Inventories and optimal monetary policy.
Lubik, Thomas A. ; Teo, Wing Leong
It has long been recognized that inventory investment plays a large
role in explaining fluctuations in real gross domestic product (GDP),
although it makes up only a small fraction of it. Blinder and Maccini
(1991) document that in a typical recession in the United States, the
fall in inventory investment accounts for 87 percent of the decline in
output despite being only one half of 1 percent of real GDP. A lot of
research has been trying to explain how this seemingly insignificant
component of GDP has such a disproportionate role in business cycle
fluctuations. (1) However, surprisingly few studies have focused on the
conduct of monetary policy when firms can invest in inventories. In this
article we attempt to fill this gap by investigating how inventory
investment affects the design of optimal monetary policy.
We employ the simple New Keynesian model that has become the
benchmark for analyzing monetary policy from both a normative and a
positive perspective. We introduce inventories into the model by
assuming that the inventory stock facilitates sales, as suggested in
Bils and Kahn (2000). We first establish that the dynamics, and
therefore the monetary transmission mechanism, differ between the models
with and without inventories for a given behavior of the monetary
authority. Monetary policy is then endogenized by assuming that
policymakers solve an optimal monetary policy problem.
First, we compute the optimal Ramsey policy. A Ramsey planner
maximizes the welfare of the agents in the economy by taking into
account the private sector's optimality conditions. In doing so,
the planner chooses a socially optimal allocation. While this does not
necessarily bear any relationship to the typical conduct of monetary
policymakers, it provides a useful benchmark. Subsequently, we study
optimal policy when the planner is constrained to implement simple
rules. That is, we specify a set of rules that lets the policy
instrument (the nominal interest rate) respond to target variables such
as the inflation rate and output. The policymaker chooses the respective
response coefficients that maximize welfare. Optimal rules of this kind
may be preferable to Ramsey plans from an actual policymaker's
perspective since they can be operationalized and are easier to
communicate to the public.
Our most interesting but surprising finding is that Ramsey-optimal
monetary policy deviates from full inflation stabilization in our model
with inventories. This stands in contrast to the standard New Keynesian
model. In the New Keynesian model, perfectly stable inflation is optimal
since movements in prices represent deadweight costs to the economy.
Introducing inventories potentially modifies that basic calculus for the
following reasons. First, we assume that a firm's inventory
holdings are relevant for its sales only in relative terms, that is,
when they deviate from the aggregate inventory stock. This presents an
externality, which a Ramsey planner may want to address. Second,
inventories change the economy's propagation mechanism as they
allow firms to smooth sales over time with concomitant effects on
consumption; that is, output and consumption need no longer coincide,
which has a similar effect as capital in that it provides future
consumption opportunities. Changes in prices serve as the equilibrating
mechanism for the competing goals of reducing consumption volatility and
avoiding price adjustment costs. The inventory specification therefore
contains something akin to an inflation-output tradeoff. Consequently,
the optimal policy no longer fully stabilizes inflation. The second
important finding concerns the efficacy of implementing simple rules.
Similar to most of the optimal policy literature, we show that simple
rules can come exceedingly close to the socially optimal Ramsey policy
in welfare terms.
Our article relates to two literatures. First, the amount of
research on optimal monetary policy in the New Keynesian framework is
very large already, and we do not have much to contribute conceptually
to the modeling of optimal policy. Schmitt-Grohe and Uribe (2007) is a
recent important and comprehensive contribution. A main conclusion from
this literature is that optimal monetary policy will choose to almost
perfectly stabilize inflation. In environments with various nominal and
real distortions, this policy prescription becomes slightly modified,
but nevertheless perseveres. We thus contribute to the optimal policy
literature by demonstrating that the results carry over to a framework
with another, previously unconsidered modification to the basic
framework in the form of inventories.
The study of inventory investment has a long pedigree, to which we
cannot do full justice here. Much of the earlier literature, as surveyed
in Blinder and Maccini (1991), was concerned with identifying the
determinants of inventory investment, such as aggregate demand and
expectations thereof, or the opportunity costs of holding inventories.
Most work in this area was largely empirical using semi-structural
economic models, with West (1986) being a prime example. (2) Almost in
parallel to this more explicitly empirical literature, inventories were
introduced into real business cycle models. The seminal article by
Kydland and Prescott (1982) introduces inventories directly into the
production function. More recent contributions include Christiano
(1988), Fisher and Hornstein (2000), and Khan and Thomas (2007). The
latter two articles especially build a theory of a firm's inventory
behavior on the micro-foundation of an S-s environment. The focus of
these articles is on the business cycle properties of inventories, in
particular the high volatility of inventory investment relative to GDP
and the countercylicality of the inventory-sales ratio, both of which
are difficult to match in typical inventory models. In an important
article, Bils and Kahn (2000) demonstrate that time-varying and
countercyclical markups are crucial for capturing this co-movement
pattern.
This insight lends itself to considering inventory investment
within a New Keynesian framework since it features interplay between
marginal cost, inflation, and monetary policy, which might therefore be
a source of inventory fluctuations. (3) Recently, several articles have
introduced inventories into New Keynesian models. Jung and Yun (2005)
and Boileau and Letendre (2008) both study the effects of monetary
policy from a positive perspective. The former combines Calvo-type price
setting in a monopolistically competitive environment with the approach
to inventories as introduced by Bils and Kahn (2000). The use of the
Calvo approach to modeling nominal rigidity allows these authors to
discuss the importance of strategic complementarities in price setting.
Boileau and Letendre (2008), on the other hand, compare various
approaches to introducing inventories in a sticky-price model. This
article is differentiated from those contributions by its focus on the
implications of inventories as a transmission mechanism for optimal
monetary policy.
The rest of the article is organized as follows. In the next
section we develop our New Keynesian model with inventories. Section 2
analyzes the differences between the standard New Keynesian model and
our specification with inventories. We calibrate both models and compare
their implications for business cycle fluctuations. We present the
results of our policy exercises in Section 3, which also includes a
robustness analysis with respect to changes in the parameterization.
Section 4 concludes with a brief discussion of the main results and
suggestions for future research.
1. THE MODEL
We model inventories in the manner of Bils and Kahn (2000) as a
mechanism for facilitating sales. When firms face unexpected demand,
they can simply draw down their stock of previously produced goods and
do not have to engage in potentially more costly production. This
inventory specification is embedded in an otherwise standard New
Keynesian environment. There are three types of agents: monopolistically
competitive firms, a representative household, and the government. Firms
face price adjustment costs and use labor for the production of finished
goods, which can be sold to households or added to the inventory.
Households provide labor services to the firms and engage in
intertemporal consumption smoothing. The government implements monetary
policy.
Firms
The production side of the model consists of a continuum of
monopolistically competitive firms, indexed by i [member of] [0, 1]. The
production function of a firm i is given by
[y.sub.t](i) = [z.sub.t][h.sub.t] (i), (1)
where [y.sub.t] (i) is output of firm i, [h.sub.t] (i) is labor
hours used by firm i, and [z.sub.t] is aggregate productivity. We assume
that it evolves according to the exogenous stochastic process
1n[z.sub.t] = [[rho].sub.z] ln [z.sub.[t-1]] + [[epsilon].sub.zt],
(2)
where [[epsilon].sub.zt] is an i.i.d. innovation.
We introduce inventories into the model by assuming that they
facilitate sales as suggested by Bils and Kahn (2000). (4) In their
partial equilibrium framework, they posit a downward-sloping demand
function for a firm's product that shifts with the level of
inventory available. As shown by Jung and Yun (2005), this idea can be
captured in a New Keynesian setting with monopolistically competitive
firms by introducing inventories directly into the Dixit-Stiglitz
aggregator of differentiated products:
[s.sub.t] = [([[integral].sub.0.sup.1][([[a.sub.t](i)/[a.sub.t]]).sup.[[mu]/[theta]]] [S.sub.t][(i).sup.[([theta]-1)/[theta]]di).sup.[[theta]/([theta]-1)], (3)
where [s.sub.t] are aggregate sales; [s.sub.t], (i) are
firm-specific sales; [a.sub.t] and [a.sub.t] (i) are, respectively, the
aggregate and firm-specific stocks of goods available for sales; [theta]
> 1 is the elasticity of substitution between differentiated goods;
and [mu] > 0 is the elasticity of demand with respect to the relative
stock of goods. Holding inventories helps firms to generate greater
sales at a given price since they can rely on the stock of previously
produced goods when, say, demand increases. Note, however, that a
firm's inventory matters only to the extent that it exceeds the
aggregate level. In a symmetric equilibrium, having inventories does not
help a firm to make more sales, but it affects the firm's
optimality condition for inventory smoothing.
Cost minimization implies the following demand function for sales
of good i:
[s.sub.t](i) = [([[a.sub.t](i)/[a.sub.t]]).sup.[mu]]
[([[P.sub.t](i)/[P.sub.t]]).sup.-[theta]][S.sub.t], (4)
where [P.sub.t] (i) is the price of good i, and [P.sub.t], is the
price index for aggregate sales [s.sub.t]:
[P.sub.t] = [([[integral].sub.0.sup.1][([[a.sub.t](i)/[a.sub.t]]).sup.[mu]] [P.sub.t][(i).sup.[1-[theta]] di).sup.[1/(1-[theta])]. (5)
A firm's sales are thus increasing in its relative inventory
holdings and decreasing in its relative price. The inventory term can
alternatively be interpreted as a taste shifter, which firms invest in
to capture additional demand (see Kryvtsov and Midrigan 2009). Finally,
the stock of goods available for sales [a.sub.t] (i) evolves according
to
[a.sub.t](i) = [y.sub.t](i) + (1 - [delta])([a.sub.[t-1]](i) -
[s.sub.[t-1]](i)), (6)
where [delta] [member of] (0, 1) is the rate of depreciation of the
inventory stock. It can also be interpreted as the cost of carrying the
inventory over the period.
Each firm faces quadratic costs for adjusting its price relative to
the steady state gross inflation rate [pi]: [[empty set]/2]
[([P.sub.t](i)/[pi][P.sub.t]-1.sup.(i)]-1).sup.2] [s.sub.t], with [empty
set] > 0, and [pi] [greater than or equal to] 1, the steady state
gross inflation rate. Note that the costs are measured in units of
aggregate sales instead of output since [s.sub.t] is the relevant demand
variable in the model with inventories. Firm i's intertemporal
profit function is then given by
[E.sub.t] [[infinity].summation over ([tau]=0)] [[rho].sub.[t,
t+[tau]]] [[[[P.sub.[t+[tau]](i)
[s.sub.[t+[tau]]](i)]/[P.sub.[t+[tau]]]] -
[[[W.sub.[t+[tau]]][h.sub.[t+[tau]]](i)]/[P.sub.[t+[tau]]]] - [[empty
set]/2] [([[P.sub.[t+[tau]]](i)/[[pi][P.sub.[t+[tau]-1]](i)]] -
1).sup.2][s.sub.[t+[tau]]]], (7)
where [W.sub.t] is the nominal wage and [[rho].sub.t,[t+[tau]] is
the aggregate discount factor that a firm uses to evaluate profit
streams.
Firm i chooses its price, [P.sub.t](i), labor input, [h.sub.t](i),
and stock of goods available for sales, [a.sub.t](i), to maximize its
expected intertemporal profit (7), subject to the production function
(1), the demand function (4), and the law of motion for [a.sub.t](i)
(6). The first order conditions are
[empty set] ([[P.sub.t](i)/[[pi][P.sub.[t-1]](i)]] - 1)
[[s.sub.t]/[[pi][P.sub.[t-1]](i)]] = (1 - [theta])
[[s.sub.t](i)/[P.sub.t]] + [E.sub.t][[rho].sub.[t, t+1]] [[empty
set]([[P.sub.[t+1]](i)/[pi][P.sub.t](i)] - 1)
[[[s.sub.[t+1]][P.sub.[t+1]](i)]/[pi][P.sub.t.sup.2](i)] + (1 -
[delta])[theta] [[s.sub.t](i)/[P.sub.t](i)][mc.sub.[t+1]](i)] (8)
[[W.sub.t]/[P.sub.t]] = [z.sub.t][mc.sub.t](i), (9)
and
[mc.sub.t](i) = [mu][[P.sub.t](i)/[P.sub.t]]
[[s.sub.t](i)/[a.sub.t](i)] + (1 - [delta]) (1 -
[mu][[s.sub.t](i)/[a.sub.t](i)]) [E.sub.t][[rho].sub.[t,
t+1]][mc.sub.[t+1]](i), (10)
where [mc.sub.t] (i) is the Lagrange multiplier associated with the
demand constraint (4). It can also be interpreted as real marginal cost.
Equation (8) is the optimal price-setting condition in our model
with inventories. It resembles the typical optimal price-setting
condition in a New Keynesian model with convex costs for price
adjustment (e.g., Krause and Lubik 2007), except that marginal cost now
enters the optimal pricing condition in expectations because of the
presence of inventories. In this model, the behavior of marginal cost,
mc, can be interpreted from two different directions. As captured by
Equation (9), it is the ratio of the real wage to the marginal product
of labor, which in the standard model is equal to the cost of producing
an additional unit of output. Alternatively, it is the cost of
generating an additional unit of goods available for sale, which can
either come out of current production or out of (previously) foregone
sales. This in turn reduces the stock of goods available for sales in
future periods, which would eventually have to be replenished through
future production. This intertemporal tradeoff between current and
future marginal cost is captured by Equation (10).
Household
We assume that there is a representative household in the economy.
It maximizes expected intertemporal utility, which is defined over
aggregate consumption, (5) [c.sub.t], and labor hours, [h.sub.t]:
[E.sub.0] [[infinity].summation over ([t=0])] [[beta].sup.t]
[[[zeta].sub.t] ln [c.sub.t] - [[h.sub.t.sup.[1+n]]/1 + [eta]]], (11)
where [eta] [greater than or equal to] 0 is the inverse of the
Frisch labor supply elasticity.
[[zeta].sub.t] is a preference shock and is assumed to follow the
exogenous AR(1) process
In [[zeta].sub.t] = [[rho].sub.[zeta]] ln [[zeta].sub.[t-1]] +
[[epsilon].sub.[[zeta], t]], (12)
where 0 < [[rho].sub.[zeta]] < 1 and
[[epsilon].sub.[[zeta],t]] is an i.i.d. innovation.
The household supplies labor hours to firms at the nominal wage
rate, [W.sub.t], and earns dividend income, [D.sub.t], (which is paid
out of firms' profits) from owning the firms. It can purchase
one-period discount bonds, [B.sub.t], at a price of 1/[R.sub.t], where
[R.sub.t] is the gross nominal interest rate. Its budget constraint is
[P.sub.t][c.sub.t] + [[B.sub.t]/[R.sub.t]] [less than or equal to]
[B.sub.[t-1]] + [W.sub.t][h.sub.t] + [D.sub.t]. (13)
The first-order conditions for the representative household's
utility maximization problem are
[h.sub.t.sup.[eta]] = [[[zeta].sub.t]/[c.sub.t]]
[[W.sub.t]/[P.sub.t]], and (14)
[[[zeta].sub.t]/[c.sub.t]] = [beta][R.sub.t][E.sub.t]
([[[zeta].sub.[t+1]]/[c.sub.[t+1]]] [[P.sub.t]/[P.sub.[t+1]]]). (15)
Equation (14) equates the real wage, valued in terms of the
marginal utility of consumption, to the disutility of labor hours.
Equation (15) is the consumption-based Euler equation for bond holdings.
Government and Market Clearing
In order to close the model, we also need to specify the behavior
of the monetary authority. The main focus of the paper is the optimal
monetary policy in the New Keynesian model with inventories. In the next
section, however, we briefly compare our specification to the standard
model without inventories in order to assess whether introducing
inventories significantly changes the model dynamics. We do this
conditional for a simple, exogenous interest rate feedback rule that has
been used extensively in the literature:
[[~.R].sub.t] = [rho][[~.R].sub.[t-1]] +
[[psi].sub.1][[~.[pi]].sub.t] + [[psi].sub.2][[~.y].sub.t] +
[[epsilon].sub.[R, t]], (16)
where a tilde over a variable denotes its log deviation from its
deterministic steady state, [[psi].sub.1] and [[psi].sub.2] are monetary
policy coefficients and 0 < [rho] < 1 is the interest smoothing
parameter. [[epsilon].sub.[R,t]] is a zero mean innovation with constant
variance; it is often interpreted as a monetary policy implementation
error. Finally, we impose a symmetric equilibrium, so that the
firm-specific indices, i, can be dropped. In addition, we assume that
bonds are in zero net supply, [B.sub.t] = 0. Market clearing in the
goods market requires that consumption, together with the cost for price
adjustment, equals aggregate sales:
[s.sub.t] = [c.sub.t] + [[empty set]/2] [([[[pi].sub.t]/[pi]] -
1).sup.2][s.sub.t]. (17)
2. ANALYZING THE EFFECTS OF MONETARY POLICY
The main focus of this article is how the introduction of
inventories into an otherwise standard New Keynesian framework changes
the optimal design of monetary policy. However, we begin by briefly
comparing the behavior of the model with and without inventories to
assess the changes in the dynamic behavior of output and inflation,
given the exogenous policy rule (16). The standard New Keynesian model
differs from our model with inventories in the following respects.
First, there is no explicit intertemporal tradeoff in terms of marginal
cost as in equation (10). This implies, secondly, that the driving term
in the Phillips curve (8) is current marginal cost, as defined by
equation (9). Finally, in the standard model, consumption, output,
sales, and goods available of sales are first-order equivalent. We note,
however, that the standard specification is not nested in the model with
inventories; that is, the equation system for the latter does not reduce
to the former for a specific parameterization.
Calibration
The time period corresponds to a quarter. We set the discount
factor, [beta], to 0.99. Since price adjustment costs are incurred only
for deviations from steady-state inflation, its value is irrelevant for
first-order approximations of the model's equation system but plays
a role when we perform the optimal policy analysis. We therefore set
[pi] = 1.0086 to be consistent with the average post-war,
quarter-over-quarter inflation rate. In the baseline calibration, we
choose a fairly elastic labor supply and set [eta] = 1, which is a
common value in the literature and corresponds to quadratic disutility
of hours worked. We impose a steady-state markup of 10 percent, which
implies [theta] = 11. The price adjustment cost parameter is then
calibrated so that [eta]([theta] - 1 )/[empty set] = 0.1, as in Ireland
(2004). This is a typical value for the coefficient on marginal cost in
the standard New Keynesian Phillips curve. (6) The parameters of the
monetary policy rule are chosen to be broadly consistent with the
empirical Taylor rule literature for a unique equilibrium. That is,
[[psi].sub.1] and [[psi].sub.2] are set to 0.45 and 0, respectively,
while the smoothing parameter is set to [rho] = 0.7. This choice
corresponds to an inflation coefficient of 0.45/0.3 = 1.5 that obeys the
Taylor principle. We specify the policy rule in this manner since it
allows us to analyze later the effects of inertial and super-inertial
rules with [rho] [greater than or equal to] 1.
The persistence of the technology shock and the preference shock
are both set to [[rho].sub.z] = [rho].sub.[zeta]] = 0.95. The standard
deviation of the productivity innovation is then chosen so as to match
the standard deviation of HP-filtered U.S. GDP of 1.61 percent. This
yields a value of [[sigma].sub.z] = 0.005. We set the standard deviation
of the preference shocks at three times the value of the former, which
is consistent with empirical estimates from a variety of studies (e.g.,
Ireland 2004). In the same manner, we choose a standard deviation of the
monetary policy shock of 0.003. The parameters related to inventories,
[mu], and [delta], are calibrated following Jung and Yun (2005);
specifically the elasticity of demand with respect to the stock of goods
available for sales is [micro] = 0.37, while the depreciation rate of
the inventory stock is [delta] = 0.01.
Do Inventories Make a Difference?
To get an idea how the introduction of inventories changes the
model dynamics, we compare the responses of some key variables to
technology, preference, and monetary policy shocks for the specification
with and without inventories. The impulse responses are found in Figures
1-3, respectively. In the figures, the label "Base" refers to
the responses under the specification without inventories, while
"Inv" indicates the inventory specification. The key
qualitative difference between the two models is the behavior of labor
hours. In response to a persistent technology shock, labor increases in
the model with inventories, while it falls in the standard New Keynesian
model before quickly returning to the steady state. (7) In the New
Keynesian model, firms can increase production even when economizing on
labor because of the higher productivity level. There is further
downward pressure on labor since the productivity shock raises the real
wage. Higher output is reflected in a drop in prices, which are drawn
out over time due to the adjustment costs, and marginal cost falls
strongly.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
The presence of inventories, however, changes this basic calculus
as firms can use inventories to take advantage of current low marginal
cost. With inventory accumulation firms need not sell the additional
output immediately, which prompts them to increase labor input.
Consequently, output rises by more than in the standard model and the
excess production is put in inventory. The stock of goods available for
sales thus rises, whereas the sales-to-stock ratio, [[gama].sub.t]
[equivalent to] [[s.sub.t]/[a.sub.t]], falls. This is also reflected in
the (albeit small) fall in marginal cost, which is, however, persistent
and drawn out. In other words, firms use inventories to take advantage
of current and future low marginal cost. Inflation moves in the same
direction as in the standard model, but is much smoother, as the
increased output does not have to be priced immediately. This behavior
is just the flip side of the smoothing of marginal cost.
In response to a preference shock, hours move in the same direction
in both models. However, the response with inventories is smaller since
firms can satisfy the additional demand out of their inventory holdings,
which therefore does not drive up marginal cost as much. Compared to the
standard model, firms do not have to resort to increases in price or
labor input to satisfy the additional demand. Inventories are thus a way
of smoothing revenue over time, which is also consistent with a smoother
response of inflation. The dynamics following a contractionary policy
shock are qualitatively similar to those of technology shocks in terms
of co-movement. Sales in the inventory model fall, but output and hours
increase to take advantage of the falling marginal cost. All series are
again noticeably smoother when compared to the standard model.
We now briefly discuss some business cycle implications of the
inventory model. (8) Table 1 shows selected statistics for key
variables. A notable stylized fact in U.S. data is that production is
more volatile than sales. We find that our inventory model replicates
this observation in the case of productivity shocks, that is, output is
30 percent more volatile. This implies that consumption, which is equal
to sales in our linearized setting, is also less volatile than GDP. The
introduction of inventories is thus akin to the modeling of capital and
investment in breaking the tight link between output and consumption
embedded in the standard New Keynesian model. However, the model has
counterfactual implications for the co-movement of inventory variables.
Sales are highly negatively correlated with the sales-inventory ratio,
whereas in the data the two series co-move slightly positively and are
at best close to uncorrelated. This finding can be overturned when
either preference or policy shocks are used, both of which imply a
strong positive co-movement. However, in the case of policy shocks,
sales are counterfactually more volatile than output. When all shocks
are considered together, we find that co-movement between the inventory
variables are positive, but not unreasonably so, while sales are
slightly more volatile than output.
Table 1 Business Cycle Statistics
Moments Technology Preference Policy All Shocks
Standard Deviation (%)
Output 1.61 1.93 0.23 2.52
Sales 1.18 2.37 0.74 2.80
Hours 0.25 1.93 0.23 2.02
Correlation
(Sales, [Sales/Inventory]) -0.85 0.87 0.51 0.49
([Sales/Stock], 0.95 0.90 0.72 0.49
Marginal Cost
The model also has implications for inflation dynamics. Most
notably, inflation is less volatile in the inventory specification than
in the standard model. In the New Keynesian model, inflation is driven
by marginal cost; hence, the standard model predicts that the two
variables are highly correlated. In the data, however, proxies for
marginal cost, such as unit labor cost or the labor share, co-move only
weakly with inflation. This has been a challenge for empirical studies
of the New Keynesian Phillips curve. Our model with inventories may,
however, improve the performance of the Phillips curve in two aspects.
First, marginal cost smoothing translates into a smoother and thus more
persistent inflation path; second, the form and the nature of the
driving process in the Phillips curve equation changes, as is evident
from equations (8) and (10). The latter equation predicts a relationship
between marginal cost and the sales-to-stock ratio, [gamma], which
changes the channel by which marginal cost affects inflation dynamics.
(9)
We can tentatively conclude that a New Keynesian model with
inventories presents a modified set of tradeoffs for an optimizing
policymaker. In the standard model optimal policy is such that both
consumption and the labor supply should be smoothed and price adjustment
costs minimized. In the inventory model, these objectives are still
relevant since they affect utility in the same manner, but the channel
through which this can be achieved is different. Inventories allow for a
smoother adjustment path of inflation, which should help contain the
effects of price stickiness, while the consumption behavior depends on
the nature of the shocks. We now turn to an analysis of optimal policy
with inventories.
3. OPTIMAL MONETARY POLICY
The goal of an optimizing policymaker is to maximize a welfare
function subject to the constraints imposed by the economic environment
and subject to assumptions about whether the policymaker can commit or
not to the chosen action. In this article, we assume that the optimizing
monetary authority maximizes the intertemporal utility function of the
household subject to the optimal behavior chosen by the private sector
and the economy's feasibility constraints. Furthermore, we assume
that the policymaker can credibly commit to the chosen path of action
and does not re-optimize along the way. We consider two cases. For our
benchmark, we assume that the monetary authority implements the
Ramsey-optimal policy. (10) We then contrast the Ramsey policy with an
optimal policy that is chosen for a generic set of linear rules of the
type used in the simulation analysis above.
We can alternatively interpret the policymaker's actions as
minimizing the distortions in the model economy. In a typical New
Keynesian setup like ours, there are two distortions. The first is the
suboptimal level of output generated by the presence of
monopolistic-ally competitive firms. The second distortion arises from
the presence of nominal price stickiness, as captured by the quadratic
price adjustment cost function, which is a deadweight loss to the
economy. In the standard model, the optimal policy perfectly stabilizes
inflation at the steady-state level. Introducing inventories can change
this basic calculus in our model, as the sales-relevant terms are
relative inventory holdings that present an externality for a Ramsey
planner. We will now investigate whether this additional wedge matters
quantitatively for optimal policy.
Welfare Criterion
We use expected lifetime utility of the representative household at
time zero, [V.sub.0.sup.a], as the welfare measure to evaluate a
particular monetary policy regime, a:
[V.sub.0.sup.a] [equivalent to] [E.sub.0] [[infinity].summation
over (t=0)] [[beta].sup.t] [[[zeta].sub.t] ln [C.sub.t.sup.a] -
[[([h.sub.t.sup.a]).sup.[1+[eta]]]/1 + [eta]]]. (18)
As in Schmitt-Grohe and Uribe (2007), we compute the expected
lifetime utility conditional on the initial state being the
deterministic steady state for given sequences of optimal choices of the
endogenous variables and exogenous shocks. Our welfare measure is in the
spirit of Lucas (1987) and expresses welfare as a percentage [THETA] of
steady-state consumption that the household is willing to forgo to be as
well off under the steady state as under a given monetary policy regime,
a. [THETA] can then be computed implicitly from
[[infinity].summation over (t=0)] [[beta].sup.t] [[zeta] ln [(1 -
[[THETA]/100])c] - [[h.sup.[1+[eta]]]/1 + [eta]]] = [V.sub.0.sup.a],
(19)
where variables without time subscripts denote the steady state of
the corresponding variables. (11) Note that a higher value of [THETA]
corresponds to lower welfare. That is, the household would be willing to
give up [THETA] percent of steady-state consumption to implement a
policy that delivers the same level of welfare as the economy in the
absence of any shocks. This also captures the notion that business
cycles are costly because they imply fluctuations that a
consumption-smoothing and risk-averse agent would prefer not to have.
Optimal Policy
We compute the Ramsey policy by formulating a Lagrangian problem in
which the government maximizes the welfare function (18) of the
representative household subject to the private sector's
first-order conditions and the market-clearing conditions of the
economy. The optimality conditions of this Ramsey policy problem can
then be obtained by differentiating the Lagrangian problem with respect
to each of the endogenous variables and setting the derivatives to zero.
This is done numerically by using the Matlab procedures developed by
Levin and Lopez-Salido (2004). The welfare function is then approximated
around the distorted, non-Pareto-optimal steady state. The source of
steady-state distortion is the inefficient level of output due to the
presence of monopolistically competitive firms.
In our second optimal policy case, we follow Schmitt-Grohe and
Uribe (2007) and consider optimal, simple, and implementable interest
rate rules.
Specifically, we consider rules of the following type:
[[~.R].sub.t] = [rho][[~.R].sub.[t-1]] +
[[psi].sub.1][E.sub.t][[~.[pi]].sub.[t+i]] +
[[psi].sub.2][E.sub.t][[~.y].sub.[t+i]], i = -1, 0, 1. (20)
The subscript i indicates that we consider forward-looking (i = 1),
contemporaneous (i = 0), and backward-looking rules (i = -1). Following
the suggestion in Schmitt-Grohe and Uribe (2007), we focus on values of
the policy parameters [rho], [[psi].sub.1], and [[psi].sub.2] that are
in the interval [0, 3]. Note that this rule also allows for the
possibility that the interest rate is super-inertial; that is, we assume
[rho] can be larger than 1. In order to find the constrained-optimal
interest rate rule, we search for combinations of the policy
coefficients that maximize the welfare criterion. As in Schmitt-Grohe
and Uribe (2007), we impose two additional restrictions on the interest
rate rule: (i) the rule has to be consistent with a locally unique
rational expectations equilibrium; (ii) the interest rate rule cannot
violate 2[sigma].sub.R] < R, where [[sigma].sub.R] is the
unconditional standard deviation of the gross interest rate while R is
its steady-state value. The second restriction is meant to approximate
the zero bound constraint on the nominal interest rate. (12)
Ramsey-Optimal Policy
A key feature of the standard New Keynesian setup is that
Ramsey-optimal policy completely stabilizes inflation. Price movements
represent a deadweight loss to the economy because of the existence of
adjustment costs. (13) An optimizing planner would, therefore, attempt
to remove this distortion. This insight is borne out by the impulse
response functions for the standard model without inventories in Figure
4. Inflation does not respond to the technology shock, nor do labor
hours or marginal cost as per the New Keynesian Phillips curve. The path
of output simply reflects the effect of increased and persistent
productivity. The Ramsey planner takes advantage of the temporarily high
productivity and allocates it straight to consumption without feedback
to higher labor input or prices. The planner could have reduced labor
supply to smooth the time path of consumption. However, this would have
a level effect on utility due to lower consumption, positive price
adjustment cost via the feedback from lower wages to marginal cost, and
increased volatility in hours. The solution to this tradeoff is thus to
bear the brunt of higher consumption volatility.
[FIGURE 4 OMITTED]
The possibility of inventory investment, however, changes this
rationale (see Figure 4). In response to a technology shock, output
increases by more compared to the model without inventories, while
consumption, which is first-order equivalent to sales, rises less.
Ramsey-optimal policy can induce a smoother consumption profile by
allowing firms to accumulate inventories. Similarly, the planner takes
advantage of higher productivity in that he induces the household to
supply more labor hours. Inflation is now no longer completely
stabilized as the lower increase in consumption leads to an initial
decline in inflation. Inventories thus serve as a savings vehicle that
allows the planner to smooth out the impact of shocks. The planner
incurs price adjustment costs and disutility from initially high labor
input. The benefit is a smoother and more prolonged consumption path
than would be possible without inventories. The model with inventories
therefore restores something akin to an output-inflation tradeoff in the
New Keynesian framework.
The quantitative differences between the two specifications are
small, however. Table 2 reports the welfare costs and standard
deviations of selected variables for the two versions of the model under
Ramsey-optimal policy. The welfare costs of business cycles in the
standard model are vanishingly small when only technology shocks are
considered and undistinguishable from the specification with
inventories. The standard deviation of inflation is zero for the model
without inventories while it is slightly higher for the model with
inventories. This is consistent with the evidence from the impulse
responses and highlights the differences between the two model
specifications. Note also that consumption is less volatile in the model
with inventories than in the standard model, which reflects the
increased degree of consumption smoothing in the former. (14)
Table 2 Welfare Costs and Standard Deviations under Ramsey-Optimal
Policy
Technology Preference All Shocks
Panel A: Model without Inventories
Welfare Cost ([THETA]) 0.0000 -0.0521 -0.0521
Standard Deviation (%)
Output 1.60 2.40 2.89
Inflation 0.00 0.00 0.00
Consumption 1.60 2.40 2.89
Labor 0.00 2.40 2.40
Panel B: Model with Inventories
Welfare Cost ([THETA]) 0.000 -0.0529 -0.0529
Standard Deviation (%)
Output 1.73 2.28 2.86
Inflation 0.02 0.04 0.04
Consumption 1.45 2.60 2.97
Labor 0.24 2.28 2.29
Panel C: Full Inflation Stabilization
Welfare Cost ([THETA]) 0.000 -0.0528 -0.0528
Standard Deviation (%)
Output 1.73 2.29 2.87
Inflation 0.00 0.00 0.00
Consumption 1.45 2.61 2.99
Labor 0.24 2.29 2.30
Figure 5 depicts the impulse responses to the preference shock
under Ramsey-optimal policy. Inflation and marginal cost are fully
stabilized in the standard model, which the planner achieves through a
higher nominal interest rate that reduces consumption demand in the face
of the preference shock. At the same time, the planner lets labor input
go up to meet some of the additional demand. In contrast, Ramsey policy
for the inventory model can allow consumption to increase by more since
firms can draw on their stock of goods for sale. Consequently, output
and labor increase by less for the inventory model. Similarly to the
case of the technology shock, optimal policy does not induce complete
inflation stabilization as it uses the inventory channel to smooth
consumption. This is confirmed by the simulation results in Table 2,
which show the Ramsey planner trading off volatility between inflation,
consumption, and labor when compared to the standard model.
[FIGURE 5 OMITTED]
Interestingly, eliminating business cycles and imposing the
steady-state allocation is costly for the planner in the presence of
preference shocks that multiply consumption. This is evidenced by the
negative entries for the welfare cost in both model specifications. In
other words, agents would be willing to pay the planner 0.05 percent of
their steady-state consumption not to eliminate preference-driven
fluctuations. This stems from the fact that, although fluctuations per
se are costly in welfare terms for risk-averse agents, they can also
induce co-movement between the shocks and other variables that have a
level effect on utility. Specifically, preference shocks co-move
positively with consumption due to an increase in demand. This positive
co-movement is reflected in a positive covariance between these two
variables. In our second-order approximation to the welfare functions,
this overturns the negative contribution to welfare from consumption
volatility.
When we consider both shocks together, the differences between the
two specifications are not large in welfare terms and with respect to
the implications for second moments. Inflation and consumption are more
volatile in the inventory version, while labor is less volatile compared
to standard specification. We also compare Ramsey-optimal policy with
inventories to a policy of fully stabilizing inflation only (as opposed
to using the utility-based welfare criterion from above). Panel C of
Table 2 shows that the latter is very close to the Ramsey policy. The
welfare difference between the two policies is small--less than 0.001
percentage points of steady-state consumption. The effects of
inventories can be seen in the slightly higher volatility of consumption
and labor under the full inflation stabilization policy. Inventory
investment allows the planner to smooth consumption more compared to the
standard model, and the mechanism is a change in prices. Although price
stability is feasible, the planner chooses to incur an adjustment cost
to reduce the volatility of consumption and labor.
Optimal Policy with a Simple and Implementable Rule
Ramsey-optimal policy provides a convenient benchmark for welfare
analysis in economic models. However, from the point of view of a
policymaker, pursuing a Ramsey policy may be difficult to communicate to
the public. It may also not be operational in the sense that the
instruments used to implement the Ramsey policy may not be available to
the policymaker. For instance, in a market economy the government cannot
simply choose allocations as a Ramsey plan might imply. The literature
has therefore focused on finding simple and implementable rules that
come close to the welfare outcomes implied by Ramsey policies (see
Schmitt-Grohe and Uribe 2007).
Therefore, we investigate the implications for optimal policy
conditional on the simple rule (20). Panel A of Table 3 shows the
constrained-optimal interest rate rules for the model without
inventories with all shocks considered simultaneously. The rule that
delivers the highest welfare is a contemporaneous rule, with a smoothing
parameter [rho] = 1 and reaction coefficients on inflation [[psi].sub.1]
and output [[psi].sub.2] of 3 and 0, respectively. (15) This is broadly
consistent with the results of Schmitt-Grohe and Uribe (2007), where the
constrained-optimal interest rate rule also features interest smoothing
and a muted response to output. Without interest rate smoothing the
welfare cost of implementing this policy increases, which is exclusively
due to a higher volatility of inflation.
Table 3 Optimal Policy with a Simple Rule
[rho] [[psi].sub.1] [[psi].sub.2]
Panel A: Model without Inventories
Ramsey Policy
Optimized Rules
Contemporaneous (i = 0)
Smoothing 1.00 3.00 0.00
No Smoothing 0.00 3.00 0.00
Backward (i = -1)
Smoothing 1.00 3.00 0.00
No Smoothing 0.00 3.00 0.00
Forward (i = 1)
Smoothing 1.00 3.00 0.00
No Smoothing 0.00 3.00 0.00
Panel B: Model with Inventories
Ramsey Policy
Optimized Rules
Contemporaneous (i = 0)
Smoothing 1.00 3.00 0.00
No Smoothing 0.00 3.00 0.00
Backward (i = -1)
Smoothing 1.00 3.00 0.00
No Smoothing 0.00 3.00 0.00
Forward (i = 1)
Smoothing 1.00 3.00 0.00
No Smoothing 0.00 3.00 0.00
Welfare Cost [[sigma].sub.[pi]] [[sigma].sub.y]
([THETA])
Panel A: Model without Inventories
Ramsey Policy -0.0521 0.00 2.89
Optimized Rules
Contemporaneous
(i = 0)
Smoothing -0.0520 0.04 2.89
No Smoothing -0.0499 0.28 2.89
Backward (i =
-1)
Smoothing -0.0520 0.05 2.89
No Smoothing -0.0501 0.27 2.90
Forward (i = 1)
Smoothing -0.0518 0.08 2.90
No Smoothing -0.0496 0.30 2.90
Panel B: Model with Inventories
Ramsey Policy -0.0529 0.04 2.86
Optimized Rules
Contemporaneous
(i = 0)
Smoothing -0.0528 0.01 2.87
No Smoothing -0.0518 0.20 2.87
Backward (i =
-1)
Smoothing -0.0528 0.02 2.87
No Smoothing -0.0518 0.19 2.87
Forward (i = 1)
Smoothing -0.0528 0.02 2.87
No Smoothing -0.0517 0.20 2.87
On the other hand, the difference between the constrained-optimal
contemporaneous rule and the Ramsey policy is small--less than 0.001
percentage points. This confirms the general consensus in the literature
that simple rules can come extremely close to Ramsey-optimal policies in
welfare terms. The characteristics of constrained-optimal
backward-looking and forward-looking rules are similar to the
contemporaneous rule, i.e., they also feature full interest smoothing
and no output response. The welfare difference between the
constrained-optimal contemporaneous rule and the other two rules are
also small.
Turning to the model with inventories, we report the results for
the constrained-optimal rules in Panel B of Table 3. All rules with
interest smoothing deliver virtually identical results but strictly
dominate any rule without smoothing. As before, the coefficient on
output is zero, while the policymakers implement a strong inflation
response. The main difference to the Ramsey outcome is that inflation is
slightly less volatile, while output is more volatile. This again
confirms the findings in other articles that a policy rule with a fully
inertial interest rate and a hawkish inflation response delivers almost
Ramsey-optimal outcomes.
Sensitivity Analysis
We now investigate the robustness of our optimal monetary policy
results to alternative parameter values. The results of alternative
calibrations are reported in Table 4, where we only document results for
the rule that comes closest to the Ramsey benchmark. In the robustness
analysis, we change one parameter at a time while holding all other
parameters at their benchmark values. The overall impression is that in
all alternative calibrations the optimal simple rule comes close to the
Ramsey policy, and that the relative welfare rankings for the individual
rules established in the benchmark calibration are unaffected.
Specifically, inertial rules tend to dominate rules with a lower degree
of smoothing.
Table 4 Optimal Policy for the Model with Inventories: Alternative
Calibration
[rho] [[psi].sub.1] [[psi].sub.2]
Panel A: [micro] = 0.8
Ramsey Policy
Contemporaneous (i = 0) 1.0 3.0 0.0
Panel B: [delta] = 0.05
Ramsey Policy
Contemporaneous (i = 0) 1.0 3.0 0.0
Panel C: [eta] = 5
Ramsey Policy
Contemporaneous (i = 0) 1.0 3.0 0.0
Panel D: [theta] = 21
Ramsey Policy
Contemporaneous (i = 0) 1.0 3.0 0.0
Welfare [[sigma].sub.[pi]] [[sigma].sub.y]
Cost
([THETA])
Panel A: [micro] = 0.8
Ramsey Policy -0.0508 0.05 2.88
Contemporaneous -0.0507 0.01 2.89
(i = 0)
Panel B: [delta] = 0.05
Ramsey Policy -0.0557 0.09 2.85
Contemporaneous -0.0553 0.02 2.86
(i = 0)
Panel C: [eta] = 5
Ramsey Policy -0.0193 0.03 1.79
Contemporaneous -0.0190 0.01 1.80
(i = 0)
Panel D: [theta] = 21
Ramsey Policy -0.0539 0.05 2.85
Contemporaneous -0.0537 0.02 2.86
(i = 0)
We first look at the implications of alternative values for the two
parameters related to inventories: the elasticity of demand with respect
to the stock of goods available for sale, [mu], and the depreciation
rate of the inventory stock, [delta]. As in Jung and Yun (2005), we
consider the alternative value [micro] = 0.8. Since sales now respond
more elastically to the stock of goods available for sale, the inventory
channel becomes more valuable as a consumption-smoothing device and
inflation becomes more volatile under a Ramsey policy. The best simple
rule has contemporaneous timing and comes very close to the Ramsey
policy in terms of welfare. The optimal rule is inertial and strongly
reacts to inflation only. The volatility of inflation is lower than
under the Ramsey policy and closer to that of the optimally simple rule
with the benchmark calibration. This suggests that the response
coefficients of the optimal rule are insensitive to changes in
elasticity parameter [mu], and that the Ramsey planner can exploit the
changes in the transmission mechanism in a way that the simple rule
misses. The quantitative differences are small, however.
In the next experiment, we increase the depreciation rate of the
inventory stock to [delta] = 0.05. It is at this value that Lubik and
Teo (2009) find that the inclusion of inventories has a marked effect on
inflation dynamics in the New Keynesian Phillips curve. Panel B of Table
4 shows that the preferred rule is again contemporaneous, but the
differences between the alternatives are very small. Interestingly,
Ramsey policy leads to a volatility of inflation that is almost an order
of magnitude higher than in the benchmark case, which is consistent with
the findings in Lubik and Teo (2009).
The benchmark calibration imposed a very elastic labor supply with
[eta] = 1. The results of making the labor supply much more inelastic by
setting [eta] = 5 are depicted in Panel C of the table. For this value,
the differences to the benchmark are most pronounced. In particular, the
volatility of output declines substantially across the board, which is
explained by the difficulty with which firms change their labor input.
The best simple rule is contemporaneous, but the differences to the
other rules are vanishingly small. Optimal policy again puts strong
weight on inflation, with the optimal rule being inertial. Another
difference to the benchmark parameterization is that the welfare cost of
no interest smoothing is also much bigger for [eta] = 5. (16) Finally,
we also report results for calibration with a lower steady-state markup
of 5 percent, which corresponds to a value of [theta] = 21. The
qualitative and quantitative results are mostly similar to the benchmark
results.
In summary, the results from the benchmark calibration are broadly
robust. Under a Ramsey policy full inflation stabilization is not
optimal, while the best optimal simple rule exhibits inertial behavior
on interest smoothing and a strong inflation response. The welfare
differences between alternative calibrations are very small, with the
exception of changes in the labor supply elasticity. A less elastic
labor supply reduces the importance of the inventory channel to smooth
consumption by making it more difficult to adjust employment and output
in the face of exogenous shocks.
4. CONCLUSION
We introduce inventories into an otherwise standard New Keynesian
model that is commonly used for monetary policy analysis. Inventories
are motivated as a way to generate sales for firms. This changes the
transmission mechanism of the model, which has implications for the
conduct of optimal monetary policy. We emphasize two main findings in
the article. First, we show that full inflation stabilization is no
longer the Ramsey-optimal policy in the simple New Keynesian model with
inventories. While the optimal planner still attempts to reduce
inflation volatility to zero since it is a deadweight loss for the
economy, the possibility of inventory investment opens up a tradeoff. In
our model, production no longer needs to be consumed immediately, but
can be put into inventory to satisfy future demand. An optimizing
policymaker therefore has an additional channel for welfare-improving
consumption smoothing, which comes at the cost of changing prices and
deviations from full inflation stabilization. Our second finding
confirms the general impression from the literature that simple and
implementable optimal rules come close to replicating Ramsey policies in
welfare terms.
This article contributes to a growing literature on inventories
within the broader New Keynesian framework. However, evidence on the
usefulness of including inventories to improve the model's business
cycle transmission mechanism is mixed, as we have shown above. Future
research may therefore delve deeper into the empirical performance of
the New Keynesian inventory model, in particular on how modeling
inventories affect inflation dynamics. Jung and Yun (2005) and Lubik and
Teo (2009) proceed along these lines. A second issue concerns the way
inventories are introduced into the model. An alternative to our setup
is to add inventories to the production structure so that instead of
smoothing sales, firms can smooth output. Finally, it would be
interesting to estimate both model specifications with structural
methods and compare their overall fit more formally.
REFERENCES
Bils, Mark, and James A. Kahn. 2000. "What Inventory Behavior
Tells Us About Business Cycles." American Economic Review 90
(June): 458-81.
Blinder, Alan S., and Louis J. Maccini. 1991. "Taking Stock: A
Critical Assessment of Recent Research on Inventories." Journal of
Economic Perspectives 5 (Winter): 73-96.
Boileau, Martin, and Marc-Andre Letendre. 2008. "Inventories,
Sticky Prices, and the Persistence of Output and Inflation."
Manuscript.
Chang, Yongsung, Andreas Hornstein, and Pierre-Daniel Sarte. 2009.
"On the Employment Effects of Productivity Shocks: The Role of
Inventories, Demand Elasticity and Sticky Prices." Journal of
Monetary Economics 56 (April): 328-43.
Christiano, Lawrence J. 1988. "Why Does Inventory Investment
Fluctuate So Much?" Journal of Monetary Economics 21(2/3): 247-80.
Fisher, Jonas D. M., and Andreas Hornstein. 2000. "(S,s)
Inventory Policies in General Equilibrium." Review of Economic
Studies 67 (January): 117-45.
Ireland, Peter N. 2004. "Technology Shocks in the New
Keynesian Model." Review of Economics and Statistics 86(4): 923-36.
Jung, YongSeung, and Tack Yun. 2005. "Monetary Policy Shocks,
Inventory Dynamics and Price-setting Behavior." Manuscript.
Khan, Aubhik. 2003. "The Role of Inventories in the Business
Cycle." Federal Reserve Bank of Philadelphia Business Review Q3:
38-45.
Khan, Aubhik, and Julia K. Thomas. 2007. "Inventories and the
Business Cycle: An Equilibrium Analysis of (S,s) Policies."
American Economic Review 97: 1,165-88.
Khan, Aubhik, Robert G. King, and Alexander L. Wolman. 2003.
"Optimal Monetary Policy." Review of Economic Studies 70
(October): 825-60.
Krause, Michael U., and Thomas A. Lubik. 2007. "The
(Ir)relevance of Real Wage Rigidity in the New Keynesian Model with
Search Frictions." Journal of Monetary Economics 54 (April):
706-27.
Kryvtsov, Oleksiy, and Virgiliu Midrigan. 2009. "Inventories
and Real Rigidities in New Keynesian Business Cycle Models."
Manuscript.
Kydland, Finn E., and Edward C. Prescott. 1982. "Time to Build
and Aggregate Fluctuations." Econometrica 50 (November): 1,345-70.
Levin, Andrew T., and David Lopez-Salido. 2004. "Optimal
Monetary Policy with Endogenous Capital Accumulation." Manuscript.
Lucas, Robert. 1987. Models of Business Cycles. Yrjo Johansson
Lectures Series. London: Blackwell.
Lubik, Thomas A., and Wing Leong Teo. 2009. "Inventories,
Inflation Dynamics and the New Keynesian Phillips Curve."
Manuscript.
Maccini, Louis J., and Adrian Pagan. 2008. "Inventories,
Fluctuations and Business Cycles." Manuscript.
Maccini, Louis J., Bartholomew Moore, and Huntley Schaller. 2004.
"The Interest Rate, Learning, and Inventory Investment."
American Economic Review 94 (December): 1,303-27.
Ramey, Valerie A., and Kenneth D. West. 1999.
"Inventories." In Handbook of Macroeconomics, Volume 1, edited
by John B. Taylor and Michael Woodford. pp. 863-923.
Schmitt-Grohe, Stephanie, and Martin Uribe. 2007. "Optimal,
Simple and Implementable Monetary and Fiscal Rules." Journal of
Monetary Economics 54: 1,702-25.
Wen, Yi. 2005. "Understanding the Inventory Cycle."
Journal of Monetary Economics 52 (November): 1,533-55.
West, Kenneth D. 1986. "A Variance Bounds Test of the Linear
Quadratic Inventory Model." Journal of Political Economy 94
(April): 374-401.
We are grateful to Andreas Hornstein, Pierre Sarte, Alex Wolman,
and Nadezhda Malysheva, whose comments greatly improved the paper. Lubik
is a senior economist at the Federal Reserve Bank of Richmond. Teo is an
assistant professor at National Taiwan University. Lubik wishes to thank
the Department of Economics at the University of Adelaide, where parts
of this research were conducted, for their hospitality. The views
expressed in this paper are those of the authors and should not
necessarily be interpreted as those of the Federal Reserve Bank of
Richmond or the Federal Reserve System. E-mails:
thomas.lubik@rich.frb.org; wlteo@nlu.edu.tw.
(1) See Ramey and West (1999) and Khan (2003) for extensive surveys
of the literature.
(2) A more recent example of applying structural econometric
techniques to partial equilibrium inventory models is Maccini and Pagan
(2008).
(3) Incidentally, Maccini, Moore, and Schaller (2004) find that an
inventory model with regime switches in interest rates is quite
successful in explaining inventory behavior despite much previous
empirical evidence to the contrary. The key to this result is the
exogenous shift in interest rate regimes, which lines up with breaks in
U.S. monetary policy.
(4) This approach is consistent with a stockout avoidance motive.
Wen (2005) shows that it explains the fluctuations of inventories at
different cyclical frequencies better than alternative theories.
(5) Consumption can be thought of as a Dixit-Stiglitz aggregate, as
is typical in New Keynesian models. We abstract from this here for ease
of exposition.
(6) This value is also consistent with an average price duration of
about four quarters in the Calvo model of staggered price adjustment.
(7) Chang, Hornstein. and Sarte (2009) also emphasize that in the
presence of nominal rigidities labor hours can increase in response to a
persistent technology shock when firms hold inventories.
(8) This aspect is discussed more extensively in Boileau and
Lelendre (2008) and Lubik and Teo (2009).
(9) This is further and more formally empirically investigated in
Lubik and Teo (2009), who suggest that the inventory channel does not
contribute much to explain observed inflation behavior.
(10) See Khan. King, and Wolman (2003). Levin et al. (2006), and
Schmitt-Grohe and Uribe (2007) for wide-ranging and deluded discussions
of this concept in New Keynesian models.
(11) We assume that the policymaker chooses the same steady-state
inflation rate for all monetary policies that we consider. The steady
state of all variables will thus be the same for all policies.
(12) If R is normally distributed, 2[sigma].sub.R] < R implies
that there is a 95 percent chance that R will not hit the zero bound.
(13) In a framework with Calvo price setting, the deadweight loss
comes in the form of relative price distortions across firms, which lead
to the misallocation of resources.
(14) This is consistent with the simulation results reported in
Schmitt-Grohe and Uribe (2007) in a model with capital. They also find
that full inflation stabilization is no longer optimal since investment
in capital provides a mechanism for smoothing consumption, just as
inventory holdings do in our model.
(15) The reader may recall that we restricted the policy
coefficients to lie within the interval [0, 3].
(16) The welfare cost of no interest smoothing is 0.0088 for [eta]
= 5, while it is 0.0021 for the benchmark parameterization.
