Money, Inflation and Economic Growth

Transcription

1 Chapter 6 Money, Inflation and Economic Growth In the models we have presented so far there is no role for money. Yet money performs very important functions in an economy. Money is a unit of account, in terms of which prices are defined, a means of payments, which reduces transaction costs, and the most liquid store of value. In this chapter we allow for the role of money in the various exogenous growth models we have analyzed so far. We first present a representative household model, in which real money balances enter the utility function of households, and then analyze a corresponding model of overlapping generations. 1 In models with money, one can draw the distinction between real variables, as the ones we have analyzed so far, and nominal variables, such as the stock of money, the price level, inflation and nominal output, nominal wages and nominal interest rates. All nominal variables are expressed in terms of money. By assuming that money enters the utility function of households, we derive a money demand function from microeconomic foundations, as a result 1 This approach to the derivation of money demand is the so-called money in the utility function approach, and was first introduced to macroeconomics by Patinkin [1956]. This approach contrasts with an alternative approach, called cash in advance, whichemphasizes the role of money as a means of payments that reduces transaction costs, eliminating the need for a double coincidence of wants between agents ( Clower [1967] ). As shown by Feenstra [1986] both of these approaches are functionally equivalent. We postpone the examination of the cash in advance approach until Chapter 10, where we provide a fuller treatment of alternative partial and general equilibrium approaches to money demand and the money market. 173

2 174 Ch. 6 Money, Inflation and Economic Growth of the solution of an intertemporal optimization problem by households. Based on this particular approach to money demand, we show that the demand for real money balances is proportional to aggregate consumption, and depends negatively on the nominal interest rate. Since money is a nominal asset that pays no interest, the nominal interest rate measures the opportunity cost of holding real money balances. The demand for nominal money balances is proportional to the price level, a property which implies the neutrality of money. The stock of nominal money balances does not a ect any real variables in these models, but only the price level. We also analyze the determination of inflation, the nominal interest rate and other nominal variables, and the intertemporal e ects of the rate of growth of the money supply on the path of economic growth. In the representative household model, the growth path of all real variables, with the exception of the stock of real money balances, is independent of the rate of growth of the money supply, which a ects inflation and nominal interest rates. The demand for real money balances, which depends negatively on the nominal interest rate, is the only real variable that is affected by the rate of growth of the money supply. This is because the rate of growth of the money supply a ects nominal interest rates, and therefore the opportunity cost of holding real money balances. To put it di erently, the rate of growth of the money supply imposes an inflation tax on the real money balances held by households. The independence of the growth path of all other real variables from the rate of growth of the money supply is known as the super neutrality of money. In overlapping generations models, the rate of growth of the money supply a ects the growth path of all real variables, as it a ects the aggregate savings rate, the accumulation of capital and the balanced growth path. The reason is that in overlapping generations models, holdings of real money balances di er among generations. Thus, when there is an increase in the rate of growth of the money supply, older generations, which hold higher real money balances, reduce their asset holdings and their consumption more than younger generations, since they pay a higher inflation tax. As a result, aggregate consumption falls, and aggregate savings rise. This leads to a higher accumulation of capital, which a ects the growth path. 2 2 The literature on money and economic growth originated with Tobin [1965]. Sidrauski [1967] first used a representative household model to demonstrate the super-neutrality of money, i.e that the rate of growth rate of the money supply does not a ect the path of real variables on the adjustment path or the balanced growth. The literature has since expanded exponentially. Weil [1987], Weil [1991] analyzed the role of money in a model of overlapping generations, and demonstrated that the super-neutrality of money does not

3 George Alogoskoufis, Dynamic Macroeconomics 175 The di erences in the e ects of the rate of growth of the money supply between representative household and overlapping generations models arise for the same reason that government debt has no real e ects in representative household models, while it has real e ects in overlapping generations models. Ricardian equivalence and the super neutrality of money are closely linked, as the rate of growth of the money supply is essentially an inflation tax on real money balances. In overlapping generations models this inflation tax, and government debt, have di erent e ects on current and future generations, and thus affect aggregate savings. An increase in public debt redistributes taxes from current to future generations, causing an increase in the consumption by current generations, while an increase in the rate of growth of the money supply redistributes taxes from future to current generations, causing a reduction in the consumption of current generations. This redistribution among generations, and its e ects on aggregate savings are thus the reason for the non existence of Ricardian equivalence and the super neutrality of money in overlapping generations models. 6.1 Private Consumption and Money Demand in a Representative Household Model Assume, as in the Ramsey model of Chapter 3, a competitive economy in which all households are identical. Households are indexed by j, wherej is uniformly distributed between zero and 1. Thus, j 2 [0, 1]. We therefore focus on the behavior of only one of them, the representative household. The number of members of each household is equal to L(t), and is growing at an exogenous rate n, which is also the rate of growth of population L(t). Labor supply and employment is equal to L(t) and the e ciency of labor h(t) grows at an exogenous rate of technical progress g. The instantaneous utility function of households depends on consumption of goods and services, and holdings of real money balances Money in the Utility Function of Households Household j selects a path of consumption and real money balances in order to maximize the following intertemporal utility function. apply in such models.

4 176 Ch. 6 Money, Inflation and Economic Growth U j = Z 1 t=0 e t u (c j (t),m j (t)) L(t)dt (6.1) The maximization takes place under the instantaneous budget constraint, ȧ j (t) =(r(t) n) a j (t)+w(t)h(t) (t)h(t) c j (t) (r(t)+ (t)) m j (t) (6.2) and the transversality condition, R t lim e (r(s) n)ds s=0 aj (t) = 0 (6.3) t!1 is the pure rate of time preference of households and u is a quasi concave instantaneous utility function that depends on consumption and real money balances (assumed positive). c j (t) denotes average consumption per person of household j at time t, m j (t) denotes average per person real money balances of household j at time t, a j (t) denotes average assets per person (non-human wealth) of household j at time t, w(t) denotes the real wage per e ciency unit of labor at time t, h(t) denotes labor e ciency per worker, and (t) denotes average taxes (minus transfers) per e ciency unit of labor at time t. r(t) is the real interest rate at time t, n is the population growth rate, equal to the growth rate of members per household, and (t) is inflation, which is equal to expected inflation. Real money balances provide utility because of their liquidity services, i.e because they facilitate payments (the exchange of goods and services) and reduce transaction costs. The instantaneous utility function is assumed to take the form, u j (t) =u (c j (t),m j (t)) = c j(t) 1 +(1 ) m j(t) 1 (6.4) 1 1 where 1/ is the intertemporal elasticity of substitution of consumption and real money balances, and the share of consumption in the utility of the household. This utility function, which is a generalization of the utility function we have assumed so far, is additively separable, in that the elasticity substitution between consumption and real money balances is equal to unity Nominal and Real Interest Rates and the Opportunity Cost of Real Money Balances Unlike other assets held by households, such as capital or government bonds, the nominal yield of money is equal to zero, because money balances do not

5 George Alogoskoufis, Dynamic Macroeconomics 177 pay interest. Moreover, when there is a positive inflation rate (t), real (inflation adjusted) money balances depreciate at the inflation rate (t). Therefore the opportunity cost of holding real money balances is equal to the sum of the real return of interest yielding assets, plus the expected inflation rate. This is defined as the nominal interest rate i(t), which is determined by, i(t) =r(t)+ (t) (6.5) This relationship between nominal and real interest rates is often called the Fisher equation, as it was first highlighted by Irving Fisher in The First Order Conditions for an Optimum Maximizing (6.1) under the constraint (6.2), by forming the relevant Hamiltonian, yields the following first order conditions, c j (t) = j (t) (6.6) (1 )m j (t) = j (t)(r(t)+ (t)) (6.7) j(t) = (r(t) ) j (t) (6.8) The asset accumulation constraint (6.2) and the transversality condition (6.3) must also be satisfied. j(t) is the current value multiplier of the relevant current value Hamiltonian, and its economic interpretation is that it measures the shadow value of marginal savings (assets) of the household. From (6.6), j(t), which is the shadow value of marginal savings, is equal to the current marginal utility of consumption. At the optimum the marginal value of savings must be equal to the marginal utility of consumption, and the household must be indi erent between exchanging consumption for savings. 3 See Fisher [1896] or Fisher [1930] Chapter 2. To quote from Fisher [1896], When prices are rising or falling, money is depreciating or appreciating relative to commodities. Our theory would therefore require high or low interest according as prices are rising or falling, provided we assume that the rate of interest in the commodity standard should not vary. (p. 58). The rate of interest in the commodity standard is the real interest rate, and rising or falling prices are expected inflation.the Fisher equation was further elaborated in Fisher [1930], where it was made even clearer that Fisher referred to expected and not actual inflation.

6 178 Ch. 6 Money, Inflation and Economic Growth From (6.7), the current marginal utility of real money balances is equal to the marginal utility of consumption j (t) times the opportunity cost of holding real money balances. At the optimum the marginal value of holding real money balances must be equal to the opportunity cost of holding money, evaluated in terms of the marginal utility of consumption. Thus, at the optimum, the household should be indi erent between substituting consumption for the utility services of money. Finally, from (6.8), the marginal utility of consumption falls at a rate which is equal to the di erence between the real interest rate and the pure rate of time preference. This is another way of saying that the expected real return on savings, including capital gains on assets, is equal to the pure rate of time preference of the household. This can be seen by re-arranging (6.8) as, r(t)+ j(t) j(t) = We can now use the first order conditions to derive the money demand function and the Euler equation for consumption The Money Demand Function Dividing (6.5) by (6.6) and solving for real money balances, we can derive a function for the demand for real money balances by the representative household. m i (t) = 1 1 (r(t)+ (t)) ci (t) (6.9) We can use (6.9) to deduce aggregate money demand. Multiplying both sides of (6.9) by L(t) we get, M(t) P (t) = 1 1 (r(t)+ (t)) C(t) (6.10) where M(t) is the aggregate nominal money supply, P (t) istheprice level, and C(t) is aggregate real consumption of goods and services. (6.10) describes the aggregate money demand function in this model. Aggregate money demand is proportional to the price level and aggregate real consumption, and depends negatively on the nominal interest rate. Put di erently, the elasticity of aggregate money demand with respect to the price level and aggregate private consumption is equal to one, while the

7 George Alogoskoufis, Dynamic Macroeconomics 179 elasticity of aggregate money demand with respect to the nominal interest rate is equal to 1/, the elasticity of intertemporal substitution. (6.10) is characterized by homogeneity of degree one with respect to the price level, because households demand money for its purchasing power. Doubling the money supply, for given aggregate real consumption and nominal interest rates, would cause a doubling of the price level. This property is known as the neutrality of money. Expressing real money balances per e ciency unit of labor, one gets, m(t) = 1 where c = C/hL and m =(M/P )/hl. h is the e 1 (r(t)+ (t)) c(t) (6.11) ciency of labor The Rate of Growth of the Money Supply and Inflation We can use (6.10) or (6.11) to determine the price level and inflation, under the assumption that the aggregate nominal money supply and its rate of growth µ are determined by the government (or a government agency such as a central bank). From (6.10) and (6.5), it follows that, M(t) P (t)c(t) = 1 1 i(t) (6.12) For given aggregate real consumption and nominal interest rates, the level of the nominal money supply determines the level of prices. From (6.12) it follows that,! Ċ(t) µ (t) = 1 i(t) (6.13) C(t) i(t) Thus, from (6.13), inflation is determined as, (t) =µ Ċ(t) C(t) + 1 i(t) i(t) (6.14) For a given rate of growth in private consumption and fixed nominal interest rates, the inflation rate?(t) is determined by the rate of growth of the money supply. For a example, in a steady state where the growth rate of consumption in equal to g + n, and inflation and nominal interest rates are constant, inflation would be determined by,

8 180 Ch. 6 Money, Inflation and Economic Growth = µ g n (6.15) (6.15) is the basis of the monetary approach to the determination of inflation. In the long run, inflation is determined by the di erence between the rate of growth of the money supply and the long run growth rate of aggregate output and consumption The Euler Equation for Consumption Turning to the first order conditions for consumption, from (6.6) and (6.8), it follows that, ċ j (t) = 1 (r(t) ) c j(t) (6.16) (6.16) is the standard Euler equation for consumption in a representative household model. It does not di er in this monetary model from the corresponding equation in a real model, such as the one examined in Chapter 3. The reason that real money balances, or the nominal interest rate, do not appear in (6.16) is the assumption of additively separable preferences of the household over consumption and real money balances. With additively separable preferences the marginal utility of consumption in (6.6) does not depend on real money balances, or the determinants of their demand, such as the nominal interest rate. It is straightforward to show that without additively separable preferences the marginal utility of consumption in (6.6) depends on real money balances as well, and (6.16) does not generally follow. For example, with a non additively separable CES instantaneous utility function of the form, u(c j (t),m j (t)) = c i (t) ( 1)/ +(1 )m i (t) ( 1)/ /( 1) where is the elasticity of substitution between consumption and real money balances, the first order condition (6.6) would take j = cj (t) ( 1)/ +(1 )m j (t) ( 1)/ c j(t) 1/ = j (t) With an elasticity of substitution which is di erent from unity, i.e non additively separable preferences, the marginal utility of consumption, depends on real money balances as well.

9 George Alogoskoufis, Dynamic Macroeconomics 181 In any case, in what follows we shall continue assuming additively separable preferences between consumption and real money balances. In this case, (6.16) holds. From (6.16), the evolution of aggregate consumption C and consumption per e ciency unit of labor c are determined by, Ċ(t) = 1 (r(t) + n) C(t) (6.17) ċ(t) =(r(t) g) c(t) (6.18) where g is the rate of exogenous technical progress. From (6.17) and (6.18), the rate of growth of aggregate consumption or consumption per e ciency unit of labor only depends on real variables and not the money supply or its rate of growth. 6.2 Aggregate Capital Accumulation in a Ramsey Model with Money We next turn to the determinants of capital accumulation, output and other real variables such as the real interest rate and real wages The Production Function, the Real Interest Rate and the Real Wage We assume as in the previous models that output per e is determined by, ciency unit of labor y(t) =f(k(t)) (6.19) where f is a neoclassical production function, with all the usual properties. Once the capital stock per e ciency unit of labor is determined, output is determined through the production function (6.19). We have already assumed an exogenous rate of growth of population n and an exogenous rate of technical progress g. The depreciation rate of the capital stock will be assumed equal to,where,1 > >0. In a competitive equilibrium, assuming that firms maximize profits, the real interest rate r(t) and the real wage per e ciency unit of labor w(t) will be determined by the usual marginal productivity conditions, r(t) =f 0 (k(t)) (6.20)

10 182 Ch. 6 Money, Inflation and Economic Growth w(t) =f(k(t)) k(t)f 0 (k(t)) (6.21) The Inflation Tax and the Accumulation of Capital We shall assume that the assets of household j in equation (6.2) consist of capital, government bonds and money. a j (t) =k j (t)+d j (t)+m j (t) (6.22) where k j, d j and m j denote physical capital, real government bonds and real money balances held by the average member of household j. Replacing (6.22) in (6.2), after multiplying by L(t), we get an accumulation equation for total household assets in the economy. K(t)+Ḋ(t)+ M(t) P (t) = r(t) T (t) C(t) (r(t)+ (t)) M(t) P (t) K(t)+D(t)+ M(t) P (t) The rate of growth of real money balances is given by µ will hold that, M(t) P (t) =(µ (t)) M(t) P (t) + w(t)h(t)l(t) (6.23) (t). Thus, it (6.24) Substituting (6.24) in (6.23), and solving for the accumulation of physical capital and real government bonds, we get, K(t)+Ḋ(t) =r(t)k(t)+w(t)h(t)l(t)+r(t)d(t) T (t) µm(t) C(t) P (t) (6.25) The aggregate accumulation of capital and real government bonds by households depends on the di erence between aggregate household disposable income from aggregate household consumption. The disposable income of households consists of their total asset and labor income, minus taxes T (t) and the inflation tax imposed by the government through the rate of growth of the money supply. Government revenue from the monopoly of issuing money, usually referred to as seigniorage, is equal to µ(m(t)/p (t)) and comprises a tax on real money balances held by households.

11 George Alogoskoufis, Dynamic Macroeconomics 183 (6.25) describes the budget constraint of households. The government budget constraint in an economy is which the government has the monopoly of money creation is described by, Ḋ(t) =C g (t)+r(t)d(t) T (t) µ M(t) P (t) (6.26) (6.26) suggests that the government accumulates government debt to the extent that primary government expenditure C g, plus the real interest expenditure on existing debt rd, exceeds total taxes (minus transfers) T, plus the inflation tax µ(m/p ). Substituting the government budget constraint (6.26) in the household budget constraint (6.25), we end up with the well known equation for aggregate capital accumulation, K(t) =r(t)k(t)+w(t)h(t)l(t) C g (t) C(t) (6.27) Only primary government expenditure, and not the way it is financed, appears in the aggregate capital accumulation equation. Debt, taxes and seigniorage revenue do not a ect the accumulation of capital. Expressing both sides of (6.27) per e ciency unit of labor, i.e dividing by h(t)l(t), we get, k(t) =r(t)k(t)+w(t) c(t) c g (t) (n + g)k(t) = f(k(t)) c(t) c g (t) (n + g + )k(t) (6.28) The economy accumulates capital per e ciency unit of labor when total savings per e ciency unit of labor exceed the investment required to maintain a constant capital stock per e ciency unit of labor. This is the same as the capital accumulation equation in the Ramsey model without money. The existence of money and money demand does not a ect the accumulation of real capital either. 6.3 The E ects of the Rate of Growth of the Money Supply in the Ramsey Monetary Model The evolution of real variables in the Ramsey model with money is determined by the Euler equation for consumption (6.18) and the capital accumulation equation (6.28). These two determine the paths of private consumption and the capital stock.

12 184 Ch. 6 Money, Inflation and Economic Growth All other real variables are functions of the capital stock (per e ciency unit of labor) and/or private consumption. Output is determined by the production function (6.19). The real interest rate and the real wage are determined by the marginal productivity conditions which are only a function of capital per e ective unit of labor k. The demand for real money balances is determined by the money demand function (6.11), while the nominal interest rate is determined by the Fisher equation (6.5), and inflation is determined by equation (6.14). As for the government, we shall make similar assumptions to the ones we made in Chapter 5. We shall assume that the government chooses a constant level of primary expenditure c g and government debt d per e ciency unit of labor, and uses taxes to satisfy the government budget constraint. The government also chooses, through the central bank, a constant rate of growth µ for the money supply. All seigniorage revenue accrues to the government budget. Under these assumptions, private consumption and the accumulation of capital are determined by the pair of di erential equations, ċ(t) = 1 f 0 (k(t) g c(t) (6.29) k(t) =f(k(t)) c(t) c g (n + g + )k(t) (6.30) As one can see from (6.29) and (6.30) neither the stock of money, nor the rate of growth of the money supply a ect the evolution of private consumption (savings) or the accumulation of capital (investment). The determination of consumption and the capital stock takes place in the same way as in a model without money. The only government policy variable that appears to a ect the economy is real primary government expenditure. The model is exactly the same as the Ramsey model without money in Chapter 5. The rate of growth of the money supply a ects inflation, nominal interest rates and the demand for money, but no other real variables, such as output, consumption, the capital stock, real wages and real interest rates. This can be seen by concentrating on the balanced growth path The Balanced Growth Path in the Ramsey Model with Money As in the Ramsey model without money, it is straightforward to prove that the economy possesses a unique balanced growth path and a unique saddle path leading to the balanced growth path.

13 George Alogoskoufis, Dynamic Macroeconomics 185 On the balanced growth path, all variables that have been defined per e ciency unit of labor will remain constant, and the same applies to interest rates (real and nominal) and inflation. From (6.29), on the balanced growth path, the real interest rate must be equal to the pure rate of time preference, plus the exogenous rate of technical progress multiplied by the inverse of the inter temporal elasticity of substitution. Therefore, steady state capital per e ciency unit of labor is determined by the condition, f 0 (k )= + g + (6.31) Steady state consumption per e the steady state version of (6.30) as, ciency unit of labor is determined by c = f(k ) c g (n + g + )k (6.32) Primary government expenditure has a one to one negative e ect on private consumption, and does not a ect the steady state capital stock or steady state output. Real output per e ciency unit of labor on the steady state is determined by the production function (6.19), as, y = f(k ) (6.33) The steady state real interest rate and the real wage per e ciency unit of labor are determined by the marginal productivity conditions as, r = + g (6.34) w = f(k ) k f 0 (k ) (6.35) (6.31), (6.32), (6.33), (6.34) and (6.35) determine the evolution of all real variables on the balanced growth path, with the exception of real money balances. Capital, output, the real wage and consumption per e ciency unit of labor are constant on the balanced growth path, as is the real interest rate. All per capita variables grow at the exogenous rate of technical progress g. In this model, neither the method of financing government expenditure, nor the money stock or the rate of growth of the money supply a ect the balanced growth path. Thus, both Ricardian equivalence and the neutrality as well as the so called super neutrality of money hold in a representative household model.

14 186 Ch. 6 Money, Inflation and Economic Growth The Super Neutrality of Money and Inflation The super neutrality of money was first analyzed by Sidrauski [1967], who demonstrated that, in a representative household model, the growth rate of the money supply does not a ect real variables on the balanced growth path. On the balanced growth path, the inflation rate is determined by the di erence between the growth rate of the money supply from the long-term growth rate n+g. This can be seen from the inflation determination equation (6.14). Assuming a constant rate of growth of the money supply, the inflation rate on the balanced growth path equals, = µ (n + g) (6.36) where n+g is the steady state rate of growth of total private consumption (and output). Moving from (6.14) to (6.36) we have assumed that the nominal interest rate is constant on the balanced growth path. From the Fisher equation (6.5), the steady state nominal interest rate is indeed constant and equal to, i = r + = + g + µ (n + g) = +( 1)g n + µ (6.37) Finally, real money balances per e ciency unit of labor are also constant on the balanced growth path. They are determined from the money demand equation (6.11) and are given by, m = c i = c +( 1)g n + µ (6.38) Note that semi elasticity of money demand with respect to the nominal interest rate is equal to the elasticity of intertemporal substitution 1/. The higher the growth rate of the money supply, given the other structural parameters of the model, the higher the rate of inflation and the nominal interest rate, and the lower the stock of real money balances on the balanced growth path. A permanent increase in the growth rate of the money supply by 5 percentage points causes an increase in inflation by 5 percentage points, and an increase in nominal interest rates by 5 percentage points as well. It also causes a decrease in the demand for real money balances. In a representative household model, real money balances is the only real variable a ected by

15 George Alogoskoufis, Dynamic Macroeconomics 187 the growth rate of the money supply on the balanced growth path. This has been termed as the super neutrality of money. It also follows that the neutrality of money also holds in this model, as the level of the money supply only a ects the price level, and no real variables. From the definition of real money balances per e ciency unit of labor, M(t) P (t) = m h 0 L 0 e (g+n)t (6.39) where h 0, L 0 are the e ciency of labor and the labor force at time 0. Real money balances are also growing at a rate g + n on the balanced growth path. From (6.35), the only e ect of a rise in the money stock is an equiproportionate rise in the price level. Thus, the neutrality of money also holds in this model The Welfare Costs of Inflation in a Representative Household Model Does the superneutrality of money mean that inflation does not matter in a representative household model? Far from it. Inflation implies a distortion in that the demand for real money balances, and, therefore, the liquidity services of money, are lower the higher the rate of inflation and nominal interest rates. Hence, the welfare of the representative household goes down with the rate of inflation, as the representative household economizes on real money balances and enjoys lower liquidity services. The analysis of the welfare costs of inflation can be simplified by using the steady state money demand function (6.38). This is depicted geometrically in Figure 6.1, for given steady state consumption per e ciency unit of labor. We compare two equilibria. In one of them inflation is zero, and the nominal interest rate is equal to the real steady interest rate + g. Inthe second there is a positive inflation rate. The welfare cost of inflation is measured by the area of the gray triangle, which is the loss of consumer surplus that does not translate into a higher inflation tax. This is the net welfare cost of inflation, which measures the reduction in the welfare of the representatitve household that does not translate into higher government revenue from inflation. One can easily show that the higher the inflation rate, the higher the welfare cost of inflation. Hence, inflation matters in the representative household model, in that it causes a reduction in the demand for real money balances, and thus lower

16 188 Ch. 6 Money, Inflation and Economic Growth i m(c*) i 1 i 1 =r*+π =ρ+θg+π Welfare Cost of Inflation i 0 i 0 =r*=ρ+θg m 1 m 0 M Figure 6.1: The Demand for Real Money Balances and the Welfare Cost of Inflation liquidity services from money. To the extent that the liquidity services of money yield utility, this implies a welfare cost. 4 We next turn to the analysis of the impact of the growth rate of the money supply in the overlapping generations model of Blanchard and Weil. 4 This approach to the welfare costs of inflation, developed by Bailey [1956] and extended by Friedman [1969], treats real money balances as a consumption good and inflation as a tax on real balances. Fischer [1981] and Lucas [1981], find the cost of inflation to be relatively low, between % of GDP. Lucas [2000] revised his estimate upward, to slightly less than 1% of GDP. The cost is low, but not insignificant, and it appears to rise significantly for high inflation economies. We shall return to this issue in Chapter 10, when we discuss alternative approaches to the demand for money and the economics of high inflation and hyperinflation.

17 George Alogoskoufis, Dynamic Macroeconomics E ects of Monetary Growth in an Overlapping Generations Model We assume, as in the Blanchard Weil model of Chapter 4, that the economy consists of overlapping generations of households born at di erent times in the past. Each generation has an infinite time horizon. nl(t) households are born at each instant t, wherel(t) is total population at time t, and n is the growth rate of the number of households and the overall population. Each household has one member and provides one unit of labor. Consequently, the growth rate of the labor force is also n. Unlike the representative household model, current generations do not internalize the welfare of future generations. We now assume that the instantaneous utility function of households depends on both consumption and real money balances, because of the liquidity services of holding money The Blanchard Weil Model with Money The household born at time j chooses a path for consumption and real money balances in order to maximize the inter temporal utility function, U j = Z 1 s=j e s u (c(j, s),m(j, s)) ds (6.40) subject to the instantaneous asset accumulation equation, ȧ(j, s) =r(s)a(j, s)+w(s)h(s) (s) c(j, s) (r(s)+ (s)) m(j, s) (6.41) and the transversality condition, lim e t!1 R t s=j r(s)ds a(j, t) = 0 (6.42) The variables and parameters are defined as in the case of the representative household model. is the pure rate of time preference of households and u is a quasi concave instantaneous utility function. c(j, s) denotes the consumption of the household born at time j at time s, m(j, s) real money balances of the household born at time j at time s, a(j, s) denotes average assets of the household born at time j at time s, w(s) denotes the real wage per e ciency unit of labor at time t, assumed to the same for all households,

18 190 Ch. 6 Money, Inflation and Economic Growth h(s) denotes labor e ciency per worker, and (s) denotes average taxes (minus transfers) per household at time s. r(s) is the real interest rate at time s, and (s) is inflation, which is equal to expected inflation. Assuming that the instantaneous utility function u takes the form of (6.4) with an intertemporal elasticity of substitution equal to unity, as in Chapters 4 and 5, we can write the instantaneous utility function as, 5 u(c(j, s),m(j, s)) = ln c(j, s)+(1 )lnm(j, s) (6.43) From the first order conditions for a maximum, assuming that the asset of households consist of physical capital, government bonds and money, we can derive the aggregate money demand function, the equation describing the evolution of aggregate consumption, and the equation describing capital accumulation, as, 6 M(t) P (t) = 1 1 C(t) (6.44) r(t)+ (t) Ċ(t) =(r(t) + n) C(t) n K(t)+D(t)+ M(t) P (t) (6.45) K(t) =r(t)k(t)+w(t)h(t)l(t) C(t) C g (t) (6.46) Expressing (6.44), (6.45) and (6.46) per e ciency unit of labor, assuming also that the government has a constant target for primary government expenditure and public debt per e ciency unit of labor, as well as a constant target for the rate of growth of the money supply, we get, m(t) = 1 1 c(t) (6.47) r(t)+ (t) ċ(t) = r(t) g n (1 ) c(t) n k(t)+ r(t)+ (t) d (6.48) k(t) =r(t)k(t)+w(t) c(t) c g (n + g)k(t) (6.49) In (6.48) we have used (6.47) to substitute for the stock of real money balances per e ciency unit of labor. 5 The reason we assume logarithmic preferences is to be able to obtain exact aggregation. This implies little loss of generality. 6 See Chapter 4 on how aggregation takes place in the Blanchard Weil model.

19 George Alogoskoufis, Dynamic Macroeconomics 191 Using the marginal productivity conditions to replace for the real interest rate and the real wage per e ciency unit of labor, (6.48) and (6.49) can be written as, ċ(t) = f 0 (k(t)) g n (1 ) f 0 c(t) (k(t)) + (t) n k(t)+ d (6.50) k(t) =f(k(t)) c(t) c g (n + g + )k(t) (6.51) (6.50) and (6.51) can be used to analyze both the balanced growth path and the adjustment path in terms of the exogenous parameters of the model and the policy variables describing government expenditure, government debt and the rate of growth or the money supply The Real E ects of the Rate of Growth of the Money Supply From (6.50) and (6.51), on the balanced growth path we shall have, c n = f 0 (k ) g n (1 ) f 0 (k ) +µ (n+g) k + d (6.52) c = f(k ) c g (n + g + )k (6.53) (6.52) and (6.53) jointly determine real private consumption and real capital per e ciency unit of labor on the balanced growth path, as a function of parameters of technology, household preferences, the population growth rate and the rate of exogenous technical progress, as well as the parameters describing fiscal and monetary policy. The balanced growth path, and the relevant unique saddle path, are shown diagrammatically in Figure 6.2. ċ = 0 is the steady state consumption function (6.52) and k = 0 is the steady state capital accumulation function (6.53). The balanced growth path and the adjustment path have the usual properties that characterize the model of Blanchard and Weil. The balanced growth path is a saddle point. The new element here is the impact of the growth rate of the money supply on private consumption. 7 A permanent increase in the growth rate of the money supply µ leads to a permanent increase in inflation and nominal interest rates. This in turn leads 7 See Weil [1987], Weil [1991]).

20 192 Ch. 6 Money, Inflation and Economic Growth c c=0 c * E k=0 k * -c g k Figure 6.2: The Balanced Growth Path and the Adjustment Path in the Blanchard Weil Model with Money to a reduction in the demand for real money balances. In this model, the decline in real money balances leads to a corresponding reduction in private consumption by current generations, thus increasing aggregate savings and causing the accumulation of physical capital. The relevant analysis is in Figure 6.3. A previously unanticipated permanent increase in the rate of growth of the money supply, which leads to an increase in inflation, results in a heavier inflation tax for the current generations, who hold higher real money balances compared to future generations who do not hold real money balances. This causes a reduction in their current consumption, an increase in aggregate savings and the initiation of a process of accumulation of physical capital. At the time of the implementation of the policy consumption falls from c E to c 0. The economy adjusts towards a new balanced growth path E 0, which implies higher capital, output and consumption per e ciency unit of labor, due to the substitution towards physical capital caused by higher inflation. In this model, the super neutrality of money does not apply, because the inflation tax impacts di erent generations di erently, and the redistribution it causes in favor of future generations reduces current consumption and increases savings by current

21 George Alogoskoufis, Dynamic Macroeconomics 193 c c=0 c Ε c Ε E E' c 0 k=0 k Ε k Ε -c g k Figure 6.3: Dynamic E ects of an Increase in the Rate of Growth of the Money Supply in the Blanchard Weil Model with Money generations. Thus, as with Ricardian equivalence, the super neutrality of money does not apply in an overlapping generations model, like the Blanchard and Weil model, because in this model current generations do not internalize the welfare of future generations A Dynamic Simulation of the E ects of a Rise in the Rate of Growth of the Money Supply in a Calibrated Blanchard Weil Model In order to gain some insights into the quantitative impact of the rate of growth of the money supply in the overlapping generations model of Blan-

22 194 Ch. 6 Money, Inflation and Economic Growth chard Weil, we shall simulate the model for specific parameter values, assuming the usual Cobb Douglas production function of the form, y(t) =Ak(t) (6.54) Consequently, the model we simulate consists of discrete time versions of, (6.50), for the evolution of private consumption, (6.51), for the accumulation of capital, (6.54), for the production function, (6.47), for the money demand function, the marginal productivity conditions (6.20) and (6.21) for the real interest rate and real wages, the Fisher equation (6.5) for the nominal interest rate and (6.14) for inflation. Where the marginal product of capital or labor appears, this is derived from the production function (6.54). Consequently, the model we simulate consists of (6.50), for the evolution of private consumption, (6.51), for the accumulation of capital, (6.54), for the production function, (6.47), for the money demand function, the marginal productivity conditions (6.20) and (6.21) for the real interest rate and real wages, the Fisher equation (6.5) for the nominal interest rate and (6.14) for inflation. Where the marginal product of capital or labor appears, this is derived from the production function (6.49). In the simulations we use the usual parameter values, the same as?in Chapters 3, 4 and 5. For, the share of consumption in the utility function we use a value equal to 97.5%. With regard to the parameters of fiscal policy it is assumed, that, c g =0.5 and d =0.5. Thus, the paramerer values we use are, A = 1, =0.333, =0.02, n =0.01, g =0.02, =0.03, =0.975, c g =0.5, d =0.5. In Figure 6.4 we present the dynamic e ects of a permanent change in the growth rate of the money supply from 5% to 10%. in the calibrated Blanchard Weil model. We assume that this change is accompanied by corresponding reductions in taxes, which are continuously equal to the increase in the inflation tax on real money balances. There is thus no impact on primary government expenditure or government debt. The change in the growth rate of the money supply by five percentage points, from 5% to 10%, reduces private consumption expenditure immediately, due to the reduction of the real money balances of current generations. This causes an increase in savings and initiates a process of capital accumulation that leads the economy towards a new balanced growth path, with a higher capital stock, higher output, higher private consumption and higher real wages per e ciency unit of labor. On the other hand, the real interest rate falls. Inflation rises by five percentage points, the same as the rise in

23 George Alogoskoufis, Dynamic Macroeconomics 195 Figure 6.4: Dynamic Simulation of an Increase in the Rate of Growth of the Money Supply in a Calibrated Blanchard Weil Model with Money the rate of growth of the money supply, and roughly the same happens to nominal interest rates. However, it is worth noting that the impact of a change in the growth rate of the money supply on the real economy is extremely small. A doubling of the rate of growth of the money supply from 5% to 10%, leads to an increase in steady state real per capita income (and real wages) by only 0.02%, and a reduction in the real interest rate by only percentage points (i.e from 4.239% to 4.234%). The overall savings rate in the balanced growth path rises from 27.69% to 27.70%, again a very slight increase. On the other hand, inflation rises by five percentage points, from 2% to 7%, and nominal interest rates from 6.24% to 11.24%. The rise in the nominal interest rate is due to the rise in inflation as the fall in the real interest rate in negligible. The rise in nominal interest rates results in a fall of the demand for real money balances by 44.5%. This is the only significant real e ect of the doubling of the rate of growth of the money supply. All other real e ects are miniscule.

24 196 Ch. 6 Money, Inflation and Economic Growth We see therefore that, as with deviations from Ricardian equivalence, deviations from the super-neutrality of money in models of overlapping generations, such as the Blanchard Weil model, are quantitatively limited. This is due to the fact that these deviations depend on the product of two quantitatively small parameters: the rate of growth of population, which determines the rate of entry of new generations in the economy, and the pure rate of time preference of households, which determines the percentage of total household wealth that is consumed. With the assumptions we have made, for a population growth rate of 1% per annum and a pure rate of time preference of 2%, their product is equal to just 0.02%. 6.5 Conclusions In this chapter we have analyzed the impact of money and the rate of growth of the money supply in optimizing models of economic growth. Initially we analyzed the role of money in a representative household model, in which money enters the utility function of the representative household. We then analyzed the role of money in a corresponding model of overlapping generations. In growth models with money, one can analyze the determination of both real and nominal variables, such as the price level, inflation and nominal interest rates and examine the dynamic impact of the rate of growth of the money supply. This is a major advantage of monetary over real models. In the representative household model, the growth rate of the money supply has virtually no real e ects, apart from reducing the demand for real money balances, since money does not pay interest. The balanced growth path of all other real variables is independent of the growth rate of the money supply, which only a ects inflation, nominal interest rates and the demand for real money balances by households. This result is known as the super neutrality of money. In a model of overlapping generations, the growth rate of the money supply has real e ects, as it has a di erent impact on the holdings of real money balances and on the level of consumption of di erent generations. When there is an increase of the growth rate of the money supply, older generations, which hold higher real money balances, pay a higher inflation tax than younger generations. Therefore, current aggregate consumption falls, and savings increase. This leads to a higher accumulation of capital and a transition to a balanced growth path with higher capital per e ciency unit of labor. However, it is worth noting that dynamic simulations suggest

25 George Alogoskoufis, Dynamic Macroeconomics 197 that these deviations from the super neutrality of money are quantitatively small for plausible parameter values. The di erences in the e ects of the growth rate of the money supply between the two categories of models are due to the same reasons that government debt has real e ects in overlapping generations models, while it does not have real e ects in a representative household model. In the representative household model, neither government debt nor the growth rate of the money supply cause a redistribution of the tax burden among generations. On the other hand, in an overlapping generations model, both government debt and the rate of growth of the money supply are associated with a redistribution of the tax burden among generations. An increase in government debt redistributes the tax burden from current to future generations, causing an increase in consumption by current generations and a fall in savings and the accumulation of capital. An increase in the rate of growth of the money supply redistributes the tax burden from future to current generations, causing a reduction of aggregate private consumption, and an increase in savings and the accumulation of capital. As is the case with the e ects of government debt, the quantitative significance of the e ects of the rate of growth of the money supply on real variables is small in overlapping generations models with competitive capital markets, because they depend on the product of two quantitatively small parameters, the rate of growth of population and the pure rate of time preference. In conclusion, both representative household and overlapping generations models imply that the rate of growth of the money supply mainly a ects long run inflation and nominal interest rates, and has quantitatively insignificant e ects on per capita real output, consumption and the capital stock. The only real variable that is a ected significantly is the stock of real money balances which depends negatively on the nominal interest rate. These conclusions are in accordance with the evidence on the e ects of the rate of monetary growth across countries that we discussed in Chapter 1 (Figures 1.12 and 1.13), where it was shown that whereas there is a very strong empirical association between monetary growth and inflation, there is an insignificant association between monetary and real growth. As we shall see in Chapter 10, these conclusions are in accordance with a wide variety of other monetary models as well.