We establish conditions under which indeterminacy can occur in a small open economy oil-in the production RBC model with lump sum tariff revenue transfers. The indeterminacy would require that the steady state tariff rates be in an open interval. This means that as long as the government revenues are exogenous, our indeterminacy result will be robust to the usage of the government revenue.Indeterminacy; Endogenous Tariff Rate; Small Open Economy; Lump Sum Transfers

Zhang, Yan

English

Research Papers in Economics

Tari⁄and Equilibrium Indeterminacy￿ (II)
Yan Zhang￿
Department of Economics
New York University, 7th Floor, 269 Mercer Street, NY, 10003, USA
June 16, 2008
Abstract
We establish conditions under which indeterminacy can occur in a small open
economy oil-in the production RBC model with lump sum tari⁄ revenue transfers.
The indeterminacy would require that the steady state tari⁄ rates be in an open
interval. This means that as long as the government revenues are exogenous, our
indeterminacy result will be robust to the usage of the government revenue.
Key Words: Indeterminacy, Endogenous Tari⁄Rate, Small Open Economy, Lump
Sum Transfers
JEL Classi￿cation Number: Q43, F41
￿Chapter 2 of my Ph.D thesis. This paper has bene￿ted from suggestions and feedback of Jess
Benhabib, Jushan Bai, Martin Uribe and Paul Dower. Send Correspondence to: Yan Zhang; Tel: 1-
212-992-9777 (USA), 86-531-88364000 (China). E-mail address: laurencezhang@yahoo.com(Y. Zhang).
11. Introduction
It is well understood by now that under some conditions open economy RBC models can
be subject to indeterminacy, in the sense that there exist a continuum of equilibrium
trajectories converging to a steady state. The literature on indeterminacy in open econ-
omy emphasizes di⁄erent channels of generating indeterminacy. Weder (2001), Meng
and Velasco (2003, 2004) prove that indeterminacy is easier to obtain for a small open
economy due to perfect or nearly perfect world capital markets that keep interest rate
more or less constant. Wen and Aguiar-Conraria (2005, 2006 henceforth WAC) supply
another way of generating indeterminacy through importing oil as a third production
factor, in which it is easier for them to have indeterminacy.
Those early models relied on increasing returns or external e⁄ects to generate inde-
terminacy. Benhabib and Farmer (1999) provide ￿ve sources of indeterminacy in closed
and open economies.1 Tari⁄ as a kind of transaction costs in international trade be-
longs to the second category which they mentioned. Schmitt-Grohe and Uribe (1997,
in short SGU) prove that within a standard neoclassical growth model (under closed
economy), a balanced budget rule can make expectations of higher tax rates self ful￿ll-
ing if the ￿scal authority relies on changes in labor income taxes to eliminate the short
run ￿scal imbalances. In Zhang (2008a), we prove that in the open economy, tari⁄ and
factor income taxes share similar channel of generating indeterminacy in the form of
1See Benhabib and Farmer (1999) page 390.
2endogenizing rates and making the government revenue exogenous. The intuition for
endogenous factor income taxes and tari⁄ to generate indeterminacy is that both of
them are countercyclical with respect to the output.
One remaining issue in our work is that although we show that factor income taxes
and tari⁄are channel equivalent to generate indeterminacy, we didn￿ t check if our result
is robust to the usage of the government revenue. SGU (1997) in their paper mentioned
(pp 985):
" On the other hand, the assumption that all government expenditures con-
sist of purchases of goods is not important for our indeterminacy result. It
can be shown that if all taxes revenues were returned to the public in the
form of lum-sum transfers, indeterminacy would still occur for steady-state
tax rates greater than sk and ... La⁄er curve."
In this paper we extend our research on indeterminacy to a small open economy
RBC model, in the way of relaxing the assumption that all tari⁄revenues are consumed
by the government. Ask the similar question as SGU and bring back this feature into
the picture. We let all of the revenues returned to the agent in the form of lump-sum
transfers and validate that the indeterminacy result is robust to this extension as long
as the government revenue is exogenous.
SGU (1997) explicitly solve the upper bound of the indeterminate region for the
steady state labor income tax rate, which is 0.5, if they assume that the government
transfers the income tax revenue to the agent (see page 985). In our model, we can￿ t
3do that since relaxation of the assumption that the government consumes the revenue
will make the determinant of the Jacobian matrix in my former model become more
complicated.
This paper is also a realistic extension of SGU and related work in the literature,
in that we incorporate the energy taxes or tari⁄ on the imported production factor
to an otherwise standard Ramsey model of a small open economy. SGU modify the
Benhabib and Farmer (1994) structure by replacing the production externality with
labor income taxes, we modify WAC model by replacing the production externality
with tari⁄s. Remember that in our model tari⁄is imposed on the energy income (otpo),
for example, we can imagine that it is a special kind of factor income taxes in open
economies.
2. The One-Sector Open Economy With Lump-Sum Transfers
Consider a modi￿ed small open economy version of Benhabib and Farmer (1994) com-
petitive model without production externality. A representative agent maximizes the
intertemporal utility function
Z 1
0
e￿￿t(logct ￿ bnt)dt (1)
4where ct is consumption of the single goods which is the numeraire and tradeable,
nt labor supply and ￿ 2 (0;1) is the subjective discount rate in the continuous time
model. Assume that the economy is open to importing oil so that the agent can use the
tradeable goods to buy oil. The oil price is assumed to be exogenous as many authors
do, for instance, Rotemberg and Woodford (1996), Wen and Aguiar-Conraria (2005,
2006). The oil supply from the rest of the world is assumed to be perfectly elastic.
On the production side, there is a single good produced with a Cobb-Douglas produc-
tion technology with three inputs￿ capital (kt), labor (nt) and non-reproducible natural
resources (ot) :
yt = k
ak
t nan
t o
a0
t (2)
where the third factor in the production, non-reproducible natural resources, say oil (ot),
is imported, and the technology displays the constant returns to scale ( ak+an+a0 = 1).
Assuming the ￿rms are price takers in the factor markets, the pro￿ts of the ￿rms are
given by
￿ = y ￿ (r + ￿)k ￿ wn ￿ po(1 + ￿)o (3)
5where (r +￿) denotes the user cost of renting capital2, w denotes the real wage, and po
denotes the real price of oil (the imported goods). ￿ is the tari⁄ rate imposed on the
imported oil, which is uniform to all ￿rms.3 Perfect competition in factor and product
markets implies that factor demands are given by:
wt = an
yt
nt
rt + ￿ = ak
yt
kt
and
po(1 + ￿t) = a0
yt
ot
Since we assume that the foreign input is perfectly elastically supplied, the factor
price, po, is independent of the factor demand for o, we can substitute out o in the
production function using
2￿ 2 (0;1) denotes the depreciation rate of capital, rt is the rental rate of capital.
3Here the tari⁄ rate can be endogeneous. We can also see the endogenous tari⁄ rate in Loewy (2004)
and Mourmouras (1991) in a two-country open economy endogenous growth model and a small open
economy OLG model respectively. This approach originates from Ramsey (1927).
6ot = a0
yt
po(1 + ￿t)
to obtain the following reduced-form production function:
yt = Ak
ak
1￿a0
t n
an
1￿a0
t (4)
where A = ( a0
p0(1+￿t))
a0
1￿a0 .
The agent budget constraint is
:
kt = rtkt + wtnt ￿ ct + G
here G = po￿tot =
￿ta0yt
(1+￿t) is the exogenous revenue collected by the government through
imposing tari⁄s on the oil.4 We assume that the government transfers the revenue to
the agent in the form of lump-sum. The ￿rst order conditions become
1
ct
= ￿t
4As we see in Zhang (2008a), the exogenous government revenue will require the endogenous tari⁄
rate to be contercyclical with respect to the output since p
o￿tot = G =
￿t￿a0yt
(1+￿t) implies
@￿
@y < 0.
7b = ￿tw
:
￿t = (￿ ￿ rt)￿t
where ￿t denotes the marginal utility of income.
Market clearing requires that aggregate demand equal aggregate supply, that is,
ct +
:
kt + ￿kt + otpo = yt (4￿ )
Note that the international trade balance is always zero. Foreigners are paid in
goods. This is clear in equation (4￿ ), according to which domestic production is divided
between consumption, investment and imports (ct+it+po
tot = yt, it = kt+1￿(1￿￿)kt).
So part of what is produced domestically is used to pay for the imports.
When we replace the consumption with 1
￿t, transform wage rate and rental rate
into functions of capital and labor, the equilibrium conditions can be reduced to four
equations:
b = ￿tanAk
ak
1￿a0
t n
an
1￿a0
￿1
t (5)
8:
￿t
￿t
= ￿ + ￿ ￿ akAk
ak
1￿a0
￿1
t n
an
1￿a0
t (6)
:
kt = (1 ￿
a0
1 + ￿t
)yt ￿ ￿kt ￿
1
￿t
(7)
and
G =
￿ta0yt
(1 + ￿t)
, yt = Ak
ak
1￿a0
t n
an
1￿a0
t (8)
We claim that the number of the steady state tari⁄ rate that generates enough
revenue to ￿nance a given level of government revenue can be 0, 1 or 2.5
Claim 1. The steady state in the continuous-time dynamic system (5)-(8) exists, given
the proper level of government expenditure.
We can derive steady state k
n = (
￿+￿
akA)
1￿a0
￿an , ￿ = b
anA(
￿+￿
akA)
ak
an , k =
anA
b (
￿+￿
akA)
￿
ak
an
[
1￿
a0
1+￿
ak
(￿+￿)￿￿]
,
G = ￿
[
1￿
a0
1+￿
ak
(￿+￿)￿￿](1+￿)
an+a0
an
cons = F(￿), constant = (a0
po)
a0
an
a0(￿+￿)an(
￿+￿
ak
)
￿
ak
an
akb . We
can see F(￿) is non-monotone and the number of the steady state tari⁄ rate that gen-
5SGU (1997) show that the revenue maximizing tax rate is the least upper bound of the set of taxes
rate for which the rational expectations equilibrium is indeterminate. But in our endogenous tari⁄ rate
case, this property doesn￿ t hold.
9erates enough revenue to ￿nance a given level of government purchases can be 0, 1 or
2.
Example 2. We give an example for a0 = 0:21, an = 0:7, ak = 0:09, ￿ = 0:025,
￿ = 0:065. ￿ is taken as Benhabib and Farmer (1994). Other parameters are taken from
WAC (2005). We can see that given the proper level of the government revenue, the
number of the steady state tari⁄ rate usually is 0 or 2.
Consider the log linear approximation of the equilibrium conditions (5)￿ (8) around
the steady state. Let
^
kt,
^
nt,
^
￿, ￿t denote the log deviations of kt, nt and ￿, ￿t from
10their respective steady states.6 The log linearized equilibrium conditions then are
0 =￿t ￿
￿ss
^
￿
1￿a0
a0 (1 + ￿ss)
+
ak
1 ￿ a0
(
^
kt ￿
^
nt) (9)
:
￿t = (￿ + ￿)[
an
1 ￿ a0
(
^
kt ￿
^
nt) +
￿ss
^
￿
1￿a0
a0 (1 + ￿ss)
] (10)
:
^
kt = [(1￿a0)
(￿ + ￿)
1 ￿ a0(1 + ￿ss)
￿￿]
^
kt+
an(￿ + ￿)(1 ￿ a0)
ak[1 ￿ a0(1 + ￿ss)]
^
nt+f￿￿+
[1 ￿ a0
(1+￿ss)]
ak
(￿ + ￿)g￿t
(11)
^
yt = ￿
1
1 + ￿ss
^
￿ =
ak
1 ￿ a0(1 + ￿ss)
^
kt +
an
1 ￿ a0(1 + ￿ss)
^
nt
7 (12)
Combining the (9) and (12), we can imply
^
nt =
￿t
ak
1￿a0 ￿ ￿ss
1￿a0
a0
an
1￿a0(1+￿ss)
+
ak
1￿a0(1+￿ss)
ak
1￿a0 ￿ ￿ss
1￿a0
a0
an
1￿a0(1+￿ss)
^
kt
6￿ss is the steady state tari⁄ rate.
7Note that
ak+an
1￿a0(1+￿ss) > 1, the increasing returns to scale comes from the endogenous tari⁄ rate.
11Using this expression to eliminate the
^
ntin the (10) and (11) results in the following
system:
2
6 6
4
:
￿t
:
^
kt
3
7 7
5 =
2
6 6
4
J11 J12
J21 J22
3
7 7
5
2
6 6
4
￿t
^
kt
3
7 7
5;J =
2
6 6
4
J11 J12
J21 J22
3
7 7
5
where
J11 = ￿(￿ + ￿)
an
1￿a0 + ￿ss
1￿a0
a0
an
1￿a0(1+￿ss)
ak
1￿a0 ￿ ￿ss
1￿a0
a0
an
1￿a0(1+￿ss)
J12 = (￿ + ￿)f[
an
1 ￿ a0
￿
￿ss
1￿a0
a0
ak
1 ￿ a0(1 + ￿ss)
] ￿
[ an
1￿a0 + ￿ss
1￿a0
a0
an
1￿a0(1+￿ss)]
ak
1￿a0(1+￿ss)
ak
1￿a0 ￿ ￿ss
1￿a0
a0
an
1￿a0(1+￿ss)
g
J21 = f￿￿ +
[1 ￿ a0
(1+￿ss)]
ak
(￿ + ￿)g +
an(￿+￿)(1￿a0)
ak[1￿a0(1+￿ss)]
ak
1￿a0 ￿ ￿ss
1￿a0
a0
an
1￿a0(1+￿ss)
12J22 = [(1 ￿ a0)
(￿ + ￿)
1 ￿ a0(1 + ￿ss)
￿￿] +
ak
1￿a0(1+￿ss)
ak
1￿a0 ￿ ￿ss
1￿a0
a0
an
1￿a0(1+￿ss)
an(￿ + ￿)(1 ￿ a0)
ak[1 ￿ a0(1 + ￿ss)]
After some tedious algebra, we can have, J11 = ￿(￿ + ￿) an
ak￿a0￿ss, J22 = (￿ +
￿) 1￿a0
ak￿a0￿ss ￿ ￿, J12 = (￿ + ￿) ￿￿ssa0
ak￿a0￿ss, J21 =
(￿+￿)
ak
(1￿a0)2+￿ss[(1￿a0)2￿ana0]￿￿2
ssa0
(ak￿a0￿ss)(￿ss+1) ￿ ￿.
Proposition 3. The equilibrium is indeterminate i⁄ trace(J) = J11 + J22 < 0 <
J22J11 ￿ J12J21 = det(J), or, ￿2 < ￿ss < ￿3, where
ak
a0 < ￿2 < ￿3, ￿2, ￿3 are de-
termined by the system parameters.
The indeterminacy requires that trace(J) =
ak
ak￿a0￿ss(￿ + ￿) ￿ ￿ < 0 if and only if
￿ss >
ak
a0
After some manipulations, the determinant of the Jacobian can be written as
det(J) =
(￿ + ￿)
ak ￿ a0￿ss
￿
where
13￿ = ￿an(￿+￿)
1 ￿ a0
ak ￿ a0￿ss
+￿an+a0￿ssf
(￿ + ￿)
ak
(1 ￿ a0)2 + ￿ss[(1 ￿ a0)2 ￿ ana0] ￿ ￿2
ssa0
(ak ￿ a0￿ss)(￿ss + 1)
￿￿g
The positive det(J) requires that ( conditional on ak ￿a0￿ss < 0) ￿ < 0. We de￿ne
G(￿ss) = ￿1￿3
ss + ￿2￿2
ss + ￿3￿ss + ￿4
where ￿1 = a2
0[￿ ￿
(￿+￿)
ak ] < 0, ￿2 = ￿￿a0an + a0f
(￿+￿)[(1￿a0)2￿ana0]
ak + ￿a0 ￿ ￿akg,
￿3 = ￿￿a0an + [￿(￿ + ￿)an(1 ￿ a0) + ￿anak] +a0
(￿+￿)
ak (1 ￿ a0)2 ￿ ￿ak, ￿4 = ￿(￿ +
￿)an(1 ￿ a0) + ￿anak. ￿ < 0 is equivalent to G(￿ss) > 0.
We can easily ￿nd that G(0) < 0, as ￿ = 0, G(
ak
a0) = 0. As ￿ > 0 but close to zero,
G(
ak
a0) < 0. There are three roots for G(￿ss) = 0. Let us order them ￿1 < 0 < ￿2 < ￿3.
We can see that
ak
a0 < ￿2, as ￿2 < ￿ss < ￿3, G(￿ss) > 0, indeterminacy arises in the
tari⁄ model.
2.1. Calibrated example
The purpose of this section is to illustrate the main result of the proposition￿ that
indeterminacy in fact occurs with the empirical tari⁄rate-by one numerical experiment.
14We adopt the following "standard" values in RBC models: a0 = 0:21, an = 0:7, ak =
0:09, ￿ = 0:025, ￿ = 0:0658.
Case 1: ￿ss =
import tari⁄
import price =
15:6$=bbl
26$=bbl = 0:6 which is the optimal tari⁄ rate of oil
from David Newbery (2005), consistent with the one in EU (2002).
We draw G(t) graph for the numerical experiment and see that ￿1 = ￿0:9353,
ak
a0 = 0:4286 < ￿2 = 0:4341, ￿3 = 2:7605. As ￿2 < ￿ss = 0:6 < ￿3, G(￿ss) > 0.
Di⁄erent from SGU (1997), we cannot explicitly get the indeterminate region because
we suppose that the government transfers the revenue to the agent in a lump-sum way.
Both of the two bounds for the indeterminate region change since relaxation of the
8The factor weights are taken from WAC (2005). They are of the country, Netherlands, based on
input-output tables from OECD (1995) reports. ￿ = 0:065, see Benhabib and Farmer (1994).
15assumption that the government consumes the revenue will make the determinant of
the Jacobian matrix become more complicated, up to a third order polynomial. But
the indeterminacy result generated by the endogenous tari⁄ rate is still robust to the
usage of the government revenue.
3. Discussion and extensions
It has been shown that an otherwise standard one-sector oil -in the production real
business cycle model may exhibit indeterminacy and sunspots under a balanced-budget
rule that consists of ￿xed and ￿wasteful￿government spending (or lump-sum transfers)
and endogenous tari⁄ rate. However, the economy always displays saddle-path stability
and equilibrium uniqueness if the government ￿nances endogenous public expenditures
with a constant tari⁄ rate. We may extend this paper by allowing for productive or
utility-generating government purchases in either of these speci￿cations. It may turn out
that the earlier determinacy results are overturned when public expenditures generate
su¢ ciently strong production or consumption externalities.
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Tariff and Equilibrium Indeterminacy--(II)

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