Notes on Models of Money and Exchange Rates

Notes on Models of Money and Exchange Rates Alexandros Mandilaras University of Surrey May 20, 2002 Abstract This notes builds on seminal contributions on monetary policy to discuss exchange rate regimes and currency crises. An emphasis is placed on mathematical derivations to establish well-known results. This review paper largely and shamelessly follows and explains Obstfeld and Rogoff (1997) [4]. Contents 1 A Simple Monetary Model 1 1.1 The General Setting in Continuous Time... 1 1.2 A Note on Seignorage... 4 2 A Monetary Model of Exchange Rates 5 2.1 The General Setting in Discrete Time... 5 2.2 Fixing the Exchange Rate... 6 2.3 Currency Crises in a Continuous Time Framework... 7 3 Summary 9 1 A Simple Monetary Model 1.1 The General Setting in Continuous Time This section derives a basic monetary framework. I begin by solving Sargent and Wallace s (1973) [5] model to show that an announced increase in money growth implies a jump in prices today. Demand for real balances is assumed to depend only on expected inflation (note the difference with the Keynesian formulation where it depends positively on real income and negatively on the nominal interest rate). Increases in expected inflation increase the opportunity cost of holding money and,

hence, demand for real balances falls. Assume a money demand function of the following form: M t log e = ηπ t (1) where M t denotes money supply in period t, is the price level in period t, η is the semielasticity of demand for real balances with respect to expected inflation π t = E t (log +1 ). Defining actual inflation ṗ t = dlogpt this perfect foresight model dt (meaning π t = dlogpt ) can be written dt ṗ t = 1 η p t 1 η m t. (2) Multiplying this differential equation by ds and also by e 1 η s, and rearranging, we get dp s 1 η s ds 1 η p te 1 η s ds = 1 η m te 1 η s ds. (3) ds e Solving this forward 1 yields τ lim τ The LHS can be written which is equal to d[p s e 1 η s ]= 1 τ η lim m t e 1 η s ds. (4) τ 1 lim [p τe η τ p t e 1 η t ] (5) τ p t e 1 η t (6) as the first term tends to zero as τ. Multiplying (6) by e 1 η t gives p t. Now, multiplying the RHS of (4) by e 1 η t yields 1 η Thus, the solution to the differential equation is p t = 1 η m s e 1 η (t s) ds. (7) m s e 1 η (t s) ds. (8) This solution says that the price level at time t is a discounted average of expected values of the future money supply. 2 So, what happens to the price level today if the central bank announces that it will raise money supply at a specific point in the future? 1 See Klein (1998) [3], chapter 14. 2 It can be shown that the discrete-time non-stochastic version of (1), i.e. m t p t = η(p t+1 p t ), can be solved to yield the equilibrium price level p t = 1 1+η ( η extends to a stochastic environment, where the solution is p t = 1 1+η 2 ) s t 1+η ms. This solution easily ( ) s t η 1+η Et m s.

Initially, money supply is expected to be M 0, but at time t the CB announces that the money stock will jump at a known point in time θ. So, money supply can be described as follows: { M 0,s<t+ θ M s = (9) δm 0,s t + θ, δ > 1 We can write (8) as p t = 1 m s e 1 η (s t) ds. (10) η Now, for a t such that t t t + θ we have p( t) = 1 η which is equal to t+θ t e 1 η (s t) m 0 ds + 1 η t+θ e 1 η (s t) (log e δ + m 0 )ds (11) m 0 [e 1 η (t+θ t) e 0 ] (log e δ + m 0 )[0 e 1 η (t+θ t) ] (12) which, after cancelling out terms, finally yields m 0 + e 1 η (t+θ t) log e δ. (13) We can see the implications in Figure 1. Before the announcement is made p = m 0. As soon as the CB credibly announces its intention to increase the money supply permanently at time t + θ, the price level jumps to m 0 + e θ η loge δ as t = t. The size of the jump depends on the magnitude of the money supply change δ but also on parameters θ and η. Increases in θ (i.e. the further away is the time when the change in money supply will be realised), the lower the value of e θ η and the smaller the jump. The higher the semielasticity of demand for real money balances w.r.t. expected inflation, the higher the jump. As t moves away from t, prices increase, until t takes the value t + θ and prices equal m 0 + log e δ. Let s examine now some properties of (10). Assume that the money supply is growing at the constant rate µ. The rule of integration by parts is b df (s) g(s)ds = a ds [f(b)g(b) f(a)g(a)] b dg(s) f(s) ds. Applying this rule where f(s) 1 a ds = e η (s t) (placing the parameter 1 df (s) in the integral implies that = 1 1 η ds η e η (s t) ) and g(s) =m s yields p t = 1 e 1 η (s t) m s ds = m t + e 1 η (s t) ṁ s ds = m t + ηµ. (14) η t t In other words, if money supply is growing at a rate µ, then the price level is also growing at a rate µ. 3 3 The same result can be reached in a discrete time environment. Details available from me. 3

Price level p= m0 + loge δ p= m + e η 0 loge δ θ p = m 0 t t + θ Time Figure 1: The effects of a CB announcement 1.2 A Note on Seignorage Seignorage can be thought of as the printing of money from the CB to purchase government bonds held by private investors (e.g. banks). Here, we expand on Obstfeld and Rogoff s (1998) analysis, who in turn build on Cagan (1956) [1]. Cagan analysed hyperinflations, and the need for seignorage revenue is regarded as an important element of such episodes. Seignorage = M t M t 1. (15) Finding the optimal level of seignorage is the next step. Printing money results to higher expected inflation, which, in turn, reduces demand for real money balances. So, the marginal revenue from money growth can be negative. If we multiply and divide (15) by M t we get Seignorage = M t M t 1 M t Mt. (16) It can be seen that the second fraction of the RHS could fall and, thus, there is a single rate of inflation for which revenues from seignorage are maximised. Exponentiating (1) we obtain ( ) η M t Pt+1 =. (17) We denote the constant growth rate of money as 1+µ = M t M t 1 = 4 1. (18)

Substituting into (15) we obtain µ 1+µ (1 + µ) η = µ(1 + µ) η 1. (19) Maximising with respect to µ gives Dividing both sides with (1 + µ) η 1 yields (1 + µ) η 1 µ(η + 1)(1 + µ) η 2 =0. (20) 1 µ(η + 1)(1 + µ) 1 = 0 (21) We easily find that µ = 1 η. (22) This shows that the revenue maximising rate of money growth depends inversely on the semielsaticity of real balances with respect to inflation. Often seignorage is confused with the inflation tax. The inflation tax is given by 1 Mt 1 1. (23) Seignorage equals inflation-tax proceeds plus the change in real money holdings. If the economy grows the government can print money to satisfy increased demand for real money balances and not yield higher inflation. Seignorage in this case is greater than the inflation tax. You can see this from M t M t 1 = ( Mt M t 1 1 ) ( Pt 1 + ) Mt 1. (24) 1 2 A Monetary Model of Exchange Rates 2.1 The General Setting in Discrete Time Obstfeld and Rogoff (1998) consider the following demand for money in a discretetime model: m t p t = ηi t+1 + φy t, (25) where i = log(1 + i). Purchasing Power Parity implies that Taking logs Uncovered interest parity implies = E t P t. (26) p t = e t + p t. (27) 1+i t+1 =(1+i t+1)e t E t+1 E t, (28) 5

or in logs i t+1 = i t+1 + E t e t+1 e t. (29) Substituting the PPP and UIP relations into the money equation we obtain (m t φy t + ηi t+1 p t ) e t = η(e t e t+1 e t ) (30) In analogy to the stochastic Cagan model solution presented in footnote 2, we have e t = 1 1+η ( ) s t η E t (m s φy t + ηi t+1 p 1+η s) (31) An increase in money supply raises the price level which in turn raises the value of the exchange rate (depreciation of the domestic currency). An increase in domestic real GDP increases demand for real money balances. With constant money supply, the price level must fall and this reduces the exchange rate and appreciates the domestic currency. Increases in the foreign interest rate lead to capital outflow and depreciation of the home currency. Increases in the foreign price level lead to an appreciation of the domestic currency. The model can be manipulated to show that instability in money supply could lead to greater variability in the exchange rate (see O&R pp 529). 2.2 Fixing the Exchange Rate When the government decides to fix the exchange rate, it abolishes control over its monetary policy. To see this, consider equation (30), where y, i, and p are normalised to zero. We have m t e t = η(e t e t+1 e t ) (32) Now, assume that the government wishes to fix the nominal exchange rate at ē. Substituting e t = e t+1 =ē yields or m t ē = η(ē ē) (33) m t = m =ē (34) This implies that once the exchange rate has been fixed then the money supply has also been fixed. O&R explain that under fixed exchange rates the UIP implies that i = i. Prices are determined through PPP and output is assumed to be at full-employment level. Hence, money demand (as a function of prices, interest rates, and output) is determined exogenously, which leaves money supply to adjust and clear the markets. 6

2.3 Currency Crises in a Continuous Time Framework Consider the following continuous-time equation: m t e t = ηe t (35) If the log exchange rate is fixed at ē the log money supply must remain fixed at m =ē (36) Flood and Graber (1984) [2] assume that the CB monetises government debt, and this function is of first priority when compared to the objective of maintaining the exchange rate value fixed. The CB s balance sheet is depicted in M t = B H,t + ĒB F,t. (37) We have assumed that government debt is monetised. The CB then is required to be constantly increasing its holdings of domestic debt, e.g. at rate µ. This implies that b H = µ. (38) To keep the money supply (and the exchange rate) fixed the CB sells foreign currency bonds. As money supply does not change (Ṁ = 0), this implies Ē B F = B H (39) At some point the CB will run out of foreign bonds to sell and the exchange rate regime shifts (as we have assumed that government finance takes precedence over keeping the exchange rate fixed). The model predicts that the exchange rate will be abandoned before the CB runs out of reserves. Recall that this is a perfect-foresight setting. If there is no speculative attack, the foreign reserves are depleted and the money supply will have to increase (at rate µ) to finance government deficits. This results in a jump in the expected rate of depreciation and price inflation equal to µ. The demand for real money balances drops by ηµ, the semielasticity of money demand with respect to changes in the exchange rate times the expected rate of depreciation. Check that if ṁ = µ then e t = m t + ηµ (an analogous result to (14). But if this outcome is known to agents (and it is) they will not wait for the exchange rate to jump by ηµ after the foreign reserves of the CB have been depleted. They will want to get rid of the domestic currency they hold and acquire the CB s reserves at an earlier stage making an instant profit. This is when the fixed exchange rate is abandoned and the regime shifts to flexible exchange rates. But when exactly? To find the exact timing we define the shadow exchange rate as ẽ t = b H,t + ηµ (40) (recall that m t ē t = ηµ; assuming m t = b H,t we get the above equation.) The shadow exchange rate is the exchange rate that would freely prevail after the attack. 7

Log exchange rate ~e e B A C T Time Figure 2: Timing of a Speculative Attack The speculative activity takes place exactly when the fixed exchange rate intersects with the shadow exchange rate. Note what happens in Figure 2. At point B the abandonment of the fixed rate and the adoption of a floating one means that the exchange rate would predictably experience a downward jump (a currency appreciation). But then speculators anticipating this, would not be attacking the CB in the first place for this would trigger big losses for them. At point C the exchange rate appreciation is expected and hence each speculator has an incentive to sell the domestic currency and buy foreign reserves before the collapse materialises. The speculators will try to move first and get hold of the CB s reserves. Applying this logic backwards in time, the moment when the speculators will hit is at point A: there, the transition from fixed to floating does not imply a perfectly foreseen exchange rate jump either way. Time T can be easily calculated. Assuming that B H,t grows at rate µ, domestic government debt can be written as b H,t = b H,0 + µt. (41) Taking into account conditions at time T (i.e. the shadow rate equals the fixed rate) and substituting in (40) we get ē = b H,0 + µt + ηµ. (42) The solution to this is T = ē b H,0 ηµ. (43) µ Now, before the collapse it holds that ē = log(b H,t + B F,t ). We can rewrite the last 8

equation as T = log(b H,0 + B F,0 ) b H,0 ηµ, (44) µ which says that the time of the attack is more imminent, the higher the degree of monetisation µ, the higher the (log) government debt b H,0 at the initial date, and the smaller the initial foreign reserves B F,0. 3 Summary Basic elements of models of money and exchange rates have been presented in this paper. Building on the intuitions provided here, progress can be made in understanding models of currency crises the second main discussion topic of the Macro Study Group. References [1] Phillip Cagan. The Monetary Dynamics of Hyperinflation. In Milton Friedman s Studies in the Quantity Theory of Money, 1956. [2] Robert P. Flood and Peter M. Garber. Collapsing Exchange Rate Regimes: Some Linear Examples. Journal of International Economics, 17:1 13, 1984. [3] Michael W. Klein. Mathematical Methods for Economics. Addison Wesley, second edition, 2002. [4] Maurice Obstfeld and Kenneth Rogoff. Foundations of International Macroeconomics. The MIT Press, 1997. [5] Thomas Sargent and Neil Wallace. The Stability of Models of Money and Perfect Foresight. Econometrica, 41:1043 1048, 1973. 9