Money Demand ECON 40364: Monetary Theory & Policy Eric Sims University of Notre Dame Fall 2017 1 / 37 Readings I Mishkin Ch. 19 2 / 37 Classical Monetary Theory I We have now defined what money is and how the supply of money is set I What determines the demand for money? I How do the demand and supply of money determine the price level, interest rates, and inflation? I We will focus on a framework in which money is neutral and the classical dichotomy holds: real variables (such as output and the real interest rate) are determined independently of nominal variables like money I We can think of such a world as characterizing the \medium" or \long" runs (periods of time measured in several years) I We will soon discuss the \short run" when money is not neutral 3 / 37 Velocity and the Equation of Exchange I Let Yt denote real output, which we can take to be exogenous with respect to the money supply I Pt is the dollar price of output, so Pt Yt is the dollar value of output (i.e. nominal GDP) I Define velocity as as the average number of times per year that the typical unit of money, Mt , is spent on goods and serves. Denote by Vt I The \equation of exchange" or \quantity equation" is: Mt Vt = Pt Yt I This equation is an identity and defines velocity as the ratio of nominal GDP to the money supply 4 / 37 From Equation of Exchange to Quantity Theory I The quantity equation can be interpreted as a theory of money demand by making assumptions about velocity I Can write: 1 Mt = Pt Yt Vt I Monetarists: velocity is determined primarily by payments technology (e.g. credit cards, ATMs, etc) and is therefore close to constant (or at least changes are low frequency and therefore predictable) −1 I Let k = Vt and treat it as constant. Since money demand, d Mt , equals money supply, Mt , our money demand function is: d Mt = kPt Yt I Money demand proportional to nominal income; k does not depend on things like interest rates I This is called the quantity theory of money 5 / 37 Money and Prices I Take natural logs of equation of exchange: ln Mt + ln Vt = ln Pt + ln Yt I If Vt is constant and Yt is exogenous with respect to Mt , then: d ln Mt = d ln Pt I In other words, a change in the money supply results in a proportional change in the price level (i.e. if the money supply increases by 5 percent, the price level increases by 5 percent) 6 / 37 Money and Inflation I Since the quantity equation holds in all periods, we can first difference it across time: (ln Mt − ln Mt−1) + (ln Vt − ln Vt−1) = (ln Pt − ln Pt−1) + (ln Yt − ln Yt−1) I The first difference of logs across time is approximately the growth rate I Inflation, pt , is the growth rate of the price level I Constant velocity implies: M Y pt = gt − gt I Inflation is the difference between the growth rate of money and the growth rate of output I If output growth is independent of the money supply, then inflation and money growth ought to be perfectly correlated 7 / 37 8 / 37 9 / 37 10 / 37 Nominal and Real Interest Rates I The nominal interest rate tells you what percentage of your nominal principal you get back (or have to pay back, in the case of borrowing) in exchange for saving your money. Denote by it I There are many interest rates, differing by time to maturity and risk. Ignore this for now. Think about one period interest rates { i.e. between t and t + 1 I The real interest rate tells you what percentage of a good you get back (or have to pay back, in the case of borrowing) in exchange for saving a good. Denote by rt I Putting one good in the bank ) Pt dollars in bank ) (1 + i )P dollars tomorrow ) purchases (1 + i ) Pt goods t t t Pt+1 tomorrow 11 / 37 The Fisher Relationship I The relationship between the real and nominal interest rate is then: Pt 1 + rt = (1 + it ) Pt+1 I Since the inverse of the ratio of prices across time is the expected gross inflation rate, we have: 1 + it 1 + rt = e 1 + pt+1 e I Here pt+1 is expected inflation between t and t + 1 I Approximately: e rt = it − pt+1 12 / 37 The Natural Rate of Interest I Over the medium to long run, the real interest rate is an equilibrium construct which balances the supply and demand for savings and investment I We sometimes refer to this as the \natural rate of interest" after Knut Wicksell I Simple theory based on the consumption Euler equation with log utility: Ct+1 P = b(1 + rt ) Ct P I r is the natural rate of interest, or the real interest rate consistent with \potential output." Take logs, approximate, and treat consumption growth as equal to output growth: P Y rt = gt+1 − ln b I Intuition based on supply and demand for savings and investment 13 / 37 Money, Inflation, and Interest Rates I Over the medium to long run, the natural rate of interest just depends on output growth and attitudes about saving, captured by b. Independent of monetary factors. Think of this as constant. I Over the medium to long run, we should also expect expected e inflation to equal realized inflation, pt+1 = pt I From the Fisher relationship, this means that nominal interest rates and inflation ought to move together 14 / 37 16 14 Correlation = 0.74 1981 12 1980 1982 10 1979 1984 1983 8 1989 1974 19851990 1973 1978 1988 19691970 6 198620001987 1975 1995 1991 1968 1977 19971996 1976 19981999 19662006 199420071967 1971 4 1965 1972 1964 1959 20011992 1957 Three Month Treasury Bill Rate 1963 2005 19621960 1993 1961 1956 2 19551958 200220082004 2003 0 20092015201020162013201420122011 0 2 4 6 8 10 Inflation Rate 15 / 37 Problems with the Quantity Theory I The quantity theory seems to provide a pretty good theory of inflation and interest rates over the medium to long run I What about the short run? I Problems with the quantity theory: I The shorter term relationships between money growth and both inflation and nominal interest rates are weak I Velocity is not constant and has become harder to predict, particularly since the early 1980s 16 / 37 17 / 37 18 / 37 19 / 37 20 / 37 Moving Beyond the Quantity Theory I The key assumption in the quantity theory is that the demand for money (i.e. velocity) is stable (or at least predictable) I Doesn't seem to be the case, particularly in last several decades I Liquidity preference theory of money demand posits that the demand for real money balances, m = Mt , is an increasing t Pt function of output, Yt , but a decreasing function of the nominal interest rate, it : Mt = L(it , Yt ) Pt − + I But then velocity: Pt Yt Yt Vt = = Mt L(it , Yt ) 21 / 37 Money in the Utility Function I Suppose that there is a representative household who receives utility from consuming goods and holding real money balances, m = Mt . Flow utility: t Pt Mt Mt U Ct , = ln Ct + y ln Pt Pt I Flow budget constraint: Pt Ct + Bt − Bt−1 + Mt − Mt−1 ≤ Pt Yt − Pt Tt + it−1Bt−1 I Bt−1 and Mt−1: stocks of bonds and money household enters t with I Both enter as stores of value. Difference being that bonds pay interest I Household discounts future utility flows by b 2 [0, 1) 22 / 37 Optimality Conditions I Plugging constraints in and taking derivatives yields: 1 1 Pt = bEt (1 + it ) Ct Ct+1 Pt+1 Pt 1 1 Pt y = − bEt Mt Ct Ct+1 Pt+1 I Government's budget constraint with Gt = 0 (similar to above): Pt Tt = (1 + it )BG,t−1 − BG,t − (Mt − Mt−1) I Market-clearing: BG,t = Bt , so Ct = Yt 23 / 37 Money Demand Function I Making use of market-clearing and combining the FOC yields: −1 1 it ymt = Yt 1 + it I Re-arranging: 1 + it mt = yYt it I Demand for real balances: (i) increasing in Yt , (ii) decreasing in it I Zero lower bound: must have it ≥ 0 to get non-negative real balances. At it ! 0, demand for real balances goes to infinity 24 / 37 Baumol-Tobin I You need to spend Y over the course of a period (say, a year) I You keep wealth in the bank earning nominal interest it I You need to determine how many trips you take to bank I Each trip incurs a cost (\shoeleather cost") of K I Let mt denote average real balances holdings over the period. Opportunity cost of holding money is it mt I Each time you withdraw money, you withdraw 2mt dollars. Total trips to bank is Y 2mt I Objective is to pick mt to minimize: Y min it mt + K mt 2mt 25 / 37 Money Demand Function I Use calculus to get first order condition: r KYt mt = 2it I Or re-arranging: 1 KY 2 − 1 m = t i 2 t 2 t I Demand for real balances again increasing in Yt and decreasing in it I There is again a zero lower bound: it ≥ 0 for demand for real balances to be positive 26 / 37 Friedman Rule I Milton Friedman argued that optimal monetary policy in the medium to long run would target a nominal interest rate of zero I With a positive natural rate of interest, this would require deflation I Basic intuition: a positive nominal interest rates dissuades people from holding money by increasing the opportunity cost of liquidity relative to bonds, whereas the marginal cost of producing (fiat) money is essentially zero I At a social optimum, want to equate private cost of holding money (interest rate) to the public cost of producing money (zero) I Holds in both the MIU model (i = 0 maximizes utility) and the B-T model (i = 0 minimizes the cost of holding money) I Why don't central banks follow Friedman rule? Because of the zero lower bound and short run stabilization policy I Does help us understand desire for low interest rates, however 27 / 37 Optimality of i = 0 MIU Model B-T Model 2 4 1.5 m 3 Y 2 K m n + l 1 2 A m " 0.5 i 1 0 0 0 2 4 6 0 2 4 6 i i 28 / 37 Instability of Velocity: Movement Away from Focusing on Monetary Aggregates I Paul Volcker and the Fed experimented with targeting monetary aggregates in the early 1980s I This brought inflation down from the 1970s, but led to high and variable interest rates I Most monetary economists concluded that the demand for money is not in fact stable, i.e.