Inflation, Nominal Interest Rates, and the Variability of Output
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

This paper examines the distribution of output around capacity when money demand is a nonlinear function of the nominal interest rate such that nominal interest rates cannot become negative. When fluctuations in output result primarily from disturbances to the money market, the variance of output is shown to be an increasing function of the trend inflation rate. When they result from disturbances to the goods market, the variance of output is a decreasing function of the trend inflation rate. When both disturbances are significant, there exists, in general, a critical non-zero trend inflation rate that minimizes the variance of output.

Abstract

This paper examines the distribution of output around capacity when money demand is a nonlinear function of the nominal interest rate such that nominal interest rates cannot become negative. When fluctuations in output result primarily from disturbances to the money market, the variance of output is shown to be an increasing function of the trend inflation rate. When they result from disturbances to the goods market, the variance of output is a decreasing function of the trend inflation rate. When both disturbances are significant, there exists, in general, a critical non-zero trend inflation rate that minimizes the variance of output.

I. Introduction

A notable achievement of macroeconomic policy has been the sustained reduction of inflation rates in many of the industrial countries. The average inflation rate in the G7, for example, declined from around 10 percent during 1974-83 to 4 percent during 1984-95. It is widely acknowledged that these gains toward a potential eventual objective of price stability led to declines in nominal interest rates. 1/ 2/ Cyclical pressures then pushed short-term nominal interest rates on occasion even lower, with rates declining in one instance--Japan--to virtually zero.

These events have prompted interest in the implications of the long-run or trend inflation rate, and the associated level of nominal interest rates, for the business cycle and the conduct of monetary policy. One of the issues that has received some attention has been the implications of a zero floor on nominal interest rates. It is widely presumed that nominal interest rates naturally have a floor of zero because at a zero interest rate, money, or central Bank money, would completely dominate interest bearing assets such as bonds, as money would continue to provide liquidity services but without foregoing interest. 3/ Summers (1991) argues that since nominal interest rates cannot become negative, a policy of zero inflation would constrain real interest rates to always be positive. Summers notes that during the postwar period in the U.S., ex post short-term real interest rates have been negative about a third of the time. 4/ If historical experience is viewed as representing equilibrium outcomes, and equilibrium occurred at negative real interest rates, then a policy of zero inflation that precluded the negative orthant for real interest rates as an equilibrium may lead to the rationing of capital. The consequences for economic stability could potentially be severe. Fuhrer and Madigan (1994) assess the constraint that nominal interest rates remain positive places on the ability of monetary policy to cushion output in the face of negative shocks to aggregate demand. 1/

In contrast to the traditional countercyclical role for monetary policy in affecting the business cycle emphasized in the literature, this paper focusses on effects of the choice of the long-run or trend inflation rate for the magnitude and form of the business cycle. We investigate the effects of the choice of the trend inflation rate for the distribution of output around potential or capacity, when money demand is a nonlinear function of the nominal interest rate such that equilibrium nominal interest rates are bounded by a zero floor. We show that when fluctuations in output result primarily from disturbances to the money market, the variance of output is an increasing function of the trend inflation rate. When fluctuations in output result primarily from disturbances to the goods market, however, the variance of output is a decreasing function of the trend inflation rate. If both disturbances are significant, there exists, in general, a critical non-zero trend inflation rate that minimizes the variance of output.

The response of output to positive and negative shocks is asymmetric. Contractions in output are more severe than expansions, but shorter-lived in that output returns to capacity at a faster speed during contractions than in expansions. As a result, the distribution of output is skewed, with the direction of skewness depending on the relative importance of the two effects. The extent of asymmetry in the behavior of output is shown to be a decreasing function of the trend inflation rate.

Simulations with representative parameter values for the U.S. economy suggest that at very low rates of inflation, increases in the inflation rate could significantly lower the variance of output. Over this range of inflation rates, therefore, there is a tradeoff between price stability and the stability of output. Improvements in price stability come only at the expense of increases in the variability of output. Continued increases in the inflation rate lead eventually, however, to increases in the variance of output so that, in this range, there is no tradeoff between price stability and output stability.

Section II focusses on the relationship between the long-run rate of inflation and, therefore, the average level of nominal interest rates, and the variance of output. Section III examines asymmetries in the response of output to positive and negative disturbances. Empirical implications for the U.S. economy are evaluated in section IV. Section V offers some concluding observations.

II. The Long-Run Rate of Inflation and the Variability of Output

This section illustrates the links between the long-run rate of inflation and the variance of output around capacity in a traditional demand-driven sticky-price model where money demand is assumed to be a nonlinear function of the nominal interest rate such that, in equilibrium, nominal interest rates cannot become negative. For expository purposes, it is useful to think of the nonlinearity in money demand as having two distinct effects, a general or average effect and a local effect. First, the nonlinearity implies that the level of the long-run or steady-state interest rate determines the average responsiveness of money demand to changes in the interest rate. This can be thought of as the general or average effect. Second, the nonlinearity implies that, starting from any steady-state nominal interest rate, the responsiveness of money demand to the interest rate varies with the magnitude and direction of changes in the interest rate. This can be thought of as the local effect of the nonlinearity. So as to establish some basic analytical results linking the variability of output and the rate of inflation, this section focusses on the general or average effects of the nonlinearity in money demand. This is accomplished by considering a linear approximation of the money demand curve around a long-run level of interest rates. The linearized system ignores the local effects of the nonlinearity and yields a system with symmetric responses. The next section discusses the form of asymmetries induced by the local effects of the nonlinearity.

There are, in principle, a variety of functional forms for aggregate money demand that can be used to capture the fact that nominal interest rates cannot become negative in equilibrium. We adopt a functional form that is closely related to traditional specifications, and is particularly parsimonious with regard to the parameters required to characterize it. In contrast to traditional specifications of the demand for money that posit the logarithm of the demand for money to be a decreasing function of the nominal interest rate, we specify the logarithm of the demand for money to be a decreasing function of the logarithm of the nominal interest rate

Ln(Money Demandt)=αLn(it)+φYt,(1)

where Ln denotes the natural logarithm of a variable, it denotes the nominal interest rate, and Yt denotes the logarithm of output. α then represents the elasticity of the demand for money with respect to the nominal interest rate. 1/ The constant elasticity specification implies that equal successive declines in the nominal interest rate result in successively larger increases in the logarithm of money demand. The money demand function is, therefore, convex in the nominal interest rate-logarithm of money demand plane, and approaches the horizontal axis asymptotically as the nominal interest rate falls to zero. 1/

The equilibrium condition in the money market, which is referred to as the LM curve, is given by

MtPt=αLn(it)+φYt,(LM)(2)

where Mt denotes the logarithm of the money supply, and Pt the logarithm of the aggregate price level. For simplicity, the (logarithm of the) money supply is assumed to evolve according to

Mt=Mt-1+μ+εt,(3)

where εt is the unanticipated innovation in the level of the money supply and is assumed to be iid with zero mean and variance σε2. μ represents the constant drift in the money supply, that is, the constant expected growth rate of the money supply in each future period. Keeping in mind the money market equilibrium condition in (2), it is worth emphasizing that εt can be thought of as either a positive innovation in the money supply or a negative innovation in money demand. Alternatively, if, as is often argued is the case, the money supply is viewed as endogenous in that it responds automatically to partly accommodate shocks in money demand, then, as long as the offset is not perfect, εt can be thought of as the “net” innovation in the money market. Returning to the interpretation of εt as a money supply shock in (3), note that the specification of the money supply as a random walk with drift implies that all unanticipated innovations in the money supply are interpreted as permanent in that they are expected to persist for ever. Again, this specification is adopted only for simplicity. A specification of the money supply as a general ARMA process with drift would in no way affect qualitatively our results.

The level of output is determined by demand. Aggregate demand is specified to be a decreasing function of the real interest rate and to be affected directly by a real demand shock term

Yt=Ytd=A[it(EtPt-1Pt)]+St,(IS)(4)

where the superscript d on output, Yt, is used to denote demand; Et denotes the mathematical expectations operator conditional on information available at time t; St represents the real demand shock term. So that linear approximations of money demand around long-run levels of nominal interest rates are meaningful, we impose that the real (and hence the nominal) interest rate are mean reverting by specifying that St is a mean-reverting process. In particular we assume that

St=S¯+Ωt,whereΩt=ρΩt-1+ηt,0ρ1,(5)

so that the demand shift term follows an AR(1) process around a mean long-run level of S¯. 1/ ηt is assumed to be iid with zero mean and variance ση2. For simplicity, it is further assumed that εt and ηt are independent.

The price level is assumed to be sticky in that it is completely predetermined at a point in time and adjusts only slowly to equilibrate the goods market. The dynamic adjustment process for the price level is assumed to be given by a form of the Barro-Grossman rule, which posits adjustment in disequilibrium to be a function of both the extent of disequilibrium in the goods market and the change in equilibrium prices. The particular rule employed is a version of that posited by Mussa (1981a, b)

Pt-1Pt=EtΠ¯+β(YtY¯)(6)

so that inflation at time t+1 responds to excess demand at time t. 2/ For the remainder of the analysis, capacity output, Y¯, is normalized to zero. 3/

The term EtΠ¯ in (6) is used to represent the time t expectation of the equilibrium change in prices. We define the equilibrium changes in prices as the change in the flexible price level in the long run, in the absence of further shocks to the economy. In principle, the term EtΠ¯ could be defined in several ways. There are two reasons why we adopt our particular definition. First, since prices are only adjusted gradually it is natural to posit that price setters adjust only to permanent or longer-term movements in the equilibrium price level. 4/ Second, note that the nonlinearity in the model fundamentally alters the distribution and expected value of output when prices are sticky and when they are flexible. If prices are flexible, output equals capacity with certainty in each period. If prices are sticky, however, the nonlinearity in money demand implies asymmetric movements in output around capacity, so that the distribution of output is skewed, and the expected value of output is not, in general, equal to capacity output. Note from equation (6) that if output were at capacity and EtΠ¯ were defined as the expected change in the flexible price level, then inflation in the sticky-price model would equal inflation in the flexible price model. However, the inflation rate in the flexible price model is that consistent with an expected value of output equal to capacity. Therefore, to avoid imposing that the expected value of output equals capacity in a nonlinear economy with sticky prices, we assume that EtΠ¯ equals the inflation rate expected to prevail in the long run in the absence of further shocks. This assumption ensures that, in the absence of shocks, output does indeed return to capacity, but does not constrain the long-run expectation of output to equal capacity.

It is straightforward to establish from equations (3)-(4) that starting at any point in time, t, under the assumption of zero growth in capacity output, and in the absence of further shocks to the economy, the equilibrium or flexible-price rate of inflation tends in the long run to the expected rate of money growth, μ, which can then be substituted for EtΠ¯ in (6). 1/ As a result, in the sticky-price economy, in the absence of further shocks to the economy, the rate of inflation tends in the long run to μ, while the real interest rate tends to S¯/A, and, therefore, the nominal interest rate tends to

i¯=S¯A+μ.(7)

Taking a first-order linear approximation of money demand around i¯, the equilibrium condition in the money market can be written as

MtPt=καi¯it+φYt,(8)

where κ represents a constant determined by the long-run level of money balances. 2/ Using (4) and (8), output can be expressed as a function of the price level, expected inflation, and the exogenous variables of the system. Substituting this expression for output into equation (6), noting that prices are predetermined at a point in time, yields a first order difference equation for the price level which can be written as 3/

Pt=λPt-1+(1λ)[Mt-1κ+αi¯μ+αi¯St-1]+μ.(9)

The current aggregate price level is thus a weighted average of last period’s price level and the exogenous variables of the system, plus a drift term representing the long-run rate of inflation. 1/ The weight on the lagged price level is

λ=1Aβα(μ+(s¯/A))(1Aβ)+Aφ.(10)

It is assumed that

Aβ1.(11)

(11) is an intuitively reasonable condition. Note from equation (4) that aggregate demand is an increasing function of expected inflation with a responsiveness of A. Inflation is in turn posited to be an increasing function of excess demand in equation (6), with a responsiveness of β. The condition that Aβ be less than 1 simply ensures that aggregate demand is a stable function of itself. 2/ This condition is sufficient to ensure that λ is less than 1. It is further assumed that

Aβα(μ+(S¯/A))(1Aβ)+Aφ1,(12)

so that the characteristic root of the difference equation describing the dynamic solution of the price level in equation (9) is strictly between 0 and 1, and the dynamics of the economy are well behaved. 3/ Both conditions (11) and (12) are easily satisfied for reasonable parameter values for the U.S. economy, which are discussed in section IV. Note that the characteristic root, λ, in equation (10) is, under condition (11), a decreasing function of μ, the long-run rate of inflation, that is

λμ0.(13)

The reasons for this relationship are elaborated below.

The dynamic solution for the price level in equation (9) can be rewritten as

Pt=λ[Pt-1+μ]+(1λ)[Et-1P¯t]+(1λ)θ[Et-1P^Et-1P¯t],(14)

where Pt denotes the flexible price level in period t, P¯t denotes the permanent component of the flexible price level, and θ is a constant. 1/ The first two terms on the right hand side of (14) then imply that the price level is a weighted average of last period’s price level, adjusted for the long-run drift in prices, μ, and last period’s expectation of the long-run component of the flexible price level. The third term arises from short run or temporary movements in flexible price level, and implies that when the flexible price level, Pt, differs from the long run flexible price level, P¯t, there are short run effects on the price level. Short run divergences between Pt and P¯t arise from the temporariness of movements in the aggregate demand shift term, St.

It is useful to consider the case where there are no temporary movements in the flexible price level. 2/ If there are no temporary movements in the flexible price level, P^t equals P¯t and the third term in equation (14) drops out. The price level is then simply a weighted average of last period’s price level and the long-run flexible price level. Then, recalling (13), note that the effect of an increase in μ, the long-run rate of inflation, is to reduce persistence or inertia in the price level. An increase in μ reduces λ, the weight given by the current price level to last period’s price level and increases the weight on the expectation of the long-run flexible price level. If prices were perfectly flexible with an exactly one period lag, that is prices were revised at the end of each period, then the coefficient on the lagged price level would be zero and the coefficient on the time t-1 expectation of the long run price level would be unity. In this sense, an increase in μ, by reducing the average responsiveness of money demand to the interest rate, causes faster adjustment of the aggregate price level to both real and monetary shocks.

The solution for output can be expressed as

Yt=λYt-1+(1λ)β[εt+αA(μ+(S¯/A))(StSt-1)].(15)

Since short run deviations of output from capacity result from the stickiness of prices, the solution for output in equation (15) is analogous to that for the price level. In particular the characteristic roots of the difference equations describing the solutions for the price level and output are the same. It follows that, as in the case of prices, the effect of an increase in μ is to lower λ and thus reduce the persistence in consecutive levels of output or, alternatively, to increase the speed with which output returns to capacity, regardless of whether the shock emanates in the money market or the goods market.

The above has established that a higher long-run rate of inflation results in a faster speed of adjustment of prices and output. To see why this is the case it is useful to consider the dynamic path of the economy traced out in the nominal interest rate-output plane. Note that if the price adjustment equation in (6) is used to substitute for expected inflation in the IS equation, goods market equilibrium can be written such that output is simply a decreasing function of the nominal interest rate and an increasing function of the real demand shock term. Equilibrium at a point in time is given by the intersection of the negatively sloped IS and positively sloped LM curves. If output exceeds capacity, in the absence of further shocks or changes in the demand shock term, prices will rise by β times the output gap. 1/ The increase in the price level shifts the LM curve up, while the IS curve remains unchanged. The magnitude of the upward shift of the LM curve is, from equation (8), [(μ+S¯/A)/α] times the change in the price level. A higher μ, therefore, implies a larger upward shift. At a higher trend inflation rate, the interest responsiveness of money demand is lower and, therefore, an increase in the price level requires a larger change in the nominal interest rate for equilibrium to be restored in the money market. The larger upward shift of the LM curve translates to a larger decline in output, that is a faster adjustment of output back to capacity.

Consider now the impact effects. It is apparent from (15) that the impact on output of a monetary or real shock is in the same direction as the shock. However, the relationship between the magnitude of the impact effect and the long run rate of inflation depends on whether the shock originates in the money market or the goods market.

The impact effect of a monetary shock from (15) is [(1-λ)/β]εt. A higher rate of inflation implies, from (13), a smaller λ, and thus a larger impact effect on output of an innovation in the money supply. The reason for the larger impact effect is that at a higher μ and higher nominal interest rates, money demand is less responsive to interest rates. Any given shock to the money market, therefore, requires a larger adjustment in nominal interest rates for equilibrium to be restored. In the real interest rate-output plane this translates to a larger shift in the LM curve as μ rises. 2/ With an IS curve that is unaffected directly by changes in μ, the larger shift in the LM curve translates into a larger equilibrium impact on output.

The impact effect of a real shock is, from (15) and (10)

(1λ)Aβα(μ+(S¯/A))ηt=αα(1Aβ)+Aφ(μ+(S¯/A))ηt,(16)

so that at a higher long-run rate of inflation the impact effect on output is smaller. At a higher μ and higher nominal interest rates, money demand is less responsive to interest rates. As money demand rises with increases in output, the required adjustment in interest rates to equilibrate the money market increases, that is to say the LM curve gets steeper with increases in μ. Consequently, at a higher μ, a shock to the IS curve results in a larger change in interest rates and a smaller change in output.

Figure 1 plots the impulse responses of output to positive monetary and goods market shocks for alternative long run rates of inflation. In the case of the goods market shock, output eventually overshoots its long run value, as prices adjust to lagged excess demand while excess demand dissipates in addition because of the temporary nature of the shock. In summary, for monetary shocks, a higher rate of inflation implies a larger impact effect on output of a shock but also a faster speed of adjustment of output back to capacity. For shocks to the goods market, on the other hand, a higher rate of inflation implies a smaller impact effect on output and a faster speed of adjustment in response to shocks.

Figure 1.
Figure 1.

Impulse response of Output to Positive Monetary and Goods Market Shocks at Alternative Long Run Rates of Inflation.

Citation: IMF Working Papers 1996, 109; 10.5089/9781451853162.001.A001

The solution for output in (15) can be rewritten as

Yt=(1λ)βΣi=0λiεt-1+(1λ)Aβα(μ+(S¯/A))Σi=0λi(St-iSt-1-i),(17)

so that output can be expressed as a function of the current and all the past shocks to the economy. Given the assumed independence of the processes generating εt and ηt the variance of output is

V(Y)=1β2(1λ)2(1λ2)σε2+α2A2β2[2(1+ρ)](1λ)2(1λ2)1(μ+(S¯/A))2ση2,(18)
orV(Y)=V(YM)+V(Ys),(19)

where V(YM) and V(YS) are used to denote the components of the variance of output due to monetary and real aggregate demand shocks, respectively. We now examine the dependence of each of these components of the variance of output on the long-run rate of inflation.

For monetary shocks, it was noted earlier that with a higher rate of inflation, the impact effect: on output of a monetary shock is larger, so the variance of output should be higher, but at a higher rate of inflation, the speed of adjustment to a shock is faster, so the variance of output should be lower. These opposing effects of an increase in μ on the variance of output due to monetary shocks are captured in the first term of equation (18). A higher p lowers A, increasing the numerator, which captures the impact effect, but also increases the denominator, which results from the speed effect. Thus, in principle, a higher μ has an ambiguous effect on the variance of output due to monetary shocks. It is straightforward to show, however, that the impact effect dominates because

V(YM)=1β2(1λ)2(1λ2)σε2=1β2(1λ)(1+λ)σε2,(20)

which is an unambiguously decreasing function of λ and hence an increasing function of μ. The variance of output due to monetary shocks is, therefore, an increasing function of the long run rate of inflation.

For shocks to the goods market, a higher long run rate of inflation reduces the magnitude of the impact effect on output of such shocks, and increases the speed with which output returns to capacity. The variance of output due to shocks to the goods market is, therefore, an unambiguously decreasing function of the long run rate of inflation. This is evident when V(YS) is rewritten using the expression in equation (16) as

V(Ys)==α2[21+ρ]ση2[1α(1Aβ)+Aφ(μ+(S/A))]21(1λ2).(21)

To establish the nature of the relationship between the total variance of output and μ when there are both monetary and aggregate demand shocks note from equation (18) that as μ approaches the negative of (S¯/A), that is as the long run rate of inflation approaches the negative of the long run real interest rate, and the long run nominal interest rate approaches zero, V(YS) will dominate V(YM). As the long run nominal interest rate approaches zero, regardless of the relative magnitudes of the variances of the actual shocks, σε2 and ση2, the relative importance of V(YS) in V(Y) grows without bound. 1/ Since V(YS) is a decreasing function of μ this implies that the total variance of output must decline with increases in μ for some range of μ.

Figure 2 plots the variance of output, and its components, as functions of the long run rate of inflation, μ. It shows that there exists, in general, a critical non-zero long-run rate of inflation which minimizes the aggregate variance of output.

Figure 2.
Figure 2.

Variance of Output as a Function of the Long Run Rate of Inflation.

Citation: IMF Working Papers 1996, 109; 10.5089/9781451853162.001.A001

III. Asymmetries in Output

This section examines asymmetries in the behavior of output resulting from the nonlinearity in money demand, and how these are affected by the choice of the long run rate of inflation. Equations (2)-(6) can be combined to yield an implicit function for output and the price level. Substituting in the price adjustment rule in (6), and upon some manipulation, the solution for output can be represented by the nonlinear difference equation

Yt=[Yt-1(St-1+Aμ)(1-Aβ)].exp[(1α)[(βφ)Yt-1+φYtεt]]+(St+Aμ)1Aβ.(22)

Appendix I shows that the conditions in equations (11) and (12), which we continue to assume hold, are sufficient to establish that

0YtYt-1|Y=01,and2YtYt-12|Y=00,(23)

so that equation (22) can be plotted as the convex curve labelled YY in Figure 3, when the forcing variables are at their long-run expected values. Starting from any initial level of output given by history, in the absence of further shocks to the economy, the intrinsic dynamics for output can then be represented by the arrows in Figure 3 and indicate that the solution for output is unique and locally stable around capacity output (the origin). 1/ Note that the slope of YY at any point represents the extent of inertia in the system so that the steeper the curve, the slower the rate at which output returns to capacity. It should be immediately apparent that the convexity of YY around capacity output (the origin) implies that the speed at which output returns to capacity is faster for negative deviations of output from capacity than for positive deviations. That is to say, if one were to compare an expansion and a contraction characterized by the same absolute initial deviation of output from capacity, the contraction would be shorter lived.

Figure 3.
Figure 3.

Intrinsic Dynamics of Output During Expansions and Contractions.

Citation: IMF Working Papers 1996, 109; 10.5089/9781451853162.001.A001

Appendix I shows further that

μ(YtYt-1)0,andμ(2YtYt-12)0,(24)

where the derivatives are evaluated at the origin. Equation (24) implies that an increase in the trend rate of inflation reduces the slope and curvature of YY to a curve like Y’Y’ in figure 3. Therefore, an increase in μ increases the speed with which output returns to capacity following any shock. In addition, an increase in μ reduces the extent of the asymmetry in the speeds at which output returns to capacity for expansions and contractions.

From equation (22), as Appendix I shows, the impact effect of a monetary shock is

Ytεt=1φ+α(1Aβ)Ait0.(25)

An increase in the trend inflation rate, μ, that raises the average level of interest rates lowers the numerator in (25), and thus increases the magnitude of the impact of a monetary shock. Equation (25) shows that for a given μ, the impact effect of a positive shock to the money supply, which lowers nominal interest rates, is smaller than the absolute value of the effect of a negative shock which raises interest rates. The convexity of the money demand function implies that as interest rates fall with an increase in the money supply, the responsiveness of money demand increases, so that interest rates do not need to fall by as much. When the money supply contracts, however, as interest rates rise, the responsiveness of money demand decreases, so interest rates need to rise by more. The LM curve, therefore, shifts up by more than it shifts down for the same absolute value of a shock to the money supply. With an unchanged IS curve, contractions are, therefore, more severe.

The impact effect of a shock to aggregate demand is determined by

Ytηt=1(1Aβ)+Aφαit0,(26)

so that a higher trend inflation rate, by increasing the average nominal interest rate, increases the denominator and reduces the magnitude of the impact effect. Again, at any level of μ, the response of output to positive and negative shocks of the same magnitude is asymmetric, with a negative shock having a larger absolute impact. Since the LM curve in the interest rate-output plane gets steeper as interest rates rise, a positive shock to the IS curve has a larger impact on interest rates and a smaller impact on output than a negative shock.

The asymmetry in the response of output to positive and negative shocks stems from the asymmetry in the response of money demand to positive and negative changes in interest rates. Since the functional form for money demand adopted implies that the asymmetry in money demand increases as interest rates decline, it follows that asymmetries in the behavior of output are accentuated at low trend rates of inflation. An implication of picking a very low trend inflation rate is then that contractions tend to be more severe than expansions. However, contractions are relatively shorter-lived. 1/ Asymmetry in the response of output to positive and negative shocks may, therefore, result in a distribution of output that is skewed. It should be noted, however, that while the choice of a low rate of inflation increases asymmetry in both the impact response of output and the speed with which it returns to capacity, because the two effects have opposing effects on the distribution of output, the net effect on the direction of skewness is, in general, ambiguous. 1/ We turn now to an empirical evaluation of the effects of alternative trend inflation rates for the expected value, variance, and skewness of output using parameter values for the U.S. economy.

IV. Empirical Relevance

The purpose of this section is to examine the empirical implications of alternative long run rates of inflation when money demand is given by the nonlinear function in equation (1). This is done by simulating the nonlinear model described by equations (2)-(6) for representative parameter values for the U.S. economy. Table 1 summarizes the parameter values employed. Appendix II details their derivation and the sources from which they were obtained. A time period is defined as a quarter and short-run elasticities are used in arriving at the parameter values. In addition to the parameter values in Table 1, the mean of the aggregate demand shock term, S¯, was parameterized to yield a mean quarterly real interest rate, S¯/A, of 1 percent per annum, and the autoregressive parameter in the demand shift term, ρ, was assumed to be 0.9. 2/3/

Table 1.

Parameter Values

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Note: See Appendix II for details of derivation and sources.

Using the parameter values in Table 1, the impulse responses of output in equation (22) were simulated for once-and-for-all 1 percent positive and negative innovations in the money supply and the real demand shock term, for annual inflation rates of zero and ten percent. The results are reported in Figure 4 where the use of logarithms implies the movements in output measured on the vertical axis are interpretable as percentage deviations of output from capacity.

Figure 4.
Figure 4.

Simulated Effect of 1 Percent Positive and Negative Shocks at Annual Trend Inflation Rates of 0 and 10 Percent.

Citation: IMF Working Papers 1996, 109; 10.5089/9781451853162.001.A001

Note: The value of µ reported corresponds to the annualized value of µ actually employed.

Two observations on Figure 4. First, note that the difference between the impulse responses of output at 0 and 10 percent inflation rates indicates that changes in the trend inflation rate can have a profound effect on the magnitude of the impact effect on output of shocks and the speed with which output returns to capacity. Therefore, quantitatively, the effects on the variance of output of changes in μ are likely to be significant. Second, the plotted impulse responses of output at a zero inflation rate reveal the asymmetry in the response of output to positive and negative shocks. For monetary shocks, the dashed line representing the response of output to a negative shock indicates that the impact effect of a negative shock is larger. 1/ However, the subsequent return of output to capacity is faster. For the goods market shock, the contractions is also more severe, but the return to capacity takes longer. 2/ The closeness of the dashed and solid lines indicates, however, that the quantitative effect of the asymmetries is small. In fact, as the inflation rate rises to 10 percent the solid and dashed curves representing the impulse responses of output to positive and negative shocks, respectively, coincide almost perfectly, so the asymmetry is not discernible. The measured impact on the expected value and skewness in the distribution of output resulting from the asymmetries is, therefore, likely to be minuscule. These impressions on the likely behavior of the moments of output are borne out by Monte Carlo simulations of the model.

The model described by equations (2)-(6), configured for representative parameter values for the U.S. economy in Table 1 was simulated for 10,100 quarters assuming that the economy was in equilibrium in the period prior to the first simulation period. The first 100 observations were then discarded before calculating the statistics reported in Table 2 to reduce any “startup” bias. The shocks εt and ηt were each generated independently from normal distributions with zero means and standard deviations of 1/2 of 1 percent. 1/ So as to focus purely on the effects of changes in μ on movements in output, simulations for each value of μ reported in table 2 were carried out for the same draw of shocks. Similarly, to distinguish the effects of monetary and goods market shocks, at each rate of inflation, the model was first simulated with the draw of monetary and aggregate demand shocks; then the model was simulated with the same draw of monetary shocks, but with the aggregate demand shocks set at zero; and, finally, vice versa.

Table 2.

Summary Statistics for (the Logarithm of) Output in Stochastic Simulations at Alternative Trend Inflation Rates

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Note: The values of µ reported correspond to the annualized values of µ actually employed. The model was simulated for 10,000 quarters of at alternative inflation rates for the same draws of money supply and aggregate demand shocks.

The mean, median, standard deviation, and coefficient of skewness of output from the simulations are reported in Table 2. Consider first the variance of output. The (quarterly) standard deviation of output due to monetary shocks almost doubles from 1.6 percent to 3.13 percent as the annual inflation rate increases from 0 to 10 percent. The standard deviation of output due to real shocks declines dramatically from 3.72 percent to 0.79 percent as the inflation rate increases from 0 to 10 percent. The standard deviation of output when both shocks are present declines from 4.01 percent at 0 inflation to reach a low of 3.03 percent at an inflation rate of 3 percent, and then rises gradually with increases in the inflation rate.

Regarding the quantitative impact of the asymmetries in output on the first and third moments, note that the expected value of output measured by the mean or median is, for all practical purposes zero. Similarly, the coefficients of skewness indicate that while there is skewness, and that the extent of skewness is systematically affected by the rate of inflation, the magnitude is small. This is borne out by a visual examination of the distributions of output obtained from the simulations presented in Figure 5 which show the dramatic effects of changes in inflation on the distribution of output but display few, if any, signs of skewness. 1/

Figure 5.
Figure 5.

Distribution of Output around Capacity.

Citation: IMF Working Papers 1996, 109; 10.5089/9781451853162.001.A001

V. Conclusion

The success of monetary policy in reducing inflation to low levels in several of the industrial countries has brought to the fore issues related to the objective of zero inflation as a long-term goal of monetary policy. The traditional role for monetary policy in the business cycle arises from the possibility that monetary policy may be able to react to information that becomes available after some private sector decisions have been made. In the staggered nominal (wage or) price contracting models of Fischer (1977), Taylor (1979, 1980a), Phelps (1979), and Calvo (1983), for example, nominal prices are preset for discrete intervals of time, with the levels determined by expectations of events during the interval. If monetary policy can react to shocks that occur since--at least some--prices were set, and before they are revised, it can reduce the variability of output. The long-run predictable rate of money growth, however, which determines the long run rate of inflation, once incorporated into private sector expectations, can play no role in affecting deviations of output from capacity. In such a framework, therefore, if the monetary authority’s objective function is decreasing in inflation and deviations of output from capacity, the trend or long-run rate of inflation, which plays no role in affecting the deviations of output from capacity, would be set at zero. The models of stabilization policy developed by Kydland and Prescott (1977), Barro and Gordon (1983), and others, employ a similar underlying framework, and focus on the strategic interaction between the central bank and the private sector. In these models the equilibrium rate of inflation is positive if the level of output targeted by the central bank exceeds capacity. If the level of output targeted by the central bank equals capacity, however, in equilibrium the rate of inflation would be zero.

In sharp contrast to these models where the trend inflation rate has no implications for the business cycle, this paper has highlighted a channel whereby the perfectly predictable component of money growth, by determining the long run rate of inflation and nominal interest rates, affects the distribution of output around capacity. It was shown that in general there exists a critical non-zero trend inflation rate that minimizes the variance of output. Further, the choice of the long-run inflation rate influences not only the variance but also other moments of the distribution of output.

APPENDIX I Characteristics of the Solution for Output in the Nonlinear Model

This appendix establishes several propositions employed in the text.

a. Existence of a unique stable solution

Differentiating the solution for output in equation (22)

YtYt-1=1+(φβα)(St-1+Aμ1AβYt-1)exp(γ)+(φα)(St-1+Aμ1AβYt-1),whereγ=(1α)[(βφ)Yt-1+φYt1εt].(A1)

Evaluating the derivative at Yt = Yt-1 = 0, St = S¯, and εt = 0

YtYt-1|Y=0=1+(φβα)(S¯+Aμ1Aβ)1+(φα)(S¯+Aμ1Aβ)=1Aβα(μ+(S/A)(1Aβ)+Aφ=λ.(A2)

Therefore, if the conditions in equations (11) and (12), which ensure that A lies between 0 and 1, hold, the slope of YY in figure 3, drawn for values of the exogenous variables at their long-run expected values, is positive and less than one at the origin. The second derivative, evaluated at the origin is, noting, (A2)

2YtYt-12|Y=0=βα(2+2φβαS¯+Aμ1Aβ)[1+(φα)(S+Aμ¯1Aβ)]3=βα(1+λ)[1+φα(S¯+Aμ)1Aβ]20(A3)

So equation (22) is convex at the origin. For any initial level of output given by history, in the absence of further shocks to the economy, the intrinsic dynamics for output can be represented by the arrows in figure 3 and indicate that output is locally stable. The convexity of YY at the origin suggests that it is likely to intersect the 45 degree line again at some value of Y, in the positive orthant at, say Yc. Yc then represents a critical level of output above which the intrinsic dynamics are unstable. We now establish that the existence of Yc is irrelevant to the dynamics of output in the model in that once exogenous variables are at their long-run values, the initial condition for output never equals or exceeds Yc, and hence output is always locally stable. Two observations. First, note that the curve YY has been drawn for constant values of the forcing variables and, in particular, we have focussed on the case where the forcing variables are at their expected long run values. Shocks to the economy shift the YY curve, so that it temporarily has a non-zero intercept. Second, note that when the forcing variables are at their long-run values, YC=(S¯+Aμ)/(1Aβ).

Now, starting from a long-run equilibrium where Yt-1=0 in period 0, consider a once-and-for-all positive shock to the money supply in period 1. This shifts the YY curve up for exactly one period. The intercept with the vertical axis provides the solution for the impact effect, which is

Y1=(S¯+Aμ1Aβ)(1exp[1α(φY1e1)])=YcYc.exp[1α(φY1e1)]Yc.(A4)

In period 2 the YY curve shifts back down to pass through the origin. The initial condition for output in period 2 is Y1 which is less than Yc. The dynamics of output in response to a money shock are, therefore, stable.

For a shock to the goods market, it is also straightforward, though somewhat more tedious, to prove that the existence of Yc is irrelevant for stability. We, therefore, do this here only for the simple case where ρ = 0 and the shock to the goods market lasts exactly one period. Note from equation (22) that both St and St-1 affect YY and do so in opposite directions--while a higher St shifts YY up, a higher St-1 shifts YY down. Therefore, if there is a shock to S in period 1 that lasts exactly one period, YY shifts up in period 1, it then shifts down in period 2, and in period 3 it returns to its original position. Now,

Y1=(S¯+Aμ1Aβ)exp(φαY1)+S¯+Aμ1Aβ+η11Aβ=YcYcexp[φαY1]+η11Aβ,(A5)

which can exceed Yc. The solution for Y2, employing (A5), can be expressed as

Y2=YcYcexp(βαY1+φαY2)Yc.(A6)

The initial condition for output in period 3, when YY returns to its long-run position, is, therefore, always below Yc.

In summary, equations (A4) and (A6) have established that capacity output represents the unique locally stable equilibrium.

b. Speed of adjustment

An increase in μ decreases the slope of YY, as can be noted directly from equation (A2). Then keeping this in mind, that is that dλ/dμ < 0, it follows that an increase in μ reduces the numerator in (A3) and increases the denominator, so that the convexity of YY also decreases with increases in μ. These two results are summarized as

μ(YtYt-1)0,andμ(2YtYt-12)0,(A7)

where the derivatives are evaluated at the origin.

c. Impact effects

Differentiating equation (22) with respect to εt and ηt, respectively, substituting in equations (2), (4) and (6), the responses of output on impact of monetary and goods market shocks are

Ytεt=1φ+[α(1Aβ)St-1+Aμ(1Aβ)Yt-1]exp(γ)=1φ+α(1Aβ)Ait0,(A8)
Ytηt=1(1Aβ)+(St-1+Aμ(1Aβ)Yt-1)φαexp(γ)=1(1Aβ)+Aφαit0.(A9)

APPENDIX II Parameter Values for the U.S. Economy

This appendix details the derivation of the parameter values employed in Table 1. A time period is defined as a quarter. The elasticities used are short run values reported by Friedman (1978) (page 604, Table 2, left hand column). The short run elasticity of the demand for money with respect to the nominal interest rate is reported by Friedman to lie between -0.074 and -0.064 for M1. We employ an intermediate value of -0.069, so a is set at 0.069. Friedman reports a short run income elasticity of money demand of 0.15 and this is the value used for φ. The short-run elasticity of real spending with respect to the interest rate is reported by Friedman to be -0.0948. The parameter A represents a semi-elasticity. The average three-month commercial paper rate for the sample period 1984,1 to 1978,4 was 3.81 percent per annum, while the six-month rate was 4.47 percent. These imply a quarter to quarter interest rate between 0.009 and 0.11. Using an intermediate value of 0.01, A = (0.0948)/(0.01) = 9.48.

The parameter β represents the responsiveness of the rate of inflation to excess demand. The estimate of 0.087 is taken from Taylor (1980b). It should be noted, though, that Taylor estimates the responsiveness of “new” wages or prices to excess demand, that is the responsiveness of the subset of wages or prices that are revised in any particular period. A responsiveness of 0.087 for the aggregate price level may, therefore, be a little high. It is worth noting, however, that the implied annualized response of the price level in response to a 1 percent output gap, allowing for the one quarter lag, of 25 percent is slightly lower than estimates of 28.5 percent obtained from annual data by Chadha, Masson, and Meredith (1992) for the G-7 countries (page 409, Table 1, line 5 reports the “preferred” estimate). On balance, therefore, 0.087 appears to represent a reasonable quarterly response.

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1/

Bankim Chadha is with the International Monetary Fund. Daniel Tsiddon is with the Hebrew University of Jerusalem, Israel, and the CEPR. We would like to thank Peter Clark, David Coe, Herschel Grossman, Mohsin Khan, Doug Laxton, Steve Symansky, and David Weil for comments and helpful discussions. We are solely responsible for any errors and all opinions expressed.

1/

We use the term “price stability” to mean, as has become traditional in the macroeconomic literature, a zero trend in the price level. See, for example, the conference volume Price Stability (1991). “Price stability” does not explicitly refer to a lowering of the short run variability of prices around trend.

2/

A direct comparison of the average level of nominal interest rates in the last two decades with the average inflation rate can be misleading because of the substantial increase in real interest rates from the 1970s to the 1980s.

3/

The floor can be likened to a form of Keynes’ liquidity trap. However, Keynes’ liquidity trap is usually described as arising from speculative motives. If interest rates fall to a level at which it is perceived that they can only rise, purchasing bonds would inevitably yield a capital loss. The demand for bonds, therefore, falls to zero. While such speculative factors may become more important as nominal interest rates fall to low levels, we do not pursue this interpretation here.

4/

Negative real rates could be a consequence of the risk associated with risky assets and the low rate of return on safe assets.

1/

They argue that for small-short-lived and large-permanent negative shocks to the goods market, monetary policy’s ability to cushion output is only modestly constrained at low (zero) inflation. For large shocks that persist a few quarters, however, at low inflation (zero), there can be significant effects.

1/

We focus on the behavior of the logarithm of the demand for money rather than the demand for money itself because the rest of the model--as is traditional--is specified to be log-linear.

1/

The functional form in (1) could be used to capture any exogenous floor, iF, on nominal interest rates, by replacing it with (it-iF).

1/

As the solution for the variance of output makes clear, our results on the relationship between the variance of output due to real shocks and the long run rate of inflation are qualitatively independent of ρ and well defined even when ρ=1. See equation (18) below.

2/

Obstfeld and Rogoff (1984) discuss the appropriateness of alternative sticky goods price adjustment rules. For a previous application of the rule in (6) in a similar context see Chadha and Prasad (1993).

3/

The term St can also be interpreted as representing a negative shock to capacity output.

4/

Gradual aggregate price adjustment is often posited to capture the fact that at the firm level, individual prices are revised only periodically and asynchronously. For empirical evidence on price adjustment at the microeconomic level see Cecchetti (1986) and Lach and Tsiddon (1992).

1/

By the method of undetermined coefficients, the solution for the flexible price level in the long run in the absence of further shocks to the economy is P¯t=Mt+αLn(μ+(S¯/A)).

2/

κ=α[1Ln(μ+(S¯/A))].

3/

Since the one-period-ahead price level is set on the basis of information available today, its expectation today must equal its realized value.

1/

The reduced form solution for the aggregate price level as a consequence of adopting (5) is exactly the solution in Chadha (1987) where a more appealing two part price setting price adjustment rule based on Calvo’s (1983) rational staggered prices model is employed. It corresponds closely to the reduced form solution in Rotemberg (1982), where the aggregate price adjustment rule is derived from an explicit microeconomic model.

2/

While this condition is not a necessary condition for stability in the model, note from equation (10) that were money demand independent of output, that is ϕ = 0, then Aβ < 1 would be a necessary condition for a unique stable solution to exist.

3/

Stability requires only that λ be less than 1 in absolute value. The stronger condition imposed here rules out cyclical behavior associated with a possible negative root.

1/

θ=(Aμ+S¯+α)(Aμ+S¯+α(1ρ))/Aαρ2.

2/

This would be the case if St were a random walk, for example.

1/

In addition to the trend movement of μ.

2/

We refer to the IS and LM curves drawn alternatively in (i) the real interest rate-output plane; and (ii) the nominal interest rate-output plane. In the real interest rate-output plane the LM curve is drawn for a (any) given level of inflationary expectations. While in the nominal interest rate-output plane the IS curve is drawn for a given level of inflationary expectations. The solutions for the model, of course, represent the case where the expectations of inflation are derived through the dynamic equation describing the adjustment of prices.

1/

Factoring the expression in equation (18), the relative weights in the total variance of output of σε2 and ση2, are 1 and [(α/A)2(2/(1+ρ)]/(μ+S¯/A)2, respectively. Clearly as μ approaches (S¯/A) the relative importance of the variance of output due to goods market shocks increases.

1/

The convexity of YY at: the origin suggests that it intersects the 45 degree line again at some value of Y, Yc, which exceeds capacity output. Appendix 1 establishes that once exogenous variables are at their long-run expected values, the initial condition for output is always less than Yc, so that the existence of Yc is irrelevant for the stability and uniqueness of output around capacity.

1/

The predictions of the model on the form and direction of asymmetries in output are, as DeLong and Summers (1986) point out, what have been argued to be empirical regularities of the business cycle, particularly by Mitchell (1927) and Keynes (1936).

1/

It follows from equation (6) that the asymmetric behavior of output is mirrored in the cyclical component of inflation. For discussions of asymmetric adjustments of nominal prices see Tsiddon (1993) and Ball and Mankiw (1994).

2/

The average 3 month t-bill expost real interest rate for the U.S. during 1947-90 was 0.82 percent per annum, while that on 6 month commercial paper was 1.5 percent.

3/

The value of the autoregressive parameter of 0.9 implies a half-life for the shock of 6.6 quarters, so that the average length of time for which the shock lasts is a little over three years--the estimated average length of a business cycle in the U.S.

1/

The response of output to negative shocks is measured on the right axis. Alternatively, absolute values of output can be read off the left axis.

2/

The duration for which output remains away from capacity depends on both the initial impact and the speed with which output returns to capacity.

1/

The simulations were carried out on PC-TROLL, version 14, 1994, developed by Intex Solutions, Inc, Needham, Massachusetts.

1/

This empirical prediction of the model agrees with DeLong and Summers (1986) finding of little evidence of skewness in the distribution of U.S. output.

Inflation, Nominal Interest Rates, and the Variability of Output
Author: Mr. Bankim Chadha and Daniel Tsiddon
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    Impulse response of Output to Positive Monetary and Goods Market Shocks at Alternative Long Run Rates of Inflation.

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    Variance of Output as a Function of the Long Run Rate of Inflation.

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    Intrinsic Dynamics of Output During Expansions and Contractions.

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    Simulated Effect of 1 Percent Positive and Negative Shocks at Annual Trend Inflation Rates of 0 and 10 Percent.

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    Distribution of Output around Capacity.