Optimality of Inflation and Nominal Output Targeting

Transcription

1 Optimality of Inflation and Nominal Output Targeting Julio Garín Department of Economics University of Georgia Robert Lester Department of Economics University of Notre Dame First Draft: January 7, 15 Please Do Not Cite or Distribute: Incomplete and Preliminary Abstract Is nominal income targeting superior to inflation targeting? We address this question within a DSGE model with nominal price and wage rigidities. In the case of a model without capital accumulation, the answer depends entirely on the relative rigidity of prices and wages. If prices are sufficiently stickier than wages, inflation targeting is preferred; otherwise the optimal policy calls for nominal income targeting. After going over the intuition in this stylized case, we estimate a medium scale DSGE model and show how the optimal rule changes based on the parameter values and frictions in the economy. JEL Classification: E31; E7; E5; E58. Keywords: Optimal Policy; Nominal Targeting; Monetary Policy. address: jgarin@uga.edu. address: rlester@nd.edu.

2 1 Introduction What rule should the central bank follow in the formation of monetary policy? Despite extensive research about this topic, it remains an open question. In this paper we compare two often proposed rules: nominal GDP targeting and inflation targeting. Both rules adjust the interest rate to a nominal anchor, but the nominal GDP targeting rule implicitly includes real activity as well. We explore this question within the context of a relatively standard dynamic stochastic general equilibrium (DSGE) model. Although there is some disagreement on which type of rule central banks should follow, economists agree on several principles in the design of monetary policy. First, rules are preferred to discretion. Rules allow households to anchor expectations which improves the inflation-output gap tradeoff. This is true in models of either ad hoc Phillips curves as in Barro and Gordon (1983) or in microfounded Phillips curves as in Woodford (3). Second, the central bank faces information constraints which should be taken into account in the formation of monetary policy. Responding to precisely measured variables is superior to responding to imprecisely measured variables or variables that are hypothetical constructs of a model. Finally, the policy objectives of central banks should be understandable to the public. As argued by Bernanke and Mishkin (1997), this requires the central bank to be more accountable. Furthermore, even if monetary policy follows some strict rule, forming expectations is difficult if households do not understand the rule. Nominal GDP targeting and inflation targeting satisfy all three of these conditions. By definition, they are both rules. While it is difficult to argue that nominal GDP and inflation are measured with precision, they are both found in the data rather than being hypothetical constructs. Finally, both concepts are easy to explain to the public. Bernanke and Mishkin (1997) argue that inflation targeting is superior to nominal GDP targeting on the last two principles. However, we are not aware of any work that systematically studies the precision of inflation versus nominal GDP measurement. On the other hand, there are theoretical reasons to believe that nominal GDP targeting dominates inflation targeting. Sumner (1) outlines the basic logic in an aggregate demand - aggregate supply framework. After a negative aggregate demand shock, the price level and real output decline. Monetary policy in either regime lowers the nominal interest rate to combat the shock. Hence, if aggregate demand shocks were the only shocks driving the dynamics of the economy, then the choice of monetary policy rule would not matter. However, if there is a negative aggregate supply shock, the price level increases and real output decreases. The interest rate rises under the inflation targeting regime, reducing the price level and inflation. If nominal wages are sticky, real wages rise above their flexible level and unemployment rises.

3 The interest rate under the nominal GDP targeting regime may either rise or fall depending on the details of the model, but in any case unemployment will not rise by as much. While an exact variance decomposition of shocks to the economy will depend on the details of the data and model used, it is plausible that productivity and oil price shocks (aggregate supply shocks) and shocks to velocity (aggregate demand shocks) play prominent roles over the business cycle. One of our contributions is to compare the two rules in an estimated model that includes several structural shocks which map into reduced form aggregate demand or aggregate supply shocks. In the next section, we discuss a basic linearized New Keynesian model which includes nominal price and wage rigidity as in Erceg et al. (). Our results confirm Sumner (1) s intuition. If nominal wages are very sticky compared to nominal prices, nominal GDP targeting is preferred to inflation targeting. If nominal prices are sufficiently sticky relative to nominal wages, inflation targeting is preferred. Following this, we go to a more realistic model that includes capital accumulation and several other variables that are empirically important. We then perform the same exercise as in the simpler model. Our paper is related to several strands in the literature. The relative merits of nominal GDP targeting versus inflation targeting has been recently revived by Billi (1) and Woodford (1) who discuss the rules within the context of the zero lower bound. Cecchetti (1995) and Hall and Mankiw (199) show in counterfactual simulations that nominal GDP targeting would lower the volatility of real and nominal variables. Neither of these latter two papers have a structural model to conduct the welfare analysis, which limits their ability to make definitive judgments. Jensen () compares the two rules in a linearized New Keynesian model with price stickiness and Kim and Henderson (5) compare the two rules in a model of wage and price stickiness. While both of these papers include structural models similar to our own, they conduct their analysis within a stylized environment, while ours is more empirically realistic. Our paper is also related to Schmitt-Grohe and Uribe (7) who compare simple and implementable rules in a New Keynesian model with capital accumulation. However, they do not explicitly consider nominal income targeting. Finally, since we estimate a DSGE model, out paper is related to Smets and Wouters (7) and Christiano et al. (5) who estimate New Keynesian models with U.S. data. The paper proceeds as follows. Section describes a basic New Keynesian and Section 3 presents a medium scale model. 3

4 The Basic New Keynesian Model This section presents the New Keynesian model with nominal wage and price rigidity. In comparing the welfare results of nominal GDP targeting versus inflation targeting, we follow Erceg et al. () in taking a quadratic approximation of the utility function and linearizing the policy functions. The forward looking IS equation results from log linearizing the household s Euler equation and is given by Ỹ t = E t Ỹ t+1 1 σ (ĩ t E t π p t+1 ). (1) Output today, Ỹ t, is an increasing function of expected future output and a decreasing function of the real interest rate, ĩ t E t π p t+1. The parameter σ is the coefficient of relative risk aversion from a utility function that is separable in consumption and leisure. The price and wage inflation Phillips curves are π p t = κ p ω t + β E t π p t+1 () π w t = κ w ((σ + η) X t + ω t ) + β E t π w t+1. (3) Price and wage inflation, given by π p t and π w t respectively, are forward looking functions of expected future inflation. The price Phillips curve is also a function of the real wage gap, ω t, which is the gap between the equilibrium real wage and the real wage in the flexible price and wage economy. Similarly, wage inflation is a function of the real wage gap and the real output gap, Xt, which is the difference between equilibrium real output and real output in the flexible price and wage economy. η is the inverse elasticity of substitution of labor supply and β is the household s discount factor. Under Calvo (1983) pricing, κ p = (1 θp)(1 θpβ) θ p and κ w = (1 θw)(1 θwβ) θ w(1+ɛ wη) where θ p is the probability a firm cannot adjust their price and θ w is the probability a worker cannot adjust their price. Also, ɛ w is the elasticity of substitution between workers. Observing equations () and (3) show that there is a tradeoff between balancing the two gaps and the two inflation rates. In general circumstances, they cannot all be simultaneously eliminated. The wage setting process leads to an evolution of the real wage gap given by ω t = ω t 1 + π w t ã t + ã t 1 π p t () where ã t is TFP which follows the exogenous stochastic process ã t = ρã t 1 + ɛ t. One can show analytically that the output gap is defined by X t = Ỹt 1+σ σ+ηãt. The model is closed by specifying a nominal interest rate rule. In the next section, the interest rate is the solution

5 to the Ramsey problem. In the following sections we compare this optimal interest rate to Taylor rules that target price inflation and nominal GDP..1 Interest Rate Rules As a baseline, we compute the optimal interest rate under commitment. The quadratic loss function is a weighted sum of squared values of the output gap, price inflation, and wage inflation. The objective function of the policy maker is to choose π p t, π w t, and X t to minimize 1 t= β t [(σ + η) X t + ɛ p κ p ( π p t ) + ɛ w κ w ( π w t ) ] subject to equations () (). The first order conditions for the problem are given by X t = κ w ξ,t (5) ξ 1,t ξ 1,t 1 = ξ 3,t + ɛ p λ p π p t () ξ,t ξ,t 1 = ξ 3,t + ɛ w λ w π w t. (7) If either wages or prices were flexible, the flexible price equilibrium could be restored. When both are sticky, the Ramsey rule balances the welfare losses due to price stickiness and those due to wage stickiness. Given equations (5) through (7), the Ramsey interest rate can be inferred from equation (1). The Ramsey interest rate serves as the benchmark we make welfare comparisons to. However, since the Ramsey rule relies on the central bank observing theoretical constructs like the output gap, it is more realistic to consider some variation on the Taylor rule (Taylor (1993)). In particular, we consider two versions of the Taylor rule. One strictly targets price inflation and the other targets nominal GDP. They are given by ĩ t = ρ i ĩ t 1 + (1 ρ i )φ π π p t (8) and ĩ t = ρ i ĩ t 1 + (1 ρ i )φ y (Ỹ nom t Ỹ nom t 1 ). (9) Note that equation (9) says that the nominal interest rate responds to the the growth rate of nominal GDP. An alternative is to make the nominal rate respond to level differences instead of growth rates. Although we consider both of these rules in the paper, we choose (9) as a baseline. 5

6 . Quantitative Analysis While we estimate the model with capital in the next section, we consider a standard parametrization for now. We set β =.99 implying an annual risk free interest rate of approximately four percent. Preferences are log over consumption and the Frisch elasticity is one implying σ = 1 and η = 1. The elasticities of substitution for intermediate goods and workers, ɛ p and ɛ w, are set equal to 1, implying a little more than a ten percent price and wage markup in steady state. We consider various parameter values for the price and wage stickiness parameters, θ w and θ p. Since these parameters are probabilities, they are between zero and one. Prices (wages) are stickier as θ p (θ w ) approaches one. We consider the following quantitative experiments. For a given choice of θ w and θ p we find the Ramsey interest rate rule and the welfare loss associated with it. Next we simulate the model twice using the inflation targeting rule, equation (8), in the first and nominal GDP targeting, equation (9), in the second. For each of these rules we choose the value of φ π and φ y that minimizes the loss function. We experiment with different values of ρ which governs the exogenous persistence of the interest rate. Table 1 shows the results when ρ =. With the exception of the case when θ w =.1 and θ p =.9, nominal GDP targeting results in a smaller welfare loss than inflation targeting. The exception corresponds to a case when wages are close to fully flexible and prices are quite sticky. This makes sense in light of Clarida et al. (1999) and Blanchard and Gal? (7) who show that strict inflation targeting implements the first best in the basic New Keynesian model with only sticky prices and technology shocks. The idea is that an interest rate rule that responds sufficiently strongly will eliminate price inflation and the output gap. Since those are the only two sources of welfare loss in that simpler model, the first best is restored. As emphasized by Sumner (1), when the economy is hit with an aggregate supply shock and nominal wages are sticky, unemployment increases and output decreases. Over the long run, this results in a welfare loss and that is exactly what the results in Table 1 show. Table 1: TFP Shock φ p =.1 φ p = /3 φ p =.75 φ p =.75 φ p =.9 φ w =.9 φ w =.75 φ w =.75 φ w = /3 φ w =.1 Ramsey Rule Inflation Response NGDP Response To explain.

7 To obtain some intuition for exactly what is going on in the model, consider an unexpected negative technology shock. In the language of aggregate demand - aggregate supply, this shock shifts aggregate supply to the left. Figure 1 shows the impulse responses following this shock when wages and prices are equally sticky (θ p = θ w =.75). An inflation targeting rule sharply raises the nominal interest rate which results in a large decrease in real GDP from its flexible price level (a decrease in the output gap). Since price inflation is also negative on impact, the real rate rises. RGDP.5 x 1 3 Wage Inflation x 1 Price Inflation x 1 3 Eq Real Rate 8 x 1 3 Nom rate.1 Output Gap Ramsey Inf Targ NGDP Targ Figure 1: Wages and Price Equally Sticky (θ p = θ w =.75) On the other hand, the Ramsey rule increases the nominal rate only slightly and real output actually exceeds the flexible price/wage level of output. Note that a fall in real output need not imply that it falls below its flexible price/wage level. The reason is that after a contraction in technology, output would fall even if all prices and wages were flexible. When carrying out the welfare calculations, it is deviations from the flexible price/wage level that matters. Monetary policy can try and correct for this inefficiency, but once it is removed the equilibrium is Pareto optimal. The nominal GDP rule actually keeps the nominal interest rate fixed following the shock. One can see why this is algebraically by substituting the lagged value of equation (1) into (9): ĩ t = φ y ( p t p t 1 + Ỹt Ỹt 1) = φ y ( p t p t 1 + ĩ t 1 + E t 1 π p t) = φ y ( π t E t 1 π pt + i t 1 ). To the extent the forecast error is small, this term is zero (don t actually know why/if this 7

8 is true). The result is that real GDP, inflation, and the output gap rise by more than under the Ramsey rule. On the whole, the nominal GDP rule produces impulse responses that are closer to the Ramsey rule. Consequently, the impulse response analysis corroborates the findings of Table. To gain some intuition, Figures and 3 show the results in two extreme cases. Prices are extremely sticky and wages are nearly flexible (φ p =.1, φ w =.9) in Figure. Both the nominal GDP rule and the Ramsey rule minimize price inflation and the output gap. The optimal coefficient in the inflation target is very small since most of the nominal rigidity and welfare loss is coming from wage inflation. On the other hand, the Calvo parameters are reversed in Figure 3 and inflation targeting performs better than nominal GDP targeting. Inflation targeting produces less wage and price inflation and a smaller output gap. This visually confirms the divine coincidence result discussed earlier. x 1 3 RGDP 1 x 1 3 Wage Inflation 5 x 1 Price Inflation x 1 3 Eq Real Rate 8 Ramsey Inf Targ NGDP Targ 1 x 1 3 Nom rate x 1 3 Output Gap Figure : Sticky Prices and Nearly Flexible Wages (θ p =.1, θ w =.9) 8

9 RGDP 1 x 1 Wage Inflation x 1 3 Price Inflation x 1 3 Eq Real Rate x 1 Nom rate 8 1 x 1 3 Output Gap 8 Ramsey Inf Targ NGDP Targ Figure 3: Sticky Wages and Nearly Flexible Prices (θ p =.1, θ w =.9) 3 Medium Scale Model While the previous section allows one to understand some of the intuition in the inflation versus nominal GDP, it did so within the context of a very simplified model. In this section, we use model with capital accumulation, variable utilization, investment adjustment costs, and several other features in addition to nominal wage and price rigidities. Such a model has been shown to capture the dynamic effects of monetary policy and the most salient business cycle facts Production Production takes place in two phases. A representative final goods firm buys a continuum of inputs produced by intermediate good firms distributed on the unit interval. Each intermediate good firm produces one unique input. The inputs are imperfect substitutes and are combined with a constant elasticity of substitution (CES) production function with an elasticity of substitution equal to ɛ p. Indexing the inputs with j, the profit maximization problem for the final goods firm is max p t ( {y j,t } 1 ɛp 1 ɛp yj,t dj) ɛp ɛp 1 1 p j,t y j,t. 1 See Christiano et al. (5) as an example of the former and Smets and Wouters (7) for the latter. 9

10 The demand equation for input j is given by where y t = ( 1 ɛp 1 y ɛp j,t y j,t = y t ( p ɛ p j,t ) (1) p t ɛ p/(ɛ p 1) dj). Therefore, the output of firm j is increasing in total output and decreasing in it s Using the assumption of perfect competition, substituting equation (1) into the objective function shows that the aggregate price index isp t = ( 1 p1 ɛp j,t ) 1 1 ɛp. Intermediate good firms hire utilization adjusted capital and labor. Because they produce differentiated goods, intermediate good firms have some pricing power. We solve their profit maximization problem in two steps. The first step is to minimize costs. The constrained optimization problem is subject to min w t n j,t + R tˆkj,t {n j,t,ˆk j,t } y j,t A tˆkα j,t n 1 α j,t. Here w t and R t are the rental rates for labor, n j,t, and utilization adjusted capital, ˆk j,t, respectively. Denoting the multiplier by mc j,t, the first order conditions are: w t = mc j,t A t ( ˆk α j,t ) n j,t R t = mc j,t A t ( n 1 α j,t ). ˆk j,t Combining the first order conditions show firms hire capital and labor in the same ratio and that marginal costs are constant across firms. Turning to the pricing problem, each firm has a probability of 1 θ p of updating its price. A firm who last updates in period t can charge a price of Π ζp t 1,t+s 1 p j,t. The parameter ζ p, governs the degree of indexation. If ζ p = there is no indexation; if ζ p = 1 there is full indexation. The firm rebates profits back to households and therefore discounts dividends by the household s stochastic discount factor, Λ t. The profit maximization problem is max E t {p j,t,,y j,t+s } s= Λ t+s Π ζp t 1,t+s 1 θj [ p j,t p t+s y j,t+s mc t+s y j,t+s ] 1

11 subject to y j,t+s = y t+s ( p ɛ t+s ). p j,t Note that this is the problem for the firm conditional on updating in period t. The firm sets its price knowing that there is a probability that it will not be able to in future periods. Substituting the constraint into the objective function and taking the first order condition gives the symmetric pricing rule written recursively as ɛ p p # t = ɛ p 1 X 1,t X,t X 1,t = Λ t mc t y t p ɛp t + βθ p (1 + π p t ) ζpɛp E t X 1,t+1 X,t = Λ t y t p ɛp 1 t + βθ p (1 + π p t ) ζp(1 ɛp) E t X,t+1. If prices were completely flexible, the reset price would simply be a markup over nominal marginal cost. 3. Households There is a continuum of households indexed by h [, 1]. Much like the function of final goods firms, households sell their labor to a labor packer who bundles the different sorts of labor into a final aggregate labor input according to the function n t = ( 1 n ɛw 1 ɛw h,t where ɛ w is the elasticity of substitution between different types of workers. The labor packer buys the differentiated labor at a nominal wage of W h,t and sells the labor to the intermediate good firms at a nominal wage rate of W t. Profit maximization yields the labor demand equation n h,t = n t (W h,t /W t ) ɛw. The aggregate wage index is given by W t = ( dh) ɛw ɛw W 1 ɛw 1 ɛw h,t ). As in Erceg et al. (), there is perfect insurance across households and utility is additively separable in consumption and leisure. investment, consumption, bonds, wages, and hours. Households choose capital, utilization, Because of complete insurance, all households choose the same value of every variable except for hours and wages. Therefore, 11

12 we drop the h subscript except where necessary. The household s problem is to maximize subject to E t t= β t (c t bc t 1 ) 1 σ 1 1 σ c t + I t + B t+1 B t P t + [χ 1 (u t 1) + χ (u t 1) ] k t Z t B t ψ n1+η h,t 1 + η P t i i 1 + R t u t k t + W h,t P t k t+1 = Z t [1 τ ( I t I t 1 1) N h,t + Π t P t + T t ] I t + (1 δ)k t The first constraint says that the sum of real consumption, real investment, change in real bond holdings and utilization cost cannot exceed the sum of interest income from bonds, utilization adjusted income from capital, profits from owning the intermediate good firms and lump sum transfers from the government. The second constraint is the capital accumulation equation, which takes investment adjustment costs into account. The variable Z t is an investment specific technology shock. When Z t increases, a given amount of final goods will produce more capital. The parameter b [, 1) governs the degree of habit persistence. The first order conditions are λ t = (c t bc t 1 ) σ bβ E t (c t+1 bc t ) σ R t = [χ 1 + χ (u t 1)] 1 Z t λ t+1 λ t = β(1 + i t ) E t 1 + π p t+1 λ t = µ t Z t [1 τ ( I t 1) ] µ t Z t τ ( I t 1) I t + β E t [µ t+1 Z t+1 τ ( I t+1 1) ( I t+1 ) ] I t 1 I t 1 I t 1 I t I t µ t = β E t {R t+1 u t+1 λ t+1 [χ 1 (u t+1 1) + χ (u t+1 1) ] λ t+1 Z t + (1 δ)µ t+1 } The next step is solving for the optimal reset wage and hours conditional on being able to adjust wages in period t. Like intermediate good firms, households can partially index their nominal wages to inflation so that W h,t+s = Π ζw t 1,t+s 1 W h,t. In any given period, a household cannot adjust their nominal wage with probability 1 θ w. The household maximization problem is max E t {n h,t+s,w h,t } t= (βθ w ) s (c t+s bc t+s 1 ) 1 σ 1 1 σ ψ n1+η h,t+s 1 + η 1

13 subject to Π ζw t 1,t+s 1 n h,t+s = n W h,t t W t c t + I t + B t+1 B t P t + [χ 1 (u t 1) + χ (u t 1) ] k t Z t B t P t i i 1 + R t u t k t + Πζw ɛ w t 1,t+s 1 W h,t P t N h,t + Π t P t + T t Substituting the first constraint into the second constraint and the objective function and then taking the first order condition gives the optimal real reset wage ɛ w (w # t )1+ɛwη = ɛ w 1 H 1,t H,t H 1,t = ψw ɛw(1+η) t N 1+η t + θ w β(1 + π p t ) ζwɛw(1+η) E t (1 + π t+1 ) ɛw(1+η) H 1,t+1 H,t = λ t w ɛw t N t + θ w β(1 + π p t ) ζw(1 ɛw) E t (1 + π t+1 ) ɛw 1 H,t+1 If households can reset their wages every period then the real wage is a constant markup over the household s marginal rate of substitution. 3.3 Shocks, Policy, and Market Clearing The government consumes a time varying share of total output given by g t = ω t y t where ω t = (1 ρ g )ω+ρ g ω t 1 +ɛ g,t. The government runs a balanced budget in every period implying g t = T t. The nominal interest rate follows the Taylor rule i t = ρ i i t 1 + (1 ρ i ) [i + φ π (π p t π) + φ y (y t /y t 1 1)] + ɛ i,t. If φ y =, this is a strict inflation targeting rule; if φ π = φ y, this is a nominal GDP targeting rule. The processes for TFP and investment specific technology are given by A t = (1 ρ a ) + ρ a A t 1 + ɛ a,t Z t = (1 ρ z ) + ρ z Z t 1 + ɛ z,t respectively. Integrating total output across all intermediate good firms give sy t = A tn 1 α t ˆk t α ν t where ν t = 1 (p j,t/p t ) ɛp dj is the deadweight loss due to price dispersion. Finally, the market 13

14 clearing condition is y t = c t + I t + g t + [χ 1 (u t 1) + χ (u t 1) ] k t Z t. 1

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16 Smets, F. and R. Wouters (7): Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach, The American Economic Review, 97, 58. Sumner, S. (1): Nominal GDP Targeting: A Simple Rule to Improve Fed Performance, Cato Journal, 3, Taylor, J. B. (1993): Discretion versus policy rules in practice, Carnegie-Rochester Conference Series on Public Policy, 39, Woodford, M. (3): Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. (1): Methods of policy accommodation at the interest-rate lower bound, Tech. rep. 1