Money Demand and Nominal Debt: An Equilibrium Model of the Liquidity Effect*
Recently, strong empirical evidence has been found to support the existence of a liquidity effect. Existing models of this effect, however, are not completely satisfactory. We attempt to overcome the shortcomings in previous theories by modeling money demand as a function of both ex- penditures on current output and transactions linked to nominal debt. Combined with several money supply processes, we show that such a money demand specification can plausibly explmn a persistent liquidity effect. Unlike most other models of the liquidity effect, ours assumes prices are flexible and monetary policy has no impact on the real mterest rate or output.
1. Introduction A n u m b e r of recent studies have provided support for the longstanding
view that there is an inverse relationship be tween monetary shocks and nom- inal interest rates. However, despite much research, the nature and cause(s) of the liquidity effect remain in dispute. Among the many models that at- t empt to explain this relationship is the standard sticky-price IS-LM model. Recently, Barro (1989), Lucas (1990), Christiano and Eichenbanm (1992), and others, have modeled this relationship in an equilibrium context.
Research indicates that the liquidity effect is not confined to short- terra interest rates. Surprisingly, changes in the money supply also appear to have a negative impact on implied long-term forward nominal interest rates. We show that it is difficult to reconcile existing models of the liquidity effect with the observed relationship be tween monetary shocks and forward interest rates.
*We thank the participants of the Bentley College Department of Economies Workshop and two anonymous referees for very helpful comments.
In Section 2, we describe a flexible-price model of the liquidity effect where money is used for both current expenditures on goods and services, as well as for expenditures made in settlement of non-indexed nominal debt contracted during previous periods. Non-indexed nominal debt is generated by the purchase of goods and services on credit, examples of which include fixed rate mortgages, automobile and other installment loans, and rental contracts. Modeled in this fashion, money demand is a function of not just current expenditures on goods and services but also of previous credit pur- chases of goods and services. In Sections 3 and 4 we discuss other approaches to modeling the liquidity effect, as well as some empirical evidence that we believe supports our approach.
In Section 5 we formally model the liquidity effect by combining the model of money demand discussed in Section 2, with various money supply processes. This model is: 1) able to generate a persistent liquidity effect; 2) the model most consistent with the observed relationship between money announcements and implied forward interest rates and commodity prices; 3) capable of explicitly identifying the impact of monetary policy on the term structure of interest rates; 4) able to provide an alternative explanation of why in conventional partial adjustment money demand models lagged money is statistically significant and money demand is not homogenous with respect to the price level.
We also cast doubt on the commonly held assumption that the exis- tence of a liquidity effect requires the presence of price stickiness. Our model differs from other models of the liquidity effect in that it is based on neither price stickiness nor costs of adjusting money holdings. Although we employ a flexible-price equilibrium model that dichotemizes real and nom- inal variables, we are able to link the current level of money demand to macroeconomic conditions several years earlier.
2. An Equilibrium Model of the Liquidity Effect In Irving Fisher's "transactions" approach to modeling money demand,
the relevant transactions include "real estate, and to some extent wages, retail prices, and securities" (1963, 225.) Friedman (1974, 6) notes that Fisher's "emphasis on transactions . . . suggests dividing total transactions into cate- gories of payments for which payment periods or practices differ." Unfor- tunately, due to the perception that transactions in financial assets generate relatively little money demand, Fisher's approach has been largely ignored in the subsequent literature. Helpman and Razin (1985, 100), however, note that "although the velocity of circulation of money in financial transactions is very large, the volume of financial transactions is very large too, so that
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the absorption of money might be significant (we know of no empirical evi- dence on this issue)."
The term "financial transactions" is often used in the restricted sense of transactions invoMng securities in organized financial markets. But from an economic perspective, all nominal payments that have been contracted in previous periods represent a form of nominal (or non-indexed) debt. Thus, the most useful distinction is not between the purchase of securities and the purchase of goods and services. Rather, it is between payments linked to nominal debt, and transactions (financial or otherwise) that do not involve nominal debt. In the U.S. economy, many mortgages, ear loans, and rental contracts involve the payment of fixed nominal sums at periodic intervals in exchange for predetermined quantities of goods. It is this contractual pre- determination of output which distinguishes these payments from economic prices (sticky or otherwise).
Our model implies that the appropriate scale variable in a money de- mand funetion is not gross national expenditure/ineome, but rather gross national trausactior~s (GNT). That is, all payments of money during a given period generated by purchases of goods in that period, or in previous pe- riods. 1 This model differs from sticky-price IS-LM models of the liquidity effect in both its assumptions and its implieations. Beeause we employ a flexible-priee model with the transactions demand for money incorporating payments linked to non-indexed debt, there is no need to assume output effects associated with unanticipated price shocks, as occur in a stieky-price model. Although all transactions prices are flexible, by incorporating nominal debt in the model we are able to mimic the price level and exchange rate overshooting of a sticky-price model. And, since nominal contracts can ex- tend over a period of many years, our approach can model a long-term liquidity effect with much greater plausibility than a model which relies on short-run priee inflexibility.
Unlike traditional (sticky-price) models of the liquidity effect, our model is fully consistent with equilibrium models that dichotomize the real and monetary sectors of the economy. Referring to models where monetary policy has real effects, Barro (1989, 6) notes "it is unclear that any of these
1A simple example helps differentiate GNP and GNT. For a new car purchased on credit in the previous period, principal payments on the outstanchng ear loan are included in current GNT, but not current GNP. This logic can be applied to other types of contracts as well. Consider the standard 12-month academic employment contract. These prowde for 12 prede- termined nominal payments in exehange for 12 months of labor. (Note that these types of contracts are fundamentally different from a "sticky-wage" contract where hourly nominalwages are fixed for a period of time but total monthly wages vary as the number of hours worked are changed at the discretion of the employer).
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approaches have isolated quantitatively important effects of money on real variables." Following Barro we will simplify the analysis by assuming that monetary policy has no impact on real interest rates or real output.
Our utilization of a flexible price model should not be construed as a rejection of the IS-LM approach to macroeeonomics. 2 Rather, this modeling strategy has the advantage of parsimony in that the usual suspects employed in explaining the liquidity effect are assumed away. Using a flexible price model allows us to highlight the distinctive implications of using GNT rather than GDP as the scale variable in a money demand function. In our model, monetary shocks impact both short- and long-term implied forward interest rates. In addition, this modeling technique accounts for the observed non- homogeneity of money demand with respect to prices. We are not ruling out the possibility that monetary policy has real effects. We are doubtful, however, that these effects are likely to be of sufficient duration to impact implied long-term forward interest rates. In a later section, we briefly con- sider the implications of adding price stickiness to our model.
Since the inclusion of transactions linked to nominal debt contracts is the distinctive feature of our model, we must show that such contracts gen- erate a significant share of monetary transactions. Table 1 presents the 1991 end-of-year installment-type liabilities of various sectors of the U.S. econ- omy. As can be seen, the amounts are not trivial. To obtain a conservative estimate of the annual payments associated with only mortgage debt in Table 1, suppose that the average time-to-maturity is ten years and the average interest rate is 7%. Use of these figures yields 8576 billion in annual payments.
The existence of adjustable rate mortgages and refinancing options complicates the estimation of nominal debt outstanding, and thus the pre- vious figures may represent an overestimate of the theoretically relevant aggregate, a However, the figures above do not include payments linked to other sectors, such as rental housing. For example, there were 30,490,535 renter-occupied housing units in 1990 with a median monthly rent of $374 (U.S. Bureau of the Census). These figures translate into approximately $136 billion in annual rental payments. In addition, our estimates exclude trans- actions involving nominal business debt. Although our estimates are impre- cise, they clearly indicate that payments linked to nominal debt contracts are a significant fraction of transactions and have the potential to play an em-
2Indeed, empirical ewdence has been found in support of nominal price stickiness. See Carl- ton (1986). Further, money has been shown to have short-term real effects, such as in Bernanke and Blinder (1992).
3Due to transactions costs and asymmetric incentives, not all changes in nominal interest rates will induce individuals to refinance.
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TABLE 1. Installment Type-Liabilities, by Sector, Year-End, 1991
Type of Liability
Installment Sector Mortgages Credit Bonds Other Total
Households, Personal Trusts, and Nonprofits $2997.8 792.4
Private Financial Institutions 2.0 Total 4046.6 792.4
270.4 4060.6 59.4 143.1
327.5 1172.8
1060.3 893.6 2071.7
561.2 121.4 684.6 1621.5 1672.3 8132.8
NOTE: All amounts are in billions of current dollars. "Other" consists of various types of loans, mostly bank loans not elsewhere classified.
Source: Balance Sheets for the U.S. Economy, 1960-1991. March, 1992. Publication C.9, Board of Governors of the Federal Reserve System.
pirically significant role in money demand. Less conservative assumptions would yield even larger estimates of annual payments, strengthening our point that nominal debt has an important influence on money demand.
3. Previous Models of the Liquidity Effect Prior to detailing our model, we first review empirical and theoretical
analyses of the liquidity effect, which we define as the inverse relationship between monetary shocks and nominal interest rates. 4 Recent studies by Bernanke and Blinder (1992), Christiano and Eichenbaum (1991a, 1991b), Strongin (1991), Eichenbaum (1992), and Sims (1992), have all found evi- dence of a liquidity effect. Despite this new evidence, the findings of Leeper and Gordon (1992) suggest that the existence of a liquidity effect remains an unresolved issue. Reichenstein (1987) reviews previous literature on the subject and concludes there is little evidence for the existence of a liquidity
4Some authors define the liquidity effect as a negative relationship between monetary shocks and real interest rates. Our approach does not require that monetary policy have any real effects.
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effect. He also provides evidence against the existence of a liquidity effect during the period 1975-1983. More recently, Pagan and Robertson (1995) point out that strong evidence of a liquidity effect is found only when non- borrowed reserves are used as the policy indicator.
Given the difficulty in pinning down the existence of a liquidity effect, it is not surprising that modeling the liquidity effect has met with limited success to date. The most widely accepted model of the liquidity effect is the IS-LM model. This model posits that, due to short-run price stickiness, the price level initially responds less than proportionately to changes in the money supply. If the nominal money stock increases, the real money stock increases as well, and the nominal interest rate must temporarily decrease in order for the public to be willing to hold larger real balances.
Grossman and Weiss (1983) and Rotemberg (1984) developed equi- librium models of the liquidity effect in which money injections occur through open market operations with the banking system. These papers have generated a class of"limited-participation" (LP) models in which individuals enter credit markets at discrete intervals. In these models, nominal money demand does not immediately adjust to the new money supply at the original level of interest ratesP Dotsey and Ireland (1995, 1455) argue that "these constraints may be appropriate in models where the period length is inter- preted as one day or one week, but at the quarterly horizon, agents are likely to have access to a more flexible transactions technology." Thus, unless asset transactions are extremely infrequent, this type of model is unlikely to gen- erate a liquidity effect in the implied long-term forward interest rate. Avery, et al. (1987) find that in 1986, the median interval between adjustments in cash holdings by individuals was only seven days.
Christiano and Eichenbaum (1992) are able to generate a persistent liquidity effect by assuming that households face costs in adjusting transac- tions balances in response to monetary shocks. They argue that even tiny adjustment costs (i.e., a few minutes of time per household) can generate a large and persistent liquidity effect. However, the fact that individuals do adjust their money holdings relatively frequently, combined with the pre- sumption that the marginal cost of complete adjustment is extremely low, suggests that individuals would fully adjust their money holdings soon after a monetary shock.
To summarize, most previous models of the liquidity effect have dif- ficulty accounting for a persistent liquidity effect. In the next section, we use
5Lucas (1990), Christiano (1991), and Christiano and Eichenbaum (1992) also construct LP models. Positive monetary shocks cause the volume of liquidity in financial markets to increase, which reduces the real interest rate. However, Lueas and Christiano observe that their models cannot generate persistent liqmdRy effects.
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evidence from the forward interest rate markets to show the existence of a highly persistent liquidity effect, perhaps extending five to ten years into the future. Nei ther IS-LM nor LP-type models appear capable of explaining a persistent liquidity effeet. The model we develop in Section 5 is eapable of doing so. 6
4. Money Announcements and the Liquidity Effect The unanticipated money announcements (UMA) literature offers a
rich source of empirical findings regarding the relationship between mone- tary policy and interest rates. There are five key findings of the money an- nouncements literature that examines the 1979-82 money targeting pe r iod ] First, the existence of a positive relationship between UMAs and nominal interest rates during the money targeting period is now firmly established. 8 Second, Shiller, Campbell and Sehoenholtz (1983) find a positive and sig- nificant impact of UMAs on 3 to 5 year and 5 to 7 year implied forward interest rates, and Loeys (1985) even finds evidence of a positive response in the 7 to 30 year forward rate. The estimated size of the response of long- term forward rates is smaller than for short-term forward rates. Third, Engel and Frankel (1984) and Frankel and Hardouvelis (1985) show that during the 1980-82 period, near-term commodity futures prices reacted negatively to UMAs. 9 Fourth, these two studies also find a negative relationship be- tween UMAs and foreign exchange rates. Finally, Cornell and French (1986) find that throughout the 1977-84 period, the spread between far- and near- term commodity prices (the basis) reacted positively to UMAs.
Sheehan (1985) provides a summary of the various hypotheses of why UMAs affect interest rates. He contends that the empirical findings of the
6The apparent negative relationship between monetary shocks and interest rates need not imply the type of causal relationship inherent in liquidity effect models. Barro (1989) develops an equilibrium model of interest rate targeting where the monetary authority attempts to min- imize both nominal interest rate fluctuations and unanticipated price level changes. Under the optimal policy rule, the money stock would respond positively to current money demand shocks and negatively to lagged money demand shocks. A positive money demand shock would si- multaneously raise the current money supply and reduce the expected money growth rate. Because Barro uses an equilibrium model, the price level overshoots and the increase in the money stock reduces the expected inflation rate, and, since the real interest rate is assumed to be unaffected by" monetary pohcy, the nominal interest rate. In Barro's model, a one-time, permanent monetary shock has no impact on nominal interest rates.
vIt is important to note that the model developed below does not depend on this (or any) particular type of monetary policy.
SSee for example, Strongin and Tarhan (1990), Frankel and Hardouvelis (1985), Shdler, Campbell, and Sehoenholtz (1983), and Cornell (1982).
9They use near-term commodity futures prices as a proxy for spot commodity prices.
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money announcements literature are best explained by the "expected li- quidity effect" hypothesis. According to this hypothesis, during the money targeting period, a positive UMA leads the public to expect the Fed to adopt a more eontractionary policy in the near future. The subsequent rise in nominal rates is interpreted as an anticipation of the liquidity effect. 1°
The positive impact of UMAs on nominal interest rates, when com- bined with the negative impact on commodity prices, has been interpreted by Hardouvelis (1987) and others as evidence that the liquidity effect extends to real interest rates. Cornell and French (1986), however, show that positive UMAs increase the basis in commodity futures prices. They interpret this finding as evidence that positive UMAs increase the expected rate of infla- tion. The reason their conclusions differ from those of Frankel and Hardou- velis (1985) is that Cornell and French look at the spread between far and near term commodity prices, rather than the levee of near term prices. When the findings of the two papers are combined, it would appear that commodity prices overshoot their long-term equilibrium values. Thus, a positive UMA reduces both spot and futures commodity prices, with spot prices falling by a larger amount. The findings of Cornell and French suggest that the li- quidity effect observed by Frankel and Hardouvelis need not extend to the real interest rate. However, because the basis may reflect carrying costs of storable commodities, it may represent a flawed estimate of inflation expectations.
Frankel and Hardouvelis (1985, 437) claim that "the observation that interest rates and foreign exchange prices (or commodity prices) react to money announcements in opposite directions constitutes a rejection of the special case of flexible prices in favor of the more general sticky-price model." They cite Dornbusch (1976) on exchange rate overshooting as an example of such a model and then argue that the responses of various mar- kets to UMAs during 1980-82 can only be reconciled in a model that com- bines both price stickiness and the expected liquidity hypothesis.
We believe that Frankel and Hardouvelis are correct that the various market responses to UMAs do support the expected liquidity hypothesis. That is, the positive response of nominal interest rates to a positive UMA is due to an anticipation of an imminent, offsetting policy change by the mon- etary authority. However, in one key respect the findings of the money an-
l°Sheehan also discusses the inflation premium hypothesis, which argues that positive money shocks will be followed by additional positive shocks, causing inflation expectations to increase. He notes that the fact that U.S. exchange rates reacted positively to UMAs is inconsistent with this hypothesis. An alternative hypothesis suggests that positive UMAs signal increased money demand generated by expectations of increased future economic activity. Sheehan notes that this hypothesis is inconsistent with the negative response of stock pnces to UMAs.
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nouncements literature are not consistent with the sticky-price version of the liquidity effect.
As mentioned, several previous studies have found the interest rate response to be smaller for implied long-term forward rates as compared to implied near-term forward rates. Even so, the existence of any significant impact of UMAs on implied long-term forward rates is difficult to reconcile with the conventional IS-LM model because it is not plausible that prices would remain sticky long enough for UMAs to impact real interest rates more than a few years forward. The sticky-price version of the expected liquidity hypothesis may be able to explain changes in short-term interest rates, commodity prices, and foreign exchange rates. But neither the sticky- price approach, nor any other existing model of the liquidity effect, can explain the behavior of implied long-term forward interest rates.
We conclude that the models discussed in the previous section have not fully accounted for the stylized facts outlined above. In particular, it has proven difficult to model the finding that UMAs are associated with a highly persistent liquidity effect.
5a. A Model Featuring Three-Period Nominal Debt The evidence presented in Section 4 on the relationship between
UMAs and the implied long-term forward interest rate implies that monetary shocks generate a persistent liquidity effect. In a two-period model, we would be limited to a single maturity, where debt is issued in period one and matures in period two. Thus, in order to examine the impact of monetary shocks on both current and forward interest rates, it is necessary to use a model with a minimum of three periods.
Consider a one good economy exhibiting complete price flexibility where a cash purchase of the good in period t results in payments of Pt in period t. A credit purchase in period t results in either a payment of Pt(1 q- it) in period t + 1, a payment of Pt(1 + it)*(1 + Et(it+l)) in period t + 2, or a linear combination of the two. Pt is the price of the good (and therefore, the price level), and it is the nominal interest rate in period t. Since we assume that monetary policy has no effect on either real output or the real interest rate, with no loss of generality we can assume that both real output and the real interest rate are fixed. 1l
Let the following represent the natural log of the nominal demand for (base) money in period t:
liThe model ignores wealth effects generated by changes in file expected rate of inflation. These effects could theoretically" affect both the real interest rate and the level of output. Cornell and French (1986) find some evidence of money shocks affecting real rates.
r is the ex ante real interest rate, Pt is the price level in period t, a, and b are constants such that a, bl, b2 > 0, and Et(') is the expected value in period t.
Tile constant a is the interest elasticity of money demand and b2/(1 + bl + b2), bl/(1 + bl + b2), 1/(1 + bl + b2) are the fractions of expen- ditures made in period t - 2, period t - 1, and period t that are paid for in period t. The variable Tt represents the log of GNT, the nominal level of cash payments in period t. Although it bears a superficial resemblance to the log of expenditures in a conventional money demand model, its inter- pretation is much different. In a model using GNP as the scale variable, the assumption of fixed output implies a direct correspondence between changes in prices and nominal spending. This correspondence breaks down, however, when we replace GNP with the nominal level of transactions (GNT). When GNT is used as the scale variable, the transactions demand for money is a function of both cash purchases of goods in the current period and the settlement of nominal debts incurred by the purchase of goods during pre- vious periods. Only those transactions involving the purchase of goods sold in period t should be included in the price index Pt. With output assumed to be fixed, nominal money demand is homogeneous of degree one with respect to GNT, but not with respect to p.12
The problem in modeling the money supply process is that we are not able to directly observe the decision rule employed by the monetary au- thority, which in any case may change over time. For the present, assume that the log of the money supply follows a random walk:
M~ = Mr-1 + u~, (3)
where ut is a serially uncorrelated random error term with zero mean and finite variance. Mt is known to individuals who make purchases in period t, and thus all payments in period t + 2 will be made with knowledge of the money supply in period t.
12Many empirical studies find that money demand is not homogeneous of degree one with respect to prices. See for example Gold£eld and Siehel (1990).
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To estimate the effect of a monetary shock in period t on the time path of the price level, we must incorporate the fact that inflation expecta- tions will impact the current price levelJ a Thus, we must solve for the impact of a monetary shock on both the current and future expected price levels. With the imposition of suitable boundary conditions for the price level and the assumption of rational expectations (i.e. that expectations are consistent with the predictions of the model), money is neutral in the long run and the effect of a monetary shock in period t on the expected price level in period t + 9 is 14
AEt(ln Pt+~) = ut. (4)
Because all transactions in period t + 2 occur at prices contracted by indi- viduals with full knowledge of Mr, period t + 2 represents the long run. Surprisingly, however, the current price level (P~) will overshoot (in the sense described by Dornbusch 1976), is despite the assumption of complete price flexibility. To understand the intuition behind dais result, suppose instead that the price level had simply risen in proportion to the money supply. In that case, the sluggish adjustment of nominal debt contracts implies that total nominal spending would have risen less than proportionally to the money supply and price level. But this would result in money demand growth falling short of the change in the money supply. The resulting excess money balances would then put upward pressure on the price level, and hence, result in overshooting.
Because prices fully adjust after two periods, we can recursively solve
13In Mussa's model (1982, 85), "[s]ince behavior is assumed to depend only on the expected values of random variables, and since the model is linear in all random variables, we may determine the expected future path of the economy, conditional on information available at a given date, by treating the expected values of all variables as if they were known with perfect certainty and solving for the perfect foresight path of the economy." Regarding the issue of finearity, see footnote 16.
14Sargent and Wallace (1973) develop boundary conditions by assuming that with a constant money supply, individuals would not expect a permanent condition of ever-aceeleratinginflation or deflation. In the presence of a once and for all change in the money supply, the existence of a stable equilibrium price level requires that the expected rate of inflation in period t + 1 be zero. (The price level in period t + 1 would only respond more than proportionately to a positive monetary shock in period t if the real demand for money decreased, but this would require expectations of still further price increases.)
15Overshooting can be defined in several alternative ways. Dornbuseh (1976) defines over- shooting relative to the long-term equilibrium value. Mussa (1982) uses a dynamic approach where overshooting occurs relative to the (unobserved) moving equilibrium. Kitchen and Den- baly (1987) study this issue and find commodity price overshootang to be most consistent with Mussa's approach. This distinction, however, is only meaningful in the context of a sticky-price model.
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the model by calculating the change in the expected price level and interest rate in period t + 1. To do so, we must use the change in the current expectation of the money supply in period t + 1. Since equilibrium in the money market requires g t = Mr, the change in the log of next period's expected price level relative to its expected value in period t - 1 can be found by combining Equations (1) and (3):
AEt (Mt+I ) = AE t (Z t+ l ) - a[AEt( i t+l ) ] . (5)
Current monetary shocks will not affect money demand arising from cash payments contracted in earlier periods. Thus, AEtT,+ 1 will be approx- imately 16 equal to the product of the change in the (log of the) expected price level in period t + 1 and the proportion of transactions that involve the payment of money for purchases made during period t or later:
and kl represents the fraction of total payments in period t + 1 that were contracted with knowledge of the monetary shock in period t. Since bj and b2 are greater than zero, kz is less than one. Combining Equation (5) with Equation (6) yields
A E t ( M t + I ) = klAEt( In Pt+l) - a[AEt(ln P,+2) - AEt(ln Pt+I)] • (7)
From Equations (3) and (4) we know that
AEt(M,+I) = ut + Et(ut+l) = ut = AEt(ln Pt+z) • (S)
Once agents observe the monetary shock at time = t, their current expec- tation of next period's money supply is revised by ut, and their expectation
16The model developed in this section is not linear in either P or ln(P). However, the model approaches linearity as the size of the monetary shock (ut) approaches zero. As such, the findings of this paper should be viewed as reasonable approximations for "small" monetary shocks, k 1 represents the derivative of T~+I with respect to ln(Pt+l) in a continuous time model. Since our model is in discrete time, k 1 is an approximation.
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of the long-run price level (ln Pt+2) is also revised by u t. Substituting u t into Equation (7) yields
ut = k lAEt( ln e t + l ) - a [ u t - AEt(ln Pt+l)] • (9)
SoMng for AEt(ln Pt+ i ) ,
AEt(ln Pt+i) = [(1 + a)/(k 1 + a)]ut . (10)
The coefficient on ut is greater than one, indicating that the price level in period t + 1 is expected to overshoot its long-run (period t + 2) equilibrium value. Iv Working back to period t, the current price level will also overshoot:
AEt(Mt) = AEt(Tt) - a[AEt(it)]
= koAEt( In Pt) - a[AEt(ln P t + l ) - - AEt(ln Pt)] , (u)
where
ko = PJ[Pt + b lE t - l (P t )* ( 1 + r) + b2Et_2(Pt)(1 + r) 2]
and ko represents the fraction of total payments in period t that were con- tracted with knowledge of the monetary shock in period t. Substituting Equations (3) and (10) into Equation (11) yields
Because ko and k 1 are both less than one, the numerator is larger than the denominator and the current price level will overshoot.
Since the real interest rate is assumed to be constant, the change in the nominal interest rate will be the difference between the change in the current expectation of next periods' price level and the change in the current expectation of this periods' price level, is The change in the expected rate of inflation between period t and period t + 1 and thus, i t, is derived by sub- tracting (12) from (10):
17We do not need to acbaeve a closed form solution in Equation (10) since we are only interested in showing that the coefficient on u t is unambiguously" greater than one.
1sit is important to point out that while we know Pt wi th certainty in period t, we are interested m the change in Pt relative to what was expected in period t - 1.
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Since 1 > kz > k0, the coefficient on u~ is negative and the expected rate of inflation moves inversely to a one-time change in the money supply. Since the real interest rate is fixed, the nominal interest rate will also move inversely to the money supply (Equation 13). Monetary shocks can thus appear to produce a "liquidity effect" without there being any change in the real interest rate. Note that if bl = b2 = 0, then ko = kl = 1, and AEt(i t) in Equation (13) would be equal to zero and there would be no liquidity effect.
The impact of the monetary shock on the expected rate of inflation between period t + 1 and period t + 2, and therefore, on the nominal interest rate, can be derived by subtracting Equation (10) from u~:
AEt(it+ 1) = AEt(ln Pt+z) - AEt(ln Pt+l) = [ ( k l - 1)/(a + kl)]u t . (14)
The coefficient on ut in Equation (14) is negative, and, if (1 - kl) <- (kl - k0), it will be smaller in absolute value than the coefficient in Equation (13). 19 This suggests that if the real interest rate is fixed, the liquidity effect should be more pronounced for the current interest rate than for the one- period forward interest rate.
In a model featuring N period contracts, we can define K~ as the pro- portion of total payments in period t + n that were contracted with knowl- edge of the monetary shock in period t. In this model, a sufficient condition for a monetary shock to affect current interest rates by a greater amount than implied forward interest rates is for AK~ to be a non-increasing function of n. This condition is actually very plausible.
For example, in an economy where nominal contracts consist solely of overlapping 30-year fixed-rate mortgages, and where the price level had been stable for the previous 30 years, Kz would equal 1/360 and Ka60 would equal 360/360. Even in this type of an economy (where all transactions involve payments linked to 30-year mortgages), AK~ would remain constant (at 1/360) as n increases. In a more realistic model featuring cash goods, as well as short, medium, and long-term debt, AKn would decline as n increased. Thus, our analysis need only assume that in period t there are more payments for goods purchased in period t - n than for goods purchased in period t - - n - - 1 .
We have shown that in a model with a money-demand equation in- eluding nominal debt, monetary shocks cause both the price level and the nominal interest rate to overshoot their long-run equilibrium values. The intuition behind this result is as follows. Money demand is modeled as a
19See Appendix 1 for a proof of this inequality.
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function of transactions associated with nominal debt contracts. At a given interest rate, a positive monetary shock will increase money demand asso- ciated with current cash purchases proportionally. However, money demand associated with debt contracts, and thus total money demand, does not in- crease in proportion to the monetary shock. Equilibrium is re-established by both an increase in the price level (Equation 12), and a decrease in nominal interest rates (Equation 13). And, although foreign exchange rates are not formally considered in the model, adding the assumption of pur- chasing power parity would clearly result in foreign exchange rate overshoot- ing as well. 2°
5b. Permanent Changes in the Money Growth Rate Now assume that the money supply follows the type of process sug-
gested by the inflation premium hypothesis--that positive UMAs lead to expectations of a permanent increase in the growth rate of the monetary base:
AMt = AM~ 1 + v t , (15)
where vt is a serially uncorrelated random error term with zero mean and finite variance. The interpretation of Equation (15) differs in two respects from that of Equation (3). Changes in the growth rate in the money supply are expected to be permanent, rather than temporary. Also, the inflation premium hypothesis requires that vt be positively correlated with the UMA, whereas the expected liquidity hypothesis assumes that ut is inversely related to the UMA.
In this section we are only concerned with the current interest rate. Therefore, we employ a model featuring two-period debt for the purpose of simplicity. The change in the current expectation of next period's price level is determined by the changes in the current expectation of next period's money supply and money demand. Because the shock to the growth rate of the money supply is expected to be permanent, the expected change in next period's money supply is 2vt. This permanent shock to the expected growth rate of the money supply will permanently change the expected inflation and interest rates, which will change money demand by -a(Ait+]). Assuming
e°Sheehan and Wohar (1995, 712) find evidence that positive UMAs lead to "increases in spot exchange rates. These increases, however, are expected to be quickly offset since the futures response is much smaller."
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suitable boundary conditions, ~1 the long-run change in the expected inflation rate, and thus, the expected interest rate, is vt. Therefore, the effect of a shock to the growth rate in the money supply in period t on the log of the expected price level in period t + i is
aEt(ln Pt+t) = 2vt - [ - a ( a i t + l ) ] = (2 + a)v t . (16)
The coefficient of vt represents the sum of the impact of the current mon- etary shock (which is expected to occur in period t + 1 as well), and the effect of the permanent increase in the expected rate of inflation, on money demand. To find the impact on the expected current price level, we start with the equality between changes in the money supply and money demand:
v t = AEt (T t - air). (17)
In the two period case, K represents the fraction of transactions in period t that were contracted with knowledge of the money supply in period t, i.e. [(ln Pt)/Tt]. The expected change in nominal transactions can then be defined by the following equation:
AEtTt = K[AEt(ln 5)] • (18)
Combining Equations (16), (17), and (18) yields the impact of a monetary shock in period t on the log of the price level in period t:
vt = KAEt(ln I t ) - a[(2 + a)vt - AG(ln Pt)]. (19)
SoMng for AEt(ln Pt) yields:
AEt(ln Pt) = [(1 + 2a + a2)/(a + K)]v t . (20)
The positive impact of UMAs on the current price level is in sharp contrast to the negative response occurring under the expected liquidity hypothesis as modeled by Frankel and Hardouvelis (1985). The unantici-
21Here, the boundary conditions are established by assuming that a permanent change in the growth rate of the money supply leads to an equal increase in the expected long-term rate of inflation. After period t + 1, the existence of a stable inflation rate requires the expected rate of inflation to be equal to the expected rate of growth in the money supply. This seems reason- able because a long-term inflation rate that is greater (less) than the money supply growth rate would imply continual decreases (increases) in real money demand, and equilibrium m the money market could then only be sustained by an ever accelerating (decelerating) rate of infla- hon.
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Money Demand and Nominal Debt
pated portion of the change in the expected rate of inflation (and the nominal interest rate) is equal to: 22
AEt(ln Pt+l - In P,) = [(2K + aK - 1)/(a + K)Jvt. (21)
Thus, the impact of a UMA on the expected interest rate is ambiguous. For example, if the expected inflation effect is stronger than the liquidity effect, then the impact of a UMA on the interest rate is positive.
It is important to note that the responses of expected inflation to mon- etary shocks shown in Equations (13), (14), and (21) follow from the specific money supply processes postulated in each case. While it is unlikely that either equation fully captures the actual money supply process, these equa- tions do incorporate the key attributes of the expected liquidity and inflation premium hypotheses. 2a The implication that the response of interest rates to money shocks is sensitive to the money supply process is consistent with the findings of numerous studies that show that the liquidity effect is not stable over different monetary regimes (Belongia, Haler, and Sheehan 1988 for example).
In principle, it is possible to reeursively solve our model for nominal debt of any finite maturity. In Section 6, we present simulations of a model featuring mnltiperiod debt.
6. Simulations of the Model The model of Section 5 is limited in that it features debt with a maturity
of only two periods. The equations become highly complex when generalized to include longer-term debt. Since actual nominal debt contracts often ex- tend over many periods, it is interesting to examine the effects of a monetary shock on the price level and nominal interest rate using a model with longer- term debt. Simulations offer a way to accomplish this task.
In the previous section, we employed backward induction to solve for changes in the expected price level. This technique allows us to develop an
22The change in the expected rate of inflation (in period t) is defined relative to its previous expected value (in period t - 1). The expectation in period t - 1, of the rate of inflation between t and t + 1, is not generally equal to zero.
2air, for instance, the growth rate of the money supply were to follow a first-order autore- gressive path,
AMt = q*(AMt_l) + vt , where 0 < q < 1 ,
then price level overshooting would be more likely to occur. Increasing the proportion of trans- actions involving current output would reduce the probability of overshooting.
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Scott Sumner, O. David Gulley and Ross Newman
algorithm for analyzing debt over any number of periods. For a given mon- etary shock, we need only specify the interest elasticity of money demand and K, where K = kl . . . kN. As defined in Seetion 5, kn represents the fraction of total payments in period t that were contracted with knowledge of the money supply in period t - n.
We use interest elasticities of 0.5 and 1, values that are within the range of elasticities found in the empirical money demand literature. We specify two vectors for K to reflect varying payment patterns. The first vector (K1) is constructed using the relatively simple assumption that all transac- tions involve amortized payments on nominal debt eontracts with maturities of 30 periods. For simplicity, we set kl = 1/30. Thus, 1/30 of all payments are contracted with knowledge of the monetary shock in period t = 1. Therefore, ks0 is 30/30 beeanse all payments in period t = 30 are eontraeted with knowledge of the shock in period t = 1. The price level and interest rate reach their long-run equilibrium levels after 30 periods.
The second vector (K2) is eonstrncted under the more realistic as- sumption that one-half of all transactions are cash rather than credit pur- chases, and all credit purchases involve amortized nominal debt with a uni- form distribution of maturities from one to thirty periods. If each amortized bond is decomposed into its component zero-coupon bonds, then the quan- tity outstanding of these zero-coupon bonds is negatively related to their maturity. Thus, k 1 = 0.5 and k approaches one as n approaehes 30:k2 = 0.5322 . . . k2s = 0.9967, k29 = 0.9989. This vector is intended to better reflect the actual payment patterns of consumers.
If there exist fixed rate mortgage contracts with maturities of up to thirty years, then one could view a "period" as representing one year. How- ever, it seems likely that this formulation overemphasizes the importance of nominal debt contracts with maturities extending more than ten years. Loeys (1985) found only weak evidence of a liquidity effect for (forward) maturities beyond the three to ten year range. Although thirty year mortgages are common, due to the prevalence of mortgage refinancing, frequent home sales, and shorter term car loans, the empirical relevance of these very long term nominal contracts is questionable. If nominal debt contracts extending beyond seven to ten years are relatively unimportant, then the simulated periods used here might best be interpreted as representing one quarter rather than one year.
Figure 1 presents the responses of the expected price levels in re- sponse to a unit monetary shock assuming that the interest elasticity of money demand (a) is equal to one. The initial price levels are normalized to zero. P1 and P2 give the responses for vectors K1 and K2. The greater the fraction of payments associated with debt eontraets, the more the price level
284
6.4
Money Demand and Nominal Debt
5.6
48
"t0
t,,q I.-- 3.2
2.4
16
08
O0 2 6 10 14 18 22 26 30 34 38 42
TIME
Figure 1. Change in Price Level (a = 1)
overshoots its long-run equilibrium. At any period, K1 </42 and therefore, £1 > P2. Over time, as existing debt contracts mature, the price level returns to its new long-run equilibrium level of one. Figure 2 shows that the price level is more responsive to the monetary shock when the interest elasticity is smaller, which is consistent with traditional models of the liquidity effect.
Figures 3 and 4 present the response of forward nominal interest rates to a unit monetary shock. The initial interest rates are normalized to zero. I1 and I2 are the responses associated with the vectors K1 and K2. As was shown in Section 5, the lower the value of K, the more the price level over- shoots. Thus, the liquidity effect is larger when there is a higher fraction of transactions associated with nominal debt. The simulations also demonstrate that in a model featuring long-term debt, the liquidity effect is strongest for short-term interest rates, as implied by the model in Section 5. In the long run, the nominal interest rate will return to its initial equilibrium level. Fi- nally, Figure 4 shows that the size of the liquidity effect is inversely related to the interest elasticity of money demand.
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Scott Sumner, O. David GuUey and Ross Newman
9 7
5
E D 4
21 f,---. I "
2 fi 10 14 18 22 26 30 34 38 42
TIME
Figure 2.
Change in Price Level (a = 0,5)
7. Summary and Conclusions The model we developed in Section 5 has implications that are con-
sistent with the observed effects of unanticipated money shocks on short- and long-term interest rates, and spot and future commodity (or foreign exchange) prices, both before and after the October 1979 monetary policy shift. In particular, our model explains the impact of monetary shocks on the term structure of interest rates. We show that a one-time unexpected increase in the money supply will lower both current and implied forward interest rates, with the forward rates falling by less than current rates.
Our model of monetary policy is consistent with the expected liquidity hypothesis interpretation of the 1979-89. money targeting period. Unlike previous versions of the expected liquidity hypothesis, our model can also account for the response of implied long-term forward interest rates to money announcements. The inflation premium hypothesis has also been shown to have implications at variance with some of the key findings of the money announcements literature. In our model changes in nominal interest rates are generated by changes in expected inflation, consistent with the
286
t j l
Z
1 O0
0 75
050
0 25
0 O0
-0 25
-0 50
-0 75
-1.00
-1 25
Money Demand and Nominal Debt
--- 7
3 7 1 I 15 19 23 27 31 35 39 43
TIME
Figure 3. Interest Rate (a = 1)
inflation premium hypothesis. Yet our model is better able to explain the various stylized facts of the money announcements literature than is the inflation premium hypothesis.
The most important assumption in our model, that money demand is linked to transactions involving nominal debt, has the virtue of being highly plausible. Although many would find the assumption of complete price flex- ibility to be rather implausible, prices almost certainly become flexible over a period of time relevant for the determination of implied long-term forward interest rates. Further, by separating the real and monetary sectors, we avoid placing restrictions on the model that are incompatible with equilibrium maeroeeonomic models.
As noted earlier, most attempts to explain the liquidity effect have utilized IS-LM or other sticky-price models. Although our model also in- corporates a sluggish adjustment mechanism, our findings contradict the widely held view that the existence of the liquidity effect necessarily consti- tutes a rejection of flexible price models. Nevertheless, nothing in our model rules out the possibility that some prices are sticky. It would be possible to
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Scott Sumner, O. David GuUey and Ross Newman
u]
05
0.0
-O.S
-I0
-I.S
-20
-2.5
-30
-3.5
F2---
3 7 11 15 19 23 27 31 35 39 43
T IME
Figure 4. Interest Rate (a = 0.5)
develop a hybrid model that featured both sticky prices and long-term nom- inal debt. We chose a flexible price framework to show that the assumption of price stickiness is not necessary to explain the response of short-term interest rates to UMAs and because price stickiness is not sufficient to explain the response of implied long-term forward interest rates to UMAs.
Most models of the liquidity effect assume that negative responses of interest rates to money supply shocks are due to movements in the real interest rate. For instance, Coehrane (1989, 75) argues that "the existence of a liquidity effect implies that (expected) real returns vary over time." We have shown that monetary shocks can generate inverse movements in nom- inal interest rates without necessarily being able to affect real interest rates.
Our work suggests that gross national transactions (GNT) is the pre- ferred scale variable in a money demand equation. We know of no previous attempt to incorporate GNT into money demand equations. Possible exten- sions of the model could allow for monetary policy to affect real variables or to affect the structure of nominal debt contracts. For example, if the cost of renegotiating nominal contracts is relatively low, then monetary shocks could
288
Money Demand and Nominal Debt
impact the vector K, and these shocks would have less impact on long-term nominal interest rates than is implied by our model. Empirical work should focus on the importance of nominal debt contracts in economic transactions.
Finally, our model also offers an alternative explanation as to wily lagged money is statistically significant in many partial adjustment-type money demand equations. The most common explanation for the signifi- cance of lagged money is that transactions costs exist in adjusting money holdings. However, empirical studies often find that the coefficient on lagged money implies a very slow adjustment in money holdings (see Goldfeld and Sichel 1990 for discussion). We have shown that in a model where nominal income is employed as the scale variable, lagged money should be statistically significant and the speed of adjustment should be relatively slow.
Received. July 1995 Final version. January 1997
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Appendix 1 This appendix demonstrates the conditions under which a monetary
shock in period t will affect current interest rates by a greater amount than
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Scott Sumner, O. Dav id Gulley and Ross N e w m a n
implied forward interest rates. From Equations (13) and (14) we know that the impact of a monetary shock in period t on the current and forward interest rates is equal to
AEtit = ([k0 + ako - (a + kz)]/[kokl + a(ko + ka) + a2]}ut ; (A1)
Since both interest rates respond negatively to monetary shocks, we are interested in the conditions under which the absolute value of the change in the eurrent interest rate is greater than the absolute value of the change in the forward interest rate. This will occur if, and only if
-[ko + ako - (a + kl)] > -[kokl + a k l - (ko + a)] (A3)
or (kl - ko - ako) > (ko - kokl - akl) . (A4)
Since ( -ako) is greater than ( - ak l ) , a sufficient condition for Equa- tion (A4) to hold is
(kl - ko) > ko(1 - kl) • (A5)
Since ko is assumed to be less than one, a sufficient condition for Equation (AS) to hold is
(kl - ko) > (1 - k~). (A6)
The right side of inequality (A6) represents the share of total payments that were eontraeted two periods earlier. The left side represents the share of total payments that were contracted one period earlier. As noted in Sec- tion 5, it seems likely that at any given point in time the later term would be larger than the former term.
Appendix 2
L = natural log of the nominal demand for (base) money. T = transactions demand for (base) money. a = interest elasticity of money demand. i = nominal interest rate.
292
E p =
bt, b2 = M =
Ut, ?)t ~-
k o =
k 1 ~--
K1 and/(2 = P1 and P2 = I1 and I2 =
Money Demand and Nominal Debt
expectations operator. price of the good. ex ante real interest rate. constants such that bl, b2 > O. natural log of the nominal (base) money supply. serially uncorrelated error terms. fraction of total payments in period t + 1 contracted with knowledge of the monetary shock in period t. fraction of total payments in period t contracted with knowledge of the monetary shock in period t. proportion of total payments in period t + n that were contracted with knowledge of the monetary shock in period t. vectors specifying how/~ evolves. time paths of P given K1 and K2. time paths of i given K1 and K2.
