International capital mobility and aggregate volatility: the case of credit-rationed open economies, P Pintus

Tags: externalities, steady state, human capital, indeterminacy, physical capital, open economies, labor supply, international credit, tax rate, elasticity of intertemporal substitution, J. Econ, saddle-point, labor income, capital accumulation, negative values, progressive income tax, elasticity, regressive taxes, open economy, equilibrium conditions, international, income tax rates, income tax rate, differential equations
Content: International Capital Mobility and Aggregate Volatility: the Case of Credit-Rationed Open Economies Patrick A. Pintus September 10, 2004 The author would like to thank, without implicating, Amartya Lahiri and Helene Rey for stimulating discussions that raised my interest in the field, as well as Jean-Olivier Hairault, Jean Imbs, Jonathan Parker, Franck Portier, Mike Woodford, seminar participants at Princeton University, University of Paris I (Eurequa), University of Toulouse I (GREMAQ) for suggestions that helped me to improve the exposition of the paper. First draft: may 23, 2003. Universite de la Mediterranee Aix-Marseille II and GREQAM. Address correspondence to: GREQAM, Centre de la Vieille Charite, 2 rue de la Charite, 13002 Marseille, France. Tel: (33) 4 91 14 07 48 (Sec: 27 or 70), Fax: (33) 4 91 90 02 27. E-mail: [email protected] 1
2 International Capital Mobility and Aggregate Volatility: the Case of Credit-Rationed Open Economies Patrick A. Pintus Abstract This paper studies how international capital mobility affects aggregate volatility by considering the case of imperfect financial markets such that only physical capital serves as collateral for international borrowing, whereas human capital cannot. We find that credit-constrained, small open economies may be destabilized by expectation-driven fluctuations, even with small externalities, provided that the share of human capital is low enough. It follows that opening the capital account may push a saddle-point stable economy on a volatile path. Moreover, economies that highly borrow on international credit markets to finance capital accumulation are more susceptible, through a financial accelerator effect, to expectation-driven fluctuations. On the contrary, tighter constraints on external borrowing may protect the economy against expectation-driven volatility. In contrast with existing results relying on the assumption of perfect financial markets, both the elasticity of labor supply and the elasticity of intertemporal substitution in consumption, though less traditionally, condition the set of parameter values associated with expectation-driven fluctuations. Finally, we extend the basic model and show, first, that our main results do not specifically depend on the presence of externalities, and, second, that tax progressivity (resp. regressivity) may protect (resp. expose) the economy against (resp. to) expectation-driven volatility. Keywords: international financial markets, endogenous borrowing constraints, progressive taxation, indeterminacy and expectation-driven fluctuations. Journal of Economic Literature Classification Numbers: E44, E62, F34, F41. 1 Introduction For decades, the potential adverse effects of financial market globalization have been a recurrent topic in economic analysis, not to mention the intellectual and public debates. Although not new, some concerns
3 that financial integration might increase growth volatility and inequality have resurfaced in the 1990's, while some consequences of financial liberalization appeared strikingly. However, what are the actual forces at work and their interplay remains a much debated question (see, e.g., Obstfeld [42], Tirole [50]). In the present paper, we focus on endogenous credit constraints as market imperfections that may magnify volatility in open economies. More specifically, we develop a simple framework which models the working of a credit-constrained, small open economy: we assume that its population of infinitely-lived agents is not allowed to use human capital as collateral for international borrowing, while the stock of physical capital determines a limit on what debtors can borrow from international credit markets. Therefore, human capital accumulation is financed domestically while international debt finances physical capital accumulation. The major result of this paper is that local indeterminacy and sunspot equilibria drive the dynamics of a small open economy when social returns to scale are close to constant - externalities are small - provided that the share of human capital is low enough (see Proposition 3.1). Therefore, this paper highlights a distinct channel through which (short-term) international capital mobility may contribute to increase aggregate volatility. It can be roughly described as follows. Other things equal, economies with low shares of human capital and high shares of physical capital (in total capital) face loose borrowing constraints and may, therefore, highly borrow on international credit markets. These economies are more likely to be subject to self-fulfilling expectations: essentially, optimistic expectations drive up current consumption, labor supply, output and investment, which is perfectly consistent with rational expectations, intertemporal equilibria. In our credit-constrained economy, an additional feature of the boom is that the physical capital stock increases, thereby allowing foreign debt to go up. The latter mechanism, a financial accelerator effect related to collateral-backed loans, is key in that it makes more likely self-fulfulling waves of optimism and pessimism even in the presence of small externalities, as long as human capital has a low share in output. Eventually, increasing interest payments will negatively affect output, so that consumption, investment and debt will go down, and this will give rise to a (temporary) recession (see Corollary 4.2). Section 4 also shows that a saddle-point stable, closed economy may, for moderate externalities, be pushed
4 on a volatile path if it would opt for financial integration (see Corollary 4.1). This may happen if financial globalization drastically reduces the share of capital accumulation that has to be financed domestically, i.e. if the economy has a low (resp. high) share of human (resp. physical) capital. On the contrary, however, tighter borrowing constraints (when the economy has low collateral) may protect the economy against expectationdriven volatility: when the share of human capital in total capital becomes larger that some threshold value, the economy is then saddle-point stable and, therefore, immunized against fluctuations driven by selffulfilling expectations. This shows that imposing, quite reasonably, some limits on international borrowing does not prevent open economies from being exposed to "animal spirits", which may seem to contrast with one's intuition. Therefore, one of our predictions is that economies deciding to postpone financial integration until some level of development is attained (until the share of human capital exceeds some threshold) should experience less volatile growth paths, as they attempt to sufficiently weaken the financial accelerator mechanism before going to international credit markets. In that sense, the mechanism at the core of our results may be seen as more relevant for middle-income and low-income countries (moreover, we will see that "rich" countries are not credit-constrained, in the model, so that they jump to the steady state and experience no volatility). Our chosen specification for modeling credit market imperfection may be seen as attractive for three main reasons. Most importantly, it is simple and tractable enough to be easily incorporated into a benchmark business-cycle model (the Ramsey-Cass-Koopmans framework). However, it captures in a parsimonious way the stylized fact that to reduce the consequences of hidden knowledge (both adverse selection and moral hazard), a significant proportion of loans to firms and households take the form of secured, or collateralbacked, debt. One prominent implication of such debt contracts is that it creates a feedback effect from the state of the economy to the (endogenous) credit constraint. This feedback is a key element of our model. One interpretation of our assumption is that the borrowers pledge only up to the level of their transferable asset. In other words, imperfect commitment rules out the possibility that net worth be negative in the debtor's balance sheet. Second, our assumption seems to accord well to the actual working of international financial
5 markets, which implies that many countries belonging to the lower end of the world income distribution do not borrow internationally, presumably because they have low collateral. On the other hand, it is also in agreement with the observation that some countries coping with a high external debt have a large share of physical capital in ouput and are also highly volatile (as, for instance, Burkina Faso, Cote d'Ivoire, Jordan or Peru, defined as "highly indebted" countries by the statistical appendix of World Bank's Global Development Finance 2003, tables A51 and A52, pp. 232-5; see section 4). Moreover, the predictions of this paper are broadly consistent with the observation that financial integration and aggregate volatility (of output and income) may be positively correlated (this was the case in the 1990s, according to Kose, Prasad and Terrones [34, Table 3]). Finally, our simple assumption describes what we think is a robust mechanism leading to both the amplification of shocks and endogenous volatility (the financial accelerator effect). Checking for robustness is in fact the purpose of two appendices. Appendix B confirms our results regarding the impact of endogenous credit constraints in an economy with constant returns to scale, in the absence of externalities, when the government follows a policy of financing a given stream of public expenditures by distortionary income taxes. Therefore, the presence of externalities and increasing returns (or any other form of nonconvexities) is not essential for our results. Appendix C, on the other hand, shows that to be operative, the financial accelerator does not require the human capital share to be as small as in Sections 3-4, when regressive taxes are imposed on factor incomes. In short, at the evident cost of abstracting from determinants of credit market imperfections (most notably hidden information), constraining the debt-to-capital ratio to be less than unity turns out to be a simple, yet plausible, formulation that captures in a parsimonious way the interplay between endogenous debt limits and the state of the economy. To quote Matsuyama [39, pp. 860-1], "the reader may thus want to interpret this formulation as a black box, a convenient way of introducing the credit market imperfection in a dynamic macroeconomic model, without worrying about the underlying causes of imperfections". The rest of the paper is organized as follows. Section 2 relates and compares our study to the existing literature. Section 3 presents the open-economy model, derives its dynamics and conditions leading to
6 local indeterminacy. Section 4 discusses, more specifically, how the tightness of international borrowing constraints affects the likelihood of local indeterminacy. Finally, some concluding remarks and directions for future research are gathered in Section 5 and three appendices contain some proofs and extensions. 2 Related Literature The present paper makes an attempt to combine two strands of the existing literature. The first approach has examined the implications of imperfect credit markets on aggregate outcomes, in closed economies, starting from early contributions by Scheinkman and Weiss [47], Bernanke and Gertler [16](e.g., Greenwood and Jovanovic [22], Bencivenga and Smith [10], Marcet and Marimon [37], Galor and Zeira [21], Kocherlakota [33], Kiyotaki and Moore [31], Aghion and Bolton [1], Azariadis and Chakraborty [4], Mastuyama [38]). The second set of contributions studies the consequences of financial globalization on volatility and inequality (e.g. Boyd and Smith [17], Aghion, Bacchetta and Banerjee [2], Matsuyama [39], Lahiri [36], Weder [51], Nishimura and Shimomura [41], Meng and Velasco [40]). However, Lahiri [36], Nishimura and Shimomura [41], Meng and Velasco [40] consider the borderline case of perfect capital markets which, as intuition suggests, may be more conducive to instability as it imposes no limit to international capital flows in and out of the economy. On the other hand, Weder [51, pp 351-6] incorporates imperfect credit markets by assuming that the international interest rate is an increasing function of the debt/capital ratio. He then shows that indeterminacy may be reconciled with almost constant returns provided that the interest rate is not too responsive to the debt/capital ratio. However interesting this specification is, it leads to a four-dimensional dynamical system that is not easily amenable to analytical study. Our assumption on the debt-to-capital limit has been used, in a different context, by Barro, Mankiw and Sala-i-Martin [8], Cohen and Sachs [20]. Another group of notable and related contributions include Boyd and Smith [17], Aghion, Bacchetta and Banerjee [2], Matsuyama [39]. In a spirit that differs from the papers cited above, Boyd and Smith [17] and Matsuyama [39] show how imperfect financial markets (subject to
7 costly-state verification or potential default, respectively) may result in a polarized world with rich and poor countries and, therefore, explain persistent inequality among nations. On the other hand, Aghion, Bacchetta and Banerjee [2] focus on the impact of credit constraints (due to moral hazard) on endogenous volatility of open economies. However, their framework is designed to describe more adequately financially-based crises, rather than business cycles occurring at higher frequency. In our context, endogenous volatility is naturally interpreted as driving high-frequency business cycles in indebted economies, as opposed to causing unfrequent regime switches (with capital inflows followed by capital outflows, as for instance in Aghion, Bacchetta and Banerjee [2]). In contrast with Boyd and Smith [17], Matsuyama [39] who consider overlapping generations economies, our framework models agents as being infinitely-lived, working and consuming in each period. Moreover, the latter authors introduce indivisible investment, which we do not. Finally, Boyd and Smith [17], Aghion, Bacchetta and Banerjee [2] assume that two distinct classes of agents, named borrowers and lenders, are present in constant proportion and that only borrowers are able to invest in production (in the model studied by Matsuyama [39], the proportion of borrowers is endogenous). In contrast, we make no such distinction. In contrast with existing results relying on the assumption of perfect credit markets, we show that both the elasticity of labor supply and, though less traditionally, the elasticity of intertemporal substitution in consumption condition the set of parameter values associated with endogenous fluctuations. More precisely, we find that indeterminacy requires labor supply to be elastic enough but, quite surprisingly, intertemporal substitution to be weak enough. In fact, indeterminacy may be ruled out when intertemporal substitution is almost perfect: this stands also in contrast with results obtained in closed-economy models; see, for instance, Bennett and Farmer [15], Harrison [27], Hintermaier [28], Pintus [43]. Compare to results by Aghion, Bacchetta and Banerjee [2], who find that financial crises cannot occur when the supply of country-specific input is highly elastic, while some of their simulations suggest that regime switches are more frequent when intertemporal substitution is weak enough. On the other hand, Weder [51, pp 351-6] assumes logarithmic utility and, therefore, does not examine this issue, while Boyd and Smith [17] and Matsuyama [39] abstract
8 from such a discussion. In addition, we also consider the case of elastic labor supply, in contrast with Lahiri [36], Weder [51], Nishimura and Shimomura [41] or Boyd and Smith [17], Matsuyama [39]. Meng and Velasco [40] assume elastic labor but conclude that, in a two-sector economy, labor supply elasticity is not relevant, although their analysis relies on a related, but borderline hypothesis, as labor can be costlessly reallocated from one sector to the other. In contrast, we show that labor supply elasticity plays a key role. More traditionally, we finally assume that firms benefit from productive externalities that generate social increasing social returns to scale (as in Benhabib and Farmer [11, 12]). Therefore, our analysis does not rely on either the questionable assumption of negative externalities or the implicit presence of nonstationary fixed costs implying zero profits (as in Weder [51], Nishimura and Shimomura [41], Meng and Velasco [40]). Again, the presence of externalities is not critical, as we just said. We also combine other mechanisms in our model, which do not appear in the studied cited above. Most importantly, we impose, in Appendix C, distortionary taxes on income such that the tax rate may be either progressive - that is, increasing with the tax base - or regressive (Guo and Lansing [24]). Although in contrast with the usual assumption of a flat tax rate, this is in agreement with income tax schedules prevailing both in OECD countries, e.g. in the U.S. or European countries (see, for instance, the Statistical Abstracts of the United States, or Tax and the Economy: A Comparative Assessment of OECD Countries, OECD Tax Policy Studies n. 6, 2002) and in developing countries. In the real world, a number of mechanisms introduce regressivity in otherwise progressive income tax schedules, most notably through proportional consumption taxes, upper bounds that are imposed on social security contributions, absent or low capital income taxes (especially in developing countries; see, e.g., Tanzi and Zee [49]). Therefore, one cannot rule out the case of effective taxes being indeed regressive. Finally, we should emphasize that this paper is also related to recent studies focusing on exclusion-based, self-enforcing debt constraints (e.g. Kehoe and Levine [30], Kocherlakota [32], Alvarez and Jermann [3], Kehoe and Perri [29], Azariadis and Lambertini [6], Azariadis and Kaas [5]). For example, Azariadis and Lambertini [6], Azariadis and Kaas [5] argue that a mechanism leading to multiple equilibria is dynamic
9 complementarity between current and future debt constraints. In this paper, endogenous debt limits also originate a related dynamic complementarity effect, as discussed above: optimistic expectations not only lead to higher output, consumption, investment and debt tomorrow, but also to a higher capital stock and, therefore, to more generous lines of credit today. In other words, sunspot shocks have, through debt constraints, an indirect effect when triggering boom-bust dynamics, which makes clear why externalities may be kept small.
3 Indeterminacy in a Credit-Rationed Open Economy This section presents our benchmark model, which combines insights by Benhabib and Farmer [11], Barro et al. [8].
3.1 Model and Laws of Motion
The economy produces a tradeable good Y by using physical capital K, human capital H, and raw labor L, according to the following technology:
Y = AK HL1--,
(1)
where A is total factor productivity, 0, 0 and + < 1.
The Ramsey households have preferences represented by:
Z
e-t{ C(t)1-
-
L(t)1+ }dt,
(2)
0
1- 1+
where C is consumption, 0 is the inverse of the elasticity of intertemporal substitution in consumption,
0 is the inverse of the elasticity of labor supply with respect to the real wage, and 0 is the discount
factor. The representive consumer owns the three inputs and rent them to firms through competitive markets.
10
Therefore, we can write down, for sake of brevity, the consolidated budget constraint as:
K + H - D = Y - (K + H) - rD - C,
(3)
where D is the amount of net debt, 0 1 is the depreciation rate for both types of capital, and r 0 is
the world interest rate given to our small open economy. The initial stocks K(0) > 0, H(0) > 0, D(0) and
the labor endowment L are given to the households. Labor and human capital are not mobile.
We also assume that international borrowing is limited to the amount of physical capital, that is, D K.
This hypothesis is, for instance, fulfilled when only physical capital is accepted as collateral for international
credit, whereas human capital is not. We focus, as is usual, on the case r = , the interest rate that
would prevail in the corresponding closed economy (that is, when D = 0). Moreover, we assume that
K(0) + H(0) - D(0) < H holds at the outset of the period, where H is the steady state value of human
capital stock. In other words, the initial net asset position is small enough so that the credit constraint
binds, that is, D = K. On the contrary, the economy jumps immediately to the steady state when the latter
inequality does not hold: if the economy is "rich" enough, at the initial date, it is able to attain the steady
state by using international credit markets (this would also be evidently true in small economies having
access to perfect capital markets).
Equating the return on physical capital to r + yields K = Y /(r + ), and, therefore, by using equation
(1),
Y
=
1 BA 1-
H a Lb,
(4)
where B = [/(r + )]/(1-), a = /(1 - ), b = 1 - a, with 0 a < + . Therefore, a is simply the share of human capital relative to the share of effective labor, that is, human capital and raw labor.1 Our next assumption is that firms benefit from productive externalities through human capital and labor (for instance thick-market externalities). More precisely, we postulate that total factor productivity is such 1It is easily shown that a similar, reduced-form, production function (4) can be obtained in the case of a CES technology.
11
that:
A = (HL1--),
(5)
where 0 is a measure of externalities. In fact, direct inspection of equations (1) and (5) shows that social returns to scale are measured by 1 + (1 - ) and are increasing whenever is positive. Excluding physical capital from the external effects somewhat conforms to the "short-term" interpretation of the model and also to some evidence offered by Benhabib and Jovanovic [14], who find capital externalities to be nonsignificant. As discussed below and in Appendix B, our analysis does not depend in an important way on the presence of externalities. Moreover, we will show that local indeterminacy may occur, in the model of this section, for small values of . The budget constraint (3) may be written as the following, observing that interest payments are proportional to output, that is, (r + )D = Y when D = K:
H = (1 - )Y - H - C.
(6)
Finally, households decisions follow from maximizing (2) subject to the budget constraint (6), using equa-
tion (4), given the initial stock H(0) < H. Straightforward computations yield the following first-order
conditions:
C
Y = a(1 - )
- - ,
L C
=
b(1
-
Y )
.
(7)
C
H
L
We may rewrite the budget constraint, from (6), as:
H H
=
(1
-
)
Y H
--
C. H
(8)
Equations (7)-(8), together with equations (4)-(5), characterize the dynamics of intertemporal equilibria. It is not difficult to check that the associated transversality constraint is met in the following analysis, as we consider orbits that converge towards the interior steady state: both the human capital stock and the co-state variable converge. Direct inspection of equations (7)-(8) reveals that the dynamics arising in our credit-rationed economy are somewhat similar to the laws of motion describing closed economies, as studied
12 by Benhabib and Farmer [11] and many others since. The major difference is that human (instead of total) capital is here the key state variable. Note that the Euler condition in equations (7) implies that the steady state marginal return on human capital equals + , the marginal return of physical capital.
3.2 Local Indeterminacy when Externalities are Small
Appendix A linearizes equations (7)-(8), (4)-(5), around the interior steady state which is shown to be
unique. Straightforward computations yield the following expressions for trace T and determinant D of the
Jacobian matrix of (the dynamical system derived from) equations (7)-(8), (4)-(5) (see equations (14) in
appendix A):
T
=
+ a
[1
+
2
a
]
+
+(1-a) a
,
D
=
(
+
)(
+(1-a) a
)[1
+
2],
(9)
with 1 = (b(1 + ) + ( + 1)[a(1 + ) - 1])/[ + 1 - b(1 + )] and 2 = [-b(1 + )][ + 1 - b(1 + )].
We focus, as most papers in the literature do, on the case for which the steady state is locally a sink - that is, asymptotically stable - and derive conditions on parameter values such that T < 0 and D > 0. We then show that these conditions may include small values of (that is, configurations with small externalities): indeterminacy is then compatible with almost constant returns to scale. In our discussion, a key parameter is the share of human capital in total labor a = /(1 - ), as we now show. Direct inspection of equations (9) shows that the sign of D is given by the sign of 1 + 2. Moreover, the numerator of 1 + 2 is negative only if ( + 1)(1 - a(1 + )) (1 - )(1 - a)(1 + ). The latter inequality is met if one assumes that a(1 + ) < 1 - that is, externalities are not large enough to ensure endogenous growth - and 1 - that is, the elasticity of intertemporal substitution in consumption is smaller than unity. Moreover, the denominator of D is negative when + 1 < (1 - a)(1 + ), that is, when externalities are large enough or > c ( + a)/(1 - a). On the other hand, T is a increasing function of , tends to + when tends, from below, to c. When increases from c, T increases from - and tends to a negative value.
13 The immediate consequence is that D > 0 and T < 0 when is larger than c. Proposition 3.1 (Indeterminacy with Small Externalities) In our credit-constrained open economy, the dynamics of consumption C and human capital H, given by equations (7)-(8), together with equations (4)-(5), have a unique positive steady state (C, H). Assume that a(1 + ) < 1 (that is, externalities are not large enough to ensure endogenous growth) and 1 (that is, the elasticity of intertemporal substitution in consumption 1/ is smaller than one). Then the steady state (C, H) is, locally: (i) a saddle (locally determinate) when < c ( + a)/(1 - a) (that is, when externalities are small enough). (ii) a sink (locally indeterminate) when > c. In particular, local indeterminacy of the steady state (C, H) occurs for values of that are close to zero (small externalities), provided that both is close to zero (that is, labor supply is almost perfectly elastic) and a is close to zero (that is, the share of human capital is close to zero). One important implication of Proposition 3.1 is that stationary sunspot equilibria are expected in neighborhoods of the indeterminate steady state, that is, when is larger than c. Appendix B shows that indeterminacy is also likely to occur in version of the model without externalities (returns to scale are then constant), when government finances a constant level of public expenditures by using distortionary taxes. In that sense, our specific assumption that external effects generate increasing returns is not critical for our main conclusion that credit-rationed open economies may be susceptible to expectation-driven fluctuations (when a is small enough). On the other hand, Appendix C extends the basic model and shows that adding tax regressivity implies that indeterminacy may be compatible with (small externalities and) values of a that are larger than in the model without taxes of this section. We note that some of our conclusions are somewhat different from existing results. First, in contrast with
14 the results of Meng and Velasco [40], the parameter constellations leading here to indeterminacy depend on labor supply elasticity (that is, 1/), as c increases with . Therefore, indeterminacy is more likely when labor supply is highly elastic - when is small. Moreover, note that the elasticity of intertemporal substitution in consumption - that is, 1/ - plays also a role, in contrast with the results in Lahiri [36], Weder [51], Nishimura and Shimomura [41], Meng and Velasco [40]. This is not surprising, as consumption is not perfectly smoothed out in our small open economy facing borrowing constraints (the return on human capital accumulation differs from the international interest rate, out of steady state). In fact, one corollary of our analysis is that indeterminacy may be ruled out, quite surprisingly, when is close enough to zero, i.e. when intertemporal substitution in consumption is almost perfect, as we now show.
Proposition 3.2 (Indeterminacy and Strong Intertemporal Substitution in Consumption) Local indeterminacy of the steady state (C, H) is ruled out, in the limit, when goes to zero, that is, when the elasticity of intertemporal substitution in consumption 1/ tends to infinity.
Proof: From equations (9), one can show that, here again, T < 0 implies that > c ( + a)/(1 - a),
as T does not depend on . However, > c implies that D < 0 when is close enough to zero. To
prove this, remember that D has the same sign as that of 1 + 2. But the denominator of 1 + 2 =
[( +1)(a(1+)-1)+(1-a)(1+)]/[ +1-(1-a)(1+)] is negative when > c, while its numerator is pos-
itive when (1-a)/(1+a). It is straightforward to see that c ( +a)/(1-a) > > (1-a)/(1+a),
so that D < 0 when > c and is close to zero. Therefore, the steady state is then a saddle if > c, and
a saddle or a source, when < c and is close enough to zero.
2
Note that, indeed, most estimates of fall within a large interval, including (0, 2) (see, e.g., Hansen and Singleton [26], Campbell [19]). Therefore, the evidence does not seem to provide clear-cut conclusions.
15 4 Imperfect International Credit Markets and Indeterminacy In this section, we discuss how the tightness of international borrowing constraints affects the likelihood of local indeterminacy and sunspots. 4.1 When does financial integration lead to indeterminacy? As a benchmark, it is useful to consider the case of closed economies deciding to open their capital account. We know, by modifying results from Benhabib and Farmer [11, p. 30] to incorporate human capital, that in the absence of international borrowing, a closed economy is saddle-point stable (the steady state is then determinate) if and only if + 1 > (1 - - )(1 + ) or equivalently < cclosed ( + + )/(1 - - ). On the other hand, Proposition 3.1 has shown that indeterminacy occurs only when is close to, but larger than c ((1 - ) + )/(1 - - ) in the credit-constrained, open economy. As c < cclosed always holds when > 0, there are parameter values such that the closed economy is saddle-point stable while indeterminacy prevails in the open economy: this is the case when is such that c < < cclosed. Corollary 4.1 (Comparing Closed and Open Economies) Suppose, under the assumptions of Proposition 3.1, that the level of externalities belongs to the interval (c ((1 - ) + )/[1 - - ], cclosed ( + + )/[1 - - ]). Then indeterminacy prevails in the credit-constrained, open economy, while the corresponding closed economy is saddle-point stable. Therefore, for moderate externalities (values of as low as 0.1, when a is small enough, well within the standard errors of available estimates), our credit-constrained economy would experience sunspots while the corresponding closed economy would be saddle-point stable. On the other hand, remember that our analysis also suggests that "rich" countries - that is, countries such that K(0) + H(0) - D(0) H - are not credit-
16 constrained. Therefore, such countries jump immediately to the steady state and, hence, do not experience volatility (this would also be evidently true in small economies having access to perfect capital markets). To illustrate Corollary 4.1, let us now examine what would happen to economies deciding to open their capital account. To fix ideas, assume that labor supply is indivisible ( = 0) and = 0.3. Then open economies exhibit indeterminacy for small externalities, provided that the share of human capital is low enough: has to be lower than 0.7/(1 + ). Therefore, indeterminacy in open economies requires < 0.03 when = 0.05, < 0.06 when = 0.1, and < 0.12 when = 0.2. In contrast, closed economies would be saddle-point stable in all three cases, as = 0.3 > /(1 + ) - . In other words, financial integration may push an economy with a low share of human capital on a volatile path because it drastically reduces the proportion of capital accumulation that has to be financed domestically. Therefore, one prediction of the model is that economies deciding to postpone financial integration until some level of development is attained (until the share of human capital exceeds some threshold) should experience less volatile growth paths, as they attempt to sufficiently weaken the financial accelerator mechanism before going to international credit markets. 4.2 Tightness of Credit Rationing and Indeterminacy The purpose of this section is to develop a sensitivity analysis with respect to a key parameter of our model: the share of physical capital in GDP - that is, - which also measures the share of interest payments in GDP. This parameter highlights the effect of openness: when = 0, our open economy has no collateral and, therefore, evolves like a closed economy in which only human capital is accumulated and generates growth; on the contrary, economies with large 's highly borrow on international credit markets. As a natural indicator of the looseness of the credit constraint, we consider the share of collateral in total capital, i.e. /( + ), given the share of total capital in output + . In other words, we fix the share of labor (given by 1 - - ) and vary the share of physical capital . The lower , the higher the share of
17 human capital , which is not accepted as collateral, and the less open the economy to international credit markets (and consequently to foreign trade), as D/Y = /(r + ) is then smaller. In contrast, the higher , the easier the access to international borrowing and the higher openness for the corresponding economy. The main specific question that we want to address in this section is the following: how does the severity of the credit constraint affects the likelihood of indeterminacy and endogenous fluctuations in open economies? When + is fixed, varying affects a = /(1 - ) in the following way. In fact, fixing + < 1 implies that 1 - a = (1 - - )/(1 - ) increases with so that a decreases with . In view of Proposition 3.1, we know how conditions leading to indeterminacy are affected by a. First, note that the condition > c - that is, + 1 < (1 - a)(1 + ) - in Proposition 3.1 is more likely to be met when a is small enough. Therefore, indeterminacy is more likely when a is not too large, in the sense that c is then smaller and this is enough to prove the following corollary. Corollary 4.2 (Indeterminacy and Tightness of Credit Constraint) Indeterminacy of the steady state (C, H) is more likely, as it requires lower externalities, when the share of human capital a = /(1 - ) is small. It follows that, given the share of total capital + , economies that have a large share of physical capital , and are, therefore, less heavily credit-constrained, are more susceptible to indeterminacy and expectationdriven volatility than economies that borrow little on international credit markets. To illustrate Corollary 4.2, we now consider benchmark parameter values: indivisible labor supply (that is, = 0), = 0.065 and = 0.1 (as in, for instance, Benhabib and Farmer [11]). For sake of brevity, we study two configurations depending on the share of total capital: + = 0.8 (the main case considered by Barro et al. [8]) and + = 0.4, that we summarize in the following table. Our numerical experiments confirm that the critical lower bound c is small (externalities are small) when a is small. Therefore, our numerical cases illustrate that when the credit constraint is getting tighter, that is when decreases for given
18
+ = 0.8 = 0.70, that is, a 0.33 c(1 - ) 0.15 = 0.78, that is, a 0.09 c(1 - ) 0.02 + = 0.4 = 0.20, that is, a = 0.25 c(1 - ) 0.27 = 0.39, that is, a 0.02 c(1 - ) 0.01 Table i: the lower bound on externalities (c(1 - )) associated with indeterminacy when the share of collateral varies. + , the human capital share a increases and indeterminacy is becoming less likely. On the contrary, open economies that face loose credit constraints - that is, have small a's - are more susceptible to indeterminacy and expectation-driven fluctuations. On the other hand, computing cclosed(1 - ) for the values of Table i (from top to bottom) yields 1.20, 0.88, 0.53, 0.41. Therefore, access to international credit may highly reduce the minimal level of externalities compatible with indeterminacy and sunspots. The predictions of our model are arguably consistent with some evidence that financial integration and aggregate volatility (of output and income) may be positively correlated (this was the case in the 1990s, according to see Kose, Prasad and Terrones [34, Table 3]). Also note that indeterminacy may be associated with larger values of a when regressive taxes on labor income are introduced into the model, as shown in Appendix C.
4.3 Interpretation
The intuition for why, in open economies with access to foreign borrowing, indeterminacy implies that the human capital share cannot be too large may be easily grasped if one looks at the labor market, following Benhabib and Farmer [11]. More precisely, one may rewrite the static condition in equations (7), in logs, as:
ls + c = cst + a(1 + )h + [(1 - a)(1 + ) - 1]ld,
(10)
19 where lowercase letters are logs of uppercase variable (so that, for instance, l ln L) and ls, ld denote, respectively, labor supply and labor demand (in logs). From equation (10), we immediately see that labor demand has a slope that is smaller than the slope of labor supply when + 1 > (1 - a)(1 + ). The latter condition may be rewritten as < c. But Proposition 3.1 has shown that < c rules out local indeterminacy, as the steady state is then a saddle (see case (i) in the statement of that proposition). On the contrary, local indeterminacy of the steady state is possible, as is usual in one-sector models, when externalities are large enough ( > c). More importantly, our analysis also shows that indeterminacy is possible, with almost constant returns to scale (small values of ), when and a are small: this generates a labor market configuration such that both labor demand and labor supply are "flat" enough and labor demand has a higher slope than that of labor supply. In essence, the basic mechanism is then that optimistic expectations may be fulfilled, in equilibrium, because agents willing to increase current consumption work harder today: this increases, in fact, the real wage, production, investment, and decreases the returns on human capital, so that optimistic expectations are compatible with intertemporal equilibrium. In our credit-constrained economy, an additional feature of the boom is that the physical stock increases, thereby allowing foreign debt to go up. Eventually, increasing interest payments will negatively affect output, so that consumption, investment and debt will go down: pessimistic expectations will then prevail during the temporary recession. In short, a traditional financial accelerator effect may exacerbate volatility, through international credit, in open economies.2 In other words, this mechanism is stronger the easier the access to international borrowing for open economies with enough collateral, in the case of constrained foreign credit. As we just discussed, the key condition for indeterminacy, that is, + 1 < (1 - a)(1 + ) (or > c), is more likely to be met when, other things equal, a is small enough. We have seen that the latter condition may be interpreted as implying that the share of physical capital, which serves as collateral, in total capital should be large enough. However, 2Again, this effect may lead to indeterminacy with small externalities in economies with low shares of human capital: in such economies, international borrowing reduces the proportion of capital accumulation that has to be financed with internal ressources.
20 one can also study the impact of a on the occurrence of indeterminacy when the share of physical capital is fixed. Then one concludes, from Proposition 3.1, that the lower , the more likely indeterminacy. Other things equal, economies with a low share of human capital may be subject to the basic mechanism described earlier and, for that reason, more susceptible to expectation-driven fluctuations. Our analysis also suggests that indebted countries having different shares of total capital in output ( + ) may be more or less likely to be perturbed by expectation-driven fluctuations, depending on their shares of human and physical capital. For example, the first and last lines in Table i suggest that for a given ratio /( + ) = 0.975, indeterminacy requires lower externalities when both the human capital share and the physical capital share are small. In the real world, different indebted countries have different physical and human capital shares. For instance, Burkina Faso, Cote d'Ivoire, and Zambia are classified, in the statistical appendix of World Bank's Global Development Finance 2003, as "severely indebted, low-income" countries (see tables A51 and A52, pp. 232-5), but these nations differ in their physical capital share, the average of which was, respectively, 76%, 66%, and 55% for the period 1960-903, and also perhaps differ in their share of human capital. However, the three countries are much more volatile than OECD countries: the standard deviation of output growth was, over the period 1960-90, 1.9 times larger for Burkina Faso than for the US, and 2.7 times larger for Cote d'Ivoire and Zambia than for the US (see Breen and Garcia-Pen~alosa [18, Table A1]; see also Ramey and Ramey [45]). Within the class of "severely indebted, middle-income" countries, as classified by the World Bank, one can find even more volatile economies: ouput volatility in Peru, Jordan and Guyana was, respectively, 2.7, 4.1 and 4.8 times larger than ouput volatility in the US, for the period 1960-90 (the average capital share of Peru, Jordan and Guyana was about 66%, 65% and 30%, respectively, for the period 1960-90, again from UN data). 3I thank Emilie Daudey for providing me with the latter figures, computed from UN data.
21 5 Conclusion This paper has studied a benchmark open-economy growth model with endogenous constraints on international borrowing and it has shown that imposing, quite realistically, some borrowing limits does not necessarily rule out the occurrence of expectation-driven volatility in open economies that have access to international credit markets. We have found that credit-constrained, small open economies may be destabilized by expectation-driven fluctuations, even in the presence of small externalities, provided that the share of human capital is low enough. It follows that opening the capital account may push a saddle-point stable economy on a volatile path. Moreover, economies that highly borrow on international credit markets to finance capital accumulation are more susceptible to expectation-driven fluctuations. The main mechanism through which fluctuations are exacerbated is the presence of a financial accelerator effect that increases the levels of collateral and foreign debt during booms. On the contrary, tighter constraints on external borrowing may protect the economy against expectation-driven volatility. In contrast with some existing results in the literature, we have shown that the elasticity of labor supply and the elasticity of intertemporal substitution in consumption both condition the set of parameter values associated with expectation-driven fluctuations. Quite surprisingly, indeterminacy requires, under some usual conditions, intertemporal substitution to be weak enough. A notable lesson that may be drawn from our analysis is that the easier the access to international borrowing (the looser the credit constraint, given the share of total capital), the more likely indeterminacy and endogenous fluctuations. However, we also have shown, in an extension of the model, that international capital mobility may or may not lead to local indeterminacy, depending also on the level of tax progressivity. In the real world, countries seem to differ significantly with respect to the progressivity of their tax system, for various reasons including distribution concerns (see, for example, Benabou [9]). Therefore, an interesting application would be to confront the level of tax regressivity compatible with indeterminacy to the corresponding estimates, so as to assess the plausibility of endogenous volatility. Moreover, our analysis
22 has shown that the level of human capital plays a key role: indeterminacy is more likely in economies with a low share of human capital. Put together, these results seem in agreement with the observation that a number of indebted, volatile countries are similar in that they have low incomes, but differ in their physical and human capital shares. However, it remains to be seen how the present model is able to match the observed correlations associated with actual business cycles that unfold in those economies. It would also be interesting to test whether indebted, volatile countries tend to have low human capital shares and regressive tax systems, which may sound plausible. On the theoretical side, a relevant extension of our analysis would be to assume, in agreement with existing tax schedules, that labor income taxes and capital income taxes have different progressivity features. Moreover, it would also be relevant to take into account the fact, documented by some studies (e.g. Krusell et al. [35]), that skilled labor and physical capital are complements while raw labor and physical capital are substitutes. In view of some results by Barro et al. [7, pp. 23-28], one expects that the stronger complementarity between physical and human capital, the higher the concavity in production, in which case the occurrence of indeterminacy and expectation-driven fluctuations should be more likely. I hope this gives directions for future and potentially fruitful research.
A Linearized Dynamics
This appendix derives and linearizes, around the steady state, the dynamical system describing intertemporal equilibria which consists of equations (7)-(8), together with equations (4) and (5). The first step is to rewrite, from the static condition in (7), the following equation:
[ + 1 - b(1 + )]l = cst + a(1 + )h - c,
(11)
23
where lowercase variables are logs of uppercase variables (so that, for instance, l = ln L), using the fact that y = cst + a(1 + )h + b(1 + )l from taking logs in equations (4) and (5). This yields, by using equation (11):
y - h = 0 + 1h + 2c,
(12)
where
0
=
ln(B)
+
b(1+)[ln(b(1- )B )] +1-b(1+)
,
1
=
b(1+)+(+1)[a(1+)-1] +1-b(1+)
,
2
=
-b(1+) +1-b(1+)
.
By rewriting equations (7)-(8) in logs and using equations (13), it is easy to get:
c = a(1 - )e0+1h+2c - - , h = (1 - )e0+1h+2c - - ec-h.
(13) (14)
It is straightforward to show that, under our assumptions, the Differential Equations (14) admit a unique interior steady state. More precisely, c = 0 yields, from the first equation of system (14):
(1 - )e0+1h+2c = ( + )/a,
(15)
On the other hand, h = 0 then yields, from the second equation of system (14):
ec = [( + )/a - ]eh.
(16)
One can then easily solve the two latter equations for c and h, provided that ( + )/a > .
Straightforward computations lead to the following expressions of T and D, respectively the trace and
determinant of the Jacobian matrix associated with equations (14), evaluated at the unique steady state
c, h:
T
=
+ a
[1
+
2
a
]
+
+(1-a) a
=

+
(+)(+1) +1-(1-a)(1+)
,
D
=
(
+
)(
+(1-a) a
)[1
+
2].
(17)
24 B Indeterminacy with Constant Returns: Predetermined Public Spending
In this appendix, we show how indeterminacy results may also be obtained in our credit-rationed openeconomy model without productive externalities - that is, with constant returns - when the government finance a predetermined amount of public expenditures G by using distortionary taxes on labor income. Following Schmitt-Grohe and Uribe [48], we assume that:
G = (t)(t)L(t),
(18)
where is the marginal productivity of raw labor and is no longer given by equation (26): the tax rate
is now determined by the balanced-budget rule in equation (18), given the constant flow of government
spending, the real wage and labor supply.
We further assume, for simplicity, that labor is indivisible (as in Hansen [25] and Rogerson [46])) - that is,
= 0 - and that consumption utility is logarithmic - that is, = 1. The representative agent then maximizes:
Z
e-t{ln C(t) - L(t)}dt,
(19)
0
subject to:
H = Y - H - C - G.
(20)
Assuming that externalities are absent - that is, = 0 - production is now given by:
Y = BHaLb,
(21)
where B = (1 - )[/(r + )]/(1-), a = /(1 - ), b = 1 - a, and it generates a flow of income such that:
Y = uH + L,
(22)
where u is the rental on human capital. If we define as the marginal utility of income, that is, = 1/C,
then straightforward computations lead to the analogs of the first-order conditions in equations (7):
= + - u, 1 = (1 - ).
(23)

25
Putting together equations (23), the market clearing condition:
H + H + 1 + G = Y,
(24)

the balanced-budget rule in equation (18), the definition of production in equation (21), and the usual conditions that input rentals equal marginal productivities - that is, L = bY and uH = aY - one gets equilibrium conditions that coincide with the four equations (5) - (8) derived in Schmitt-Grohe and Uribe [48, pp. 979-80]. Consequently, we can borrow from the results established by Schmitt-Grohe and Uribe [48, p. 983] and conclude that a necessary and sufficient condition for local indeterminacy is that the steady-state tax rate belongs to the interval bounded below by a and above by the tax rate that maximizes tax revenues. In this version of the model, therefore, local indeterminacy arises under the assumption of constant returns to scale. Further borrowing from Schmitt-Grohe and Uribe [48], we could also establish that the occurrence of indeterminacy is robust to some extensions of the model. In particular, allowing, more realistically, for the presence of capital income taxes, of income-elastic government expenditures, or of public debt confirms that indeterminacy is compatible with values of income tax rates that are observed for the United States and other OECD countries.
The main lesson from this exercise is that credit-rationed open economies are, despite the imposed constaint on international borrowing, susceptible to expectation-driven fluctuations even in the absence of externalities and with an a priori flat tax rate, which extends both existing results in the literature and our results of section 3. However, this conclusion relies on the assumption that a is small enough, just as in our main analysis (see sections 3 and 4 above). In fact, actual tax rates on labor income probably belong to (0.2; 0.5) in most OECD countries. Focusing on the lower end of this interval, we conclude that a = /(1 - ) has therefore to be lower than 0.2: the share of human capital must be less than 20% of the share of total labor. This means, if we accept the usual value of 1 - = 0.7 for the share of total labor, that indeterminacy occurs only if human capital has a share below 0.14. On the other hand, fixing the total share of physical
26 and human capital + means that small values for imply high values for : indeterminacy is then more likely to occur in economies that have easily access to international borrowing and, therefore, have a large debt/GDP ratio. Therefore, our main conclusions stated in sections 3 and 4 remain valid in this version of the model.
C Indeterminacy and Progressive Taxes In this appendix, we extend the basic model presented in the main text and show two results: when the tax rate on labor income is regressive, indeterminacy is more likely, while, on the contrary, progressive taxes may protect the economy against endogenous volatility.
C.1 Model and Laws of Motion
The consolidated budget constraint is now changed to:
K + H - D = (1 - )(Y - RK K) + (1 - K)RK K - (K + H) - rD - C,
(25)
where K 0 is the constant tax rate on physical capital income RKK, and 0 is the labor income tax rate to be specified below. Government is assumed to finance public spending G that do not affect private decisions by taxing output, i.e. G = Y . To keep things simple, we assume that the tax schedule is as follows:
= 1 - Y -,
(26)
where 0 < < 1 - is a parameter that plays only a minor role in the following analysis and < 1. For instance, one may interpret as depending on the base level of income - the steady state level for instance - that is taken as given. The important feature here is that the tax rate increases (resp. decreases) with the tax base when is positive (resp. negative). Therefore, this tax system exhibits progressivity when is
27 positive, and regressivity when is negative. As in Guo and Lansing [24], we assume that households take into account how the tax rate will affect their earnings. Note that we assume only for simplicity that capital income tax rate is flat (Pintus [44] shows, in a related framework with population growth and technical progress, that including progressive taxes on capital income yield a similar model). So as to ensure the existence of a positive steady state, we also assume that taxes are not too regressive, that is, > , where is a negative lower bound to be determined below (see Proposition C.1)
Equating the return on physical capital to r + yields K = (1 - K)Y /(r + ), and, therefore, using equation (1), this gives equation (4) where, here, B = [(1 - K )/(r + )]/(1-). Therefore, the budget constraint (25) may be written as the following, by observing that interest payments are proportional to output, that is, (r + )D = (1 - K)Y when D = K:
H = (1 - )(1 - )Y - H - C.
(27)
Households decisions follow from maximizing (2) subject to the budget constraint (27), using equations (4)-(5) and (26), given the initial stock H(0) < H. The first-order conditions are:
C
=
Y 1- a(1 - )(1 - )
- - ,
L C
Y 1- = b(1 - )(1 - ) .
(28)
C
H
L
We may rewrite the budget constraint, from (27) and (26), as:
H
Y 1-
C
= (1 - )
-- .
(29)
H
H
H
Equations (28)-(29), together with equations (4) and (5), characterize the dynamics of intertemporal equilibria. Note that after-tax income is concave in H whenever a(1 + )(1 - ) < 1, that is, > 1 - 1/(a(1 + )).
28 C.2 Local Indeterminacy when Externalities are Small
Below, we linearize equations (28)-(29) and (4)-(5), around the interior steady state which is shown to be unique. For a steady state to be feasible, we need the additional condition that + > a(1 - ) to be met (see, below, the proof of Proposition C.1), which imposes a negative lower bound on , that is, > 1 - ( + )/(a). In other words, the tax rate cannot be too regressive.
Straightforward computations yield the following expressions for trace T and determinant D of the Jacobian
matrix of the dynamical system derived from equations (7)-(8), (4) and (5) (see equations (34) below):
T
=
+ a(1-)
[1
+
2
a(1-)
]
+
+(1-a(1-)) a(1-)
,
D
=
(
+
)(
+(1-a(1-)) a(1-)
)[1
+
2],
with
1
=
b(1+)(1-)+(+1)[a(1+)(1-)-1] +1-b(1+)(1-)
and
2
=
-b(1+)(1-) +1-b(1+)(1-)
.
(30)
We focus, as most papers in the literature do, on the case for which the steady state is locally a sink, that is, asymptotically stable, and we derive conditions on parameter values such that T < 0 and D > 0. We then show that these conditions include arbitrarily small values of (that is, configurations with arbitrarily small externalities): indeterminacy is compatible with almost constant returns to scale. In our discussion, a key parameter is the level of tax progressivity , the empirical estimate of which is not easily measured and, therefore, quite uncertain.
Direct inspection of equations (30) shows that the sign of D is in fact the sign of (1-a)(1+)(1-)- -1 (the latter term is the opposite of the denominator of 1 + 2) under the assumptions that < 1, 1 and a(1+)(1-) < 1. The latter assumption is equivalent to > 1-1/(a(1+)), where > . On the other hand, T is a decreasing function of , the denominator of which vanishes when (1 - a)(1 + )(1 - ) = + 1, i.e. when = c 1 - ( + 1)/((1 - a)(1 + )). Therefore, T tends to - when tends to c from below and decreases from + (to a positive value) when increases from c to one. The immediate consequence is that D > 0 and T < 0 when is slightly smaller than c. More precisely, we can in fact show that the
29 following statements hold.
Proposition C.1 (Indeterminacy, Arbitrarily Small Externalities, and Progressive Taxes) In our credit-constrained open economy, the dynamics of consumption C and human capital H, given by equations (28)-(29) and (4)-(5), have a unique positive steady state (C, H) if 1 - ( + )/(a) < < 1. Assume that 1 (that is, the elasticity of intertemporal substitution in consumption 1/ is smaller than one) and, moreover, that a( + 1)[( + ) + ] < (1 - a) (that is, the elasticity of labor supply 1/ is large enough, or externalities, measured by , are small enough). Then the steady state (C, H) is, locally: (i) a saddle (locally determinate) when c 1 - ( + 1)/((1 - a)(1 + )) < < 1 (that is, when the tax rate is progressive enough). (ii) a sink (locally indeterminate) when h < < c, undergoes a Hopf bifurcation at = h, and is a source (locally unstable) when < < h, with h 1 - ( + 1)[( + ) + ]/((1 - a)(1 + )) > 1 - 1/(a(1 + )) > . In particular, local indeterminacy of the steady state (C, H) occurs for values of that are arbitrarily close to zero (arbitrarily small externalities), provided that is close enough to, but smaller than c. Positive but small values of imply, in turn, that c is then negative. In summary, local indeterminacy occurs when externalities are small and returns to scale are almost constant, provided that the tax rate is regressive and elastic enough.
Proof: We first derive and linearize, around the steady state, the dynamical system describing intertemporal equilibria which consists of equations (28)-(29), together with equations (4)-(5). The first step is to rewrite, from the static condition in (28), the following equation:
[ + 1 - b(1 + )(1 - )]l = cst + a(1 + )(1 - )h - c,
(31)
where lowercase variables are logs of uppercase variables (so that, for instance, l = ln L), using the fact that
30
y = cst + a(1 + )h + b(1 + )l from taking logs in equations (4) and (5). This yields, by using equation (31):
(1 - )y - h = 0 + 1h + 2c,
(32)
where
0
=
ln(B 1- )
+
b(1+)(1-)[ln( b(1- )(1-)B 1- )] +1-b(1+)(1-)
,
1
=
b(1+)(1-)+(+1)[a(1+)(1-)-1] +1-b(1+)(1-)
,
2
=
-b(1+)(1-) +1-b(1+)(1-)
.
By rewriting equations (28)-(29) in logs and using equations (33), it is easy to get:
(33)
c = a(1 - )(1 - )e0+1h+2c - - , h = (1 - )e0+1h+2c - - ec-h.
(34)
It is straightforward to show that, under our assumptions, the differential equations (34) admit a unique interior steady state. More precisely, c = 0 yields, from the first equation of system (34):
(1 - )e0+1h+2c = ( + )/(a(1 - )),
(35)
On the other hand, h = 0 then yields, from the second equation of system (34):
ec = [( + )/(a(1 - )) - ]eh.
(36)
One can then easily solve the two latter equations for c and h, provided that ( + )/(a(1 - )) > , i.e. that > 1 - ( + )/(a).
Straightforward computations lead to the following expressions of T and D, respectively the trace and
determinant of the Jacobian matrix associated with equations (34), evaluated at the unique steady state
c, h:
T
=
+ a(1-)
[1
+
2
a(1-)
]
+
+(1-a(1-)) a(1-)
,
D
=
(
+
)(
+(1-a(1-)) a(1-)
)[1
+
2].
(37)
Note that expressions (37) slightly differ from the corresponding formula appearing in Guo and Lansing
[24, p. 488] (once we take into account that the latter authors impose = 1).
31
Direct inspection of equations (33) and (37) shows that D has the same sign as that of 1 + 2, when < < 1, with 1 - ( + )/(a). But 1 + 2 = [(1 - )(1 - a)(1 + )(1 - ) + ( + 1)(a(1 + )(1 - ) - 1)]/[ +1-(1-a)(1+)(1-)] has a negative numerator when 1 and > 1-1/(a(1+)), so that its sign is the opposite of the sign of its denominator. Therefore, D is positive, under the assumptions just stated, if and only if + 1 < (1 - a)(1 + )(1 - ), or equivalently if and only if < c 1 - ( + 1)/((1 - a)(1 + )). The steady state is then a saddle if and only if > c, which proves (i). When < c, the steady state may be a sink if T < 0 or a source if T > 0, and it may undergo a Hopf bifurcation at T = 0. Straightforward computations from equations (33) and (37) yield:
( + )( + 1)
T = + + 1 - (1 - a)(1 + )(1 - ) .
(38)
Therefore, equation (38) shows that T is a decreasing function of . In particular, T increases from a positive value to + when decreases from 1 to c, as the denominator of the fraction in equation (38) tends to zero from above while its numerator is positive. Moreover, T increases from - to zero when decreases from c toward h 1 - ( + 1)[( + ) + ]/((1 - a)(1 + )), where h is the value such that T = 0. Therefore, T > 0 if < h, T = 0 when = h, and T < 0 when h < < c (cases covered by (ii)). It is straightforward to check that 1 > c > h and that > under the assumptions on our primitive parameters. Moreover, h > when a( + 1)[( + ) + ] < (1 - a). Note that the latter assumption implies that externalities cannot be too large ( must be small enough), that labor supply has to be elastic enough ( must be small enough), and that a has to be small enough.
Finally, our last result shows that local indeterminacy of the steady state (when D > 0 and T < 0) may be
associated with arbitrarily small externalities (values of close to zero), when a( +1)[(+)+] < (1-a)
is met. In fact, c and h(< c) respectively increases and decreases with , and h tends to c when tends to zero. Therefore, indeterminacy occurs, for small 's, when is close enough to, but smaller than
c. Finally, one observes that c = 1 - ( + 1)/((1 - a)(1 + )) is negative when is close enough to zero,
as ( + 1)/(1 - a) > 1 when a < 1.
2
32 The main result of Proposition C.1 is that local indeterminacy is associated with values of that belong to (h, c). Therefore, indeterminacy is a priori compatible with progressive or regressive taxes, that is, with positive or negative values of . In fact, c is negative (and h is also negative and close to c) when is close to zero, but this value increases with and is positive if > ( + a)/(1 - a). Therefore, indeterminacy implies regressive taxation when externalities are arbitrarily small. Appendix B shows that indeterminacy is also likely to occur in a version of the model without externalities, when government finances a constant level of public expenditures by using distortionary taxes. In that framework, imposing to the public budget to be balanced implies that the tax rate is regressive, which is reminiscent of the results established in Proposition C.1. Therefore, our specific assumptions that the tax rate is elastic to the tax base and that external effects generate increasing returns are not critical for our main conclusion that credit-rationed open economies may be perturbed by expectation-driven fluctuations. Note that indeterminacy is ruled out when the tax rate is progressive enough (when is close enough to one), independent of whether externalities are small or large (as we just noted, large values of are associated with a large threshold value c, above which the steady state is saddle-point stable), in contrast with results by Guo and Harrison [23]. Finally, note that the elasticity of intertemporal substitution in consumption that is, 1/ - plays also a role here, in contrast with the results in Lahiri [36], Weder [51], Nishimura and Shimomura [41], Meng and Velasco [40]. Proposition C.2 (Indeterminacy and Strong Intertemporal Substitution in Consumption) Assume that = 0 (that is, labor is indivisible). Then indeterminacy of the steady state (C, H) is ruled out, in the limit, when goes to zero (that is, when the elasticity of intertemporal substitution in consumption 1/ tends to infinity).
33
Proof: From equations (30), one can show that, here again, T < 0 implies that < c 1 - ( +
1)/((1 - a)(1 + )), as T does not depend on . However, < c implies that D < 0 when = 0 and
is close enough to zero. To prove this, remember that D has the same sign as that of 1 + 2, when
> 0, > 0, a < 1, < < 1, with 1 - ( + )/(a). But when < c, the denominator of
1 + 2 = [( + 1)(a(1 + )(1 - ) - 1) + (1 - a)(1 + )(1 - )]/[ + 1 - (1 - a)(1 + )(1 - )] is negative,
while its numerator is positive when = 0 and = 0 (implying 2 = 0). Therefore, the steady state is then
a saddle if < c, and a saddle or a source when > c.
2
Note that the condition a( + 1)[( + ) + ] < (1 - a) in Proposition C.1 is more likely to be met when a is small enough. Moreover, both and tend to - when a goes to zero: the lower bounds on are small when a is small. In fact, one has h < c < when a is greater than 0.5. Second, manipulating the corresponding expressions appearing in Proposition C.1 lead one to conclude that both h and c decrease with a, although the size of the range c - h increases with a (however, the size of c - h is proportional to /(1 + ) and is, therefore, small when is close to zero). Therefore, indeterminacy is more likely when a is not too large, in the sense that h and c are then not too negative (when is close to zero). This shows that Corollary 4.2 still holds in this economy. To illustrate, in the present context, Corollary 4.2, we now consider benchmark parameter values: indivisible labor supply (that is, = 0), small externalities ( = 0.001), = 0.065 and = 0.1 (as in, for instance, Benhabib and Farmer [11]). These values imply that condition a( + 1)[(+ )+ ] < (1 - a) in Proposition C.1 is now simply a < 0.3935. Moreover, note that all cases considered below imply that (1 - ) < 0.001. For sake of brevity, we study two configurations depending on the share of total capital: + = 0.8 (the main case considered by Barro et al. [8]) or 0.4, that are summarized in the following table. Our numerical experiments confirm that although the range (h, c) is quite small when is small (externalities are then negligible), indeterminacy is compatible with slightly negative values of when a is
34 + = 0.8 = 0.7, that is, a 0.33 (h, c) (-0.502, -0.499) = 0.75, that is, a = 0.2 (h, c) (-0.251, -0.249) + = 0.4 = 0.2, that is, a = 0.25 (h, c) (-0.335, -0.332) = 0.35, that is, a 0.08 (h, c) (-0.084, -0.083) Table ii: indeterminacy range (h, c) when the share of collateral varies. small enough, that is, when is large: the tax rate need not necessarily be highly regressive. Therefore, our numerical cases illustrate that when the credit constraint is getting tighter, that is when decreases for given + , the human capital share a increases and indeterminacy is becoming less likely. On the contrary, open economies that face loose credit constraints - that is, have small a's - are more susceptible to indeterminacy and expectation-driven fluctuations. Let us, again, consider the case of closed economies. We know, by modifying results from Guo and Lansing [24, p. 488] to incorporate human capital, that in the absence of international borrowing, a closed economy is saddle-point stable (the steady state is then determinate) if and only if + 1 > (1 - - )(1 + )(1 - ), that is, if and only if > cclosed 1 - ( + 1)/((1 - - )(1 + )). On the other hand, Proposition C.1 has shown that indeterminacy occurs only when is close to, but smaller than c 1 - ( + 1)/((1 - /(1 - ))(1 + )) in the credit-constrained, open economy. Then there are parameter values such that the closed economy is saddle-point stable while indeterminacy prevails in the open economy: this is the case when c > cclosed, that is, if 1 - - < 1 - /(1 - ). It is not difficult to check that the latter condition is always met when + < 1, which we have assumed. Therefore, we can state the following result. Corollary C.1 (Comparing Closed and Open Economies) Suppose that the level of tax progressivity belongs to (cclosed, c). Then indeterminacy prevails in the credit-constrained, open economy, while the corresponding closed economy is saddle-point stable.
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