The effects of monetary policy on input inventories

Tiantian Dai; Xiangbo Liu; Wei Sun

doi:10.1515/bejm-2017-0251

Article

The effects of monetary policy on input inventories

Tiantian Dai , Xiangbo Liu and Wei Sun

Published/Copyright: April 8, 2019

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From the journal The B.E. Journal of Macroeconomics Volume 20 Issue 1

Abstract

This paper explores both the long-run and short-run effects of monetary policy on input inventories in a search model with monetary propagation and two-stage production. Inventories arise endogenously due to search frictions. In the long run, we analytically show that an increase in the money growth rate has hump-shaped real effects on steady-state input inventory investment, input inventory-to-sales ratio as well as sales. These effects are driven by both the extensive and intensive margins in the finished goods market. We then calibrate the model to the US data to study the short-run effects of monetary policy. We first show that our model can reproduce the stylized facts of input inventories quite well and then find that input inventories amplify aggregate fluctuations over business cycles.

Keywords: business cycles; input inventories; monetary policy; monetary search

JEL Classification: E32; E52; G31; D83

Acknowledgments

Xiangbo Liu acknowledges the research support by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant No. 19XNI002). We are grateful to Allen Head and Thorsten Koeppl for their enormous guidance and encouragement. We also would like to thank Jonathan Chiu, Oleksiy Kryvtsov, Lealand Morin, Shouyong Shi, Hongfei Sun, Wen Yi and seminar participants at the Western Economic Association International – Annual Meeting 2015, the Canadian Economics Association – Annual Meeting 2011, 2013, and the 2013 China Meeting of Econometric Society for comments and discussions.

A Appendix

In this section, we prove that the model economy exists at least one steady state, which satisfies (λf;Ωa)>0. Denote the steady state values with an asterisk, which can be rewritten by the dynamic system:

(46)zf(sf∗)ξ=γ−ββωf⋆U′(c∗)−wf⋆,

(47)Ωi∗=βφ′(qi∗),

(48)(1−β(1−δn))βk(v∗)={σzfBf(sf∗)ξωf∗−σφf+[(1−zfBf(sf∗)ξ)σ(1−δi)β−σ]φ′(qi∗)},

(49)Φf′(sf∗)=zf(sf∗)ξ−1[U′(c∗)−ωf∗]qf∗,

(50)v∗μ(v∗)=δnqf∗,

(51)qi∗={1−(1−δi)[1−zfBf(sf∗)ξ]}qf∗,

(52)c∗=apfBfzf(sf∗)ξqf∗.

As demonstrated in Section 2, the steady state system can be reduced to two equations which are repeated here for future use:

zf[sf(ωf∗,qf∗)]ξ=γ−ββωf∗U′(c(ωf∗,qf∗))−wf∗,(1−β(1−δn))k(v(qf∗))+βσφf=β{σzfBf(sf(ωf∗,qf∗))ξωf∗+[(1−zfBf(sf(ωf∗,qf∗))ξ)σ(1−δi)β−σ]φ′(qi(ωf∗,qf∗))}.

Similarly to Shi (1998), above two equations give a relationship between ω^f and q^f, denote qf∗=qf1(ωf∗) and qf∗=qf2(ωf∗). The steady state value ωf∗ is a solution to Qf1(ωf∗)=Qf2(ωf∗). To ensure λ^f > 0, the solution must satisfy U′(c∗)≥ωf∗+Δ, where Δ > 0 is an arbitrarily small number. That is, we require qf∗≤qf(ωf∗,Δ),^[18] where qf(ωf,Δ) is defined by:

(53)U′(c(ωf,qf(ωf,Δ)))=ωf+Δ.

Using Lemma 3.2 as explained in Shi (1998), we can prove that the function Qf(ωf,λ) is well defined and has the following properties for sufficiently small Δ > 0: Qωff(ωf,Δ)<0, Qf(∞,Δ)=0, and limΔ→0Qf(0,Δ)=∞. The function qf1(ωf) satisfies qf1′(ωf)<0,qf1(0)=∞ and qf1(∞)=0. Furthermore, the two curves qf1(ωf) and Qf(ωf,Δ) have a unique intersection at a level denoted ω1f(Δ) which satisfies limΔ→0ωf1(Δ)=0.^[19]

In order to prove the uniqueness, we also need to know the properties of qf2. Although the properties of qf2 are the same as what is described in Lemma 3.3 in Shi (1998), the proof is not the same due to different function forms.

We are going to prove that qf2 has the following properties: qf2(0)=0, qf2(∞)=0, and qf2′(ωf)<0 for sufficiently large ω^f. The two curves qf2(ωf) and Q(ωf,Δ) have a unique intersection at a level denoted ωf2(Δ) which approaches infinity when Δ approaches zero.

First, let us show qf2(0)=0 by rearranging equation (36):

(54)φ′(qi(ωf∗,qf∗))=β[σzfBf(sf(ωf∗,qf∗))ξωf∗−σφf]−(1−β(1−δn))k(v(qf∗))β[1−(1−zfBf(sf(ωf∗,qf∗))ξ)σ(1−δi)β]σ≤σzfBf(sf(ωf∗,qf∗))ξωf∗1−(1−zfBf(sf(ωf∗,qf∗))ξ)σ(1−δi)β]σ≤σzfBfωf∗{[1−σ(1−δi)β]/(sf(ωf∗,qf∗))ξ+zfBfσ(1−δi)β}σ

The right-hand side of (54) approaches zero as ω^f approaches zero, becuase limωf→0numerator=0 and limωf→0denominator=zfBfσ(1−δi)β. Since φ′(⋅)>0, equation (54) implies limωf→0qi(ωf∗,qf∗)=0. Finally, limωf→0qf2(ω)=0 is implied by the steady state equation qi∗={1−(1−δi)[1−zfBf(sf∗)ξ]}qf∗.

Second, let us prove that the two curves qf2(ωf) and Q(ωf,Δ) have a unique intersection. By plugging the equation of c^∗ into the fourth equation of the steady state system, we can get a useful equation:

(55)Φf′(sf∗)sf∗=[U′(c∗)−ωf∗]c∗apfBf.

As the definition of Qf(ωf,Δ), set ωf=u′(c)−Δ. Then equation (55) implies that s^f is a function of (c, Δ): Φf′(sf∗)sf∗=Δc∗/(apfBf). Denote the solution for s^f as sf(c,Δ). Because Φf′(0)=0, Φf′(⋅)>0 and Φf′′(⋅)>0, we can get sf(c,0)=0, sf(∞,Δ)=∞, sf(0,Δ)=0 and scf(c,Δ)>0. By rearranging equation (55), we can prove that c/(s(c,Δ))ξ is an increasing function of c.

Rearranging the steady state equation of c^∗, it is easy to see that qf(c,Δ) is also an increasing function of c: qf(c,Δ)=c/apfBfzf[sf(c,Δ)]ξ. Similarly, qⁱ can be rewritten as a function of (c, Δ): qi(c,Δ)={1−(1−δi)[1−zfBf(sf(c,Δ))ξ]}qf(c,Δ). Since both sf(c,Δ) and qf(c,Δ) are increasing in c, qi(c,Δ) is an increasing function of c.

Now we are ready to prove that the two curves qf2(ωf) and Q(ωf,Δ) have a unique intersection. Rewrite equation (36) in terms of (c,Δ):

(56)LHS(36)=(1−β(1−δn))k(v(qf(c,Δ)))+βσφf

(57)RHS(36)=β{σzfBf(sf(c,Δ))ξωf∗+[(1−zfBf(sf(c,Δ))ξ)σ(1−δi)β−σ]φ′(qi(c,Δ))}

The left-hand side of (36) is an increasing function of c, and the right-hand side of (36) is a decreasing function of c, because sf(c,Δ), qf(c,Δ) and qi(c,Δ) are increasing in c, k′(v)>0 and φ′(qi)>0. Moreover, since qf(0,Δ)=0, k(v(qf(0,Δ)))=0, qf(∞,Δ)=∞ and k(v(qf(∞,Δ)))=∞ it is easy to see that lim_c→0 LHS(36) = 0 and limc→∞LHS(36)=∞. Similarly, the right-hand side has the following properties. lim_c→0 RHS(36) = ∞, because qi(0,Δ)=0, sf(0,Δ)=0 and limc→0cu′(c)=∞. And limc→∞RHS(36)=−∞. because qi(∞,Δ)=∞, sf(∞,Δ)=∞ and limc→∞cu′(c)=0.

Given these properties of (36), there is a unique solution for c to (36). Denote this solution by c(Δ), then ωf2(Δ)=u′(c(Δ))−Δ is unique. Thus there must be a unique intersection between the two curves qf2(ωf) and Q(ωf,Δ).

Third, we are going to prove that limΔ→0ωf2(Δ)=0, qf2′(ωf)<0 and qf2(∞)=0. For fixed c, limΔ→0LHS(36)=∞ and limΔ→0RHS(36)=−∞, because limΔ→0s(c,Δ)=0 and limΔ→0qf(c,Δ)=∞. Thus (36) is satisfied only when limΔ→0c(Δ)=0 and limΔ→0ωf2(Δ)=∞. Next, qf2′(ωf)<0 since qf2(c,Δ) is an increasing function of c and ωf2′(c(Δ))<0. This can be proved by plugging ωf2(c(Δ)) into qf2(ωf) and analyzing qf2(ωf2(c(Δ))).

Now, we are going to prove qf2(∞)=0. Because Qf(0,Δ) is a positive constant and qf2(0)=0 is proven, qf2(0)<Qf(0,Δ) and the curve qf2(wf) must cross the curve Qf(ωf,Δ) from below if the two have a unique intersection. Moreover, because Qf(∞,Δ)=0 is proven in Lemma 1 and 0≤qf2(ωf)<Qf(ωf,Δ) for ωf<ωf2(Δ), 0≤qf2(∞)<0 for ωf<ωf2(Δ). Then qf2(∞)=0 since qf2(ωf) is continuous and only has one intersection with Qf(ωf,Δ).

Finally, given the properties of equations (35) and (36) proven, there exists at least one steady state for the model.

B Appendix

B.1 Proof of proposition 1

Now we are going to prove that the long run effect of money growth on q_f is not monotonic. Since Equation (36) is independent of γ, while equation (35) will be shifted to the right as γ → β and to the left as γ → ∞. Since qf2(0)=0, qf2(∞)=0 and qf2′(ωf)<0 for sufficiently large ω^f, equation (36) is hump-shaped. Thus steady state q^f decreases with γ if γ is high, but increases with γ if it is low.

To prove the production of intermediate goods abiqi is nonmonotonic, we take derivative of that:

(58)∂i∂γ=∂i(qf(ωf))∂qf∂qf∂ωf∂ωf∂γ,

where ∂ωf/∂γ<0, and ∂qf/∂ωf first increases with γ then decreases with large value of γ. ∂i/∂qf>0 will be proved in the next proposition. Since n=qi with Leontief production function, employment also increases with γ if it is low, but decreases with γ if it is high.

Take derivative of the final sales (sfξabfqf):

(59)∂FS∂γ=zfξabfqfsfξ−1∂sf∂γ+zfabfsfξ∂qf∂γ.

Since ∂sf/∂γ>0, final sales are not monotonic in γ as q^f does.

B.2 Proof of proposition 2

Since the difference between steady state net input inventory investment and steady state inventory level is just a constant multiplier apfδi, we only prove the long run response of input inventory investment. Equation (37) implies that the steady state NIII is the difference between the quantity of goods per match and the final sales discounted at a proper rate, namely,

(60)NIII∗=apfδii=(1−δi)δiapfqf[1−zfBf(sf)ξ].

The derivative of i with respect to q^f can be derived from this equation:

(61)∂i∂qf=(1−δi)[1−zfBf(sf)ξ]−(1−δi)qfξzfBf(sf)ξ−1∂sf∂qf>0.

∂i/∂qf>0 because ∂sf/∂qf<0. Since i is a function of qf(ωf), the effects of money growth on input inventory investment can be studied by taking the derivative of apfδii(qf(ωf)) with respect to γ:

(62)apfδi∂i(qf(ωf))∂γ=∂i∂qf∂qf∂ωf∂ωf∂γ>0,if γ if low;<0,if γ if high.

We can conclude that ∂NIII∗/∂γ>0 if γ is low and ∂NIII∗/∂γ<0 if γ is high. This is because that ∂qf/∂ωf<0 when γ is low, ∂qf/∂ωf>0 when γ is high as implied by Proposition 1. Moreover, ∂i/∂qf>0 and ∂ωf/∂γ<0.

Since both final sales and NIII are nonmonotonic in γ, it is clear that GDP is also increases with γ if it is low, and decreases with γ if it is high.

B.3 Proof of proposition 4

By rearranging equation (39), we can get a expression for steady state inventory-to-sales ratio:

(63)IISR=apfi∗apfBfzf(sf∗)ξqf∗=(1−δi)[apf(qf∗)apfBfzf(sf∗)ξqf∗−1],=(1−δi)[1Bfzf(sf∗)ξ−1]

The effects of money growth on the inventory-to-sales ratio can be studied by taking the derivative with respect to γ:

(64)∂IISR∗∂γ=−(1−δi)Bfzfξsfξ−1[Bfzf(sf∗)ξ]2∂sf∂qf∂qf∂ωf∂ωf∂γ,>0,if γ if low;<0,if γ if high.

Evaluating at ω=ω∗(γ), the inventory-to-sales ratio has a hump-shaped long run response to the money growth rate across steady states, because ∂qf/∂ωf<0 when γ is low; and ∂qf/∂ωf>0 when γ is high as implied by Proposition 1. Moreover, ∂ωf/∂γ<0 and ∂sf/∂qf<0.

C Appendix C

The household’s new decision problem is altered as follows. The representative household taking the sequence {q^ti,m^ti,q^tf,m^tf,W^t}t≥0 and initial conditions {M0,i0,n0} as given, chooses {Ct,at,sti,stf,Δt+1,Mt+1,it+1,vt,nt+1}t≥0 to maximize its expected lifetime utility:

(65)max∑t=0∞βtE−1[U(ct)−gstiapiφ(q^ti)−apn^tφf−abiΦi(sti)−abfΦf(stf)−apfK(vt)]

subject to the following constraints for all t ≥ 0:

(66)ct≤(1−FI)stfgbtfabfq^tf,

(67)(1−Δt+1)Mt+1abf≥m^t+1f,∀ Fbt+1∗

(68)qtf=Atatαnt1−α,

(69)at≤it+1apfstigbtiabiq^ti,∀Fpt∗

(70)qtf≥q^tf,∀Fpt∗

(71)Δt+1Mt+1abi≥m^t+1i,∀Ibt∗

(72)Mt+1≤Mt+τt−stigbtiabim^ti+gstiapim^ti+apfn^tPt^W^t−stfgbtfabfm^tf+gstfapfm^tf−Pt^apfW^tnt,

(73)0≤apf[(1−δn)nt+vtμt−nt+1],

(74)apfit+1≤(1−δi)[apfit+stigbtiabiq^ti−gstfapfat].

Functions U(⋅), Φ^f(⋅) and K(⋅) have the same properties as in the benchmark model. Φi(sti) is a buyer’s disutility of searching in the intermediate goods market. The function Φⁱ satisfies Φi′>0 and Φi′′>0 for si>0, and Φi(0)=Φi′(0)=0. Ibt∗ (with measure stigbtiabi) is the set of matched intermediate goods buyers in period t. Moreover, as discussed in Shi (1998)^[20]. We modify the model to incorporate fixed investment which is a constant fraction of aggregate sales.

Denote the multipliers of money constraint (67) and (71) by Λtf and Λti, respectively. All of the multipliers of the rest conditions are the same as in the benchmark model.

The terms of trade in the intermediate goods market are determined by Nash bargaining. As in the benchmark model, we assume the intermediate goods buyers and sellers have the same bargaining powers, and the terms of trade can be pinned down by the following two equations:

(75)Pti(Ω¯Mt+Λ¯ti)=Ω¯at+(1−δi)Ω¯it,

(76)φ′(qti)=PtiΩMt.

Then, we can write the dynamic system

(77)nt+1=(1−δn)nt+vtμ(vt),

(78)apfit+1=(1−δi)[apfit+stigbtiabiqti−gstfapfat],

(79)it=at−stigbtiabiqti/apf,

(80)E[1−Δt+1Δt+1]=E[ωt+1fqt+1fabfφ′(qt+1i)qt+1iabi],

(81)0=βE[{ωt+1f+zf(st+1f)ξ[(1−FI)U′(ct+1−ωt+1f)]}]−E[(1−Δt+1)γtωtfqtf(1−Δt)qt+1f],

(82)Ωit=βE{(1−δi)Ωit+1+gst+1fapf[φ′(qt+1i)+λt+1i−(1−δi)Ωit+1]},

(83)k(vt)=βE[σgst+1fAt+1at+1αnt+1−αωt+1f(1−α)−σφf+(1−δn)Ωnt+1],

(84)Φi′(sti)=gbti[gstfapfλti−(1−gstfapf)φ′(qti)+(1−gstfapf)(1−δi)Ωit]qti,

(85)Φf′(stf)=zf(stf)ξ−1[(1−FI)U′(ct)−wtf]qtf,

(86)ct=(1−FI)apfBfzf(stf)ξqtf,

(87)Atα(ntat)1−αωtf=apf[φ′(qti)+λti]+(1−apf)(1−δi)Ωit.

D Appendix

Now, we are going to describe the calibration procedures. Derive the steady state equations from the dynamic system:

(88)v∗μ(v∗)=δnn∗,

(89)δiapfi∗=(1−δi)[ziabi(si∗)ξqi∗−zfBf(sf∗)ξapfa∗],

(90)a∗=i∗+ziabi(si∗)ξqi∗/apf,

(91)zf(sf∗)ξ=γ−ββωf⋆(1−FI)U′(c∗)−wf⋆,

(92)Ωi∗=βzfBf(sf∗)ξapf[λi∗+φ′(qi∗)]1−β(1−δi)(1−zfBf(sf∗)ξapf)

(93)k(v∗)=β[(1−α)σzfBf(sf∗)ξA∗(a∗)α(n∗)−αωf∗−σφf]+β(1−δn)k(v∗),

(94)Φi′(si∗)=gbi∗{[gsf∗apfλi∗−(1−gsf∗apf)[φ′(qi∗)−(1−δi)Ωi∗]}qi∗,

(95)Φf′(sf∗)=zf(sf∗)ξ−1[(1−FI)U′(c∗)−ωf∗]qf∗,

(96)A∗α(n∗a∗)1−αωf∗=apf[φ′(qi∗)+λi∗]+(1−apf)(1−δi)Ωi∗,

(97)c∗=(1−FI)apfBfzf(sf∗)ξqf∗.

The average labor participation rate (LP = 0.6445), the average unemployment rate (UR = 0.061) and the assumption api=apf can be used to pin down parameters (u,api,apf):

(98)u=LP⋅UR=0.0393.

Since the households have measure one, the labor participation rate equals u+apf(1+n)+api. Using the assumption api=apf, the number of sellers in the intermediate goods market api is equate to its counterparts in the finished goods market apf, and the steady state vacancies can be calculated as the following:

(99)apf=(LP−u)/(2+n)=0.2017,

(100)api=LP−u−apf(1+n)=0.2017,

(101)v∗=[δn⋅nμ¯(apf/u)ϕ−1]1/ϕ.

The depreciation rate of input inventory can be pinned down by matching the average input inventory to output ratio and the average input inventory investment to output ratio, which are apii∗/GDP and apiδii∗/GDP, respectively in the model.

(102)δi=NIII/GDPINV/GDP=0.0038,

where GDP = NIII + FS.

By matching the average velocity of M2 money stock (vcf∗=1.836), we can get zf(sf∗)ξ=2.4947 which can be used later:

(103)vcf∗=pfcf∗/mf=cf∗/(abfqf∗)=(1−FI)zf(sf∗)ξ

Similarly, vci∗=zi(si∗)ξ. We need two more targets to pin down B^f: the average input inventory to final sales ratio (IISR) and the intermediate inputs to final sales ratio (IIPS). In this model:

(104)IISR=apii∗/cf∗,

(105)IIPS=zfBf(sf∗)ξapfa∗BfvcfapfqfPiPf,

(106)⇒zfBf(sf∗)ξapfa∗Bfvcfapfqf∗=IPS(1+markup).

By plugging equation (90) into equation (89), We can get i∗/a∗=(1−δi)[1−zfBf(sf∗)ξ]. We also can rewrite i∗/a∗ in terms of IISR and IIPS:

(107)i∗a∗=apii∗cfBfvcf∗apfqf∗zfBf(sf∗)ξapfa∗zfBf(sf∗)ξ,=IISR⋅Bfvcf(1−FI)IPS(1+markup).

After equalizing the two equations of i∗/a∗, B^f can be calculated in terms of IISR, IIPS, vcf∗ and the markup:

(108)Bf=(1−δi){(1−δi)+IISR/[IIPS(1+markup)]}vcf∗/(1−FI),=0.1947.

Next, equations (91)–(93) are plugged into equation (96) to get rid of ωf,λi and Ω_i; we can then pin down the parameter α. Rearranging equations (88)–(97), we can get:

(109)ωf∗=βvcf∗γ∗−β+βvcf∗/(1−FI)U′(c∗),≡E⋅U′(c∗),λi=γ∗−ββzi(si∗)ξωf∗1+markup,=γ∗−ββvci∗(1+markup)E⋅U′(c∗),≡F⋅U′(c∗),

where,

(110)ωf∗ωi∗=Pf∗ΩM∗Pi∗ΩM∗⇒ωi∗=ωf∗1+markup,

and,

(111)Ωi∗=βBfvcf∗apf/(1−FI)1−β(1−Bfvcf∗apf/(1−FI))(1−δi)[ωf∗1+markup+λi∗],=βBfvcf∗apf/(1−FI)1−β(1−Bfvcf∗apf/(1−FI))(1−δi)[E1+markup+F]U′(c∗),≡G⋅U′(c∗).

Then α can be calculated by plugging the above equations into equation (96):

(112)α=apf[E/(1+markup)+F]+(1−apf)(1−δi)GEa∗q∗,≡Ha∗qf∗,=0.4156,

where a∗/qf∗=0.6822 can be caluculated by rearranging i∗/a∗=(1−δi)[1−zfBf(sf∗)ξ]:

(113)apfi∗=(1−δi)[1−zfBf(sf∗)ξ]apfa∗,⇒apfi∗Bfvcf∗apf(qf∗)=(1−δi)[1−zfBf(sf∗)ξ]apfa∗Bfvcf∗apfqf∗,⇒IISR=(1−δi)1Bfvcf∗a∗qf∗−(1−δi)zfBf(sf∗)ξapfa∗Bfvcf∗apfqf∗,⇒a∗/qf∗=[IISR+(1−δi)IPS(1+markup)]Bfvcf∗/(1−δi).

Since α is known, we can calculate a∗=0.5198 by rearranging the production function qf∗=A∗a∗αn∗(1−α):

(114)qf∗a∗=A∗a∗(α−1)n∗(1−α),⇒a∗=[qf∗a∗A∗n∗(1−α)]1/(α−1).

Then qf∗,cf∗,i∗ and abf can be calculated:

qf∗=A∗(a∗)α(nα)1−α=0.7619,cf∗=(1−FI)apfBfvcf∗qf∗=0.0546,i∗=i∗/a∗⋅a∗=0.2662,abf=Bfapf=0.0393.

Now, we can calculate ωf,λi and Ω_i using the value of cf∗:

ωf∗≡E⋅U′(c∗)=7.4215,λi≡F⋅U′(c∗)=0.4760,Ωi∗≡G⋅U′(c∗)=4.4572.

Using the seventh target, which is that the shopping time of the population is 11.17% of the working time and the working time is 30% of agents discretionary time, we can calculate the buyer’s search intensity in the finished goods market sf∗. Once sf∗ is known, z^f and z1f can be determined as follows:

sf∗=0.1117∗0.3(apf(1+n)+api)/abf=0.5162,zf=zf(sf)ξ/((1−FI)(sf∗)ξ),z1f=zf∗(Bf)1−ξ=3.0524.

Since we assume the intermediate goods market and the finished goods market are symmetric, abi,si∗, and zii can be determined in a similar way:

abi=Biapi=0.0393,si∗=0.1117∗0.3(apf(1+n)+api)/abi=1.7207,zi=zi(si)ξ/(si)ξ=vci/(si)ξ,z1i=zi(Bi)1−ξ=0.0934.

Now the quantity of intermediate goods per trade (qⁱ), the constant in the disutility function of producing intermediate goods (b) and the constant in the disutility of posting vacancies (K₀) can be calculated by using equation (90), the function of φi(qi) and the last target (K=3.72∗10−4):

qi∗=a∗−i∗vci∗abi=6.5096,b=φi′(qi∗)/qi∗,=ωi∗/qi∗=0.6706,K0=K/v∗2=5.9501e−004.

Finally, the parameters (φi,φ0i,φf,φ0f) can be determined by using the steady state relations:

φi=b2(qi)2=14.2091,φf=[β(1−α)σBfvcf∗ωf∗qf∗/((1−FI)n∗)−[1−β(1−δn)]Ωn]/(βσ),=1.6025,φ0i={zi(si∗)ξ−1[Bfapfvcf∗λi∗/(1−FI)+(1−Bfapfvcf∗/(1−FI))[(1−δi∗)Ωi∗−bqi∗]]qi∗φi(1+1/ϵi)(si∗)1/ϵi}ϵi1+ϵi=0.1104,φ0f=(zf(sf∗)ξ−1((1−FI)(cf∗)−η−ωf∗)qf∗(φf(1+1/ϵf)(sf∗)1/ϵf))ϵf/(1+ϵf)=1.8043.

E Appendix

E.1 Data sources

E.1.1 Underlying detail – NIPA tables, the bureau of economic analysis

Table 1AU. Real Manufacturing and Trade Inventories, Seasonally Adjusted, End of Period [Chained 1996 dollars, 1967–1996, SIC] (Q)
Table 1AU2. Real Manufacturing and Trade Inventories, Seasonally Adjusted, End of Period [Chained 2005 dollars, 1967–1997, SIC] (Q)
Table 1BU. Real Manufacturing and Trade Inventories, Seasonally Adjusted, End of Period [Chained 2005 dollars, 1997 forward, NAICS] (Q)
Table 2AU. Real Manufacturing and Trade Sales, Seasonally Adjusted at Monthly Rate [Chained 1996 dollars, 1967–1996, SIC] (Q)
Table 2AUI. Implicit Price Deflators for Manufacturing and Trade Sales [Index base 1996, 1967–1996, SIC] (Q)
Table 2BU. Real Manufacturing and Trade Sales, Seasonally Adjusted at Monthly Rate [Chained 2005 dollars, 1997 forward, NAICS] (Q)
Table 2BUI. Implicit Price Deflators for Manufacturing and Trade Sales [Index base 2005, 1997 forward, NAICS] (Q)
Table 4AU1. Real Manufacturing Inventories, by Stage of Fabrication (Materials and supplies), Seasonally Adjusted, End of Period [Chained 2005 dollars, 1967–1997, SIC] (Q)
Table 4AU2. Real Manufacturing Inventories, by Stage of Fabrication, Seasonally Adjusted (Work-in-process), End of Period [Chained 2005 dollars, 1967–1997, SIC] (Q)
Table 4BU1. Real Manufacturing Inventories, by Stage of Fabrication (Materials and supplies), Seasonally Adjusted, End of Period [Chained 2005 dollars, 1997 forward, NAICS] (Q)
Table 4BU2. Real Manufacturing Inventories, by Stage of Fabrication (Work-in-process), Seasonally Adjusted, End of Period [Chained 2005 dollars, 1997 forward, NAICS] (Q)

E.1.2 Databases, the Federal Reserve Bank of St. Louis

M2 Money Stock, seasonally adjusted, end of period, quarterly
Velocity of M2 Money Stock, seasonally adjusted, end of period, quarterly

E.1.3 Databases, bureau of labor statistics

Civilian Labor Force (Seasonally Adjusted) – LNS11000000
Civilian Employment (Seasonally Adjusted) – LNS12000000
Civilian Unemployment (Seasonally Adjusted) – LNS13000000
Manufacturing Employment – CES3000000001

E.1.4 Manufacturing industry productivity database, the national bureau of economic research

emp: Total employment in 1000s, 1987 SIC version
matcost: Total cost of materials in $1,000,000, 1987 SIC version
pimat: Deflator for MATCOST 1987=1.000, 1987 SIC version

References

Berentsen, Aleksander, Guido Menzio, and Randall Wright. 2011. “Inflation and Unemployment in the Long Run.” American Economic Review 101 (1): 371–398.10.1257/aer.101.1.371Search in Google Scholar

Bils, Mark. 2004. “Studying Price Markups from Stockout Behavior.” Unpublished.Search in Google Scholar

Bils, Mark, and James A. Kahn. 2000. “What Inventory Behavior Tells us About Business Cycles.” American Economic Review 90 (3): 458–481.10.1257/aer.90.3.458Search in Google Scholar

Blanchard, Oliver, and Peter Diamond. 1989. “The Beveridge Curve.” Brookings Papers on Economic Activity 20 (1): 1–76.10.2307/2534495Search in Google Scholar

Blinder, Alan S., and Louis J. Maccini. 1991. “Taking Stock: A Critical Assessment of Recent Research on Inventories.” Journal of Economic Perspectives 5 (1): 73–96.10.1257/jep.5.1.73Search in Google Scholar

Chang, Yongsung, Andreas Hornstein, and Pierre Daniel Sarte. 2006. “Understanding How Employment Responds to Productivity Shocks in a Model with Inventories.” Unpublished.10.2139/ssrn.2186160Search in Google Scholar

Christiano, Lawrence J. 1988. “Why Does Inventory Investment Fluctuate so Much?” Journal of Monetary Economics 20: 247–280.10.1016/0304-3932(88)90032-3Search in Google Scholar

Coen-Pirani, Daniele. 2004. “Markups, Aggregation, and Inventory Adjustment.” American Economic Review 94 (5): 1328–1353.10.1257/0002828043052376Search in Google Scholar

Fisher, Jonas D. M., and Andreas Hornstein. 2000. “(S, s) Inventory Policies in General Equilibrium.” Review of Economic Studies 67 (1): 117–145.10.1111/1467-937X.00124Search in Google Scholar

Herrera, Ana, and Elena Pesavento. 2005. “The Decline in U.S. Output Volatility: Structural Changes and Inventory Investment.” Journal of Business & Economic Statistics 23 (4): 462–472.10.1198/073500104000000596Search in Google Scholar

Herrera, Ana, Irina Murtazashvili, and Elena Pesavento. 2008. “The Comovement in Inventories and in Sales: Higher and Higher.” Economics Letters 99 (1): 155–158.10.1016/j.econlet.2007.06.037Search in Google Scholar

Hosios, Arthur. 1990. “On the Efficiency of Matching and Related Models of Search Unemployment.” Review of Economic Studies 57: 279–298.10.2307/2297382Search in Google Scholar

Humphreys, Brad R., Louis J. Maccini, and Scott Schuh. 2001. “Input and Output Inventories.” Journal of Monetary Economics 47 (2): 347–375.10.1016/S0304-3932(01)00046-0Search in Google Scholar

Iacoviello, Matteo, Fabio Schiantarelli, and Scott Schuh. 2011. “Input and Output Inventories in General Equilibrium.” International Economic Review 52 (4): 1179–1213.10.1111/j.1468-2354.2011.00664.xSearch in Google Scholar

Jung, Yongseung, and Tack Yun. 2006. “Monetary Policy Shocks, Inventory Dynamics, and Price-setting Behavior.” Unpublished.10.24148/wp2006-02Search in Google Scholar

Kahn, James A. 1987. “Inventories and the Volatility of Production.” American Economic Review 77 (4): 667–679.Search in Google Scholar

Kahn, James A., Margaret M. McConnell, and Gabriel Perez-Quiros. 2002. “On the Causes of the Increased Stability of the U.S. Economy.” Economic Policy Review 8: 183–202.Search in Google Scholar

Khan, Aubhik, and Julia K. Thomas. 2007a. “Explaining Inventories: A Business Cycle Assessment of the Stockout Avoidance and (S,s) Motives.” Macroeconomic Dynamics 11 (5): 638–665.10.1017/S1365100507060269Search in Google Scholar

Khan, Aubhik, and Julia K. Thomas. 2007b. “Inventories and the Business Cycle: An Equilibrium Analysis of (S,s) Policies.” American Economic Review 97 (4): 1165–1188.10.1257/aer.97.4.1165Search in Google Scholar

Kocherlakota, Narayana. 1998. “Money Is Memory.” Journal of Economic Theory 81 (2): 232–251.10.1006/jeth.1997.2357Search in Google Scholar

Kryvtsov, Oleksiy, and Virgiliu Midrigan. 2010a. “Inventories and Real Rigidities in New Keynesian Business Cycle Models.” Journal of the Japanese and International Economies 24 (2): 259–281.10.1016/j.jjie.2009.12.004Search in Google Scholar

Kryvtsov, Oleksiy, and Virgiliu Midrigan. 2010b. “Inventories, Markups, and Real Rigidities in Menu Cost Models.” Unpublished.10.3386/w14651Search in Google Scholar

Lagos, Ricardo, and Randall Wright. 2005. “A Unified Framework for Monetary Theory and Policy Analysis.” Journal of Political Economy 113 (3): 463–484.10.1086/429804Search in Google Scholar

Maccini, Louis J., Bartholomew J. Moore, and Huntley Schaller. 2004. “The Interest Rate, Learning, and Inventory Investment.” American Economic Review 94 (5): 1303–1327.10.1257/0002828043052295Search in Google Scholar

McConnell, Margaret M., and Gabriel Perez-Quiros. 2000. “Output Fluctuations in the United States: What has Changed since the Early 1980’s?” American Economic Review 90 (5): 1464–1476.10.1257/aer.90.5.1464Search in Google Scholar

Menner, Martin. 2006. “A Search-Theoretic Monetary Business Cycle Model with Capital Formation.” The B.E. Journal of Macroeconomics 6 (1): 11.10.2202/1534-6005.1384Search in Google Scholar

Ramey, Valerie A., and Kenneth D. West. 1999. Inventories. Vol. 1 of Handbook of Macroeconomics. 1 ed., J. B. Taylor and M. Woodford.Search in Google Scholar

Ramey, Valerie A., and Daniel J. Vine. 2004. “Why Do Real and Nominal Inventory-Sales Ratios Have Different Trends?” Journal of Money, Credit and Banking 36 (5): 959–963.10.1353/mcb.2004.0075Search in Google Scholar

Rogerson, Richard, and Robert Shimer. 2011. Search in Macroeconomic Models of the Labor Market. Vol. 4 of Handbook of Labor Economics. 1 ed., Elsevier.10.3386/w15901Search in Google Scholar

Shimer, Robert. 2005. “The Cyclical Behavior of Equilibrium Unemployment and Vacancies.” The American Economic Review 95 (1): 25–49.10.1257/0002828053828572Search in Google Scholar

Shi, Shouyong. 1997. “A Divisible Search Model of Fiat Money.” Econometrica 65: 75–102.10.2307/2171814Search in Google Scholar

Shi, Shouyong. 1998. “Search for a Monetary Propagation Mechanism.” Journal of Economic Theory 81: 314–352.10.1006/jeth.1998.2407Search in Google Scholar

Shi, Shouyong. 2006. “Viewpoint: A Microfoundation of Monetary Economics.” Canadian Journal of Economics 39: 643–688.10.1111/j.1540-5982.2006.00364.xSearch in Google Scholar

Stock, James H., and Mark W. Watson. 2002. “Has the Business Cycle Changed and Why?” NBER Macroeconomics Annual 2002 17: 159–230.10.3386/w9127Search in Google Scholar

Wang, Weimin, and Shouyong Shi. 2006. “The Variability of the Velocity of Money in a Search Model.” Journal of Monetary Economics 53: 537–571.10.1016/j.jmoneco.2005.01.005Search in Google Scholar

Wang, Pengfei, and Yi Wen. 2009. “Inventory Accelerator in General Equilibrium.” Unpublished.10.20955/wp.2009.010Search in Google Scholar

Wang, P., Y. Wen, and Z. Xu. 2014. “What Inventories Tell us about Aggregate Fluctuations-A Tractable Approach to (S,s) Policies.” Journal of Economic Dynamics and Control 44: 196–217.10.1016/j.jedc.2014.05.003Search in Google Scholar

Wen, Yi. 2005. “Understanding the Inventory Cycle.” Journal of Monetary Economics 52 (8): 1533–1555.10.1016/j.jmoneco.2004.10.009Search in Google Scholar

Wen, Yi. 2011. “Input and Output Inventory Dynamics.” American Economic Journal: Macroeconomics 3 (4): 181–212.10.20955/wp.2011.008Search in Google Scholar

Published Online: 2019-04-08

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Articles in the same Issue

https://doi.org/10.1515/bejm-2017-0251

Keywords for this article

business cycles; input inventories; monetary policy; monetary search