The Welfare Consequences of Nominal GDP Targeting

Transcription

1 The Welfare Consequences of Nominal GDP Targeting Julio Garín Department of Economics University of Georgia Robert Lester Department of Economics University of Notre Dame This Draft: March 7, 25 Please Do Not Cite or Distribute: Incomplete and Preliminary Abstract When is a Nominal GDP target preferred? It depends. 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: E3; E47; E52; E58. Keywords: Optimal Policy; Nominal Targeting; Monetary Policy. address: jgarin@uga.edu. address: rlester@nd.edu.

2 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 (983) or in microfounded Phillips curves as in Woodford (23). 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 (997), 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 (997) 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 (24) 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. The interest rate 2

3 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 New Keynesian model which includes nominal price and wage rigidity as in Erceg et al. (2). Our results confirm Sumner (24) 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. We also consider a Taylor rule where interest rates respond to a linear combination of inflation and output growth. 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 (24) and Woodford (22) who discuss the rules within the context of the zero lower bound. Cecchetti (995) and Hall and Mankiw (994) 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 (22) compares the two rules in a linearized New Keynesian model with price stickiness and Kim and Henderson (25) 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 (27) 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 (27) and Christiano et al. (25) who estimate New Keynesian models with U.S. data. The paper proceeds as follows. Section 2 describes a basic New Keynesian and Section 3 presents a medium scale model. See Gali (28) for a textbook exposition. 3

4 2 The Basic New Keynesian Model We follow Erceg et al. (2) by using a model of sticky prices and wages. With one exception, our model is identical to theirs and, consequently, we relegate its description to the appendix. The one exception is that we include government spending shocks. Government spending is financed by lump sum taxes and provides no utility to the agents in the economy. Government spending follows an autoregressive process. There is a continuum of workers and firms. Workers (firms) are imperfect substitutes with other workers (firms) and have some price setting power. However, workers and firms reset prices with a probability of less than one which gives rise to wage and price inflation. 2. Interest Rate Rules We consider a variety of interest rate rules. Each interest rate rule is categorized as a strict targeting rule or an instrument rule. A strict targeting rule adjusts the nominal interest rate so that an economic variable hits some target. For example, if the policy maker wants to target an inflation rate of two percent per year, the interest rate will be adjusted so that in equilibrium the two percent target is achieved. We consider an inflation target and a nominal GDP target. The inflation target can be described as π t = π. Inflation at any time, t, is equal to some preset target, π. Another rule is a nominal GDP target. There are two seemingly different variants of nominal GDP targets. The first targets nominal GDP growth, Y t Y t = + g NGDP where Y t is nominal GDP at time t and gngdp is the targeted rate of growth for nominal GDP. The second targets the level of nominal GDP, Y t = Y t ( + g NGDP ) t. The level rule makes up for nominal GDP growth falling short in previous periods whereas the growth rule ignores shortfalls outside the current period. To see why this matters, consider an example. Suppose g =.5. If nominal GDP is expected to grow at five percent this year, but only grew at two percent last year, the growth rule would not change interest rates, but the levels rule would so as to make up the gap. Another way of thinking about this is to say the growth rule lets bygones be bygones, whereas the level rule makes up for past 4

5 indiscretions. An interesting result emerges from these strict rules, however. A central bank committing to either rule at t will adjust nominal interest rates in exactly the same way for all subsequent periods. If a central bank adjusts the nominal interest rate so that nominal GDP growth is gngdp in every period, then there is no gap to make up for. The only time time these rules would prescribe different nominal interest rate rules is when there is a regime change at time j > in which the previous regime had some other target. The central bank following a nominal GDP target would then need to decide if they should follow the targeting rule from j forward or if they should make up the gap. Outside this regime switching exercise, it is surely easier for the central bank to commit to the growth rate rule over the levels rule. The latter requires that the public believes that the central bank would make up a nominal GDP gap if one ever emerged. This belief is required so that no gap in fact emerges in equilibrium. However, since the same allocations can be achieved with the central bank committing to a period-by-period growth rate, then this accountably problem is not as severe. To a first order approximation, the growth rate of nominal GDP is equal to the growth rate of real GDP plus inflation. Importantly, however, the growth rates of real GDP and prices are considered together. The nominal GDP rule only pays attention to the combined growth not the growth of the individual components. This means that interest rates move in the same direction and magnitude if real GDP growth is gngdp +. and inflation is. or vice versa. It is not clear if this rule would be preferred to some other rule g NGDP that targets the components separately. To explore the possible optimality of the interest rate responding to several economic variables, we introduce an instrument rule which specifies the interest rate as a function of economic variables. The specific rule we use is a Taylor (993) rule, log( + i t ) = ρ i log( + i t ) + ( ρ i ) [φ π log ( + π t + π ) + φ ylog ( y t y t )] where y t is real GDP. We consider several different Taylor rules. In one, we choose ρ i, φ π and φ y to maximize welfare. In another we consider a strict inflation Taylor rule in which the interest rate is only a function of deviations of inflation from its steady state. By specifying a Taylor rule that strongly responds to inflation, we can directly compare inflation targeting in its strict sense as described in the previous paragraphs and an instrument rule that responds strongly to inflation. In principle, we could also consider more complicated rules which set the nominal interest rate to minimize a quadratic loss function. A key element though of the Taylor, nominal 5

6 GDP, and inflation targeting rules are that they are easily understood by the public, at least on a relative basis. This means it is much easier to hold the central bank accountable to one of these rules than a more complicated rule. We think this accountability issue is essential to policy debates and therefore exclusively focus on these proposed rules. 2.2 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. The Frisch elasticity is one implying η =. The elasticities of substitution for intermediate goods and workers, ɛ p and ɛ w, are set equal to, 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. Since we are interested in monetary policy over the business cycle, we make trend growth and trend inflation equal to. In terms of the stochastic processes, we consider TFP shocks and shocks to the share of government expenditure. The share of government expenditure is taken as the ratio of government consumption expenditures plus investment to GDP. The share is detrended using an HP filer with a smoothing parameter of 6 and we estimate an AR on the cyclical part of the series. The autoregressive term and standard deviation of the residuals of the process are.898 and.5 respectively. The autoregressive term and standard deviation of the residuals of the TFP process are.97 and. respectively. The parameters are described in Table. Table : Value of Parameters Parameter Description Value β Discount rate.99 η Frisch elasticity ɛ w Elasticity of substitution Workers ɛ p Elasticity of substitution Firms θ w Wage stickiness.75 θ p Price stickiness.75 σ a Standard deviation TFP. σ g Standard deviation Government spending.5 ρ a Persistence TFP.97 ρ g Persistence Government spending.97 6

7 We consider the following quantitative experiments. For a given choice of price and wage stickiness parameters, we calculate the optimal coefficients in the Taylor rule and compute welfare under the Taylor rule and compare it to welfare under nominal GDP and inflation targeting. We also consider simple Taylor rules where we choose coefficients. First, we choose φ π =.5, φ y =.25 and ρ i and call this rule TR w/o smoothing. The Taylor rule w/smoothing has the same coefficients on inflation and output, but ρ i =.7. Finally, the Strict inflation TR has φ π = and φ y = ρ i =. With these rules in place we use the flexible price and wage equilibrium as the benchmark and compute the percent of consumption needed to make the economic agent indifferent between living in the flexible price and wage equilibrium (where monetary policy is irrelevant) to the equilibrium under sticky prices and wages under one of the monetary policy regimes. The regime with the smallest compensating variation is the most preferred. Under our assumed utility function, the compensating variation is given by λ = [exp(e W flex E W ) ] where E W flex is the expected present discounted value of utility under flexible prices and wages, E W is the expected discounted value of utility under some regime of sticky prices and wages denoted by. Table 2 shows the results when both shocks are included. 2 The thing that immediately jumps off the page is that strict inflation targeting is an extremely poor policy choice unless nominal wages are flexible. This is because the central bank essentially threatens to blow up the economy if any price inflation emerges. To keep price inflation at, the output gap must spike causing large losses in welfare. However, if prices are the only source of stickiness, then inflation targeting implements the first best. This result is emphasized in Blanchard (27) and Clarida et al. (2). Once wage stickiness is introduced, however, nominal GDP targeting becomes the optimal rule. Notice that the strict inflation TR performs much better on the whole than the strict inflation rule. This is because the instrument rule still allows for some price inflation. The optimally chosen Taylor rules deliver the smallest welfare losses. This is not surprising given that there are three degrees of freedom - the weight on inflation, the weight on output growth, and the inertia - associated with this policy rule. However, the welfare differences between the optimal Taylor rule and nominal GDP targeting are small. 2 The blank in the first column of the inflation rule is due to a very poor approximation. 7

8 Table 2: Both Shocks θ p =. θ p =.5 θ p =.75 θ p =.75 θ p =.75 θ w =.75 θ w =.75 θ w =.75 θ w =.5 θ w =. NGDP Inflation Strict inflation TR TR w/o Smoothing TR w/ Smoothing Optimal Taylor Rule To explain. The results in Table 2 indicate that the optimality of inflation versus nominal GDP targeting depends largely on the degree of price stickiness relative to wage stickiness. Figure makes this explicit. The panel on the left compares the strict inflation targeting rule to the NGDP targeting rule and the panel on the right compares the strict inflation TR to the NGDP targeting rule. Price stickiness is plotted on the vertical axis and wage stickiness is plotted on the horizontal axis. The shaded area is where NGDP targeting is the preferred rule. In the northwest region, where price stickiness is much higher than wage stickiness, the inflation targeting rules are preferred to the NGDP targeting rules. As wage stickiness increases and price stickiness falls, NGDP targeting becomes the preferred rule. Also, note that the region in which the strict inflation Taylor rule is preferred to NGDP targeting is bigger than the region where strict inflation targeting is preferred to NGDP targeting. This is because the Taylor rule allows for some leeway in letting inflation rise, whereas the strict inflation targeting rule does not. On the whole, however, NGDP targeting is most often preferred. 8

9 θp.5 θp θ w (a) NGDP Targeting vs. Inflation Targeting θ w (b) NGDP Targeting vs. TR Inflation Targeting Figure : Area Where NGDP Targeting is Preferred Tables 3 and 4 show that most of the welfare losses come from TFP rather than government spending shocks. Moreover, the tables show that the disparity in terms of the welfare costs comes from the TFP shocks. The welfare losses under the government spending shocks are more or less the same under any rule and, in fact, inflation targeting replicates the flexible price and wage equilibrium. This can be demonstrated in the model laid out in the appendix. The log-linear Phillips curve is π t = ϕmc t + β E π t+ where ϕ is a function of the structural parameters. Forcing inflation to in every period eliminates movements in inflation from steady state and marginal cost, mc t from steady state. Real wages are a function of two variables - marginal cost and TFP. Since neither marginal cost nor TFP are deviating from their steady state, real wages stay constant. Constant real wages plus zero price inflation implies that nominal wage inflation is also zero. Hence, inflation targeting eliminates both price and nominal wage inflation. 9

10 Table 3: TFP Shocks θ p =. θ p =.5 θ p =.75 θ p =.75 θ p =.75 θ w =.75 θ w =.75 θ w =.75 θ w =.5 θ w =. NGDP Inflation Strict Inflation TR TR w/o Smoothing TR w/ Smoothing Optimal Taylor Rule To explain. Table 4: Government Spending Shocks θ p =. θ p =.5 θ p =.75 θ p =.75 θ p =.75 θ w =.75 θ w =.75 θ w =.75 θ w =.5 θ w =. NGDP Inflation..... Strict Inflation TR..... TR w/o Smoothing TR w/ Smoothing Optimal Taylor Rule..... To explain. To further understand the implications of the different monetary policy rules, Figures 2 and 3 shows the responses of endogenous variables to a TFP shock. The most conspicuous observation from Figure 2 is that the nominal interest rate is held constant after a TFP shock. This comes directly from the Euler equation. In log-linear terms, the Euler equation is c t = E c t+ i t + E π t+. Since we are considering a TFP shock, government consumption is held fixed implying that the percent deviation of consumption from its steady state equals the percent deviation of output from its steady state giving us y t = E y t+ i t + E π t+

11 y t E y t+ = i t + E π t+. The nominal GDP targeting rule requires y t y t = π t every period, so it must hold in expectation. This requires i t = implying that the interest rate does not deviate from steady state. A second observation is that the responses are of a similar magnitude, but nominal GDP targeting actually raises the nominal rate whereas the Taylor rules lower the nominal rate. Since a favorable technology shock tends to raise real GDP and lower inflation, the direction of the interest rate change in the Taylor rule depends entirely on the relative weights on inflation and output. Following empirical estimates, we have φ π > φ y which causes the interest rate to fall on impact. 3 Also, under a Taylor rule, the price level, and hence nominal GDP, is not stationary. While real GDP returns to trend in all specifications, a temporary productivity boom permanently lowers the price level. Figure 3 shows that if the central bank implements an inflation target, the nominal rate must fall dramatically so that prices do not decrease in equilibrium. A consequence to this is that real GDP rises by two orders of magnitude more on impact compared to the other interest rate rules. These massive movements in the interest rate and real GDP provide a clue as to why the welfare losses are so large under inflation targeting Real GDP -3 Nominal Interest Rate Nominal GDP Inflation -3 8 Wage -3 TFP NGDP Targeting Taylor Rule - No Inertia Taylor Rule - Inertia Figure 2: NGDP Targeting and Taylor Rules TFP Shock (θ p = θ w =.75) 3 There is also an issue of equilibrium determinacy with the Taylor rules. To have a saddle path equilibrium, the Taylor principle must be satisfied which requires φ pi to be almost one regardless of the value of φ y.

12 .5 Real GDP Nominal Interest Rate.2 Nominal GDP Inflation x 3 Wage x 3 TFP Figure 3: Inflation Targeting With TFP Shock (θ p = θ w =.75) Turning to the government spending shock, Figures 4 and 5 show that the nominal interest rate increases regardless of the rule. Real GDP also increases since an increase in government spending causes an expansion of labor supply through a wealth effect. Note that whether or not inflation rises depends on the interest rate rule. This shows why the AD-AS intuition is dependent on the rule the central bank follows. Finally, note that by eliminating movements in inflation and marginal cost, inflation targeting also eliminates movement in the real wage since w t = mct a t and TFP is not moving Real GDP -3 Nominal Interest Rate Nominal GDP NGDP Targeting Taylor Rule - No Inertia Taylor Rule - Inertia Inflation -4 2 Wage.4 Government Spending Figure 4: NGDP Targeting and Taylor Rules With Government Spending Shock (θ p = θ w =.75) 2

13 5 x 3 Real GDP x Nominal 4 Interest Rate x 3 Nominal GDP.5 2. Inflation. Wage Government Spending Figure 5: Inflation Targeting with Government Spending Shock (θ p = θ w =.75) 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 } ɛp ɛp yj,t dj) ɛp ɛp p j,t y j,t. 4 See Christiano et al. (25) as an example of the former and Smets and Wouters (27) for the latter. 3

14 The demand equation for input j is given by where y t = ( ɛp y ɛp j,t y j,t = y t ( p ɛ p j,t ) () p t ɛ p/(ɛ p ) 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 () into the objective function shows that the aggregate price index isp t = ( p ɛp j,t ) 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 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 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 θ p of updating its price. A firm who last updates in period t can charge a price of Π ζp t,t+s p j,t. The parameter ζ p, governs the degree of indexation. If ζ p = there is no indexation; if ζ p = 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,t+s θj [ p j,t p t+s y j,t+s mc t+s y j,t+s ] 4

15 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 X,t X 2,t X,t = Λ t mc t y t p ɛp t + βθ p ( + π p t ) ζpɛp E t X,t+ X 2,t = Λ t y t p ɛp t + βθ p ( + π p t ) ζp( ɛp) E t X 2,t+. If prices were completely flexible, the reset price would simply be a markup over nominal marginal cost. 3.2 Households There is a continuum of households indexed by h [, ]. 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 = ( n ɛw 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 ɛw ɛw h,t ). As in Erceg et al. (2), there is perfect insurance across households and utility is additively separable in consumption and leisure. Households choose capital, utilization, investment, consumption, bonds, wages, and hours. Because of complete insurance, all households choose the same value of every variable except for hours and wages. Therefore, 5

16 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 ) σ σ c t + I t + B t+ B t P t + [χ (u t ) + χ 2 2 (u t ) 2 ] k t Z t B t ψ n+η h,t + η P t i i + R t u t k t + W h,t P t 2 k t+ = Z t [ τ 2 ( I t I t ) N h,t + Π t P t + T t ] I t + ( 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 [, ) governs the degree of habit persistence. The first order conditions are λ t = (c t bc t ) σ bβ E t (c t+ bc t ) σ R t = [χ + χ 2 (u t )] Z t λ t+ λ t = β( + i t ) E t + π p t+ λ t = µ t Z t [ τ 2 ( I 2 t ) ] µ t Z t τ ( I t ) I t + β E t [µ t+ Z t+ τ ( I t+ ) ( I 2 t+ ) ] I t I t I t I t I t µ t = β E t {R t+ u t+ λ t+ [χ (u t+ ) + χ 2 2 (u t+ ) 2 ] λ t+ Z t + ( δ)µ t+ } 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,t+s W h,t. In any given period, a household cannot adjust their nominal wage with probability 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 ) σ σ ψ n+η h,t+s + η 6

17 subject to Π ζw t,t+s n h,t+s = n W h,t t W t c t + I t + B t+ B t P t + [χ (u t ) + χ 2 2 (u t ) 2 ] k t Z t B t P t i i + R t u t k t + Πζw ɛ w t,t+s 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 )+ɛwη = ɛ w H,t H 2,t H,t = ψw ɛw(+η) t N +η t + θ w β( + π p t ) ζwɛw(+η) E t ( + π t+ ) ɛw(+η) H,t+ H 2,t = λ t w ɛw t N t + θ w β( + π p t ) ζw( ɛw) E t ( + π t+ ) ɛw H 2,t+ 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 = ( ρ g )ω + ρ g ω t + 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 + ( ρ i ) [i + φ π (π p t π) + φ y (y t /y t )] + 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 = ( ρ a ) + ρ a A t + ɛ a,t Z t = ( ρ z ) + ρ z Z t + ɛ z,t respectively. Integrating total output across all intermediate good firms give sy t = A tn α t ˆk t α ν t where ν t = (p j,t/p t ) ɛp dj is the deadweight loss due to price dispersion. Finally, the market 7

18 clearing condition is 4 Quantitative Analysis 4. Estimation y t = c t + I t + g t + [χ (u t ) + χ 2 2 (u t ) 2 ] k t Z t. Following Smets and Wouters (27) we estimate the model using Bayesian estimation techniques. Our data series include: the growth rate of real GDP, the growth rate of the implicit GDP deflator, the growth rate of real investment, and the Fed Funds Rate. 5 The data goes from Prior to estimation, we calibrate a number of parameters to long run data. β =.995 is set to target an annual risk free rate of two percent and steady state government consumption, ω, equals 2 percent which approximately matches the average over the time period. Capital s share of income is set to target the long run average of one third, implying α = /3 and depreciation is set to match its long run average in the National Income and Products Account, implying δ =.25. All other parameters are estimated. Table 6 contains the results. These results are broadly consistent with Smets and Wouters (27) and Justiniano et al. (2). Investment shocks play a critical role in business cycles, as they have nearly twice the standard deviation of TFP shocks. On the other hand, government spending and nominal interest rate shocks play a relatively minor role. The weight on inflation in the nominal interest rate rule is much higher than on the output gap which is consistent with the aforementioned papers, as well as, Taylor (993). In the next section, we use these estimated values to compare the performance of alternative interest rate rules. 5 Investment is defined as the sum of spending on consumer durables, residential fixed investment, and nonresidential fixed investment. 8

19 Table 5: Estimated Parameters Prior Posterior Parameter Dist. Mean SE Mode Mean SE 9% Probability Interval b Beta [.294,.342] τ Normal [.782,.44] θ p Beta [.789,.7967] θ w Beta [.3692,.6242] φ π Normal [.489,.79] φ y Normal [.263,.377] ζ p Beta [.7,.94] ζ w Beta [.539,.7542] η Normal [.675,.727] ρ z Beta [.96,.943] ρ a Beta [.9566,.984] ρ i Beta [.872,.8644] ρ g Beta [.95,.9772] ɛ z Inv. Gamma [.44,.2] ɛ a Inv. Gamma [.85,.3] ɛ i Inv. Gamma [.8,.25] ɛ g Inv. Gamma [.48,.66] To explain. Table 6: Welfare Under Different Parameter Values Panel A: th Percentile of Estimated Values Target Baseline b τ θ p θ w ζ p ζ w η Inflation NGDP Panel B: 9th Percentile of Estimated Values Target Baseline b τ θ p θ w ζ p ζ w η Inflation NGDP To explain. 9

20 4.2 Interest Rate Rule Performance As in Section 2, we compute the compensating variations using the flexible price and wage equilibrium as our benchmark. Carrying out this exercise implies a.462 percent loss for strict inflation targeting and a.295 percent loss for NGDP targeting. The smaller loss associated with NGDP targeting emerges despite price stickiness being higher than wage stickiness. However, the welfare losses of following strict inflation targeting are not nearly as large as in the simpler model. Table?? shows how sensitive the results are to parameter changes. For each estimated parameter, except for the ones governing the stochastic processes, we compute the welfare losses when the parameter is moved to the upper-bound of the 9th percent confidence interval and when the the parameter is moved to the lower-bound of the 9th percent confidence interval. When moving one parameter to the boundary of a confidence interval, all other parameters are held to their estimated value. Although the medium scale model is much more complicated than the simple model we first considered, only the wage stickiness parameter significantly affects the results. The welfare losses of price inflation targeting at the upper-bound of wage stickiness is more than three times larger than at the lower-bound of the wage stickiness parameter. Consequently, over the range of parameters we consider in the sensitivity analysis, inflation targeting is only preferred when wage-stickiness is at its lower-bound. In summary, the medium scale model reduces some of the huge magnitudes associated with inflation targeting, but the basic implications for monetary policy are similar to the simpler model. 5 Conclusion Despite extensive research, there is no general consensus on what constitutes the ideal monetary policy rule. In this paper, we consider three often proposed rules-nominal GDP targeting, inflation targeting, and a Taylor rule- and find that a Taylor rule with optimally chosen coefficients maximizes welfare, but nominal GDP targeting comes very close to achieving the same results. Alternatively, inflation targeting is usually an extremely bad policy unless wages are almost entirely flexible. If the choice comes done to nominal GDP targeting and a Taylor rule, there are probably other things to consider besides the structural parameters of the economy. Questions like: Which policy is more implementable?, Which policy hold central bankers more accountable?, and Which policy is more robust to mismeasurement or incomplete information become important from a theoretical and policy perspective. We leave these for future 2

21 research. 2

22 References Barro, R. J. and D. B. Gordon (983): Rules, discretion and reputation in a model of monetary policy, Journal of Monetary Economics, 2, 2. Bernanke, B. S. and F. S. Mishkin (997): Inflation Targeting: A New Framework for Monetary Policy? Journal of Economic Perspectives,, Billi, R. M. (24): Nominal GDP Targeting and the Zero Lower Bound: Should We Abandon Inflation Targeting? Tech. Rep. 27, Sveriges Riksbank Working Paper Series. Cecchetti, S. G. (995): Inflation Indicators and Inflation Policy, NBER Working Papers 56, National Bureau of Economic Research, Inc. Christiano, L. J., M. Eichenbaum, and C. L. Evans (25): Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy, Journal of Political Economy, 3, 45. Erceg, C. J., D. W. Henderson, and A. T. Levin (2): Optimal monetary policy with staggered wage and price contracts, Journal of Monetary Economics, 46, Gali, J. (28): Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework, Princeton University Press. Hall, R. E. and N. G. Mankiw (994): Nominal Income Targeting, Tech. rep. Jensen, H. (22): Targeting Nominal Income Growth or Inflation? American Economic Review, 92, Justiniano, A., G. E. Primiceri, and A. Tambalotti (2): Investment shocks and business cycles, Journal of Monetary Economics, 57, Kim, J. and D. W. Henderson (25): Inflation targeting and nominal-income-growth targeting: When and why are they suboptimal? Journal of Monetary Economics, 52, Schmitt-Grohe, S. and M. Uribe (27): Optimal simple and implementable monetary and fiscal rules, Journal of Monetary Economics, 54, Smets, F. and R. Wouters (27): Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach, The American Economic Review, 97,

23 Sumner, S. (24): Nominal GDP Targeting: A Simple Rule to Improve Fed Performance, Cato Journal, 34, Taylor, J. B. (993): Discretion versus policy rules in practice, Carnegie-Rochester Conference Series on Public Policy, 39, Woodford, M. (23): Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. (22): Methods of policy accommodation at the interest-rate lower bound, Tech. rep. 23

24 A Basic New Keynesian Model This is the model of Section 2. We use the same notation as in Section 3. A. Households Households are indexed by h [, ] and provide an imperfectly substitutable labor input. There is perfect insurance across households and utility is separable implying that consumption is equalized and that we can maximize the utility of one aggregate household. The optimization problem is to choose consumption and bond holdings to maximize subject to E t t= β t (c t ) σ σ ψ n+η h,t + η c t + B t+ B t P t B t i i + W h,t N h,t + Π t + T t P t P t P t Households sell their labor to a labor bundler. The aggregate bundle of labor is n t = ( n ɛw ɛw h,t dh) ɛw ɛw. The demand for each type of labor is n h,t = n t (W h,t /W t ) ɛw, where the aggregate wage index is given by W t = ( W ɛw ɛw h,t ). Just as in Section 3, households reset their wages with probability φ w which gives rise to the reset equation ɛ w (w # t )+ɛwη = ɛ w H,t H 2,t H,t = ψw ɛw(+η) t N +η t + θ w β E t ( + π t+ ) ɛw(+η) H,t+ H 2,t = λ t w ɛw t N t + θ w β E t ( + π t+ ) ɛw H 2,t+. 24

25 A.2 Firms Firms are indexed by j [, ] and have the production function y j,t = A j,t n j,t. Firms sell their products to a final goods firm whose production function is y t = ( ɛp ɛp yj,t dh) ɛp ɛp This gives rise to the demand equations y j,t = y t ( p j,t p t ) ɛp P t = ( P ɛp ɛp j,t ).. where Intermediate good firms maximize profits by choosing price and output subject to their demand curves. Each firm has a probability of φ p of changing its price. The price reset equation is A.3 Shocks ɛ p p # t = ɛ p X,t X 2,t X,t = c σ t mc t y t p ɛp t + βθ p E t X,t+ X 2,t = c σ t y t p ɛp t + βθ p E t X 2,t+. There are two shocks - TFP and government spending. They follow exogenous processes given by: A t = ( ρ a ) + ρ a A t + ɛ a,t ω t = ( ρ g )ω + ρ g ω t + ɛ g,t where ω t is the share of government spending. 25