Download Investment Efficiency and the Welfare Gain from

InvestmentEfficiencyandtheWelfareGainfrom
InternationalFinancialIntegration*
PhilipL.Brock
UniversityofWashington
September2016
Abstract
I analyze international financial integration in a Ramsey growth model with capital adjustment
costs that affect the efficiency of investment. The welfare gain relative to financial autarky is
measuredbytheincreaseinthevalueofthecapitalstockatthetimeoffinancialintegration.My
model offers an analytical structure that is consistent with four stylized facts of financial market
liberalizationsinemergingmarketeconomies:thecostofcapitalfalls,stockmarketindexesrise,
aggregate investment increases, and growth rates increase at rates that depend on country
characteristics. The size of the welfare gain depends on structural parameters and is nonmonotonicinthedegreeofconvexityofthecapitaladjustmentcosts.
JELClassificationNumbers:F43,F63,O16,O19
Keywords:CapitalAdjustmentCosts,InternationalFinancialIntegration,Tobin’sq
____________________________________________
*This is a substantially revised and expanded version of an earlier paper, circulated as “Growth
RateAccelerationsandtheGainsfromInternationalFinancialIntegration”. The paper has improved
as a result of comments at the 2016 Society for Nonlinear Dynamics and Econometrics Conference
and the 2015 and 2016 Infiniti International Finance Conferences.IespeciallythankSebastianSosa,
CharlesEngel,ManfredKremer,andJennyBerrillforextensivecommentsonearlierdrafts.
Contactdetails:[email protected]
1Introduction
Whatisthenatureofthewelfaregainsthatarecreatedbyfinancialintegrationofasmalleconomy
with world capital markets? There are two important approaches to calculating the welfare gain
frominternationalfinancialintegrationthatisassociatedwithcapitalaccumulation.Onestresses
the gain due to the faster rate of capital accumulation induced by the lower world interest rate
relativetotheautarkicinterestrate.Thesecondstressesthepotentialincreaseintherateofreturn
tocapitalthatmaybeinducedbycontemporaneouschangesintotalfactorproductivity.Thispaper
follows the first approach by extending the Ramsey growth model to include capital adjustment
costsinfinancialautarkyaswellasininternationalfinancialintegration.
In the theoretical literature, there appears to be little support for the proposition that one-
sectoraggregategrowthmodelscangenerateasubstantialwelfaregainduetoareductionofthe
interestratefromautarkictotheworldrateforasmalleconomy.GourinchasandJeanne(2006),in
particular,showedthattheRamseygrowthmodelimpliesasmallwelfaregainfrominternational
financial integration due to the fast rate of the economy’s convergence toward steady-state in
financialautarky.1
In contrast to the welfare implications of the Ramsey model, a body of empirical evidence—
originating in the work of Henry (2003, 2007) and Bekaert, Harvey, and Lundblat (2005)—
demonstrates that there are empirically observed increases in the stock market value of firms as
wellasgrowthrateaccelerationsthatfrequentlyaccompanymeasurestopromotetheinternational
financialintegrationofaneconomy;thesesuggestawelfaregainarisingfromfinancialintegration.2
This paper revisits the theoretical hypothesis that a substantial part of the welfare gain from
international financial integration may be attributable to the faster rate of capital accumulation.
The analysis extends the Ramsey growth model by including capital adjustment costs; that is, it
1SeealsotherelatedpapersbyAoki,Benigno,andKiyotaki(2010);Mendoza,Quadrini,andRíos-Rull(2009);
AngeletosandPanousi(2011);Corneli(2011);Hoxha,Kalemli-Ozcan,andVollrath(2013);AntunesandCavalcanti
(2013);andCoeurdacier,Rey,andWinant(2013)forcontributionstothisliterature.
2SeealsoQuinnandToyoda(2008);anddeNicolòandJuvenal(2014).
2
generalizes the investment technology. With no adjustment costs, as in the Ramsey framework
used by Gourinchas and Jeanne, the gains from integration are very small. Gains from financial
integrationinthispaperaretiedtotheexistenceofcapitaladjustmentcosts—andthewelfaregain
frominternationalfinancialintegrationisgreaterforeconomiesthathavelargercapitaladjustment
costsandgreaterinitialcapitalscarcity(relativetothesteady-statecapitalstock).
An important aspect of the analytical model is that adjustment of the capital stock in the
financially integrated economy with non-zero adjustment costs is faster (but finite) than
adjustmentinthefinanciallyautarkiceconomy.Thismeansthatthesizeofthewelfaregainfrom
financialintegrationcanbeinferredfromthegrowthrateaccelerationthatfollowsintegration.In
particular, given estimated parameter values of the initial relative capital scarcity, elasticity of
intertemporal substitution, and convexity of capital adjustment costs (along with other standard
parameters), the welfare gain can be estimated as a function of the observed growth rate
acceleration.
ThisfeatureofthemodelallowsittoaddressthefollowingconcernraisedbyGourinchasand
Jeanne(2006):
Henry(2003)andBekaert,HarveyandLundblad(2005)findthatopeningthestock
markettoforeigninvestorsboostsgrowthby1-2%forfiveyearsinarow.Sucha
result, however, is not obvious to translate in terms of domestic welfare. How
persistent is the impact of capital account opening on growth? What share of the
output increase is transferred to foreign investors? These questions are crucial in
assessingthewelfareimpactofcapitalaccountopeningandcanbeaddressedonly
bylookingatthedatathroughthelensesofamodel.[italicsadded]
The lens of the model in this paper shows that the persistence of the impact of international
financial integration depends on capital adjustment costs and other parameters that affect the
speedofconvergence.Thewelfaregaincanbemeasuredbytheincreaseinthestockmarketvalue
ofthecapitalstockorbythegrowthrateaccelerationatthetimeoffinancialintegration.
Thepaper’splanisasfollows.Section2developsthemodelusedtoaddresstheissueofthe
importance of capital accumulation for the welfare gain from international financial integration.
Section3derivesthealgebraicexpressionforthewelfaregainfromfinancialintegration.Section4
3
calibrates the model and does a sensitivity analysis for Tobin’s q, growth rate acceleration, and
welfaregaininrelationtothesizeoftheadjustmentcostparameter.Section5derivesamethodfor
calculatingthewelfaregainfrominternationalfinancialintegrationbasedontheobservedgrowth
rate acceleration. Section 6 extends the model to show that the proportional welfare gain from
integrationisnon-monotonicwithrespecttoincreasingadjustmentcosts.Thesectionalsoderives
thedynamicsassociatedwithanticipatedfinancialintegration.Section7concludes.
2TheModel
ThissectionmodifiesthestandardRamseymodeltoincludecapitaladjustmentcosts.Thismodel
was first developed by Abel and Blanchard (1983), and has subsequently been generalized by
Rappaport(2006)fortheanalysisofbottleneckdevelopmentpaths.
A representative infinitely-lived agent produces a single good with a linearly homogeneous
production function Af (k) . A is the total factor productivity (TFP) of capital and labor inputs in
theproductionfunction.FollowingUzawa(1969),LucasandPrescott(1971),andHayashi(1982),
capital accumulation (dk dt) is subject to linearly homogeneous costs: γφ (i,k) . The parameter γ measurestheefficiencyoftransforminginvestmentintonewcapital;inthissenseitbearsaclose
resemblancetoTFP.3Themaximizationproblemcanbeconsideredwitheitheraclosedoranopen
capitalaccount.Theclosedcapitalaccountwillbereferredtoasfinancialautarky.
2.1
Financialautarky
Therepresentativeagent’sutility-maximizationprobleminfinancialautarkyisthefollowing:
∞
V (k0 ,0,∞) = max ∫ u(ct )e− ρt dt subjectto
i
(2.1)
0
Af (kt ) = ct + it (2.2)
3Insection4,thespecificfunctionalformsfortheproductionfunctionandtheadjustmentcostfunctionwillbe
adopted: Af (k) ≡ Ak α and γφ (i,k) ≡ (γ 2)(i k − δ )2 k .
4
dk
= i − δ kt − γφ (it ,kt )
dt t
k0 given (2.3)
TheHamiltonianforthisproblemis:
e ρt H (t) = u ⎡⎣ Af (kt ) − it ⎤⎦ + q!t ⎡⎣ it − δ kt − γφ (it ,kt ) ⎤⎦ (2.4)
Theoptimalityconditionisgivenby(2.5):
u ′(ct ) = q!t ⎡⎣1− γφi (it ,kt ) ⎤⎦ (2.5)
where q! istheshadowpriceofcapitalmeasuredintermsofthemarginalutilityofconsumption.
Theshadowpriceofcapitalevolvesovertimeaccordingtotheequationofmotiongivenby(2.6):
dq!
= − u ′(ct ) Af ′(kt ) + ⎡⎣ ρ + δ + γφk (it ,kt ) ⎤⎦ q!t dt
(2.6)
Thetransversalityconditionis
lim q!t kt e− ρt = 0 (2.7)
t→∞
The linearized dynamics of consumption and the capital stock around a steady state k and i are
(seetheappendixforthederivation):
⎡
⎡ di ⎤ ⎢0
⎢ dt ⎥ ⎢
⎢ ⎥ ⎢
⎢ ⎥=⎢
⎢ dk ⎥ ⎢
⎢ ⎥ ⎢
⎣ dt ⎦ ⎢
⎣1
⎤
⎥ ⎡i − i ⎤
σ
+ γφii (i,k) ⎥ ⎢
⎥
c
⎥⎢
⎥ ⎥⎢
⎥
⎥⎢
⎥
⎥ ⎣k − k ⎦
⎥⎦
ρ
Af ′′(k)
(2.8)
where c = −cu ′′(c) u ′(c) is the inverse of the agent’s elasticity of intertemporal substitution in
consumption(EIS).
ThedynamicadjustmentissaddlepathstableandisillustratedinFigure1.Thehorizontalline
at δ = i k corresponds to dk dt = 0 from equation (2.3), while the d(i k) dt = 0 locus is
negativelysloped.TheinterpretationofFigure1isintuitivelyappealing:whentherelativepriceof
capital is greater than one, the rate of investment is greater than the rate of depreciation, so the
5
capitalstockgrows.Onthesaddlepaththedeclineintherateofinvestmentleadstheeconomyto
thestationarystate.
Theconvergencespeedofthecapitalstockinfinancialautarkyis:
2
⎛ ρ⎞
ρ
Af ′′(k)
µ= − ⎜ ⎟ −
(2.9)
σ
2
⎝ 2⎠
+ γφii (i,k)
c
Notethattheconvergencespeeddependsnegativelyontheconvexity( γφii )oftheadjustmentcost
functionaswellasontheEIS.
2.2TheFinanciallyIntegratedEconomy
The economy of 2.1 can be analyzed under the assumption of an open capital account, where
borrowing and lending on international capital markets is permitted. The model extends the
widely-employed representative agent small open economy model with adjustment costs for
capital [e.g., Blanchard (1983), Cohen and Sachs (1986), Matsuyama (1987)] to include a limited
commitmentconstraint.4
In the open economy capital accumulation takes place as in the autarkic economy. The
resourceconstraintismodifiedtoallowforborrowing( d )oninternationalcapitalmarkets.
The representative agent is able to pledge the incremental value of the capital stock, qt (kt − k0 ) ,
againstforeignborrowingwhere qt istherelativepriceoftheinvestmentgoodtotheconsumption
good.
Formally,theagentmaximizesutilityfromconsumptionsubjecttothedebtaccumulation
constraint,thecapitalaccumulationconstraint,andthelimitedcommitmentconstraint:
∞
V ( k0 ,d0 ,0,∞ ) = max ∫ u(ct )e− ρt dt
i
0
(2.10)
4SeealsoFrankelandRodriguez(1975)foranearlierversionofthisframework.
6
dd
= ct + it + rdt − Af (kt )
dt
dk
= i − δ kt − γφ (it ,kt )
dt t
dt ≤ qt (kt − k0 )
(2.11)
k0 given
(2.12)
d0 = 0 (2.13)
Inordertofocusonthewelfaregainfromfinancialintegrationarisingfromtransitionaldynamics,I
assumethattheworldinterestraterisequaltorateoftimepreference( r = ρ ).
TheHamiltonianforthisproblemis:
e ρt H (t) = u(ct ) − λt ⎡⎣ ct + it + rdt − Af (kt ) ⎤⎦ + q!t ⎡⎣ it − δ kt − γφ (it ,kt ) ⎤⎦ + ω! t ⎡⎣ dt − qt (kt − k0 ) ⎤⎦ (2.14)
where q! = λ q and ω! = λω are,respectively,theshadowvaluesofinstalledcapitalandthelimited
commitmentconstraint.Necessaryandsufficientconditionsforanoptimumare:
u ′(ct ) = λt (2.15)
qt = 1 ⎡⎣1− γφi (it kt ) ⎤⎦ (2.16)
dλ
= λt ( ρ + ω t − r) dt
(2.17)
dq
= − Af ′(kt ) + ⎡⎣ r + δ + γφk (it ,kt ) ⎤⎦ qt dt
(2.18)
Thetransversalityconditionsare:
lim λt dt e− ρt = 0 (2.19)
lim λt qt kt e− ρt = 0 (2.20)
Linearizationof(2.18)and(2.12)aroundthesteadystategivesthelocaldynamicsoftheprice
t→∞
t→∞
ofcapitalandthecapitalstock:5
5Thelinearizationisessentiallythesameasthatforsimilartextbookmodels,e.g.BlanchardandFischer(1989,
Chapter2).
7
⎡ di ⎤ ⎡ r
⎢ dt ⎥ ⎢
⎢ ⎥ ⎢
⎢ ⎥= ⎢
⎢ dk ⎥ ⎢
⎢ ⎥ ⎢
⎣ dt ⎦ ⎢⎣1
Af ′′(k) ⎤
⎡ i − i! ⎤
γφii (i,k) ⎥ ⎢
⎥
⎥
⎢
⎥ ⎥
⎥
⎥⎢
⎥ ⎢ k − k! ⎥
⎦
⎥⎦ ⎣
0
(2.21)
Thedynamicadjustmentissaddlepathstable.Theconvergencespeed ( µ * ) inthefinanciallyopen
economyis:
2
⎛ r⎞
r
Af ′′(k)
µ = − ⎜ ⎟ −
2
⎝ 2 ⎠ γφii (i,k)
Acomparisonof(2.9)with(2.22)indicatesthattheconvergencespeedinthefinanciallyopen
*
(2.22)
economy does not depend on the agent’s intertemporal elasticity of substitution because
investmentexpenditurecanbefullyfinancedbyforeignborrowing,giventhelimitedcommitment
constraint dt ≤ qt (kt − k0 ) .Asaresult,therateofinvestmentishigherintheopeneconomythanin
the financially closed economy. Figure2 illustrates this difference between financial autarky and
opennessbythedottedsaddlepaths,correspondingtofinancialautarky,andthesolidsaddlepaths,
correspondingtofinancialintegration.
In the financially autarkic economy consumption equals production minus investment:
ct = Af (kt ) − it . Steady-state consumption in the autarkic economy is c = Af (k ) − δ k . Under
limited commitment, the capital stock is financed by external borrowing. From (2.11) and a
bindingconstraint(2.13),consumptioninthefinanciallyintegratedeconomycanbesolvedforas:
ct =
dd
− i − rqt (kt − k0 ) + Af (kt )
dt t
where
dd
dk
dq
= qt
+ (kt − k0 )
dt
dt
dt
Using(2.12)and(2.18)for
(2.23)
dk
dq
,andcollectingterms,consumptionintheopeneconomyis
and
dt
dt
8
equaltotherealwage, w(kt ) = Af (kt ) − Af ′(kt )kt ,pluscapitalgainsincomefromtheinitialcapital
stock:
⎛
1 dq ⎞
ct = w(kt ) + qt k0 ⎜ r −
qt dt ⎟⎠
⎝
(2.31)
Steady-state consumption (c * ) is equal to autarkic steady-state consumption minus interest
paymentsonexternaldebt:
c * = c − rd = c − r(k − k0 ) (2.32)
Figure 3 compares the adjustment paths of consumption in financial autarky and financial
openness. The limited commitment constraint allows the agent to smooth consumption by
substitutingforeignfordomesticsavingtofinanceinvestmentinthecapitalstock.Thequasi-rents
generated by the time-zero installed capital stock raise consumption above the wage and
additionallyflattenthetimepathofconsumptioninthefinanciallyopeneconomy.
3TheWelfareGainfromFinancialIntegration
The welfare gain from financial integration generally refers to the gain associated with moving
from financial autarky to financial openness. Calculating this gain in this model involves making
two assumptions. First, that the initial stock of debt at the time of integration is zero (d0 = 0) .
Second,thatintegrationfollowsaone-offopeningofthecapitalaccount,referredtoas“aonce-andfor-all unanticipated regime change” by Mendoza, Quadrini, and Ríos-Rull (2009) and as an
“unexpectedandirreversible”reformbyAngeletosandPanousi(2011).Thismeansthatfinancial
integrationismodeledasaswitchattime0fromaperfectforesightpathunderfinancialautarkyto
aperfectforesightpathwithanopencapitalaccount.6Thispapermakesanadditionalassumption
6AsHall(2011)hasnoted,“experiencehasshownthattheseperfectforesightpathsgivereasonablyaccurate
measurementsoftheresponseofafullystochasticmodel,providedthattheinteractionswithuncertaintyarenot
important.”
9
oflimitedcommitment, dt ≤ qt (kt − k0 ) .Thismeansthattheagentcancommitcurrentandfuture
capitalincome,butnotwageincome.
The welfare gain from capital account opening is the difference in the maximized present
discountedvaluesofconsumptioninthefinanciallyintegratedandfinanciallyautarkiceconomies,
as given by equations (2.10) and (2.1). In autarky, the economy’s resource constraint can be
writtenas:
ct = w(kt ) + Af ′(kt )kt − it (3.1)
where is w(kt ) is the real wage. The value function (2.1) can then be linearized around a steadystateas:
∞
∫
∫
V (k) = u ⎡⎣ w(k ) ⎤⎦ e− ρt dt + u ′ ⎡⎣ w(k ) ⎤⎦ ⎡⎣ Af ′(kt )kt − it ⎤⎦ e− ρt dt 0
(3.2)
(
)
From (2.3) and (2.6), the second term of (3.2) is equal to q!o k0 , where q! = u ′ ⎡⎣ w(k) ⎤⎦ 1− γφi (i,k) .
Denoting q = q! u ′ ⎡⎣ w(k) ⎤⎦ , the value function (2.1), linearized around w(k ) , can be approximated
by
∞
V (k) = ∫ u ⎡⎣ w(k ) ⎤⎦ e− ρt dt + u ′ ⎡⎣ w(k ) ⎤⎦ q0 k0 (3.3)
0
Followingfinancialintegration(i.e.,openingofthecapitalaccount),thevalueoftheinstalledcapital
stock increases from q0 k0 to q0* k0 . The value function (2.10) can be linearized—using equations
(2.11),(2.12),andabindingcommitmentconstraint(2.13)--as:
∞
∫
V (k,d) = u ⎡⎣ w(k ) ⎤⎦ e− rt dt + u ′ ⎡⎣ w(k ) ⎤⎦ q0* k0 *
(3.4)
0
Thewelfaregainfrominternationalfinancialintegration(assuming ρ = r )istherefore:
(
)
ΔW! = V * (k,d) − V (k) = u ′ ⎡⎣ w(k ) ⎤⎦ q0* − q0 k0 Theappendixshowsthatthisgaincanalsobeapproximatedas:
10
(3.5)
(
ΔW! ≈ u ′ ⎡⎣ w(k ) ⎤⎦ γ µ * − µ
)( k − k ) ≥ 0 (3.9)
0
where µ and µ * are the convergence speeds of the capital stock in the autarkic and integrated
economies.Expressedasafractionofthesteady-statecapitalstock,thewelfaregainis:
) (
)
⎛ k − k0 ⎞
ΔW
= q0* − q0 = γ µ * − µ ⎜
≥0
k0
⎝ k0 ⎟⎠
(
(3.10)
where W = W! u ′ ⎡⎣ w(k ) ⎤⎦ . The size of the welfare increase is proportional to the adjustment cost
parameter (γ ) , the relative adjustment speeds along the transition paths, and the relative capital
scarcity (k0 k ) at the time of the capital account opening. It is worth noting that in the limiting
caseofzeroadjustmentcosts (γ = 0) ,thereis(approximately)nowelfaregainfromcapitalaccount
liberalization,aresultthatmirrorsthefindingofGourinchasandJeanne(2006).
4Calibration
This section provides numerical estimates of the importance of capital adjustment costs to
convergence speed, the price of capital (Tobin’s q), the rate of per capita income growth, the
current account, and welfare. As derived in the appendix, the speed of convergence around a
steady state in financial autarky can be expressed in terms of the model’s parameters as the
following7:
2
⎛ ρ⎞
ρ
(1− α )(r + δ )
µ= − ⎜ ⎟ −
ασ
2
⎝ 2⎠
+γ
ρ + (1− α )δ
(4.1)
In the financially open economy the adjustment speed does not depend on the elasticity of
intertemporalsubstitution:
7In(4.1)
ρ istherateoftimepreference, α istheshareofcapitalinproduction, δ istherateofdepreciation, σ is
thecoefficientofrelativeriskaversion,and γ isthecapitaladjustmentconvexityparameter.
11
2
⎛ r ⎞ (1− α )(r + δ )
r
µ = − ⎜ ⎟ +
2
γ
⎝ 2⎠
*
(4.2)
Note that with r = ρ , the steady-state capital stock k is the same in a financially integrated
economy as in autarky. This is because Af ′(k ) = ( ρ + δ ) in autarky and Af ′(k ) = (r + δ ) in a
financiallyopeneconomy.With ρ = r ,thedifferencebetweenthetwoeconomiesisonlythespeed
ofadjustmenttowardthesteady-statecapitalstock.
TheresultsforTobin’sq,growthacceleration,debtaccumulation,andwelfaregaindependon
the ratio of the capital stock to the steady-state capital stock at the time of the capital account
liberalization. The results also depend importantly on the assumed value of the convexity
parameter (γ ) for capital adjustment costs. Two studies using annual data estimate a capital
installationcostfunctionwith γ = 2 :
Hall(2002):approximately2fortheUnitedStatesusingannualdata(1958-1997)
Garcia-Cicco,Pancrazi,andUribe(2010):2.3-3forArgentinausingannualdata
(1900-2005)
Onestudythatestimatesavalueof γ ≈ 0.5 (onanannualizedbasis)isMendoza(2010)forMexico
usingquarterlydata(1993:1-2005:2).
Figures 4, 5, and 6 plot Tobin’s q, the growth acceleration at the time of the capital account
opening,andthewelfaregainfromcapitalaccountopeningfor γ = 0.5, 1, 2. Therearetwogeneral
resultsthatthefiguresillustrate:
1)LargeradjustmentcostsareassociatedwithhighervaluesofTobin’s“q”andsmallergrowth
accelerations, as shown in Figures 4 and 5. Although larger values of “q” are usually associated
withfastergrowth,thepriceofcapitalisdeterminedbyboththepresentdiscountedvalueoffuture
rentalsoncapitalandbyfuturemarginaladjustmentcosts.Anincreaseintotalfactorproductivity
will raise the price of installed capital and increase the growth rate, while an increase in capital
adjustmentcostswillraisethepriceofinstalledcapitalandlowerthegrowthrate.
12
2) The relative welfare gain from the opening of the capital account is greater for economies
that have larger capital adjustment costs, as shown in Figure 6. This result does not imply that
larger capital adjustment costs are better than small adjustment costs. It does mean that, in
economies for which the marginal costs of adjusting the capital stock are high, capital account
openingwillproduceasubstantialrevaluationoftheinitialcapitalstock.
The welfare gain from capital account opening also depends on the agent’s elasticity of
intertemporal substitution η = 1 σ . Most of the literature on the welfare gain from international
financial integration makes the assumption that the EIS is one, following the specification of
GourinchasandJeanne(2006).8AlargeliteraturesuggeststhattheEISmaybesubstantiallylower,
with a recent meta-analysis by Havranke, Horvath, Irsova, and Rusnak (2013) of 169 studies
covering104countriesfindingsubstantialheterogeneityinestimates,butwithameanestimateof
0.5. Table 1 shows the sensitivity of the welfare gain for combinations of three initial capital
scarcity ratios, three adjustment cost parameters, and three intertemporal rates of substitution.
Taking as a benchmark η = 1, k0 k = .83, and γ = 1 , the welfare gain from opening the capital
accountwouldbeabout2.8percentofthevalueofthecapitalstockwitharangeof2.3–3.1percent
for 0.5 ≤ γ ≤ 2 ; 2.2 – 3.3 percent for 2 ≥ η ≥ 0.5 ; and 1.4 – 4.2 percent for 0.91 ≥ k0 k ≥ 0.77 .
Varying all three parameter values simultaneously generates a range of welfare gains from 1.0
percent of the initial capital stock (k0 k = 0.91, γ = 0.5, η = 2) to 5.9 percent of the value of the
initialcapitalstock (k0 k = 0.77, γ = 2, η = 0.5) .
8SeeAntunesandCavalcanti(2013),Hoxha,Kalemli-Ozcan,andVollrath(2013),Corneli(2010),and
AngeletosandPanousi(2011).Mendoza,Quadrini,andRíos-Rull(2009a,b)use η = 1 2 ,while
Couerdacier,Rey,andWinant(2013)use η = 1 4 .ItshouldbenotedthatAPandCRWbothspecify
Epstein-Zinpreferencestodistinguishriskaversionfromintertemporalsubstitution.
13
5GrowthRateAccelerations
Aretherealternativewaystomeasurethewelfaregainfromfinancialintegration?Onewayisto
try to estimate directly the magnitude of the increase in quasi-rents following a financial market
liberalization. This literature, which was influenced by the work of Henry (2000), measures the
gain from international financial market integration by calculating excess returns on capital
associated with stock market liberalizations. Henry (2000) used a 12-country sample 9 of
liberalizationstoconstructaneight-monthpanelwindowpriortotheofficialimplementationofa
stockmarketliberalizations.10Hefoundacumulativerevaluationof38percent,withanestimated
26percentrevaluationafteradjustingfortheeffectsofforeignstockmarketfluctuationsandother
concurrentdomesticeconomicliberalizationprograms.
Kim and Singal (2000) used data on estimated stock market openings for 18 countries to
comparethechangeinmeanstockmarketreturnsfortheone-yearperiodaftertheopeningwith
two years prior to the opening.11 Using this window they estimated a cumulative increase of 12
percentacrossthe18countries.
Bekaert and Harvey (2000) estimated duration windows of excess stock market returns for
capital market liberalizations in 20 countries.12 For duration windows of twelve months (six
months prior to a stock market liberalization and six months following), they found cumulative
excessreturnsofabout15percent.13
Takenatfacevalue(andinthecontextofthispaper’smodel),thesefiguressuggestawelfare
gain in the range of 10-20 percent of the value of the initial capital stock due to international
financialmarketintegration.Butitisclearthatfirmslistedonthestockmarketsoftheliberalizing
countriesinthesethreestudiesformarestrictedsampleofcompanies;inaddition,thestudiesdo
9Argentina,Brazil,Chile,Colombia,India,Korea,Malaysia,Mexico,thePhilippines,Taiwan,Thailand,andVenezuela.
10Stockmarketliberalizationsaredefinedasthemonththeliberalizationwasimplemented,basedoninformation
takenfromseveralsourcesdocumentedbyHenry.
11ThecountriesareHenry’sminusMalaysiaplusGreece,Indonesia,Jordan,Pakistan,Portugal,Turkey,and
Zimbabwe.
12ThecountriesareHenry’splusGreece,Indonesia,Jordan,Nigeria,Pakistan,Portugal,Turkey,andZimbabwe.
13Theseresultswereconsistentacrossstockmarketliberalizationsbasedeitheronofficialliberalizationdatesoron
empirically-baseddatesfromstructuralbreaksinthedata.
14
notdifferentiatebetweencompaniesthathadincreasedaccesstointernationalcapitalmarketsand
thosethatremainedshutoutfromexternalfinancing.
Gozzi, Levine, and Schmukler (2008) directly compute a measure of (average) Tobin’s q for
about9,000firmsin74countriesduring1989-2000.Inthismicroeconomicstudyoftheeffectof
internationalizationonmarketcapitalization,theempiricalworkindicatesthat,forafirmthatgains
access to international capital markets, “q” rises about twelve percent in the two-year period
precedingtheinternationalization,andthenfallsbacktoitsinitiallevelduringthenextthreeyears.
Theempiricalmeasureof“q”isthefollowingratio:
q = ( market value of equity plus book value of debt ) ( book value of assets ) Abreakdownofthebehaviorofthenumeratoranddenominatorindicatesthattheincreasein“q”
priortointernationalizationisdrivenbyprimarilybyanincreaseinthestockmarketvalueofthe
firm, which remains high after the internationalization. The decline in “q” following the
internationalization is primarily driven by the denominator, suggesting that firms expand rapidly
bypurchasingothercompanies.
In terms of this paper’s model, the increase in “q” is consistent with the revaluation of quasi-
rentstoexistingcapitalatthetimeoffinancialintegration.Therapiddecreasein“q”forindividual
firmssuggestsverysmallcapitaladjustmentcosts,whichwouldbethecaseatamicrolevelfora
firm that can purchase capital assets from other domestic firms. However, the results do not
addresstheimportanceofcapitaladjustmentcostsattheaggregate(economy-wide)level.
A different approach is to look at growth accelerations that have followed financial market
liberalizations.Henry(2003)findsthatcapitalaccountliberalizationsareassociatedwithgrowth
rateaccelerationsofabouttwopercent(roughlystartingfromtwopercentperyearandthenrising
to four percent per year). Bekaert, Harvey, and Lundblad (2005) find that equity market
liberalizations lead to a growth acceleration for five years of about one percent, a finding that “is
robust to alternative definitions of liberalization and does not reflect variation in the world
businesscycle.”
15
Figure7takestheinformationfromTable1(plusotherinterpolatedpoints)andgraphsthem
as iso-welfare curves. Each curve represents the different combinations of the adjustment cost
technology (γ ) andinitialcapitalscarcity (k0 k ) thatgeneratetheequivalentrelativewelfaregain
(
)
fromopeningthecapitalaccount,expressedasapercentoftheinitialcapitalstock ΔW k0 .For
any given capital adjustment cost technology, the welfare gain from capital account liberalization
increaseswithcapitalscarcity.Asadjustmentcostsbecomelinear,thewelfaregainassociatedwith
capital account opening becomes smaller. There is no welfare gain from international financial
integration when the economy is initially at its steady-state capital stock (the vertical axis,
k0 k = 1 )orwhentherearenocapitaladjustmentcosts(thehorizontalaxis, γ = 0 ).
Figure 7 also plots two growth rate acceleration curves. These curves comprise the sets of
points k0 k and γ that generate an acceleration of the output growth rate by either one or two
percentage points at the time of the capital account opening. The upward slopes of the curves
reflectthe propertythat anincreaseinadjustmentcostslowerstheoutputacceleration,whilean
increaseininitialcapitalscarcityraisestheoutputacceleration(thecurvesaredrawnforEIS=1).
Thetwopercentcurveistotherightoftheonepercentcurvebecause,foranyinitiallevelofcapital
scarcity,lessconvexadjustmentcostsraisethegrowthrate.
Figure 7 shows that the welfare gain from capital account opening will depend on the initial
levelofcapitalscarcityandtheconvexityofadjustmentcosts,givenotherparametersofthemodel.
Gourinchas and Jeanne (2006) estimate that initial capital scarcity may be approximately 70
percentonaveragefornon-OECDcountries.Foracountrywithinitialcapitalscarcity k0 k = 0.71 ,
adjustment cost convexity parameter γ = 1.8 , and EIS=1, a one percent growth acceleration
following a capital account liberalization would imply a welfare gain of about 6.0 percent of the
value of the initial capital stock. An observed one-percent growth rate acceleration is also
consistent with other parameter values. For example, k0 k = 0.8 and γ = 1 are consistent with a
onepercentgrowthrateaccelerationandimplyawelfaregainofabout3.5percentofthevalueof
16
theinitialcapitalstock.Totheextentthatthesetwoparametervalues k0 k and γ areproblematic
to estimate, the example given here illustrates the difficulty in attempting to make precise
estimatesofthewelfaregainattributabletothegenerationofquasi-rentsontheinitialcapitalstock
resultingfrominternationalfinancialintegration.14
6Extensions
6.1HigherValuesof γ Sections4and5haveworkedwithvaluesoftheefficiencyofinvestmentparameter γ between0.5
and 2.0, in line with several results reported for the U.S. and Argentina, using annual data over
extended periods. However, other studies have estimated substantially higher values of γ . For
example,CooperandHaltiwanger’s(2006)estimateforaggregateU.S.investment(usingquadratic
adjustmentcosts)was γ = 5 for1972-1988.GilchristandHimmelberg(1995)estimated γ = 12 for
theirfullsampleofU.S.firmsduring1979-1989.Hayashi(1982)estimated γ = 24 foraggregate
U.S.investment1953-1976,whileSummersestimated γ = 8 forU.S.investmentbetween1932and
1978. In an open economy setting, Neumeyer and Perri (2005) simulated Argentinian business
cyclesbetween1983and2001withthreealternative(annualized)valuesofthecapitaladjustment
costparameter: γ = 2, 6, 10 .
With regard to the substantial differences among estimated values of the capital adjustment
parameter,KhanandThomas(2008)haveemphasizedthat:
Throughoutthehistoryoftheiruse,theprimarypurposeofadjustmentcostshasbeentoreduce
thedistancebetweenmodel-generatedandactualeconomictimeseries.Becausetheylargely
representimplicitcostsofforgoneoutput,wehavelittleabilitytodirectlymeasureadjustment
frictions.Thus,whenweadoptthemtoenhancetheempiricalperformanceofourmodels,the
resultingimprovementsare,insomesense,ameasureofourignorance….Thisraisesanobvious,
14Itisevenpossiblethatanobservedtwo-percentagepointgrowthrateaccelerationrepresentsasmallerrelative
welfaregainthanaone-percentagepointacceleration.Forexample,arelativecapitalscarcityofabout0.8combined
withaconvexityparameterof0.5correspondstoawelfaregainofabout2percentofthevalueoftheinitialcapital
stock,givenanobservedtwo-percentagepointgrowthrateacceleration.Arelativecapitalscarcityofabout0.7with
aconvexityparameterof1.0correspondstoawelfaregainofabout3.5percentoftheinitialcapitalstock,givenan
observedone-percentagepointgrowthrateacceleration.
17
butsometimesforgotten,point.Adjustmentcostsderivedwithinagivenclassofmodelmaybe
quiteinappropriateinasecond,distinctclassofmodel.
Inthispaper’smodel,asadjustmentcostsbecomemoreconvextherelativewelfaregainfrom
internationalfinancialintegrationfirstrisesasafunctionof γ ,asemphasizedinsections4and5.
Forsufficientlyhighvaluesof γ therelativewelfaregainisstillpositive,butdeclining.Figures8
and9illustratethisforaninitialcapitalscarcityof k0 k = 0.8 .Therelativewelfaregainfrom
!
internationalfinancialintegrationrisesrapidlyfromzerowhen γ = 0 toabout3.5percentofthe
valueofthecapitalstockwhen γ = 4 .Forhighervaluesof γ therelativewelfaregaindeclines,
fallingtoalittlelessthan2.5percentofthevalueofthecapitalstockwhen γ = 64 .
Thenon-monotonicityresultisduetothesupplyanddemandfactorsdeterminingthepriceof
capital.Internationalfinancialintegrationlowersthecostofcapital,therebyraisinginvestment
demandandthepriceofcapital,asshowninFigure8.Buttheefficiencyofinvestment—thatis,
theconvexitycoefficient γ --alsoplaysaroleinthedeterminationofthepriceofcapital.When
investmentisverycostly,thepriceofcapitalishighbeforefinancialintegration(asshownin
Figure8),butthegainfromintegrationissmall,asshowninFigure9,sincethehigherinvestment
demandonlyincreases q byasmallamount.
6.2AnticipatedInternationalFinancialIntegration
Internationalfinancialintegrationisgenerallyagradualprocess,asemphasizedbyHenry
(2000b),BekaertandHarvey(2000),Bekaert,Harvey,andLundblad(2001,2003,2005),Quinn
andToyoda(2008),Gozzi,Levine,andSchmukler(2008),andBalakrishnan,Vashishtha,and
Verrecchia(2014).Bekaert,Harvey,andLundblad(2003)tracethefinancialintegrationprocess
ofBrazilandKoreaoveratwelveyearperiod(1989-2001)toshowthedifficultyofchoosinga
singleimportantdateatwhichintegrationtookplace.
18
Inadditiontothegradualnessoftheprocessofintegration,importantequitymarket
liberalizationdateswereoftenannouncedinadvance.BekaertandHarvey(2000),andBekaert,
Harvey,andLundblad(2005)constructtimeseriesofliberalizationdatesfordevelopingcountry
equitymarketsthatemployindicatorvariablesforthe“firstsigns”ofliberalizationaswellasan
“official”liberalizationindicator.Thefirstsignsindicatoristhefirstofoneofthefollowingthree:
“alaunchingofacountryfund,anAmericanDepositaryReceipt(ADR)announcement,andan
OfficialLiberalization.”(BHL2005,p.12).Threeexamplesoftheestablishmentofcountryfunds
precedingofficialliberalizationareChile(1989countryfund,1992officialliberalization),India
(1986countryfund,1992officialliberalization),andKorea(1984countryfund,1992official
liberalization).15BekaertandHarvey(2000)findsignificantexcessreturnsinequitymarkets
followingfirstsigns,priortoofficialliberalizations.Henry(2000b)reportsexcessreturnof3.3
percentintheeightmonthspriortoaninitialstockmarketliberalizationforasampleof12
emergingmarketeconomies.
Inrelatedresearch,Gozzi,Levine,andSchmukler(2008)examinethebehaviorofTobin’sqfor
asampleof9,000firmsfrom74countriesduringthetimeperiod1989-2000.Usingasetof
annualdummyvariables,theyfindthat“qstartstorisetwoyearsbeforefirminternationalization,
whichisconsistentwiththemarketanticipatingthebeneficialeffectsofinternationalization.”
Theynotethat“theevolutionofqhasadistincthump-shapedpattern….risingbefore
internationalizationandevenfurtherduringtheyearofinternationalization,andthen
relinquishingthesegainsafterinternationalization,”asshowninFigure10(takenfromtheir
paper).
Figure11graphsthepriceofcapitalbeforeinternationalfinancialintegrationalongthe
saddlepathfrom (k0 ,q0 ) to (k ,1) .Iftheintegrationoffinancialmarketsisannouncedattime0,
aheadoftheactualtimeofintegration,theanticipationofafuturedeclineinthecostofcapital
15Aclosed-endcountryfund“isaninvestmentcompanythatinvestsinaportfolioofassetsinaforeigncountrybut
issuesafixednumberofsharesdomestically.(BHL2003).
19
leadstoanimmediateupwardjumpinthepriceofcapitalfrom q0 to q0* and,correspondingly,to
anincreaseinwealthfrom q0 k0 to q0*k0 .Thepricecapitalcontinuestoriseastheintegrationdate
(withitslowercostofcapital)approaches.Withinternationalfinancialintegrationtheeconomy
convergestoitsnewsaddlepathtrajectoryat (k fi ,q fi ) ,leadingtothesteady-statecapitalstock k .
Comparedtotheinitialsaddlepathtrajectory,thespeedofconvergenceishigherduetothelower
costofcapital.16
Correspondingtothebehaviorofthepriceofcapital,thegrowthrateacceleratesuponthe
announcementoftheplannedfinancialintegration.AsshownintheAppendix,thegrowthrateof
thecapitalstockisthefollowingfunctionofthepriceofcapital:
1 dk 1 ⎛ q − 1⎞
=
k dt γ ⎜⎝ q ⎟⎠
(6.1)
ThusinFigure11theaccelerationofthegrowthratepeaksatthemomentofthefinancial
integrationandthendeclinesmonotonicallyastheeconomygrowstowardthesteady-statecapital
stock k .
7Conclusion
ThispaperhasanalyzedthewelfaregainattributabletocapitalaccumulationinaRamseygrowth
model that is augmented by the inclusion of capital adjustment costs. The paper provides an
analytical framework that addresses the concern of Gourinchas and Jeanne (2006) regarding the
measurement of the welfare gain from transitional dynamics following international financial
integration. With no adjustment costs, as in the standard Ramsey model, the gains from
16ThisanalysisfollowsthatinAbel(1982)andBrock(1988).Moreformally,theannouncementofafuturedatefor
financialintegrationdoesnotchangethesteady-statecapitalstockorpriceofcapital( q = 1 ).Duringtheperiod
followingtheannouncementandpriortointegration,thedynamicsaredrivenbyboththenegativeeigenvalue µ in
(2.9)aswellasthecorrespondingpositiveeigenvalue.Followingintegrationthedynamicsaredrivenbythe
negativeeigenvalue µ * in(2.22).Ano-arbitrageconditiondeterminesthatthepriceofcapitalatthetimeof
integration( q fi )cannotjump.Giventheseconstraints,thesizeofthejumpinthepriceofcapitalatthe
announcementtimecanbeshowntobesmallerthefartherintothefutureistheactualliberalizationdate.
20
international financial integration are very small. Gains from financial integration are tied to the
existenceofcapitaladjustmentcosts.
Thepapershowsthatthesizeofthewelfaregainisrelatedtotheincreaseinthevalueofthe
installed capital stock at the time of international financial integration. Even if this increase in
value cannot be explicitly measured, the welfare gain from integration can be inferred from the
growthrateaccelerationthatfollowsanopeningofthecapitalaccount.
Inarelatedcontext,Bekaertet.al.(2005)conjecturethat“[T]hegrowtheffectshoulddepend
on two factors: how much additional investment the reforms generate (e.g., because the cost of
capital goes down) and the efficiency of new investments.” They add that “Countries with a
relatively high physical and human capital stock, relatively efficient financial markets, good legal
institutions,andsoon,mightseehighlyefficientinvestmentandalargegrowthresponse.”
If one takes γ (the convexity of the adjustment cost function) as a reduced-form measure of
investment-specific productivity, then more efficient investment (lower values of γ ) will, as
suggested by Bekaert et. al. (2005) result in a larger growth response to international financial
integration,asshowninFigure5.Buttheproportionalincreaseinwelfareisnon-monotonicinthe
efficiencyofinvestment,asshowninFigure9.Atlowvaluesof γ ,welfareinproportionaltermsis
increasingintheconvexityofadjustmentcostsduetothegreaterrevaluationofthecapitalstock
(duetohigherquasi-rents)thataccompaniesthereductioninthecostofcapital.Athighlevelsof
γ , the proportional increase in welfare is declining in the convexity of adjustment costs: even
though the capital stock revaluation effect still operates, the supply response of investment
declines.
21
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24
Table 1
Welfare Gain from International Financial Integration
⎛ k − k0 ⎞
ΔW
= q0* − q0 = γ µ * − µ ⎜
k
⎝ k ⎟⎠
(
0
k k0
k0 k
η=2
γ = 0.5
η =1
η = .5
)
η=2
0
γ =1
η =1
η = .5
η=2
γ =2
η =1
η = .5
1.1
0.91
.010
.012
.013
.011
.014
.016
.011
.015
.020
1.2
0.83
.020
.023
.026
.022
.028
.033
.022
.031
.039
1.3
0.77
.029
.035
.039
.033
.042
.049
.033
.046
.059
2
⎞
γ⎛i
γ is the capital adjustment convexity parameter: γφ (i,k) = ⎜ − δ ⎟ k
2⎝k
⎠
η is the intertemporal elasticity of substitution: −
u ′(c)
cu ′′(c)
Figure1
TransitionDynamicsinFinancialAutarky
dk
=0
dt
25
26
1.08
1.02
1.00
1.8
1.06
1.04
1.2
0.8
.80
0.4
0.0
.90
1.00
.80
.90
1.00
.08
.06
.04
.02
.00
.80
.90
1.00
Inallthreeofthefiguresthehorizontalaxisistheratiooftheinitialcapitalstocktothesteady-statestock.
isameasureoftheefficiencyofinvestment—itparameterizestheconvexityofthecapitaladjustment
costfunction.
InFigure4“ ”isthepriceofinstalledcapitalafterfinancialintegration.
InFigure5
isthepercentagepointincreaseinthegrowthratefollowingfinancialintegration.
InFigure6thewelfaregainfrominternationalfinancialintegrationisexpressedasafractionoftheinitial
capitalstockandtheefficiencyofinvestment.
27
Figure7
TheGainfromInternationalFinancialIntegration:
WelfareandGrowthRateAccelerationCurves
28
Figure8
ThePriceofInstalledCapital
1.40
1.30
1.20
1.10
0.4
0.5
0.7
1
1.1
2
1.6
4
2.2
8
2.8
16
3.5
32
4.2 ln(1+γ)
64 γ
Note:Thegraphshowstherelativepriceofinstalledcapitalattimezero,bothinfinancial
autarky( )andwithinternationalfinancialintegration( )foraninitialrelativecapital
scarcityof
.Thevaluesof
convexityparameter(rangingfrom
areplottedagainstvaluesoftheadjustmentcost
to
29
).
.04
Figure9
TheRelativeWelfareGainfromInternationalFinancialIntegration
.03
.02
.01
0.4
0.5
0.7
1
1.1
2
1.6
4
2.2
8
2.8
16
3.5
32
4.2 ln(1+γ)
64 γ
Note:Thegraphshowstherelativewelfaregainfrominternationalfinancialintegration
foraninitialrelativecapitalscarcityof
.Thewelfaregainisplotted
againstvaluesoftheadjustmentcostconvexityparameter(rangingfrom
30
to
).
Figure10
DynamicAdjustmentofTobin’sq
for3,351FirmsthatInternationalizedBetween1989and2000
Source:Gozzi,Levine,andSchmukler(2008)
31
Figure11
AnticipatedInternationalFinancialIntegration
andthePriceofCapital
32
Appendix:DerivationofEquations
1.DerivationofEquation(2.13)
Fromequation(2.5)(withtimesubscriptssuppressed):
u ′(c) = q! ⎡⎣1− γφi (i,k) ⎤⎦ weobtainthefollowing:
u ′′(c)
dc di
di
dq!
= − q!φii + ⎡⎣1− φi ⎤⎦ di dt
dt
dt
dq! q!φii − σ c di
=
dt
1− φi dt
(a)
Theco-stateequation(2.6)isthefollowing:
dq!
= − u ′(c) Af ′(k) + ⎡⎣ ρ + δ + γφk (i,k) ⎤⎦ q! dt
(b)
Substituteout
dq!
in(a)and(b)toget(c):
dt
(c)
2
⎫
⎧⎪
⎡⎣ ρ + δ + γφk (i,k) ⎤⎦ ⎫⎪ ⎧⎪
⎡⎣1− φi (i,k) ⎤⎦
di
⎪
= − ⎨ Af ′(k) −
⎬⎨
⎬ dt
1− φi (i,k)
⎪⎩
⎪⎭ ⎪⎩ ⎡⎣1+ φi (i,k) ⎤⎦σ c + φii (i,k) ⎪⎭
Linearizing(c)aroundasteadystategives:
di
Af ′′(k)
=
(k − k ) dt σ
+ φii (i,k)
c
Thisgivesthefirstlinearizationusedinequation(2.8).
Thesecondlinearizationin(2.8)followsdirectlyfromthecapitalaccumulationconstraint(2.3)in
conjunctionwiththeeconomy’sresourceconstraint(2.2).
2.Derivationoftheconsumptionequation(2.32):
dd
dk
dq
= qt
+ (kt − k0 )
(limited commitment constraint) dt
dt
dt
33
dd
= ct + it + rdt − Af (kt )
dt
(current account) {
}
qt ⎡⎣ it − δ kt − γφ (it ,kt ) ⎤⎦ + kt − Af ′(kt ) + ⎡⎣ r + δ + γφk (it ,kt ) ⎤⎦ qt − k0
dq
dt
= ct + it + rqt (kt − k0 ) − Af (kt )
*
Cancellingoutcommontermsanddenotingopeneconomyconsumptionby ct leaves:
ct* = Af (kt ) − Af ′(kt ) − k0
dq
+ rqt k0
dt
⎛
1 dq ⎞
ct* = w(kt ) + qt k0 ⎜ r −
qt dt ⎟⎠
⎝
c * = w(k ) + rk0
3.Derivationoftherelativepriceofcapital(q):
From(2.5)and(2.16):
1 dk qt − 1
=
(*) kt dt
γ qt
Aroundasteady-stateequilibrium,thelinearizeddynamicsaregivenby:
*
kt = k + (k0 − k )e µ t
dkt
= µ * (k0 − k )e µt = µ * (kt − k ) dt
⎛k −k⎞
1 dkt
= µ* ⎜ t
(**)
kt dt
⎝ kt ⎟⎠
Substitutingoutthegrowthrateofcapitalaccumulationin(*)and(**)gives
1
qt =
⎛ kt − k ⎞
1− γµ ⎜
⎝ k ⎟⎠
t
34
⎛k −k⎞
qt ≈ 1+ γµ ⎜ t
⎝ kt ⎟⎠
4.Derivationofthenumericalexample:
Let f (k) = Ak α , f ′(k) = α Ak α −1 = r + δ aroundasteadystate, f ′′(k) = α (α − 1) Ak α −2 .Then
kf ′′(k) = α (α − 1) Ak α −2 = (α − 1)(r + δ ) .
γ (i − δ k)2
2
k
γ
φi (i, k) = (i − δ k) k
γ
φii (i, k) =
k
Let φ (i, k) =
Atasteadystateinaclosedeconomythefollowingholds:
Then
c f (k)
=
−δ
k
k
α Ak α −1
=
−δ
α
ρ +δ
=
−δ
α
ρ + (1− α )δ
=
α
f ′′(k)
kf ′′(k)
(α −1)( ρ + δ )
=
=
σ
σ
ασ
+ φii k + kφii
+γ
c
c
ρ + (1 − α )δ
Thetransitionspeedinfinancialautarkywiththesespecificfunctionalformsfortheutility,
installation,andproductionfunctionsis:
2
⎛ ρ⎞
⎛ ρ⎞
(1− α )( ρ + δ )
µ=⎜ ⎟− ⎜ ⎟ +
ασ
⎝ 2⎠
⎝ 2⎠
+γ
ρ + (1− α )δ
Thecorrespondingconvergencespeedinafinanciallyopeneconomyis:
2
⎛ r⎞
⎛ r ⎞ (1− α )(r + δ )
µ =⎜ ⎟− ⎜ ⎟ +
where r = ρ
γ
⎝ 2⎠
⎝ 2⎠
*
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