Economics 302 - Macroecononomic Theory II Winter 2011 Jean-Paul Lam Lecture 4 Money1 Introduction Monetary economics is concerned with the effects of monetary institutions and policy actions on important economic variables such as inflation, output, interest rates, unemployment, commodity prices, etc. The study of monetary economics as a special field is important as money, although one of many commodities, plays a special role in the economy. Money (fiat money) facilitates transaction between different parties. Moreover, monetary policy and decisions by central banks has an overwhelmingly impact on the economic life of a nation (good or bad). Some of the basic questions we will try to answer in this lecture are: what is the nature and functions of money? why is money important and why do people want to hold money? In this lecture, we will provide a definition of money and show how the existence of money facilitates transactions and reduces search and transaction costs by solving the double coincidence of wants problem. We begin this lecture by defining what money is and what its functions are. 2 Definitions and functions of money What is money? Money is usually a commodity (worthless) that can be used to purchase goods and service or an asset. It is generally accepted as a means of payment. For example, coins, banknotes, cheques written, e-money, plastic cards, travellers cheques and so on. Typically a commodity should possess the following characteristics before being used as money: durable, easily transportable, divisible in small parts, units of standard value and worthless as a commodity. Money serves three main functions: 1. Medium of exchange 2. Medium of account 3. Store of value 3 Money as a Medium of Account Money as a medium of account facilitates transactions by simplifying the calculation of relative prices. Define the relative price (real price) as the ratio at which exchanges can take place for each pair of goods. For example, the relative price of good X in terms of good Y is the number of units of Y that will be exchanged for a unit of good X. 1 there is a strong tendency for the medium of exchange to be used as the only medium of account (or numeraire). For example if J = 100. Money usually serves as the common numeraire or medium of account and thus reduces the complexity of transaction. all exchanges could be based on credit and with multilateral credit and a complete set of markets. For example assume three goods: • J1 = pears • J2 = oranges • J3 = apples Then we need to know only three prices to tell us how to exchange one good for another. if we have J = 100. In our case. 3. the number of relative prices is 100000∗99999 = 499. the value of money is simply the inverse of the price level. .999 commodities in terms of money.999 money prices is all that is needed or useful for transactions (compared to 499. 000..j where j is money. Thus. Note that if everyone in the market could be fully trusted.. A very large 2 number to remember if we don’t have a common medium of account (imagine the supermarket cashier trying to figure out the relative price of a good with such a system). In this case knowledge of the 99. then the relative price of goods J = 1. 000 commodities. Since money is the common medium of exchange in a monetary economy. However. 3. 2. Hence if we have J commodities goods where J = 1. 000. All the other goods can be expressed in terms of that numeraire.If an economy has J distinct commodities.. 2. we can express the prices of each of the other 99. . if we have money as the numeraire and medium of account. j is simply: P1 = P2 = . money would not be needed. • p1 = number of units of pears required to buy 1 apple • p2 = number of units of pears required to buy 1 orange • p3 = number of units of apples required to buy 1 orange This is easy with 3 goods but imagine an economy with thousands or millions of goods. 2 . Not having one common numeraire can make the calculation of relative prices confusing when performing transactions.. therefore (relative) prices are quoted in terms of money.000 in the barter economy). in our case money.. there are J(J−1) different relative prices denoting the 2 ratios at which exchanges can be made for each pair of goods. Pj = = pj =1 pj p1 pj p2 pj (1) (2) (3) (4) Hence pj is the number of units of money that must be given up in order to acquire one unit of a given good.950. In fact.. 950. 1 On the other hand. The literature usually distinguishes between two types of barter: (i) fairground barter where a fair at a specific location. that is when the price level increases. The use of money as a medium of exchange implies a smaller amount of resources (in terms of search) and hence a lower social cost. This is because in a monetary economy. that is the opportunity costs of holding money is the interest rate that someone forgoes on a bond for example. It does not pay any interest. as a store of value. that have a small amount of goods and services. For barter to operate. she can acquire money and hold it. There are many other assets or commodities that serves as a store of value (bonds. and its value is quickly eroded by increases in the price level. 1 3 . These store of value are typically better than money. is a way to transfer purchasing power from the present to the future. in a monetary economy. 3. To understand the medium of exchange function. trade arrangements in a barter economy can be very inefficient as individuals are forced to spend a large amount of their time and resources in the activity of searching for suitable partners for trade. this trade arrangement does not present many challenges and problems. 4 Money as Medium of exchange The medium of exchange function of money is the most important function of money. the split into separate transactions not only imply the separation of wants but also the separation of transactions in time. if the price level is rising very rapidly (hyperinflation). Put simply. money is required as a means of payment. we do not care whether money takes the form of a commodity or simply is fiat money. Money is a store of value since it enables individuals to transfer wealth across different genrations. This is why individuals usually hold less money when the price level is rising. time and date is held for the exchange of goods and (ii) trading post barter where an individual would set up a trading post where goods and services could be exchanged. In extreme cases. If somebody wishes to save part of its current income for use at a future date. At this stage. stocks. For economies. its value falls. for large economies. I must find somebody who not only has a Blackberry but also wants an ipod in exchange). uncertainty and the unwillingness of traders to extend credit. In a monetary economy. Money is a poor store of value for many reasons. Moreover. double coincidence of wants must prevail (if I want an ipod and I have a Blackberry in exchange. consider two possible economies: a barter economy where individuals exchange goods (or commodities) for goods (or commodities) 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. money can cease to be a store of value.1 Money as a Store of Value Money. the single transaction of a barter economy can be decomposed into several transactions of sale and purchases. money facilitates transactions since individuals can use money to perform transactions without engaging in a lengthy and costly search process. the need for the double coincidence of wants is eliminated. commodities). However. Clower (1971) has labelled this as a liquidity or cashin-advance constraint. the opportunity cost of a system of exchange by barter is very high.because of asymmetric information. If we again assume that there is a search cost µ. I need to meet someone who not only has what I want but also needs what I have. We start with a very simple example and assume that J = 3. then in a barter economy. the need for double coincidence of wants must prevail. good a can be exchanged for good b and c. Hence with money. Hence in this case. On the other hand. that is there are three commodities: (a. If J is a large number. If there are J = 3 goods. if J = 1000000. If we assume that there is a search cost µ. we can calculate the probability on any given attempt that someone is going to make a successful exchange under the barter economy. Hence. in a monetary economy. the match is simply (100000021 −1000000) = 1. In other words. the probability of finding the right −12 . then this search cost under a monetary economy is simply µJ and this is less than the search cost under a barter economy which was given by µ(J 2 − J) when J > 2. it takes on average J searches to find a successful trade. indeed a very small number. In Table 1 below x represents permissible exchanges while o represents non-permissible exchanges Table 1: Exchanges in a a b a o x b x o c x x Barter Economy c x x o For example.As an example consider the following two hypothetical economies J different types of goods. This probability is unambiguously less than the probability of success under the barter economy. then this search cost under a barter economy is simply µ(J 2 − J). that is money can be used (good a) to buy good b and/or good c. the probability of finding a match where the double coincidence of want is satisfied is simply 1 (J 2 −J) . then the probability of making a successful exchange becomes very small. it is clear that the number of possible exchanges is given by J − 1 = 2. It is clear from table 1. we can approximate the number of possible to be simply J. trade occurs by exchanging goods. It is to be noted that the saving in time and energy provided the medium of exchange—money— 4 . The medium of exchange function is generally regarded as the distinguishing function of money. In the barter economy. Given this information. since goods cannot be exchanged for other goods and assuming no transaction costs and good a is money. we have the following: Table 2: Exchanges in Ca Ca o o Cb Cc o a Monetary Economy Cb Cc x x o o o o We will assume for now that the money we are talking about is imply fiat money. the inverse of the probability of success on any single try. the probability of finding what you need in 1 exchange for money will be simply J . b.0e average number of attempts before finding a double coincidence of wants is J 2 − J. that the number of possible exchanges with 3 goods is simply J 2 −J = 6. If J is large. Recall that under the barter system. as we should expect in a modern economy. For example. c). 3. In period t. Thus in period 1. 5 . The second feature of money as a medium of exchange is that is facilitates transactions and in some cases make transactions possible. the good is perishable and thus cannot be stored from one period to another. To illustrate this point. each individual receives an endowment of the consumption good in the first period of life and this amount is denoted by y. Hence in each period t. You can also interpret this endowment as labour. we assume that there is only one good in this economy and the good is homogeneous. there will be N2 young individuals and N1 old individuals. The economy begins in period 1 with N0 members of the initial old. are called the future generations of the economy.does not depend on whether that commodity has value or not. Each individual lives for two periods and individuals are endowed with y when young and 0 units when old. For simplicity.. The pattern of endowments is illustrated in Figure 1.1 2 Preferences One of the strengths of the OLG model is that it is highly tractable and easy to use. whereas in the second period. generation t is born. they become old and die at the end of that period. For example in period 2. Individuals when old.. Table 3: Pattern of endowments in the OLG model Period 1 Period 2 Period 3 Period 4 Period 5 Period 6 0 y 0 y 0 y 0 y 0 y 0 y Period 7 Generation Generation Generation Generation Generation Generation Generation 0 1 2 3 4 5 6 0 5.. N1 individuals are born.. In period 1. and so on. In any given period. Nt individuals are born.2 The economy begins in period 1.2. The individual uses this labour endowment and earn income y. The only requirement is that it is acceptable as a medium of exchange. where time is index by the subscript t. N2 individuals are born. as a medium of exchange. individuals live for two periods. they are born and are described as being young. in period 2. 5 An Overlapping Generations Model In the OLG model. that is when they are in their second period of their life receives no endowment. In the basic set-up. The individuals born in period 1. More importantly. The overlapping generations model will show how the existence of money. We refer to those in the second period of their life as old. improves welfare and facilitates trade. there are Nt young individuals and Nt−1 old individuals. we will use an overlapping generations model (OLG). In each period t ≥ 1. we first begin by laying out the assumptions of our model. there is one generation of young people and one generation of old people. that is those who were born in period 1 and who are now old in period 2. • The individual is able to rank consumption bundles over time in order of preference. Suppose we reduce the second period consumption (c2 ) of the individual by 2 units. In other words. 5. we have to compensate the individual by increasing the first period consumption by the 2 units also. • The individual has diminishing marginal rate of substitution. 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. they need to acquire goods in the second period (remember they don’t have any endowment in the second period) and must find a way to do so. Thus given two bundle of goods.2 Optimal consumption choice Before we solve for the optimal choice of the consumer. that is the consumption of the old generation born at time t. To illustrate this point. The indifference curve tells us that to keep the individual’s utility constant. This moves the individual to point B.t+1 represents consumption that occurs in t+1 period by individuals born at time t. the individual prefers to smooth consumption over time and does not want to consume all her endowment in one period only. This implies that individuals have to be compensated more to part with or are less willing to give up a consumption good if they have little of them in the first place. The problem facing the future generations of this economy is very simple. we introduce some notation. we will denote the first and second period consumption of an individual as c1 and c2 respectively. they need to consume a positive amount of goods in both periods. 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).Members of future generations in the OLG model consume both when young and when old. bundles A and B. Similarly c2. Now suppose we reduce the second period consumption of the individual by another 2 units. They want to maximize their utility subject to their resource constraint and to maximize utility. An individual’s utility therefore will depend of how much she consumes when young and when old. We will examine this problem from two perspectives. Note that c2. suppose we start at point A. It is clear that there are diminishing marginal rate of substitution. 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. Hence these assumptions imply that the individual will have indifference curves that are convex to the origin. A typical indifference 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. Therefore.t . The first will be from a central planner’s view where a benevolent social planner will maximize the utility of future generations on their 6 . Given that there is no storage technology.t+1 denotes the amount of goods the same individual consumes in the second period of her life. then the question is how can the future generations acquire goods in the second period. We denote the amount of the good that is consumed by an individual in her first period of life by c1. When the time period is not important (as in steady-state or equilibrium). the consumer can say whether she prefers bundle A over bundle B or vice-versa or is indifferent between the two. t ≤ N y Dividing by N on both sides.t ≤ Nt y (9) For simplicity. Total consumption at time t is thus the sum of the consumption of young and old at time t. we have: Nt c1. The existence of a medium of exchange. that is the central planner treats the young persons as identical and the old persons as identical.3 Optimal allocation under the central planner Imagine for a moment that the central planner has complete knowledge of and total control of the economy. Total amount of consumption good = Nt y (5) Suppose that for the purpose of equity.t (8) This represents the total demand for consumption at time t.t + c2. He will do so by allocating the economy’s resource between the young and the old. The second solution. therefore given that the supply of good at time t is simply equation 5.t + N c2. Recall that Nt of young people are born or endowed with y. Since in equilibrium demand must equal to supply. recall that the number of old people at time t are those who were born at time t-1. In this case. their utility is maximized.t + Nt−1 c2.t ≤ y (11) (10) 7 .behalf. Therefore the total consumption of old people at time t is given by equation 7. we have: c1.t + Nt−1 c2. We begin with the central planner.t . each member of generation t is given the same amount of consumption. we assume that the population is constant (or alternatively there is one person of each generation at each point in time). thus Nt = N . We then compare the two solutions and examine which one offers the consumer the highest utility. that is money will permit trades to take place. a decentralized or competitive solution will allow individuals to trade with each other thereby acquiring goods when old. Thus the total amount of resources in the economy (or consumption good) at any point in time is simply the number of young people times their endowment. total consumption by the young generation in period t is: Total young consumption = Nt c1. we can rewrite equation 9 as: N c1. This is given by: Total consumption = Nt c1. The objective of the central planner is to allocate goods among the young and old such that at each point in time. 5.t and the total consumption by the old generation in period t is: Total old consumption = Nt−1 c2.t (7) (6) To make the notation clear again. There are Nt−1 of them and the consumption of each old person in period t is given by c2. In this case. The third equation is simply the resource constraint.. c2 ). Note that if we substitute the first order condition for c1 into the first order condition for c2 . Thus using the first order conditions we have derived. She has to maximize the utility of future generations on their behalf subject to a resource constraint.. . We have the same expression for consumption of the good in period 2. u1 (c1 . c2 ) − λ = 0 (16) ∂c1 ∂ℓ = u2 (c1 . t = 1. c2 ) c1 . A stationary allocation is one that gives members of the same generation the same consumption pattern over time. we have: u1 (c1 . 3. ℓ = u(c1 . Note that any allocation on or below the yy line is a feasible set of allocation.We are concerned with stationary or equilibrium allocations.c2 (13) (14) subject to c1 + c2 ≤ y Thus we can set up this problem using a simple Lagrangean and denoting λ as the Lagrange multiplier. that is the benefit of an additional unit of consumption good. c2 ) (19) This implies that the marginal utilities of consumption must be equal across all the periods. More formally.. For example if u(c1 .t = c2 for every period. Thus with a stationary allocation. In other words. Total demand c1 + c2 must equal to total supply. Note that the stationary allocation does not imply that c1 = c2 . must equal the shadow price of that consumption good .t = c1 and c2. 2. c2 ) = ln c1 + β ln c2 . The point A is not feasible since it is unattainable given the resources of the economy. c1. under a stationary allocation.4 The golden rule allocation The objective of the central planner is simple. 5. we can rewrite equation 11 as: c1 + c2 ≤ y (12) The budget or resource constraint is shown in Figure 2. c2 ) − λ = 0 (17) ∂c2 ∂ℓ = c1 + c2 − y = 0 (18) ∂λ The first condition implies that the marginal utility of consumption in period one. c2 ) + λ (c1 + c2 − y) (15) The first order conditions for this simple maximization problem is: ∂ℓ = u1 (c1 . we have: 1 ∂ℓ = −λ = 0 (20) ∂c1 c1 β ∂ℓ = −λ = 0 (21) ∂c2 c2 ∂ℓ = c1 + c2 − y = 0 (22) ∂λ 8 . c2 ) = u2 (c1 . that is the cost of consuming that extra unit. we have: max u(c1 . Point D is preferable to point A. but it is unattainable given the existing resource constraint. To achieve the 3 Try the same example with u(c1 . it does not maximize the welfare or utility of the initial old. This would be represented by point E in Figure 3. Note that although the golden rule allocation maximizes the welfare of all individuals. that is where the slope of the indifference curve is equal to the slope of the resource constraint.Using the first order conditions for c1 and c2 . c∗ ). This stationary allocation where all the goods are allocated to the initial old and none to the young would of course not maximize the utility of future generations since they prefer the more balanced allocation bundle (c∗ . and influenced by the fact that there is one generation of initial old and an infinite number of future generations. we have: 1 β = c1 c2 or βc1 = c2 Using the above in the resource constraint. In our example. Nevertheless. For example points B and C are feasible but they are on a lower indifference curve. on subjective grounds. If the central planner’s goal were to maximize the utility of the initial old . β 1+β 6 Decentralized economy In the previous section. To see this. The golden rule is achieved at point A which offers the individual the consumption bundle (c∗ . we found that the feasible and optimal bundle of consumption goods that a central planner would choose to allocate between the young and old generation. the golden rule of allocation is thus c∗ = 1+β y. c∗ ). The golden rule allocation is shown in Figure 3. Faced with this conflict between the initial old and the future 1 2 generations. she would give as much of the consumption as possible to the initial old. This combination of (c∗ . 1−θ 2 9 . we have: c∗ = 1 1 y. c∗ ) yields the highest feasible set of utility given the resource constraint 1 2 and the preferences of the individual. Note that this is achieved at the point of 1 2 tangency between the indifference curve and the resource constraint. c∗ = 2 1 y. recall that the initial old’s utility depends solely and directly on the amount of the good they consume in their second period (and last period) of their life. c∗ = 2 1+β β y 1+β (25) (24) (23) Thus the social planner maximizes the utility of the future generations by allocating consumption according to the optimal allocation we found above3 The point at which consumption is maximized 1 is known as the golden rule. c2 ) = 1 c1−θ 1−θ 1 + 1 c1−θ . an economist cannot choose between them on objective ground. we tend to pay more attention to the golden rule allocation and think that this allocation is more realistic and fair. Note that so far we have not said anything about the existence or role of money. This would allocate y units (all of the units) of consumption goods to the initial old and none to the young. In our example and model. For example. 2 1 These are strong assumptions about the power and wisdom of the central planner. The old generation would like to have what the young generation have. In our simple model. a young knows that if she gives part of her endowment to the old person at time t. All markets clear. The resulting equilibrium is autarkic. the central planner must know the exact utility function of the individuals. In other words. In our economy. This implies that the actions of a single individual cannot influence prices and wages. Trading with the old generation does not make sense also. From Table 1. in order to determine c∗ and c∗ . there are Nt young people who are born with an endowment and Nt−1 old people who do not have any endowment. 2. can we achieve this result? The answer to this question is clearly no in an economy without money. each old generation wants what the young generation has but does not have what the young generation wants.βy optimal allocation. This lack of possible trade is the manner in which the OLG model captures the absence of double coincidence of wants. the old generation will disappear or die. Without the central planner. Each individual maximizes utility given her budget or resource constraint. Unable to make mutually beneficial trade and unable to carry 10 . 6. No such trades are in fact possible without money. Recall that at time t. this implies that the supply of goods equals the demand for goods and the supply of money equals the demand for money. then she will never see the endowment again since in period t+1. we first define what we mean by competitive equilibrium. the central planner has to take away c∗ = 1+β from each young person and 2 give this amount to each old person. The utility of these old 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. we can see that a young person in period t can potentially trade with two types of persons: other young people of the same generation and old people of the previous generation (people born in period t-1 ). that is individuals have no economic interactions with each other such that no trade occurs. they will do so by engaging in mutually beneficial trades with other individuals. A competitive equilibrium has the following properties: 1. that is one in which the economy reaches the optimal allocation through mutually beneficial trades conducted by the individuals themselves. It is clear that trade with fellow young people would not be mutually beneficial since this would involve only swapping the same good and they like him have none of the consumption good when old.1 Equilibrium without money Let us first consider the nature of a competitive equilibrium in an economy that do not use money or where money does not exist. This leads us to ask if there is some way where we can achieve this optimal allocation in a more decentralized manner. but they do not have what the young would like since they will not be around the following period. Each individual is a price taker and also a wage taker. Furthermore. Such a redistribution requires that the central planner have the ability to reallocate endowments costlessly between the generations. that is a positive endowment. can we let the market do the work instead of the central planner? That is can the market achieve a (competitive) equilibrium that is as good as the central planner? Before we answer this question. 3. For fiat money to have value. its worth will depend on one’s view of its value in the future. If individuals believe that money will be accepted for the exchange of goods in the future and its value will not change. Moreover. if it is believed that money will have no value in the future. each individual therefore consumes her entire endowment when young and nothing when old. The presence of money opens up trading possibilities. We begin our analysis of the monetary economy with an economy with a fixed stock of m perfectly divisible units of fiat money. However. Hence. trade can take place and this would be Pareto improving.2 Equilibrium with money To open up a trading opportunity that can allow individuals to move away from the autarkic equilibrium. Because individuals derive no utility from holding money. We assume that each of the initial old begins with an equal number m of these units. Hence money will have no value today. money is valuable only if it enables individuals to trade for something they want to consume.4 6. 6. As a result. This is discussed in the next section. if people are willing to give up some of the consumption good in exchange for fiat money and vice versa. we mean that the fiat money can be traded for some of the consumption good. 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. that is money. by permitting trade. then individuals will simply choose not to hold money today. money can be costlessly stored from one period to another and it is costless to exchange. in this autarkic equilibrium. Thus the existence of money as a medium of exchange. 4 11 . Both the future generations and the initial generation is worse off in this case compared to the central planner’s case. the existence of trade would make each generation better off. A monetary equilibrium is a competitive equilibrium in which there is a valued supply of fiat money. if individuals An allocation is Pareto improving if at the new allocation. we assume that a central bank can produce fiat money costlessly and no one else can produce (or even counterfeit) it. in other words. everybody can be made better off without making someone else worse off. By valued. by introducing a commodity. Fiat money is a worthless commodity that is nearly costless to produce and that cannot itself be used in consumption or in production.3 The demand for fiat money Trade will take place only if money is valued. In other words. For an exchange to take place. As a result no trading will take place. people must believe that money will have a value in the future in the sense that it can be exchanged for goods. then they will be willing to hold money today. utility is low. A young person N can now sell part of her endowment of goods to an old person in exchange for money. Because the value of fiat money is intrinsically useless.her endowment in the next period (the good is perishable and cannot be stored). will be Pareto improving. For the purpose of our example. we now introduce fiat money in our simple economy. its supply must be limited (remember what happens to inflation and hence the value of money when you have too much money in circulation) and it must be impossible or very costly to counterfeit. hold that money until the next period and then trade it for goods when they are old with a member of the young generation. Assuming money is valued.t + pt+1 c2.t + pt+1 c2. This implies that there is an inverse relationship between the price level and the value of money. Similarly. we can now characterize the monetary equilibrium. The price level denotes how many units of fiat money an individual has to exchange for one unit of the consumption good.t and the acquisition of money. By similar reasoning. Given the set of prices the budget constraint of an individual is given by: pt c1. this is achieved via the use of money. period T. this implies that the value of money has fallen. That is one way an individual can transfer her endowment from the first period to the second period and consume it in the second period (when old) is to save part it. To see this consider. In the second period of her life. when old the individual can now buy goods by using the money she acquired in the previous period or when young (that is she uses her savings). we can move back in time and show that money will have no value today if it has no value at time T.know that at any future date T money will have no value. The above equation implies that the real demand for money by a young generation is simply: mt = (y − c1. If money is worthless in period T. we have: pt c1. In the second period of her life. Hence her budget constraint is: pt+1 c2. then individuals in period T-1 will not be willing to hold any fiat money. The right hand side is simply the total sources of resources or the value of the endowment the individual has and this is equal to pt y. Denote the number of dollars acquired by an individual (by giving up some consumption goods) at time t by mt .t ) pt (27) You can view this as savings.t+1 ≤ pt y Dividing by pt on both sides. we need to find an expression for pt+1 pt 12 . we have the budget constraint for a young c1. the individual has an endowment y of goods. mt . the individual does not receive any endowment. However. Recall that in her first period. The individual can do two things: she can consume them completely or sell them in exchange for money.t + mt ≤ pt y (26) The left hand side of equation 26 is simply the individual’s total uses of resources which consists of the value of consumption pt c1.t+1 ≤ mt or mt ≥ pt+1 c2.t+1 ≤ y pt (30) (29) (28) We have thus everything set up to answer our question: has the introduction of money made people better of? Before we do so.t+1 worth of consumption goods. the money acquired when young will purchase pt+1 c2. In our simple model. individuals in period T-2 will also not want to hold and demand money since they know it is going to be worthless in period T-1. Denote the price of the consumption good at time t by pt . Thus if more fiat money is needed in exchange of a unit of the consumption good.t+1 Substituting equation 28 into equation 26 for mt . Money is neutral in the long-run since it affects prices only and does not affect any real variable. c2 ) c1 . We can set up the individual’s problem as: max u(c1 . we have: pt+1 = pt mt+1 Nt+1 (y−c1. Hence the competitive monetary equilibrium will be the same as the planner’s solution and the optimal allocation in this case also 13 . that is mt = mt+1 .t+1 ) (33) Thus if we divide equation 33 by equation 32.c2 (37) (38) subject to c1 + c2 ≤ y This is exactly the same problem as in the social planner’s case. we have: mt = pt Nt (y − c1. this implies that the quantity theory holds in the long-run and the classical dichotomy applies to this model. that is the money market must clear.t ) (see equation 26) times the number of young individuals which is simply Nt . this implies that the price level is also constant and equal to one. we have: pt+1 = mt+1 Nt+1 (y − c1. we can rewrite the individual budget constraint as (that is equation 30) c1 + p c2 ≤ y or c1 + c2 ≤ y p (36) The objective of the individual is thus simple.t+1 ) mt Nt (y−c1.t ) (34) If we assume that the population is constant. that is Nt = N for all t and c1. The individual will want to maximize her utility u(c1 . that is pt (y − c1. we have: pt = mt Nt (y − c1. c2 ) subject to the budget constraint given by 36.t ) or mt = Nt (y − c1. This consists of the total amount of money demanded by a single young individual. If we rearrange equation 31. Assuming that the money supply is constant. we have: mt+1 pt+1 = (35) pt mt The above equation implies that prices will grow at the same rate as money in equilibrium.t ) (32) and writing this equation one period forward. mt is simply the total nominal money supplied and the right hand side is simply the total amount of money demanded (nominal).t+1 = c1 .6. In other words.t ) pt (31) where the left hand side.4 Optimal allocation under a stationary equilibrium and constant population Since the total supply of fiat money must be equal the demand for fiat money.t = c1. With pt+1 = 1 with a constant money supply and a stationary pt equilibrium. that is a monetary standard. In equilibrium trade occurs. that is the young generation would consume their entire endowment when young and nothing when old. Real money demand in that case is simply: 2 1 md y βy t = [y − c∗ ] = y − = 1 pt 1+β 1+β (39) Recall that in the absence of money. Most monetary system (if not all) are fiat money system nowadays. unlike the monetary standard. E-money is already gaining importance around the Try the case when the population is not constant. there have been two types of monetary standards: • Commodity standard in which a commodity preferably valuable and in limited in supply acts as a basis for a monetary system. As shown. c∗ = 1+β . carries no promise whatsoever of being backed by a commodity such as gold or silver presently or in the future. With technology. In such a system. Young generations are willing to give up part of their endowment and acquire money which they can use to acquire goods when old. Where money exists. it is not surprising that most economies have adopted some form of money. In the future. we may see another type of monetary standard which is e-money. the existence of money and trade makes everybody better. Any individual could technically exchange or redeem its unit of currency for a given piece of gold. For example gold during the Gold standard acted as the basis for the monetary system. 5 14 . The optimal solution is no longer autarkic. Moreover. that is where all payments are done electronically without the exchange of fiat money as we know it today. c2 ) = ln c1 + β ln c2 . In terms of a diagram. It is clear that the introduction of money has made everybody better off. The fiat money system. however. Allocations that were not attainable before are now possible with money. there are certain rules and regulations that govern its circulation. the optimal allocation was autarkic. This allocation was inferior compared to the social planner’s allocation. society may become a cashless economy. Put simply. That is c∗ = 1+β . a country’s currency (minted coins and paper currency) was backed by the amount of reserves of gold. the solution which was depicted in Figure 3 for the planner’s solution applies to the monetary competitive solution also. that is the golden rule allocation is achieved. throughout history. the introduction of money is Pareto improving. that is the golden rule allocation. The value of money or currency depended on the price of gold. opened the possibility of trade between generations. Broadly speaking. the competitive monetary equilibrium is exactly the same as the social planner’s solution.5 7 Monetary standards Given the inefficiency of the barter system. • Fiat money system (paper money standard or token coins) where the circulating medium of exchange is made of material that has extremely low costs of production and therefore having no commodity value whatsoever (or a very low one).is simply the golden rule. Thus if we have u(c1 . that is Nt+1 = nNt . we get exactly the same solution as in the planner’s βy y case. The introduction of money. Is the social planner solution the same as the competitive non-monetary and monetary solution? Does money improves welfare again?. and S. it is unlikely that fiat money and banks will disappear in the future. Finally. Moreover. Freeman. how would the central bank exert a control on something there is no demand for? Many economists such as Charles Goodhart argue that this is unlikely to happen since money offers anonymity in transactions. Money as a medium of exchange facilitates transactions and reduces the cost of transactions. Modeling Monetary Economies. he argues that people because of this anonymity feature of money and the fact that banks play unique and important roles in our modern society. B. money is also a store of value. we may have other private institutions that can act as custodian of wealth that do not issue any money or credit and do not hold account at central banks. For example. Cambridge Press. This debate remains open among central bankers. 8 Conclusions We presented int his lecture the various functions of money. For example. 9 References Champ. again making transactions simpler. One of the main issues regarding the advent of e-money is whether or not central banks will lose their monopoly power over the issuance of money and hence their ability to conduct monetary policy. In this case. 2001. Second Edition. money as a a unit of account simplifies the price system. enabling individuals to transfer wealth from one period to the other. However. something that e-money does not.world. what would happen to monetary policy as we know it. the latter will be able to conduct monetary policy as we know it and influence interest rate. they can exert control over the supply of money and hence implements monetary policy by controlling its supply (for the sake of argument. the growing importance of e-money can fundamentally alter and undermine the very foundations of monetary policy. we will assume for now that central banks conducts monetary policy by controlling the money supply). Since central banks are the sole issuers of money (base money). As long as individuals demand fiat money (or settlement balances) that are under the direct control of central banks. if there is a private issuer of money and if individuals demand the money issued by the private issuer rather than settlement balances issued by the central bank. On the other hand. [CF] 15 . instead of having private banks that have deposits at central banks. Figure 1: Diminishing Marginal Rate of Substitution Figure 2: Feasible Allocations 16 . Figure 3: Optimal Allocations 17 .