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Modelling Oil Prices in a Dynamic GeneralEquilibrium Model with Free Entry∗
[VERY PRELIMINARY]
Ippei Fujiwara†
Bank of JapanNaohisa Hirakata‡
Bank of Japan
July 6, 2006
Abstract
In this paper, we first set up a model that combines the New TradeTheory initiated by Krugman (1980) with the New Open Economy Macro-economics introduced by Obstfeld and Rogoff (1995). We then inquire intothe cause of soaring oil prices, and their transmission and implications onwelfare as well as stabilization policy with such a dynamic general equi-librium model. Both standard contemporaneous shocks and expectationshock, namely anxiety about future oil demand and supply, are also ex-amined. Our conclusion is that for a monetary authority to stabilize theeconomy it is of great importance for the authority to identify correctlythe shocks that cause oil price hikes.
JEL Classification: E52; F10; F41
Keywords: Oil Price; Dynamic General Equilibrium; Free Entry;Endogenous Variety∗We would like to thank Makoto Minegishi for his invaluable inputs. The views expressed
in this paper should not be taken to be those of the Bank of Japan nor any of its respectivemonetary policy or other decision-making bodies. Any errors are solely the responsibility ofthe authors.
†Email: [email protected]‡Email: [email protected]
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North Sea Brent
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2000 2001 2002 2003 2004 2005 2006
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Figure 1: North Sea Brent
1 IntroductionOne notable feature of recent world economic developments is soaring oil prices.As shown in Figure 1, the price of North Sea Brent is now more than twice asmuch as that of three years ago. Yet, the cause of this hike and therefore itstransmission mechanism are not very obvious. Some argue that the increasingpresence of developing countries in the world economy is the cause of oil pricehikes. Particularly, the recent rapid economic growth among the BRICs, namelyBrazil, Russia, China, and India, must have resulted in an increase in the de-mand for raw materials. Since those countries have very large populations, itseems quite possible that oil prices are skyrocketing due to the massive increasein the demand for crude materials. Higher demand for oil in the future is alsoexpected in line with this argument. On the other hand, some point out thatexpected geographical risks in the Middle East will constrain future oil produc-tion. It seems possible that not all but some of the recent soaring oil prices isreflecting those supply-side developments and therefore market speculation. Itis not very easy to find a definitive answer. Indeed, in the recent press releaseby OPEC on 1 June 2006, “141st (Extraordinary) Meeting of the OPEC Con-ference,” it is written that “The Conference also noted that, similarly, worldcrude oil prices continued to remain high and volatile as a consequence of abid-
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ing concern over the lack of effective global oil refining capacity, in the shortand medium term, coupled with anxiety about the ability of oil producers tomeet anticipated, future oil demand. This price volatility is being exacerbatedby geopolitical developments and speculation in the oil futures markets.”Even if we know that recent oil price developments are due either to de-
mand side or to supply side, there exist various candidates for demand andsupply factors. For example, as for demand factors, oil price may increasethanks to increased working population ignited by moving from rural to urbanareas, changes in preference, and technology growth in the final or intermediategoods sector. For the policy making institutions, it is of great importance toacknowledge the source of economic fluctuations, including oil price develop-ments, so that they can conduct proper stabilization policy. We will clarify thecauses of soaring oil prices as well as their theoretical transmission and theirimplications on economic welfare since the oil price is not exogenous variablesin a multi-country dynamic general equilibrium modelSo far, there have been various studies concerning the effects of changes in
oil price, but very few exist to inquire into the possible cause of it in detailin a dynamic general equilibrium model, where oil prices are not exogenousvariables. In this paper, we will show the theoretical mechanism of soaringoil prices, its transmission and welfare implications, that have not been fullyanalyzed in detail as endogenous mechanisms. For this purpose, we construct adynamic general equilibrium model with free entry. The reason why we employthe recently developed dynamic general equilibrium model based on the recentdevelopments in the New Trade Theory as in Melitz (2003) than the standardNew Open Economy Model with fixed entry is, as written in Bergin and Corsetti(2006), “Firstly, we see strong empirical evidence that entry dynamics comovewith the business cycle, a stylized fact that will be discussed below. Secondly,entry has the potential to serve as an amplification and propagation mechanismfor real shocks, and to affect the transmission mechanism for monetary policy.Thirdly entry may have notable welfare effects, to the degree that householdsderive utility from greater variety, or to the degree that the entry of new firmsraises competition in a market.” Besides the above, since the aim of this paper istheoretically to inquire into the possible scenario of soaring oil prices in detail, amodel with plausible mechanism for current oil prices hike is most desirable. Webelieve that for the scenario of future increase in oil demand, the endogenousvariety model is the most useful since it can analyze the effect of increasedpopulation on oil demand as well as the home market effect. Furthermore, wecan examine various shocks as decreased entry fixed cost for introducing foreigndirect investment in the developing economies, which also results in soaring oilprices. This is the main reason for employing the endogenous variety model inthis paper.This paper is organized as follows. In Section 2, we introduce our model for
the analysis on oil price developments, namely the dynamic general equilibriummodel with free entry. Then, Section 3 shows the impulse response analysisagainst various kinds of shocks that affect oil prices. One contribution of thispaper is to show impulse responses not only against the usual contemporaneous
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shocks but also expectation shock on future technology or demand condition.We will show how we can express the realistic scenario, “anxiety about theability of oil producers to meet anticipated, future oil demand” in a dynamicgeneral equilibrium model by employing the expectation shock introduced byBeaudry and Portier (2004) Jaimovich and Rebelo (2005) and Christiano, Mottoand Rostagno (2006). Finally, Section 4 summarizes the findings in this paper.
2 The ModelThe model used in this paper is based on recent literature, which combines theNew Open Economy Macroeconomics initiated by Obstfeld and Rogoff (1995)and the New Trade Theory advocated by Krugman (1980), for example, Ghi-roni and Melitz (2005), Bilbiie, Ghironi and Melitz (2005), Corsetti, Martin andPesenti (2005), and Bergin and Corsetti (2006). Since heterogeneous technologylevel among firms is not considered, our model can be interpreted as a dynamicextension of Corsetti, Martin and Pesenti (2005) or a multi-country extensionof Bilbiie, Ghironi and Melitz (2005) and Bergin and Corsetti (2006). Further-more, our model incorporates nominal rigidities in price and wage settings, anddynamic adjustment costs and dynamic entry/exit decision with time to buildconstraints. Although the model incorporates free entry condition for endoge-nous variety as a new dynamic feature, dynamic parts are mostly based on theGlobal Economy Model (GEM) by Laxton and Pesenti (2002).The model is a two-country (economy) model, which consists of home and
foreign countries. Agents in each country are households, firms, and the mon-etary authority as a sole institution in the government. Households maximizetheir welfare from consumption of final goods C and leisure after differentiatedlabor supply l to domestic firms. The number of households in domestic coun-try is L, while that of in foreign country is L∗, where superscript ∗ denotes thevariables in foreign countries. Both are exogenous. They own domestic firmsand therefore receive profits as a dividend.1
The goods market is monopolistically competitive. Each firm produces dif-ferentiated products. The number of domestic goods is n, while that of foreigngoods is n∗. Unlike other standard papers in new open economy macroeco-nomics, they are endogenously determined in this paper, which is its main con-tribution. When entering the market, each firm needs to incur fixed entry costs.Therefore, entry occurs when the net present value of future profits exceeds en-try cost. This eventually determines the macro production level as well as thenumber of goods, namely variety. On the exit side, a certain proportion δ offirms exits each period.The production structure of this model can be well understood from the
concept chart as shown in Figure 2. Oil production using labor and exogenouslysupplied land takes place only in the foreign country. Intermediate goods are
1We can easily extend our model to incorporate capital stock by assuming that householdsrent capital to firms. For the simplicity of analysis, however, we abstract capital formation inthis paper. The conclusions in this paper will not be affected by incorporation of capital.
4
C C*
I Q M M* Q* I*
Y Y*
F Li Li* F*
O*
Lo* LAND*
Oil
Domestic Country Foreign Country
Final Goods
Intermediate Goods
Figure 2: Production Structure
produced using oil and labor. They are used as intermediate inputs in finalgoods production2 either in the domestic or foreign country or as a fixed costto enter the market, namely a cost to start up a firm.3 Final goods producedas such are all consumedBelow, we first derive structural equations from firms’ and then households’
optimization behavior. In this model, j ∈ [0, Lt] denotes the index of domestichouseholds and j∗ ∈ [0, L∗t ] the index of foreign households, while h ∈ [0, nt] theindex of domestic firms and f ∈ [0, n∗t ] the index of foreign firms.
2.1 Firms
2.1.1 Final goods production
The final goods consumed by household j, namely Ct (j), are produced by fol-lowing CES technology using a basket of home goods Qt (j) and a basket offoreign goods Mt (j):
Ct(j) =hν1εQt (j)
1− 1ε + (1− ν)
1ε Mt (j)
1− 1ε
i εε−1
, (1)2As showin in Bilbiie, Ghironi and Melitz (2005), considering empirical problems associated
with increasing returns to specialization and a C.E.S. production function, it may be betterto model that the household consumes the basket of goods defined over a continuum of goods.Neigher specification, however, makes a difference in simulations conducted in this paper.
3This cost is considered to be an investment although there is no endogenous capitalformation. Therefore, we use the notation I in Figure 1.
5
where ε denotes the elasticity of substitution between home goods and importedgoods, and ν is the home bias parameter. By minimizing total expendituredefined as the sum of PQ,tQt (j) and PM,tMt (j), where PQ,t is the aggregateprice index for domestic goods and PM,t is that for foreign goods, subject toequation (1), we can obtain demand for Qt (j) and Mt (j):
Qt (j) = ν
µPQ,tPt
¶−εCt (j) , (2)
and
Mt (j) = (1− ν)
µPM,tPt
¶−εCt (j) , (3)
and the utility-based consumer price index Pt as a Lagrange multiplier on theconstraint:
Pt =hνP 1−εQ,t + (1− ν)P 1−εM,t
i 11−ε
.
Furthermore, baskets for home and foreign goods are also expressed as the CESaggregator of each good provided by different firms indexed by h and f :
Qt (j) ≡ AQ,t∙Z nt
0
Qt (h, j)1− 1
θ dh
¸ θθ−1
, (4)
and
Mt (j) ≡ A∗Q,t
"Z n∗t
0
Mt (f, j)1− 1
θ∗ df
# θ∗θ∗−1
, (5)
where θ (θ∗) > 1 denotes the elasticity of substitution among intermediate home(foreign) goods and AQ,t and A∗Q,t determine degree of taste for variety and takethe forms of:
AQ,t ≡ (nt)γ−θ
θ−1 ,
andA∗Q,t ≡ (n∗t )
γ∗− θ∗θ∗−1 ,
where γ denotes the degree of taste for good variety. As shown in Benassy(1996), 1 − γ denotes the marginal utility (productivity) gain for increasing agiven amount of consumption on a basket that includes one additional goodvariety. If γ = θ
θ−1 (γ∗ = θ∗
θ∗−1), equations (4) and (5) collapse to standardDixit-Stiglitz aggregator. As analyzed in Corsetti, Martin and Pesenti (1995)and Bergin and Corsetti (2006), effects through taste for variety itself are veryintriguing. Furthermore, as emphasized in Ghironi and Melitz (2005) and Bil-biie, Ghironi and Melitz (2005), we can obtain realistic price developments byusing average price instead of utility-based measure even with some taste forvariety. In this paper, however, our focus is solely on the oil price determinationprocess and its transmission. Therefore, we abstract taste for variety by settingγ and γ∗ equal unity.
6
Each household takes the prices of the home differentiated goods pt (h) asgiven and minimizes the total expenditure expressed as
R nt0pt (h)Qt (h, j) dh
subject to equation (4). The cost-minimizing price of one unit of the homegoods basket, PQ,t, obtained from this optimization problem is:
PQ,t =1
AQ,t
∙Z nt
0
p (h)1−θ dh
¸ 11−θ
.
Similarly, that of the Foreign goods basket, PM,t is defined as:
PM,t =1
A∗Q,t
"Z n∗t
0
p (f)1−θ∗
df
# 11−θ∗
.
As is the case of final goods production, we can obtain the demand for domesticdemand for each domestically produced good:
Qt (h, j) = Aθ−1Q,t
µpt(h)
PQ,t
¶−θQt (j)
= νAθ−1Q,t
µpt(h)
PQ,t
¶−θ µPQ,tPt
¶−εCt (j) ,
as well as that for each foreign good:
Mt (f, j) =¡A∗Q,t
¢θ∗−1µpt(f)PM,t
¶−θ∗Mt (j)
(1− ν)¡A∗Q,t
¢θ∗−1µpt(f)PM,t
¶−θ∗ µPM,t
Pt
¶−εCt (j) ,
The same derivation holds in the foreign country. Therefore, we can derive:
Q∗t (f, j∗) = ν∗Aθ−1
Q,t
Ãp∗t (f)
P ∗Q,t
!−θ µP ∗Q,tP ∗t
¶−εC∗t (j
∗),
and
M∗t (h, j∗) = (1− ν∗)
¡A∗Q,t
¢θ∗−1µp∗t (h)PM,t
¶−θ∗ µP ∗M,t
P ∗t
¶−ε∗C∗t (j
∗).
2.1.2 Intermediate goods production
Production technology Intermediate goods are produced in a monopolisti-cally competitive market. Production Yt requires two factors, labor lIt and oilOt.4 Each domestic firm h has a CES production function:
Yt(h) = Zt
h(1− α)
1ξ lIt (h)
1− 1ξ + α
1ξOt(h)
1− 1ξ
i ξ
ξ−1 , (6)4We can incorporate capital in our standard model. However, to focus on the oil price
developments, we abstract capital formation in this paper.
7
where ξ is the elasticity of substitution and Zt is the Hicks-neutral technology,5
and tries to minimize the total cost as the sum of Wt
PtlIt (h) and
PO,tPtOt(h), where
Wt
Ptdenotes the real wage and P o
t
Ptis the real oil price, subject to equation (6).
We can then derive the factor demands for labor and capital:
lIt (h) = (1− α)
µWt
PtΨtZt
¶−ξYt(h)
Zt, (7)
Ot(h) = α
µPO,tPtΨtZt
¶−ξYt(h)
Zt. (8)
where Ψt is the Lagrange multiplier to equation (6), the marginal cost of pro-ducing one unit of intermediate goods. Ψt can be computed by substitutingequations (7) and (8) into equation (6):
Ψt =1
Zt
"(1− α)
µWt
Pt
¶1−ξ+ α
µPO,tPt
¶1−ξ# 11−ξ
. (9)
In addition to this production cost, firms incur sunk entry costs of fE,t unitof final goods prior to entry. From equations (7) and (8), we can derive factordemands required for entry:
lE,t = (1− α)
µWt
ΨtZt
¶−ξfE,tZt,
OE,t = α
µPO,tΨtZt
¶−ξfE,tZt.
Thus, aggregate factor demands for labor and oil by firm h are expressed aslE,t + l
It (h) and OE,t +Ot(h) respectively.
Price setting Each incumbent firm h must set two prices, pt (h) in the homemarket and p∗t (h) in the foreign market so that the present discounted valueof profit is maximized. Each firm h takes into account the demand in homeQt (h, j) as well as foreign marketM∗t (h, j
∗). We assume that there are sluggishprice adjustment costs measured in terms of total profits, namely the Rotemberg(1982) type adjustment cost. Thus the maximization problem of each incumbentfirm h to set its prices is expressed as follows:
maxpτ (h),p∗τ (h)
Et∞Xτ=t
(1− δ)τ−tDt,τ (j)Πτ (h) ,
5At the moment, we have not incorporated trend growth into this model.
8
where Dt,i (j) is the stochastic discount factor between t and i, and the profitΠt(h) is defined as:
Πt (h) ≡ [pt (h)−Ψt]Z Lt
0
Qt (h, j) dj [1− ΓQ,t(h)] (10)
+ [Etp∗t (h)−Ψt (1 + τ)]
Z L∗t
0
M∗t (h, j∗) dj∗
£1− Γ∗M,t(h)
¤.
Et is home currency per unit of foreign currency,6 τ is the transportation costof foreign goods, and ΓQ,t (h) and Γ∗M,t (h) are the Rotemberg (1982) type ad-justment costs defined as:
ΓQ,t(h) ≡φQ2
∙pt (h) /pt−1 (h)
PQ,t−1/PQ,t−2− 1¸2,
Γ∗M,t(h) ≡φ∗M2
"p∗t (h) /p
∗t−1 (h)
P ∗M,t−1/P∗M,t−2
− 1#2,
where φQ and φ∗M define the size of adjustment costs. By solving for the firstorder condition with respect to pt(h), we can obtain a price-setting relation fordomestically consumed goods:
0 = [1− ΓQ,t (n)] [pt (h) (1− θ) + θΨt]
− [pt (h)−Ψt]φQpt (h) /pt−1 (h)
PQ,t−1/PQ,t−2
∙pt (h) /pt−1 (h)
PQ,t−1/PQ,t−2− 1¸
+Et (1− δD)Dt,t+1 [pt+1(h)−Ψt+1]
×R Lt+10
Qt+1 (h, j) djR Lt0Qt (h, j) dj
φQpt+1 (h) /pt (h)
PQ,t/PQ,t−1
∙pt+1 (h) /pt (h)
PQ,t/PQ,t−1− 1¸.
Similarly, with respect to p∗t (h), the price-setting equation for exported goods
0 =£1− Γ∗M,t (h)
¤[Etp∗t (h) (1− θ) + θΨt (1 + τ t)]
− [Etp∗t (h)−Ψt (1 + τ t)]φ∗Mp
∗t (h) /p
∗t−1 (h)
P ∗M,t−1/P∗M,t−2
"p∗t (h) /p
∗t−1 (h)
P ∗M,t−1/P∗M,t−2
− 1#
+Et (1− δD)Dt,t+1£Et+1p∗t+1 (h)−Ψt+1 (1 + τ t)
¤×R L∗t+10
M∗t+1 (h, j∗) dj∗R L∗t
0M∗t (h, j
∗) dj∗
φ∗Mp∗t+1 (h) /p
∗t (h)
P ∗M,t/P∗M,t−1
"p∗t+1 (h) /p
∗t (h)
P ∗M,t/P∗M,t−1
− 1#.
Under the flexible price equilibrium, where φQ and φ∗M are zero, firms set prices
that reflect the markup θ/ (θ − 1) over marginal cost. Therefore, prices in thesteady state become:
pt (h) =θ
θ − 1Ψt, (11)
6Therefore, the intermediate goods firms conduct producer currency pricing.
9
and
Etp∗t (h) =θ∗
θ∗ − 1Ψt (1 + τ) .
With these equations, if θ = θ∗, we can derive the following relationship:
Etp∗t (h) = pt (h) (1 + τ) .
This implies that the law of one price does not hold due to the transportationcost, even if prices are fully flexible and no difference exists in price markups intwo countries.
Free entry and value of firms We model firms’ entry/exit decision follow-ing Ghironi and Melitz (2005). Merit of entry is dependent on the net presentvalue of profit after entry, namely {Πτ (h)}∞τ=t+1 discounted by stochastic dis-count factor7 since firms are eventually owned by households. At the same time,firm h faces such a shock that it needs to exit the market with constant positiveprobability δ. Such exiting probability also needs to be considered when dis-counting future profits. Expected profit $t (h) of firm h at t is now expressedas follows:
$t (h) = Et∞Xτ=t
(1− δ)τ−t
Dt,τ (j)Πτ (h) .
Firms enter the market until the sunk cost becomes equal to the expected profit.Hence, free entry condition is obtained as:
$t(h) = fE,tΨt. (12)
A firm that enters the market at t can only start producing for profit at t + 1due to time to build constraints. At the beginning of t, there already exist ntfirms and during t, nE,t new firms entry. As a result, we get n0t ≡ nt + nE,t inperiod t. At the same time, production takes place with only nt firms. However,a shock then comes that makes δ firms exit from the market at the end of eachperiod. Therefore, at the end of period t as well as at the beginning of periodt + 1, the number of firms is (1 − δ)(nt + nE,t). This also defines the numberof firms that distribute profits to households, since profits are assumed to bedistributed at the beginning of each period. δnE,t firms that enter at t exitmarket without any production. This can be understood from Figure 2 below.Forward iteration of the equation for share holdings and absence of specula-
tive bubbles yield the asset price solution:
$t (h) = (1− δ)Dt,t+1 (j) [Πt+1 (h) +$t+1 (h)] .
In the steady state,
$ (h) =(1− δ)D(j)Π(h)
1− (1− δ)D(j).
7Dt,t+1(j) is defined below when households’ optimization behavior is explained.
10
Period t+1Period t
Beggining of period t; firms.tn
,
,
New entry of firms;' firms.
Only firms produce
E t
t t E t
t
nn n n
n= +
( )After realization of a shock;only 1 ' firms survive.D tnδ−
( )1 D
Beggining of period t+1;= 1- ' firms.t tn nδ+
Figure 3: Timing of Entry and Exit
From free entry condition in equation (12), we can derive:
fEΨ =(1− δ)D(j)Π(h)
1− (1− δ)D(j). (13)
The corporate profit in equation (10) becomes in the steady state with equation(11):
Π(h) =Ψ
θ − 1Y (h). (14)
sinceEtp∗t (h) = pt (h) (1 + τ) ,
and the resource constraint of the product of firm h:
Y (h) =
Z Lt
0
Qt (h, j) dj + (1 + τ)
Z L∗t
0
M∗t (h, j∗) dj∗.
By plugging equation (14) into (13), we can obtain:
Y (h) = (θ − 1) fEZ1− (1− δ)D (j)
(1− δ)D (j).
In the steady state, the production level of each firm is determined by degree ofeconomics of scale fEZ and product differentiation θ in addition to the discountrate 1−(1−δ)D(j)
(1−δ)D(j) .
11
2.1.3 Oil production
Oil is produced in a perfectly competitive market. To simplify arguments, weassume the existence of a continuum of firms in the unit mass in the rest of theworld
O∗t (s) = ZO∗t
h(1− αO∗)
1
ξO lO∗t (s)1− 1
ξO∗ + α∗ 1
ξO LAND∗t1− 1
ξO∗i ξO∗
ξO∗−1 . (15)
2.2 Household
A consumer j receives utility from goods consumption C (j) and disutility fromlabor supply l (j). Consumer j maximizes the lifetime expected utility as follows:
Et∞Xτ=t
βτ−t {Uτ [Cτ (j)]− Vτ [lτ (j)]} , (16)
where,
Ut [Cτ (j)] = ZU,t(1− bC)σ [Ct (j)− bCCt−1 (j)]1−σ − 1
1− σ, (17)
Vt [lτ (j)] = ZV,t(1− bl)−ζ [lt (j)− bllt−1 (j)]1+ζ
1 + ζ. (18)
bC and bl are habit formation parameters in consumption and labor supplyrespectively. σ and ς determine the intertemporal elasticity of substitution inconsumption and labor supply.
2.2.1 Budget constraint
Since we incorporate the dynamic entry behavior of firms, the number of firmsis determined endogenously. Household income depends on the number of newentry and incumbent firms. Firm’s operating profit Πt (h) is paid to householdsas a dividend income. Furthermore, households recognize the value of shareposition, some of which are carried into the next period. That dividend income— the value of selling and purchasing shares — also relies on the number of firms.During period t, the representative home household buys xt+1 share in a
mutual fund of n0t ≡ nt + nE,t home firms (those already operating at time tand the new entrants). Only nt+1 = (1− δ)n0t firms produce and pay dividendsat time t + 1. As explained above, the dynamics of nt are already defined asfollows:
n0t ≡ nt + nE,t, (19)
nt+1 = (1− δ)n0t,
and thereforent+1 = (1− δ) (nt + nE,t) .
12
Reflecting these dynamics, the budget constraint of the household j now be-comes:
EtBF,t+1 (j) +BH,t+1 (j) + xt+1 (j)Z n0t
0
$t (h) dh (20)
≤ (1 + i∗) [1− ΓB,t (j)] EtBF,t (j) + (1 + it)BH,t (j)+Wt (j) lt (j) [1− ΓW,t (j)]
+xt (j)
Z nt
0
[Πt (h) +$t (h)] dh
−PtCt (j) .
Several adjustment costs are assumed in this budget constraint. First, followingthe GEM, we assume that each household faces the following cost when tradingbond in foreign currency:
ΓB,t (j) = φB1
exphφB2
EtB∗F,t(j)Pt
− ZB0i− 1
exphφB2
EtB∗F,t(j)Pt
− ZB0i+ 1
+ ZB,t. (21)
Since household j is the monopolistic supplier of differentiated labor supply, ithas wage setting power. Similar to price setting by firms, wage adjustment costis a Rotemberg type, as below:
ΓW,t (j) =φW2
∙Wt (j) /Wt−1 (j)
Wt−1/Wt−2− 1¸2. (22)
The consumer’s optimization problem is to maximize equation (16) subject toequations (17) to (22) with respect to Ct (j) , Wt (j) , BH,t+1 (j) , BF,t+1 (j) ,and xt+1.
2.2.2 Euler equation: bonds
The Euler equation below is obtained by differentiating the objective with re-spect to home BH,t+1 and foreign bond holding BF,t+1.
1 = (1 + it)EtDt,t+1 (j) = (1 + i∗t ) [1− ΓB,t+1 (j)]Et
∙Dt,t+1 (j)
Et+1Et
¸,
where Dt,τ (j) is the stochastic discount factor:
Dt,τ (j) ≡ βτ−tPtU
0 [Cτ (j)]
PτU 0 [Ct (j)].
2.2.3 Euler equation: shares
Concerning stock share, the first order condition below is obtained:
$t (h) = (1− δ)EtDt,t+1 (j) [Πt+1 (h) +$t+1 (h)] .
This and the above equation equates returns from holding of BH,t+1, BF,t+1,and xt+1.
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2.2.4 Wage equation (labor supply)
Wage setting by households is expressed as:
ψtV 0t (j)
U 0t(j)
PtWt (j)
= (ψt − 1) [1− ΓW,t (j)] +∙Wt (j)
∂ΓW,t (j)
∂Wt (j)
¸+Et
∙Dt,t+1 (j)
lt+1 (j)
lt (j)Wt+1 (j)
∂ΓW,t (j)
∂Wt (j)
¸,
where
V 0t (j) = ZV,t
∙lt (j)− bllt−1 (j)
1− bl
¸ζ.
Each household j supplies differentiated labor l (h, j) at wageW (j). We assumethat firm h has CES aggregator of differentiated laborl (j):
lt (h) = Al,t
"Z Lt
0
lt (h, j)1− 1
ψt dj
# ψtψt−1
,
where
Al,t ≡ Lγl−
ψt1−ψt
t .
Cost minimization implies that firm h’s demand for labor input l(h, j) is afunction of the relative wage:
lt (h, j) = Aψt−1l,t
∙Wt (j)
Wt
¸−ψtlt (h) ,
whereW (j) is the nominal wage paid to home labor input j and the wage indexW is defined as:
Wt =1
Al,t
"Z Lt
0
Wt(j)1−ψtdj
# 11−ψt
.
2.2.5 Other
We can define the holdings of net foreign asset as follows:
Ft(j) = (1 + i∗) [1− ΓB,t (j)] EtBF,t (j) .
Then, and exchange rate:
EtDt,t+1Lt+1Ft+1(j) = LtFt(j) + (1 + i∗t−1)ΓB,t−1(j)EtLtBF,t(j)
+EtP ∗M,tL∗tM∗t (j∗)− PM,tLtMt(j)
−POM ,t
∙Z nt
0
OM,t(h)dh+
Z nE,t
0
OM,t(e)de
¸.
14
2.3 Government
There is no tax collection. Therefore, only the central bank exists for stabi-lization to facilitate households’ consumption smoothing motives and works asa nominal anchor to obtain a unique rational expectation equilibrium. Themonetary authority follows this simple instrument rule:
(1 + it+1)4 − 1 = ωi
h(1 + it)
4 − 1i+ (1− ωi)
h(1 + Etit+1)
4 − 1i
+ω1Et
µPt+τPt+τ−4
−Πt+τ¶.
2.4 Market clearing
2.4.1 Industry equilibrium
To close the model, we need to clear goods markets globally, which requirestotal revenue equal to world expenditure:Z nt
0
"pt (h)
Z Lt
0
Qt (h, j) dj + Etp∗t (h)Z L∗t
0
M∗t (h, j∗) dj∗
#dh = µt
Z Lt
0
Et (j) dj
+(1− µ∗t ) EtZ L∗t
0
E∗t (j∗) dj∗.
LHS is total industry revenue in the home and foreign markets, while the RHSis the expenditure for the industry in each country. Et(j) denotes the totalspending by consumer j., which is defined as:
Et(j) = PtCt(j). (23)
µQ,t is home consumer j’s expenditure share of home country goods, and µ∗Q,t
is foreign consumer j∗’s expenditure share of foreign country goods, which aredefined by using equations (2) and (3) as:
µt =PQ,tQt(j)
Et(j)=PQ,tQt (j)
PtAt (j)= ν
µPQ,tPt
¶1−²QM,
µ∗t =P ∗Q,tQ
∗t (j)
E∗t (j∗)
=P ∗Q,tQ
∗t (j)
P ∗t A∗t (j∗)= ν∗
µP ∗Q,tP ∗t
¶1−²∗QM.
Furthermore, this relationship can be transformed so as to derive the numberof firms in each industry:
nt =µQR Lt0Et(j)dj + (1− µ∗Q)L∗tEtE∗t (j∗)
pt(h)LtQt(h, j) + Etp∗t (h)L∗tM∗t (h, j∗).
15
2.4.2 General equilibrium
In a general equilibrium, spending for goods in each period must the equal factorincome of labor, plus net income by bonds plus investment income excludingthe cost of investing in new firms. Thus,Z Lt
0
Et (j) dj =
Z Lt
0
(1 + i∗t ) [1− ΓB,t (j)] EtBF,t (j) dj +Z Lt
0
(1 + it)BH,t (j) dj (24)
−EtZ Lt
0
BF,t+1 (j) dj −Z Lt
0
BH,t+1 (j) dj +
Z Lt
0
Wt (j) lt (j) [1− ΓW,t (j)] dj
+
Z nt
0
Πt (h) dh−Z nE,t
0
$t (h) dh.
In addition, all goods, factors and bonds markets satisfy following market clear-ing conditions:
Yt(h) =
Z Lt
0
QDt (h, j)dj + (1 + τ)
Z L∗t
0
M∗Dt (h∗, j)dj.
The factor market clearing conditions can be written as:
lt(j)dj =
Z nt
0
lIt (h, j)dh+
Z 1
0
lOt (s, j) ds+ nE,t
Z nE,t
0
lE,t (e, j) de,
These equations imply that labor and capital market clearing conditions arerequired so that the total supply of the factor should be equal to be the factordemand for production and entry. Market clearing in the bond market requires:Z Lt
0
BH,t(j)dj = 0,
Z Lt
0
BF,t(j)dj +
Z L∗t
0
B∗F,t(j∗)dj∗ = 0.
The market clearing condition for the raw materials, namely oil in this paper is:
O∗t (s∗) = n∗tO
∗Q,t(f) + n
∗E,tO
∗Q,t(e
∗) + (1 + τ∗t ) [ntOM,t(h) + nE,tOM,t(e)] .
The model is solved under the assumption of symmetric equilibrium. Detailsare shown in the appendix.
2.5 Calibration
In this subsection, we show the calibration of major parameters. In the sim-ulations below, symmetry between the home and foreign country is assumed.Basically, parameters are set following previous researches using the GEM. Table1 shows the values of major parameters.
16
Table 1Parameter Values
Parameter Value Description and Definitionsε 1.5 Elasticity of substitution between domestic and imported goodsζ 3.0 Inverse of Frisch elasticityψ 6.00 ψ/(ψ − 1) is Wage markupθ 6.00 θ/(θ − 1) is Price markupφQ 400 Adjustment cost for price setting in domestic marketφM 400 Adjustment cost for price setting in export marketφW 400 Adjustment cost for wage settingγ 1 Degree of taste for variety in goodsγl 1 Degree of taste for variety in laborα 0.3 Scale parameter that determines law materials shareαO 0.8 Scale parameter that determines land’s share in raw material productionξ 0.75 Elasticity of Substitution between raw material and laborξO 0.75 Elasticity of Substitution between land and laborβ 1.03−0.25 Discount rateδ 0.025 Firm exit shockν 0.5 Home bias parameterφB1 0.05 Transaction-cost parameter in the bond marketφB2 0.1 Transaction-cost parameter in the bond marketbc 0.83 Habit persistence parameter in consumptionbl 0.0 Habit persistence parameter in labor1/σ 0.80 Intertemporal elasticity of substitution
The parameters that determine the nominal rigidity (φQ,φM ,φW ), elasticityof substitution in production of intermediate goods and raw materials (ξ, ξO)and transaction cost parameters in the bond market (φB1,φB2) are set followingLaxton and Pesenti (2003). Elasticity of substitution between domestic andimported goods (ε) are set 1.5 according to Smets and Wouters (2002) andChari, Kehoe and McGrattan (2002). And we set Inverse of Frisch elasticity (ζ),habit persistence parameter in consumption (bc) and intertemporal elasticity ofsubstitution (1/σ) following Julliard,Karam Laxton and Pesenti (2005).In steady state, we set the size of domestic country (L) and foreign countries
(L∗), 0.05 and 1.00 respectively. Furthermore, we assume that there is no homebias.8
8Therefore, ratio of imports over aggregate output becomes large. Since the aim of thispaper is to understand the effects through soaring oil price rather theoretically than empiri-cally. We eliminate the effects of home bias so that we can understand spill-over effects fromoil price hikes. There, however, exist almost no qualitative differences between in the casewith and without home bias.
17
3 Various Scenarios for Soaring Oil PriceIn this section, we reproduce soaring oil prices as impulse responses againstshocks in a dynamic general equilibrium model, where both supply and demandshocks are examined.9 We show only responses of major variables determined inthe model. One is the variable comparable to the GDP concept. In a symmetricequilibrium, budget constraints in equation (24) with equation (23) are writtenas:
LtPtCt + nE,t$t (h) + [EtLtBF,t+1 − Lt (1 + i∗t ) EtBF,t]+ [Lt (1 + i
∗t )ΓB,tEtBF,t + ΓW,tLtWtlt]
= LtWtlt + ntΠt.
This is the macro budget constraint. On the LHS of the equation, the first termis macro consumption, the second is mutual fund purchase for newly establishedfirms and considered to be investment, the third is net exports, and the fourthis adjustment costs on trading bonds and labor. On the other hand, on theRHS, the first term is labor income while the second is profit generated frominvestment. We define the total of variables on the RHS, namely therefore theLHS as well, “aggregate income,” which will be shown in the impulse responsesanalysis below.At the same time, we also show the responses of natural rate of interest,
namely the Vicksellian rate both in domestic and foreign countries in order tounderstand implications of soaring oil prices on the monetary policy stance.Unlike the case with the small open economy, computing the natural rate ofinterest is not very trivial with the multi-country model. For analytical sim-plicity, in this paper, we compute it by assuming that there is no nominal priceand wage rigidity all over the world. Therefore, we can define this is the naturalrate that is considered to be faced by the international organization.10
First, we show the impulse responses on soaring oil prices against standardcontemporaneous shocks, then those against expectation shock, such as a per-ception about future technology and demand conditions.
9To be exact, distinction about supply and demand shocks are innocuous in a dynamicgeneral equilibrium model, especially in open economies. For example, technology shock inthe rest of the world acts like a demand shock to the domestic country.10On the other hand, we can compute the natural rate of interest which the central banks
face. It is, however, a bit tricky to be computed since the central bank should takes whathappened in the rest of the world as exogenous event that cannot be controlled by itself.Therefore, we need to compute it as follows. First, we compute impulse responses against acertain shock in multi countries. Then, by keeping the impulse responses in the foreign countryas exogenous variables, we simulate the model where nominal price and wage rigidities areeliminated in the domestic country with the same shock. We have found almost no significantdifferences in the natural rate of interest computed this way from that in this paper.
18
3.1 Standard shocks
We conduct five simulations against (1) decreased labor disutility, (2) increasedworking population, (3) increased technology, (4) reduced fixed cost, and (5)reduced oil production technology. All shocks are contemporaneous and occurin the rest of the world. (1) and (2) aim at capturing the situation wheredemand for oil is increasing due to the expanding economic activities in emergingeconomies. (3) and (4) are similar experiments but increased oil demand inthe rest of the world is due to supply side improvement, unlike the demandimprovement in (1) and (2). Shocks in (1) to (4), however, are considered tobe demand shocks for the domestic economy. (5) is not a reality as of now.This scenario is examined to compare the realistic scenario examined in theexpectation shock simulation, “world crude oil prices continued to remain highand volatile as a consequence of abiding concern over the lack of effective globaloil refining capacity, in the short and medium term, coupled with anxiety aboutthe ability of oil producers to meet anticipated future oil demand.” (2) and(4) are unique simulation only available with the dynamic general equilibriummodel with free entry. We will see how similar simulations can result in differenttransmission as well as welfare implication.
3.1.1 Decreased labor disutility
Decreased relative labor disutility is a very popular exercise as a demand shock.A positive shock is given to ZU,t in the rest of the world version of equation(17).11 Impulse responses are shown in Figure 4. The domestic country is thesmall oil importing country whose share of world production is about 5 percent.12 Bold lines show responses of domestic variables while thick lines arethose of variables in the rest of the world. Decreased labor disutility in the restof the world naturally results in the higher labor supply and consumption in theforeign countries. Therefore, aggregate income as well as inflation rates becomehigher. Naturally, due to the elasticity of substitution between labor and oil,demand for oil becomes higher in the intermediate goods sector. This resultsin higher oil price inflation. On the other hand, to meet the higher demand inthe foreign country, more labor is required to produce more tradeables in thedomestic country with less domestic consumption. Yet, its degree is higher forthe rest of the world. Therefore, welfare is better off in the domestic countryagainst soaring oil prices caused by decreased disutility of labor in the rest of theworld and welfare is worse off in the rest of the world where disutility of laboris decreased Investments and number of firms decrease although they show asmall increase right after the shocks since more should spend on consumptionto satisfy substitution between labor and consumption.As for monetary policy implications, in both countries, the natural rate of
interest becomes higher. Therefore, with no changes in nominal interest rates,11Almost equivalent results can be obtained by adding a negative shock to ZV,t in the rest
of the world version of equation (18).12The parameters, however, are not set according to the Japanese stylized facts.
19
Decreased Labor Disutility
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Foreign countriesDomestic country
Oil Price Inflation
0 10 20 30 40
0.00
0.05
0.10
Consumption
0 10 20 30 40
-0.0
50.
050.
150.
25
Investment
0 10 20 30 40
-0.5
0.0
0.5
Aggregate Income
0 10 20 30 40
0.00
0.05
0.10
0.15
Inflation rate
0 10 20 30 40
0.00
000.
0010
0.00
20Labour Supply
0 10 20 30 40
0.00
0.10
Number of firm
0 10 20 30 40
-0.1
5-0
.05
0.00
Welfare
0 10 20 30 40
-0.0
4-0
.02
0.00
Nominal Interest Rate
0 10 20 30 40
0.00
20.
006
0.01
0
Terms of Trade
0 10 20 30 40
-0.0
3-0
.01
0.01
Natural Interest Rate
0 10 20 30 40
0.00
20.
006
0.01
0
Figure 4: Decreased Labor Disutility
20
monetary policy loosens across the world.
3.1.2 Increased working population
This is a unique experiment that cannot be examined in the standard dynamicgeneral equilibrium model with fixed varieties. With the fixed variety, an in-crease in population only results in a proportionate increase in all detrendedvariables by population. On the other hand, with the endogenous variety model,an increase in working population increases the labor supply so that the num-ber of firms, which equals the number of goods since each firm produces its owngoods, becomes larger. Although we have not taken taste for variety into ac-count in this paper, we can analyze the effects of increased population on othereconomic variables through the creation of firms. Furthermore, since there is atrade cost for exporting τ , the country that experiences larger population canenjoy benefits from increased variety and firms more than the other countriescan. This is the so-called “Home Market Effect.”There may be suspicion that increased working population is an inappropri-
ate simulation for current soaring oil prices via increased oil demand in emergingeconomies. We believe that this is, for example, what is happening in Chinabehind the expanding domestic economy. It is supported by massive popula-tion shifts from rural areas to the industrial cities, as Japan experienced in the1950 and 1960s. Although people in the rural areas are also naturally con-ducting economic activities, they are considered home production. Therefore,increased working population shifts from rural areas to industrial cities are nat-urally considered to be the most appropriate simulation for the current increasein domestic demand in the BRIC countries.Figure 5 shows the responses when working population is increased by 1
per cent in the rest of the world. Although the degree of oil price hikes isquite similar to Figure 4, the mechanism behind it is quite different. Since theresource constraint in equation (24) is expanded in this case, consumption aswell as investment on new firms’ establishment are raised. On the other hand, inthe domestic country, responses are very similar to the above case since nothinghas happened in the domestic country.As a result, the contrast in welfare between domestic and foreign countries
become more distinct. Because there is no home bias in final goods production,per capita consumption and therefore welfare decrease in the rest of the world.13
As more imported goods are required to produce final goods, there exist verylarge positive spillover from the foreign countries to the domestic country. Thesedevelopments result in improving terms of trade and welfare in the small oilimporting country.At the same time, thanks to the above-mentioned improvements in demand
conditions, natural rate of interest becomes much higher in the foreign countries.13Therefore, with very high home bias, results here are reversed.
21
Increased Working Population
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Foreign countriesDomestic country
Oil Price Inflation
0 10 20 30 40
-0.2
0.2
0.4
0.6
0.8 Consumption
0 10 20 30 40
0.0
0.4
0.8
Investment
0 10 20 30 40
-3-2
-10
12
Aggregate Income
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
Inflation rate
0 10 20 30 400.
000
0.00
40.
008
Labour Supply
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
Number of firm
0 10 20 30 40
-0.2
0.0
0.1
0.2
Welfare
0 10 20 30 40
-0.0
010.
002
0.00
4
Nominal Interest Rate
0 10 20 30 40
0.00
00.
015
0.03
0 Terms of Trade
0 10 20 30 40
-0.3
-0.2
-0.1
0.0
0.1
Natural Interest Rate
0 10 20 30 40
0.00
0.02
0.04
0.06
Figure 5: Increased Working Population
22
3.1.3 Increased technology
Increased technology in the rest of the world is a candidate factor for currentsoaring oil prices. This is indeed a supply shock to the rest of the world, butworks like a demand shock as a better external economic activity to the domesticcountry. Responses against this shock are shown in Figure 6. In the foreigncountry, labor supply is increased thanks to the positive technology shock14.This results in higher consumption as well as investment thanks to the expandedbudget constraints. Although oil prices are increased due to higher demand,inflation rates in aggregate are reduced due to lower marginal costs. On the otherhand, in the domestic country, inflation rates are higher thanks to improvedexternal demand conditions. In the medium run, more labor inputs are neededto satisfy the demand from foreign countries. Therefore, welfare becomes worsein the domestic country and naturally better in the rest of the world.Reflecting those developments in supply and demand conditions, the natural
rate of interest is higher in the domestic country but lower in the rest of theworld. Qualitatively, a stabilization policy on inflation rates does a good job inneutralizing the effects from exogenous shocks.
3.1.4 Reduced fixed cost
Reduced fixed cost is naturally interpreted as an improved technology sincefewer goods are needed to establish a firm, namely for production.15 Similarlyto the case with increased working population, this is a unique simulation to theendogenous variety model. Figure 7 shows the responses against reduced fixedcost in the rest of the world.This simulation is again on improvement technology, but significant differ-
ences exist. When the fixed cost is reduced, more resources are directed toestablish new firms. Therefore, consumption decreases in the rest of the world.Even though consumption becomes lower, inflation rates become higher thanksto the higher employment for creating new firms. Oil prices are also raised dueto the increase in demand to establish new firms. On the other hand, worseningterms of trade reduce domestic consumption and goods varieties in the domesticcountry. In the rest of the world, more resources are directed to establish newfirms. Hence, consumption and imports from the small oil importing countryare decreased. These developments result in the worsening terms of trade in thedomestic country. As a result, welfare is reduced in the domestic country whileimproved in the rest of the world.Reflecting those economic conditions, the natural interest rate is higher in
the rest of the world but becomes temporally lower in the domestic country.14Although sticky price is incorporated, labor is increased after the positive technology
shock. This is somewhat contrary to the results in Gali (1999) and Gali and Rabanal (2004).Positive responses of hours against technolgy shock are, however, produced with sticky wagesin Altig, Christiano, Eichenbaum and Linde (2005) and with habit formation in consumptionin Vigfusson (2004).15 In this paper, however, taste for variety is not considered.
23
Increased Technology
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Foreign countriesDomestic country
Oil Price Inflation
0 10 20 30 40
-20
24
6
Consumption
0 10 20 30 40
0.0
0.2
0.4
Investment
0 10 20 30 40
-40
020
4060
Aggregate Income
0 10 20 30 40
-40
24
68
Inflation rate
0 10 20 30 40-0
.02
0.00
Labour Supply
0 10 20 30 40
-50
510
Number of firm
0 10 20 30 40
-10
12
3
Welfare
0 10 20 30 40
-0.0
100.
000
0.01
0
Nominal Interest Rate
0 10 20 30 40
-0.1
0-0
.04
0.00
0.04
Terms of Trade
0 10 20 30 40
-1.0
-0.6
-0.2
0.2
Natural Interest Rate
0 10 20 30 40
-0.0
10.
010.
03
Figure 6: Increased Technology
24
Reduced Entry Cost
0 10 20 30 40
-1.0
-0.6
-0.2
Foreign countriesDomestic country
Oil Price Inflation
0 10 20 30 40
-1.0
0.0
1.0
Consumption
0 10 20 30 40
-0.1
00.
000.
05
Investment
0 10 20 30 40
-15
-55
15
Aggregate Income
0 10 20 30 40
-2-1
01
2
Inflation rate
0 10 20 30 400.
000
0.01
0Labour Supply
0 10 20 30 40
-2-1
01
2
Number of firm
0 10 20 30 40
-0.4
0.0
0.4
0.8
Welfare
0 10 20 30 40
-0.0
040.
000
Nominal Interest Rate
0 10 20 30 40
0.00
0.02
0.04
0.06
Terms of Trade
0 10 20 30 40
-0.1
00.
000.
10
Natural Interest Rate
0 10 20 30 40
-0.0
50.
05
Figure 7: Reduced Entry Cost
25
3.1.5 Reduced oil-producing technology
Reduced oil-producing technology is not the reality for the current soaring oilprices since no clear evidence of damaged oil plant technology has been reported.Impulse responses here are, however, very useful to understand truly supply sideeffects on oil prices and the contrast between the expectation shock about futuredeterioration of oil production, which will be examined in the next subsection.Responses to such a shock are shown in Figure 8.Negative technology in oil production naturally increases oil prices. This
would result in lower consumption, investment, labor supply and welfare in bothcountries. On monetary policy implication, the natural rate of interest becomesmuch lower in the domestic country, where a more expansionary monetary policyis needed.
3.2 Expectation shocks
Consider the compelling story on the current oil price hikes that “world crudeoil prices continued to remain high and volatile as a consequence of abidingconcern over the lack of effective global oil refining capacity, in the short andmedium term, coupled with anxiety about the ability of oil producers to meetanticipated, future oil demand.” We need to test whether expectation aboutfuture demand improvements as well as anxiety on the future oil-producingcondition would result in what we observe from the data of major economicvariables recently.In this subsection, we first show how we can draw impulse responses against
such expectation shocks. Then, impulse responses against future expectationfor larger working population in the rest of the world, as well as that for dete-riorating oil producing technology, are demonstrated.
3.2.1 How to simulate expectation shock
First, we will explain a general solution of the rational expectation model. Wethen show how to incorporate expectation shock.16 Generally, a rational expec-tation model can be represented as17
α0Et (Zt+1 − Z∗) + α1 (Zt − Z∗) + α2 (Zt−1 − Z∗) (25)
+β0 (St+1 − S∗) + β1 (St − S∗) = 0.
The solution that we would like to obtain is
Zt = Z∗ +A (Zt−1 − Z∗) +B (St − S∗) , (26)
andSt = S
∗ + P (St−1 − S∗) + Cεt. (27)16The contents in this subsection are based on Christiano, Motto and Rostagno (2006).17This is just a difference version of Christiano (2000). Since in our model, there are
variables whose steady state value is zero, we use difference rather than log-difference.
26
Reduced Oil Producing Technology
0 10 20 30 40
-1.0
-0.6
-0.2
Foreign countriesDomestic country
Oil Price Inflation
0 10 20 30 40
0.0
0.2
0.4
0.6
Consumption
0 10 20 30 40
-0.0
4-0
.02
Investment
0 10 20 30 40
-4-3
-2-1
0
Aggregate Income
0 10 20 30 40
-0.5
-0.3
-0.1
Inflation rate
0 10 20 30 400.
0000
0.00
10Labour Supply
0 10 20 30 40
-0.6
-0.4
-0.2
0.0
Number of firm
0 10 20 30 40
-0.2
0-0
.10
0.00
Welfare
0 10 20 30 40-1 e
-03
0 e
+00
Nominal Interest Rate
0 10 20 30 40
0.00
10.
003
0.00
5
Terms of Trade
0 10 20 30 40
-0.0
050.
010
0.02
0
Natural Interest Rate
0 10 20 30 40
-1 e
-03
5 e
-04
Figure 8: Reduced Oil-Producing Technology
27
By substituting, equations (26) and (27) into (25), we can obtain:
α0A2 + α1A+ α2 = 0,
and(β0 + α0B)P + (β1 + α1B + α0AB). (28)
Matrix A and B in solutions in equations (26) and (27) are computed by solvingthe above equations. Especially whether we can obtain unique A is dependenton the usual Blanchard and Kahn (1980) condition.Simulation with expectation shock can be materialized by making adjust-
ment to β0 and β1 so that we can obtain a new B matrix. For simplicity ofargument, let us consider a very simple technology shock process s, which iscomparable to equation (27),18 as follows:
(st − s) = ρ (st−1 − s) + bεt−p + bξt.With this shock process, we can express a news shock for higher future pro-ductivity. As a simple example, here we suppose a situation that we receive anews that “productivity is raised in period 2” today, but it turns out to be falsewhen period 2 actually comes.19 The above equation is represented as followsin canonical form as :⎛⎝ st − sbεtbεt−1
⎞⎠ =
⎛⎝ ρ 0 10 0 00 1 0
⎞⎠⎛⎝ st−1 − sbεt−10
⎞⎠+⎛⎝ bξtbεt
0
⎞⎠ (29)
If we add a news shock, , at period zero,⎛⎝ s0 − sbε0bε−1⎞⎠ =
⎛⎝ ρ 0 10 0 00 1 0
⎞⎠⎛⎝ s−1 − sbε−1bε−2⎞⎠+
⎛⎝ 0bε00
⎞⎠ .ε0 will not affect bs0 and bs1, but shock on technology at period 2 expected inperiod zero is now:
E0
⎛⎝ s2 − sbε2bε1⎞⎠ =
⎛⎝ ρ 0 10 0 00 1 0
⎞⎠2⎛⎝ s0 − sbε0bε−1⎞⎠ =
⎛⎝ ρ2 1 ρ0 0 00 0 0
⎞⎠⎛⎝ s0 − sbε0bε−1⎞⎠ .
Hence,E0 (s2 − s) = bε0.
Therefore, the shock on technology at period 2 expected in period zero indeedbecomes ε0. If the expectation is actually materialized, the simulation is con-ducted using appropriate S vector as above. On the other hand, once period 218For simplicity, s = 1.19 In simulations below, we also show the case when the initial guess turns out to be true.
28
comes, such a positive shock does not happen actually. ε0 is offset by ξ2, sinceξ2 = −ε0. This is depicted as:
⎛⎝ s2 − sbε2bε1⎞⎠ =
⎛⎝ ρ 0 10 0 00 1 0
⎞⎠⎛⎝ s1 − sbε0bε−1⎞⎠+
⎛⎝ bξ200
⎞⎠ =
⎛⎝ 000
⎞⎠ .Thus, we can generate such a shock as at period zero and one, technology shockis expected to happen at period 2, but it turns out to be a bubble expectationin period 2.This canonical form is exactly equation (27) and shock vector S now incorpo-
rates expectational shock terms as bεt. Therefore, if we appropriately arrange β0and β1 by adding zero vectors to columns corresponding to bεt in S and rewriteP , we can compute new B matrix by using equation (28). We can then obtainimpulse responses under expectation shock with equations (26) and (27).
3.2.2 Expected increase in working population
Further economic expansion in emerging economies, such as the BRIC countries,are expected. Some analysts have pointed out this is the reason why we arefacing the current oil price hikes. Here, we simulate a situation of an increasein economic activities due to increased working population next year. Figure9 shows responses when such expectation is actually materialized, and Figure10 demonstrates those when the expectation turns out to be false.20 By theway, there are no differences in responses up until the date the expected event issupposed to happen. The reason why we show cases when the expectation doesnot materialize is for our understanding the results from a bubble expectation.Expecting a future increase in working population, investment level and
therefore labor for establishing new firms are raised to prepare for the higherdemand in the future, since establishing a new firm incurs adjustment costs inthe form of time to build constraint. Oil prices are raised due to the necessity toestablish more new firms. Since no significant expansion of resource constraintexists, consumption is reduced up until the expectation is realized. Consump-tion jumps in Figure 9 because this is the response of total consumption, whichis per capita consumption multiplied by working population. Therefore, con-sumption is smoothed at the individual level. Even in the small oil importingcountry, labor and investment are increased to prepare future high demand ondomestic tradable goods. Reflecting those demand developments, inflation ratesrise in both countries. As is the case with the contemporaneous shock on theworking population, welfare is greatly improved in the domestic country whileit deteriorates in the rest of the world.20As obvious from Figure 10, even with shocks which turns out to be false, economic vari-
ables show very volatile movements. This is consistent with the OPEC statement in theintroduction.
29
Increase Working Population
0 10 20 30 40
0.0
0.4
0.8
Foreign countriesDomestic country
Oil price inflation
0 10 20 30 40-0
.05
0.05
0.15
0.25
Consumption
0 10 20 30 40
0.0
0.4
0.8
Investment
0 10 20 30 40
-20
24
Aggregate Income
0 10 20 30 40
0.0
0.2
0.4
0.6
Inflation rate
0 10 20 30 40
-0.0
010.
002
0.00
5
Labour Supply
0 10 20 30 40
-0.2
0.0
0.2
0.4
Number of firm
0 10 20 30 40
-0.1
0.0
0.1
0.2
0.3 Welfare
0 10 20 30 40
-0.0
010.
001
0.00
3
Nominal Interest Rate
0 10 20 30 40
0.00
00.
010
0.02
0
Terms of Trade
0 10 20 30 40
-0.2
-0.1
0.0
Figure 9: Expected Increase in Working Population (materialized)
30
Increase Working Population
0 10 20 30 40
-1.0
-0.5
0.0
0.5
1.0
Foreign countriesDomestic country
Oil price inflation
0 10 20 30 40-0
.6-0
.20.
2
Consumption
0 10 20 30 40
-0.0
4-0
.02
0.00
0.02
Investment
0 10 20 30 40
-4-2
02
4
Aggregate Income
0 10 20 30 40
-0.4
0.0
0.4
Inflation rate
0 10 20 30 40
-0.0
060.
000
0.00
4
Labour Supply
0 10 20 30 40
-0.8
-0.4
0.0
0.4
Number of firm
0 10 20 30 40
0.00
0.05
0.10
0.15
Welfare
0 10 20 30 40
-0.0
010.
001
Nominal Interest Rate
0 10 20 30 40
-0.0
10.
010.
02
Terms of Trade
0 10 20 30 40
-0.3
-0.2
-0.1
0.0
Figure 10: Expected Increase in Working Population (not materialized)
31
3.2.3 Expected reduction in oil producing technology
As documented in the OPEC statement, the biggest concerns are geopoliticaldevelopments and speculation in the oil futures markets. Here, to monitor howsuch concern affects current as well as future developments of major economicvariables, we draw impulse responses when people expect that oil-producingtechnology is reduced by 1 per cent next year in the rest of the world. Fig-ure 11 shows the case when such expectation is materialized while Figure 12demonstrates the case when that turns out to be false.Anticipation about future deteriorating oil-producing technology increases
the value of oil and therefore oil prices rise. At the same time, since futurelower technology21 is expected, it is optimal for households to increase the laborsupply due to negative wealth effects. Furthermore, consumption is reducedgradually thanks to consumption smoothing to prepare for lower income in thefuture. As a result, investment is increased to satisfy the resource constraint. Itfirst seems controversial to see the increase in labor and oil prices but a decreasein inflation rates. As was explained above, initially after the news about futurelower technology on oil production is received, the labor supply curve shiftsoutward. This, then, reduces real wages. Therefore, aggregate inflation ratesdecrease all over the world. To materialize higher inflation across the world afteroil price hikes due to anxiety about future oil supply, we need to introduce verystrong labor adjustment costs so that the substitution effects for smoothinglabor supply dominate. This will keep the labor supply curve from shiftingoutward.22
4 Conclusion• We have examined several mechanisms which induce soaring oil prices.
• It is of great importance to acknowledge the source of economic fluctua-tions, including oil price developments, so that they can conduct properstabilization policy.
• Even with similar magnitude of oil price hikes, effects on inflation, termsof trade and welfare are quite different. Particularly, cases with increasedtechnology, reduced fixed cost, and increased working population are in-triguing.(1) With increased technology, marginal cost decreases and therefore in-flation rates are lowered in the foreign countries. On the other hand,inflation rates rise due to more demand on the domestic goods. Thereforeterms of trade improves in the domestic country.(2) A reduction in fixed costs induces investment to create new firms.
21Technology on oil production works as a standard technology in the aggregate productionfunction.22This is a similar mechanism to what is mentioned on capital formation in Christiano,
Motto and Rostagno (2006).
32
Reduced Oil Producing Technology
0 10 20 30 40
-1.0
-0.6
-0.2
Foreign countriesDomestic country
Oil price inflation
0 10 20 30 400.
00.
20.
4
Consumption
0 10 20 30 40
-0.0
20-0
.010
Investment
0 10 20 30 40
-3-2
-10
12
Aggregate Income
0 10 20 30 40
-0.4
-0.2
0.0
0.2
Inflation rate
0 10 20 30 40
-4 e
-04
2 e
-04
Labour Supply
0 10 20 30 40
-0.4
0.0
0.2
0.4 Number of firm
0 10 20 30 40
-0.0
50.
05
Welfare
0 10 20 30 40
-4 e
-04
2 e
-04
8 e
-04
Nominal Interest Rate
0 10 20 30 40
-0.0
010.
001
Terms of Trade
0 10 20 30 40
0.00
00.
005
0.01
0
Figure 11: Expected Reduction in Oil-Producing Technology (materialized)
33
Reduced Oil Producing Technology
0 10 20 30 40
-1.0
-0.5
0.0
0.5
1.0
Foreign countriesDomestic country
Oil price inflation
0 10 20 30 40-0
.2-0
.10.
00.
1
Consumption
0 10 20 30 40
-0.0
100.
005
0.02
0
Investment
0 10 20 30 40
0.0
1.0
2.0
Aggregate Income
0 10 20 30 40
0.00
0.10
0.20
0.30
Inflation rate
0 10 20 30 40
-1 e
-03
-4 e
-04
Labour Supply
0 10 20 30 40
0.0
0.1
0.2
0.3
0.4 Number of firm
0 10 20 30 40
0.00
0.05
0.10
0.15
Welfare
0 10 20 30 40
-2 e
-04
2 e
-04
Nominal Interest Rate
0 10 20 30 40
-0.0
04-0
.002
0.00
0 Terms of Trade
0 10 20 30 40
-0.0
100.
000
0.01
0
Figure 12: Expected Reduction in Oil-Producing Technology (not materialized)
34
Since the number of firms cannot be altered immediately, inflation ratesrise in the foreign country. On the other hand, more resources are directedto investment in the rest of the world. Therefore, exports from the do-mestic country decreases with worsening of terms of trade.(3) With increased working population, marginal cost as well as inflationrates rise in the foreign country similar to the case with reduced fixedcost thanks to the increase in demand. Terms of trade in the domesticcountry, however, improves since there exists more demand for domesticgoods. This is quite contrary to the case with reduced fixed entry cost.
• Following OPEC statement, we simulate with expectation on future in-crease in oil demand and anxiety for future deterioration of oil producingfacility. Both scenarios result in soaring oil prices, but in the latter, ag-gregate inflation rates across the world decrease. We need to inquire intothe role of expectation shock on soaring oil prices.
35
References[1] to be added
36
Appendix
Model Equations
Final Goods Production
Ct(j) Ct(j) =hν1ε
QQt(j)1− 1
ε + (1− νQ)1εMt(j)
1− 1ε
i εε−1
C∗t (j∗) C∗t (j
∗) =h(ν∗Q)
1ε∗ (Q∗t (j
∗))1−1ε∗ + (1− ν∗Q)
1ε∗ (M∗t (j
∗))1−1ε∗i ε∗ε∗−1
Demand for Intermediate Goods
Household’s demand for aggregated intermediate goods
Qt(j) Qt(j) = νQ
³PQ,tPt
´−εCt(j)
Q∗t (j∗) Q∗t (j
∗) = ν∗Q
³P∗Q,tP∗t
´−ε∗C∗t (j
∗)
Mt(j) Mt(j) = (1− νQ)³PM,t
Pt
´−εCt(j)
M∗t (j∗) M∗t (j
∗) = (1− ν∗Q)³P∗M,t
P∗t
´−ε∗C∗t (j
∗)
Household’s demand for aggregated intermediate goods producedby each firm
Qt(h, j) Qt(h, j) = Aθ−1Q,t
³pt(h)/PtPQ,t/Pt
´−θQt(j)
Q∗t (f, j∗) Q∗t (f, j
∗) = (A∗Q,t)θ∗−1
³p∗t (f)/P
∗t
P∗Q,t/P∗t
´−θ∗Q∗t (j
∗)
Mt(f, j) Mt(f, j) =¡A∗Q,t
¢θ∗−1 ³pt(f)/PtPM,t/Pt
´−θ∗Mt(j)
M∗t (h, j∗) M∗t (h, j
∗) = (AQ,t)θ−1
³p∗t (h)/P
∗t
P∗M,t/P∗t
´−θM∗t (j
∗)
AQ,t AQ,t ≡ nγ− θ
θ−1t
A∗Q,t A∗Q,t ≡ (n∗t )γ∗Q− θ∗
θ∗−1
Demand for Raw Materials
Incumbents
OQ,t(h) OQ,t(h) = νOQ
³POQ,t/Pt
PO,t/Pt
´−²OQMOt(h)
O∗Q,t(f) O∗Q,t(f) = ν∗OQ
µP∗OQ,t/P
∗t
P∗O,t/P∗t
¶−²∗OQMO∗t (f)
OM,t(h) OM,t(h) = (1− νOQ)³POM,t/PtPO,t/tP
´−²OQMOt(h)
O∗M,t(f) O∗M,t(f) = (1− ν∗OQ)³P∗OM,t/P
∗t
P∗O,t/P∗t
´−²∗OQMO∗t (f)
37
Entrants
OQ,t(e) OQ,t(e) = νOQ
³POQ,t/Pt
PO,t/Pt
´−²OQMOt(e)
O∗Q,t(e∗) O∗Q,t(e
∗) = ν∗OQ
µP∗OQ,t/P
∗t
P∗O,t/P∗t
¶−²∗OQMO∗t (e
∗)
OM,t(e) OM,t(e) = (1− νOQ)³POM,t/PtPO,t/Pt
´−²OQMOt(e)
O∗M,t(e∗) O∗M,t(e
∗) = (1− ν∗OQ)³P∗OM,t/P
∗t
P∗O,t/P∗t
´−²∗OQMO∗t (e
∗)
Relative price of intermediate goods
Aggregate price
PQ,tPt
PQ,tPt
= 1AQ,t
n1
1−θt
pt(h)Pt
P∗Q,tP∗t
P∗Q,tP∗t
= 1A∗Q,t
(n∗t )1
1−θ∗ p∗t (f)P∗t
PM,t
Pt
PM,t
Pt= 1
A∗Q,t(n∗t )
11−θ∗ pt(f)
Pt
P∗M,t
P∗t
P∗M,t
P∗t= 1
AQ,tn
11−θt
p∗t (h)P∗t
Price Setting
pt(h)Pt
0 = (1− ΓQ,t(h))h(1− θ)pt(h)Pt
+ θMCt(h)Pt
i−hpt(h)Pt− MCt(h)
Pt
ipt(h)Γ
0Q,t(h)
−Eth(1− δD)Dt,t+1(j)πt+1
nH,t+1Qt+1(h,j)nH,tQt(h,j)
³pt+1(h)Pt+1
− MCt+1(h)Pt+1
´Γ0Q,t+1(h)
ip∗t (h)P∗t
0 = (1− Γ∗M,t(h))h(1− θ∗)
p∗t (h)P∗t
EtP∗tPt
+ θ∗MCt(h)Pt
(1 + τ t)i
−hp∗t (h)P∗t
(1− θ∗)EtP∗tPt− θ∗MCt(h)
Pt(1 + τ t)
ip∗t (h)Γ
∗0M,t(h)
−Eth(1− δD)Dt,t+1(j)πt+1
n∗H,t+1M∗t+1(h,j
∗)
n∗H,tM∗t (h,j
∗)
×hp∗t+1(h)
P∗t+1
Et+1P∗t+1Pt+1
− MCt+1(h)Pt+1
(1 + τ t)ip∗t+1(h)Γ
∗0M,t+1(h)
ipt(f)Pt
0 = (1− ΓM,t(f))hpt(f)Pt(1− θ) Pt
EtP∗t+ θ
MC∗t (f)P∗t
(1 + τ∗t )i
−hpt(f)Pt
PtEtP∗t
− MC∗t (f)P∗t
(1 + τ∗t )ipt(f)Γ
0M,t(f)
−Eth(1− δ∗D)Dt,t+1
nH,t+1Mt+1(f,j)nH,tMt(f,j)
×hpt+1(f)Pt+1
Pt+1Et+1P∗t+1
− MC∗t+1(f)
P∗t+1(1 + τ∗t )
ipt+1(f)Γ
0M,t+1(f)
ip∗t (f)P∗t
0 = (1− Γ∗Q,t(f))h(1− θ∗)
p∗t (f)P∗t
+ θ∗MC∗t (f)P∗t
iPH_ RW −
hp∗t (f)P∗t− MC∗t (f)
P∗t
ip∗t (f)Γ
∗0Q,t(f)
−Eth(1− δ∗D)D
∗t,t+1
n∗H,t+1Q∗t+1(f,j
∗)
n∗H,tQ∗t (f,j
∗)
³p∗t+1(f)
P∗t+1− MC∗t+1(f)
P∗t+1
´p∗t+1(f)Γ
∗0Q,t+1(f)
iRelative price of raw materials
38
POQ,tPt
POQ,tPt
= MCt(s)Pt
P∗OQ,tP∗t
P∗OQ,tP∗t
=MC∗t (s
∗)P∗t
POM,t
Pt
POM,t
PtPtEtP∗t
=MC∗t (s
∗)P∗t
(1 + τ∗t )P∗OM,t
P∗t
P∗OM,t
P∗t
EtP∗tPt
= MCt(s)Pt
(1 + τ t)
Price adjustment costs for differentiated intermediategoods
ΓQ,t(h) ΓQ,t(h) ≡φQ2
³πt(h)πQ,t−1
− 1´2
Γ0Q,t(h)£pt(h)Γ
0Q,t(h)
¤= φQ
πt(h)πQ,t−1
³πt(h)πQ,t−1
− 1´
Γ0Q,t+1(h)£Dt,t+1pt+1(h)Γ
0Q,t+1(h)
¤= −Dt,t+1φQ
πt(h)2
πQ,t−1
³πt(h)πQ,t−1
− 1´
Γ∗Q,t(f) Γ∗Q,t(f) ≡φ∗Q2
³π∗t (f)π∗Q,t−1
− 1´2
Γ∗0Q,t(f)£P ∗Q,tΓ
∗0Q,t(f)
¤= +φ∗Q
π∗t (f)π∗Q,t−1
³π∗t (f)π∗Q,t−1
− 1´
Γ∗0Q,t+1(f)£D∗t,t+1P
∗Q,t+1Γ
∗0Q,t+1(f)
¤= −D∗t,t+1φ∗Q
π∗t (h)2
π∗Q,t−1
³π∗t (h)π∗Q,t−1
− 1´
ΓM,t(f) ΓM,t(f) ≡ φM2
³πt(f)πM,t−1
− 1´2
Γ0M,t(f)£PM,tΓ
0M,t(f)
¤= φM
πt(f)πM,t−1
³πt(f)πM,t−1
− 1´
Γ0M,t+1(f)£Dt,t+1PM,t+1Γ
0M,t+1(f)
¤= −Dt,t+1φM
πt(h)2
πM,t−1
³πt(h)πM,t−1
− 1´
Γ∗M,t(h) Γ∗M,t(h) ≡φ∗M2
³π∗t (h)π∗M,t−1
− 1´2
Γ∗0M,t(h)£p∗t (h)Γ
0M,t(h)
¤= φ∗M
π∗t (h)π∗M,t−1
³π∗t (h)π∗M,t−1
− 1´
Γ∗0M,t+1(h)£D∗t,t+1P
∗M,t+1Γ
∗0M,t+1(h)
¤= −D∗t,t+1φ∗M
π∗t (h)2
π∗M,t−1
³π∗t (h)π∗M,t−1
− 1´
Marginal costs
Intermediate goods producers
Wt
Pt
MCt(h)Pt
= 1ZQ,t
∙(1− γQ)
³Wt
Pt
´1−ξ+ γQ
³PO,tPt
´1−ξ¸ 11−ξ
W∗tP∗t
MC∗t (f)P∗t
= 1Z∗Q,t
∙(1− γ∗Q)
³W∗tP∗t
´1−ξ∗+ γ∗Q
³P∗O,tP∗t
´1−ξ∗¸ 11−ξ∗
Raw material goods producers
MCt(s)Pt
MCt(s)Pt
= 1Zs,t
∙(1− γs)
³Wt
Pt
´1−ξs+ γs
³PL,tPt
´1−ξs¸ 11−ξs
MC∗t (s∗)
P∗t
MC∗t (s∗)
P∗t= 1
Z∗s,t
∙(1− γ∗s)
³W∗tP∗t
´1−ξ∗s+ γ∗s
³P∗L,tP∗t
´1−ξ∗s¸ 11−ξ∗s
39
Labor market
Taste for the variety of labor input
Al,t Al,t = Lγl−
ψψ−1
t
A∗l,t A∗l,t = (L∗t )γ∗l−
ψ∗ψ∗−1
Aggregate wage
Wt(j)Pt
Wt
Pt= 1
Al,tL
11−ψt
Wt(j)Pt
W∗t (j∗)
P∗t
Wt∗P∗t
= 1A∗l,t
(L∗t )1
1−ψ∗ W∗t (j∗)
P∗t
Wage for each household
V 0t (j) ψV 0t (j)U 0t(j)
PtWt(j)
= (ψ − 1) (1− ΓW,t(j)) +Wt(j)Γ0W,t(j)
+Et
hDt,t+1
lt+1(j)lt(j)
Wt+1(j)Γ0W,t+1(j)
iV ∗0t (j
∗) ψ∗V ∗0t (j∗)U∗0t (j
∗)P∗t
W∗t (j∗) = (ψ
∗ − 1)¡1− Γ∗W,t(j∗)
¢+W ∗t (j
∗)Γ∗0W,t(j)
+Et
hD∗t,t+1
l∗t+1(j∗)
l∗t (j∗) W
∗t+1(j
∗)Γ∗0W,t+1(j)i
Utility functions
U 0t(j) U 0t(j) = ZU,t³Ct(j)−bCCt−1(j)
1−bC
´−σU∗0t (j) U∗0t (j
∗) = Z∗U,t
³C∗t (j
∗)−b∗CC∗t−1(j∗)1−b∗C
´−σ∗lt(j) V 0t (j) = ZL,t
³lt(j)−bllt−1(j)
1−bl
´ζl∗t (j
∗) V ∗0t (j∗) = Z∗V,t
³l∗t (j∗)−b∗l l∗t−1(j∗)1−b∗l
´ζ∗Wage adjustment costs
ΓW,t(j) ΓW,t(j) =φW2
³πW,t(j)πW,t−1
− 1´2
Γ0W,t(j)£Wt(j)Γ
0W,t(j)
¤= φW
πW,t(j)πW,t−1
³πW,t(j)πW,t−1
− 1´
Γ0W,t+1(j)£Dt,t+1Wt+1(j)Γ
0W,t+1(j)
¤=
−Dt,t+1φWπW,t+1(j)
2
πW,t
³πW,t+1(j)
πW,t− 1´
Γ∗W,t(j∗) Γ∗W,t(j
∗) =φ∗W2
³π∗W,t(j)
π∗W,t−1− 1´2
Γ∗0W,t(j∗)
£W ∗t (j)Γ
∗0W,t(j)
¤= φ∗W
π∗W,t(j)
π∗W,t−1
³π∗W,t(j)
π∗W,t−1− 1´
Γ∗0W,t+1(j∗)
£D∗t,t+1W
∗t+1(j)Γ
∗0W,t+1(j)
¤=
−D∗t,t+1hφ∗W
π∗W,t+1(j)2
π∗W,t
³π∗W,t+1(j)
π∗W,t− 1´i
40
Demand for labor input by an intermediate goods pro-ducer
Incumbents
lt(h, j) lt(h, j) = Aψ−1l,t
³Wt(j)/PtWt/Pt
´−ψlt(h)
lt(h) lt(h) = (1− γQ)³
Wt/PtZQ,tMCt(h)/Pt
´−ξ ³Yt(h)ZQ,t
´l∗t (f, j
∗) l∗t (f, j∗) = (A∗l,t)
ψ∗−1³W∗t (j
∗)/P∗tW∗t /P
∗t
´−ψ∗l∗t (f)
l∗t (f) l∗t (f) = (1− γ∗Q)³
W∗t /P∗t
Z∗Q,tMC∗t (f)/P∗t
´−ξ∗ ³Y ∗t (f)Z∗Q,t
´Entrants
lt(e, j) lt(e, j) = Aψ−1l,t
³Wt(j)/PtWt/Pt
´−ψlt(e)
lt(e) lt(e) = (1− γQ)³
Wt/PtZQ,tMCt(h)/Pt
´−ξ ³fE,tZQ,t
´l∗t (e
∗, j∗) l∗t (e∗, j∗) = (A∗l,t)
ψ∗−1³W∗t (j
∗)/P∗tW∗t /P
∗t
´−ψ∗l∗t (e
∗)
l∗t (e∗) l∗t (e
∗) = (1− γ∗Q)³
W∗t /P∗t
Z∗Q,tMC∗t (f)/P∗t
´−ξ∗ ³ f∗E,tZ∗Q,t
´Demand for labor input by a raw-material producer
lt(s, j) lt(s, j) = Aψ−1l,t
³Wt(j)/PtWt/Pt
´−ψl(s)
lt(s) lt(s) = (1− γO)³
Wt/PtZs,tMCt(s)/Pt
´−ξ ³Ot(s)Zs,t
´l∗t (s
∗, j∗) l∗t (s∗, j∗) = (A∗l,t)
ψ∗−1³W∗t (j
∗)/P∗tW∗t /P
∗t
´−ψ∗l∗t (s
∗)
l∗t (s∗) l∗t (s
∗) = (1− γ∗O)³
W∗t /P∗t
Z∗s,tMC∗t (s)/P∗t
´−ξ∗ ³O∗t (s)Z∗s,t
´Demand for raw materials by an intermediate goods
producer
Incumbents
Ot(h) Ot(h) = γO
³PO,t/Pt
ZQ,tMCt(h)/Pt
´−ξ ³Yt(h)ZQ,t
´O∗t (f) O∗t (f) = γ∗O
³P∗O,t/P
∗t
Z∗Q,tMC∗t (f)/P∗t
´−ξ∗ ³Y ∗t (f)Z∗Q,t
´Entrants
Ot(e) Ot(e) = γO
³PO,t/Pt
ZQ,tMCt(h)/Pt
´−ξ ³fE,tZQ,t
´O∗t (e
∗) O∗t (e∗) = γ∗O
³P∗O,t/P
∗t
Z∗Q,tMC∗t (f)/P∗t
´−ξ∗ ³ f∗E,tZ∗Q,t
´41
Demand for raw materials by a raw-material producer
PL,tPt
Lt(s) = γs
³PL,t/Pt
Zs,tMCt(s)/Pt
´−ξ ³Ot(s)ZQ,t
´P∗L,tP∗t
L∗t (s∗) = γ∗s
³P∗L,t/P
∗t
Z∗s,tMC∗t (s∗)/P∗t
´−ξ∗ ³O∗t (s)Z∗s,t
´Aggregate price of raw material
PO,tPt
PO,tPt
=
∙αO
³POQ,tPt
´1−ξO+ (1− αO)
³POM,t
Pt
´1−ξO¸ 11−ξO
P∗O,tP∗t
P∗O,tP∗t
=
∙α∗O
³P∗OQ,tP∗t
´1−ξ∗O,t+ (1− α∗O)
³P∗OM,t
P∗t
´1−ξ∗O,t¸ 11−ξ∗
O
Profits and share prices
Euler equations of share
$t(h)Pt
$t(h)Pt
= (1− δD)Dt,t+1(j)πt+1
hΠt+1(h)Pt+1
+ $t+1(h)Pt+1
i$∗t (f)P∗t
$∗t (f)P∗t
= (1− δ∗D)D∗t,t+1(j
∗)π∗t+1
hΠ∗t+1(f)
P∗t+1+
$∗t+1(f)
P∗t+1
iFree entry conditions
MCt(h)Pt
$t(h)Pt
= fE,tMCt(h)Pt
MC∗t (f)P∗t
$∗t (h)P∗t
= f∗E,tMC∗t (f)P∗t
Profits
Πt(h)Pt
Πt(h)Pt
=³pt(h)Pt− MCt(h)
Pt
´nH,tQt(h, j)][1− ΓQ,t(h)]
+hp∗t (h)P∗t
EtP∗tPt− MCt(h)
Pt(1 + τ t)
in∗H,tM
∗t (h, j
∗)[1− Γ∗M,t(h)]
Π∗t (f)P∗t
Π∗t (f)P∗t
=³p∗t (f)P∗t− MC∗t (f)
P∗t
´n∗H,tQt(f, j)][1− Γ∗Q,t(f)]
+hpt(f)Pt
PtEtP∗t
− MC∗t (f)Pt
(1 + τ∗t )inH,tMt(f, j)[1− ΓM,t(f)]
Dynamics of the number of firms
nt nt = (1− δ)(nt−1 + nE,t−1)n∗t n∗t = (1− δ∗)(n∗t−1 + n
∗E,t−1)
Exchange ratesEtP∗tPt
EtDt,t+1Lt+1Ft+1(j)Pt+1
πt+1 = LtFt(j)Pt
+ (1 + i∗t−1)ΓB,t−1(j)EtP∗tPtLt
BF,t(j)Pt−1
1πt
+EtP∗tPt
P∗M,t
P∗tL∗tM
∗t (j∗)− PM,t
PtLtMt(j)−
POM,t
Pt[ntOM,t(h) + nE,tOM,t(e)]
42
Financial assetsFt(j)Pt
Ft(j)Pt
= (1 + i∗t )(1− ΓB,t(j)) 1π∗tEtP∗tPt
BF,t(j)P∗t−1
Government
it (1 + it+1)4 − 1 = ω1
£(1 + it−1)
4 − 1¤+ (1− ω1)
£(1 + it,neut)
4 − 1¤
+ω2Et (πt+τ − πtar)i∗t (1 + i∗t+1)
4 − 1 = ω∗1£(1 + i∗t−1)
4 − 1¤+ (1− ω∗1)
£(1 + i∗t,neut)
4 − 1¤
+ω∗2Et¡π∗t+τ − π∗tar
¢it,neut it,neut = π0.25t+3 /β
i∗t,neut i∗t,neut =¡π∗t+3
¢0.25/β∗
Bond market
Dt,t+1 1 = (1 + it)EtDt,t+1D∗t,t+1 1 = (1 + i∗t )EtD
∗t,t+1
ΓB,t(j) 1 = (1 + i∗t )(1− ΓB,t+1(j))Et³Dt,t+1
Et+1P∗t+1Pt+1
PtEtP∗t
πt+1π∗t+1
´BF,t+1(j)
P∗tΓB,t(j) = φB1
exp³φB2
EtP∗tPt
BF,t+1(j)
P∗t−ZB0,t
´−1
exp³φB2
EtP∗tPt
BF,t+1(j)
P∗t−ZB0,t
´+1+ ZB,t
B∗F,t+1(j)
Pt
B∗F,t+1(j∗)
Pt= −Lt
L∗t
BF,t+1(j)P∗t
Market clearing
Labor market
nE,t lt(j) = ntlt(h, j) + nE,tlt(e, j) + lt(s, j)n∗E,t l∗t (j
∗) = n∗t l∗t (f, j
∗) + n∗E,tl(e∗j∗) + l∗t (s
∗, j∗)
Raw materials market
Ot(s) Ot(s) = ntOQ,t(h) + nE,tOQ,t(e) + (1 + τ t)£n∗tO
∗M,t(f) + n
∗E,tO
∗M,t(e
∗)¤
O∗t (s∗) O∗t (s
∗) = n∗tO∗Q,t(f) + n
∗E,tO
∗Q,t(e
∗) + (1 + τ∗t ) [ntOM,t(h) + nE,tOM,t(e)]
Intermediate goods market
Yt(h) Yt(h) = LtQt(h, j) + (1 + τ t)L∗tM∗t (h, j
∗)Y ∗t (f) Y ∗t (f) = L
∗tQt(f, j
∗) + (1 + τ∗t )L∗tMt(f, j)
Inflation rates
Final goods
43
πt Dt,t+1 ≡ βU 0t+1
πt+1U 0t
π∗t D∗t,t+1 ≡ βU∗0t+1
π∗t+1U∗0t
Aggregate intermediate goods
πQ,t πQ,t =PQ,t/Pt
PQ,t−1/Pt−1πt
π∗Q,t π∗Q,t =P∗Q,t/P
∗t
P∗Q,t−1/P∗t−1
π∗t
πM,t πM,t =PM,t/Pt
PM,t−1/Pt−1πt
π∗M,t π∗M,t =P∗M,t/P
∗t
P∗M,t−1/P∗t−1
π∗t
A intermediate good by each firm
πt (h) πt (h) =pt(h)/Pt
pt−1(h)/Pt−1πt
π∗t (f) π∗t (f) =p∗t (f)/P
∗t
p∗t−1(f)/P∗t−1
π∗t
πt (f) πt (f) =pt(f)/Pt
Pt−1(f)/Pt−1πt
π∗t (h) π∗t (h) =p∗t (h)/P
∗t
p∗t−1(h)/P∗t−1
π∗t
44
