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University of Waterloo
ECON 304
Jean- Paul Lam

Economics 304: Monetary Economics Jean-Paul Lam Lecture 2 Money Contents 1 Introduction 2 2 What is money? 2 2.1 Functions of money . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3. . . ..... 3 Money as Medium of exchange 3 3.1 Search costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . .... 3.2 Medium of exchange in an OLG model . . . . . . . . . . . . . . . . . . . . 5..... 3.2.1 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . . . . .. 3.2.2 Optimal consumption choice . . . . . . . . . . . . . . . . . . . . .7. ..... 3.2.3 Equilibrium in a barter economy . . . . . . . . . . . . . . . . . . 8 ...... 3.2.4 Equilibrium with money . . . . . . . . . . . . . . . . . . . . . . . .9. . .... 3.3 Monetary equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . .0. . ...... 1 3.3.1 Budget constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . ... 1 3.4 Optimal allocation under a stationary equilibriu. . . . ...........1... 1 4 Money as a Medium of Account 15 4.1 Money as a Store of Value . . . . . . . . . . . . . . . . . . . . . . . . .6. . . ..... 1 5 Monetary standards 16 6 Empirical measures of money 18 7 Conclusions 19 8 References 19 1 noi tcu dort1nI In this lecture, we explore the properties, functions and importance of money. Money as we will see is a special commodity that plays a very important role in the economy. We begin by defining what money is, then we explain in details (using models) the functions of money and we conclude by providing an overview of how money is measured and the di▯erent monetary standards across time. In subsequent lectures, we will look at the importance of money and its relationship with output and inflation in the short and long-run. ?yenms tah Money can mean di▯erent things to di▯erent people. Many people use the term money to describe wealth, income or how much one earns per year. Economists heav a very di▯erent view of what money is. In fact, economists have a more precise definition ofoy. nyiantil bonds and equities but it has many special characteristhat these other assets do not. Money is defined as a worthless commodity (money does not pay any inter est) that can be used to purchase goods, services or other assets. Money for many economists represent a claim individuals (holders of money) have on the assets of the government.1 If money is a commodity that is worthless, then why do individuals or societies value money? Is it because we, collectively believe that it has value and it will always be accepted as a means of exchange?it Ibsecause, it provides special functions that none of none of the other assets can provide? iibuewebiehthe government will guarantee that money has value as it represents on claim on some government real assets? To obtain answers to the following (and other) questions, we need to study the functions and characteristics of money. Money is generally accepted as a means of payment. For example, coins, banknotes, cheques written, e-money, plastic cards, traveller’s cheques annaegellpdnicne for goods and services. The monetary base is money issued by the central bank and consists of notes and coins that is currency in circulation. The rest is k nown as reserves which are accounts that banks hold at central banks. Typically a commodity should possess the following charaeristics before being used as money: 1. Durable: money should retain its shape and form after even ao gpidf. or example, coins and notes should last for a long period. 2. Easily transportable: Money should be light and convenient to transport. 3. Divisible in small parts: Money should be available in di▯erent denominations to facilitate transactions. 4. Units of standard value: Money over time must look the sameand should be easily recognized. This allows for easy transactions. 5. Worthless as a commodity: Notes and coins should not have avalue that is higher than their actual denomination or face value. 1Money can be viewed as short-term government debt. This debtis valuable since it represents a claim on future taxes that the government can raise to pay back holders of thatdebt. 2 6. Cannot be copied or duplicated: Counterfeiting should notbsl.oritoapn, money has to contain a lot of security features (holograms, watermarks, microscopic features, etc...) and should be issued by a single entity, currentlernmt.o 2.1 Functions of money Money serves three main functions: 1. Medium of exchange 2. Medium of account 3. Store of value We describe these functions in details. egnahcxe muid sye3Mo The medium of exchange function of money is the most importantufcinfmn. Mnyi amm i atipdasamasofymtfrhro ods and services. Money is a special asset because of this function. Other assets, sutksndbd,antbeedo buy goods and services. To fully understand the medium of exchange function, ec ronwsidpossible economies: a barter economy where individuals exchange goods (or commodities) for goods(orcommodities) and a monetary economy where there exists a certain durable and transportable commodity that is generally acceptable in exchange for any other good or service, which we can call money. At this stage, we do not care whether money takes the form of a commodity or simply is fiat money. For barter to operate, double coincidence of wants mu ailtTrevt is for barter to operate I must have something you want and you must have something I wta .nIf I want a litre of milk and Iveapudofhe,hnImutfidorahfromoew ho has a litre of milk but who also wants a pound of cheese in exchange. For economies, that have a small amount of goods and s ,sreinmetos not present many challenges and problems. However, for larg eeconomies,tsinaarrangemen barter economy can be very ine▯cient as individuals are forc ed to spend a large amount of their time and resources in the activity of searching for suitable partners for trade. Put simply, the search costs under barter are very high and the probability of success, that is solving the double coincidence of wants is low. On the other hand, in a monetary economy,m oeyisdocndttnai.M ny facilitates transactions since individuals can use monefmtantiswotegnig in a lengthy and costly search process. This is because in a monetary economy, the need for the double coincidence of wants is eliminated. In a monetary economy, an individual can simply search for theo/vshewtso purchase and then exchange the money for the goods she desire s. In a monetary economy, trade no longer requires the double coincidence of wants to prevai l. Everything is paid with a medium of exchange, that is money. The use of money as a medium of exchange implies lower search and transactions costs. 3 3.1 Search costs As an example consider the following two hypothetical economies (barter and a monetary economy) with J di▯erent types of goods. We start with a very simple example andassumethat J =3,that is there are three commodities: (a,b,c). In the barter economy, trade occurs by exchanging goods. In Table 1 below, the symbol x represents permissible exchanges (exchanges that are allowed) while the symbol orepresents non-permissible exchanges (exchanges that are not allowed). As a reminder, in a barter economy, goods are exchanged for goods ,whereasinthemonetary,conom money is exchanged for goods. Table 1: Exchanges in a Barter Economy C a Cb Cc C a ox x C b xo x C c xx o For example, we can see from the table that a gcond be exchanged for good b and c.G d b can be exchanged for good a and good c and good c can be exchanged for good a and good b. Individuals will not exchange the same good for the same one ast iwuliolehnnige same good for another one while incurring a cost. Given this information, we can also calculate the probability on any given attempt that someone is going to make a successful exchange under the barter economy. Recall that under the barter system, the need for double coincidence of wants must prevail. I need to meet someone who not only has what I want but also needs what I have. 2 It is clear from table 1, that the number of possible exchangeswith3goodsissimply J ▯J =6. Hence, the probability of finding a match where the double coincidence of want is satisfied is simply 1 1 (J ▯J) ntesehegdhibiiis 6.f J is a large number, (as it is the case in amodrncnm ,thnheprbtyfmakngascssf ul exchange becomes very small. For example, iJ f =1 ,000,000, then in a barter economy, the probability of finding the right 1 12 trade partner is given by (1000000 ▯1000000)1 .0e ,niedaveym lumb. ntrwod, the average number of attempts before the double coincidenceo fwssfidivnby 2 J ▯ J = J(J ▯ 1), the inverse of the probability of success on any single try. If we assume that there is a search cost µ,thaterytimousehforapotialpartner, you have to pay a costµ,thenthetotalsearchcostwhenthereare j goods under a barter economy is simply µ(J ▯ J). If J is large, in our example, J =1 ,000,000, and µ =$0 .01 or one cent, then the total search cost is equal to(1000000 ▯ 1000000) ▯ 0.01 = $1.0e .10 On the other hand, in a monetary economy, money is used to buy goods and goods cannot be exchanged for other goods. Assuming that commoditya is money, we have the following permissible exchanges: We will assume for now that the money we are talking aboau ttionFeiy. If a is money, then we can use money to buy goodb and good c.I feae J =3goods,itisclearthatthenumberof possible exchanges is given by J ▯1=2,thatismoneycanbeused(good a)tobuygood b and/or good c.Iteseheodiclahesffii gasuitabletradepartnerina monetary economy is lower and the probability of success is higher. 4 Table 2: Exchanges in a Monetary Economy C a C b C c C a ox x C b oo o C c oo o If we have a large number of J goods, we can easily calculate the probability of finding a suitable partner in a monetary economy. IN the case of J goods, it is clear that the number of possible exchanges is J ▯1whihisapximatelyequalto J if the number of goods is large. Hence in this case, the probability of having a successful trade is simJ▯1 ▯ J1 if J is large. This probability is unambiguously lower than the probability of success under the barter econ- omy. Hence with money, it takes on average J searches to find for a successful trade whereas it takes on average J(J ▯ 1) searches in a barter economy to find a partner. If we again assume that there is a search cost µnteolhctuerameycomyine by 2µJ (because there is a cost for buying and selling). This search cost is unambiguously lower than under a barter economy (2µJ < µJ(J ▯ 1) if J> 3. The medium of exchange function is generally regarded as the distinguishing function of money. It is to be noted that the saving in time and energy provided theimofxhne—mon— does not depend on whether that commodity has value or not. Th eo nlqumetihatii acceptable as a medium of exchange. The second feature of money as a medium of exchange is that is facilitates transactions and in some cases make transactions possible. To illustrate thisoiewlenolig generations model (OLG). The overlapping generations modelwillshowhowtheexistenc,ofmoney as a medium of exchange, improves welfare and facilitates trade. We first begin by laying out the assumptions of our model. We then explain why the existence ofm nyaisrdendwhy it makes everyone in the economy better o▯. 3.2 Medium of exchange in an OLG model In the OLG model, individuals live for two periods. In period 1, they are born and are described as being young, whereas in the second period, they become oldand die at the end of that period. We refer to those in the second period of their life as olcd e.aHennts in the OLG model are heterogenous and age is the only source of heterogeneity. Each agent has thesameprcs and face the same constraints. The economy begins in period 1. In each period t ▯ 1, N tndividuals are born, where time is index by the subscript t.i,1 N 1ndividuals are born, in period 2, N 2ndividuals are born, and so on. The economy begins in period 1 with N m0mbers of the initial old.T e individuals born in period 1,2,3,.... are called the future generations of the economy. Hence in each period t,heae N young individuals andN old individuals. For example t t▯1 in period 2, there will be2N young individuals andN1old individuals, that is those who were born in period 1 and who are now old in period 2. For simplicity, we assume that there is only one good in cotomisyeand the good is homo- geneous. More importantly, the good is perishable and thus cannot be stored from one period to another. In other words you cannot store of save the good for future consumption. 5 In the basic set-up, each individual receives an endowment (or gift) of the consumption good in the first period of life (when young) and this amount is denotedby y. Individuals when old, that is when they are in their second period of their life receives no endowment. The pattern of endowments is illustrated in Figure 1. In period t,nain t is born. Each individual lives for two periods and individuals are endowedwh y when young and 0 units when old. In any given period, there is one generation of young people and one generation of old people. Thus two generations overlap each other, hence the term overlapping generations. Table 3: Pattern of endowments in the OLG model Period 1 Period 2 Period 3 Period 4 Period 5 Period 6 Period 7 Generation 0 0 Generation 1 y 0 Generation 2 y 0 Generation 3 y 0 Generation 4 y 0 Generation 5 y 0 Generation 6 y 0 At any given period in time, there will be one young and one ollidving at the same time. The young individual is born with good y while the old individual has no good. The next period, the young individual becomes old , the old individual dies andsa dipears and a new young generations is born. 3.2.1 Preferences All individuals in the OLG model consume goods when young and when old. An individual’s total utility therefore will depend of how much she consumes when y oung and when old. We make the following assumptions about the individual’s preferences about consumption: • The individual’s utility is higher if she can consume in both periods rather than in only a single period. In other words, the individual prefers to smooth consumption over time by consuming when young and old and does not want to consume all her endowment in one period only and nothing in the other. • The individual has diminishing marginal rate of substitution. Recall that the marginal rate of substitution (MRS) of good X for good Y is simply the number of good Y one must give up to obtain additional units of good X in order to maintain the same level of utility. With diminishing marginal utility, an individual is prepared to forgo less and less of a good Y to have more and more units of good X. Put di▯erently, to give up the same amount of good Y over time, the individual has to compensated by more and mo orfegood X over time. The MRS of good X for good Y is given by the ratio of the marginal utility of good X over the marginal utility of good Y . • The individual is able to rank consumption bundles over time in order of preference. Thus given two bundle of goods, bundlesA and B,theconsumercansaywhethersheprefersbundle 2 You can also interpret this endowment as labour. Theuailvuisdes this labour endowment and earn income y. 6 A over bundleB or vice-versa or is indi▯erent between the two. Assumptions regarding transitivity holds also. Hence these assumptions imply that the individual will have indi▯erence curves that are convex to the origin. A typical indi▯erence curve is shown in Figure 1 where the amount of consumption of good 1 is on the x-axis and the amount of consumption of good 2 is on the y-axis.teIilc that there are diminishing marginal rate of substitution. To illustrate this point, suppose we start at p Ao.ipewedeteedpeid consumption (c 2 of the individual by 2 units. The indi▯erence curve tells us that to keep the individual’s utility constant, we have to compensate the individual by increasing the first period consumption by the 2 units also. This moves the individual to point B.N wsupsewedce the second period consumption of the individual by another 2 units. It is clear that this time the consumer has to be compensated by 6 more units of consumption in the first period (from 5 units to 11 units). Thus individuals are less willing to give up (or have to be compensated more) consumption good if they have little of them in the first place. 3.2.2 Optimal consumption choice We are going to analyze the outcome of such an economy uo ndpsrftom. Onewhere money does not exist and exchanges take place under the barters yma dteoriwhe money exists as a medium of exchange and money can be used to purchase goods. Before we do so, we introduce some notation. 1. c1,temontheodatoumdbyniilie rfirstperiodoflife(when young) 2. c2,t+1nshemutfeodhemeiilums in the second period of her life (when old) Note that c 2,t+1represents consumption that occurs in t+1 period by individuals born at time tttitomoiftelnitnatite oiit important (as in steady-state or long-run equilibrium), we will denote the first and second period consumption of an individual as c1and c 2espectively. The problem facing any individual born at time t is very simple. Each individual wants to maximize her utility subject to the resource constraint. To maximize utility, they need to consume apiimontfods in both periods. Consuming a positive amount in both periods is always better than consuming in only one period. Moreover, we assume that each individual is impatient. Thisimplies that given the opportunity, each individual would rather consume more today than in the future. If we denote utility by u, then we can denote the utility of an individual born at time t by u(c1,tc2,t+1 (1) Each individual born at time t faces a problem. The individual born at time t is endowed with good y that is unfortunately not storable. The individual when old does not have any endowment. To consume in the second period, that is when old, they m dastyoncuieosG ne 7 that there is no storage technology, then the question is how can the a young individual born at time t acquire goods at time t +1whensheisold? To answer this question and to illustrate the basic fuo nfctonney as a medium of exchange, we will consider two economies as before: (i) a barter econo ywheosaehndfr goods and (ii) a monetary economy, where goods are exchangedfor a commodity named money. 3.2.3 Equilibrium in a barter economy Let us first consider the nature of a (competitive) equilibrium in an economy where money does not exist and only barter can take place. . Recall that at time t,here N toung people who are born with an endowment anN dt▯1 old people who do not have any endowment. The utility of theseold can be increased if they can somehow give up some of their endowment when they are young in exchange for some goods when they are old. Can they achieve such an objective? The answer to this question is clearly NO in an economy without money. In the example we have, no such trades are in fact possible without money. From Table 1, we can see that a young person in period t can potentially trade with two types of persons: other youngpeople of the same generation and old people of the previous generation (peoplebrni oid t-1). It is clear that trade with another young person would not be mutually beneficial since this would involve only swapping the same good but after incurring a cost. Trading with the old generation does not make sense alsoo .ldTeneration would like to have what the young generation have, that is a part of their en dowment. But the old does not have anything to o▯er in exchange and does not have what the young is looking for. The old cannot borrow from the young since the latter knows that he will not b eeaihenxtpidbase the old will not be around the next period. For example, a youngk nwshtiheldsatof her endowment to the old person at time t,thnsewllerehendwmntgi iei period t+1,theoldgenerationwilldisappearordie.Thisisclearlyno abeneficialarrangementfor the young. This lack of possible trade is the manner in which the OLG modelc peseaneof double coincidence of wants. In our example and model, each o ld generation wants what the young generation has but does not have what the young gener ant wants. Thus within this barter economy, no trade occurs. Each generation then consumes itsendowment. The young consumes y and the old nothing. This is clearly not a great outcome. An economy where individuals consume their own endowment andw henoaeeli known as an autarh ccciiIe such that no trade occurs. Since no one is able to carry her endowment in thenextperiodand/oreannotmak atradethatwouldbemutuallybeneficia,iotheotherparty nthebarterec,theyoungends up consuming all her endowment when young and nothing when ol d. As a result, in this autarkic equilibrium, utility is low. It is clear that each member of the future generation would be glad to give up some of their endowment when young and consume when old while the initial old would like to consume a positive amount. In other words, the existence of trade would make each generation better o▯. In this world where barter is the only way individuals can trade, there is noyfrti oapp. rtde 3 We define a competitive equilibrium where individuals izexmeir utility and the market for goods clear (demand equals supply) 8 to happen, we need to introduce a commodity, money. We will seethatthetroductiono,money as a medium of exchange will make everyone better o▯. In other words, the introduction of money 4 is Pareto improving. 3.2.4 Equilibrium with money To open up a trading opportunity that can allow individ omualyot tui equilibrium, we now introduce money in our simple economy. Wec naumeateteof money introduced is simply fiat m ylsimseyoiF costless to produce and that cannot itself be used in consumption or in production. For the purpose of our example, we assume that a centralan bprokduce fiat money costlessly and no one else can produce (or even counterfeit) it. Moreover, money can be costlessly stored from one period to another and it is costless to exchange. Because individuals derive no utility from holding fiat money, money is valuable only if it enables individuals to trade for something they want to consume. A monetary equilibrium is a competitive equilibrium in which there is a valued supplyo f fiat money that clears the money market. By valued, we mean tha tt hefitmoeyanbeadd for some of the consumption good. For fiat money to have value, people must believe that money will have and hold its value in the future, its supply must be limited and it must be extremely di▯cult to counterfeit. We begin our analysis of the monetary economy with an ecw onihmayfixed stock of M perfectly divisible units of fiat money. We assume that the in itial stock of money is M and the money is divided equally among the N old individuals. Thus at time 1, each of the initial old (born M at time 0) is allocatedN units of money. Each young person born at time 1 is endowed wit hgood y. Recall that each person lives for two periods (young in the first period and old in the second) and would like to consume a positive amount of goods in both periods. How can the introduction of money improve the welfare of all individuals in this economy? The presence of money will simply open up trading possibilities that never existed under barter. Ay ungprncnnowslprtfrenwmentfoodoan old person in exchange for money. The old person now receives some goods to consume in exchange of money. The young person consumes a fraction of her endowment and exchanged th eoercifrmny. e young person can then, when she becomes old use the money she cquired in exchange for goods when young to buy goods when old. As a result, each young personb yaigoosormny can now consume a positive amount of goods when young and old. The existence of money as a medium of exchange, by permitting trade (goods for money), allows individuals to consume a positive amount in both periods of their lives. Money, as a medium of exchange, allows trade to take place and by doing so makes everyone better o▯. In other words, the existence of money is Pareto improving. This is discussedinthenextsection. Note that if we lived in a world where we had perfect record keeping and perfect credibility and trust, then money would not be needed as we could write IOU’s and trade IOU’s. Unfortunately we An allocation is Pareto improving if at the new allocation, e verybody can be made better o▯ without making someone else worse o▯. The same argument can be made with commodity money 9 have information frictions, limited memory and thus imperf ect record keeping. Money is memory since it overcomes all these information frictions and does not require perfect record keeping. 3.3 Monetary equilibrium In the introduction, we argued that money is valued if and onlys i opl iehtm nyw lli be accepted as a medium of exchange in the future and if it reta ins its value. In the example we just described, trade will take place only if money is valued,ni hrwd,fiplewillig to give up some of the consumption good in exchange for fiat money and vice versa. Because the value of fiat money is intrinsically useless, its worth weip lledd on one’s view of its value in the future. If individuals believed that money will have no valuen ihefr,eniiilwlli simply choose not to hold money today. As a result no trading will take place. For an exchange to take place, people must believe thatw milhaeye a value in the future in the sense that it can be exchanged for goods. Hence money willhave no value today, if individuals know that at any future date T money will have no value. To see this consider, an economy that lasts for only T periods. If money is worthless in period T,hnniili eid T-1 will not be willing to demand and hold any fiat money since they know that no one will accept money in period T in exchange for goods. Similarly, individuals in period T-2 will also not want to hold and demand money since they know that money is going to be worthless in period T-1.B yiir reasoning, we can move back in time and show that money will have no value today if it has no value at timeT or at any other time before T for that matter. Assuming money is always valued, we can now characterize the monetary equilibrium more formally using the OLG model. To do so, we will introduce some notation and set-up the model to solve for the monetary equilibrium. We first describe the objective of each individual and then derive her budget constraint. With this information, we can derive the optimal consumption of each individual when young and old in equilibrium as well as the set of prices that clears the money and goods market. In our set-up, in her first period, each young individual has ane nowmnt y of goods. The individual can do two things: (i) she can consume them completely or (ii) sell part of the endowment in exchange for money. The existing old generation is endowedwithallthemoneystockeassume for simplicity that there is one young person and one old person. Denote the nominal money stock at timet by M ,thtpricelevelattime t by pt,theconsumption of a young person at time t by c and the consumption of that same person when old at time 1,t t+1by c 2,t+1.Asehetuciinmpipest he generations (1=young, 2=old) and the second superscript represents the time period. The price level simply denotes how many units of fiat money one must give up in exchange for one unito fonsuin. usione needs to give up more fiat money in exchange of a unit of the cons umption good because the price of the unit of consumption has increased, this indicates thatt healefmnyhasl. hi implies that there is an inverse relationship between the price level and the value of money. 3.3.1 Budget constraint Given the set-up, we can easily derive the constraints that anindividualfaceinthefirstandsecond period of her life. This is given by: pt 1,t M =tp y t (2) 10 The left hand side of equation 2 is simply the individual’s total uses of resources which consists of the value of consumption p ct 1,twhen young (price times quantity of goods consumed when young) and the acquisition of money, M (ytu can view this as savings). The right hand side is simply the total resources or the value of the endowment the individual has (wealth). This this is equal to pty (price times the amount of the good). Thus given the value of her endowment, the individual when young can allocate herendowment to buying goods p c t 1,t or acquiring money M ,tt htisvi,htshenuseoaqieoodswnol. The above equation implies that the real demand for money ( pt)byayuggnoiis simply: t M t =( y ▯ c 1,t (3) pt You can view this as savings. That is one way an individu traalsfer her endowment from the first period to the second period and consume it in the second p eriod (when old) is to save part of it. In our simple model, this is achieved via the use of money. In the second period of her life (when she is old), the individual does not receive any endowment. However, when old the individual can now buy goods by using them oyseariihe previous period, when she was young. In the second period of h er life, the money acquired when young will be used in exchange for goods. Thus the budget co trsint that an individual faces when old is given by: pt+1 2,t+1= M t (4) This implies that that value of her consumption in period t+1 must not exceed the amount of money she carried from the pervious period. In other words, the individual when old cannot consume more than her savings. We can combine the budget constraint of the individualow unhgwiyth her budget constraint when old. Doing so, we obtain the consolidated budget constraint or intertemporal budget constraint. To obtain the intertemporal budget constraint,w eubieeuin4noeuain 2fr M tDoio,ea: pt 1,t p t+1 2,t+1= p t (5) Dividing by p tn both sides, we have the budget constraint for a young ▯ pt+1▯ c1,t+ c2,t+1= y (6) pt This budget constraint simply states that the value of consumption when young (in real terms) plus the value of consumption when old (in real terms) must equal tot herlvueofhendwme. Thus in this economy, individuals cannot consume more than their endowment. We have thus everything set up to answer our question: h iatsrhuection of money as a ▯ediu▯ of exchange made people better of? Before we do so, we need to find an expression for pt+1 pt 3.4 Optimal allocation under a stationary equilibrium ▯ ▯ To obtain the expression pt+1 ,wexiheiahti iimheoluylfo fiat pt money must be equal the demand for fiat money, that is the money market must clear. Doing so, 11 we have: M )= p N (y ▯ c ) or 7 t = N (y ▯ c ) ( t t t 1,t pt t 1,t where the left hand side, M ts simply the total nominal money supplied at time t and the right hand side (p N (y ▯ c )) is simply the total amount of money demanded in nominal terms. This t t 1,t consists of the total amount of money demanded by a single young individual, that is p ty ▯ c1,t (see equation 2) times the number of young individuals which is simply N . t If we rearrange equation 7, we have: M t pt= (8) N ty ▯ c1,t and writing this equation one period forward, we have: M t+1 pt+1= N (y ▯ c ) (9) t+1 1,t+1 Thus if we divide equation 9 by equation 8, we have: M t+1 pt+1 Nt+1(y▯c1,t+1 = (10) pt N (y▯c ) t 1,t This is the expression we needed to derive. If we assume that the population is constant, that is Nt= N t+1 = N and that in equilibrium c = c = c ,quin10i lso” 1,t 1,t+1 1 pt+1 = M t+1 (11) pt M t The above equation implies that prices will grow at the same rate as money in equilibrium. To see this more clearly, subtract one on both side. Doing so, we obtain: pt+1 pt+1 ▯ pt M t+1 M t+1▯ M t p ▯ 1= p = M ▯ 1= M (12) t t t t Thus in equilibrium, that is in the long-run the rate at which prices grow is exactly equal to the rate at which money grows. Since inflation is the rate at which prices grow, this implies that the inflation rate in the long-run will depend on the rate at which money grows. This result is essentially a replication of Friedman’s assertion that money is always and everywhere a monetary phenomenon. The above implies that in the long-run,t eqaihyofmnyndheil-i chotomy holds. The classical dichotomy implies that in th ongl-run, nominal variables (money) a▯ect nominal variables (prices) and do not a▯ect real variables (output, employment, etc..). In other words, money is neutral in the long-run. Assuming that the money supply is constant, that is M = t t+1= M,t hiilhthe price level is also constant and equal to one. With pt+1 =1,astmnyul da pt stationary equilibrium (that is markets clear and demand equals supply in all markets), we can rewrite the individual budget constraint as (that is equati on 6) ▯ p c1+ c2▯ y or c 1 c ▯2y (13) p 12 The intertemporal budget constraint when the money supply isc ntadppini constant is thus given by c1+ c2▯ y (14) Note that since we are dealing with stationary equilibrium, we can drop the time subscripts since ▯ in equilibrium c1,t= c 1,t+1= c 1,t+2= ... nire1vii the utility function of each individual, we can easily derivhepiloaii amoey economy. Each individual wants to maximize her utility u(c ,1 )2 ubtotheudtcnaitin by 14. We can set up the individual’s problem as: ) 5 1 maxu(c ,1 )2 ( c1,2 subject to c1+ c 2 y (16) In terms of Figure 2, the utility function of each individual is given by the indi▯erence curves while the budget constraint is given by the straight blue line yyhedalaiiilt when the indi▯erence curve is tangent to the indi▯erence curvd aciih A in Figure 2. At the point of tangency between the indi▯erence curve and the resource constraint, the slope of the indi▯erence curve is equal to the slope of the resource constraint, that is the MRS (slope of the indi▯erence curve) is equal to the ratio of priclpefhebuti)e We can also solve this problem mathematically. The abosv ie ileaoptimization problem that we can be solved in many ways. We can either use the fact that at the tangency point, the MRS t,t+1= pt or solve this problem using a Lagrangean. pt+1 The MRS of consumption when young for consumption when old, denoted by MRS t,t+1is simply given by MU t where MU it the marginal utility of consuming in period t and MU t+1 is MU t+1 the marginal utility of consuming in period t+1. Thus the consumer maximizes utility where: MU 1 pt MRS 1,2= MU = p (17) 2 t+1 If we use the Lagrangean method, denoting ▯ as the Lagrange multiplier, we can set up the problem of the consumer as: ) 8 ▯ = u(c 1,c2)+ ▯(y ▯ c 1 c )2 ( The first order conditions for this simple maximization problem is: ▯▯ ▯c = u 1c1,c2) ▯ ▯ =0 (19) 1 ▯▯ ▯c = u 2c1,c2) ▯ ▯ =0 (20) 2 ▯▯ ▯▯ = y = c 1 c 2 (21) The first condition implies that the marginal utility of consumption in period one, that is the benefit of an additional unit of consumption good, u 1c 1c 2, must equal the shadow price of that consumption good, that is the cost of consuming that extra unit. We have the same expression for consumption of the good in period 2. The third equation is simply the resource constraint. Total demand c 1 c m2st equal to total supply. 13 Note that if we substitute the first order condition for c in1o the first order condition for c , 2 we have: ) 2 2 u 1c 1c 2= u (c2,c1) 2 ( This implies that the marginal utilities of consumption mustbeequalacrossalltheperiods. Ifthey were not, we would try to shift consumption from one period t aonother. As an example, let us assume thatu(c ,c1)=2n c +ln1 .T 2 uheMRS 1,2nthiscaseisgivenbyMU c1 1/c MU 1= 1/c If we assume that p1 =1,thenwehave c2=1or c = c .Snec c + c =
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