Download 5. Dynamics of a Small Open Economy y p y

Prof. Dr. Thomas Steger
Advanced Macroeconomics II | Lecture| SS 2014
5. Dynamics
y
of a Small Open
p Economyy




Introduction
A open-economy Ramsey
An
R
model
d l
Behavior of a small open economy
World equilibrium
 Credit constraints
 Capital adjustment costs
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
I t d ti
Introduction
 An open-economy
open economy version Ramsey model is considered
considered.
 There is international trade in final output goods and there is
borrowing and lending.
 It is shown that the open
open-economy
economy version of the Ramsey model leads
to a number of paradoxical conclusions.
 We consider two extensions that can be employed to avoid these
shortcomings
 imperfections of world credit markets
investment adjustment costs
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
A open-economy Ramsey
An
R
model
d l (1)
 Basic assumptions
 The world comprises a large number of small open economies.
 Each
E h off these
h
countries
i represents a Ramsey-type
R
economy.
 There is international trade in goods, such that GDP may diverge from domestic
expenditures (C+I)
(C+I).
 The intertemporal aspects of international trade are considered, whereas the
implications for the specialization are neglected
neglected.
 Domestic and foreign claims on capital are perfect substitutes as stores of value.
p y the same rate of return (ROR)
(
) r.
Therefore,, both must pay
 Labor is immobile, domestic residents cannot work abroad and foreigners cannot
work in the domestic economy.
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
A open-economy Ramsey
An
R
model
d l (2)
 Setup of the model
Economy i has assets per person ai and capital per person ki.
 Net claims by foreigners on the domestic economy are di=ki-ai.
 Assets per person equal domestic capital per person less the foreign debt per person: ai=ki-di.
Aggregate foreign debt reads Di=Lidi.
 The current account balance (CA) may be written as -Di.
 The p
per capita
p CA ((-Di//Li) for economyy i equals
q
−
 An economy that runs a positive CA (exports
exceed imports) accumulates claims against
the rest of the world.
 Hence, the net debt decreases: CAi=-Di.
D i
L d + Li di
=− i i
= − ( ni di + di )
Li
Li
The household’s budget constraint in country i is given by ai = wi + (r − ni )ai − ci
 Continuum off llength
h L off
identical agents, every
agent is endowed with one
unit of time per period.
A = wL + rA − C
Consumption will follows the well-known Keynes-Ramsey rule ci =
ci
σi
Factors are rewarded according to marginal productivities
f ′(ki ) = r + δ i ; f (ki ) − ki f ′(ki ) = wi
(r − ρi )
a :=
A
 aˆ = Aˆ − Lˆ
L
aˆ =
wL + rA − C ˆ
−L
A
a = w
L
C
ˆ
a + ra − a − La
A
A
C
a = w + (r − Lˆ )a −
L
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
A open-economy Ramsey
An
R
model
d l (3)
 Reduced form of the model
 The
Th a-equation
i may be
b expressed
d as ffollows
ll
ai =
w
i
+ ( r − ni ) ai − ci
=f ( ki ) − ki f ′ ( ki )
ai = f (ki ) − ki f ′(ki ) + (r − ni ) ai − ci

=r +δ i
ai = f (ki ) − (r + δ i ) ( ki − ai ) − (ni + δ i )ai − ci
 The
Th ffollowing
ll i dynamic
d
i system
t
( ith b
(with
boundary
d
conditions)
diti ) d
describes
ib th
the evolution
l ti off th
the economy
(1) ci =
ci
σ
(r − ρi )
(2) ai = f (ki ) − (r + δ i ) ( ki − ai ) − (ni + δ i ) ai − ci
((3))
f ′( ki ) = r + δ i
(4) di = ki − ai
 Notice that ki is determined, given r, by
equation (3): the capital stock in a small
open economy is determined by the
exogenous interest rate.
q
((1)) and ((2)) determine the
 Given ki equations
dynamics of ci and ai.
 Given ki and ai equation (4) determines di.
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
B h i off a smallll open economy (1)
Behavior
 Further assumption
 It is assumed that r is fixed (time invariant), i.e. the world is in steady state.
 Moreover, we assume that r≤ρi (if r>ρi, the domestic economy would accumulate enough
assets to violate the small-country assumption).
 We also assume that r>ni. The world interest rate exceeds the steady state growth rate that
would apply in the closed economy. Otherwise, the PDV of wages will turn out to be infinite
and, hence, the attainable utility will be unbounded.
 Counterfactual implication #1: capital stock
 If r is
i constant,
t t th
then f’(ki)=r+δ
) δi determines
d t
i
th
the steady
t d state
t t level
l l off k,
k denoted
d
t d as kiopen.
 Starting from any ki(0)<kiopen, capital flows into the economy such that, an instant in
time after opening up the economy, ki=kiopen. The speed of convergence is infinite (i.e.
the economy jumps into the steady state).
 If, on the other hand, ki(0)>kiopen, capital flows out of the economy such that ki=kiopen.
 This counterfactual prediction of an infinite speed of convergence for ki is one of the
problematic implications of the open-economy version of the Ramsey model.
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
B h i off a smallll open economy (2)
Behavior
 Output and wages
 The steady state level of ki, which follows from ff’(k
(ki)=r+δi, determines GDP per capita yi=f(ki).
)
 The steady state level of ki also determines the wage rate wi=f(ki)-kf’(ki).
 Both wi and yi converge to their respective steady state values at an infinite speed.
 Consumption
 Provided that (r-ρi)/σ=0, ci remains constant.
 If,
If on the other hand,
hand (r-ρ
(r ρi)/σ<0,
)/σ<0 ci decreases exponentially and approaches zero as t→∞.
t→∞
 Domestic residents borrow to enjoy a high consumption early on (“impatience in the sense ρi>r”) but
have to pay the price in the form of low consumption levels later on.
 Counterfactual implication #2: wealth (impatient economy)

w iopen 
ai (t ) =  ai (0) +
e

r − ni 

 r − ρi 

t
 σi 
∞
w iopen
−
r − ni
>0

open − ( r − ni ) t
i
 w
0
−
e
dt =
w iopen − ( r − ni ) ∞  w iopen − ( r − ni )0  w iopen
−−
e
e
=
r − ni
 r − ni
 r − ni
 This solution results from equations (1) and (2) (cf. Advanced Macroeconomics Exercises: Chapter 5).
 The final term is the “PDV of wages” (notice that both w and r are constant).
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
B h i off a smallll open economy (3)
Behavior
 Counterfactual implication #2: wealth (cont’d)
a
100
The time path of wealth in an “impatient economy” (r<ρi).
80
 r − ρi 
t
σi 

w iopen  
ai (t ) =  ai (0) +
e
r − ni 

60
40
w iopen
−
r − ni
20
20
40
60
80
100
t
-20
-40
ki = d i
ai (∞ ) = −
w iopen
r − ni
 For r<ρi the first term on the RHS diminishes and approaches zero (if, on the other hand, r=ρi, ai=const.).
 Provided that ai(0)>0, ai sooner or later equals zero. When ai=0, di=ki (the net debt to foreigners equals
the domestic capital stock).
 Subsequently,
b
l ai becomes
b
negative. The
h d
domestic country b
becomes a d
debtor
b not only
l in the
h sense off
not owning its domestic capital but also of borrowing against the PDV of future wage income as
collateral (“Pfand”).
 Asymptotically,
Asymptotically ai approaches -w
wiopen/(r-n),
/(r n) so that di=ki-a
ai=kiopen+wiopen/(r-n).
/(r n)
 An impatient country asymptotically mortgages all its capital and labor income (“mit Hypothek
belasten”) (counterfactual implication #2).
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
B h i off a smallll open economy (4)
Behavior
Terminology
Economy 1 is the most patient economy if ρ1<ρ2<…
Economy 1 is called impatient if ρ1>r.
 World Equilibrium
ρ3
ρ2
r > ρ 1 : country1’s consumption would eventually exceed world output
ρ1
r = ρ1
r < ρ 1 : most patient economy would end up being indebted
1
2
3
economies
 Suppose the world consists of a set of countries indexed by i∈{1,…,M}, which can be ordered in terms
of their time preference rates, ρi, with country 1 having the lowest value (“country 1 is most patient”).
 If ρ1>r,, c1→0 aand
d a1→
→-w1open/(
/(r-n),
), i.e.
.e. tthee most
ost pat
patient
e t eco
economy
o y ends
e ds up being
be g indebted.
debted.
 If ρ1<r, c1 and a1 would rise forever and country 1’s consumption would eventually exceed world
output (“country 1 experiences the highest growth rate!”).
 Neither ρ1>r nor ρ1<r can be an equilibrium (world capital stock must be owned by someone and
consumption of one country cannot exceed world output).
 Hence, only the constellation is ρ1=r (“world interest rate equals time preference rate of most patient
country”)) is compatible with an equilibrium in the global economy
country
economy.
 In this case, country 1 owns all the wealth (asymptotically) in the sense of the claims on capital and
the PDV of wage income in all countries (counterfactual implication #3).
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C dit constraints
Credit
t i t (1)
 It has been shown that the open-economy Ramsey model leads to several
paradoxical conclusions
 infinite adjustment
j
to the steadyy state
 consumption of all but the most patient economy goes to zero
patient economyy ends up
p owningg all the wealth
 most p
 The credit constraint model: basic assumptions
p
 There are two types of capital, i.e. physical capital and human capital.
 Physical capital serves as security on loans since the creditor can take possession of the
object in case of default (“Zahlungsunfähigkeit”,
(“Zahlungsunfähigkeit” “Zahlungseinstellung”)
“Zahlungseinstellung”).
 In contrast, human capital and future wage income cannot serve as collateral
(“Sicherheit”).
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C dit constraints
Credit
t i t (2)
Model setup
 The
Th production
d ti technology
t h l
i given
is
i
b
by
Y = AK α H η L1−α −η

y = Ak α hη =: f (k , h)
 It is assumed that 0<α+η<1 such that there are diminishing returns to “broad capital”.
 Thi
This iis b
basically
i ll a one-sector
t model
d l in
i th
thatt output
t t off th
the fi
final-output
l t t sector
t can be
b used
d ((onefor-one) for consumption, investments to physical capital or additions to human capital.
 Wealth is given by a=k+h-d and the budget constraint now reads
a = Akk α hη − (r + δ )(k + h − a ) − (n + δ )a − c
(*)
 The derivation is analogous to the previous chapter; notice that: w=f(k,h)
w=f(k h)-ffkkk-ffhh.
h
11
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C dit constraints
Credit
t i t (3)
 The closed economy
 In the closed economy we have d=0 and hence a=k+h.
 In the steady state the interest rate r is constant. Moreover, investors equate the
MP of each type of capital to r+δ,
r+δ i.e.
i e fk(.)=α(y/k)=r+δ
( )=α(y/k)=r+δ and fh(.)=η(y/h)=r+δ
( )=η(y/h)=r+δ such
that
k* α
=
*
h η
 In the steady state the quantities of k and h are constant.
p types
yp is ggiven byy preceding
p
g equation.
q
 The ratio of both capital
 It can be shown that the (local) speed of convergence is the same as for the
standard Ramsey model with a capital share equal to α+η.
α+η
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C dit constraints
Credit
t i t (4)
 The open economy (1)
The distinction between two types of capital becomes more interesting when considering the open
economy under credit constraints.
It is assumed that the amount of foreign debt d cannot exceed k, i.e. d≤k: physical capital can be
used
d as collateral
ll
l on fforeign
i lloans, b
but h
human capital
i l and
d llabor
b iincome cannot!!
!!
The world interest rate is assumed fixed at r=ρ (“steady state interest rate that would apply in a
closed economy”).
Provided that k(0)-d(0)≥h*-h(0), the borrowing constraint (d≤k) is not binding and the economy
jumps into the steady state k=k* and h=h*.
 In this case,, initial k is high
g enough
g to serve as collateral for loans to finance required
q
h-investment.
 Notice that k-investment can be financed without restriction since k serves as security.
If, on the other hand, k(0)-d(0)<h*-h(0), the borrowing constraint (d≤k) is binding.
There are two institutional settings…
 Domestic residents own the physical capital stock but obtain part of the financing for this stock by issuing
bonds to foreigners.
 The result would be the same if we allowed for FDI, in which case the foreigners would own part of the
physical capital stock rather than bonds.
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C dit constraints
Credit
t i t (4
(4a))
 Case #1
k (0) − d (0) ≥ h* − h(0)
10
0
10
5
10 ≥ 5
 The desired investment of Δh=5 can be financed by running into debt where a fraction
of k(0)=10 is used as collateral; d≤k is not binding!
 Case #2
k (0) − d (0) < h* − h(0)
10
10
10
5
0<5
 The desired investment of Δh=5 cannot be financed byy runningg into debt and usingg k(0)
( )
as collateral; d≤K is binding!
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C dit constraints
Credit
t i t (5)
The open economy (2) (d≤k is binding)
 Since k serves as collateral fk=α(y/k)=r+δ.
=α(y/k)=r+δ Hence,
Hence one gets k=αy/(r+ δ).
δ)
 When k serves as collateral, the stock of physical capital is increased immediately such that fk=r+δ.
 This does not apply to human capital. Since h does not serve as collateral, it cannot be increased immediately
such that fh=r+δ.
r δ. Along the transition fh>r+δ
r δ might apply. But h is increased up to the point where d
d=k.
k.
 Plugging k=αy/(r+δ) into y=Akαhη gives
α
 αy  η
y = A
 h  y=A
r
+
δ


1
1−α
α
1−α
 α 


 r +δ 
h
η
1−α
 ε
 y = Ah
1
1−α
with A := A
α
1−α
 α 


 r +δ 
and ε :=
η
1−α
The condition 0<α+η<1 implies 0<ε<α+η<1.
0<ε<α+η<1
 Noting d=k, a=h, α(y/k)=r+δ, the budget constraint (*) can be written as
GDP

 ε − α Ah
 ε − (δ + n) h − c
h = 
Ah
(**)
GNP
 The term αAhε corresponds to rental payments on physical capital. Since d=k this term corresponds to net
factor payments to foreigners.
15
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C dit constraints
Credit
t i t (6)
The open economy (3)
 We
W assume that
th t households
h
h ld produce
d
goods
d di
directly
tl ((producer-consumer
d
approach).
h)
Households then maximize utility ∫u(c)e -ρtdt subject to the budget constraint (**) and
h(0)=h0.
∂GNP


∂h





c 1 
ε
−
1
 Consumption then follows the well-known KRR
= (1 − α ) 
Aε
h − δ − ρ 
c σ

∂GDP
=f h ( =
)
∂
h


 Because we assume that r=ρ, the steady state is the same as for the closed economy (in
steady state fk=ffh=r+δ
r δ applies).
 The opportunity to borrow on the world capital market affects only the transition
towards the steady state.
 The open economy under credit constraints converges smoothly towards the steady state (i.e.
there is no jump).
 The credit-constrained open economy exhibits a higher rate of convergence compared to the
closed economy. (“k is relatively high at the outset in an open economy because the availability
of foreign financing makes is easy to acquire capital quickly.”)
16
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C it l adjustment
Capital
dj t
t costs
t (1)
 Capital adjustment cost (CAC) represent an alternative and complementary
mechanism to avoid some of the counterfactual implications outlined above.
 CAC do aalso
so lead
ead to
o aan instructive
s uc e eexplanation
pa a o o
of the
e de
demand
a d for
o investment
est e t goods.
goods
 The standard neoclassical growth model assumes that there are no capital adjustment
costs. Whatever is saved can be added to the capital stock at no cost.
 This implies that investment is purely passive
passive, i.e.
i e investment is determined residually as
output not consumed.
 Put differently, within this setup saving decisions cannot be separated from investment
decisions.
decisions
 This means that there is no well-defined demand for investment goods.
 To keep things simple, we assume that the interest rate is fixed.
 This applies from the perspective of an individual firm or
 can be considered as a short cut for analyzing the small open economy case (however,
(however we
do not model intertemporal resource trade and the accumulation of external debt).
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Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C it l adjustment
Capital
dj t
t costs
t (2)
Firms (1)
 The production function (assumed to satisfy Inada conditions) of final-output firms is
given by
 Inada conditions are assumed to hold:
Y = F ( K , L)
 FK(.)>0
 FKK(.)<0
 limK→0FK (.)
(.)=∞
limK→∞FK (.)=0
 Firms own the capital stock and households own the firms and hence have a claim on the
firm’s net cash flow ((or entrepreneurial
p
residual income).
)
 The change in the firm’s capital stock is given by
K =
I
−δ K
gross
investment
 The cost of investment is ggiven byy

 I 
Cost of investment = I 1 + φ   
 K 

where φ(0)=0, φ‘(.)>0, and φ‘‘(.)≥0.
 Why do we use φ(I/K) instead of φ(I)?
 This modeling assumption avoids the
(implausible) implication that CAC per unit
of I increase as the economyy grows.
g
 The fact that CAC per unit of I decrease with
K represents a learning by doing effect.
18
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C it l adjustment
Capital
dj t
t costs
t (4)
Firms (3)
 The firm
firm’ss net cash flow (or entrepreneurial residual income) is given by

 I 
Net Cash Flow = F ( K , L) − wL − I 1 + φ   
 K 

 The value of outstanding equity shares at time 0 is determined on a stock market to be
V(0);
( ); byy normalizingg the number of shares to one,, V(0)
( ) ggives the p
price of each share.
 The net cash flow is paid out as dividends to shareowners and hence V(0) equals the PDV
of future cash flows.
 The firm is assumed to maximize the value of the firm V(0), i.e.
∞


 I   
− rt 
max V (0) =  e  F ( K , L) − wL − I 1 + φ    dt 
I ,L
 K   



0
s.t. K = I − δ K , K (0) = K 0
19
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C it l adjustment
Capital
dj t
t costs
t (4
(4a))
Firms (4)
 The (current value) Hamiltonian function is given by

 I 
J = F ( K , L) − wL − I 1 + φ    + q ( I − δ K )
 K 

 q is the current-value shadow price of installed capital K; q has units of goods per unit of
capital at time t (hence “current value”).
 The present-value shadow price is then v=qe-rt.
 The first-order conditions are given by (plus a standard TVC: limt→∞vK=0)
∂J
= FL (.) − w = 0  FL (.) = w
∂L
∂J
I

 I 
I 1
= − 1 + φ    − Iφ ′   + q = 0  q = 1 + φ (.) + φ ′ (.)
K
∂I
 K 
K K

2
∂J


 I 
I 
+ rq ⇔ q = −  FK (.) − Iφ ′(.)  − 2  − δ q  + rq  q = ( r + δ )q − FK (.) −   φ ′(.)
q = −
∂K
 K 
K


20
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C it l adjustment
Capital
dj t
t costs
t (5)
Firms (5)
 The first
first-order
order conditions for the above problem in intensive form read
(1)
(2)
(3)
f (k ) − f k (k )k = w
i i i
q = 1 + φ   + φ′ 
k k k 2
i
i
q = (r + δ )q − f k (.) −   φ ′( )
k
k
Equ. (1), the FOC for optimal L, states that the wage rate must equal the MPL.
Equ. (2) , the FOC for optimal i, shows that the shadow price of installed capital q>1 if i>0.

Equ.
( ) can be rewritten as
(3)
RHS: ROR resulting from investing one unit of
LHS: market rate of return
2
1
q
i ′ i 
r =  f k (.) +   φ ( )  − δ +
q
k 
q
k
output in physical capital (when the price of
physical capital is q, one unit of output buys
/q units of p
physical
y
capital);
p ); three
1/q
components:
(i) MPC plus the marginal reduction in CAC
(due to increase in k); (ii) capital depreciation;
(iii) rate of capital gain.
 This is a standard Fisher-type, no-arbitrage condition of the form: q+payoff=rq (2 assets: bond paying r and
“2nd asset” with price q paying “payoff”)
 If adjustment costs were absent this boils down to r=fk(.)-δ.
21
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C it l adjustment
Capital
dj t
t costs
t (6)
Firms (6)
 Equ.
E
(2) which
(2),
hi h shows
h
a monotonically
t i ll iincreasing
i relation
l ti b
between
t
q and
d i/k,
i/k can be
b
inverted to yield
 i  i ′ i 
q = 1 + φ   + φ    i = kψ (q )
k k k
with
ψ ′(q ) > 0
This relation gives the demand for investment goods: demand for investment is an
increasing function of q, the value of installed capital.
In empirical applications one usually uses, following a suggestion by Brainard and Tobin
(1968), the ratio of firm’s market value to the capital stock, i.e. V/K, as proxy for q.
V/K is the average q,
q whereas the shadow price of installed capital (as in the model
considered here) is called marginal q.
Hayashi (1980) has shown that both average and marginal q coincide provided that (i)
the technology exhibits CRS and (ii) the stock market is efficient.
22
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C it l adjustment
Capital
dj t
t costs
t (7)
Steady state and transitional dynamics (1)
 We assume the following specific adjustment cost function
i
bi
φ   =
; b>0
k
2k
(***)
so that φ‘(i/k)=b/2>0.
 The parameter b>0 expresses the sensitivity of CAC to the total amount invested.
invested A
higher value of b implies more CAC per unit of i/k.
 Plugging
ugg g (***)
( ) into
to equ. ((2)) ggives
es i=k((q-1)/b).
((q )/b). Thiss equat
equation
o toget
together
e with
t K=I-δK
δ implies
p es
q −1

− (n + δ )  k
k = i − ( n + δ )k = 
 b

 Moreover, equ. (3) together with i=k((q-1)/b) gives
This constitutes a twodimensional dynamic
system in k and q.
(q − 1) 2
q = (r + δ )q − f k (.) −
2b
23
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C it l adjustment
Capital
dj t
t costs
t (8)
Steady state and transitional dynamics (1)
 The k=0 locus is horizontal
at q
q*.. The q=0
0 is downward
sloping (around the steady
state).
 Recall that q is the market
value of a unit of installed
capital.
stab e saddle
sadd e pat
path iss
 Thee stable
downward sloping
throughout.
k=0
 Hence,, for low values of k,,
q>q* applies. In this case,
the transition exhibits
monotonic increase in k
and decease in q.
k
24
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
C it l adjustment
Capital
dj t
t costs
t (9)
 Summary
S
 CAC lead to a well-defined investment demand articulated by firms.
 Under capital adjustment costs the adjustment of the capital stock is stretched
over time.
 Hence, the adjustment to the steady state is sluggish, even in an open economy
where an “infinitely
infinitely amount of resources”
resources is available to add to the capital stock
stock.
25
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
N t ti
Notation
0<α,η<1
ai
A
a=da/dt
b
ci
C/C
di
fh (.)
fk (.)
H
h H/L
h:=H/L
h*
i
i
J
K
k:=K/L
constant technology parameters
wealth (per capita) in country i
technology parameter
first derivative w.r.t. time
technology parameter
consumption per capita in country i
growth rate of C
net debt per capita in country i
(domestic)
partial derivative of f(.) w.r.t. h
partial derivative of f(.) w.r.t. k
human capital
h
human
capital
i l per capita
i
steady state value of h
country index (section “open-economy
Ramey model
model”))
investment (section “capital adjustment
cost”)
Hamiltonian function
physical capital
physical capital per capita
k*
kiopen
L
ni
q
r
t
V
v
w
wiopen
x
x:=dx/dt
d /d
Y
y:=Y/L
δ
ρ
σ
φ(.)
ψ(.)
steady state value of k
steady state value of ki in open economy
labor
population growth rate in country i
shadow price of installed capital
(world) interest rate
time index
firm’s market value
shadow p
price of installed capital
p
wage rate
steady state value of wi in open economy
output
f
first
d
derivative off x(t)
( ) w.r.t. time
output
output per capita
capital depreciation rate
time preference rate
elasticity of marginal utility w.r.t.
consumption
capital adjustment cost function
investment function
26
Dynamics of a Small Open Economy
Institut für Theoretische Volkswirtschaftslehre
Makroökonomik
Abb i ti
Abbreviations
CA
CAC
CRS
FDI
FOC
GDP
GNP
KRR
MP
MPC
MPL
PDV
ROR
s.t.
TVC
w.r.t.
current account balance
capital adjustment costs
constant returns to scale
foreign direct investment
first-order condition
gross domestic product
gross national product
Keynes-Ramsey rule
marginal product
marginal product of capital
marginal product of labor
present discounted value
rate of return
subject to
transversality condition
with respect
p to
27