Lecture notes for Macro 2

The main contents of this chapter include all of the following: Money demand (and some supply), the price of money, derivations of the pricing relations, money and sticky prices, money in models of monopolistic competition, money and price setting, monetary policy, empirical measures of the effect of money on output.

Contents Lecture Notes for Macro 2001 (first year PhD course in Stockholm) Paul Săoderlind1 June 2001 (some typos corrected later) Money Demand (and some Supply) 1.1 Money Supply 1.2 Overview of Money Demand 1.3 Money Demand: A General Equilibrium Model with Money in the Utility Function 1.4 The Mechanics of Money Supply∗ The Price of Money 2.1 UIP, Fisher Equation, and the Expectations Hypothesis of the Yield Curve 2.2 The Price Level as an Asset Price: Cagan’s (1956) Model with Rational Expectations 2.3 A Simple Model of Exchange Rate Determination A Derivations of the Pricing Relations A.1 A Real Bond A.2 A Nominal Bond A.3 A Nominal Foreign Bond A.4 Real Effects of Money? A.5 Empirical Evidence on the Pricing Relations University of St Gallen and CEPR Address: s/bf-HSG, Rosenbergstrasse 52, CH-9000 St Gallen, Switzerland E-mail: Paul.Soderlind@unisg.ch Document name: MacAll.TeX Money and Sticky Prices: A First Look 3.1 Basic Models of the Effects of Monetary Policy Surprises 3.2 “Money and Wage Contracts in an Optimizing Model of the Business Cycle,” by Benassy 3.3 “Money and the Business Cycle,” by Cooley and Hansen 4 16 24 24 25 31 38 40 40 41 41 42 45 45 46 56 3.4 X Sticky Wages or Sticky Prices? 61 Money in Models of Monopolistic Competition 4.1 Monopolistic Competition 63 63 Money and Price Setting 5.1 Dynamic Models of Sticky Prices 5.2 Aggregation of One-Sided Ss Rule: A Counter-Example to M → Y∗ 69 69 A Summary of Solution Method for Linear RE Models A.1 Summary A.2 Special Case: Scalar Second Order Equation A.3 An Alternative for the Scalar Second Order Equation: tion Method The Factoriza 88 88 91 97 Empirical Measures of the Effect of Money on Output 7.1 Some Stylized Facts about Money, Prices, and Exchange Rates 7.2 Early Studies of the Effect of Money on Output 7.3 Early Monetarist Studies of the Effect of Money on Output 7.4 Unanticipated or Anticipated Money∗ 7.5 VAR Studies 7.6 Structural Models of Monetary Policy 124 124 125 125 126 126 84 81 81 82 85 A Derivations of the Aggregate Demand Equation Money and Prices in RBC Models Money and Monopolistic Competition Sticky Prices Monetary Policy Empirical Measures of the Effect of Money on Output The Transmission Mechanism from Monetary Policy to Output 80 B Calvo’s Model: An Alternative Derivation Monetary Policy 6.1 The IS-LM Model 6.2 The Barro-Gordon Model 6.3 Recent Models for Studying Monetary Policy 0.3 0.4 0.5 0.6 0.7 0.8 104 106 106 107 108 115 116 119 Reading List 123 0.1 Money Supply and Demand 123 0.2 Price Level and Nominal Assets 123 are used in many different models, for instance as the LM curve is IS-LM models Mt in (1.1) is often a money aggregate like M1 or M3 In most of the models on this course, we will assume that the central bank have control over this aggregate Money Demand (and some Supply) Main references: Romer (1996) (Romer), Blanchard and Fischer (1989) (BF), Obstfeldt and Rogoff (1996) (OR), and Walsh (1998) 1.1 Money Supply References: Burda and Wyplosz (1997) 9, OR 8.7.6 and Appendix 8B, and Mishkin (1997) The really short version: the central bank can control either some monetary aggregate or an interest rate or the exchange rate How they that is typically not very important for most macroeconomic questions Still, this is discussed in Section 1.4 Why they it, that is, the monetary policy, is much more important—and something we will discuss at length later 1.2 Overview of Money Demand 1.2.1 Money in Macroeconomics Reference: Goldfeld and Sichel (1990) Applied money demand equations often take the form ln Mt−1 Pt Mt = b0 + b1 ln Yt + b2 i t + b3 ln + b4 ln + ut Pt Pt−1 Pt−1 (1.2) (1.1) References: Romer 5.2, BF 4.5, OR 8.3, Burda and Wyplosz (1997) The standard money demand equation Mt = constant + ψ ln Yt − ωi t Pt Applied money demand equations where Mt is nominal money holdings, Pt the price level, Yt some measure of economic activity, and i t the net nominal interest rate (like 0.07) The inclusion of Mt−1 /Pt−1 and Pt /Pt−1 is thought to capture partial adjustment effects due to adjustment costs of either nominal (b4 = 0) or real money balances (b4 = 0) For instance, the estimate for Germany (69:1-85:4) reported by Goldfeld and Sichel (1990) is {b1 , b2 , b3 , b4 } = {0.3, −0.5, 0.7, −0.7} (They interest rate used in their estimation is in percentages, that is, like instead of the 0.07 used here, so I have scaled their b2 = −0.005 by 100.) Traditional money demand equations ln Money Demand and Monetary Policy There are many different models for why money is used The common feature of these models is that they all generate something pretty close to (1.1) But why is this broader money aggregate related to the monetary base, which the central bank may control? Short answer: the central bank creates a demand for narrow money by forcing banks to hold it (reserve requirements) and by prohibiting private substitutes to narrow money (banks are not allowed to print bills) The idea behind central bank interventions is to affect the money supply However, most central banks use short interest rates as their operating target In effect, the central bank has monopoly over supply over narrow money which allows it to set the short interest rate, since short debt is a very close substitute to cash In terms of (1.1), the central bank may set i t , which for a given output and price level determines the money supply as a residual 1.2.4 Roles of money: medium of exchange, unit of account, and storage of value (often dominated by other assets) Money is macro model is typically identified with currency which gives no interest The liquidity service of money ( medium of exchange) is emphasized, rather than store of value or unit of account 1.2.2 1.2.3 In general, this type of equation worked fine until 1975, overpredicted money demand during the late 1970s, and underpredicted money demand in the early 1980s Financial innovations? (1.2) has been refined in various ways Various disaggregated money measures have been tried, a wealth of different interest rates and alternative costs have been used, the income variable has been disaggregated, and fairly free adjustment models have been tried (error correction models) Single equation estimation of (1.2) presumes that this is a true demand function, with monetary authorities setting the interest rate, and with the other right hand side variables being predetermined we introduce leisure or credit goods Shopping-time models typically have a utility function is terms of consumption and leisure 1.2.5 Different Ways to Introduce Money in Macro Models 1.3 Reference: OR 8.3 and Walsh (1998) 2.3 and 3.3 The money in the utility function (MIU) model just postulates that real money balances enter the utility function, so the consumer’s optimization problem is ∞ β t u Ct , max∞ {Ct ,Mt }t=0 t=0 Mt Pt (1.3) ∞ β s U (Ct , − lt − n t ) , (1.5) s=0 where lt is hours worked, and n t hours spent on shopping (supposed to give disutility) The latter is typically modelled as some function which is increasing in consumption and decreasing in cash holdings Money Demand: A General Equilibrium Model with Money in the Utility Function Reference: BF 4.5; OR 8.3; Walsh (1998) 2.3; and Lucas (2000) 1.3.1 Model Setup The consumer’s optimization problem is One motivation for having the real balances in the utility function is that having cash may shopping save time in transactions The correct utility function would then be u Ct , L¯ − L t , ∞ {Ct ,Mt }t=0 shopping is a decreasing function of Mt /Pt where L t Cash-in-advance constraint (CIA) means that cash is needed to buy (some) goods, for instance, consumption goods Pt Ct ≤ Mt−1 , (1.4) β t u Ct , max∞ t=0 Mt Pt (1.6) subject to the real budget constraint K t+1 + Mt Mt−1 = (1 + rt ) K t + + wt − Ct − Tt , Pt Pt (1.7) where Mt−1 was brought over from the end of period t − Without uncertainty, this restriction must hold with equality since cash pays no interest: no one would accumulate more cash than strictly needed for consumption purposes since there are better investment opportunities In stochastic economies, this may no longer be true The simple CIA constraint implies that “money demand equation” does not include the nominal interest rate If the utility function depends on consumption only, then all rates of inflation gives the same steady state utility This stands in sharp contrast to the MIU model, where the optimal rate of inflation is minus one times the real interest rate (to get zero nominal interest rate) However, this is not longer true if the cash-in-advance constraint applies only to a subset of the arguments in the utility function For instance, if where rt is the (net) real interest rate (from investing in t − and receiving the return in t), and wt the real wage rate Labor supply is normalized to one The consumer rents his capital stock to competitive firms in each period Tt denotes lump sum taxes Production is given by a production function with constant returns to scale Yt = F (K t , L t ) , (1.8) and there is perfect competition in the product and factor markets The firms rent capital and labor from the households and make zero profits I assume perfect foresight in order to simply the algebra somewhat It is straightforward to derive the same results in a stochastic setting, at least if we assume that variances and covariances not depend on the level of the other variables Under perfect foresight, (1.12) can then be written it Mt = u M/P Ct , + it Pt 1.3.2 Optimal Consumption and Money Holdings Use (1.7) in (1.6) to get the unconstrained problem for the consumer ∞ max β t u (1 + rt ) K t + { K t+1 ,Mt }∞ t=0 t=0 Mt−1 Mt Mt + wt − Tt − K t+1 − , Pt Pt Pt (1.9) /u C Ct , Mt Pt , (1.14) which highlights that the nominal interest rate is the relative price of the “money services” we get by holding money one period instead of consuming it Note that (1.14) is a relation between real money balances, the nominal interest rate, and an activity level (here consumption), which is very similar to the LM equation The first order condition for K t+1 is u C Ct , Mt Pt = (1 + rt+1 ) βu C Ct+1 , Mt+1 , Pt+1 (1.10) which is the traditional Euler equation for real bonds (with uncertainty we need to take the expected value of the right hand side, conditional on the information in t) It would also hold for any other financial asset The first order condition for Mt is u C Ct , Mt Pt = u M/P Ct , Mt Pt + βu C Ct+1 , Mt+1 Pt+1 Pt Pt+1 (1.11) If money would not enter the utility function, then this is a special case of (1.10) since the real gross return on money is Pt /Pt+1 It is not obvious, however, that we get an interior solution to money holdings unless money gives direct utility The left hand side of (1.11) is the marginal utility lost because some resources are taken from time t consumption, and the right hand side is the marginal utility gained by having more cash today and the extra consumption this allows tomorrow (cash provides utility and is also a form of saving, whose purchasing power depends on the inflation) Substitute for βu C (Ct+1 , Mt+1 /Pt+1 ) from (1.10) in (1.11) and rearrange to get uC Mt Ct , Pt Pt 1− + rt+1 Pt+1 = u M/P + i t = Et (1 + rt+1 ) Pt+1 , Pt Mt Ct , Pt (1.12) The Fisher equation is Example (Explicit money demand equation from Cobb-Douglas/CRRA.) Let the utility function be = 1−γ Mt u Ct , Pt Ctα Mt Pt 1−α 1−γ , in which case (1.14) can be written Mt − α + it = Ct , Pt α it which is decreasing in i t and increasing in Ct This is quite similar to the standard money demand equation (1.1) Take logs and make a first-order Taylor expansion of ln [(1 + i t ) /i t ] around i ss ln Mt = constant + ln Ct − it Pt i ss (1 + i ss ) Compared with the money demand equation (1.1), ψ ln Yt is replaced by ln Ct and ω = 1/ [i ss (1 + i ss )] If i ss = 5%, then ω ≈ 20, which appears to be very high compared to empirical estimates Example (Explicit money demand equation from Lucas (2000) Let the utility function be −1 1−γ Ct Mt = Ct + B , u Ct , Pt 1−γ Mt /Pt (1.13) where the convention is that the nominal interest rate is dated t since it is known as of t 1.3.4 in which case (1.14) can be written it = + it Ct Mt /Pt Two definitions: B, or Mt it = Ct B 1/2 Pt + it • Neutrality of money: the real equilibrium is independent of the money stock −1/2 , • Superneutrality of money: the real equilibrium is independent of the money growth rate which can be written (approximately) on the same form as (1.1) ln Mt it ≈ constant + ln Ct − Pt i ss (1 + i ss ) This gives a value of the ω in the money demand equation (1.1) which is only half of that in Example 1.3.3 Steady State Let the money growth rate be σ , so Mt /Mt−1 = + σ A steady state implies that inflation, consumption, and real money balances are constant: Pt /Pt−1 = + π, Ct /Ct−1 = 1, and (Mt /Pt ) / (Mt−1 /Pt−1 ) = This implies that π = σ The first order condition for K , (1.10), can then be simplified as + rss = 1/β Combining with (1.16) gives that steady state capital stock must solve General Equilibrium with MIU We now add a few equations to close the MIU model in a closed economy The government budget is assumed to be in balance in every period (not restrictive since Ricardian equivalence holds in this model) Ms − Ms−1 −Ts = , Ps (1.15) so the seigniorage (right hand side) is distributed as lump sum transfers (negative taxes) Note that this is taken as given by each individual agent Competitive factor markets, constant returns to scale, and a fixed labor equal to one (recall that is was not part of the utility function) give ∂ F (K t , 1) , and ∂K ∂ F (K t , 1) wt = F (K t , 1) − K t ∂K rt = (1.16) (This follows from that w = ∂ F/∂ L and r = ∂ F/∂ K and that constant returns to scale implies F = L∂ F/∂ L + K ∂ F/∂ K ) In general, the price level is determined jointly with the rest of the dynamic equilibrium In special cases, as with log utility and complete depreciation of capital (as in the model of B´enassy (1995)) there is a closed form solution However, in most cases, the equilibrium must be computed with numerical methods 10 FK (K ss , 1) = rss = 1/β − 1, (1.17) which depends only on the technology and the real discount rate, not on the money stock or growth In steady state, the capital stock is constant so Css = F (K ss , 1), so consumption in steady state is uniquely determined by the real side of the economy: in the steady state of this model, money is neutral and superneutral Note that this is not true for the dynamics around the steady state unless marginal utility of consumption is independent of the real money balances (see Walsh (1998) 2.3 for a textbook treatment) With a value for steady state consumption, we can solve for the steady state real money balances, Mss /Pss , by combining the first order Mss i ss = u M/P Css , + i ss Pss /u C Css , Mss Pss , (1.18) and the Fisher equation + i = (1 + rss ) Pt+1 /Pt = (1 + rss ) (1 + π ) (1.19) Using (1.19) in (1.18) gives an equation in Mss /Pss and known model parameters 11 1.3.5 The Welfare Cost of Inflation The welfare cost of inflation is typically analyzed for the steady state, since we can then make use of the superneutrality of money The growth rate of money, and therefore the inflation rate and nominal interest rate, can then be changed without affecting the real equilibrium The welfare loss from a higher nominal interest rate is often measured as the extra consumption needed in order to achieve the same utility as in the case with lower interest rate The approach is typically to find the money demand function which expresses real money balances as a function of the consumption level and the nominal interest rate M P = f (i, C), and calculate utility as the value of the period utility function u [C, f (i, C)] If C = in steady state, a certain interest rate i gives the utility u 1, f i , The welfare loss from another nominal interest rate, i , is the value of C which solves Fraction of consumption, % Money in utility function: cost of i>3% 1.5 Cobb−Douglas, α=0.99 Lucas, B=0.0018 0.5 Nominal interest rate, % 10 Figure 1.1: Utility loss, in terms of consumption, of inflation in two MIU models u 1, f i , = u C, f i , C (1.20) To get the same utility as with C = and i = 3% consumption must be This value of C is the compensation that the consumers need to be as well off with the interest rate i as with i Note that C − can be interpreted as the percentage change in consumption needed to compensate for the higher nominal interest rate Example (Welfare loss with Cobb-Douglas/CRRA.) From Example 1, the utility at the nominal interest rate i is u [C, f (i, C)] = 1−γ Cα C 1−α1+i α i 1−α 1−γ Consider a steady state where C = and suppose that inflation is zero, so i = r For instance, to get the same utility as at C = and i = 3%, u [1, f (0.03, 1)], then consumption must be 1.03/ (1 + i) 1−α = C 0.03/i Example (Welfare loss from Lucas (1994).) From Example we get 1/2 −1 1−γ  i 1+i + B 1/2 0.03 1/2 1.03 = C Figure 1.1 illustrates the result for B = 0.0018 (Lucas’ point estimate) Also a cash-in-advance model (see, for instance, Cooley and Hansen (1989)) can generate welfare costs of inflation (more precise: of a non-zero nominal interest rate) if the cash-in-advance restriction applies only to a subset of the arguments in the utility function A positive inflation acts like a tax on those goods that must be paid in cash, and thereby creates a distortion Friedman’s Rule for Optimal Money Supply Reference: Romer 9.8, BF 4.5 Friedman suggested a money rate growth which would set the nominal interest rate to zero and thereby saturate money demand The idea is that bills are (virtually) costless to print and it has a (utility) value for agents, so why not give them as much as they would possibly would like to have? Figure 1.1 illustrates the result for α = 0.99   it u [C, f (i, C)] = C + B 1/2 1−γ + it 1/2 + B 1/2 12 13 We know that the steady states of the real variables is unaffected by the inflation rate (see above), so if we concentrate attention to the steady state, then (1.14) tells us that consumers are satiated with real money balances if u M/P = 0, that is, if i = By the Fisher equation (1.13) this means that the monetary policy should set (1 + rt ) Pt+1 /Pt = 1, so the rate of deflation should equal the real interest rate In this way, holding cash gives the same return as a real bond, so savers will be happy to keep large real money balances and to get the utility out of it In steady state, inflation equals the money growth rate, so a deflation requires a shrinking money supply, which means that seigniorage is negative—see (1.15) In this setting, this is compensated by lump-sum taxes, which highlights the assumption that the government revenues from the inflation tax is either wasted or can be raised in other, nondistortive, ways If, instead, a certain revenue must be raised and the alternative taxes are distortive, then it may no longer be optimal with a zero inflation rate See Walsh (1998) If the utility function is separable in consumption and real money balances, then this result hold in general, not just in steady state The Welfare Cost of Inflation - Other Arguments Reference: Fischer (1996), Romer 9.8, Driffil, Mizon, and Ulph (1990), and Walsh (1998) 4.5-4.6 Inflation raises the effective capital income tax (subsidy), since the nominal return (loss) is taxed (part of which is just compensation for inflation) The real net of tax return is r net = (1 − τ ) i − π = (1 − τ ) r − τ π, (1.21) where the Fisher relation gives i = r + E π and we assume that π = E π This distorts the savings decision Some calculation for the US (Feldstein, NBER, 1996) suggest that this effect is large (twice as large as the effect on government revenues) Counter-argument: a lower inflation and therefore lower government revenues from capital income taxation is likely to bring higher tax rates Costs of price adjustments and indexation 14 Some empirical evidence that really high inflation is bad for growth It is (both theoretically and empirically) unclear if zero inflation is better for growth than 5% inflation Seigniorage is low for most OECD countries (less than one percent of GDP, see OR 8.2) Low inflation means that it will be hard to drive down the real interest really low to stimulate output (The nominal interest rates cannot be negative since the nominal return on cash is zero.) Variable inflation may lead to large inflation surprises which redistribute wealth, increases uncertainty (affects savings in which way?), and increases the information costs 1.3.6 The Relation to Traditional Macro Models Equation (1.14) is a money demand equation, which in many cases can be approximated by ln Mt − ln Pt = γ1 ln Ct − γ2 i t , (1.22) which is a traditional LM equation When the utility function is separable in consumption and real money balances, then the optimality condition for consumption (1.10) can often be approximated by −γ ln Ct = ln (1 + rt+1 ) + γ ln Ct+1 , (1.23) where consumption growth is related to the real interest rate From the Fisher equation, we can replace ln (1 + rt+1 ) by i t −Et (ln Pt+1 − ln Pt ) This is clearly reminiscent of an IS equation 1.3.7 The Price Level The price level is determined simultaneously with all other variables, and there is typically no closed form solution In the special case where the utility function in (1.6) is separable, so the Euler equation for consumption (1.10) is unaffected by real money balances, and where money supply 15 is exogenous is might be possible to arrive at an analytical expression for the price level In this case, we can solve for the real equilibrium (consumption, real interest rates, etc) without any reference to money supply The price level can then be found by solving (1.11) and information about money supply This is an example of a classical dichotomy i a Interest rule supply M Example (Solving for the price level.) Use the approximate Fisher equation, i t =Et ln Pt+1 − ln Pt + rt+1 , in the approximate money demand equation in Example i b Money rule supply ln Mt − ln Pt = a + ln Ct − (Et ln Pt+1 − ln Pt + rt+1 ) , i ss (1 + i ss ) and rewrite as ln Pt M i ss (1 + i ss ) + 1 = −a − ln Ct + ln Mt + rt+1 + Et ln Pt+1 , i ss (1 + i ss ) i ss (1 + i ss ) i ss (1 + i ss ) c Mixture i supply which is a forward looking difference equations for ln Pt in terms of the “exogenous” variables ln Ct , ln Mt , and rt+1 M 1.4 The Mechanics of Money Supply∗ References: Burda and Wyplosz (1997) 9, OR 8.7.6 and Appendix 8B, and Mishkin (1997) The short version: the central bank can control either some monetary aggregate or an interest rate or the exchange rate This section is about how they that, even if this is not particularly important for most macroeconomic issues Why they it, that is, the monetary policy, is much more important—and something we will return to later 1.4.1 Operating Procedures of the Central Bank Suppose demand for the monetary base is decreasing in the nominal interest rate Suppose the central bank does no interventions at all Shifts in the demand curve for money will then lead to movements in the nominal interest rate Alternatively, suppose the central bank announces a discount rate where any bank can lend/borrow unlimited amounts This will fix the interest rate and any shifts in the demand curve leads to movements in the monetary base (as the banks are free to borrow reserves and currency at the fixed rate) Finally, it is possible to strike a compromise between these two extremes letting banks 16 Figure 1.2: Partial equilibrium on money market lend at increasing interest rates This effectively creates an upward sloping supply curve for the monetary base This is illustrated in Figure 1.2 1.4.2 Money Supply and Budget Accounting Money supply has a direct effect on government finances Consider the consolidated government sector (here interpreted as treasury plus central bank) The real budget identity is Mt−1 Bt Mt Bt−1 Gt + = Tt + + , (1.24) (1 + i t−1 ) + Pt Pt Pt Pt where G t and Tt are real government expenditures revenues, respectively, Bt is nominal debt, i t the nominal interest rate, Mt is the monetary base (the central bank liabilities), and Pt the price level 17 1.4.4 Assume the Fisher equation holds, so the nominal interest rate is + i t−1 = Et−1 (1 + rt ) Pt , Pt−1 (1.25) where rt is the real interest rate The convention is that the nominal interest rate is dated t − since it is known as of t − To simplify, assume rt is known in t − We then get from (1.25) that the real debt in t is Bt Bt−1 Pt−1 Pt Mt − Mt−1 = G t − Tt + (1 + rt ) Et−1 − Pt Pt−1 Pt Pt−1 Pt (1.26) Consider the case where real government expenditures and tax revenues are unaffected by monetary policy (money is neutral), and where the central bank increases money supply, Mt > Mt−1 This drives down the real value of government debt, Bt /Pt in two ways First, the real revenues from money creation (printing), called seigniorage, is (Mt − Mt−1 )/Pt Second, the money supply increase will probably increase the price level If this increase is unanticipated, then actual inflation exceeds expected inflation, Pt−1 /Pt Et−1 (Pt /Pt−1 ) < 1, so the real value of government debt brought over from t − decreases 1.4.3 Money Aggregates and the Balance Sheet of the Central Bank The liabilities of the central bank are currency (Cu) plus banks reserves (Re) deposited in the central bank, the assets are the foreign exchange reserve, the holding of domestic bonds, and perhaps gold Balance Sheet of Central Bank Liabilities Assets Domestic bonds Currency (Cu) Foreign currency Reserve deposits (Re) Foreign bonds Net worth Gold 18 Reserve Requirements and Deposits Money stock, M, is currency, Cu, plus deposits, D, (also called “inside money” since it is generated inside the private banking system) M = Cu + D (1.27) Suppose that private banks (because of reserve requirements or prudence) hold the fraction r of deposits, D, in reserves, Re This means that an increase in reserves, Re, allows the bank to increase deposits with the reserve multiplier, 1/r , D= Re r (1.28) These new deposits may be lent to someone (an thereby bring in profits to the bank, assuming the lending rate is above the deposit rate) Note that if r goes to zero, then the central bank cannot control the creation of new deposits by affecting the availability of reserves Reserve requirements mean that a private bank must hold a fraction (usually a few percentage) of the (checkable) deposits in either cash (in the vault) or as reserves with the central bank This fraction is often specified as an average over some period (two weeks in the US) Suppose a bank needs to get more reserves (maybe depositors withdrew money during the preceding week) It can then either sell some assets, borrow from other banks (“federal funds” market in the US), or borrow from the central bank (at the Fed’s “discount window” in the US) Note that borrowing from the central bank is effectively a decrease (as long as the loan lasts) in the reserve requirement Therefore something has to be done to make the reserve requirements bite in spite of the possibility to borrow from the central bank This is typically a penalty rate for these loans, or some kind of administrative rationing of loans It is the fact that r < that makes banks different from other financial institutions To see why, suppose r = Then all deposits would have to be kept as reserves and couldn’t be used for lending Consequently, any lending has to be done from the banks own capital In this sense, the bank is not an intermediary any more and cannot “create money.” The money stock deposits discussed above can be interpreted/measured in different ways The most common monetary aggregates are: M1 (currency, travellers’ checks, 19 This system of expectational difference equations (with stable and unstable roots) can be solved in several different ways For instance, a decomposition of A − B F in terms of eigenvalues and eigenvectors will work if the latter are linearly independent Otherwise, other techniques must be used (see, for instance, Săoderlind (1999)) A necessary condition for a unique saddle path equilibrium is that A − B F has as many stable roots (inside the unit circle) as there are predetermined variables (that is, elements in x1t ) To solve the model numerically, parameter values are needed The following values have been used in most of Figures 6.2-6.4 (exceptions are indicated) β δ φ γ τπ τ y υ χ λ y λi 0.99 2.25 2/7 0.5 0.5 0.5 1.5 0.5 Suppose the central bank’s loss function is ∞ β s L t+s , where (6.34) s=0 L t+s = πt+s − π ∗ + λ y yt+s − y ∗ + λi i t+s − i ∗ π y i 2 0 −2 −2 −2 period c Large output coefficient Optimal Monetary Policy Et b Large inflation coefficient (6.35) A particularly straightforward way to proceed is to optimize (6.34), by restricting the policy rule to be of the simple form discussed above, (6.25) Optimization then proceeds 100 −2 period Persistent price shock: simple policy rule The choice of δ implies relatively little price stickiness The choice of φ means that a 1% increase in aggregate demand leads to a desired increase of the relative price of 2/7% The choice of the relative risk aversion γ implies an elasticity of intertemporal substitution of 1/2 The υ and χ are those advocated by Taylor The loss function parameters (see next section) means that inflation is twice as important as output, and that the policy maker does not care about fluctuations in the nominal interest rate The first subfigure in Figure 6.2 illustrates how the model with the policy rule (6.25) works An inflation shock in period t = increases inflation The policy maker reacts by raising the nominal interest even more in order to increase the real interest rate This, in turn, has a negative effect on output and therefore on inflation via the “Phillips curve.” The central bank creates a recession to bring down inflation The other subfigures illustrates what happens if the coefficients in the reaction function (6.25) are changed 6.3.2 a Baseline model −2 −2 period Figure 6.2: Impulse responses to price shock; simple policy rule as follows: guess the coefficients υ and χ, solve the model, use the time series representation of the model to calculate the loss function value Then try other coefficients υ and χ, and see if they give a lower loss function value Continue until the best coefficients have been found The unrestricted optimal commitment policy and the optimal discretionary policy rule are a bit harder to find Methods for doing that are discussed in, among other places, Săoderlind (1999) Figure 6.3 compares the equilibria under the simple policy rule, unrestricted optimal commitment rule, and optimal discretionary rule, when it is assumed that π ∗ = y ∗ = It is clear that the optimal commitment rule achieves a much more stable inflation and output, in spite of a less vigorous increase in the nominal interest rate This is achieved by credibly promising to keep interest rates high in the future (and even raise further), which gives expectations of lower future output and therefore future inflation This, in turn, 101 a Simple policy rule b Commitment policy a Simple policy rule b Commitment policy 4 2 0 0 −2 −2 −2 −2 π y i −2 period c Discretionary policy −2 period −2 0 −2 −2 period period 8 −2 period −2 period Persistent demand shocks −2 π y i c Discretionary policy Persistent price shocks 4 Figure 6.3: Impulse responses to price shock: simple rule, optimal commitment policy, and discretionary policy Figure 6.4: Impulse responses to positive demand shock: simple rule, optimal commitment policy, and discretionary policy gives lower inflation and output today The discretionary equilibrium is fairly similar to the simple rule in this model Note that there is no constant “inflation bias” when target levels are at their natural levels (zero) as they are in these figures The discretionary rule is still different from the commitment rule (they are, after all, outcomes of different games) The intuition is that there is a time-varying “bias” since the conditional expectations of output and inflation in the next periods (their “conditional natural rates”) typically differ from the target rates (here zero) Figure 6.4 makes the same type of comparison, but for a positive demand shock, −ε yt In this case, both optimal rules “kill” the demand shock, which is seen almost directly from (6.24): any shock ε yt could be met by increasing i t by γ ε yt In this way output is unaffected by the shock, and there will then be no effect on inflation either, since the only way the demand shock can affect inflation is via output (see (6.23)) This is very similar to the static model discussed above: the demand shock drives both prices and output in the same direction and should, if possible, neutralized Of course, the result hinges on the assumption that the policy maker is not averse to movements in the nominal interest rate, that is, λi = in (6.35) (It can be shown that this case can be approximated in the simple policy rule (6.25) by setting the coefficients very high.) Many studies indicate that central banks are unwilling to let the nominal interest rate vary much This is sometimes interpreted as a concern for the banking sector, and sometimes as due to uncertainty about the state of the economy and/or the effect of policy changes on output/inflation In any case, λi > is often necessary in order to make this type of model fit the observed variability in nominal interest rates 102 103 A Derivations of the Aggregate Demand Equation King, R G., 1993, “Will the New Keynesian Macroeconomics Resurrect the IS-LM Model?,” Journal of Economic Perspectives, 7, 67–82 The period utility function is U (Ct ) = At 1−γ C , 1−γ t (A.1) Obstfeldt, M., and K Rogoff, 1996, Foundations of International Macroeconomics, MIT Press where At is a taste shift parameter The Euler equation for optimal consumption is ∂U (Ct+1 ) ∂U (Ct ) = βEt Q t+1 , ∂Ct ∂Ct+1 Mishkin, F S., 1997, The Economics of Money, Banking, and Financial Markets, Addison-Wesley, Reading, Massachusetts, 5th edn (A.2) where Q t+1 is the gross real return The marginal utility of Ct is Romer, D., 1996, Advanced Macroeconomics, McGraw-Hill Rotemberg, J J., 1987, “New Keynesian Microfoundations,” in Stanley Fischer (ed.), NBER Macroeconomics Annual pp 69–104, NBER ∂U (Ct ) −γ = At C t , Ct (A.3) Săoderlind, P., 1999, Solution and Estimation of RE Macromodels with Optimal Policy,” European Economic Review, 43, 813–823 so the optimality condition can be written At+1 = βEt Q t+1 At Ct+1 Ct = βEt exp (ln Q t+1 + Walsh, C E., 1998, Monetary Theory and Policy, MIT Press, Cambridge, Massachusetts −γ ln At+1 − γ ln Ct+1 + γ ln Ct ) (A.4) Assume that ln Q t+1 , ln At+1 , and ln Ct+1 are jointly normally distributed (Recall Eexp (x) = exp (Ex + Var (x) /2) is x is normally distributed.) Take logs of (A.4) and rewrite it as = ln β + Et ln Q t+1 + Et ln At+1 − γ Et ln Ct+1 + γ ln Ct + Vart (ln Q t+1 + ln At+1 − γ ln Ct+1 ) /2, or 1 Et ln Ct+1 = ln Ct + Et ln Q t+1 + Et z t+1 , γ γ (A.5) where Et z t+1 = ln β+Et ln At+1 +Vart (.) The most important part of Et z t+1 is Et ln At+1 If ln At+1 = ρ ln At + u t+1 , then Et ln At+1 = (ρ − 1) ln At , so the AR(1) formulation carries over to the expected change, but the sign is reversed if ρ > Bibliography Blanchard, O J., and S Fischer, 1989, Lectures on Macroeconomics, MIT Press 104 105 The monetary contraction in many countries after the 1929 crash in the US can probably be regarded as exogenous shifts Countries which left the gold standard early had weaker recessions (OR Fig 9.10) Empirical Measures of the Effect of Money on Output Reference: Romer 5.6, Walsh (1998) 1, Mishkin (1997) 25, Isard (1995), Meese (1990), and Obstfeldt and Rogoff (1996) 9.1 7.1 Some Stylized Facts about Money, Prices, and Exchange Rates The correlation between long run inflation and money growth is almost one across countries (BW Fig 8.9a) The correlation between short run inflation and money growth is more uncertain (BW Fig 10.10) There is no clear long run correlation between inflation and the growth of real output or between money growth and the growth of real output Money stock innovations and output innovations are correlated Money stock changes seems to lead output changes Most of this correlation is between output and “inside money” (created within the banking system, for instance, deposits) PPP does not hold, except possibly for long horizons Realignments seem to have long lasting (although perhaps not permanent) effects on the real exchange rate (BW Fig 8.9b) Countries with weak current accounts, and rapidly expanding money supply often experience depreciation of the exchange rate Real exchange rates are much more volatile under flexible exchange rate regimes than under fixed exchange rates Real and nominal exchange rates are very strongly correlated (this is evidence of monetary non-neutrality only if we can prove existence of important nominal shocks) (Isard Fig 3.2 and 4.2) 10 Most contracts are written in nominal terms, and prices are typically changed fairly seldom—even at the fairly high inflation rates of the late 1970s This seems to change as we move into very high inflation rates The way it changes is by indexation or “dollarization.” 11 Sharp exogenous monetary contractions (or a sudden and unexpected increase in the short interest rates by the central bank) seems to have an effect on output and employment which may last for years (Walsh Fig 1.3) 7.2 Early Studies of the Effect of Money on Output Early Keynesians (until the 1960s, say) thought that money has little effect on output There were several reasons for this First, nominal interest rates (on high grade bonds) were very low during the Great Depression, but that did not appear to boost output Second, investment and consumption regressions showed very little effect of nominal interest rates The idea that the Great Depression was a period loose monetary policy was heavily challenged by Friedman and Schwartz (1963a) They showed that the decline in money supply was the largest ever in US history (A possible counter argument to this is that the monetary base changed fairly little, and that most of the change in the money aggregates were due to the fact that the public chose to hold more cash and less deposits, and that banks chose to hold more reserves.) Another weak spot of the early Keynesian interpretation of the Great Depression is that it is based on low nominal interest rates Prices were falling, so the real interest rates were actually very high, perhaps as high as 10%, which could be interpreted as a very tight monetary policy The log exchange rate behaves almost like a random walk (that is, ln St ≈ ln St−1 + u t ) Equations for predicting ln St − ln St−1 typically have R < 0.1 106 107 7.3 Early Monetarist Studies of the Effect of Money on Output 7.3.2 7.3.1 Friedman&Schwartz and St Louis Equations Friedman and Schwartz (1963a) and Friedman and Schwartz (1963b) study 100 years of US data and find that money aggregates lead output, in the sense that all recessions were preceded by declining money growth rates However, the response to money growth changes showed “long and variable lags.” They also argue that many of the money supply changes can be regarded as exogenous with respect to output (changes within the monetary sector) The St Louis equation (see, for instance, Andersen and Jordan (1968)) is a regression of output (growth) on current and lagged money (growth) and perhaps some other variables This can be thought of as a formalization of the approach of Friedman and Schwartz, although no attention is paid to whether the money supply changes are exogenous or not In its simplest form it is αs m t−s + εt , yt = On the Interpretation of Cov(yt , m t ) I: Reverse Causality The problem with a causal interpretation of the correlation between money and output, or of the St Louis equation, is that most of the correlation between money and output is between output and “inside money” (deposits which is money created within the private banking system, as opposed to the monetary base which is outside money) It is possible that banks extend more loans (which generates deposits) as the business conditions are about to pick up The correlation between money and output is then due to reverse causality as discussed in King and Plosser (1984) The idea is that money may not have any effect on output, but output affects money aggregates positively, which explains the positive correlation of output and money For this to make sense, it must be the case that the central bank cannot, or does not want to, control the broad monetary aggregates Example 33 (Regressing output on money, two-way causality.) Suppose the structural model of output and money is yt = βm t + u yt (7.1) s=0 m t = γ yt + u mt , and the αs coefficients are sometimes interpreted as the effect of the money stock, m t−s , on output, yt (Initially, St Louis equations were used to explain nominal output, but here the focus is on real output.) The St Louis equation is not a structural model; it is a reduced form Monetarists would perhaps argue that the transmission mechanism from money to output is very complex, and that it makes sense to use (7.1) since it can potentially summarize the effects Keynesians would perhaps be more “structural” in the sense that their view of the transmission mechanism is quite clear: money supply affects the interest rate (the LM equation), which in turn affects output (IS equation) Any causal interpretation of the correlation between money and output, or of the St Louis equation relies on the assumption that most movements in the money stock are due exogenous forces and not due to changes in output These exogenous forces could be policy shocks or shifts in money demand which are not caused by output 108 where the shocks are assumed to be uncorrelated Output and money are here allowed to depend on each other (in the same period) The reduced form is yt mt = 1 − βγ β γ u yt u mt Suppose we run a very simple St Louis equation yt = αm t + εt , 109 The least squares (LS) estimate of α is (in probability limit, plim) plim αˆ L S = = 7.3.3 Cov (m t , yt ) Var (m t ) Cov γ u yt + u mt , u yt + βu mt Instead of assuming that output affects money stock directly, it may be that there is a third variable which drives both output and money This will also make the interpretation of money-output correlations very murky, since it gives an omitted-variables bias Var γ u yt + u mt = γ Var u yt + βVar (u mt ) = γ Var u yt /Var (u mt ) + β γ Var Example 37 (Regressing output on money, unobservable driving force.) Suppose the structural model of output and money is u yt + Var (u mt ) γ Var u yt /Var (u mt ) + On the Interpretation of Cov(yt , m t ) II: Common Driving Force m t = z t + u mt yt = βm t + κz t + u yt Example 34 (Exogenous money supply.) If m t is exogenous, γ = 0, then plim αˆ L S = β in Example 33 and the regression coefficient captures the effect of money shocks on output This is the interpretation in Friedman and Schwartz (1963a) We get the same result if Var u yt /Var(u mt ) → 0, that is, when most of the movements in output (and money) is caused by the exogenous money shocks, so money is once again essentially exogenous Example 35 (Output shocks dominate.) Conversely, Var u yt /Var(u mt ) → ∞ gives plim αˆ L S → 1/γ in Example 33 (use l’Hˆopital’s rule to show this), so the regression coefficient merely reflects how money (and output) are both driven by output shocks (“reverse causality”) The reverse causality story can be turned on its head, however Suppose money have effect on output, but that the central bank tries to stabilize output, that is, output has a negative effect on money supply In this case, the correlation of money and output will underestimate the effect of money on output Example 36 (Countercyclical policy.) From the reduced form in Example 33, we see that output follows β u yt + u mt yt = − βγ − βγ Suppose β > 0, then a negative γ makes the effect of an output shock small This gives plim αˆ L S < β, so the St Louis equation has a negative bias in the direct effect of money on output = (κ + β) z t + u yt + βu mt where the shocks are uncorrelated with each other and with z t Money affects output if β = 0, but output does not affect money However, both money and output are driven by the common factor z t The LS estimate of α in the St Louis equation, yt = αm t + εt , is then Cov (m t , yt ) Var (m t ) Cov z t + u mt , (κ + β) z t + u yt + βu mt = Var (z t + u mt ) (κ + β) Var (z t ) + βVar (u mt ) = Var (z t ) + Var (u mt ) (κ + β) Var (z t ) /Var (u mt ) + β = Var (z t ) /Var (u mt ) + plim αˆ L S = This equals β only if z t is (effectively) a constant, Var(z t ) /Var(u mt ) = 0, or if output is not affected by z t , κ = 0, so there is no common driving force Example 38 (Unobservable driving force and “reverse causality.”) Suppose β = in Example 37, so money has no effect on output (nor has output any effect on money) In this case, the estimate of α in the St Louis equation, yt = αm t + εt , becomes plim αˆ L S = κVar (z t ) /Var (u mt ) , Var (z t ) /Var (u mt ) + which has the same sign as κ If we interpret yt as output in the next period, which thus depends on today’s z t , then we have the case of King and Plosser (1984) Their 110 111 mechanism is that z t signals high future productivity, which leads to more purchases of production factors today (needs to be accumulated in advance) This is a version of the traditional “reverse causality,” with the twist that future output affects today’s money demand Run a regression m t = λAt + ξt , where we get from the reduced form that plim λ = = Example 39 (Countercyclical policy.) Suppose Var(u yt ) =Var(u mt ) = in Example 37 and that the central bank sets κ = −β This achieves complete stabilization of output, and plim αˆ L S = in the St Louis equation In this case, the successful monetary policy makes it look as if monetary policy cannot affect output One way of getting around the problem with the common driving force is to estimate an extended St Louis equation, where output is related to both money and a vector of other variables capturing the driving force, xt , αs m t−s + yt = s=0 βs xt−s + εt (7.2) s=0 Of course, this approach assumes that we can observe the relevant variables and that they are exogenous The problem with a direct reverse causality (direct effect of output on money, possibly with leads/lags) remains in this equation, however Cov(At , m t ) Var (At ) γ Cov(At , 1−βγ u yt + 1−βγ At + 1−βγ εmt ) Var (At ) = , − βγ since At is assumed to be uncorrelated with u yt and εmt Form a fitted value of m t as mˆ t = λAt , or mˆ t = At , − βγ and use this instead of m t in the St Louis regression, yt = αm t + εt This gives a new estimate plim αˆ iv = Cov mˆ t , yt Var mˆ t Cov = 1−βγ At , u yt + β At + βεmt Var 1−βγ At = β, 7.3.4 St Louis Equation with Instrumental Variables Suppose we could get exogenous indicators of the exogenous movements in monetary policy As a first step, regress m t on the “instruments” and construct a series of fitted policy, mˆ t In a second step, use mˆ t instead of m t in a regression like (7.1) This is the IV/2SLS method, which is consistent as long as the instruments are not driven by output The “trick” in the IV/2SLS method is to discard all variation in m t which is not driven by the instruments This side-steps all movements in m t which are due to reverse causality, that is, due to the output shock The regression will then look at how output moves in response to the exogenous changes in money Example 40 (IV and the case of two-way causality.) Consider the model in Example 33, and suppose we now that u mt = At +εmt , where At are observable shifts in money supply 112 where we once again use the fact that At is uncorrelated with u yt and εmt This is the correct value Romer and Romer (1990) went through the Fed’s minutes to identify (exogenous) policy shifts They found six such policy shifts during the post war period In each case, short interest rates increased, while money aggregates and output decreased They then estimated a St Louis equation with both least squares and instrumental variables, where dummies for these episodes were used as instruments for money They found that the instrumental variables method gave larger effects of money on output, as expected if the central bank uses money in a systematic way to stabilize output The analysis of Romer and Romer (1990) is very much like a formalization of what Friedman and Schwartz (1963b) did: pick out a number of monetary contractions which look exogenous and study what happens to output after that Overall, the evidence from the historical episodes in these and other studies point in the direction that money probably 113 have an effect on output This conclusion is strengthened if we look at the reaction of output of large exchange rate realignments (another type of monetary policy) Example 41 (Summary of Romer and Romer (1990) and Romer 232-236.) Main issue: to find instruments for exogenous shifts in monetary policy The problem with possibly endogenous money is circumvented by focusing on episodes of exogenous (according to Romer and Romer) shifts in monetary policy: September 1955, December 1968, April 1974, August 1978, and October 1979 According to the authors, these monetary contractions were brought about by a desire to take down inflation, with little concern about the effects on output To investigate if there really were monetary contractions, univariate forecasting equations for log money stock where estimated 24 αs m t−s + u t mt = s=1 on monthly postwar data Dynamic forecast for each period of 36 months after the break dates indicate that the money stock were a lot lower (in data) than predicted: there seem to have been contractions Also, the interest rates show fairly consistent increases over the same 36 months, so any effect on output could be consistent with either a monetarist or a Keynesian model To see if theses contractions mattered for output, a St Louis equation was estimated 24 yt = a + bt + 24 di m t−s , ci yt−s + s=1 7.3.5 Dynamic Effects and Causality Suppose the structural model consists of a reaction function where money is predetermined p p δs m t−s + u mt γs yt−s + mt = (7.3) s=1 s=1 and a somewhat extended St Louis equation p yt = α0 m t + p αs m t−s + s=1 βs yt−s + u yt (7.4) s=1 The shocks are assumed to be uncorrelated This is a “fully recursive” system of simultaneous equations, so least squares on (7.3) and (7.4) separately is consistent The reason is that the simultaneity problem is assumed away: money is not contemporaneously affected by the output shock It is also assumed that the lagged money and output capture all relevant common driving forces, so there is no omitted-variables bias However, if we care about more than just the impact effect of money supply shocks, then money is endogenous in the economic sense since we have the following chain of effects: u mt → m t → yt → m t+1 → yt+1 → We therefore need to estimate both equations in order to trace out the effect of a monetary policy shock on output This is the VAR approach discussed below s=0 with OLS and instrumental variables (IV) The instruments were a constant, a trend, yt−s , m t−s , and {P St−s }36 s=0 , where P St = if t is one of the five policy shift dates, and zero otherwise If m t is affected (negatively - to stabilize output) by Yt , OLS on the output equation gives a downward bias in di The IV approach should give consistent estimates if the P St are not affected by output (the policy shifts were exogenous with respect to output, driven by concern about inflation, and inflation does not affect output per se), and if P St affects output via money only (probably reasonable) The result is that dˆI V is larger than dˆO L S 7.4 Unanticipated or Anticipated Money∗ Reference: Romer 6.4, Mishkin (1983) 6, and Barro (1977) Both the new monetarist (Lucas’ model for the Phillips curve ) and new Keynesians (“micro based” models of nominal rigidities) emphasize the distinction between anticipated and unanticipated policy changes The theoretical predictions, based on rational expectations, are that anticipated policy changes should have no or only small effects on output, while unanticipated changes could have large effects For simplicity, assume that the structural model for output is p αs (m t−s − Et−s−1 m t−s ) + u yt , yt = (7.5) s=0 114 115 so output depends on money stock surprises and an output shock, u yt The money supply rule is assumed to depend on past information and a policy shock; money supply is predetermined in relation to output m t = γ z t−1 + u mt ⇒ Et−1 m t = γ z t−1 (7.6) The output and policy shocks are uncorrelated The vector z t−1 typically involves lagged output, money supply, and some other variables As in (7.3) and (7.4), the econometric simultaneity problem is assumed away by making money supply predetermined in relation to the output shock Use the policy rule (7.6) in the output equation (7.5) k n αs m t−s − γ z t−s−1 + yt = s=0 θ z t−s−1 + u yt , (7.7) s=0 where θ = if only unanticipated money matters Suppose the system (7.6) and (7.7) is estimated jointly and rational expectations is imposed (restricting γ to be the same in both equations) It is then straightforward to test if θ = Mishkin’s results indicate that anticipated policy does matter Under the maintained hypothesis that only unanticipated money matters, only the αs coefficients are needed in order to study the effect on money on output This is different from (7.3) and (7.4), where we needed the whole system The reason is that in (7.5)-(7.6) the feedback from yt to m t+1 (due to an initial money supply shock u mt ) has no effect on yt+1 7.5 7.5.1 VAR Studies VAR Models and Simultaneous Equations Systems∗ The VAR approach is an attempt to capture the main time series properties of money and output (and sometimes other variables) at the same time as enough restrictions are imposed to identify exogenous policy shifts It is basically an attempt to estimate a system of simultaneous structural equations For instance, (7.3) and (7.4) is a typical VAR model In order to illustrate how VAR models are handled, we look at that simple bivariate system of output and money once 116 again The two (structural, it is assumed) equations can be written B0 mt yt = B1 m t−1 yt−1 m t− p yt− p + + B p u mt u yt + , (7.8) B0 , , B p are × matrices The parameters (B0 , , B p , and the covariance matrix of the shocks) cannot be estimated without imposing some identifying restrictions In fact, all data can tell us is the parameters of the reduced form mt yt = A1 m t−1 yt−1 + + A p m t− p yt− p + εt , (7.9) The problem is that there may be many structural forms that generate the same reduced form By (7.8), the reduced form can be written as mt yt = B0−1 B1 m t−1 yt−1 + + B0−1 B p m t− p yt− p + B0−1 u mt u yt , (7.10) so Ai = B0−1 Bi for i = 1, , p and Cov (εt ) = B0−1 Cov u mt u yt B0−1 (7.11) In the structural form (7.8), there are (1 + p) coefficients in B0 , , B p and parameters in covariance matrix of the structural shocks In the reduced form (7.10), there are p4 coefficients in A1 , , A p and parameters in covariance matrix of the reduced form shocks We therefore need to impose at least restrictions to calculate the parameters in the structural form The following is a common set of restrictions First, the structural shocks are uncorrelated (one restriction, the off-diagonal element in the covariance matrix is zero) This means that the structural shocks have an economic interpretation as money or output shocks Second, the diagonal elements of B0 are unity (two restrictions); alternatively we could assume that the variances of the structural shocks are unity Third, B0 is triangular Often (see, for instance, Sims (1980))), the assumption has been that money is not 117 contemporaneously affected by output, so B0 has the form B0 = −α0 (7.12) Using (7.12) in (7.8) gives us a system on the same form as (7.3) and (7.4) As discussed before, these equations can be estimated with LS as they stand Alternatively, the reduced form (7.9) can be estimated with LS Imposing the assumptions allows us to solve for the structural parameters (typically a matter of solving non-linear equations, but can be simplified by using some straightforward matrix decompositions like the Cholesky decomposition) This illustrates that the VAR does not allow us to escape the tricky question of endogenous money, or more generally, monetary policy The simultaneous equations bias may therefore affect the VAR estimate—if the identifying assumptions we make are wrong Including lagged money and output in both equations is an attempt to control for common movements in money and movements which are driven common factors (to get away from a potential omitted-variables bias) Once we have an estimate of the structural form (7.8), we may trace out the effect on, for instance, output of shocks to monetary policy (“impulse response function”) We may also calculate how large a fraction of the variance of the forecast error of output that is explained by the monetary policy shocks (“variance decomposition”) 7.5.2 Typical Results from VAR Studies of Money and Output The typical result for the US on quarterly data for the last 30 years or so is that output has a humped-shaped response to u mt which may last for several years However, monetary policy shocks seems to have accounted for a fairly small fraction of the variance of output, in particular after 1982 This does not, of course, mean that systematic monetary policy has not been important or that monetary policy shocks cannot be important For the latter, the impulse response function is much more informative than the variance decomposition It is unfortunate that the VAR analysis typically says very little about the importance of the systematic (feedback) part of policy, which by many is believed to be very important (partly in opposition to the idea that only unanticipated money matters) Some authors (for instance, Bernanke and Blinder (1992)) argue that M1 is not a good proxy for the monetary policy instrument, and that a short nominal interest rate should 118 be used instead First, there is evidence (see, for instance, Sims (1980)) that if output, money, and the nominal interest rate are included in a VAR, then money can no longer help predict output, but that interest rates help predict both output and money (This suggests that much of the movements in monetary aggregates are endogenous, not exogenous as often assumed in macro models of monetary policy.) Second, the impulse responses to money aggregates like US M1 is often weird: a positive innovation in M1 gives a decline in output and an increase in nominal interest rates (see, for instance, Eichenbaum (1992)) In contrast, a positive shock to the federal funds rate gives decline in output, which seems much more reasonable Third, many analysts argue that most central banks discuss and set monetary policy in terms of a short interest rate Often, the dynamic response to a policy shock (stricter policy) is that the price level increases instead of decreases: the price puzzle This is often the case when a short interest rate is taken to be the policy instrument The reason is probably that the VAR contains too little information compared to what the policy makers use Including a commodity price “solves” the puzzle The interpretation is that the central bank reacts to commodity price increases (by raising the interest rate/decreasing money supply) since they signal future inflation Suppose the monetary policy is unable to “kill” the inflation impulse completely Unless you control for the commodity prices, it will appear as if tighter monetary policy leads to higher inflation (See Walsh Fig 1.4) Critique against VAR models: too little forward looking information included (asset prices?), policy shocks differ wildly between different studies, relatively little information about the effect of systematic policy, what if policy changes? (Lucas critique) 7.6 Structural Models of Monetary Policy Reference: Fuhrer and Moore (1995), Fuhrer (1997), Săoderlind (2000), and Clarida, Gal´ı, and Gertler (1998) Another line of research is to estimate a structural economic model directly (although the requirement for the label “structural” may have shifted over time) Of course, there is a long tradition of estimating AS-AD (IS-LM) models with adaptive expectations Recently, this have gained new popularity, but we have also seen a number of papers estimating similar models while allowing for rational expectations Many central banks still use fairly large macro models, even if there is a tendency towards smaller models (in order to 119 handle rational expectations, among other things) Consider the simple model πt = βEt πt+1 + δφyt + δεπt Et yt+1 = yt + (i t − Et πt+1 ) + ε yt γ i t = χπt + υyt + εit (7.13) (7.14) (7.15) The first equation comes from a Calvo model of price setting for a firm under monopolistic competition; the second from the intertemporal optimality condition for a consumers (plus the assumption that consumption equals output); and the third is Taylor’s policy rule, which describes how the central banks sets the short nominal interest rate It is clear that this is a complicated model to estimate: there is plenty of simultaneity, and expectations about future values are needed This model can be estimated by maximum likelihood by specifying the distribution of the shocks (επ t , ε yt , and εit ) The MLE is found by iteration on the following steps (i) guess a vector of parameters (β, δφ, γ , χ, υ, and the variances of the three shocks); (ii) solve for the equilibrium and times series process (of πt , yt , and i t ); (iii) evaluate the likelihood function; (iv) improve the guess of the parameter vector This gives a set of estimates where rational (model consistent) expectations have been imposed A possible alternative is to use survey information as proxies for some or all expectations Another approach is taken by Clarida, Gal´ı, and Gertler (1998) who want to estimate the monetary policy rule (“reaction function”) for several countries They specify a policy rule for the short interest rate, i t , of the form i t = β Et πt+12 + γ Et yt + ρi t−1 + u t , (7.16) where Et yt is the best estimate of the current output gap and where πt+12 is the inflation over the next twelve months This formulation is can be seen as an approximation of the optimal policy in many models where there is some price stickiness and where the central bank wants to stabilize inflation and output, but also avoiding excessive movements in the short interest rate The equation is estimated with an instrumental variables method To see why that makes sense, not that we can rewrite (7.16) as i t = βπt+12 + γ yt + ρi t−1 + u t + εt+n , where εt+n is a linear combination of the surprise in inflation, πt+12 − Et πt+12 , and in the output gap, yt −Et yt The new residual in (7.17), u t +εt+n , is clearly not uncorrelated with the regressors, so we need to apply an instrumental variables method All information at t or earlier should be valid instruments Clearly, this approach allows us to understand the monetary policy rule only To see how a policy change, for instance, a shock u t to (7.16) affects output and inflation, we have to estimate the rest of the model Bibliography Andersen, L C., and J L Jordan, 1968, “Monetary and Fiscal Actions: A Test of Their Relative Importance in Economic Stabilization,” Federal Reserve Bank of St Louis Review, 50, 11–24 Barro, R J., 1977, “Unanticipated Money Growth and Unemployment in the United States,” American Economic Review, 67, 101–115 Bernanke, B S., and A S Blinder, 1992, “The Federal Funds Rate and the Channels of Monetary Transmission,” American Economic Review, 82, 901–921 Clarida, R., J Gal´ı, and M Gertler, 1998, “Monetary Policy Rules in Practice: Some International Evidence,” European Economic Review, 42, 1033–1067 Eichenbaum, M., 1992, “Comment on ’Interpreting the Macroeconomic Time Series Facts: The Effects of Monetary Policy’ by Christopher Sims,” European Economic Review, 36, 1001–1011 Friedman, M., and A J Schwartz, 1963a, A Monetary History of the United States, 18671960, Princeton University Press, Princeton Friedman, M., and A J Schwartz, 1963b, “Money and Business Cycles,” Review of Economics and Statistics, 45, 32–64 Fuhrer, J., and G Moore, 1995, “Inflation Persistence,” Quarterly Journal of Economics, 110, 127–159 (7.17) 120 121 Fuhrer, J C., 1997, “Inflation/Output Variance Trade-Offs and Optimal Monetary Policy,” Journal of Money, Credit, and Banking, 29, 214–234 Isard, P., 1995, Exchange Rate Economics, Cambridge University Press King, R G., and C Plosser, 1984, “Money, Credit and Prices in a Real Business Cycle Model,” American Economic Review, 74, 363–380 Meese, R A., 1990, “Currency Fluctuations in the Post-Bretton Woods Era,” Journal of Economic Perspectives, 4, 117–133 Reading List Main references: Romer (1996) (Romer), Blanchard and Fischer (1989) (BF), and Obstfeldt and Rogoff (1996) (OR) For an introduction to many of the issues, see Burda and Wyplosz (1997) (BW) Walsh (1998) (Walsh) is a more advanced text, and is recommended for further study Papers marked with (∗ ) are required reading Mishkin, F S., 1983, A Rational Expectations Approach to Macroeconometrics, NBER Mishkin, F S., 1997, The Economics of Money, Banking, and Financial Markets, Addison-Wesley, Reading, Massachusetts, 5th edn 0.1 Money Supply and Demand ∗ Lecture notes Obstfeldt, M., and K Rogoff, 1996, Foundations of International Macroeconomics, MIT Press Romer, C D., and D H Romer, 1990, “New Evidence on the Monetary Transmission Mechanism,” Brookings Papers on Economic Activity, 1, 149–213 Sims, C A., 1980, “Macroeconomics and Reality,” Econometrica, 48, 1–48 ∗ OR 8.1, 8.3.1-4 (MIU), and 9.1 (stylized facts) or ∗ Walsh 2.3.1 (MIU) ∗ Romer 9.8 (cost of inflation) Lucas (2000) (cost of inflation) BF 4.2 and 4.5 (MIU) Săoderlind, P., 2000, Monetary Policy and the Fisher Effect, Journal of Policy Modeling, Forthcoming, also available as Working Paper No 159, Stockholm School of Economics Keywords: money multiplier, central bank intervention, traditional money demand equations, money in the utility function, Friedman’s rule Walsh, C E., 1998, Monetary Theory and Policy, MIT Press, Cambridge, Massachusetts 0.2 Price Level and Nominal Assets ∗ Lecture notes ∗ OR 8.2 (seignorage) or ∗ Walsh 4.3 (seignorage) ∗ OR 8.4.1 and 8.4.3-4 (exchange rates) ∗ Meese (1990) (exchange rate regressions) BF 4.7 and 5.1 (seignorage) 122 123 Romer 9.7 (seignorage) Keywords: general equilibrium in static model of monpolistic competition (BlanchardKiyotaki), menu costs, Nash equilibria with/without sticky prices, importance of ”real rigidities.” Isard (1995) (PPP), (UIP), and (exchange rate regressions) Keywords: price level in general equilibrium, superneutrality of money, Cagan’s model with rational expectations, seignorage, exchange rate determination from money demand and UIP 0.3 0.5 Sticky Prices ∗ Lecture notes ∗ Rotemberg (1987) (adjustment costs for prices) or ∗ Walsh 5.5 (inflation persistence) Money and Prices in RBC Models ∗ Lecture notes ∗ Roberts (1995) ∗ B´enassy (1995) (Long-Plosser model with money and predetermined wages) Romer 6.5-9 (staggered prices, Caplin-Spulber) Cooley and Hansen (1995) (RBC model with money and predetermined wages) BF 8.2-4 (staggered prices, Ss, Caplin-Spulber) King and Plosser (1984) (reverse causality) Keywords: quadratic adjustment costs for prices (Rotemberg), Calvo’s model of price changes, combining with “flex price” from monopolistic competition (Advanced: Ss rules) Walsh 3.3 (CIA) and 5.3.1 (wage rigidity in MIU models) Keywords: RBC model with money and sticky prices, reverse causality, CIA 0.4 0.6 Money and Monopolistic Competition Monetary Policy ∗ Lecture notes ∗ Lecture notes ∗ OR 10.1 (GE with monopolistic competition) ∗ OR 9.4-5 (Barro-Gordon model) or ∗ Walsh 8.1-2 (Barro-Gordon) ∗ Romer (1993) (overview of New Keynesian models) ∗ Taylor (1995) (Empirical analysis of transmission mechanism) ∗ King (1993) (overview of New Keynesian models) ∗ Fischer (1996) (Costs of inflation) BF 8.1 (GE with monopolistic competition) ∗ Obstfeld and Rogoff (1995) (Mirage of fixed exchange rates) Romer 6.6 and 6.10-15 (monopolistic competition, real rigidities) Bernanke and Mishkin (1997) (inflation targeting) BF 9.5 (real rigidities) Fuhrer (1997) (inflation/output trade-off) Walsh 5.3.2 (imperfect competition and price stickiness) Romer 5.1-5 (IS-LM) and 9.6 (What can policy accomplish?) 124 125 Bibliography BF 11.2 (traditional monetary policy issues) and 11.4 (Barro-Gordon model) 10 Walsh 5.4 (macro model for policy analysis), 8.4-5 (institutions), and 10.5 (macro model for policy analysis again) B´enassy, J.-P., 1995, “Money and Wage Contracts in an Optimizing Model of the Business Cycle,” Journal of Monetary Economics, 35, 303–315 11 OR 9.2-3 (Dornbusch model) Bernanke, B., 1983, “Nonmonetary Effects of the Financial Crisis in the Propagation of the Great Depression,” American Economic Reveiw, 73, 257–276 Keywords: Mundell-Flemming model and choice of exchange rate regime, BarroGordon model of monetary policy (discretion and commitment), recent models for monetary policy, Dornbusch model 0.7 Bernanke, B S., and M Gertler, 1995, “Inside the Black Box: The Credit Channel of Monetary Policy Transmission,” Journal of Economic Perspectives, 9, 27–48 Bernanke, B S., and F S Mishkin, 1997, “Inflation Targeting: A New Framework for Monetary Policy,” Journal of Economic Perspectives, 11, 97–116 Empirical Measures of the Effect of Money on Output Blanchard, O J., and S Fischer, 1989, Lectures on Macroeconomics, MIT Press ∗ Lecture notes Burda, M., and C Wyplosz, 1997, Macroeconomics - A European Text, Oxford University Press, 2nd edn ∗ Walsh (overview) Romer 5.6 (selective overview) Keywords: interpreting money-output correlations, anticipated or unanticipated money, VAR studies, “case studies” (Romer-Romer) 0.8 The Transmission Mechanism from Monetary Policy to Output Cooley, T F., and G D Hansen, 1995, “Money and the Business Cycle,” in Thomas F Cooley (ed.), Frontiers of Business Cycle Research, Princeton University Press, Princeton, New Jersey Fischer, S., 1996, “Why Are Central Banks Pursuing Long-Run Price Stability,” in Achieving Price Stability, pp 7–34 Federal Reserve Bank of Kansas City Fuhrer, J C., 1997, “Inflation/Output Variance Trade-Offs and Optimal Monetary Policy,” Journal of Money, Credit, and Banking, 29, 214–234 Mishkin (1995) Walsh (1998) Isard, P., 1995, Exchange Rate Economics, Cambridge University Press Bernanke and Gertler (1995) King, R G., 1993, “Will the New Keynesian Macroeconomics Resurrect the IS-LM Model?,” Journal of Economic Perspectives, 7, 67–82 Bernanke (1983) King, R G., and C Plosser, 1984, “Money, Credit and Prices in a Real Business Cycle Model,” American Economic Review, 74, 363–380 Romer and Romer (1990) Stiglitz and Weiss (1981) Lucas, R E., 2000, “Inflation and Welfare,” Econometrica, 68, 247–274 Keywords: money view, lending view, credit rationing (Stiglitz-Weiss) 126 127 Meese, R A., 1990, “Currency Fluctuations in the Post-Bretton Woods Era,” Journal of Economic Perspectives, 4, 117–133 Mishkin, F S., 1995, “Symposium on the Monetary Transmission Mechanism,” Journal of Economic Perspectives, 9, 3–10 Obstfeld, M., and K Rogoff, 1995, “The Mirage of Fixed Exchange Rates,” Journal of Economic Perspectives, 9, 73–96 Obstfeldt, M., and K Rogoff, 1996, Foundations of International Macroeconomics, MIT Press Roberts, J M., 1995, “New Keynasian Economics and the Phillips Curve,” Journal of Money, Credit, and Banking, 27, 975–984 Romer, C D., and D H Romer, 1990, “New Evidence on the Monetary Transmission Mechanism,” Brookings Papers on Economic Activity, 1, 149–213 Romer, D., 1993, “The New Keynesian Synthesis,” Journal of Economic Perspectives, 7, 5–22 Romer, D., 1996, Advanced Macroeconomics, McGraw-Hill Rotemberg, J J., 1987, “New Keynesian Microfoundations,” in Stanley Fischer (ed.), NBER Macroeconomics Annual pp 69–104, NBER Stiglitz, J E., and A Weiss, 1981, “Credit Rationing in Markets with Imperfect Information,” American Economic Review, 71, 393–410 Taylor, J B., 1995, “The Monetary Transmission Mechamism: An Empirical Framework,” Journal of Economic Perspectives, 9, 11–26 Walsh, C E., 1998, Monetary Theory and Policy, MIT Press, Cambridge, Massachusetts 128 ... 7.6 Structural Models of Monetary Policy 124 124 125 125 126 126 84 81 81 82 85 A Derivations of the Aggregate Demand Equation Money and Prices in RBC... ), and Pt = Mt exp (δω) and use (2. 11) to obtain the requirement that any bubble must satisfy bt = ηEt bt+1 ⇒ Et bt+s = η−s bt (2. 14) 28 29 2. 2.5 Use this in (2. 15) to get Seignioraget = Mt−1... vt = −ψ ln Yt − ln Yt∗ + ln Mt − ln Mt∗ + ln Q t , (2. 22) and rewrite (2. 21)as Et vt + ln St or ω ω ln St = vt + ωEt ln St+1 ln St+1 = − (2. 23) Note how the exchange rate is like any other asset: