University of Adelaide Money Banking and Financial Markets Questions

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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 HOME PAGE MY TIMES TODAY'S PAPER VIDEO MOST POPULAR http://www.nytimes.com/2006/05/02/world/africa/02... Welcome, cedmond2 TIMES TOPICS Member Center Africa WORLD U.S. N.Y. / REGION BUSINESS TECHNOLOGY SCIENCE HEALTH Log Out World All NYT SPORTS OPINION ARTS STYLE TRAVEL JOBS REAL ESTATE AUTOS AFRICA AMERICAS ASIA PACIFIC EUROPE MIDDLE EAST 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. How, and where, did Bill propose to Hillary? Also in Travel: Which 17th-century watering hole was George H.W. Bush spotted in? Which four presidents were born in Boston? 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. BLOGGED SEARCHED 1. For Girls, It’s Be Yourself, and Be Perfect, Too 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 4. Parenting: Accepting Gay Identity, and Gaining Strength 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 6. Op-Ed Contributor: What Really Ruined Baseball 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 3. The DNA Age: Stalking Strangers’ DNA to Fill in the Family Tree 7. Justices Rule Against Bush Administration on Emissions 8. Shortcuts: Too Busy to Notice You’re Too Busy 9. Court Rebukes Administration in Global Warming Case 10. Ex-Aide Says He’s Lost Faith in Bush Go to Complete List » 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 Also in Business: 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. Sign up to Business Today Free daily newsletter Get set for the working day – we'll point you to all the business news and analysis you need every morning Enter your email address Sign up Privacy Notice: Newsletters may contain info about charities, online ads, and content funded by outside parties. For more information see our Privacy Policy. We use Google reCaptcha to protect our website and the Google Privacy Policy and Terms of Service apply. “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 +


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 +


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 +


u e2 +
N Pt
N Pt
Pt+1
N Pt Pt+1
15
which simplifies to
u
0

Mt−1
e1 −
N Pt


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 −

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 −

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 −

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... Purchase answer to see full attachment

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