Money, Interest and Prices. Money in General Equilibrium Models, the Price Level, Interest Rates and Inflation

Transcription

1 Money, Interest and Prices Money in General Equilibrium Models, the Price Level, Interest Rates and Inflation Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015

2 Money, the Price Level and Nominal Variables Money, the existence of which in a modern economy is usually taken for granted, performs three main funcjons. First, it is a unit of account, second it is a universally accepted means of payment, and thirdly, it is a store of value. While in models without money one can only analyze the determinajon of real variables, such as the quanjjes of goods and services produced and consumed, and their rela9ve prices, in models with money one can also determine nominal variables such as the price level, nominal income, the level of nominal wages, nominal interest rates and infla9on. These nominal variables are expressed in terms of money. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

3 Money as a Unit of Account In a monetary economy all prices are determined and quoted in terms of the monetary unit. Otherwise, economic agents would have to calculate all the relajve prices of goods and services in order to conduct their transacjons. In an economy with N goods plus money, there are N money prices. Without money, economic agents would need to calculate N(N+1)/2 relajve prices in order to make their transacjons. As the number of goods and services increases, the number of relajve prices to be calculated grows exponenjally. For example, if there are 5 goods and services, there are five money prices, and 15 relajve prices of goods between them. With 1000 goods and services, there are 1000 money prices, and 500,500 relajve prices between goods and prices. Money therefore helps to simplify the calculajon of prices and values, and thus facilitates economic transacjons through its unit of account funcjon. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

4 Money as a Means of Payment Being accepted by all, money greatly facilitates economic transacjons and drasjcally reduces their costs. Without money, in order to complete a transacjon the seller of a product or service would have to find a buyer who would be prepared to offer in return another good or service that the seller wishes to acquire. This requires that there is a double coincidence of wants in all economic transacjons. TransacJons or this kind are called barter, which implies huge costs on the part of economic agents in order to find suitable counter-parjes to their transacjons. A modern economy would immediately cease funcjoning if there was not a generally accepted medium of exchange and payments, because transacjon costs would become prohibijve. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

5 Money as a Store of Value Money is a store of value, i.e. a means of holding wealth, and is indeed the asset that is characterized by greater liquidity, as it can be used directly for payments for the acquisijon of goods and services. This is a key feature of money, because if money were not a store of value, and lost its value quickly, it would not be generally accepted as a means of payments either. Then again, since money is the only store of value which is also a means of payments, by definijon it is the most liquid store of value. However, as a means of holding wealth, money has the weakness that it does not pay interest, unlike other less liquid assets. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

6 Defining the Money Supply We define as money the sum of banknotes, coins and deposits in current accounts in commercial banks held by households and firms. This definijon of money supply is usually known as M1. It emphasizes the more liquid assets of households and firms, which usually do not yield interest. However, there are broader definijons of the money supply, that include less liquid assets such as Jme deposits and other less liquid deposits and securijes (M2, M3 etc). Deposits of credit insjtujons and other insjtujons parjcipajng in the interbank market and the foreign exchange market are not considered as part of the money supply. These deposits are not used for the transacjons of the general public. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

7 The Supply of Money Monetary condijons in modern economies are determined by central banks. The central bank may affect, through a variety of policy instruments at its disposal, both the quanjty of bank notes (and coins) in circulajon, and, indirectly, the amount of deposits in commercial banks, which are also part of the money supply. AlternaJvely, a central bank may follow an interest rate rule, intervening in the money market and pegging nominal interest rates. In this la[er case, the stock of money in the economy is determined by the demand for money, and the money supply adapts to demand in order to achieve the goal of the central bank regarding the nominal interest rate. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

8 The Roles of Central Banks Central banks are public insjtujons that manage a state s money supply, interest rates and regulate the commercial banking system. In most countries the central bank possesses a monopoly on prin9ng notes, and min9ng coins, which serve as the state s legal tender. In addijon, central banks usually act as lenders of last resort to the banking system and, in many cases, the government. Central banks can thus directly determine the circulajon of notes (and coins) and indirectly the amount of deposits in commercial banks. The determinajon of the money supply by central banks is not a simple process. It depends on the rules under which the central bank parjcipates in money and asset markets and regulates the financial system, on its relajons with the government, and on the goals envisaged in its charter. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

9 The Goals and Policy Instruments of Central Banks The main goals of a central bank are the control of infla9on, the stability of the financial system, and in some cases, the support of the general economic policies of the government. In what follows we shall ignore many of the insjtujonal details that relate to how a central bank operates, and will make two alternajve simple assumpjons. First, we shall assume that the central bank has full control of the money supply. This is an assumpjon with a long history in macroeconomic analysis, although not parjcularly realisjc, as central banks have imperfect control over the money supply. AlternaJvely we shall assume that the central bank follows a policy of pegging the nominal interest rate and commi^ng to providing unlimited credit to households, businesses and commercial banks at this rate. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

10 The Demand for Money The demand for money depends on three main factors. The first factor is the price level. The higher the level of prices, the higher will be the amount of money that households and firms would want to hold for their current and future transacjons. The demand for money is usually assumed to be proporjonal to the price level. The second factor is the volume of transacjons. When the volume of transacjons, usually measured by aggregate real output, increases, households and firms will need more money to carry out their increased transacjons. The third factor is the level of interest rates. Banknotes pay no interest. On the other hand, demand deposits and current accounts, even when they pay interest, pay a very low rate compared to the yields of less liquid assets such as Jme deposits, treasury bills or bonds. Consequently, the demand for money will depend negajvely on the nominal interest rate, as the nominal interest rate measures the opportunity cost of holding money. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

11 The Money Demand FuncJon M d = P m(y,i) where M d is the quanjty of money demanded, P is the price level, Y real aggregate output and income (GDP) and i the nominal interest rate. m is a funcjon increasing in real aggregate income and decreasing in the nominal interest rate. The demand for money is proporjonal to the price level, in the sense that an increase in the price level requires an increase in the quanjty of money by the same proporjon, in order to complete the same number of transacjons. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

12 The Demand for Real Money Balances M d P = m(y,i) Since the demand for money is proporjonal to the price level, it can be expressed as a demand for real money balances. Holding money is useful for its purchasing power. The relajonship between real money demand and the nominal interest rate is negajve, because holding money becomes more expensive as interest rates rise, since money does not pay interest. Therefore, households and firms reduce the amount of money holdings and increase holdings of securijes and other interest yielding assets. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

13 The Demand for Real Money Balances, the Nominal Interest Rate and Real Income Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

14 Short Run Equilibrium in the Money Market M s P = M d P = m(y,i) The equilibrium condijon in the money market is for the money supply to be equal to money demand by households and firms. In the short run, with prices and income given, how the money market equilibrates depends on the modus operandi of the central bank. If the central bank fixes the money supply, the equilibrajng mechanism is the nominal interest rate. If the central bank fixes the nominal interest rate, the equilibrajng mechanism is the money supply. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

15 Short Run Equilibrium in the Money Market when the Central Bank Fixes the Money Supply Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

16 An Increase in the Money Supply and Nominal Interest Rates: The Liquidity Effect Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

17 An Increase in Money Demand and Nominal Interest Rates: The Liquidity Effect Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

18 Short Run Equilibrium in the Money Market when the Central Bank Fixes the Nominal Interest Rate Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

19 Long Run Effects of the Money Supply In the longer term, the price level also adjusts. We can see the direcjon of this adjustment by rearranging the equilibrium condijon in the money market, and solving for the price level. We then get, P = M s m(y,i) In long run equilibrium, aggregate real income and the real interest rate are on their balanced growth paths. With constant inflajon, nominal interest rates are also constant. Thus, the factors affecjng the demand for money are given, and the money supply determines the price level, without affecjng the evolujon of real variables. This property is called long-run neutrality of money. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

20 The Neutrality of Money in StaJc General Equilibrium Models The neutrality of money applies to all stajc general economic equilibrium models with flexible prices. What determines the level of equilibrium real income, and other real variables are the available resources, technology, preferences, the funcjoning of markets, as well as economic insjtujons that determine total factor producjvity and the producjvity of specific factors. In stajc general equilibrium models real output and income and other real variables do not depend on the money supply. Money is merely a veil which covers the economy, simply determining nominal variables such as the price level. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

21 The Neutrality and Super-neutrality of Money in Dynamic General Equilibrium Models In dynamic general equilibrium models we usually disjnguish between the neutrality and the super-neutrality of money. The neutrality of money refers to the effects of a one off change in the money supply, and the super-neutrality of money to the effects of the rate of change of the money supply. The neutrality of money applies to all dynamic general economic equilibrium models with flexible prices. However, as the growth rate of money supply affects inflajon and long-term nominal interest rates, it thus affects real money demand. The super-neutrality of money holds in representajve household models, as money does not affect other real variables, but does not hold in overlapping generajons models. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

22 Long Run Neutrality of Money and Monetary Reforms An alternajve way to think about the neutrality of money, is to consider what would be the impact of a very radical change in the money supply. Such radical changes take place in Jmes of monetary reforms. A number of such historical examples exist, which suggest that, ader a monetary reform, the price level adjusts immediately to the new monetary standard. For example, in May 1954, there was a radical monetary reform in Greece. A new drachma was created, which amounted to 1,000 old drachmas. EssenJally this amounted to a direct reducjon in the money supply to one thousandth of the old money supply. As one would expect on the basis of our money demand funcjon, the price level in Greece fell immediately to one thousandth of the price level before the reform. Nothing else changed, other than the level of prices. Gradual increases in the money supply in the long run have effects similar to such monetary reforms. The tripling of the money supply in a decade, in the long run has the same effect as a monetary reform in which a currency unit is replaced with three units of a new currency. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

23 Dynamic General Equilibrium Models with Money The Samuelson (1958) overlapping generajons model. RepresentaJve Household Models with Money in the UJlity FuncJon (PaJnkin 1956, Sidrauski 1967) Overlapping GeneraJons Models with Money in the UJlity FuncJon (Weil 1987). Cash-in-Advance RepresentaJve Household Models (Clower 1967, Grandmont and Younes 1972, Lucas 1980, 1982, Svensson 1985). Cash-in-Advance Models of Overlapping GeneraJons (Samuelson- Lucas). Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

24 The Samuelson Overlapping GeneraJons Model: Money as a Store of Value We assume that the economy consists of successive generajons of households, each of which lives for two periods. Every household has exogenous income y 1 in the first period of life and y 2 in the second period of life. This income is in the form of a non storable good, which cannot be transferred from period to period. The only non-perishable commodity is money, which can be used as a means of holding wealth. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

25 The Inter-temporal OpJmizaJon Problem of Households Τhe household born in period t maximizes the ujlity funcjon, U t = u(c 1t ) + βu(c 2t+1 ) = lnc 1t + β lnc 2t+1 under the constraints, P t C 1t + M t = P t Y 1 P t+1 C 2t+1 = M t + P t+1 Y 2 Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

26 DefiniJons of Variables and Parameters C 1 is household consumpjon in the first period of life and C 2 consumpjon in the second period of life. u is a concave ujlity funcjon and β=1/(1+ρ) the discount factor, where ρ is the pure rate of Jme preference. M t is the money supply, carried over by the household from the first to its second period of life. The money supply is equal to the savings of households in their first period of life. P t is the money price of the consumpjon good in period t and P t+1 the money price of the consumpjon good in period t+1. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

27 ConsumpJon of Young and Old Households in Period t P t C 1t = 1 ( ) 1+ β P ty 1 + P t+1 Y 2 P t C 2t = M + P t Y 2 Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

28 Goods Market Equilibrium and Money Demand C 1t + C 2t = Y 1 + Y 2 M = 1 P 1+ β βy 1 P t+1 Y P 2 t t Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

29 Equilibrium Price Level M P * = 1 ( 1+ β βy 1 Y ) 2 P* > 0 βy 1 Y 2 > 1 The demand for money, and hence the price level, will be posijve only if the discounted first period income of households exceeds second period income. It is only then that savings, and hence money demand, will be posijve. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

30 Price Adjustment EquaJon P t+1 P* = βy 1 Y 2 (P t P*) Since the price level is a non predetermined variable, the condijon for the stability of the dynamic adjustment to the equilibrium price level P* is that the root of the difference equajon above is greater than one. Consequently, the condijon for the existence of a posijve equilibrium price level coincides with the condijon for the stability of the equilibrium. If βy 1 /Y 2 >1, then a posijve equilibrium price level exists, and in addijon the equilibrium is a saddle point, i.e. dynamically stable. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

31 ImplicaJons of the Samuelson Model The Samuelson model has a striking implicajon: Money improves welfare, because it allows households to engage in inter-temporal trade and smooth consumpjon over Jme. In the absence of money, consumpjon in each period would have to be equal to current income for all generajons. This equilibrium is clearly subopjmal compared with the equilibrium of a monetary economy which allows for consumpjon smoothing. The Samuelson model of overlapping generajons is one of the first dynamic general equilibrium models that generate a posijve demand for money as a store of value. The neutrality of money follows immediately from the money demand funcjon.moreover, in this model, since the price level is a non predetermined variable, the increase in the price level would happen immediately. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

32 Weaknesses of the Samuelson Model The first weakness is that the equilibrium we have just described, which entails a posijve demand for money, is not unique. There is a second, subopjmal, equilibrium, with zero money demand. Thus the demand for money in this model is extremely fragile. A second weakness of this model is that there is no alternajve store of value. The only way to save in this model is by holding money. However, if there is an alternajve asset which pays interest, for example bonds or capital, then money would be ostracized from this economy, because its only role is as a store of value, and money does not pay interest. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

33 MulJplicity of Equilibrium in the Samuelson Model M = M Y 2 P P βy (1+ β) ( M / P ) t+1 t 1 t Two long-run equilibria: M P * = 1 ( 1+ βy β 1 Y ) 2 and M P ** = 0 Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

34 MulJplicity of Equilibrium in the Samuelson Model Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

35 Unique Equilibrium if the Old Have no Income If Y 2 =0 then there is a unique equilibrium, M P * = β 1+ β Y 1 Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

36 Money in the UJlity FuncJon of a RepresentaJve Household There is a representajve household maximizing an inter-temporal ujlity funcjon of the form, U t = s=t β s t u(c s, M s P s ) under the constraint, C s + M s P s + B s P s = Y s T s + M s 1 P s + (1+ i s 1)B s 1 P s Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

37 Lagrange FuncJon and First Order CondiJon E t β s t u(c s, M s M ) + λ s 1 s=t s P s P s + (1+ i s 1 )B s 1 P s + Y s T s C s M s P s B s P s λ t = u C t λ = β(1+ i t )E t+1 t P t λ t P t+1 λ t = 1 u λ + βe t+1 t P t P t M t P t+1 Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

38 InterpretaJon of First Order CondiJons First, is the stajc first order condijon, according to which, the marginal ujlity of consumpjon should in any period is equal to the shadow value of marginal savings. EssenJally, the household should be indifferent at the margin between consumpjon and savings. Second is the dynamic first order condijon, according to which the total expected real return on savings should be equal to the pure rate of Jme preference of the household. Third is the dynamic first order condijon according to which the marginal ujlity of real money balances is equal to the difference of the pure rate of Jme preference from the expected real return of money, taking into account expected inflajon and expected capital gains from a change in λ. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

39 The UJlity FuncJon and the Money Demand FuncJon Assuming that the periodic ujlity funcjon u takes the form, u = ln γ C t ( ) ε + (1 γ ) M t P t ε 1 ε Imposing the goods market equilibrium condijon C t =Y t, the first order condijons imply that, M t P t = γ 1 γ i t 1+ i t 1 1 ε Ct = γ 1 γ i t 1+ i t 1 1 ε Yt Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

40 The Cash-in-Advance Constraint and Money Demand The basic idea of models in which money is the only means of payment, is that in order to complete any economic transacjon, payment must be in money, and in parjcular cash, which the buyer holds in advance of the complejon of the transacjon. This idea is due to Clower (1967), and its integrajon into general equilibrium models leads to a class of models known as cash-inadvance models. The restricjon that the transacjon must be paid with money held in advance, imposes a cost of holding money, because, alternajvely, economic agents could hold another asset, such as bonds, which pays interest. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

41 AlternaJve Cash-in-Advance Models The cash in advance restricjon can take several forms, depending on the assumpjons made about the sequencing of transacjons. A simple tradijonal way of expressing this constraint is to assume that spending cannot exceed the money balances carried over from the end of the previous period (Svensson 1985). An alternajve hypothesis is that each period consists of two different sub-periods. In the first sub-period agents visit a financial market, say a bank, where they can swap interest bearing assets with money, or borrow cash, and in the second sub-period they deal in markets for goods and services, which are liable to the cash-in-advance constraint (see Helpman 1981, Lucas 1980, 1982). Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

42 The Cash-in-Advance Constraint when Households Can Visit Financial Markets prior to Making Purchases In the first sub-period agents visit a financial market, say a bank, where they can swap interest bearing assets with money, or borrow cash. In the second sub-period they deal in markets for goods and services, which are liable to the cash-in-advance constraint. In the second sub-period, households also receive their exogenous real income Y and pay their taxes (net of transfers) Τ. A t = M t + B t P t C t M t A t+1 = M t + (1+ i t )B t + P t (Y t T t C t ) = (1+ i t )A t i t M t + P t (Y t T t C t ) Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

43 The Lagrangian of the RepresentaJve Household and the First Order CondiJons M E t β s t u(c s ) +ν t s=t t P t C t + λ t (1+ i t ) A t P t + Y t T t C t i t M t P t A t+1 P t λ +ν t t = u = u (C t ) C t = βe t (1+ i t+1 ) λ t+1 P t λ t P t+1 ν t = λ t i t Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

44 InterpreJng the First Order CondiJons of the RepresentaJve Household with a Cash-in- Advance Constraint First is the stajc first order condijon according to which the opjmal consumpjon equates the marginal ujlity of consumpjon with the shadow value of savings λ, plus the shadow value of money ν. The shadow value of money results from the restricjon for cash in advance in order to buy consumer goods. Second is the dynamic first-order condijon, according to which the total expected real return on savings, including expected inflajon and expected capital gains, should be equal to the pure rate of Jme preference of the household. Third is the stajc first order condijon according to which, the shadow value of money should be equal to the shadow value of savings Jmes the opportunity cost of holding money, which is none other than the nominal rate, since money pays no interest. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

45 The Euler EquaJon and the Money Demand FuncJon in the Cash-in-Advance RepresentaJve Household Model From the first order condijon we can derive the Euler equajon for consumpjon in a monetary economy. u (C ) u (C ) t = β(1+ i )E t+1 P t t P t t+1 The money demand funcjon is derived from the cash-in-advance constraint. M t P t = C t Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

46 The Euler EquaJon for ConsumpJon and the Nominal Interest Rate with Logarithmic Preferences Assuming logarithmic preferences, the Euler equajon for consumpjon can be wri[en as, 1 1 = β(1+ i )E PC t t P C t t t+1 t+1 From the Euler equajon above and the money demand funcjon it follows that the nominal interest rate is determined by 1 M = βe t 1+ i t M t t+1 Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

47 Cash-in-Advance in an Overlapping GeneraJons Model We finally examine the implicajons for money demand of a cash in advance constraint in a variant of the Samuelson overlapping generajons models. In this model, money funcjons both as a means of payments and a store of value. The household born in the beginning of period t lives for two periods, period t and period t +1. It receives income Y t and pays taxes T t, in the first period of life, and consumes in both periods. It maximizes an inter-temporal ujlity funcjon under the constraints that the present value of consumpjon must be equal to the present value of income net of taxes, and the cash-inadvance constraints for each period. U t = lnc 1t + β lnc 2t+1 P t C t i t P t+1 C 2t+1 = P t (Y t T t ) P t C 1t M 1t P t+1 C 2t+1 M 2t+1 Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

48 Cash-in-Advance in an Overlapping GeneraJons Model Aggregate consumpjon and aggregate money balances in each period are given by, C t = C 1t + C 2t M t = M 1t + M 2t Total assets of households are equal to A, and we assume that young households are born without assets. As a result, all assets belong to the old households. For simplicity we assume that taxes T are only paid by young households. Given that old households receive no current income, their consumpjon is equal to their assets, which, however, they need to convert into money, in order to purchase consumer goods. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

49 Cash-in-Advance in an Overlapping GeneraJons Model ConsumpJon of old households is thus given by, C 2t = A t P t Given that young households hold no assets, they need to borrow and convert their loan into money, in order to finance their consumpjon. As a result, for young households the following constraints must hold, M 1t = P t C 1t = B 1t A t+1 = M 1t + (1+ i t )B 1t + P t ( Y t T t C ) 1t = P ( t Y t T t (1+ i t )C ) 1t Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

50 Cash-in-Advance in an Overlapping GeneraJons Model Introducing assets in place of second period consumpjon in the ujlity funcjon, we find that young households will choose consumpjon in their first period of life in order to maximize, U t = lnc 1t + β ln P t ( ) Y t T t (1+ i t )C 1t P t+1 From the first order condijons it follows that, C 1t = 1 Y t T t 1+ β 1+ i t Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

51 Cash-in-Advance in an Overlapping GeneraJons Model Aggregate consumpjon in period t is given by, C t = C 1t + C 2t = 1 Y t T t + A t 1+ β 1+ i t P t From equilibrium in the market for goods and services and money it follows that, Y t = C t = 1 Y t T t + A t 1+ β 1+ i t P t M t = P t Y t These can be solved for the price level and the nominal interest rate. The neutrality of money holds since real income is exogenous. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

52 Nominal and Real Interest Rates: Money in the UJlity FuncJon With logarithmic preferences, the first order condijons for the maximizajon of the ujlity funcjon of the representajve household are given by, λ t = γ C t λ λ t = β(1+ i )E t+1 P t t P t t+1 λ t P t = 1 γ λ + βe t+1 M t P t t+1 From these condijons, it follows that, 1 1 = β(1+ i )E PC t t P C t t t+1 t+1 1 = 1 γ PC γ t t βe M t P C t t+1 t+1 Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

53 Nominal and Real Interest Rates: Money in the UJlity FuncJon The last first order condijon can be solved as. 1 = 1 γ PC γ t t s=t β s t 1 E t M s Given that C t =Y t which is exogenous, this equajon determines the equilibrium price level, as a funcjon of expectajons about the future evolujon of the money supply. SubsJtuJng in the money demand equajon for ε=0, and solving for the nominal interest rate, 1+ i t i t = γ 1 γ M t = β s t M E t PC t s=t t t M s Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

54 Nominal and Real Interest Rates: Money in the UJlity FuncJon The nominal interest rates is determined by the current money supply and expectajons about the future development of the money supply, at a rate of discount that depends on the pure rate of Jme preference of the household. Suppose the expected growth rate of the money supply is constant and equal to μ. 1+ i t 1 = β s t s=t i t 1+ µ s t = s=t 1 1+ ρ ( )( 1+ µ ) s t = (1+ ρ)(1+ µ) (1+ ρ)(1+ µ) 1 It follows that, i t = (1+ ρ)(1+ µ) 1! ρ + µ Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

55 Nominal and Real Interest Rates: Money in the UJlity FuncJon i t = (1+ ρ)(1+ µ) 1! ρ + µ The higher the growth rate of money supply μ, the higher will be the nominal interest rate i, as the expected future inflajon rate will be higher. It is worth nojng that the real equilibrium interest rate in the model is equal to ρ. For μ=0, i=ρ. In this case, because the expected future inflajon rate is equal to zero, the nominal interest rate equals the equilibrium real interest rate, i.e. the pure rate of Jme preference of the representajve household. It is worth nojng that if μ=-ρ/(1+ρ), i.e. if the money supply is reduced at this rate, the nominal interest rate is driven to zero. A zero nominal interest rate has a[racjve properjes and leads to the op9mal money demand by households. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

56 Nominal and Real Interest Rates: Cash-in-Advance In the representajve household model in which money demand results from the cash-in-advance constraint, under the assumpjon of logarithmic preferences, the nominal interest rate is determined by, 1 M = βe t t 1+ i t M t+1 = β 1 1+ µ = 1 (1+ ρ)(1+ µ) From this condijon, it follows that, i t = (1+ ρ)(1+ µ) 1! ρ + µ Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

57 The Indeterminacy of the Price Level when the Central Bank pegs the Nominal Interest Rate: Money in the UJlity FuncJon From the equilibrium condijons in the goods and money markets, if the central bank pegs the nominal interest rate at the level i 0, then the representajve household model with money in the ujlity funcjon, implies that, M t P t = 1 γ γ 1+ i 0 i 0 Y t Given that real income is exogenous, this condijon is sajsfied for an infinite number of combinajons of Μ and P. If it is sajsfied for M 0 and P 0, it is also sajsfied for λm 0 and λp 0, for any λ. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

58 The Indeterminacy of the Price Level when the Central Bank pegs the Nominal Interest Rate: Cash-in-Advance From the Euler equajon for consumpjon and the equilibrium condijon in the goods market, it follows that, P t+1 Y t+1 = β(1+ i 0 )P t Y t Given that real income is exogenous, this condijon is sajsfied for an infinite number of combinajons of P t και P t+1. If it sajsfied for P 0 and P 1, then it is also sajsfied for λp 0 and λp 1, for any λ. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

59 Interest Rate Pegging and Price Level Indeterminacy This indeterminacy was first demonstrated by Sargent and Wallace (1975) in the context of an ad hoc macro model with rajonal expectajons. This indeterminacy does not only arise in representajve household monetary models, but in money models with rajonal expectajons. The reason for the indeterminacy is that, under interest rate pegging, there is no monetary anchor which can determine the price level, as in the case where the central bank determines the money supply. Since the central bank is commi[ed to providing unlimited credit at a nominal interest rate i 0, then the money supply is determined by the demand for money. Neither the price level, nor the money supply can be idenjfied uniquely. The equilibrium condijons for private consumpjon and money demand can be sajsfied with both high prices and a consequent high stock of money, and with low prices and a consequent low stock of money, i.e. virtually for any level of prices. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

60 Dealing with Price Level Indeterminacy under Interest Rate Rules: The Wicksell and Taylor Rules One of the first answers to this problem was provided by the monetary economist who first realized its existence, namely Wicksell (1898). Wicksell proposed that, So long as prices remain unaltered, the banks rate of interest is to remain unaltered. If prices rise, the rate of interest is to be raised; and if prices fall, the rate of interest is to be lowered; and the rate of interest is henceforth to be maintained at its new level unjl a further movement of prices calls for a further change in one direcjon or the other. (p. 189). Thus, Wicksell proposed that central banks should have a price level target, and change interest rates when the price level deviates from this target. AlternaJve ways to solve the problem of price level indeterminacy when the policy instrument of the central bank is the nominal interest rate, have been proposed: InflaJon targejng rules, nominal income rules, and more recently the Taylor (1993), which is a generalizajon of Wicksell s rule. We shall examine the properjes of such rules in the lectures on aggregate fluctuajons and monetary policy. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

61 The Fiscal Theory of the Price Level One theorejcal development worth menjoning is the so-called fiscal theory of the price level (see. Leeper 1991, Sims 1994 and Woodford 1994, 1995). This theory argues that even if monetary policy is not sufficient to determine the price level, as under interest rate pegging, the price level can be determined at the level which ensures that public debt, which is defined in nominal terms, does not follow an explosive path. A path for the price level that ensures a path of nominal public debt that sajsfies the inter-temporal budget constraint of the government is sufficient in those models to determine the price level. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

62 Price Level Determinacy in Overlapping GeneraJons Models and the Pigou Effect It is also worth stressing that the problem of price level indeterminacy does not arise in overlapping generajons models. Unlike the representajve household model, where both the current and the future price level are non predetermined variables, in the overlapping generajons model, the price level is determined through the predetermined nominal financial assets of old households. These funcjon as a monetary anchor and help in determining the price level. In tradijonal monetary models, the dependence of consumpjon on the financial wealth of households was called the Pigou effect (see Pigou 1943), or the real balance effect (see PaJnkin 1956). As Sargent and Wallace (1975) had indicated in their original analysis of price level indeterminacy, in the presence of a Pigou or real balance effect, the problem does not arise even if the central bank pegs the nominal interest rate. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

63 Seigniorage and InflaJon If the growth of the money supply translates into higher inflation in the longer term, why don t governments and central banks keep the rate of growth of the money supply low and stable in order to control and eliminate inflation? The answer is that governments often have other policy motives besides the motive of tackling inflation. Perhaps the most important incentive for the issuance of new money by governments is to finance expenditure that they cannot, or do not want to, finance through other methods, such as higher taxes or higher government debt. The main cause of all the episodes of high inflation or hyperinflation appears to have been the need of governments to use revenue from money creation (seigniorage) to finance wars and war reparations, revolutions, extraordinary costs related to natural disasters or sudden reductions in their borrowing capacity from financial markets and their capacity to raise revenue from taxes and customs revenues. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

64 High InflaJon and HyperinflaJon We explore the relajonship between the growth rate of money supply, inflajon and the needs of governments to raise revenue through seigniorage. We examine both the case in which the required income from seigniorage can be raised on the balanced growth path, a situajon in which equilibrium turns out to be characterized by high infla9on, and the situajon in which the required revenue from seigniorage is so high, that it cannot be raised in steady state equilibrium, which can lead to hyperinfla9on. The generally accepted definijon of hyperinflajon is due to Cagan (1956). Cagan defined a a period of hyperinflajon as one beginning in the month in which the rise in prices exceeds 50% and as ending in the month before the monthly rise in prices drops below that amount and stays below for at least a year. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

65 Episodes of High InflaJon and HyperinflaJon The first modern episodes of hyperinflajon occurred in Europe in the adermath of World War I, as well as during and in the adermath of World War II. In the last forty years very high inflajon and hyperinflajon reappeared in some LaJn American countries, in some transijon economies ader the collapse of the Soviet Union and in some belligerent countries of Asia and Africa. Moreover, many countries, without reaching the levels of hyperinflajon, have experiences with high inflajon from 100% to 1000% per year for quite long periods. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

66 Monetary Growth, InflaJon and Seigniorage: The Demand for Money FuncJon In order to study the relajon between the rate of growth of the money supply, inflajon and revenue from seigniorage, we shall start from a linear logarithmic form of the money demand funcjon. In equilibrium, the demand for money equals the supply of money, so it follows that, M P = κye ηi κ is a constant, e the basis of natural logarithms, και η>0 the semi-elasjcity of money demand with respect to the nominal interest rate i. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

67 Monetary Growth, InflaJon and Seigniorage: Nominal and Real Interest Rates The nominal interest rate is defined by the Fisher equajon, i = r + π e where r is the real interest rate, and π e expected inflajon. Real Output Y is considered exogenous, with a growth rate g+n>0, while the rate of growth of the nominal money supply M is denoted by μ>0. Under these assumpjon, inflajon on the balanced growth path, sajsfies, π = µ (g + n) Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

68 The Demand for Money and the Rate of Growth of the Money Supply Assuming rajonal expectajons, the money demand funcjon can thus be wri[en as, M P = κye η(r+µ (g+n)) To further simplify ma[ers, we shall assume that the golden rule applies on the balanced growth path, which implies that r=g+n. Under this addijonal assumpjon, M P = κye ηµ Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

69 Seigniorage Revenue and Rate of Growth of the Money Supply Seigniorage revenue is equal to, S = M P = M M M P = µ M P = µκye ηµ As a proporjon of total output, seigniorage revenue is defined by, s = S Y = µ M PY = µκ e ηµ Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

70 Maximizing Seigniorage Revenue Seigniorage revenues as a proporjon of total output, depend on μ, according to, s = (1 ηµ)κ e ηµ µ This dependence is posijve in μ, up to the point where μ=1/η. When μ>1/ η the dependence becomes negajve. Seigniorage revenue, as a proporjon of total output is thus maximized when μ=1/η. Maximum seigniorage revenue in steady state is thus given by, s max = κ λe Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

71 The Seigniorage Laffer Curve Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

72 The Rate of Growth of the Money and Maximum Seigniorage Revenue Cagan, using annual data, esjmated that η lies between 1/2 and 1/3. Consequently, he esjmated the growth rate of the money supply that maximizes revenues from seigniorage, as a percentage of total output, and the corresponding inflajon, at between 200% and 300% per year. Assuming that κ=0.10, the maximum revenue from seigniorage as a percentage of total output is between 7-11%. This is roughly the esjmate of Cagan (1956). For the period , Sachs and Larrain (1993) esjmated actual revenue from seigniorage at about 5 to 6.5% for high inflajon countries such as Italy, Bolivia, Turkey and Peru, and much lower for a series of other countries. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

73 High InflaJon Balanced Growth Path Let us now consider a government which needs to fund a proporjon of its public spending through seigniorage. We will assume that this financing requirement, as a proporjon of total output is equal to s E, which is less than the maximum seigniorage smax that the government can achieve by se^ng the growth rate of the money supply at μ=1/η. There are two opjons to achieve revenue equal to s E. One is with a growth rate of the money supply μ E < 1/η, and the other is with a growth rate of the money supply μ E > 1/η. Assuming that the government dislikes inflajon, it will choose the lowest growth rate of the money supply that is compajble with the objecjve of raising revenue s E from seigniorage. For as long as the government needs to finance a proporjon s E of its output through seigniorage, the economy is trapped in an equilibrium with a rate of growth of the money supply equal to μ E and the corresponding high inflajon. For example, if the government wants to raise seigniorage corresponding to 6% of total output, assuming η=1/2, this implies an annual growth rate of the money supply (and corresponding steady state inflajon) equal to about 100%. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

74 A High InflaJon Balanced Growth Path Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

75 Seigniorage Revenue Outside the Balanced Growth Path: Gradual Adjustment of Money Demand or Gradual Adjustment in ExpectaJons Suppose now that the government needs to raise seigniorage which, as a proporjon of total output, is higher than the maximum that can be raised in the steady state. We assume s E > s max. Obviously there can be no balanced growth path in which the government can raise revenues from seigniorage to exceeds s max. However, for a Jme, and as the economy adjusts towards the balanced growth path, the government may be able to raise seigniorage revenues greater than smax. This could happen if for example there is gradual adjustment in the demand for money, or gradual adjustment in inflajonary expectajons. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

76 Seigniorage Revenue Outside the Balanced Growth Path: Gradual Adjustment of Money Demand Suppose that the demand for money does not adjust immediately to its steady state level ader a change in the nominal interest rate, but only adjusts gradually. Thus, when the nominal interest rate increases, money demand is temporarily higher than in the steady state. In this case, during the adjustment, the monetary base upon which the inflajonary tax is imposed is higher than the steady state monetary base. Consequently, during the adjustment, as μ increases, seigniorage revenues will exceed s max because real money balances are higher than on the balanced growth path. As the demand for money decreases gradually, the government should constantly increase the rate of monetary expansion and the consequent inflajon, to be able to have the required high revenues from seigniorage. This can lead to an explosive path for the rate of growth of the money supply, and a consequent hyperinflajon. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

77 Gradual Adjustment in Money Demand Assume that, in the short run, real money demand adjusts gradually towards its steady state value according to, d ln m(t) dt = m (t) m(t) =ψ ( ln m * ln m(t) ) where, ψ is the speed of adjustment, and, m(t) = M (t) P(t)Y (t) m* = M PY = κ e ηµ Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

78 Monetary Growth and the Gradual Adjustment in Money Demand SubsJtuJng for the determinants of steady state money demand m*, the adjustment of money demand takes the form, m (t) m(t) =ψ ( lnκ ηµ(t) ln m(t) ) In orded to achieve a constant target s E for seigniorage revenue, the rate of growth of the money supply must be equal to the rate of reducjon of the demand for real money balances s E = µ(t)m(t) and µ (t) µ(t) = m (t) m(t) Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

79 Dynamic Adjustment of the Rate of Growth of the Money Supply in order to maintain revenues from seigniorage constant as a percentage of total income at the level s E, the growth rate of the money supply must keep increasing conjnuously, at the same rate as the decline of real money demand relajve to output. From the previous equajons this implies that, µ (t) µ(t) = ψ ( lnκ ln s + ln µ(t) ηµ(t) ) E A necessary and sufficient condijon for stabilizing the rate of growth of the money supply is that, s E = µκ e ηµ s max Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

80 TransiJon to HyperinflaJon If s E >s max then the rate of growth of the money supply increases conjnuously. Ader a point it starts increasing at an increasing rate, with the result an explosive path for the rate of growth of the money supply and inflajon. The result is hyperinflajon. µ (t) ( ) µ(t) = ψ lnκ ln s E + ln µ(t) λµ(t) Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

81 High InflaJon and HyperinflaJon with ParJal Adjustment in Money Demand Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

82 Required Seigniorage Revenue, Money Supply Growth and InflaJon In the case where the financing needs of the government from seigniorage are less than or equal to the maximum possible on the balanced growth path, then the rate of growth of the money supply stabilizes at a rate that may indeed entail significant inflajon, but inflajon is stable and does not evolve into hyperinflajon. However, if the financing needs of government exceed the maximum that is sustainable on the balanced growth path, then, as the government tries to raise the necessary revenue from seigniorage, the rate of growth of the money supply gradually accelerates, in order to keep up with the declining monetary base, and the economy falls into a state of hyperinflajon. The reason is that inflajon gradually reduces the demand for money relajve to total output, and the government needs an ever increasing growth rate of the money supply in order to be able to collect the needed seigniorage revenue. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

83 Conclusions on the Determinants of High InflaJon Our basic analysis explains why, in many cases, inflajon may be driven to very high levels. This is due to the inability of a government to finance its spending from other revenue sources, such as taxajon or borrowing from the markets, and its need to use seigniorage, i.e revenue from money creajon. The analysis also explains why even though inflajon may reach very high levels, it is not necessary that it will evolve into an explosive hyperinflajon. For this to happen, the financing needs of the government must be so high that they exceed the maximum level that can be financed through seigniorage on the balanced growth path. Finally, the analysis emphasizes the central role of fiscal problems as the main root causes of both high inflajon and hyperinflajon. A significant precondijon for tackling high inflajon or hyperinflajon is to pursue reforms that address the underlying fiscal problems (Sargent 1982). Prof George Alogoskoufis, Dynamic Macroeconomic Theory,

84 Final Conclusions In this lecture we have analyzed the role and funcjons of money. Money performs three funcjons. First, it is a unit of account, second, it is a generally accepted means of payment, and, thirdly, it is a store of wealth. We first reviewed the basic funcjons of money and the factors that determine the demand for and supply of money. We analyzed the concept of short run equilibrium in the money market, assuming that the central bank follows a policy of either targejng the money supply or pegging nominal interest rates, and also defined the nojon of the long-term neutrality of money. We then focused on a number of dynamic general economic equilibrium models with money, in order to analyze the determinajon of the price level and nominal interest rates and also analyzed the long relajonship between the money supply, the price level and inflajon. Finally we examined the fiscal incenjve for increasing the money supply and its effects on inflajon. The most important mojve for sustained large increases in the money supply by governments has been the incenjve to use seigniorage in order to finance government expenditure that could not be financed by other methods, such as addijonal taxes or government bonds. Prof George Alogoskoufis, Dynamic Macroeconomic Theory,