Monetary policy and natural disasters in a DSGE model.

Keen, Benjamin D. ; Pakko, Michael R.

1. Introduction

In late August 2005, Hurricane Katrina hit the U.S. Gulf coast with a catastrophic fury that caused unprecedented damage to the region. Burton and Hicks (2005) calculate that the total damage to homes, businesses, and infrastructure was more than $150 billion, making Katrina the costliest hurricane ever. (1) That estimate, in economic terms, represents about 0.4% of the Bureau of Economic Analysis's figure for total fixed capital and consumer durables at the end of 2004. Economic activity in the Gulf coast region also was disrupted during and immediately after the hurricane. As a result, quarterly U.S. GDP growth is estimated to have declined by around 0.4% in the third quarter of 2005. (2)

The magnitude of the disaster fueled speculation by financial market participants that the Federal Open Market Committee (FOMC) might ease policy at its meeting of September 20 by postponing a widely expected 25 basis point increase in the federal funds rate. The change in expectations was widely reported by the financial press. For example, an article in the Cincinnati Post (September 7, 2005, Business Section) stated that, "Before the hurricane ... economists considered it a foregone conclusion that Fed policy makers would boost short-term interest rates by another quarter percentage point.... Now, a growing number of economists say the odds are rising that the Fed might take a pass...."

Data from federal funds futures markets also confirm this shift in expectations. Figure 1 shows that the expected average funds rate for September and October began falling on the day after Katrina's landfall. From August 29 to September 6, the expected rate derived from the September contract fell 5 basis points, whereas the expected funds rate for October fell by 11 basis points.

At the September 20 meeting, the FOMC raised the federal funds rate by 25 basis points, as was widely expected before Katrina. The press release following the meeting stated that, "While these unfortunate developments have increased uncertainty about near-term economic performance, it is the Committee's view that they do not pose a more persistent threat" (FOMC (2005). (3) Given the FOMC was expected to raise its funds rate target before Katrina, the Committee's subsequent decision to follow that course of action indicates that monetary policy did not respond to the disaster. (4)

In this article, we use a dynamic stochastic general equilibrium (DSGE) model to investigate how monetary policy should respond to catastrophic events such as Hurricane Katrina. We model infrequent catastrophic events using a two-state Markov switching process. Most of the time, the economy is in the nondisaster (or normal) state. In each period, however, a small probability exists that the economy will experience a disaster. A disaster is characterized by the destruction of a portion of the capital stock and a temporary negative technology shock that reduces output. We then analyze the effect of a disaster shock in model specifications with and without nominal price and wage rigidities. (5)

Our results indicate that the monetary authority should raise its nominal interest rate target following a disaster. This prescribed increase in the federal funds rate clearly runs contrary to the conventional wisdom following Hurricane Katrina. The press and financial markets based their beliefs on an assumption that the Federal Reserve is motivated to dampen the fall in output caused by a disaster. When conducting monetary policy within a Taylor (1993) rule framework, however, the nominal interest rate responds primarily to higher inflation rather than to lower output. (6) That finding also holds when an explicit disaster variable enters the policy rule. Using optimal control theory, as applied by Woodford (2002) and others, the monetary authority should strive to replicate the dynamics of a flexible price and wage equilibrium. Generating those dynamics requires an increase in the nominal interest rate in response to higher inflation and also to the higher real interest rate associated with depletion of the capital stock. Such a policy minimizes real distortions resulting from nominal price and wage rigidities.

[FIGURE 1 OMITTED]

The article proceeds as follows: Section 2 outlines the model, section 3 presents the specification of our disaster shock and its effects on the capital stock and output, section 4 examines the effect of a disaster when the Federal Reserve follows a standard Taylor rule, section 5 analyzes the optimal monetary policy response to a disaster, and section 6 concludes.

2. Model Framework

We examine a fully articulated DSGE model wherein firms are monopolistically competitive producers of goods and households are monopolistically competitive suppliers of labor. (7) Imperfect competition in the goods and labor markets enables us to consider models with price and nominal wage rigidities. Specifically, we consider three different specifications of our benchmark DSGE model: flexible prices and wages (flexible model), sticky prices and flexible wages (sticky price model), and sticky prices and sticky wages (sticky price and wage model).

Households

Households supply differentiated labor services to the firms in a monopolistically competitive labor market. Total aggregate labor hours, [n.sub.t], is calculated as a Dixit and Stiglitz (1977) continuum of labor hours, [n.sub.h,t], supplied by each household, h [member of] [0, 1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where -[[epsilon].sub.w] is the wage price elasticity of demand for household h's labor services. Firms' demand for household h's labor services are calculated by minimizing labor costs subject to the equation of aggregate labor hours,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [W.sub.h,t] is the nominal wage rate of household h, and [W.sub.t] is interpreted as the aggregate nominal wage rate:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Household h is an infinitely lived agent who values consumption, c,, and real money balances, ([M.sub.t]/[P.sub.t]), but dislikes labor. Household h also participates in a state-contingent securities market. That assumption enables all households to be homogeneous with respect to consumption, investment, capital, money, and bonds. The expected utility function for household h is then summarized as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[E.sub.t] is the conditional expectation at time t, and [beta] is the discount factor.

Households own the capital stock, [k.sub.t], and rent it to the firms. In each period, household h selects a level of investment, it, such that

[k.sup.'.sub.t+1] = [PHI] ([i.sub.t]/[k.sub.t],) [k.sub.t] + (1 - [delta])[k.sub.t], (6)

where [k.sup.'.sub.t+1] is the amount of capital carried into period t + 1, 4([PHI])(x) represents a Hayashi (1982) form of capital adjustment costs, and 8 is the depreciation rate. The capital adjustment costs are the resources lost in the conversion of investment to capital, [i.sub.t] - [PHI]([i.sub.i]/[k.sub.t])[k.sub.t], and are an increasing and convex function of the steady-state investment-to-capital ratio such that [[PHI].sup.'](x) > 0 and [[PHI].sup.'](x) < 0. A disaster, if it strikes, occurs at the beginning of period t before production begins. The nondestroyed capital, [k.sub.t], that is available for use in production is

In ([k.sub.t]/k) = In ([k.sup.'.sub.t]/[k.sup.']) - [kappa]ln ([D.sub.t]/D), (7)

where [D.sub.t] is the "disaster shock" variable, which is discussed in more detail in the next section, and the variables without time subscripts represent steady-state values.

Household h begins each period with an initial stock of nominal money balances, [M.sub.t-1], and receives a payment, [R.sub.t-1][B.sub.t-1], from its nominal bond holdings, [B.sub.t-1], where [R.sub.t] is the gross nominal interest rate. During the period, household h receives labor income, [W.sub.h,t][n.sub.h,t]; rental income from capital, [P.sub.t][q.sub.t][k.sub.t]; dividends from the firms, [D.sub.t]; a lump sum transfer from the monetary authority, [T.sub.t]; and a payment from the state-contingent securities markets, [A.sub.h,t], where [q.sub.t] is the real rental rate of capital. Those funds then are used to finance consumption and investment purchases and end-of-the-period bond, [B.sub.t], and money, [M.sub.t], holdings. The budget constraint for household h is represented as follows:

[B.sub.t] + [P.sub.t]([c.sub.t] + [i.sub.t]) + [M.sub.t] = [W.sub.h,t][n.sub.h,t] + [P.sub.t][q.sub.t][k.sub.t] + [D.sub.t] + [R.sub.t-1][B.sub.t-1] + [T.sub.t] + [M.sub.t-1] + [A.sub.h,t]. (8)

Finally, household h chooses a level of [c.sub.t], [i.sub.t], [k.sub.t], [B.sub.t], and [M.sub.t] that maximizes its expected utility subject to its capital accumulation and budget constraint equations.

Household h also might negotiate a wage contract that can remain in place for an unknown number of periods. The opportunity to renegotiate a wage contract follows a Calvo (1983) model of random adjustment. That is, [[eta].sub.w] is the probability that household h can set a new nominal wage, [W.sub.t.sup.*], and (1 -[[eta].sup.w]) is the probability that its nominal wage can only increase by the steady-state inflation rate, [pi]. (8) When a wage adjustment opportunity occurs, household h selects a nominal wage, [W.sub.t.sup.*], that maximizes its utility given the firms' demand for its labor,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [(1 - [[eta].sup.w]).sup.i] is the probability that another wage adjustment opportunity will not take place in the next i periods. Finally, a value [[eta].sup.w]. = 1 implies that the nominal wage is perfectly flexible.

Firms

Firms, which are owned by the households, produce differentiated goods in a monopolistically competitive market. Firm f hires labor, [n.sub.f,t], and rents capital, [k.sub.f,t], from the households to produce its output, [y.sub.f,t] according to a Cobb-Douglas production function,

[y.sub.f,t] = [Z.sub.t] [([k.sub.f,t]).sup.[alpha]] [([n.sub.f,t]).sup.(1-[alpha]]) (10)

where 0 [less than or equal to ] [alpha] [less than or equal to] 1. The productivity factor, [Z.sub.t], comprises the typical technology shock, [z.sub.t], that follows a first-order autoregressive process and an additional component related to the disaster shock variable,

ln([Z.sub.t]/Z) = ln ([z.sub.t]/z) + [zeta] ln([D.sub.t]/D). (11)

Firm f then chooses the combination of labor and capital that minimizes its production costs, [w.sub.t][n.sub.f,t ]+ [q.sub.t][k.sub.f,t], given its production function. Solving firm f's cost minimization problem yields the following factor demand equations,

[q.sub.t] = [[PSI].sub.t] [alpha], [Z.sub.t] [([n.sub.f,t]/[k.sub.f,t]).sup.(1-[alpha])] (12)

[W.sub.t] / [P.sub.t] = [[PSI].sub.t] (1-[alpha]) [Z.sub.t] [([k.sub.f,t]/[n.sub.f,t]).sup.[alpha] (13)

where [[PSI].sub.t] is the Lagrange multiplier on the cost minimization problem and is interpreted as the real marginal cost of producing an additional unit of output. The marginal cost, [[PSI].sub.t], is identical for all firms because every firm pays the same rental rates for capital and labor.

Aggregate output, [y.sub.t], is a Dixit and Stiglitz (1977) continuum of differentiated products,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where - [[epsilon].sub.p] is the price elasticity of demand for good J; Cost minimization by households yields the demand equation for firm f's good,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where [P.sub.f,t] is the price charged by firm f and [P.sub.t] is a nonlinear price index:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Each period, firm f also may have an opportunity to select a new price, [P.sub.f,t], for its product, [y.sub.f,t]. Firm price adjustment opportunities follow a Calvo (1983) model of random adjustment. That is, the probability a new price, [P.sub.t.sup.*], can be set is [[eta].sub.p], and the probability the price only can adjust by the steady state inflation rate, [pi], is (1 -[[eta].sub.p]). A price-adjusting firm selects a price, [P.sub.t.sup.*], that maximizes the discount value of its expected current and future profits subject to its factor demand and product demand equations,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

where [[beta].sup.i] [[lambda].sub.t+i] is the households' real value in period t of an additional unit of profits in period t + i, and [(1 - [[eta].sub.p]).sup.i] is the probability that the firm will not have another price-adjusting opportunity in the next i periods. (9) Finally, prices are completely flexible in this specification when [[eta].sub.p] = 1.

The Monetary Authority

The monetary authority utilizes a generalized Taylor (1993) rule. Specifically, the monetary authority's nominal interest rate target responds to changes in the inflation rate, [[pi].sub.t], the nominal wage growth rate, [DELTA][W.sub.t], the level of output, and the disaster shock variable

ln([R.sub.t]/R): [[theta].sub.[pi]] ln([[pi].sub.t]/ [pi]) +[[theta].sub.w] ln([DELTA][W.sub.t]/[DELTA]W) + [[theta].sub.y] ln([y.sub.t]/y)+ [[theta].sub.D] ln ([D.sub.t]/D) + [[epsilon].sub.R,t], (18)

where [[epsilon].sub.R,t] is a discretionary monetary policy shock which is normally distributed with a zero mean and variance of [[sigma].sup.2.sub.[epsilon]]. Finally, [DELTA]W = [pi] in the steady state because the model does not include any endogenous growth.

3. The Disaster Shock

We consider two crucial characteristics of a natural disaster like Hurricane Katrina. First, a disaster destroys an economically relevant share of the economy's productive capital stock, as shown in Equation 7. Second, a disaster temporarily disrupts production, which we model as a transitory negative technology shock in Equation 11. Because a disaster is an infrequent event, the disaster shock is modeled as a two-state Markov switching process. The negative shocks to the capital stock and to technology are specified as functions of the two-state disaster variable.

The disaster shock variable, [D.sub.t], can take on one of two states. State 1 is the "normal" or "nondisaster" state, whereas state 2 is defined as a "disaster." The two states evolve according to a transition matrix with the calibrated probabilities,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

where [p.sub.ij] = prob ([D.sub.t] = [D.sup.j] | [D.sub.t-i] = [D.sup.i]). For the given probability values, there is a 2% probability a disaster will occur, regardless of the disaster variable's state in the previous period. (10)

To map the regime-shifting framework onto the canonical difference-equation structure of the model, a log-linearized version of the Markov-switching process is expressed in the following form (11):

[[??].sub.t+1] = [[rho].sub.D] [[??].sub.t] + [[epsilon].sub.Dt+1]. (20)

It is convenient to define the baseline steady state as the unconditional expected value of the disaster shock:

ln (D) = 1 - [p.sub.22] / 2 - [p.sub.11] - [p.sub.22] ln ([D.sup.1]) + 1 - [p.sub.11] / 2 - [p.sub.11] - [p.sub.22] ln ([D.sup.2]) (21)

The composite expressions weighting the two values of [D.sub.t] in Equation 8 are the ergodic probabilities of being in each of the two states.

When [D.sub.t] is in state 1, its logarithmic deviation from the baseline steady state is

[[??].sup.1] [equivalent to] ln ([D.sup.1]) - ln (D) = 1 - [p.sub.11] / 2 - [p.sub.11] - [p.sub.22] [ln ([D.sup.1]) - ln ([D.sup.2])], (22)

and when [D.sub.t] is in state 2, it is

[[??].sup.2] [equivalent to] ln ([D.sup.2]) - ln (D) = 1 - [p.sub.22] / 2 - [p.sub.11] - [p.sub.22] [ln ([D.sup.2]) - ln ([D.sup.1])]. (23)

A useful property of a two-state Markov-switching process is that the conditional probabilities implicit in the expectation term in Equation 7 can be represented as a first-order autoregressive process. With Equations 9 and 10 and the probability transition matrix, it is straightforward to show that the autoregressive coefficient defined as [E.sub.t]([ [??].sub.t+1] | [[??].sub.t]/[ [??].sub.t]) is independent of the present state and is equal to [p.sub.11] + [p.sub.22] - 1. (12) For the linearly approximated simulations, this expression defines the value for [rho].sub.D]. (13) The sequence of disturbances placed into the model is calculated as

[[epsilon].sub.Dt] = [[??].sub.t] - [E.sub.t-1] ([[??].sub.t]) = [[??].sub.t] - (1 - [p.sub.11] - [p.sub.22]) [[??].sub.t-1] (24)

4. Simulation Experiments

Calibration

The disaster shock variable is calibrated to reflect the magnitude of Hurricane Katrina's economic impact. First, the ratio [D.sup.2]/[D.sup.1] is set to 1.004, providing a baseline magnitude for the shock's impact. The effect of [[??].sub.t] on capital and technology are calibrated to generate specific impulse responses consistent with the effect of Hurricane Katrina. In particular, the disaster variable's effect on the capital stock and technology are calibrated such that both the capital stock and output decline by 0.4% in the flexible price and wage equilibrium when the economy is in the "disaster" state. In terms of Equations 7 and 11, this requires that we set [kappa] = 1 and [zeta] = -0.58.

The existence or absence of nominal price and wage rigidities depends on the calibration of the probability of price adjustment, [eta].sub.p], and the probability of wage adjustment, [eta].sub.w] w. The probability of price adjustment equals 1 when prices are flexible and 0.25 when prices are sticky. Our calibration for the sticky price specification indicates that firms reset their price, on average, once per year, which is consistent with findings in Rotemberg and Woodford (1992). The probability of wage adjustment is set to 1 if wages are flexible and 0.25 if wages are sticky. The sticky wage calibration, which is consistent with Erceg, Henderson, and Levin (2000), suggests that nominal wage readjustment occurs, on average, once every year. Finally, Table 1 details the calibration of the model's other parameters, except the parameters in the policy rules (which are discussed below).

Taylor Rule Responses

Figure 2 illustrates the effect of a disaster on the flexible model, sticky price model, and sticky price and wage model when the central bank follows a Taylor rule with [[theta].sub.[pi]] = 1.5 and [[theta].sub.y] = 0.125. (14) We calibrate the shock to deliver a 0.4% fall in output for the flexible model. (15) The negative shock to the capital stock and productivity factor prompts firms to lower their output and raise prices. That decline in output and rise in inflation is moderated when prices are sticky because some firms cannot optimally reset their price. (16) To compensate for lost productivity and a lower capital stock, the non-price-adjusting firms must increase their labor to maintain their production levels. Conversely, price-adjusting firms reduce their labor demand as output falls. As a result, employment increases in the sticky price model and sticky price and wage model but decreases in the flexible model.

[FIGURE 2 OMITTED]

The larger output decline in the flexible model lowers household income, giving them fewer resources to invest in capital than in models with nominal rigidities. In the period after the disaster, the return of productivity to its predisaster level permits firms to increase output, which lifts households' income and enables them to increase their investment in physical capital. That process continues for a number of periods as the capital stock is slowly reconstructed. In the longer term, the protracted rebuilding of the capital stock is associated with below-trend output and persistent, above-trend paths for employment, investment, and inflation. (17)

The endogenous response of monetary policy to inflation and output, via the Taylor rule, drives the movements in the nominal interest rate. The policy reaction to higher inflation after a disaster puts upward pressure on the nominal interest rate, whereas the decline in output generates downward pressure. In the standard Taylor rule calibration, the inflation rate effect dominates, so that an increase in the nominal interest rate is the prescribed monetary policy response. The initial increase in the nominal interest rate is more than 80 basis points in the flexible model, whereas it rises only by about 30 basis points in the sticky price model and the sticky price and wage model. Destruction of the capital stock also increases future capital rental rates, which causes the equilibrium real interest rate to rise. The real interest rate rises initially by around 50 basis points in the flexible model but only by about 20 basis points in the models with nominal rigidities. Finally, the gradual reconstruction of the capital stock keeps inflation and inflation expectations above their steady states for an extended period of time in all of the models.

Both components of the disaster shock--the destruction of the capital stock and the temporary decline in technology affect key economic variables after such a shock. Table 2 decomposes the contemporaneous effect of a disaster shock on output, inflation, and the nominal interest rate into the two separate components of the models in Figure 2. Immediately after the disaster shock, the temporary reduction in technology amplifies the decline in output and the rise in inflation and the nominal interest rate caused by the destruction of the capital shock. In subsequent periods, the persistent responses of the variables shown in Figure 2 are entirely attributable to the gradual rebuilding of the capital stock.

To evaluate the optimality of policy rules, we make use of Woodford's (2002) finding that an optimal monetary policy replicates the efficient level of output. (18) The introduction of monopolistic competition in our model creates an inefficiency wedge between the efficient level of output and the flexible price and wage level of output. (19) Those markup distortions, however, are nonstochastic, so that wedge remains constant. In particular, our disaster shock--which produces a temporary decline in technology and a loss of capital--has no effect on the wedge. Accordingly, the optimal monetary policy response to a disaster shock is closely approximated by the output dynamics of the flexible model. (20) With the use of this criterion, the dynamics illustrated in Figure 2 show that this calibration of the Taylor rule is suboptimal in the presence of nominal distortions. An optimal monetary policy should enable output to reach its flexible price and wage equilibrium.

5. Optimal Monetary Policy

The specifications of the Taylor rule considered in the previous section fail to generate the optimal monetary policy response to a disaster shock, and a discretionary departure from the Taylor rule can worsen the outcome from a welfare perspective. In this section, we consider systematic optimal monetary policy rules as derived in the literature and evaluate their implications for our disaster shock model. Because such policy rules might not be feasible in practice, we also consider a constrained optimal monetary policy in which the monetary authority responds directly to the disaster shock.

An Example of Optimal Monetary Policy

The optimal monetary policy rule depends on the source of nominal rigidities in the economy. Woodford (2002) argues that stabilizing the price level is the optimal monetary policy when prices are sticky. In a Taylor rule setting, that finding indicates that the parameter on inflation, [[theta].sub.[pi]], should be set to an extremely high value. Erceg, Henderson, and Levin (2000) find that it is optimal for the monetary authority to target the wage inflation rate when wages are sticky. That is, the monetary authority should place a large weight on the wage inflation parameter, [[theta].sub.w], in the Taylor rule. In a sticky price and wage model, however, neither extreme monetary policy rule will suffice. A monetary policy rule vigorously targeting price inflation eliminates the effects of the price distortion but the wage rigidity remains. When wage inflation is the target of monetary policy, the effects of the wage distortion are removed from the model, but the effect of price stickiness remains.

The effect of a disaster shock when the monetary authority follows an extreme price inflation control policy ([[theta].sub.[pi]] = 10,000) is illustrated in the left-hand column of Figure 3. When the monetary authority aggressively responds to price inflation, monetary policy effectively prevents price inflation from changing, which eliminates distortions due to the price rigidity. That policy is optimal for the sticky price model because it enables the model to generate precisely the same output response as the flexible model. Elimination of the price distortion also causes the sticky price and wage model to resemble a model with nominal wage rigidities. The reduction in output pushes down labor demand, which produces a decline in the real wage. Because price inflation is unchanged, the nominal wage inflation rate falls after a disaster shock in all of the models.

[FIGURE 3 OMITTED]

The middle column of Figure 3 shows the effect of a disaster shock when the monetary authority pursues an extreme wage inflation target ([[theta].sub.w] = 10,000). By aggressively targeting the wage inflation rate, the monetary authority eliminates distortions caused by the nominal wage rigidity. Therefore, wage inflation is unchanged after a disaster shock in all three models. That policy causes the sticky price and wage model to resemble the sticky price model. Targeting wage inflation, however, is not optimal for either of the models with nominal rigidities because real output deviates from the flexible price and wage equilibrium. The lower real wage rate caused by lower labor demand combined with a monetary policy objective of maintaining a constant nominal wage rate forces price inflation to rise in all three models.

A monetary policy that aggressively targets both price inflation and wage inflation fails to eliminate the effects of either price or wage distortions. The right-hand column of Figure 3 shows the effect of a disaster shock for a policy that targets both price and wage inflation. In all of the models, price inflation rises, but wage inflation declines. The wage inflation decline is caused by a falling real wage rate which dominates the price inflation increase. In the Taylor rule, the downward pressure from wage inflation cancels out the upward pressure on the nominal interest rate from the rising price inflation rate. Because the effects of both nominal rigidities are still present, the output response is suboptimal in both the sticky price model and the sticky price and wage model.

Constrained Optimal Policy Responses

A monetary policy rule that vigorously responds to price or wage inflation fluctuations might be optimal in theory but might not be feasible in practice. The impulse responses illustrated in Figure 3 require that the market participants believe the monetary authority will respond with "excessive force" to any price or wage deviations, so that prices or wages will never change. Assuming such a policy is not feasible in practice, we consider an alternative strategy in which the monetary authority systematically and directly responds to a disaster shock in an otherwise standard Taylor rule.

[FIGURE 4 OMITTED]

We begin by searching for the optimal monetary policy response to the disaster shock, [[theta].sub.D], in a standard calibration of the Taylor rule ([[theta].sub.[pi]] = 1.5, [[theta].sub.y] = 0.125, and [[theta].sub.w] = 0.0). Following Woodford (2002), the optimal policy response to a disaster shock in a model with nominal rigidities is the value of [[theta].sub.D] that minimizes variation of output from its flexible price and wage equilibrium. Figure 4 displays a grid search across a range of values for [[theta].sub.D], in which a positive value for 0o implies an increase in the nominal interest rate target. (21) In both models with nominal rigidities, the optimal value for [[theta].sub.D] is positive, which suggests that the optimal conditional response to a disaster shock is a policy tightening. A comparison of the output responses in Figure 5 indicates that when [[theta].sub.D] is set equal to its optimal value, the interest rate rises more than indicated by the Taylor rule alone, and the reduction in output more closely mirrors the flexible price and wage equilibrium.

The optimal coefficient on the response of the nominal interest rate target to a disaster shock can be negative in some circumstances. Figure 6 illustrates an example of a plausible policy rule calibration ([[theta].sub.[pi]] = 5.0, [[theta].sub.y] = 0.0, and [[theta].sub.w] = 0.0) in which the optimal value of [[theta].sub.D] is found to be negative for the sticky price and wage model. In this case, the systematic response to price inflation in the policy rule now eliminates much of the distortion from the price stickiness. Inefficiencies from the nominal wage rigidity, however, remain in the sticky price and wage model. Eliminating that wage distortion requires an easing of monetary policy in direct response to a disaster shock (i.e., [[theta].sub.D] is negative). Figure 7 demonstrates that the output response to a disaster shock is much closer to the optimal output response from the flexible model when policy directly responds to the disaster shock. On the other hand, it also shows that the aggressive response of monetary policy to rising inflation still makes it optimal to raise the nominal interest rate target in response to a disaster shock. The direct response to the disaster ([[theta].sub.D] < 0) only partly mitigates the increase in the nominal interest rate.

[FIGURE 5 OMITTED]

The results from Figures 4 and 6 indicate that the optimal monetary policy response to a disaster shock, [[theta].sub.D], depends on the calibration of the other monetary policy rule parameters. Figures 5 and 7 then show that the impulse responses of output and the nominal interest rate to a disaster shock depend on the calibration of monetary policy parameters including [[theta]sub.D]. Given those results, we examine the sensitivity of both the optimal value of [[theta].sub.D] and the impulse responses of output and the nominal interest rate to different calibrations of the monetary policy rule. Table 3 displays the optimal value for [[theta].sub.D] and the contemporaneous effect of a disaster shock on output and the nominal interest rate for parameter values of [[theta].sub.[pi]] that range from 1.5 to 5.0 and [[theta].sub.y] that is either 0 or 0.125. Our results reveal that the calibration of the monetary policy has a small effect on the optimal value for [[theta].sub.D] in the sticky price model but has a more sizable effect on [[theta].sub.D] in the sticky price and wage model. Regardless of the calibration of the monetary policy rule, however, the optimal response of monetary policy to a disaster shock is an increase in the nominal interest rate target.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

6. Conclusion

Once the damage from Hurricane Katrina became apparent, the media and financial markets speculated that the Federal Reserve might ease policy by delaying an expected 25--basis point increase in the federal funds rate. Three weeks later at its next meeting, however, the Federal Reserve decided to maintain its pre-Katrina policy stance and raise the federal funds rate by 25 basis points. In this article, we examined the appropriate monetary policy response to a natural disaster such as Hurricane Katrina.

Our findings suggest that, in most circumstances, the monetary authority should increase its nominal interest rate target after a natural disaster that temporarily reduces productivity and destroys some capital stock. When monetary policy is conducted using a Taylor-style rule, the higher inflation effect dominates the lower output effect such that the endogenous policy response to a disaster is a rise in the nominal interest rate. We then apply Woodford's (2002) findings on optimal monetary policy to show that the optimal response to a disaster depends on the sources of nominal rigidities. Any direct easing after a disaster, nonetheless, is dominated by the need to tighten policy in response to higher inflation and to accommodate the increase in the real interest rate. Thus, the optimal monetary policy response to a natural disaster entails an increase in the nominal interest rate.

The reaction of monetary policy to any recurring shock should be evaluated in terms of systematic responses to infrequent events, not as discretionary responses to random shocks. Individuals observe policy actions and form expectations about similar future events. Those expectations must be endogenized within economic models to provide robust policy analysis. The findings from our disaster scenario framework demonstrate that a rigorous model-based approach to policy analysis sometimes generates prescriptions that are at odds with the prevailing public opinion expressed in the popular press.

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(1) Estimates on the economic losses from Hurricane Katrina vary. Risk Management Solutions, for example, on September 9, 2005, estimated that insured losses were between $40 and $60 billion, whereas total economic losses exceeded $100 billion.

(2) A quarterly decline of 0.4% in U.S. third quarter GDP is based on estimates by the forecasting firms Macroeconomic Advisors and Global Insight. According to Cashell and Labonte (2005), Macroeconomic Advisors lowered their third quarter GDP annual growth forecast from 4.6-3.2% and their fourth quarter forecast from 3.6-3.3%. Insight lowered its forecast for annual GDP growth in the second half of 2005 by 0.7%. Evaluating those magnitudes for a single quarter, at a non-annualized rate, suggests an impact on output of approximately 0.4%.

(3) In his lone dissent, Governor Olsen recommended no change in the federal funds rate "pending the receipt of additional information on the economic effects resulting from the severe shock of Hurricane Katrina."

(4) Although the FOMC did not respond directly to Hurricane Katrina, the Committee did take explicit action in the wake of a previous catastrophic event: the 9/11 attack. The 9/11 event differed from other disasters in that it threatened to disrupt the efficient functioning of the financial system (Ncely 2004).

(5) In related work, Wei (2003) models the effects of the 1973-1974 energy crisis as a negative shock to the capital stock in a general equilibrium framework.

(6) Our finding that a natural disaster generates a rise in prices and inflation expectations is consistent with inflation forecasts before and after Hurricane Katrina. The Philadelphia Fed's Survey of Professional Forecasters" estimate for the average 2005 inflation rate rose from 2.9% before Katrina (August 15, 2005) to 3.9% afterward (November 14, 2005).

(7) Our model is similar to that of Gavin, Keen, and Pakko (2009), except it also includes a disaster shock.

(8) One advantage of indexing to the steady-state inflation rate. as opposed to partial or full indexation to the lagged inflation rate, is that it is not a form of adaptive expectations.

(9) The parameter [[lambda].sub.t] is the Lagrange multiplier from the households' real budget constraint.

(10) A 2% disaster probability implies that a disaster occurs, on average, once every 10 years. We experimented with other calibrations for the disaster probability that are consistent with a rare event and found that our qualitative results were unchanged.

(11) See Hamilton (1994, p. 684) for a detailed description of the AR(I) representation of the two-state Markov process. This procedure of linearizing a two-state Markov process also is used in Pakko (2005).

(12) The expression [p.sub.11] + [p.sub.22] - 1 defines the stable eigenvalue of the probability transition matrix, P.

(13) Given our assumed values for the elements of the probability transition matrix, the implied autocorrelation coefficient equals 0 in this application.

(14) This calibration of the response to output is the equivalent of a coefficient of 0.5 on annualized percent changes. In addition, the coefficients on the gross wage inflation rate and the disaster shock are set to 0.

(15) Alternatively, we could calibrate the shock separately in each model to generate a 0.4% fall in output. Such a modification would require us to magnify the effect of the disaster shock on technology in Equation 11 (i.e., the absolute value of [zeta] would be higher). Because Table 2 shows that the technology component of the disaster shock simply amplifies the capital stock component's effects on key economic variables, calibrating all of the models to a 0.4% change in output will only enhance our key qualitative results.

(16) An alternative approach is to calibrate the disaster shock so that output falls by 0.4% in the models with nominal rigidities. To generate that result, the disaster shock would need to have a larger effect on the level of productivity, [Z.sub.t] (i.e., [zeta] is larger). Because we will show in Table 2 that the productivity component of the disaster shock further pushes output down and inflation and the nominal interest rate up, raising the calibration of [zeta] to generate an output decline of 0.4% in the models with nominal rigidities will strengthen our qualitative results in Figure 2.

(17) These longer term effects distinguish our disaster shock from a simple, transitory technology shock. The persistent increase in inflation is one property of the Taylor rule policy that indicates its suboptimality.

(18) Kim and Henderson (2005) also analytically derive this result in a model with one-period price and wage contracts.

(19) The efficient level of output occurs when the economy has no nominal rigidities and no distortions due to market power or taxes.

(20) Smets and Wouters (2003) also use this criterion for evaluating optimal monetary policy. It should be noted that this criterion identifies "optimal" policy as that which eliminates the distortions due to sticky prices. It does not address the welfare costs of inflation (area under the money demand carve) and the inefficiency of monopolistically competitive pricing, which remain even in the flexible price equilibrium.

(21) For each set of parameter values, the model is simulated 1000 times over a sample period of 160 quarters.

Benjamin D. Keen, Department of Economics, University of Oklahoma, 329 Hester Hall, 729 Elm Avenue, Norman, OK 73019, USA; E-mail ben.keen@ou.edu; corresponding author.

Michael R. Pakko, Chief Economist, Institute for Economic Advancement, University of Arkansas at Little Rock, 2801 South University Avenue, Little Rock, AR 72204, USA; E-mail mrpakko@ualr.edu.

The authors acknowledge the helpful comments of participants at the 82nd annual conference of the Western Economics Association International, San Diego, CA, 2007. The research on this project was conducted while Michael R. Pakko was an economist at the Federal Reserve Bank of St, Louis and Benjamin D. Keen was a visiting scholar. The views expressed in this article are those of the authors and do not necessarily reflect official positions of the Federal Reserve Bank of St. Louis, the Federal Reserve System, or its Board of Governors.

Received September 2009; accepted April 2010.

Table 1. Parameter Calibrations

Parameter                                   Symbol           Value

Depreciation rate                   [delta]                  0.025
Discount factor                     [beta]                   0.99
Leisure utility parameter           [[sigma].sub.1]          0.33
Consumption utility parameter       [[sigma].sub.2]          0.5
Capital's share of output           [alpha]                  0.33
Steady state gross quarterly        [pi]                     1.005
  inflation rate
Steady state labor supply           n                        0.3
Price elasticity of demand          [[epsilon].sub.p]        6.0
Wage elasticity of demand           [[epsilon].sub.w]        6.0
Average capital adjustment costs    [phi](.)                 i/k
  parameter
Marginal capital adjustment costs   [phi]'(.)                1.0
  parameter
Elasticity of the ilk ratio with    x = [(i/k) [phi]" (.)/   -0.2
  respect to Tobin's q                [phi]'(.)].sup.-1]

Table 2. The Contemporaneous Effect of a Disaster Shock

                                            Percent

Flexible model                     y      [pi] (a.r.)    R (a.r.)
  Full disaster shock             -0.40       0.65         0.78
  Capital only ([zeta] = 0)       -0.11       0.14         0.16
  Technology only ([kappa] = 0)   -0.29       0.51         0.62
Sticky price model                 y      [pi] (a. r.)   R (a. r.)
  Full disaster shock             -0.16       0.25         0.29
  Capital only ([zeta] = 0)       -0.09       0.13         0.15
  Technology only ([kappa] = 0)   -0.07       0.12         0.15
Sticky price and wage model        y      [pi] (a.r.)    R (a. r.)
  Full disaster shock             -0.18       0.29         0.35
  Capital only ([zeta] = 0)       -0.13       0.20         0.23
  Technology only ([kappa] = 0)   -0.05       0.10         0.11

a.r.--annualized rate.

Table 3. Direct Responses to a Disaster Shock: A Sensitivity Analysis

Sticky Price Model

      [[theta].sub.y] = 0              Contemp. Impact, %

[[theta].sub.[pi]]   [[theta].sub.D]      y     R (a.r.)

1.5                        0.40         -0.39     0.69
2.0                        0.40         -0.39     0.67
2.5                        0.40         -0.40     0.66
3.0                        0.40         -0.40     0.66
3.5                        0.40         -0.40     0.65
4.0                        0.40         -0.40     0.65
4.5                        0.40         -0.40     0.65
5.0                        0.40         -0.40     0.65

    [[theta].sub.y] = 0.125            Contemp. Impact, %

[[theta].sub.[pi]]   [[theta].sub.D]      y     R (a.r.)

1.5                        0.50         -0.39     0.78
2.0                        0.50         -0.39     0.72
2.5                        0.50         -0.40     0.70
3.0                        0.50         -0.40     0.69
3.5                        0.50         -0.40     0.68
4.0                        0.50         -0.40     0.67
4.5                        0.50         -0.40     0.67
5.0                        0.50         -0.40     0.67

Sticky Price and Wage Model

      [[theta].sub.y] = 0              Contemp. Impact, %

[[theta].sub.[pi]]   [[theta].sub.D]      Y     R (a.r.)

1.5                        0.20         -0.38     0.60
2.0                        0.15         -0.40     0.55
2.5                        0.05         -0.38     0.44
3.0                        0.00         -0.38     0.41
3.5                        0.00         -0.40     0.45
4.0                       -0.10         -0.37     0.36
4.5                       -0.05         -0.42     0.45
5.0                       -0.10         -0.41     0.43

[[theta].sub.y] = 0.125                Contemp. Impact, %

[[theta].sub.[pi]]   [[theta].sub.D]      Y     R (a.r.)

1.5                        0.40         -0.40     0.82
2.0                        0.30         -0.40     0.69
2.5                        0.20         -0.39     0.59
3.0                        0.15         -0.39     0.55
3.5                        0.10         -0.39     0.52
4.0                        0.05         -0.39     0.49
4.5                        0.05         -0.41     0.52
5.0                       -0.05         -0.39     0.45

a.r.--annualized rate.