Capital Mobility, Interest Rate Rules, and Equilibrium Indeterminacy. in a Small Open Economy

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1 Capital Mobility, Interest Rate Rules, and Equilibrium Indeterminacy in a Small Open Economy Wen-ya Chang Hsueh-ang Tsai Department and Graduate Institute o Economics, Fu-Jen Catholic University Abstract We develop a monetary model o the small open economy operating under lexible exchange rates and either with or without international capital mobility. Money is introduced into this economy through the channel o shopping-time technology. We ind that in a small open economy with capital immobility an active interest rate rule yields determinacy o equilibrium while a passive rule shows indeterminacy. In a small open economy with capital mobility, the equilibrium displays determinacy regardless o the type o interest-rate eedback rules. Keywords: Capital mobility; Interest rate rules; Indeterminacy; Small open economy JEL Classiication: F41; E5; E31 Please send all correspondence to: Proessor Wen-ya Chang Department and Graduate Institute o Economics Fu-Jen Catholic University Hsinchuang, Taipei 4, TAIWAN ecos1001@mails.ju.edu.tw Tel: Fax:

2 1. Introduction It is well known in the literature on interest rate rules that the active interest-rate rule, the policy by which the monetary authority raises nominal interest rates by more than the rise in inlation, will stabilize the economy by ensuring a unique equilibrium. This has sprouted a growing body o research to investigate its validity. 1 Among these, Benhabib et al. (001) employ a monetary endowment economy with money entering preerences and/or production technology to show that interest rate rules cause indeterminacy in equilibrium. Yet all these studies are restricted to the analysis o closed economies. As increasing integration has prevailed in international goods and asset markets in recent years, domestic macroeconomic perormance has become more sensitive to oreign shocks, especially under the ebbs and lows o oreign capitals. This act naturally inspires us to explore the relationship between the interest-rate eedback rule and macroeconomic stability rom the perspective o an open-economy ramework. This paper speciically makes a new attempt to assess the role o international capital mobility in impacting the relationship between interest-rate eedback rules and macroeconomic stability. The role o international capital mobility has long attracted much attention by international macroeconomists. Fleming (196) and Mundell (1963) assess how capital mobility aects the eectiveness o macroeconomic policies under alternative exchange rates. Frenkel and Rodríguez (198) examine the eect o capital mobility on exchange rate dynamics. Obsteld (198) urther notes that, when the economy is initially in external balance, the current-account eect o terms-o-trade shocks is dierent under perect capital mobility and imperect mobility. Turnovsky (1997) later emphasizes that a temporary policy has a permanent eect under perect capital mobility while a temporary policy only has a temporary eect under imperect capital mobility. In view o these developments, it is a worthwhile task or us to study how international capital mobility aects the link between interest-rate eedback rules and macroeconomic stability in 1 The related literature includes, or example, Leeper (1991), Woodord (1995), Taylor (1999), Clarida et al. (000), and Carlstrom and Fuerst (001). Furthermore, extended discussions o interest rate rules in both the history o economic thought and modern analysis o theory can be ound in Woodord (003). 1

3 an open economy. To address this link, we extend the ramework o Kimbrough (1986) and Ljungqvist and Sargent (000) to a continuous-time setting o a small open economy operating under lexible exchange rates and either without capital mobility or with capital mobility. We ind that, in a small open economy without capital mobility an active interest rate rule yields a determinate, and a passive rule, indeterminate equilibrium. By contrast, in a small open economy with capital mobility, the equilibrium displays determinacy regardless o interest rate rules.. A small open economy without capital mobility Assume that the country is a small open economy and operates under lexible exchange rates. We irst consider that this economy has no access to the world capital market. This implies no international capital mobility. The economy produces a single traded good whose price in terms o oreign currency is given on the world market and is normalized to be unity. That is, P= EP, (1) where P is the price level o traded goods in home currency, E is the exchange rate (deined as the home currency price o oreign currency), and P is oreign prices o traded good ( P = 1). The domestic country consists o an ininitely-lived representative agent and a government. The agent is endowed with a constant output y each instant o time and may hold two assets (domestic money and bonds). The agent s objective is to maximize the ollowing lietime utility 0 u ρ t ( c, l) e dt, () subject to a& = y + ( R π ) a Rm c τ, (3) l + s = 1, (4) where c = consumption, l = leisure, ρ = a constant rate o time preerence, a m+ b = total real wealth, m M P = real money holdings, b B P = real bond holdings, M = nominal money holdings, B = nominal bond holdings, R = nominal interest rates, π P & P = E & E = rate o inlation (rate o exchange depreciation), τ = lump-sum taxes, and s = shopping-time technology.

4 Equation () says that the representative agent maximizes the discounted lietime utility with perect oresight. The instantaneous utility unction u satisies u > c 0, u 0 l >, u cc < 0, u ll < 0, u cl > 0, and uccull ucl > 0. Equation (3) is the agent s budget constraint. Equation (4) is the constraint o time allocation. The time endowment is normalized to be unity and the agent allocates his or her time endowment to leisure and time spent transacting. We ollow Kimbrough (1986) and Ljungqvist and Sargent (000) to motivate the agent or holding money in the economy by a transaction technology with shopping time. It describes that the amount o shopping time s required or purchasing each unit o consumption depends on the ratio o real money holdings to consumption. Speciically, the shopping-time technology is s = g ( m c) c; g < 0, g 0. (5) Equation (5) states that the shopping time is an increasing unction o consumption and a decreasing unction o real balances. We now substitute equation (5) into equation (4) and let be the co-state variable and γ the multiplier o the current value Hamiltonian associated with (3) and (4). The necessary optimum conditions or the representative agent are u c ( c, l) = + γ [ g( m c) g ( m c) m c], (6a) u l ( c, l) = γ, (6b) R = γ g ( m c), (6c) & = ( ρ + π R), (6d) together with equations (3) and (4), and the transversality condition o a, t lim ae ρ = 0. (7) t Following Leeper (1991), Benhabib et al. (001), and Carlstrom and Fuerst (001), we assume that the monetary authority adopts an interest-rate eedback rule as R = ψ ( π ), ψ > 0. (8) From equation (8), monetary policy is active at an inlation rate π i ψ > 1, and is passive i ψ < 1. We urther assume that at all times the government incurs no expenditure and inances its 3

5 deicit by raising seigniorage and issuing bonds. Accordingly, the government s budget constraint in real value can be stated as a& = ( R π ) a Rm τ. (9) With equations (7) and (9), we mainly ocus on the Ricardian iscal policy. Equations (3) and (9) give the balance-o-payments equilibrium: 0= y c. (10) With no international capital mobility, the balance o payments equals the balance o trade. Because there exists only one traded commodity in the world, the balance o trade is the excess o domestic production over domestic absorption. Under lexible exchange rates, the balance o trade is in equilibrium at all times. As a consequence, the perect-oresight equilibrium o a small open economy may be expressed by equations (6a)-(6d), (4), (5), and (7)-(10). gives As equations (6a)-(6c), (4), (5), (8), and (10) hold at all times, manipulating these equations u ( y, 1 g( m y) y) = + u ( y,1 g( m y) y)[ g( m y) g ( m y) m y], (11a) c l From equations (11a) and (11b), we urther have m m where m 1 { [ ( )] = g ucl ull g g + ul g } > 0, y y π = {[( g ) u ll ul y ψ ( π ) = u l ( y,1 g( m y) y) g ( m y). (11b) g ] m m = m(), (1a) π = π (), (1b) 1 ψ} < 0. ψ Dierentiating equation (1b) with respect to time and using equations (6d) and (8), we have & π = π [ ρ + π ψ ( π )]. (13) * From equation (13) with π& = 0, the steady-state value o exchange rate depreciation π is * * ρ = ψ ( π ) π r, (14) * where r denotes the real interest rate. Linearizing equation (13) at π yields * & π = φ( π π ), 4 (15)

6 * where φ = π [1 ψ ( π )]. Because φ is the characteristic root o equation (15), the equilibrium * at π is determinate or indeterminate depending on the sign o φ. Given < 0 in equation π (1b), the sign o φ hinges on the type o monetary policy. Since the rate o exchange depreciation is a non-predetermined variable, the equilibrium shows determinacy i the monetary policy is active ( ψ > 1) while the equilibrium is under indeterminacy i the monetary policy is passive ( ψ < 1). Based on the above analysis, we come to the ollowing proposition: Proposition 1. In a small open economy without capital mobility, the active interest rate rule yields a unique perect-oresight equilibrium, while the equilibrium displays indeterminacy under a passive monetary policy rule. The intuition or our result is the ollowing. Assume that initially the inlation rate (exchange rate depreciation) is above its steady-state level. I the monetary policy is active, then a rise in the inlation rate brings an increase in the real interest rate. This in turn induces the shadow price to decrease, because & = ρ + π R < 0 is true. A lower shadow price then raises desired consumption and lowers the marginal utility o consumption. Since the balance o trade must be in equilibrium under lexible exchange rates, consumption equals endowment at all times. In order to optimally ollow the declining path o the marginal utility o consumption, agents thus must cut leisure time as a result o u cl > 0. However, less leisure time means more time spent transacting, which thereby lowers real money holdings and urther raises the inlation rate. This obviously drives the inlation rate away rom its value at the steady state. I, on the other hand, the monetary policy is passive, then a rise in the inlation rate brings about a all in the real interest rate. A all in the real interest rate will raise the shadow price and then the marginal utility o consumption. To match the higher marginal utility so as to keep the consumption constant, agents must raise leisure time. However, more leisure means less shopping time, which in turn raises the need or real balances and lowers the inlation rate. Thereore, this trajectory is consistent with an equilibrium in which inlation converges to its stationary value. 5

7 3. A small open economy with capital mobility We now relax the assumption o no international capital mobility and assume that the home economy aces an imperect world capital market as proposed by Obsteld (198) and Turnovsky (1997). In this regard, the domestic resident may hold an internationally-traded bond denominated in oreign currency in addition to domestic money and bonds. Furthermore, the cost (beneit) o borrowing (lending) aced by the domestic resident is an increasing (decreasing) unction o his or her indebtedness (creditworthiness) to the rest o the world. As a consequence, we assume that R = R ( b ), R dr db < 0, (16) where R represents oreign nominal interest rates and b is the stock o oreign bonds denominated in oreign currency. With capital mobility, the agent s budget constraint is revised as ollows a& = y+ ( R π) a+ ( R + π R) b Rm c τ, (17) where a ( M + B+ Eb ) P= m+ b+ b = total real wealth. The agent s objective then is to maximize equation () subject to equations (17) and (4). By the same calculation in section, the optimum conditions or the representative agent are equations (6a)-(6d), (17), (4), (7), and R= R ( b ) + π. (18) Equation (18) expresses the non-arbitrage condition o portolio selection between domestic and oreign bonds. As or the government sector, monetary policy is the same as equation (8). The government s budget constraint in real value is now revised as 3 a& = ( R π) a+ ( π R) b + b& Rm τ. (19) The low budget constraint o the representative agent in nominal value is given by M& + B& + Eb& = Py + RB + ER b Pc Pτ. Using equation (1) with P = 1, the deinitions o m, b, and π in the previous section, and the deinition o a in equation (17), we can obtain equation (17). 3 The low budget constraint o the government in nominal value is given by M& + B& = RB Pτ. Following the same procedure as in ootnote, we can yield equation (19). 6

8 Combining equations (17) and (19) yields the balance-o-payments equilibrium: b& = y c+ R ( b ) b. (0) Equation (0) describes that the rate o accumulation o oreign bonds is equal to the current-account surplus, which equals the trade surplus plus interest payments on oreign bond holdings. Under lexible exchange rates, the current-account surplus (deicit) is oset by the capital-account deicit (surplus) and hence the balance o payments is always in equilibrium at all times. Given the above assumptions and derivations, equations (6a)-(6d), (4), (5), (7), (8), and (18)-(0) thereore solve the perect-oresight equilibrium o a small open economy with international capital mobility. Manipulating equations (6a)-(6c), (4), (5), (8), and (18) gives u( c,1 gmc ( ) c) = + u( c,1 gmc ( ) c)[ gmc ( ) g ( mcmc ) ], (1a) c l ψ ( π ) = u ( c,1 g( m c) c) g ( m c), (1b) l ψ ( π ) = R ( b ) + π. (1c) From equations (1a)-(1c), the instantaneous relationships can be derived as: π = π( ), b (a) c= c(, b ), (b) m= m(, b ), (c) where 1 b R ( 1), {[( ) ] } 0, c g ull ulg R c b b π = ψ = + θ < = ψπ θ, c m m m m = R u u g g + u g g u g + < c c c { [ ( ) ( ) cc cl ll l ] 3 θ} 0, m m m mb = ψ π [ ( ) ( ) ] b u cc ucl g g + ull g g ul g, 3 c c c m m θ = g [ ucl ull ( g g )] ul g <0, c c 1 = ( g ) ( uccull ucl ) ul g ( ull g + ucc ucl g) > 0. c Substituting equation (18) into (6d) and equation (b) into (0), we have & = [ ρ R ( b )] =Γ(, b 7 ), (3a)

9 b& = y c(, b ) + R ( b ) b =Ω(, b ), (3b) where Γ = 0, Γ b = R > 0, Ω = c > 0, and Ω b = R + R b cb. Let α1 and α be the two characteristic roots satisying equations (3a) and (3b), we then have αα c R 1 = < 0. (4) Equation (4) indicates that the dynamic system possesses one positive root and one negative root. Because the economy has one non-predetermined variable,, and one predetermined variable,, b there exists a unique saddle-path leading to the stationary equilibrium. Thereore, we establish the ollowing proposition: Proposition. In a small open economy with international capital mobility, the steady state equilibrium displays determinacy regardless o interest-rate eedback rules. Proposition runs in sharp contrast with Proposition 1. The key to this contrast is that an integrated world capital market plays an important role in acilitating the inancing needs o a small open economy rom abroad. Hence, the non-arbitrage condition o the agents portolio choice pins down the unction o the interest-rate eedback rule. 4. Concluding remarks This paper develops a monetary model o the small open economy operating under lexible exchange rates, assumed either with or without international capital mobility. Money is introduced into the economy through the channel o shopping-time technology. Within this model, we make a new attempt to investigate the role o capital mobility in aecting the relationship between interest-rate eedback rules and macroeconomic stability. We ind that, in a small open economy without capital mobility the active interest-rate rule yields a determinate equilibrium, while the passive rule oers an indeterminate equilibrium. By contrast, in a small open economy with capital mobility, the equilibrium always displays determinacy regardless o interest-rate eedback rules. 8

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