c1,t ) + u(c2,t+1 ) by choice of consumption during youth, c1,t ,

and consumption in old age, c2,t+1 . The utility function u(c) has the properties that the

marginal utility of consumption is strictly positive, u0 (c) > 0 and there is diminishing

marginal utility of consumption u00 (c) < 0.
Suppose that money is electronic money and that individuals trade through a perfectly
competitive electronic market in which balances in their electronic money accounts are
exchanged for goods; once a trade is verified, money is tranferred between electronic
accounts and goods are arranged for delivery. Each period, there is a probability 1 ⇢ 2
(0, 1) that the electronic market stop functioning. In such a case, trade forever ceases to
occur and individuals live in autarky. Otherwise, with probability ⇢, there is no market
failure and money balances can be exchanged for goods.
As long as the electronic market has collapsed in the past, individuals have no problem
to solve; they simply eat their endowments when young and when old. If the market
has not yet failed then individuals might choose to hold on to money balances in order
to transfer resources between youth and old age. The problem of an young individual in
such a circumstance is
n
o
max
u(c1,t ) + ⇢u(cn2,t+1 ) + (1 ⇢)u(cf2,t+1 )
f
c1,t ,cn
2,t+1 ,c2,t+1
2
subject to the constraints
M1,t
Pt
M1,t
= e2 +
Pt+1
= e2 .
c1,t = e1
cn2,t+1
cf2,t+1
In words, the individual chooses how much consumption goods to exchange for money
when young in an e↵ort to maximize expected lifetime utility (welfare). Here the consumption when old is a weighted average of the utility that is obtained from consumption
when the electronic market does not fail, u(cn2,t+1 ) and the utility from consumption in
the event that the market collapses, u(cf2,t+1 ) where the respective weights are equal to
the probability that the market continues to operate, ⇢, and the market fails, 1 ⇢. As
usual, Pt is the measure of money required to obtain a single unit of the consumption
good in period t.
1. Assume that the optimal consumption trade-o↵ between consumption when young
and old for an individual who is young when the market has not yet failed to be,
u0 (c1,t ) =
Pt
⇢u0 (cn2,t+1 ).
Pt+1
Using this together with the individual’s budget constraints, provide an equation
that implicitly define the young individual’s demand for real money balances, m1,t .
(5 Marks)
2. As the population and the stock of nominal money supply is constant, suppose that
the demand for real money balances by young individuals is constant over time as
long as the electronic money market continues to operate (i.e. m1,t = m1,t 1 = m1
as long as the money market operates). Use money market clearing to derive the
value of the inflation rate, ⇡t = PPt t 1 that prevails when the electronic money market
continues to operate. (10 Marks)
3. Suppose that the utility function is u(c) = ln(c) so that u0 (c) = 1c . Derive a simple
expression for the young’s demand for real money balances, m1,t , as a function of
⇢, e1 and e2 . (10 Marks)
4. Show that the demand for real money balances changes with an increase in the
probability ⇢ that the electronic money market continues to operate. Provide the
economic intuition behind this result and show that the value of money, P1t , changes
when ⇢ increases. (5 Marks)
3
5. Read the article below and explain the di↵erence between a cryptocurrency exchange and a cryptocurrency. Viewed through the lens of the model sketched above,
provide an explanation of why the value of Bitcoin (relative to the U.S. dollar) fell
by about 25% around the days of the demise of the cryptocurrency exchange FTX.
Make reference to specific behaviour of the individual behaviour in this model and
the equilibrium determination of the value of money. (20 Marks)
4
How Bad Is Inflation in Zimbabwe? - New York Times
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How Bad Is Inflation in Zimbabwe?
By MICHAEL WINES
Published: May 2, 2006
More Articles in International »
E-MAIL
Correction Appended
PRINT
nytimes.com/travel
HARARE, Zimbabwe, April 25 — How bad is
inflation in Zimbabwe? Well, consider this: at
REPRINTS
SAVE
a supermarket near the center of this
tatterdemalion capital, toilet paper costs $417.
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No, not per roll. Four hundred seventeen
Zimbabwean dollars is the value of a single two-ply sheet. A roll
costs $145,750 — in American currency, about 69 cents.
MOST POPULAR
E-MAILED
The price of toilet paper, like everything else here, soars almost
Vanessa Vick for The New York Times
Ayina Musoni, 58, has taken in
lodgers to help with expenses, but
she can barely afford food for her
family.
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daily, spawning jokes about an impending better use for Zimbabwe's
$500 bill, now the smallest in circulation.
2. Home Cooking for Pets Is Suddenly Not So Odd
But what is happening is no laughing matter. For untold numbers of
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Zimbabweans, toilet paper — and bread, margarine, meat, even the
5. Some Hospitals Call 911 to Save Their Patients
once ubiquitous morning cup of tea — have become unimaginable
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luxuries. All are casualties of the hyperinflation that is roaring
toward 1,000 percent a year, a rate usually seen only in war zones.
Zimbabwe has been tormented this entire decade by both deep
recession and high inflation, but in recent months the economy
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seems to have abandoned whatever moorings it had left. The
national budget for 2006 has already been largely spent.
Government services have started to crumble.
nytimes.com/business
The purity of Harare's drinking water, siphoned from a lake
downstream of its sewer outfall, has been unreliable for months, and
Can Intel recover bonuses it shouldn't have paid?
dysentery and cholera swept the city in December and January. The
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city suffers rolling electrical blackouts. Mounds of uncollected
garbage pile up on the streets of the slums.
How much did the Barclays chief receive in compensation?
Which executive received a 94% pay increase?
Special report on executive pay
Zimbabwe's inflation is hardly history's worst — in Weimar
Germany in 1923, prices quadrupled each month, compared with
doubling about once every three or four months in Zimbabwe. That
said, experts agree that Zimbabwe's inflation is currently the world's
ADVERTISEMENTS
highest, and has been for some time.
Public-school fees and other ever-rising government surcharges
have begun to exceed the monthly incomes of many urban families lucky enough to find work. The
jobless — officially 70 percent of Zimbabwe's 4.2 million workers, but widely placed at 80 percent when
The Bowery, 1900
idle farmers are included — furtively hawk tomatoes and baggies of ground corn from roadside tables, an
1 of 4
4/3/07 9:19 AM
How Bad Is Inflation in Zimbabwe? - New York Times
http://www.nytimes.com/2006/05/02/world/africa/02...
occupation banned by the police since last May.
Buy Now
Those with spare cash put it not in banks, which pay a paltry 4 to 10 percent annual interest on savings,
but in gilt-edged investments like bags of corn meal and sugar, guaranteed not to lose their value.
"There's a surrealism here that's hard to get across to people," Mike Davies, the chairman of a
civic-watchdog group called the Combined Harare Residents Association, said in an interview. "If you
need something and have cash, you buy it. If you have cash you spend it today, because tomorrow it's
going to be worth 5 percent less.
"Normal horizons don't exist here. People live hand to mouth."
President Robert G. Mugabe has responded to the hardship in two ways.
Although there is no credible threat to his 26-year rule, Zimbabwe's political opposition is calling for
mass protests against the economic situation. So Mr. Mugabe has tightened his grip on power even
further, turning the economy over to a national security council of his closest allies. In addition, he has
seeded the government's civilian ministries this year with loyal army and intelligence officers who now
control key functions, from food security to tax collection.
At the same time, Mr. Mugabe's government has printed trillions of new Zimbabwean dollars to keep
ministries functioning and to shield the salaries of key supporters — and potential enemies — against
further erosion. Supplemental spending proposed early in April would increase the 2006 spending
limits approved last November by fully 40 percent, and more such emergency spending measures are all
but certain before the year ends.
On Friday, the government said it would triple the salaries of 190,000 soldiers and teachers. But even
those government workers still badly trail inflation; the best of the raises, to as much as $33 million a
month, already are slightly below the latest poverty line for the average family of five.
This will only worsen inflation, for printing too many worthless dollars is in part what got Zimbabwe
into this mess to begin with. Zimbabwe fell into hyperinflation after the government began seizing
commercial farms in about 2000. Foreign investors fled, manufacturing ground to a halt, goods and
foreign currency needed to buy imports fell into short supply and prices shot up.
Inflation, about 400 percent per year last November, edged over 600 percent in January, but began to
soar after the government revealed that it had paid the International Monetary Fund $221 million to
cover an arrears that threatened Zimbabwe's membership in the organization.
In February, the government admitted that it had printed at least $21 trillion in currency — and probably
much more, critics say — to buy the American dollars with which the debt was paid.
By March, inflation had touched 914 percent a year, at which rate prices would rise more than tenfold in
12 months. Experts agree that quadruple-digit inflation is now a certainty.
In the midst of this craziness, some Harare enclaves seem paradoxically normal. North of downtown,
where diplomats and aid workers are financed with American dollars, and generators and bottled water
are the norm, the cafes still serve cappuccino and the markets sell plump roasting chickens, albeit $1
million chickens.
Everywhere else, the hardship is inescapable.
In Glen Norah, a dense suburb of thousands of tiny homes southwest of the city, 58-year-old Ayina
Musoni and her divorced daughter Regai, 26, share their five-room house with Regai's two children and
three lodgers. The lodgers, two security guards and a teacher, pay monthly rent totaling $3 million, or
about $14.25 in American money.
Ms. Musoni's latest monthly bill for services from the Harare city government was $2.4 million. The
2 of 4
4/3/07 9:19 AM
How Bad Is Inflation in Zimbabwe? - New York Times
http://www.nytimes.com/2006/05/02/world/africa/02...
refrigerator in her closet-size kitchen is empty except for a few bottles of boiled water. Christmas dinner
was sadza, or corn porridge, with hard-boiled eggs. For Easter, there was nothing.
Mother and daughter make as much as $10 in American money each week by selling vegetables, from 7
a.m. to 6 p.m. daily. But the profits are being consumed by rising costs at the farmers' market where they
buy stock. "Like potatoes," Regai said. "I went last week, and it was $500,000 for a packet. And when I
went this weekend, it was $700,000.
Millions of Zimbabweans survive these days on the kindness of outsiders — foreigners who donate food
or medicine and, more important, family members who have fled the nation for better lives abroad.
As many as three million Zimbabweans now live elsewhere, usually in Britain, South Africa or the
United States. An economist here, John Robertson, estimates that they remit as much as $50 million a
month to their families — the equivalent of one sixth of the gross domestic product.
Ms. Musoni's is not a hard-luck story; in Harare, most people now live this way, or worse. Indeed, life
for many may be better in the nation's impoverished rural areas, where subsistence farming is the only
industry and millions of people are guaranteed free monthly rations from the United Nations and other
donors. In the cities, little is free.
Unity Motize, 64, lives with her 65-year-old husband, Simeon, in Highfield, a middle-class suburb
turned slum not far south of town. The couple occupies one room of their three-room house. The second
sleeps two sons, their wives and their two infants, all left homeless last May after riot police bulldozed
the homes of hundreds of thousands of slum-dwellers. A 23-year-old son and an unemployed daughter
sleep in the living room.
Hyperinflation is a cradle-to-grave experience here. The government recently announced that the price
of childbirth, now $7 million, would rise 463 percent by October. Funeral costs are to double over the
same period.
In rural areas, said one official of a foreign-based charity who declined to be named, fearing
consequences from the government, even the barest funeral costs at least $6 million, or about $28.50
— well beyond most families' means. The dead are buried in open fields at night, she said. Recently, she
watched one family dismantle their home's cupboard to construct a makeshift coffin.
"I'll never forget that," she said. "The incredible sadness of it all."
Critics say that Zimbabwe's rulers are oblivious to such suffering — last year, Mr. Mugabe completed his
own 25-bedroom mansion in a gated suburb north of town, close by the mansions of top ministers and
military allies.
But the government says it has a plan to revive the economy. That plan, the latest of perhaps seven in 10
years, would quickly raise billions of American dollars to end a chronic foreign currency shortage, cut
the inflation rate to double digits by year's end and an end to the recession that has gripped Zimbabwe,
halving its economic output, since 1999.
Mr. Robertson, the economist, says that is unlikely. Zimbabweans can and probably will endure greater
hardship, he says. As a whole, the nation has only now sunk to standards common elsewhere in Africa.
But the government may have reached the limit of its ability to do anything about it. Cutting spending
seems impossible, and raising taxes further is unthinkable.
That leaves one option: "much more inflation," he said. "Because this government is always going to be
printing its way out of its current difficulty."
Correction: May 5, 2006
A chart on Tuesday with an article about Zimbabwe's high inflation rate misstated the reasons for its
sharp rise and fall during 2004. The rise was caused by a flight of foreign capital, shortages and a
3 of 4
4/3/07 9:19 AM
02/04/2023, 13:10
What do we know so far about collapse of crypto exchange FTX? | Cryptocurrencies | The Guardian
Cryptocurrencies
Explainer
What do we know so far about collapse of crypto exchange FTX?
How did Sam Bankman Fried’s FTX fail and what does the
ﬁrm’s fate tell us about cryptocurrencies?
Alex Hern and Dan Milmo
Sat 19 Nov 2022 05.33 AEDT
The collapse of FTX, one of the world’s largest cryptocurrency exchanges, has unleashed another bout of volatility in the highly
speculative digital asset market. The fortune of FTX’s founder, Sam Bankman-Fried, went from nearly $16bn to zero within days
as his crypto empire ﬁled for bankruptcy protection in the US on 11 November. Here we answer some of your questions about the
story so far.
How was FTX structured and what was its business model?
In corporate terms, FTX was a chaotic web of more than 100 diﬀerent companies, all united under the common ownership of
Bankman-Fried and his co-founders, Gary Wang and Nishad Singh. In a bankruptcy ﬁling, John Ray III – an American bankruptcy
specialist who previously oversaw the collapse of Enron – described it as four main “silos”: a venture capital arm, which invested
in other businesses; a hedge fund, which traded crypto for proﬁt; and two exchanges, one supposedly ringfenced and regulated
for the US audience, and one international exchange where the rules were much freer.
The revenue streams were as diverse as the business, but the core of the group was the exchange. Most people buy
cryptocurrency by transferring money (“ﬁat”) to an exchange like FTX, which operates like a bureau de change, trading currency
pairs at a ﬂoating exchange rate. FTX’s regulated exchange oﬀered that service, and the company took a cut of every transaction,
but the big bucks were in the much more aggressive trading on the international exchange, where traders would try to proﬁt from
swings in the prices of crypto assets, borrowing money to increase their potential earnings (or losses). The more complex the
trade, the bigger the cut.
Why did it collapse?
In the short term, because of a token called FTT. This was eﬀectively a share in FTX, that the company issued itself and promised
to buy back using a portion of its proﬁts. But documents leaked to news site CoinDesk suggested that Alameda, the group’s hedge
fund, was using FTT to make risky loans – eﬀectively trading using company scrip. The revelation prompted a major holder of
FTT, rival exchange Binance, to declare it was selling its holdings, prompting a run on the exchange as other customers scrambled
to withdraw their funds.
In the medium term, it collapsed because of deeper issues to do with the link between FTX and Alameda. The exchange didn’t
have the ability to accept wire transfers, so customers would send money to Alameda, and FTX would credit their accounts. But
https://www.theguardian.com/technology/2022/nov/18/how-did-crypto-ﬁrm-ftx-collapse
1/9
02/04/2023, 13:10
What do we know so far about collapse of crypto exchange FTX? | Cryptocurrencies | The Guardian
the actual money was never passed on: three years later, Alameda had kept hold of, traded with, and frequently lost, $8bn of FTX
customer funds. When the run on the exchange started, FTX couldn’t ﬁnd the money it thought it had, because it had never
taken it.
In the long term, FTX failed because the company was a mess. “Never in my career have I seen such a complete failure of
corporate controls and such a complete absence of trustworthy ﬁnancial information as occurred here,” said Ray, the bankruptcy
specialist.
What does FTX’s fate tell us about cryptocurrencies?
Within the sector, diﬀerent conclusions have been drawn. Some have argued the collapse is a triumph for “decentralised
ﬁnance”, or DeFi, which uses computer code to build versions of ﬁnancial services that don’t rely on trust or a central party. The
head of a DeFi exchange can’t buy a $40m penthouse with customer funds because there is no head.
But outside the sector, the conclusion is plain. Cryptocurrencies are a bet on the idea that a world where government power over
money and ﬁnance is ended would be a better one: the collapse of FTX is perfect evidence that actually, government regulations
over ﬁnance are pretty useful.
Will people get their money back?
Some people will get some money back, but no one is going to get everything. Even Bankman-Fried is convinced that it would
take an $8bn injection of capital to make every depositor whole. But the accounts presented by Ray make clear that is wishful
thinking. There isn’t even a single document detailing all the company’s depositors, he says, and while the balance sheet suggests
a healthy mixture of assets and liabilities, “I do not have conﬁdence in it and the information therein may not be correct as of the
date stated”.
Robert Frenchman, a partner at New York law ﬁrm Mukasey Frenchman, said FTX customers in the US whose money is trapped
in the failed business will have to join a queue of creditors because there are no special protections for customers of unregistered
crypto ﬁrms like FTX.
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“There is no backstop here for customers in the US, unlike for bank or brokerage account holders. The
customers will have to ﬁght it out with everyone else because they have no special protections. They go into this process as
individual creditors, or as a group of creditors if they band together, who must battle it out with legions of other creditors, large
and small.”
In the meantime, the US attorney’s oﬃce for the southern district of New York is reportedly looking into the case and US treasury
secretary, Janet Yellen, has said that crypto markets need more robust oversight.
Could there be contagion within the crypto markets?
There have already been signs of a spillover eﬀect. BlockFi, a crypto lender rescued by FTX in the summer, has paused customer
withdrawals, admitting that it has “signiﬁcant exposure to FTX”. On Wednesday, the crypto exchange Genesis “made the diﬃcult
decision to temporarily suspend redemptions” from the company’s lending business after a series of withdrawals from the
service.
This week the chief executive of the Singapore-based crypto exchange Crypto.com said his ﬁrm would prove wrong all those who
said the platform was in trouble, adding that it had a robust balance sheet and took no risks. Kris Marszalek made the statement
after investors questioned the transfer of $400m-worth of ether tokens from Crypto.com to another exchange called Gate.io on 21
October. Marszalek said the transfer was an error and the ether tokens had been returned to the exchange.
Crypto market watchers expect more instability, although the core crypto asset, bitcoin, has held up this week by staying broadly
ﬂat at about $16,700.
Teunis Brosens, head of regulatory analysis at Dutch bank ING, said the crisis would “surely deepen” the latest crypto winter,
which has resulted in the value of the crypto market falling from $3tn last year to less than $1tn now.
“In terms of prices, we saw bitcoin quite stable around $19,000-$20,000 for months. I’d consider it likely that we will now be
seeking stability at lower levels – but ﬁrst, the storm has to subside, and we are deﬁnitely not there yet.”
https://www.theguardian.com/technology/2022/nov/18/how-did-crypto-ﬁrm-ftx-collapse
2/9
OLG Models with Money
Class Notes for MBFM III
The University of Adelaide
March 7, 2023
These notes are a complement to the course textbook “Modeling Monetary Economies”
written by Champ, Freeman and Haslag. Some of the results in the textbook are not
transparent as the authors avoid the mathematical derivations even though they instigate the use of formal mathematical models. These notes should provide the rigour that
enables the diligent student to follow the implications of the mathematical economic
models and also provide an appreciation of the type of theoretical modelling that is used
by academic and many practicing economists. Such modelling often lies behind applied
empirical work because the theoretical logic places restrictions on the interpretation of
empirical identification schemes - that is, the modelling provides the economic structure
that the empirical analyst is relying upon in order to interpret their econometric findings
in a sensible way.
Many diagrams are in the textbook so such figures will not be reproduced in these
notes... just the math. Also, we will use a more general setting than that used in
the textbook so that the textbook’s results will be a special case of the model in the
derivations in this note. We will also develop some models to use in lieu of those presented
in chapters towards the end of the textbook as the models presented in these notes will
provide a complete description whereas those in some chapters of the textbook lack a
proper derivision of results and, if solving properly, are no less tractable than the ones
presented in this set of notes. You can find an Appendix at the end of these notes
that provide a refresher about differentiation and might introduce you to the use of the
Method of Lagrange that we will use to solve constrained optimization problems. Please
look at the Appendix.
1
Timing in a Simple Two-Period OLG Model
The OLG model serves as one of the simplest dynamic general equilibrium models
of an economy that lasts forever, and is one of the two workhorse models in theoretical
analysis of modern macroeconomics (the other being the representative agent model).
The model features a demographic structure consisting of individuals who live for finite
horizons and whose lives overlap with individuals from other cohorts. In the simplest
1
1
2
3
4
5
t
G0
G1
G2
G3
G4
Figure 1: Overlapping Generation Structure
case of the model, individuals live for two periods. Those who are born and are young in
period t interact with those who are old and were born in period t − 1. The young who
were born in period t become old in period t + 1 and interact with the young cohort who
are born in period t + 1. Figure 1 shows how the cohorts (or generations) overlap with
each other over time. Each dot represents a period of life. Those in cohort Gt are born
at the beginning of period t, for all t ∈ {1, 2, ..., ∞}, and die at the end of period t + 1.
Another way of saying this is that an individual born in period t is young in period t and
old in period t + 1. Note that the initial old are denoted by G0 .
Going forwards in these notes, we will follow notation convention that x1,t represents
the value of variable x attached to an individual in the first period of their lives who are
living at time t. Similarly, x2,t+1 would represent the value of variable x that is attached
to an individual who is in the second period of their life at time t + 1. We typically use
lower case letters to represent values of individual level variables and upper case letters
to represent aggregate values of variables. For example, c1,t will represent the level of
consumption enjoyed by an individual in their first period of life at time t while Ct would
represent the aggregate level of consumption across the entire population of an economy
at time t.
2
An Endowment OLG Model with Social Norms
Consider a two-period endowment OLG economy. The endowment profile for each
individual is (e1,t , e2,t ) = (e1 , e2 ). Preferences are captured by a utility function that is
strictly increasing, strictly concave and satisfying an Inada condition limc→0 u0 (c) = ∞.
Each period, a social norm dictates an amount of goods that each young individual is to
hand over to an old individual. Denote this level of transfer by a young as τt . We will hold
this amount constant over time so that τt = τ . Each old individual receives an amount
2
st = s when old. As a society, assume that the total transfers by the young are pooled
and divided equally amongst the old. If there are Nt young individuals in period t and
Nt−1 old individuals, then the relationship between tranfsers is Nt τ = Nt−1 s. Resource
feasbility of this social norm scheme requires Nt τ ≥ Nt−1 s so that total transfers from
the young can completely finance the transfers received by the old in each period. Note
that as utility for all individuals is strictly increasing in consumption, it is never optimal
to dispose of resources. Hence it is assumed that under resource feasibility, the weak
inequality is replaced by an equality.
If a young individual decides not to participate in the scheme when young by deviating
from the social norm and providing τ d = 0 transfers, this information is recorded and
this deviating individual receives no transfers when old, sd = 0.1
It is possible to determine the lifetime utility of sticking with the social norm. The
individual’s lifetime welfare obtained by following the social norm is u(c1,t ) + βu(c2,t+1 )
with c1,t = e1 − τ and c2,t+1 = e2 + s. Substituting the consumption under the social
norm into the lifetime utility function we have u(e1 − τ ) + βu(e2 + s). If the individual
chooses to deviate from the prescribed social norm, the utility is u(e1 ) + βu(e2 ). Hence
the representation of the individual’s problem
max {x [u(e1 − τ ) + βu(e2 + s)] + (1 − x) [u(e1 ) + βu(e2 )]}
x∈0,1
Definition 1 A social norm equilibrium is a consumption pair (c1,t , c2,t ) and a social
norm (τ, s) such that
1. taking the social norm as given individuals are optimizing, and
2. the social norm is resource feasible.
A social norm equilibrium requies that, taking the social norm as given, individuals
are optimizing. This means that if u(e1 − τ ) + βu(e2 + s) ≥ u(e1 ) + βu(e2 ) the individual
will stick with the social norm while if the opposite is true, the individual will elect to
abandon the social norm with the understanding that they will be placed in autarky.
We can study the stationary social norm equilibrium (defined such that allocations
for young individuals are time invariant as well as for old individuals) by noting that,
resource feasibility requires Nt τ = Nt−1 s. Defining the gross population growth rate,
t
and setting n̂t = n̂ for all t = 0, 1, 2, ..., we have s = (1 + n̂)τ . For
1 + n̂t = NNt−1
individuals to elect the social norm, it must be that
u(e1 − τ ) + βu(e2 + (1 + n̂)τ ) ≥ u(e1 ) + βu(e2 )
where the requirements of the social norm’s resource feasibility have been used to replace
s as a function of τ .
1
Technically, we can define a young individual as a someone who provides τt 6= τ . If this is the case
and the individual is to receive no transfer when old then their best strategy is to provide no transfer
when young.
3
What is the value of youth transfers, τ , that maximizes each individual’s lifetime
welfare under the social norm? Identifying this value of the transfer simply requires
maximizing the following function
V (τ ) = u(e1 − τ ) + βu(e2 + (1 + n)τ ).
Notice that u(c) is strictly concave and as the sum of two strictly concave functions is
strictly concave, we know that if we can find a critical point of this problem we will be
identifying a local maximum. Thus we take the derivative of V (τ ) and set it equal to
zero, V 0 (τ ) ≡ dVdτ(τ ) = 0 .
V 0 (τ ) = −u0 (e1 − τ ) + βu0 (e2 + (1 + n̂)τ )(1 + n̂) = 0
Rearranging this optimality condition, we see that at an optimum, the choice of τ must
be such that
u0 (c1 ) = β(1 + n̂)u0 (c2 ).
(1)
Let us interpret this trade-off. Consider an social norm (τ, s) that delivers a consumption bundle (c1 , c2 ) over an individual’s lifetime. If it is optimal then it must deliver
a consumption bundle satisfying (1). Now ask whether the social norm can be improved
upon by changing τ by a little bit.
If τ is increased by just a bit (say a unit for convenience of discussion), then each
young individual must give up u0 (c1 ) units of utils due to the loss of a unit of young
consumption. The total increase in resources from youth transfers is Nt . Dividing this
pool of extra resources amongst the old means that each old individual would receive an
additional 1 + n̂ units of consumption goods. The utility gain of an old individual is then
(1 + n̂)u0 (c2 ) because the old individual values each additional unit of consumption by
u0 (c2 ) utils and there are 1 + n extra units of consumption goods. In order to compare
old utility to young utility, units of old utility are multiplied by the discount factor β in
order for the utils to be comparable between youth and old age. If the utility loss from
youth exceeds the utility gain from old age, then the transfer scheme, τ , can be improved
upon by reducing the youth transfer. On the flipside, if the youth utility loss is less than
the old age gain, then the transfer scheme can be improved upon by increasing the youth
transfer. Thus, optimality requires that the utility loss from youth be exactly offset by
the utility gain in old age yielding the condition in equation (1).
Figure 2 depicts the construction of the optimal social norm. The line with slope
−(1 + n̂) represents the feasibility constraint. For each unit of resources taken from the
current young, the current old can receive at most 1 + n̂ units of resources (assuming
resources are pooled and divided equally). The red line represents the indifference curve
the runs through the individual endowment point.2 It is the indifference curve for the
2
Recall that an indifference curve plots combinations of c1 and c2 such that the level of utility delivered
by the combinations return the same level of utility, say U .
U = u(c1 ) + βu(c2 )
4
(2)
c2
c∗2
e2
Slope = −(1 + n̂)
c1
c∗1 e1
Figure 2: The Social Norm Economy
level of utility that an individual would obtain over their lifetime if he/she opted out of
the social norm. As long as the social norm places the economy at a point that is in the
shaded region but not at the point of tangency between an indifference curve and the
feasibility constraint, the economy can do better by moving towards the point (c∗1 , c∗2 )
because it would be reaching an indifference curve at a higher level of utility and the
bundle would be feasible.
From Figure 2, the optimal choice of the social norm requires that the slope of the
indifference curve, evaluated at the consumption bundle delivered by the optimal transfer
scheme (c1 , c2 ) = (e1 − τ, e2 + (1 + n̂)τ ) be equal to the slope of the feasibility constraint,
1 + n. This can be related to the optimality condition of equation (1) as optimality
requires
u0 (c1 )
= 1 + n̂
βu0 (c2 )
As the level of utility does not change across bundles on the indifference curve yielding utility U , when
we take the total differential of equation (2),
dU = u0 (c1 )dc1 + βu0 (c2 )dc2
it must be that dU = 0 for variations (dc1 , dc2 ) to be consistent with remaining on the indifference curve
U . Setting dU = 0 and rearranging,
−βu0 (c2 )dc2 = u0 (c1 )dc1
so dividing both sides by dc1 and then dividing both sides by −βu0 (c2 ) we have
dc2
u0 (c1 )
=− 0
,
dc1
βu (c2 )
where the righthand side is the marginal rate of (intertemporal) substitution between young consumption
and old consumption.
5
which, after multiplying both sides by −1 returns the condition from the diagram. Thus,
the diagram represents the same information as does the optimal trade-off condition.
A technical point to note when looking at Figure 2. Moving along the feasibility
frontier is an exercise of moving along the combinations of consumption bundles that
can be implemented between young and old of the current period. When looking at
indifference curves, that is considering the trade-off between young consumption of the
current period and old consumption of the following period through the perspective of a
young individual from the current period. In a stationary equilibrium, the consumption
profile across individuals and time is constant which allows us to consider such a plot.
3
An Endowment OLG Model with Money
Consider a discrete-time OLG economy in which individuals live for two periods.
Time is infinite, t = 0, 1, 2, .... Each period, a cohort of size Nt is born into the economy
and there is an initial old cohort of size N−1 . Let the population growth rate in period t
be nt . For simplicity, assume that nt = n so that the population growth rate is constant
over time. In period t, each young person is endowed with e1,t units of consumption
goods and each old person is endowed with e2,t units of the consumption good. The
consumption good is non-durable and cannot be stored between periods.3
Money supply is exogenous in the baseline model. The supply of money in the economy during period t is given by Mt and it is assumed that money grows at rate ẑt in
period t so that Mt = (1 + ẑ)Mt−1 and define zt as the gross growth rate of the money
supply so that zt = 1 + ẑt and Mt = zt Mt−1 . Let A1,t denote the monetary transfer to
each young individual in period t and let A2,t denote the monetary transfer to each old
individual in period t. For these transfers to be consistent with the level of money growth
it must be that total transfers (ie. injections or withdrawls from the money supply) must
be accounted for by total transfers to young and old. Therefore, it must be that
ẑt Mt−1 = Nt A1,t + Nt−1 A2,t .
Equivalently, if we want to use zt in lieu of ẑt
(zt − 1)Mt−1 = Nt A1,t + Nt−1 A2,t .
Let us define the gross rate of inflation by π and rate of inflation by π̂, with π = 1+ π̂.
Also, define the stock of real money balances as the stock of nominal money balances,
3
This endowment set-up is equivalent to having firms that produce the consumption good using labour
with a production function Y = N where N is labour supply, and individuals are endowed with e1,t and
e2,t+1 units of labour that they supply inelastically to firms. In a perfectly competitive equilibrium of
such an economy, the wage is equal to the marginal product of labour, wt = 1 so that labour income
is equal to labour endowments and equilibrium profits equal zero. For the purposes of following the
textbook, we sidestep any issues regarding labour markets and production to focus on the effects of
money supply on inflation rates, etc.
6
t
. The initial old own the initial stock
Mt divided by the current price level Pt , mt = M
Pt
of money which is equally distributed across the intial old. Thus the each initial old
−1
individual holds M
units of money prior to receiving their monetary transfer a2,0 .
N−1
Young individuals can either purchase the consumption good out of their endowments
any monetary transfer received in youth or save money to help finance consumption in
old age. Again, endowments cannot be physically transferred into old age. Let M1,t
denote the amount of money saved by the young individual in period t. Old individuals
finance consumption out of savings and any monetary transfers received in old age.
Denote by Pt the rate of exchange of units of money for a unit of the consumption
good. Its reciprocal, vt = P1t is the number of goods that must be given up in exchange
for a single unit of money.
The timing of events within each period t is as follows:
1. Period t begins with a new cohort being born.
2. The young of period t − 1 become old and the old of period t − 1 die.
3. Monetary transfers occur.
4. Individuals enter a market to exchange money for consumption goods.
5. Consumption occurs.
6. Period t ends.
3.1
The Individual’s Problem
The problem of the individual is to choose how much to consume and save in youth.
The old individual’s problem is trivial as they will simply consume all that their wealth
can afford. Therefore, we can write the problem of the individual as
max
c1,t ,c2,t+1 ,M1,t
{u(c1,t ) + βu(c2,t+1 )}
subject to the budget constraint when young of M1,t = Pt e1,t + A1,t − Pt c1,t and the
budget constraint in old age, Pt+1 c2,t+1 = Pt+1 e2,t+1 + M1,t + A2,t+1 . We will be assuming
that the utility function is strictly increasing, strictly concave and satisfies the Inada
condition limc→0 u0 (c) = ∞ (which will allow us to rule out corner solutions as the first
infinitessimal amount of consumption in each period has infinite marginal value).
This mathematical problem can be solve in several ways. We can take a route the
does not involve using the chain-rule of differentiation by using the Method of Lagrange
to deal with the two budget constraints. Introduce two variables, λ1,t and λ2,t+1 , as
Lagrange multipliers for the problem of an individual born into period t and construct
the Lagranegan function
L(c1,t , c2,t , M1,t , λ1,t , λ2,t+1 ) = u(c1,t ) + βu(c2,t+1 ) + λ1,t [Pt e1,t + A1,t − Pt c1,t − M1,t ]
+λ2,t+1 [Pt+1 e2,t+1 + M1,t + A2,t+1 − Pt+1 c2,t+1 ] .
7
Take the partial derivatives of the Lagrangean function with respect to each of its arguments, setting each of them equal to zero in order to find the Lagrangean’s critical
points. By the Theory of Lagrange, this will return values of (c1,t , c2,t , M1,t ) that are
necessary conditions for a maximum of the constrained optimization problem. As the
utility function is assumed to be strictly increasing and concave, this should be sufficent
for a maximum in our model.
∂L(·)
=0
∂c1,t
∂L(·)
=0
∂c2,t+1
∂L(·)
=0
∂M1,t
∂L(·)
=0
∂λ1,t
∂L(·)
=0
∂λ2,t+1
: u0 (c1,t ) = Pt λ1,t
: βu0 (c2,t+1 ) = Pt+1 λ2,t+1
: λ1,t = λ2,t+1
: M1,t = Pt e1,t + A1,t − Pt c1,t
: Pt+1 c2,t+1 = Pt+1 e2,t+1 + M1,t + A2,t+1
The partial derivatives with respect to the multipliers simply return the constraints
faced by the individual. Using the first-order conditions with respect to c1,t , c2,t+1 and
M1,t , it is possible to construct the trade-off between c1,t and c2,t+1 via saving through
the use of M1,t . Use the derivative with respect to c1,t to derive an expression for λ1,t , use
the derivative with respect to c2,t+1 to obtain an expression for λ2,t+1 and then substitute
these results into the first-order condition with respect to M1,t to get
Pt
0
u0 (c2,t+1 ).
(3)
u (c1,t ) = β
Pt+1
We will refer to this condition as either the optimal consumption trade-off condition or
the intertemporal Euler equation.
∗
Thus, if (c∗1,t , c∗2,t+1 , M1,t
) were an optimum, it must be the case that
Pt
0 ∗
u (c1,t ) = β
u0 (c∗2,t+1 )
Pt+1
∗
M1,t
= Pt e1,t + A1,t − Pt c∗1,t
∗
Pt+1 c∗2,t+1 = Pt+1 e2,t+1 + M1,t
+ A2,t+1
The optimal strategy of the individual can be interpreted as follows. Consider a
∗
candidate optimal choice set for the individual (c∗1,t , c∗2,t+1 , M1,t
). This triplet must satisfy
both the period budget constraints as well as the optimal intertemporal consumption
trade-off condition of equation (3). Suppose the individual considers deviating from this
optimal triplet by a small change in young consumption. Can the individual be better off?
8
Any feasible plan still requires the individual to satisfy the two period budget constraints.
Thus, if the individual gives up a unit of youth consumption, then they will increase their
holdings of money by Pt units as it is the rate of exchange between money for one good.
The decrease in utility in youth is then the amount of foregone consumption u0 (c∗1,t ).
Consider this the marginal cost of saving through money. With an increase in money
t
extra units of consumption goods
savings of Pt dollars, the individual can now affort PPt+1
in period t + 1. As each unit of consumption in old age provides u0 (c∗2,t+1 ) units of utility
t
the total gain in period t + 1 utility is then PPt+1
u0 (c∗2,t+1 ). This is the current value of
increased consumption from the perspective of period t + 1. In order to compare this
marginal benefit from saving to the marginal cost (which was valued in date t utils), it
must be discounting by the subjective discount factor β. At the optimum, the marginal
benefit of saving must equal its marginal cost. If the marginal cost of savings is greater
than the marginal benefit, then savings would be reduced, so that money holdings would
decrease. If the marginal benefit is greater than the marginal cost then then individual
∗
could do better by increasing money savings. Ergo, to be optimal (c∗1,t , c∗2,t+1 , M1,t
) must
satisfy both the budget constraints (to be feasible) and the intertemporal consumption
trade-off condition.
Notice that if we substituted the budget constraints into the optimal intertemporal
consumption trade-off condition,
∗
∗
M1,t
Pt
A2,t+1
A1,t M1,t
0
0
−
=β
u e2,t+1 +
+
.
u e1,t +
Pt
Pt
Pt+1
Pt+1
Pt+1
∗
. As this is an optimality condition,
This gives a single equation, in a single unknown M1,t
∗
and it can serve
we can think of this equation as pinning down the optimal value M1,t
as implicitly defining the individuals money demand. Note that money demand is determined as a function of prices Pt , Pt+1 and endowments/income (e1,t , e2,t+1 ). Once optimal
money holdings is determined, the budget constraints can then be used to back out the
amount of young and old consumption that is consistent with this money demand.
It is also instructive to remember that there is a cohort of initial old who each hold
−1
an equal share of the initial money stock, M
and receive an transfer a2,0 . These initial
N−1
old simply trade all their money for as much consumption as possible. They must do
better than the utility that they would achieve by eating their endowment - if they would
be worse off by trading with money, they will simply choose not to participate in the
monetary economy.
3.2
A Monetary Equilibrium
In order to determine money demand, young individuals who are making savings
decisions must know the value of Pt and forecast the value of Pt+1 . However, to be able
to forecast the value of Pt+1 they must be able to understand how much money the young
of period t + 1 would like to hold which depend on the value of Pt+1 and their forecasts of
Pt+2 , ad infinitum. As a simplification, we have not exposed this economy to any random
9
disturbances which allows us to imagine that the individuals in this economy have perfect
foresight. Each individual understands their problem and the problems faced by everyone
else who lives or will be living in the future.
Structure on the determination of prices can now be specified.
Definition 2 A competitive monetary equilibrium is a sequence of prices, {Pt }∞
t=0 ,and a
∞
sequence of allocations {c1,t , c2,t , M1,t }t=0 such that for an initial money supply M−1 and
a sequence money injections, {A1,t , A2,t }∞
t=0 , taking prices as given,
1. individuals are optimizing, and
2. markets clear.
Implicit in this definition of market clearing prices of the dynamic economy is the
result that the sequence of prices is such that the market in which goods are exchanged
for money clears in every single period under a competitive equilibrium - there is no
excess demand for goods nor money in any period from t = 0 into the infinite future.
How does this work? Well equilibrium requires the money market to clear. That
means that quantity money demand must equal quantity money supply. The total supply of money in period t is simply Mt . This must equal quantity money demand at
equilibrium prices. We know that old people demand no money so total money demand
must be comprised from total demand of the young, Nt M1,t . Competitive equilibrium
imposes the requirement that
M1,t =
Mt
.
Nt
Under a competitive equilibrium, individuals are required to be optimizing. This
means that each individual must be satisfying their optimal money demand and their
period budget constraints when young and old. From the money demand equation,
∗
∗
M1,t
Pt
A2,t+1
A1,t M1,t
0
0
−
=β
u e2,t+1 +
+
.
u e1,t +
Pt
Pt
Pt+1
Pt+1
Pt+1
so imposing money market clearing so that all money is held by young individuals at the
end of each period, Mt = N M1,t ,
A1,t
Mt
Pt
Mt
A2,t+1
0
0
u e1,t +
−
=β
u e2,t+1 +
+
.
(4)
Pt
Nt Pt
Pt+1
Nt Pt+1
Pt+1
As we used both the period budget constraints as well as the optimal intertemporal
trade-off condition of the individual in deriving this equation as well as the money market
clearing condition, this implies that almost all the equilibrium information is incorporated
10
into equation (4) - the only equilibrium condition not employed yet is the goods market
clearing requirement.4
In this model, in any period t, there are two markets (the goods market and the
money market) and a single relative price in each period, Pt , that links these two markets
together. Thus if one of these markets clears, the other one must also clear.5
Equation (4) is the key equation in our equilibrium. Recall that we are solving for
values of c1,t , c2,t , M1,t and Pt for all periods that satisfy the equilibrium conditions.
In deriving equation (4), we have used all the individual optimality conditions and the
money market clearing condition. With two perfectly competitive markets, if one clears,
the other must also clear so this means that the goods market should clear as well and all
our equilibrium conditions are met. How can we use equation (4). Endowments (e1,t , e2,t ),
nominal money supply M t and the distribution of newly printed money (A1,t , A2,t ) are
not variables for which we solve - they are given to us as pre-determined values. Thus
equation (4) can be used to determine the path of the price of goods and services. Start
in period t = 0, and take P0 as given. Then (4) implies a value for P1 that must hold in
equilibrium. Now roll over to t = 1. As we know the value for P1 , equation (4) implies
a value that must be taken by P2 . Now roll over to period t = 2. As we know the value
of P2 equation (4) implies a value that must be taken by P3 . We can continue on in this
manner until we construct the entire sequence for {Pt }∞
t=0 . This is the key sequence to
t
. Using this information
determind. We know from money market clearing that M1,t = M
Nt
we can back out c1,t for any period t using the young’s budget constraint and we can
back out the value for c2,t using the old individual’s budget constraint.
To conclude the mathematical derivations, we need to make use of the one equilibrium
condition that we have not yet used - goods market clearing. This requires all goods
available each period are consumed - as utility is strictly increasing in consumption, not
goods will be put to waste. Working through some algebra,
Nt e1,t + Nt−1 e2,t = Nt c1,t + Nt−1 c2,t
A2,t M1,t−1
A1,t M1,t
+ Nt−1 e2,t +
−
+
= Nt e1,t +
Pt
Pt
Pt
Pt
Cancelling terms and noting that in equilibrium, all money has to be held by the young
at the end of each period, Nt M1,t = Mt , this reduces to
Mt = Mt−1 + Nt A1,t + Nt−1 A2,t
= Mt−1 + ẑt Mt−1
4
Using textbook notation, we can replace P1t with vt so that
Nt A1,t
Mt
vt+1
Mt
Nt+1 A1,t+1
u0 e1,t + vt
−
=β
u0 e2,t+1 + vt+1
−
.
Nt
Nt
vt
Nt
Nt
5
This is the simplest application of Walras’ Law which states that in a Walrasian equilibrium in which
all markets are perfectly competitive (everyone takes prices as given and makes their optimal choices)
and prices adjust to equate quantity supply and quantity demand in all markets, if there are N markets
then as long as N − 1 of these markets clear, the last one automatically clears.
11
where we exploited the fact that total money injections to current young and old must
equal the growth in the stock of money. Clearing the goods market leads to the flow
equation for the stock of nominal money balances so it verifies that goods market clearing
is consistent with money market clearing.
Dividing both sides by Pt ,
Mt−1
Mt−1
Mt
=
+ ẑt
Pt
Pt
Pt
and then multiplying and dividing Mt−1 by Pt−1 (which is the same as multiplying these
terms by one and does not change the relationship between left- and righthand side of
our equation),
Mt
Mt−1 Pt−1
Mt−1 Pt−1
=
+ ẑt
Pt
Pt−1
Pt
Pt−1
Pt
1 + ẑt Mt−1
=
π
P
t t−1
1 + ẑt
mt−1 .
mt =
πt
As this must hold for every period starting from t = 0 and this economy starts in period
0 with P0 , we can push the time subscripts to write
1 + ẑt
mt =
mt−1 .
(5)
πt
Lastly, we have to remember that there is a cohort of initial old. Under the definition
of a competitive monetary equilibrium all individuals must be optimizing in every period
of the economy. This means that the initial old must also be optimizing. The initial old’s
problem is to solve
max {u(c2,0 )}
c2,0
−1
+ a2,0 . Clearly the initial old will not
subject to the constraint P0 c2,0 = P0 e2,0 + M
N−1
want to accumulate money so any potential equilibrium that calls for the initial old to
trade goods for money cannot be feasible as no initial old would be optimizing in such a
situation.
3.3
Constructing a Monetary Equilibrium
Equation (4) and the flow equation for money, Mt = Mt−1 + Nt A1,t + Nt−1 a2,t−1 are
the outcomes of our equilibrium model that characterize the entire dynamic path of our
model economy. They can be used to determine the time path of the price level {Pt }∞
t=0 .
Once the dynamic path of endogenously determined price levels is known, it can be
12
used to construct the dynamic path of the gross inflation rate, {πt }∞
t=1 =
n
Pt+1
Pt
o∞
and
t=0
along with the exogenously specified time path for money supply {Mt }∞
t=−1 and monetary
transfers {A1,t , A2,t }∞
can
be
used
to
construct
the
time
paths
for
real
money balances,
t=0
consumption of the young and the consumption of the old, {m1,t , c1,t , c2,t }∞
t=0 using the
definition of real balances and the individual’s period budget constraints.
Let us work our way through the equilibrium dynamics. Equations (4) and the flow
equation are replicated here for convenience.
A2,t+1
A1,t
Mt
Mt
Pt
0
−
= β
u e2,t+1 +
+
.
u e1,t +
Pt
Nt Pt
Pt+1
Nt Pt+1
Pt+1
0
Mt = Mt−1 + Nt A1,t + Nt−1 a2,t−1 .
(6)
(7)
∞
An equilibrium is a sequence of prices {Pt }∞
t=0 and allocations {c1,t , c2,t , M1,t }t=0
such that, given an initial money supply M−1 and a sequence of monetary injections
{A1,t , A2,t }∞
t=0 , taking prices as given individuals optimize and markets clear. All these
conditions have been used to derive these two dynamic equations. We start with a money
supply M−1 . The sequence of monetary injections results in a known time path for the
nominal money supply satisfying Mt = (1 + ẑt )Mt−1 where ẑt Mt−1 = Nt A1,t + Nt−1 A2,t
for all t = 0, 1, 2, ..., ∞.
Construct an equilibrium as follows: Take the exogenous sequence of endowments,
monetary injections and youth population size, {e1,t , e2,t , A1,t , A2,t , Nt }∞
t=0 , the initial
stock of money and the initial old population N−1 to be used in equations (6) and (7).
1. Conjecture a initial equilibrium price level P0 .
2. Take the money supply at the beginning of the period, (M−1 in the case of t = 0)
together with the current period’s monetary injections, (A1,0 , A2,0 ) to construct the
current period’s money supply M1 . Then take the exogenous variables endowments
for cohort t = 0, (e1,0 , e2,1 , N0 ), the current period’s money supply M0 and the
monetary injections (A1,0 , A2,1 ) along with P0 and stick them into equation (6).
Find the value of P1 that makes the lefthand side of equation (6) equal to its
righthand side. Store this value of P1 .
3. Repeat the previous step for periods t = 1, 2, 3, ..., ∞, each time using the previously
calculated value of Pt along with the exogenous values for (e1,t , e2,t+1 , A1,t , A2,t+1 , Mt )
to construct Mt and Pt+1 .
4. After the previous steps, you now have a candidate equilibrium path for the price
level, {Pt+1 }∞
t=0 for an initial equilibrium price level conjecture P0 .
5. In order to pin down P0 we need to make use of the initial old’s optimal decisions.
Thus far, we have used individual optimization for all cohorts born from t = 0
onwards as well as the market clearing conditions for all periods. However, we have
13
to ensure that in the initial period, for the price level P0 , that the initial old are
willing to trade with the initial young.
If the initial old choose not to trade,
2,0 ). Participation in the
then u(c2,0 ) = u(e
+ NM−1−1P0 . As long as aP2,0
+ NM−1−1P0 >

monetary economy returns utility u e2,0 + aP2,0

0

0

0 the initial old are better off trading with the initial young.

3.4

Implications

A few things can be seen from this general set-up. First, generally, money demand will

depend on the distribution of monetary injections between the young and old individuals.

This is evident as A1,t and A2,t+1 appear in the equilibrium equation that we derived

from the individual’s money demand equation. Why does money demand depend on the

timing of monetary injections? Remember that individuals are assumed to have strictly

increasing and strictly concave utility functions. As a result, they prefer to have a smooth

time path for consumption.

To see this, from individual optimal money demand,

u0 (c1,t ) =

β

πt+1

u0 (c2,t+1 )

β

= 1. Then u0 (c1,t ) = u0 (c2,t+1 ). Given the assumption on the utility

Suppose that πt+1

function of diminishing marginal utility, u0 (c) > 0 and u00 (c) < 0, it must be that c1,t =
c2,t+1 . In other words, the individual has a preference for a smooth consumption profile.
Now when the individual gets a larger monetary injection in youth, they will want to
pass some of this wealth into the future in order to smooth consumption. On the other
hand, if the monetary injection is larger in old age, then the individual will have an
incentive to borrow in youth and repay in old age as a strategy to bring some of his/her
β
will tilt
future old age wealth into youth. From this we can also see that the ratio πt+1
β
the individual’s consumption profile away from being flat if πt+1 6= 1.
Another interesting property arises in analyzing a stationary equilibrium in which
individual consumption and real money holdings are constant across cohorts. We know
that individual money holdings equal zero for the old. Let us define m1,t as the real
money holdings of a young individual, m1,t = MP1,t
. Then in a stationary equilibrium, we
t
impose (c1,t , c2,t , m1,t ) be constant over time. What is the relationship between real money
t
holdings per young and the aggregate stock of real money balances? Well, mt = M
is
Pt
the outstanding stock of real money balances. This must be held be the young. We have
already used the fact that Nt M1,t = Mt in preceding derivations. Deflating both sides by
t
the price level, Nt MP1,t
=M
. Thus Nt m1,t = mt .
Pt
t
It has already been shown in equation (5), that in a monetary equilibrium,
1 + ẑt
mt =
mt−1 .
πt
14
so as mt = Nt m1,t ,
Nt m1,t =
1 + ẑt
πt
Nt−1 m1,t−1 .
Hence, in a stationary equilibrium with constant population growth and constant money
growth, as m1,t = m1,t−1 for all t,
1 + n̂ =
1 + ẑt
πt
so that the gross inflation rate is
π=
1 + ẑ
z
≡ .
1 + n̂
n
If there is no population growth, the inflation rate is purely driven by the rate of
money growth. However, if there is population growth, all else equal, an economy with
higher population growth will have a lower rate of inflation. This is because there is
growing demand for money balances which makes money more valuable and pushes up
the rate of return on money (the inverse of the inflation rate).
3.4.1
Example: A1,t > 0 and A2,t = 0 for all t.

Let us consider a situation with constant money growth and no population growth

so that Nt = N along with constant endowment profiles (e1,t , e2,t+1 ) = (e1 , e2 ) and

constant monetary injections. If old individuals do not get monetary transfers then the

equilibrium optimal consumption trade-off condition becomes, if we start at equation (6),

setting A2,t+1 = 0 we can write

A1,t

Mt

Pt

Mt Pt

0

0

u e1 +

−

=β

u e2 +

.

Pt

N Pt

Pt+1

N Pt Pt+1

All monetary injections must go to the young individuals so ẑMt−1 = N A1,t yielding

A1,t = ẑMNt−1 . Then we can write

ẑMt−1

Mt

Pt

Mt Pt

0

0

u e1 +

−

=β

u e2 +

.

N Pt

N Pt

Pt+1

N Pt Pt+1

The flow equation for the stock of money is Mt = (1 + ẑ)Mt−1 so ẑMt−1 = Mt − Mt−1 .

Using this

Pt

Mt Pt

Mt − Mt−1

Mt

0

0

u e1 +

−

=β

u e2 +

N Pt

N Pt

Pt+1

N Pt Pt+1

15

which simplifies to

u

0

Mt−1

e1 −

N Pt

=β

Pt

Pt+1

Mt Pt

u e2 +

.

N Pt Pt+1

0

If we multiply and divide Mt−1 terms by Pt−1 , then we are multiplying Mt−1 terms

by one. Multiplying anything by one does not change any relationships so we can use

this algebra trick to write

Mt−1 Pt−1 1

Mt Pt

Pt

1

0

0

u e1 −

=β

u e2 +

.

Pt−1

Pt N

Pt+1

Pt Pt+1 N

Pt−1

t−1

From our math relating the flows of real money balances, we know that M

=

Pt−1

Pt

Mt

1

so we can write

Pt 1+ẑ

1

1

Pt

1

Mt

Mt Pt

0

u e1 −

=β

u e2 +

.

Pt 1 + ẑ N

Pt+1

Pt Pt+1 N

0

In order to really highlight how our equilibrium equations tie together, let rewrite

this as

mt

1

mt

0

0

u e1 −

=β

u e2 +

.

(8)

zN

πt+1

πt+1 N

This shows us that the period t equilibrium condition which employed optimal individual

behaviour and money market clearing gives us a relationship between period t demand

for real money balances and the period t + 1 inflation rate.

The last piece of the equilibrium that we have not yet used is the goods market

clearing condition. This requires all goods available each period are consumed – as utility

is strictly increasing in consumption, not goods will be put to waste. Working through

some algebra,

Nt e1,t + Nt−1 e2,t = Nt c1,t + Nt−1 c2,t

M1,t−1

A1,t M1,t

−

+ Nt−1 e2,t +

= Nt e1,t +

Pt

Pt

Pt

Cancelling terms and noting that in equilibrium, all money has to be held by the young

at the end of each period, N M1,t = Mt , this reduces to

Mt = Mt−1 + Nt A1,t

= Mt−1 + ẑt Mt−1

where here we have exploited the fact that total money injections to current young must

equal the growth in the stock of money.

So far we have derived a relationship between real money balances in equilibrium and

the inflation rate. Both the level of period real money balances and the inflation rate

16

must be determined within the equilibrium system. This means that we are short of one

equation because equation (8) provides only a single equation that cannot be used to pin

down the values of two separate variables. To find our second equation to help jointly

determine the values of mt and πt+1 we continue working with the flow equation for the

stock of nominal money balances. Dividing both sides by Pt

Mt−1

Mt−1

Mt

=

+ ẑ

Pt

Pt

Pt

and then multiplying and dividing Mt−1 by Pt−1 (which is the same as multiplying these

terms by one and does not change the relationship between left- and righthand side of

our equation),

Mt−1 Pt−1

Mt−1 Pt−1

Mt

=

+ ẑ

Pt

Pt−1

Pt

Pt−1

Pt

1 + ẑ Mt−1

=

π

P

t t−1

1 + ẑ

mt =

mt−1 .

πt

As this relationship must hold for every period starting in period 0, we know that equilibrium requires

1 + ẑ

mt .

(9)

mt+1 =

πt+1

for all periods t = 0, 1, …, ∞.

Thus, the complete dynamics in this example in which only the young receive transfers, can be characterized by equations (8) and (9) which are repeated here for convenience,

mt

1

mt

0

0

u e1 −

= β

u e2 +

zN

πt+1

πt+1 N

1 + ẑ

mt+1 =

mt .

πt+1

As most of the intuition has already been gained in the general case, let’s see what

we can say if we choose a specific utility function, let’s say u(c) = ln(c). The derivative

of ln(c) = 1c so

#

”

1

1

1

mt = β

mt

e1 − zN

πt+1

e2 + πt+1

N

17

Now we can solve for mt by doing some algebra.6

mt

1

mt

e2 +

= β

e1 −

πt+1 N

πt+1

zN

mt

πt+1 N e2 + mt = βN e1 −

zN

βmt

πt+1 N e2 + mt = βN e1 −

z

z+β

mt = βN e1 − πt+1 N e2

z

zN

(βe1 − πt+1 e2 ) .

mt =

z+β

(10)

This is holds for all t so in the flow equation for real money balances,

1 + ẑ

mt+1 =

mt

πt+1

1 + ẑ

zN

zN

(βe1 − πt+2 e2 ) =

(βe1 − πt+1 e2 ) ,

z+β

πt+1

z+β

so that

βe1 − πt+2 e2 =

1 + ẑ

πt+1

(βe1 − πt+1 e2 )

which, using z = 1 + ẑ, rearranges to yield

πt+1 − z

πt+2 e2 =

βe1 + ze2 .

πt+1

(11)

e2 = 0: Suppose we look at an economy in which old get no monetary transfers and they

have no endowments. Then the aggregate demand for real money balances (equation 10)

is7

βzN

mt =

e1 .

z+β

Setting e2 = 0 in the flow equation for real money balances, equation (11) requires

πt+1 = z.

Therefore, in such an economy, the inflation rate is equal to the rate of money growth

and money demand is time-invariant. The larger the endowment of the young, the greater

6

A note of caution here: remember that the individual demand for money is M1,t so in real terms

M

. In the aggregate total money demand equals total money supply so with a constant

it is m1,t = P1,t

1

M

t

population, N m1,t = N P1,t

=M

Pt = mt .

t

mt

7

Recall that m1,t = N is the individual demand for real money balances.

18

is equilibrium money demand and also the greater is the gross growth rate of money, z,

the greater the demand for real money balances.

e1 = 0: What happens if e1 = 0 and e2 > 0? Setting e1 = 0 in the aggregate money

demand equation we see that the demand for real money balances is negative if the gross

rate of inflation is positive,

zN

mt = −

e2 πt+1 .

z+β

t

One possibility, given that mt = M

, is for the price level of goods is negative every

Pt

period with positive inflation Is this nonsensical? Obviously there is a problem. Clearly,

money demand cannot be negative as this would mean that people receive money when

they receive goods rather than giving money in exchange for goods. If the initial old

must give their money along with their goods to the initial young then the initial old will

elect not to participate in monetary exchange and the monetary equilibrium is destroyed

– the outcome is autarky.

Otherwise, the gross inflation rate must be negative forever. Unfortunately, for a

monetary equilibrium to exist, equation (11) informs us that the rate of inflation must

be equal to the rate of nominal money growth. As the nominal money supply growth

is strictly positive in this example (A1,t > 0 and A2,t = 0), this equilibrium condition

is violated so no monetary equilibrium will exist. A negative gross inflation rate means

that eventually, the price level of goods becomes negative so that someone is giving goods

along with money in order for the money to be taken in exchange. Once the economy

gets to this point, the old will not be willing to participate and equilibrium unravels.

4

Pareto Efficiency of the Monetary Equilibrium

We can now examine how the relationship between the monetary equilibrium and the

social planner’s outcome hinges on the relationship between π and n = 1 + n̂ where n̂ is

the growth rate of the population.

Let c∗ ≡ (c∗1 , c∗2 ) be the allocation under the social planner’s solution (which is unique

given our assumptions on preferences). Notice that the social planner’s outcome coincides

with the golden rule equilibrium of the decentralized economy.

The individual’s period budget constraints can be merged into a single, lifetime budget

constraint as follows. Take the two period budget constraints

M1,t = Pt e1,t + A1,t − Pt c1,t

Pt+1 c2,t+1 = Pt+1 e2,t+1 + M1,t + A2,t+1 .

The old age budget constraint can be rewritten as M1,t = Pt+1 c2,t+1 − Pt+1 e2,t+1 − A2,t+1 .

Using this substitute M1,t out of the youth budget constraint returns

Pt+1 c2,t+1 − Pt+1 e2,t+1 − A2,t+1 = Pt e1,t + A1,t − Pt c1,t .

19

Divide both sides by Pt and rearrange to find

Pt+1

A1,t A2,t+1

Pt+1

c2,t+1 = e1,t +

e2,t+1 +

+

.

c1,t +

Pt

Pt

Pt

Pt

This lifetime budget constraint states that the sum of present discounted value of consumption must equal the sum of present discounted value of endowments and monetary

transfers. Notice that because monetary transfers are all divided by Pt , these transfers

are being converted into units of consumption goods. The value of future consumption

and endowments are being “discounted” back into units that are comparable to youth

consumption with the rate of inflation serving as the discount factor. If the individual

gives up a unit of old age consumption from period t + 1, he/she gets Pt+1 dollars. Those

units of date t goods.

Pt+1 dollars can purchase a total of P Pt+1

t

Definition 3 A Pareto Efficient economy is one in which c∗1 > e1 . A Parato Inefficient

economy is one in which in which c∗1 < e1 .
Figure 3 contrasts stationary equilibrium in potential monetary economies with the
allocations chosen by the benevolent social planner. Recall that the social planner chooses
the distribution of goods across individuals in an economy subject only to the resource
feasibility of such allocations. The aggregate resource constraint for our economy is
Nt c1 + Nt−1 c2 = Nt e1 + Nt−1 e2 .
We can rewrite this constraint in per-young terms by dividing both sides by Nt and
letting n be the gross population growth rate tells us that8
Nt−1
Nt−1
c1 +
c2 = e1 +
e2
Nt
Nt
1
1
c1 +
c2 = e1 +
e2 .
n
n
The way to view representation of the aggregate resource constraint in per-young person
terms is that if the planner takes one unit of resources from each young person and
divides the pool of Nt resources amongst the current old, then each old person gets n1
units of consumption goods. This means that the social planner has a technical (gross)
rate of return equal to n1 .
Graphically, we can plot an individual agent’s indifference curves in terms of consumption when young versus when old. The budget constraint that supports autarky
has slope − π1 , whereas the set of feasible allocations is bounded by a line with slope
−n = −(1 + n̂). Importantly, when viewing Figure 3 from the perspective of the decentralized economy, the diagram represents what is feasible and optimal for the individual
8
Writing the aggregate resource constraint in per-young person terms allows for easy diagramatic
comparison to the individual’s budget constraint. Note n = 1 + n̂ where n̂ being the population growth
rate.
20
c2
c2
1
>n

π

1

1 and π1 = n then it must be that if the monetary economy is to satisfy the

golden rule allocation, π < 1 ,so that the gross inflation rate would need to be less than
one. If the gross inflation rate is less than one then the inflation rate π̂ would need
to be negative implying that deflation is a requirement for the monetary equilibrium’s
allocation to coincide with that of the Golden Rule.
We have shown that in a stationary equilibrium with growth in the nominal money
supply as well as population growth,
π=
1 + ẑ
.
n
If the monetary equilibrium is to produce the same allocations as the Golden Rule policy,
1
we want π1 = n or equivalently, πn = 1. In the stationary monetary equilibrium, πn = 1+ẑ
so to produce a Golden Rule allocation, ẑ = 0 - the growth in the nominal money supply
need be equal to zero.
23
6
The Social Planner’s Problem
Consider the problem of the social planner of an economy in which population growth
is constant over time Nt+1 = (1 + n̂)Nt (or equivalently, Nt+1 = nNt ). Endowments
are time invariant so each young individual receives e1 goods while each old individual
receives e2 units of the consumption good. The planner, like the individuals, does not
have a storage technology that allow for the transfer of goods from one period to the
next. What the planner is able to do is transfer units of goods between young and old
within a period.
In this section we will examine the planner’s optimal allocation over time to characterize the Pareto Optimal allocation. The planner discounts the lifetime utility of
individuals born into period t by a discount factor φt , φ ∈ (0, 1). The planners objective
from the perspective of period t = 0 is to maximize the sum of discounted lifetime utilities of all the individuals who will ever live in the economy subject to the period resource
feasibility constraint of every period t = 0, 1, 2, .... Formally,
max
{c1,t ,c2,t+1 }∞
t=0 ,c2,0
N−1 βu(c2,0 ) +
∞
X
φt {Nt [u(c1,t ) + βu(c2,t+1 )]}
t=0
subject to the sequence of aggregate resource constraint
Nt c1,t + Nt−1 c2,t = Nt e1,t + Nt−1 e2,t ,
t = 0, 1, 2, ..., ∞.
Letting λt be the Lagrange multiplier on the aggregate resource constraint of period t,
write the Lagrangian as
L =
∞
X
φt {Nt [u(c1,t ) + βu(c2,t+1 )] + λt [Nt e1,t + Nt−1 e2,t − Nt c1,t − Nt−1 c2,t ]}
t=0
+N−1 βu(c2,0 )
The associated first-order conditions are
c1,t : φt Nt u0 (c1,t ) = φt Nt λt
c2,t+1 : φt βNt u0 (c2,t+1 ) = φt+1 Nt λt+1
c2,0 : N−1 βu0 (c2,0 ) = N−1 λ0
for all t = 0, 1, ...
for all t = 0, 1, ...
which allow us to show that at the optimum, u0 (c1,t ) = λt and βu0 (c2,t+1 ) = φλt+1 so at
the optimum, we obtain the planner’s intratemporal trade-off between a young individual
and an old individual at time t,
λt = u0 (c1,t )
and
βu0 (c2,t+1 ) = φλt+1 ⇒ βu0 (c2,t ) = φλt
so
u0 (c1,t ) =
β 0
u (c2,t ).
φ
24
for all t = 0, 1, ...
(13)
Consider the trade-off that the social planner faces in transferring a unit of consumption
from a single young individual from cohort t to a single old individual from cohort
t − 1. The lost discounted utility to the planner is valued φt u0 (c1,t ) but the gain in
discounted utility to the planner is φt−1 βu0 (c2,t ). At the optimum, these must be equal
or else the planner has incentive to reallocate resources between these two individuals.
This characterizes the optimal intratemporal trade-off that the planner faces between
young and old who are alive in any arbitrary period t = 0, 1, .... Clearly, the optimality
conditions imply that the planner will equate consumption across all young individuals
from a given cohort as well as consumption across all old individuals within a given
cohort.
Notice that so far we have treated the social planner’s discount factor as some arbitrary number. If φ = 1 then the social planner cares about all individuals equally
within and across cohorts. As a technical condition, when φ ∈ [(1 + n̂)−1 , 1] the objective
1
.
function is not convergent.9 Consider the special (and standard) case in which φ = 1+n̂
Then our optimality conditions become
u0 (c1,t ) = β(1 + n̂)u0 (c2,t )
so that optimality in the steady state requires that the modified golden rule condition
is obtained. Therefore, the conditions for maximizing steady state consumption can be
derived as the outcome of the social planner’s optimal allocations in the case where the
social planner discounts each generation’s welfare by a factor (1 + n̂)−1 . What does it
1
mean for φ = 1+n̂
? It means that the social planner discounts the utility of each cohort
with a discount rate that exactly equals the rate of population growth. In other words,
1
the planner cares about a young individual born at time t, 1+n̂
as much as the planner
t
cares about an individual born in period t − 1. Then φ Nt = 1 under the normalization
N0 = 1.
9
There are other ways to show that the optimality conditions hold, for example, an overtaking
criterion is often cited as a way to deal with such “boundedness” problems in OLG frameworks.
25
7
A Simple OLG Model with Productive Capital
Thus far, money has been the only asset that individuals could use to pass wealth
across time. In this section we add physical capital into the model as a second avenue
for storage of wealth. Investment in physical capital will yield an intertemporal rate of
return for saving and we will see what optimal choice between multiple options for savings
implies for equilibrium rates of return on the two assets: money and capital. Following
the investigation into the equilibrium implications for rates of returns across the two
assets, we will turn our attention to the implications for money growth on production.
In this two period OLG model, young individuals are endowed with e1 units of the
endowment good and there are no old endowment goods. They can use it to consume
when young or to save for old age. Endowments that are not consumed in youth can be
convert into units of productive capital one-for-one. Denote by kt the amount of capital
created by a young individual in period t. Then in period t + 1, this capital produces
yt+1 = f (kt ) units of the consumption good for the old individual. Assume that the
production function f (k) is strictly increasing in capital so that its marginal product
of capital is everywhere positive, f 0 (k) > 0 and that the production function either

exhibits constant returns to scale, f 00 (k) = 0 (so that the slope of the production function

is constant), or that the production function exhibits diminishing marginal product of

capital, f 00 (k) < 0. Another way to say this is that f 00 (k) ≤ 0. Stored capital can be
added to the output produced in old age for consuming but disintegrates at the end of the
period that it is used in production. Population growth is not essential to the message of
this section so assume that the population is constant with N young born each period.
Individual utility functions, u(c), are assumed to have the same properties as in the
baseline model, u0 (c) > 0, u00 (c) < 0 and limc→0 u0 (c) = ∞.
With these assumptions we can write the individual’s problem as
max
c1,t ,c2,t+1 ,kt
{u(c1,t ) + βu(c2,t+1 )}
subject to the constraints
c1,t = e1 − kt
c2,t+1 = kt + f (kt ).
Setting up the Lagrangean for this problem, taking its first-order necessary conditions
and rearranging yields an intertemporal consumption trade-off condition,
u0 (c1,t ) = β(1 + f 0 (kt ))u0 (c2,t+1 ).
(14)
At the optimum, the individual must be indifferent between consuming a marginal
unit of the consumption good when young or saving this marginal unit of the endowment
good as additional capital, producing output when old with the capital and then consuming the extra output along with the saved capital. Notice that the marginal product
26
of capital, f 0 (kt ), is the rate of return from saving a bit more capital if the individual
is already saving kt units of capital. If the production function exhibits diminishing returns to capital, then as more endowments are saved in youth, c1,t decreases so u0 (c1,t )
increases. The increased savings reduces the rate of return f 0 (kt ) because f 00 (kt ) < 0 and
with the increase in consumeable capital and the additional output that is also consumed,
u0 (c2,t+1 ) decreases due to diminishing marginal utility of consumption. If we draw the
individual’s resource constraints, it is clear that the slope of the constraint in c1 −c2 space
is −(1 + f 0 (k)) which is constant if f 00 (k) = 0 and decreasing as k increases if f 00 (k) < 0.
As an aside, note that summing consumption across young and old returns the aggregate resource constraint,
N c1 + N c2 = N (e1 − k) + N (k + f (k))
N c1 + N c2 = N e1 + N f (k)
In this set-up, there is no trade because no old individual will give anything to a
young individual. Also, as all young individuals are identical they will not trade with
each other because there is no double coincidence of wants with the same holding true
for old individuals. The stationary equilibrium achieves the golden rule allocations.
7.1
Adding Money
The point of this section is to add money to an economy with productive capital. Both
can be used as a store of value so in the absence of differential savings risk, we should
see an equality in endogenous rates of returns as both are both capable of functioning as
a store of wealth. Additionally, as we have equality in equilibrium rates of return across
the two assets and as money growth reduces the rate of return on money, money growth
will also reduce the rate of return to capital. This is because a flight from money will
increase capital holdings thereby reducing its rate of return (the Tobin Effect).
Introduce money and growth in the money supply. For simplicity, assume only the
old are provided with monetary transfers, i.e. A1,t = 0. Individuals now choose how
much to consume when young, how much money to hold and how much capital to hold.
In old age, money can be exchanged for consumption goods, capital produces output
that can be consumed along with the saved capital stock. Also, to ensure the model’s
equilibrium will exist, assume that the production function has the properties f 0 (k) > 0,

f 00 (k) < 0 and limk→0 f 0 (k) = ∞. This says that the marginal product of capital is
always positive, there is diminishing returns to capital in production and that as the
capital stock converges to zero, the marginal product of capital goes to infinity (the slope
of the production function becomes vertical).
Write the individual’s problem as
max
c1,t ,c2,t+1 ,kt ,M1,t
{u(c1,t ) + βu(c2,t+1 )}
27
subject to the constraints
Pt c1,t = Pt e1 − Pt kt − M1,t
Pt+1 c2,t+1 = Pt+1 kt + Pt+1 f (kt ) + M1,t + A2,t+1 .
The associated Lagrangean is now
L(c1,t , c2,t+1 , M1,t , kt , λ1 , λ2 ) = u(c1,t ) + βu(c2,t+1 ) + λ1 [Pt e1 − Pt kt − M1,t − Pt c1,t ]
λ2 [Pt+1 kt + Pt+1 f (kt ) + M1,t + A2,t+1 − Pt+1 c2,t+1 ]
Taking the first-order conditions to identify the critical points of the Lagrangean function,
∂L(·)
=0
∂c1,t
∂L(·)
=0
∂c2,t+1
∂L(·)
=0
∂M1,t
∂L(·)
=0
∂kt
: u0 (c1,t ) = Pt λ1
: u0 (c1,t ) = Pt+1 λ2
: λ1 = λ2
: Pt λ1 = Pt+1 (1 + f 0 (kt ))λ2
Combining the FONCs with respect to c1,t , c2,t+1 and M1,t it can be shown that if the
youn individual trades a unit of date t consumption for money and uses this money to
buy consumption goods in old age, it must be that
u0 (c1,t ) = β
Pt 0
u (c2,t+1 ).
Pt+1
We have seen this intertemporal consumption trade-off through use of money before and
will not reinterpret it here.
Consumption can now be shifted over time using capital too. Combining the FONCs
with respect to c1,t , c2,t+1 and kt it can be shown that if the youn individual reduces
youth consumption by a unit for an increase in capital holdings and uses this capital to
consume and to produce in old age, it must be that at an optimum,
u0 (c1,t ) = β(1 + f 0 (kt ))u0 (c2,t+1 ).
These two optimal intertemporal trade-off conditions imply that at an optimum, the
individual will select capital holdings such that
Pt
= 1 + f 0 (kt ).
Pt+1
As the marginal product of capital is infinite when capital is zero, it would appear that
individuals will all hold some capital savings as long as the return to holding money is
finite.
28
Definition 4 A competitive equilibrium in this monetary economy with capital is a se∞
quence of allocations {c1,t , c2,t , M1,t , kt }∞
t=0 , and prices, {Pt }t=0 , such that given a sequence
for the money supply {Mt }∞
t=−1 ,
1. taking prices as given, individuals optimize, and
2. markets clear for all t = 0, 1, ..., ∞.
In the competitive equilibrium, goods market clearing implies that aggregate consumption equals aggregate supply of consumption goods. As retained the simple assumptions of the model with only capital, only the young receive endowments and there
is no population growth. Thus
Nt c1,t + Nt−1 c2,t + Nt kt = Nt e1 + Nt−1 kt−1 + Nt−1 f (kt−1 ).
As Nt = Nt−1 = N
c1,t + c2,t + kt = e1 + kt−1 + f (kt−1 ).
If we restrict ourselves to studying stationary equilibria in which all individual-level
variables are constant over time,
c1 + c2 + k = e1 + k + f (k).
or
c1 + c2 = e1 + f (k)
which is an aggregate resource constraint for the economy.
This shows that in a stationary equilibrium, capital savings by the young are offset
by capital dissavings of the old. Then consumption of young and old must be financed
out of endowments plus output produced through capital savings. Using the individual
budget constraints, in this aggregate resource constraint,
M1,t−1 A2,t
M1,t
+ kt−1 + f (kt−1 ) +
+
= e1 + f (k)
e1 − kt −
Pt
Pt
Pt
so imposing the restrictions from a stationary equilibrium,
M1,t
M1,t−1 A2,t
e1 − k −
+ k + f (k) +
+
= e1 + f (k)
Pt
Pt
Pt
or equivalently,
M1,t
M1,t−1 A2,t
=
+
Pt
Pt
Pt
29
t
As only the young hold money at the end of period t, M1,t = M
, and total money
N
At
injections are A2,t = N
Mt
Mt−1 At
=
+ .
N
N
N
Therefore, clearing the goods market and the using the fact that equilibrium requires
individuals to respect their budget constraints returns the money flow equation
Mt = Mt−1 + At
giving us confidence that the clearing the goods market implies that the money market
also clears.
As in the baseline model, we can obtain an expression relating the inflation rate to
growth in the money supply. First note that At = ẑt Mt−1 . Dividing both sides of the
money supply equation by Pt ,
Mt
Mt−1 ẑt Mt−1
=
+
.
Pt
Pt
Pt
so multiplying and dividing the terms on the righthand side by Pt−1 ,
Mt−1 ẑt Mt−1
Pt−1
Mt
=
+
.
Pt
Pt−1
Pt−1
Pt
t
t
= mt
Rewriting, with πt = PPt−1
and M
Pt
πt =
(1 + ẑt ) mt−1
.
mt
In a stationary equilibrium, mt = m so that in a stationary equilibrium, π = z. Hence we
have the usual result that inflation is a monetary phenomenon in a stationary equilibrium.
Fully characterizing the stationary equilibrium requires solving for c1 , c2 , k, m1 and π.
The system of equations that fully characterize the steady state allocations and inflation
30
rate are10
π = z
1
1 + f 0 (k) =
π
e1 + f (k) = (e1 − k − m1 ) + (k + f (k) + m1 + a2 )
ẑm1
= (e1 − k − m1 ) + k + f (k) + m1 +
π
c1 = e1 − k − m1
1
+ a2
c2 = k + f (k) + m1
π
m ẑm
1
1
= k + f (k) +
+
π
π
which is a system of 5 equations and 5 unknowns.
7.2
The Tobin Effect
The punchline of this model is that the rate of return on money (which is the inverse
of the inflation rate) is equal to the rate of return to capital savings,
Pt
= 1 + f 0 (kt ).
Pt+1
Clearly in the steady state (or stationary equilibrium) we have
1
= 1 + f 0 (kt )
π
so that when the inflation rate increases, the rate of return on capital, 1 + f 0 (k) will
decrease in equilibrium. This requires that the marginal product of capital decreases with
higher equilibrium inflation rates. How can this occur? The assumption of diminishing
marginal product of capital in the production function necessitates the capital stock per
old individual, k, increases when inflation rates increase.
10
The derivation of real money injections per old individual is
a2
=
=
=
=
ẑMt−1
N Pt
ẑMt−1 Pt−1
N Pt−1
Pt
ẑN M1,t−1 Pt−1
N Pt−1
Pt
ẑm1
.
π
31
This is the Tobin Effect of inflation - decreasing the return from holding money causes
individuals to seek better returns from saving through other assets. While individuals
push their resources into saving via capital holdings other than money, the demand for
capital pushes down its equilibrium rate of return. As a result, in equilibrium, rates of
return are equalized between capital and money. Theoretically, this suggests that higher
inflation rates can be accompanied by lower rates of return on capital but higher output
due to a larger capital stock (holding everything else constant).
What does the Tobin effect imply for the impact of money growth on production? In
this simple model, inflation is proportional to nominal money growth. That is, higher
rates of money growth results in higher inflation rates. Couple this with the fact that
higher inflation rates lead to increased capital stock and the theoretical implication is that
higher money growth rates should stimulate investment and output growth. Whether this
causal relationship is reflected in real-world data is an empirical issue that requires the
ability to disentable the Tobin effect from other factors that might affect the correlation
between inflation rates and the rate of return from physical investment.
32
8
Money - A Liquid Asset
Why to people choose to hold money if its realized rate of return is typically lower
than the realized rate of return from other assets? One argument is that money is a liquid
asset - it can be easily transformed into a form that is used for transactional purposes.
In this section, we will present an equilibrium model in which physical capital cannot
be transformed into a form that can be used for transactional purposes. The model will
provide an example in which individuals hold money partly for insurance and also for
storing wealth over time in a manner that can be used to trade for goods while some
wealth is locked up in a less liquid form that might provide a higher realized rate of
return.
8.1
The Model
This is a three-period OLG model so while the economy is infinitely lived, individuals
live for three periods. Population is constant as is the stock of money. When young,
individuals receive an endowment of e1 units of the consumption good. Out of this endowment, when young, they can choose to consume, c1,t , invest in capital, kt , or exchange
goods for money and hold on to the money balances, M1,t . Young individuals can transform the endowment good into capital at a rate of one-for-one; for each unit of capital
investment, a young individual in period t uses one unit of the endowment/consumption
good. Let the rate of exchange between money and the consumption good in period t be
Pt .
If an individual invests in kt units of capital when young, f (kt ) units of output are
received when the individual is old (in the third period of life) with probability θ while
the capital investment is a complete failure and yields nothing with probability 1 − θ.
Assume that f (k) is strictly increasing and strictly concave, f 0 (k) > 0, f 00 (k) < 0. Capital
investment is irreversible and cannot be transformed back into the consumption good.
An implication is that middle-aged consumption (the second period of an individual’s
life), c2,t+1 , must be financed out of money savings. The individual can also elect to hold
on to, M2,t+1 units of money in middle age in order to have money in old age.
In the third period of life, if the capital investment is successful, then the individual
finances consumption, cS3,t+2 out of production f (kt ) and money holdings, M2,t+1 . If the
in...
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