Devaluation, Capital Formation, and the Current Account

Transcription

1 Devaluation, Capital Formation, and the Current Account Wen-ya Chang Hsueh-fang Tsai Department of Economics, Fu-Jen Catholic University, Hsinchuang, Taipei 242, Taiwan Abstract This paper re-examines the effects of a devaluation and that of a rise in the rate of devaluation on capital formation and the current account in an optimizing monetary model of a small open economy with endogenous labor, investment with adjustment costs, and perfect capital mobility. It is shown that a devaluation leads to capital accumulation and a current-account deficit in the long run and during the adjustment process, whereas a rise in the rate of devaluation has an ambiguous impact on capital formation and the current account depending on the relationship between consumption and real balances in the utility function. If consumption and real balances are separable or complements, then a rise in the rate of devaluation produces capital decumulation and a current-account surplus in the long run as well as on the transition path. These results stand in sharp contrast with Calvo (1981). Keywords: Devaluation; Capital formation; Adjustment costs; Endogenous labor; Perfect capital mobility; The current account JEL Classification: F41 Please send all correspondence to: Dr. Wen-ya Chang, Department and Graduate Institute of Economics, Fu-Jen Catholic University, Hsinchuang, Taipei 242, TAIWAN. ecos11@mails.fju.edu.tw; Tel: ; Fax:

2 1. Introduction A popular economic tool adopted by policymakers in developing countries is the exchange-rate policy. These countries, for example the Southern Cone of Latin America, Israeli, and Southeast Asia, often use an exchange-rate policy as an instrument of stabilization. Edwards and Montiel (1989) point out that some countries conduct a stepwise devaluation while others enact a devaluation followed by a crawling peg. There are therefore two types of exchange-rate policy, a once-andfor-all devaluation of the currency and a permanent rise in the rate of devaluation. In a well cited contribution, Calvo (1981) constructs an optimizing model of a small open economy with money in the utility function and capital immobility in order to draw the important implications of a devaluation and that of a rise in the rate of devaluation on the balance of payments. He establishes that a devaluation improves, whereas a rise in the rate of devaluation deteriorates, the balance of payments. Obstfeld (1981) introduces not only capital mobility but also endogenous time preference into Calvo s model and concludes that an increase in the rate of devaluation improves the balance of payments. Michener (1984) builds a three-good (exports, non-traded goods, and imports) two-sector optimizing model without capital mobility and obtains that a devaluation improves the balance of payments. Obstfeld (1985) extends Calvo s (1981) model to include capital mobility in the first part of the paper and focuses on the effect of a pre-announced reduction in the rate of devaluation on the economy since an unanticipated permanent reduction in the rate of devaluation is neutral. Agénor and Montiel (1996) and Agénor (1998) respectively modify the Calvo (1981) model when facing an imperfect world capital market and find that Calvo s results are robust. Petrucci (23) sets up an optimizing model of a small open economy with a finite horizon, capital formation, a cash-in-advance constraint, and capital immobility. He yields that the result of devaluation is the 1

3 same as Calvo (1981), and a rise in the rate of devaluation has an ambiguous impact on the balance of payments. The goal of this paper is to re-examine the robustness of Calvo s results. As documented by Petrucci (23), Despite the simplified framework considered, Calvo s findings are very general and robust, in the sense that they are not qualitatively affected by other ways of introducing money into the economy and/or by the consideration of an endogenous capital stock, as an alternative asset to real money balances. Without depending on an endogenous time preference or finite horizon, this paper extends Calvo s (1981) framework to the Brock (1974)-like model of a small open economy. Specifically, this paper modifies the Calvo (1981) model to introduce endogenous leisure-labor discretion, capital formation with convex costs, and perfect capital mobility. Based on an amended Calvo s model, it can be shown that a devaluation leads to capital accumulation and a current-account deficit in the long run and during the adjustment process, while a rise in the rate of devaluation results in an ambiguous effect on capital formation and the current account depending on the relationship between consumption and real money balances in the utility function. If consumption and real balances are separable or complements, then an increase in the rate of devaluation definitely produces capital decumulation and a current-account surplus in the steady state as well as on the transition path. These results run in sharp contrast with Calvo (1981). The remainder of the paper is organized as follows. The structure of the model is outlined in section 2. Section 3 investigates the evolution of relevant variables following a currency devaluation shock and a rate of devaluation shock, respectively. Finally, section 4 summarizes the main findings of our analysis. 2

4 2. The Model The modeling strategy extends Calvo s (1981) framework so as to allow for endogenous labor, capital accumulation with adjustment costs, and international capital mobility. The open economy under investigation operates under fixed exchange rates and is assumed to be small in the sense that it cannot affect foreign price levels and foreign interest rates. This small open economy produces a traded good which is the same as that on the world market, i.e., the purchasing power parity holds. In addition, there is zero rate of foreign inflation, and without loss of generality the foreign price of the traded good is set at unity. Accordingly, the domestic price level ( P ) is equal to the exchange rate ( E ). Moreover, the economy can borrow or lend any amount at the exogenous world real rate of interest r in the international capital market, i.e., there is perfect capital mobility between the domestic and the international economies. Our small open economy consists of a representative household and a government. The objective of a representative household is to maximize the discounted sum of future instantaneous utilities: Max t [ U ( c, m ) V ( )] ρ l e d t, (1) where c is consumption, m ( M E ) is real cash balances, M is nominal money holdings, l is labor supplied, and ρ is a constant rate of time preference. The instantaneous utility functions U and V are assumed to be well-behaved, satisfying U c >, U m >, U cc <, U mm <, 2 U cc U mm U cm >, V <, V <, and the Inada conditions. In addition, we postulate consumption and real money balances to be normal goods, i.e., U mmu c UcmU m < and U ccu m UcmU c <. 3

5 At each instant of time, the representative household is bound by a flow constraint linking wealth accumulation to any difference between its gross income (output, government transfers, and foreign interest earnings) and its expenditure (the sum of consumption, inflation tax, and total investment cost). Moreover, output y is produced using a stock of productive capital k and a supplied labor, according to a neoclassical production function f ( k, l) that satisfies f k >, f l >, f kk <, f ll <, and the Inada conditions. In addition, this production function is assumed to be linearly homogeneous in capital and labor, implying f kl >. As a consequence, this constraint can be expressed as: b& m& = f ( k, l) R r b c C( I) πm, (2) where an overdot denotes the time derivative, b is foreign bonds holdings, R is government transfers, I is investment, and π E & E is the rate of inflation (the rate of devaluation). Following Sen and Turnovsky (1989) and Turnovsky (1997), we assume that investment involves adjustment costs. 1 The function C (I) satisfies C >, C >, C ( ) =, and C ( ) = 1. In addition, the rate of capital accumulation and investment are related by the constraint: k & = I. (3) The optimum conditions necessary for this optimization problem are: U c ( c, m) = λ, (4a) V ( l) = λfl ( k, l), (4b) q = λ C (I), (4c) 1 Based on real world experience, the stickiness in their evolution of capital seems to be more appropriate in reality. Introducing adjustment costs of investment in capital captures this fact. The early literature includes Lucas (1967) and Gould (1968). 4

6 & λ = λ( ρ r ), (4d) & λ = λ( ρ π ) U m ( c, m), (4e) & q = q ρ λ fk ( k, l), (4f) and equations (2) and (3), where λ is the shadow value of financial wealth and q is the shadow value of investment. From equations (4d) and (4e), we yield: U m ( c, m) λπ = λr. (5) Equation (5) states that the non-arbitrage condition between holding foreign bonds and holding real balances must hold in each instant of time, implying that foreign bonds and real balances are perfect substitutes. It is well known in the literature, such as Obstfeld (1983), Sen and Turnovsky (1989), and Turnovsky (1997), that with both the rate of time preference ρ and the world real interest rate r being exogenously given in the small open economy, the ultimate attainment of a steady state is possible if and only if ρ = r. It thus follows from equation (4d) that & λ = should be satisfied at all times, so that λ will at once jump to its long-run equilibrium value λ, when news or a shock hits, and will remain constant at this value along the adjustment path. It is noted that the choice of a constant for λ is not arbitrary; it is endogenously determined as a part of the equilibrium. Furthermore, we utilize the shadow value of financial wealth as the numeraire in order to simplify the exposition and hence define q q / λ as representing the market value of capital in terms of the (unitary) value of financial wealth. As a consequence, equations (4c) and (4f) can be rewritten as: q = C (I), (6a) 5

7 q& = qr fk ( k, l). (6b) Following Calvo (1981), the government s only function is to transfer its tax revenue from inflation so as to compensate the opportunity cost of holding real money balances. That is: R = πm. (7) Since foreign bonds and real balances are perfect substitutes, we define a b m as the stock of financial wealth. Combining equations (2) and (7) and using the definition of a, we have: a& = f ( k, l) r a c C( I) r m. (8) It should be noted that equation (8) limits the economy s accumulation of financial assets to its net savings, i.e., changes in financial wealth must take place over time. Because the monetary authorities must sell and buy foreign exchange in the fixed exchange rate regime, foreign bonds b can be bought and sold instantaneously for foreign exchange in the international capital market. As a consequence, the division of total assets between b and m is instantaneously reversible at any instant of time t. A similar assumption is made by Mankiw (1987), Obstfeld (1989), Sen (1991), Zou (1995), and Chang, Tsai, and Lai (22). Finally, equation (8) can be reinterpreted as the status of balance of payments. Since the monetary authorities maintain a regime of fixed exchange rates, the increase in real money supply is determined by the balance-of-payments surplus; that is: m & = f ( k, l) r b c C( I) b&. (8') The perfect foresight equilibrium of this small open economy can be described by equations (3), (4a), (4b), (5), (6a), (6b), and (8) associated with λ = λ. The evolution of the system proceeds as follows. First, it is clear from equations (4a), (4b), (5), and (6a) that the instantaneous relationships 6

8 should be satisfied at all times and can be derived as: c = c( λ, π ), (9a) m = m( λ, π ), (9b) l = l( λ, k), (9c) I = I(q), (9d) where c λ = ( U cu mm U mu cm ) / U c <, cπ = λucm / > <, m λ = ( U mu cc UcU cm ) / Uc <, mπ = λucc / <, l λ = fl /( V λfll ) >, l k = λ f k l /( V λfll ) >, I q = 1 / C >, and = 2 U ccu mm U cm >. 2 A simple intuition for equations (9a)-(9d) is as follows. An increase in the shadow value of financial wealth induces the agent to increase savings, thereby leading to a decline in consumption and real balances as well as a rise in labor supplied. A rise in the inflation rate increases the cost of money holdings and hence decreases real money balances, which in turn results in an ambiguous response in consumption depending on U cm. An increase in capital enhances the marginal product of labor, thereby leading to an increase in labor employment. A rise in the market value of capital increases the incentive of investment so that investment increases. Substituting equations (9a)-(9d) into equations (3), (6b), and (8) and expanding the resulting equations in a neighborhood of the stationary state, we next have: k & = I(q), (1a) 2 From equations (4a) and (5), the original results of c λ and m λ are as follows: cλ = [ U mm ( r π ) U cm ] and mλ = [ U cc ( r π ) U cm ]. Furthermore, using the relationship U m U c = r π from equations (4a) and (5), we obtain the expressions in the text. 7

9 q& = qr fk ( k, l( λ, k)) = H ( k, q, λ ), (1b) a & = f ( k, l( λ, k)) r ( a m( λ, π )) c( λ, π ) C( I( q)) = J ( k, q, a, λ, π ), (1c) where H k = ( fkk fkl l k ) >, 3 H = r q >, H λ = f kl l λ <, J k = fk f l l k >, J q = I q <, 4 J = r a >, = λ Jλ f l l λ cλ r m >, and J = ( ) < > π cπ r mπ. 5 Let µ be the characteristic root of the dynamic system. From equations (1a)-(1c), we can then yield: 2 ( J a µ )( µ H qµ H k Iq ) =. (11) Letting µ 1, µ 2, and µ 3 be the three characteristic roots that satisfy equation (11), it is clear from equation (11) that 1 2 ( ) / µ µ = H k Iq = fkk fkll k C < and µ 3 = J a = r >. Among the three characteristic roots of the system, two are positive and one is negative. This implies that the system displays a saddlepoint stability, which is common for perfect foresight models. For expository convenience, we assume that µ 1 < < µ 2. The general solution for k, q, and a thus can be described by: = t t k µ h A e µ h12 A2e, (12a) k 2 = t t q µ h A e µ h22 A2e, (12b) q 2 3 Substituting the result of l k from equation (9c) into fkk f k ll k, we have: fkk fkll k = fkkv ( V λ fll ) <. In deriving the above expression, we have used the assumption of linearly 2 homogeneous production function, i.e., f kk fll f =. kl 4 The original expression of J q is C ( I) I q. As the dynamic system is evaluating in a neighborhood of the steady state, I = holds in the steady state and hence C ( ) = 1. As a result, J q = Iq = 1 C <. 5 Substituting the result of c π and m π from equations (9a) and (9b) into J π, we obtain: J ( ) > π = λ Ucm r Ucc <. If U cm, then J π > is true. If U cm <, then J > π < depends on U U < r cm cc >. 8

10 µ t t t a A e 1 µ A e 2 µ = 1 2 A3e, (12c) a 3 where a bar over a variable denotes its stationary level; A 1, A 2, and A3 are as yet undetermined coefficients; h 11 = ( µ 1 r ) ( J k µ 1) <, h 12 = ( µ 2 r ) ( J k µ 2 ), h 21 = C µ 1h11 >, and h = µ. 22 C 2h12 At the long-run equilibrium, the economy is characterized by k & = q& = a& =. From equations (1a)-(1c), we can easily derive the following long-run relationships: 6 q = 1, (13a) k = k (λ ), (13b) a = a( λ, π ), (13c) where kλ = f kl l λ H k >, a = ( J k J ) r < > λ k λ λ, and aπ = Jπ r <. Two points should be noted here. First, the long-run equilibrium value of relevant variables will be definitely obtained when λ is determined in the following analysis. Second, not only the rate of devaluation but also devaluation will have a long-run effect. In particular, devaluation leads to a rise in the domestic price as a result of purchasing power parity, thereby further affecting the long-run equilibrium through a once-and-for-all jump in λ. This result stands in sharp contrast with Calvo (1981), Obstfeld (1981), and Petrucci (23). They all conclude that devaluation has no effects on the long-run endogenous variables. 6 k & = implies I =. Accordingly, from equation (6a) q = C ( ) = 1 is true. With q& = and q = 1, r = f k holds in the steady state. As an increase in the shadow value of financial wealth induces the agent to increase labor supplied and hence the marginal product of capital, capital stock must increase so as to ensure after substituting a& =, I =, and k = k (λ ) into equation (1c), we can obtain equation (13c). r = fk. In addition, 9

11 The phase diagram is illustrated in Figure 1. In the upper panel of Figure 1, it is clear from equations (1a) and (1b) that the slopes of the loci k & = and q& = are: q k k& = =, (14a) q k q& = = H k r <. (14b) The lines SS and UU represent the stable and unstable branches, respectively. As indicated by the direction of the arrows, the convergent saddle path SS is downward sloping, while the divergent branch UU is upward sloping. In the lower panel of Figure 1, from equation (1c) the slope of the locus a& = for a given value of q is: a k a& = = J k r <. (15) Since k and a are predetermined variables, as indicated by the direction of the arrows, there also exists a convergent saddle path ZZ. The line ZZ is downward sloping, but flatter than the locus a& =. 3. Dynamics of a Shock in Devaluation We are now ready to examine an unanticipated permanent devaluation and an unanticipated permanent increase in the rate of devaluation. We assume initially that the economy is in a steady state at time t =. We next consider the situation when the authorities increase the exchange rate permanently from E to E 1 at time t =. Based on the general solution reported in equations (12a)-(12c) with the long-run relationships of 1

12 equations (13a)-(13c), we can use the following equations to express the feature of such a devaluation policy: k k ( λ = ), µ ( ) 1t k λ1 h11a1 e, t = t (16a) 1, t = q = µ 1 1t h21a1 e, t (16b) a( λ =, π ), a µ (, ) 1t a λ1 π A1 e, t = t (16c) Here, and denote respectively the instant before and after the policy change, and A 1 is an undetermined coefficient. There are some supplementary explanations for the specifications of equations (16a)-(16c). First, at time, the economy is in its stationary equilibrium with π = π. With the initial stock of capital and financial assets being respectively given at k and a, the stationary values of initial λ ( = λ ) and q ( = 1) are associated with k = k ( λ ) and a = a( λ, π ) so as to satisfy both r = fk ( k ( λ ), l( λ, k ( λ ))) from equation (1b) with q& = and a = a( λ, π ) = m( λ, π ) [ c( λ, π ) f ( k, l( λ, k ))]/ r from equation (1c) with a& =. Second, from onwards, as the home currency has devalued while the rate of devaluation remains intact at π, the shadow value of financial wealth will jump to a new value λ 1 so as to satisfy the intertemporal budget constraint in a stationary equilibrium. As a consequence, the stationary value of k is associated with λ 1 and that of a corresponds to λ 1 and π. Third, the home currency devalues permanently at time, as the transversality condition requires that the economy move to a point exactly on the convergent stable branch at that instant of time. This means that the undetermined 11

13 coefficients associated with unstable eigenvalues, namely A 2 and A 3, must be set to zero from onwards. To understand the exact paths of k, q, and a, we must solve the appropriate value for A 1 and the jump direction and magnitude of λ. Both are determined by the following conditions: 7 k = k, (17a) 1 2 a a = M ( E E ) E. (17b) Equation (17a) indicates that capital stock remains intact when the authorities increase the exchange rate. Equation (17b) states that the stock of financial assets instantaneously decreases as a result of devaluation. Putting equations (16a) and (16c) into equations (17a) and (17b) gives: 8 2 A 1 = kλ M ( E1 E ) E ( kλ h11a λ ) >, (18a) 2 ( λ1 λ ) = h 11M ( E1 E ) E ( kλ h11aλ ) >. (18b) Equipped with the information of equations (13b), (13c), and (18b), we can easily infer that a devaluation leads to an increase in the steady-state capital stock and a decrease in the stock of financial assets, thereby reflecting a current-account deficit in the long run. The result for the current account obviously runs in sharp contrast with Calvo (1981), Obstfeld (1981), and Petrucci (23). 7 In response to a devaluation, the change of financial wealth is da = db 2 (dm E ) ( M E ) de, where M is the initial money balances. Since there is an instantaneous reversion between foreign bonds holdings and money holdings ( db = 2 dm E ), da = ( M E )de < will be true. 8 The term kλ h11aλ seems to be ambiguous. However, substituting the result of h 11 and a λ from equations (12a) and (13c) into kλ h11aλ and manipulating the expression will yield: k [ ( ) ( )] λ h11 aλ = µ 1kλ J k r Jλ µ 1 r r ( J k µ 1) <, since J k r = fk fl l k r = fll k >. 12

14 Furthermore, based on equations (16a)-(16c), (17a), (17b), (18a), and (18b), in what follows we can analyze the evolution of the economy. In Figure 2 the initial equilibrium, where k & = intersects q& = ( λ ), is established at point Q in the upper panel; the initial market value of capital and the initial stock of capital are 1 and k, respectively. The lower panel of Figure 2 indicates that with the locus a& = ( λ, q = 1, π ), then in response to k, the initial stock of financial assets is required to be a at point Q. Upon the shock of an unanticipated permanent rise in E, in the upper panel the q& = ( λ ) locus shifts rightward to the q& = ( λ 1 ) locus while k & = remains intact; 9 the new equilibrium is at point Q 1, where the capital stock increases from k to k. At the same time, the a& = ( λ, q = 1, π ) locus shifts downward to the a& = ( λ 1, q = 1, π ) locus in the lower panel, 1 and in association with k the new steady state is at Q 1 whereby the steady-state value of a decreases from a to a. As is evident in the upper panel of Figure 2, at the instant of currency devaluation, the market value of capital must immediately increase from 1 to q so as to place the economy exactly at Q on the stable branch SS. At the same time, in the lower panel the stock of financial assets 2 drops from a to a as a result of ( M E ) de, 11 in order to take the economy at once to point Q on the stable arm ZZ. Thereafter, q will decrease and k will increase along the SS curve towards its long-run equilibrium Q 1 in the upper panel, while in the lower panel a continues 9 As is obvious in equation (18b), a devaluation leads to a rise in λ. As a result, from equations (1a) and (1b), we have: k E q& = = H λ H k λ E > and k λ &. k= = 1 Similar to footnote 9, it is evident from equation (1c) that a E a= = J λ J a λ E < 11 Please see footnote 7. &. 13

15 to decrease in association with an increase in k along the ZZ locus towards its long-run equilibrium Q. 1 During the transition process, the economy experiences capital accumulation and a continuing current-account deficit following a currency devaluation. We now discuss the other situation whereby the authorities will increase the rate of devaluation permanently from π to π 1 at time t =. Based on the general solution reported in equations (12a)-(12c) with the long-run relationships of equations (13a)-(13c), we can use the following equations to express the feature of such a policy switch: k k ( λ = ), ( ) µ 1t k λ1 h11a1 e, t = t (19a) 1, t = q = µ 1 1t h21a1 e, t (19b) a( λ =, π ), a (, ) µ 1t a λ1 π1 A1 e, t = t (19c) where A 1 is an undetermined coefficient. There are some supplementary explanations for the specifications of equations (19a)-(19c). First, at time, the economy is in its stationary equilibrium with π = π. With the initial stock of capital and financial assets being respectively given at k and a, the stationary values of initial λ ( = λ ) and q ( = 1) are associated with k = k ( λ ) and a = a( λ, π ) so as to satisfy both r = fk ( k ( λ ), l( λ, k ( λ ))) from equation (1b) with q& = and a = a( λ, π ) = m( λ, π ) [ c( λ, π ) f ( k, l( λ, k ))]/ r from equation (1c) with a& =. Second, from onwards, as the rate of devaluation has increased from π to π 1, the shadow value of financial 14

16 wealth will jump to a new value λ 1 so as to satisfy the intertemporal budget constraint in a stationary equilibrium. As a consequence, the stationary value of k is associated with λ 1 and that of a corresponds to λ 1 and π 1. Third, the rate of devaluation increases permanently at time, as the transversality condition requires that the economy move to a point exactly on the convergent stable branch at that instant of time. This means that the undetermined coefficients associated with unstable eigenvalues, namely A 2 and A 3, must be set to zero from onwards. To understand the exact paths of k, q, and a, we must solve the appropriate value for A 1 and the jump direction and magnitude of λ. Both are determined by the following conditions: 12 k = k, (2a) a = a. (2b) Equations (2a) and (2b) indicate that the stock of capital and financial assets should remain intact when the authorities increase the rate of devaluation. Putting equations (19a) and (19c) into equations (2a) and (2b) yields the value of A 1 and the jump direction and magnitude of λ : > A1 = kλ ( λ1 λ ) h11 <, (21a) > ( λ1 λ ) = h 11aπ ( π1 π ) ( kλ h11aλ ) <. (21b) Since the sign of a π depends on the sign of U cm as stated in equation (13c) with the information of J π, the jump direction and magnitude of λ hinge crucially on the sign of U cm. If consumption 12 Due to the definition of E & π t π E, the time path of the exchange rate is E = Ee, where E denotes the initial level of exchange rate. When π increases from π to π 1 at the instant of time t =, the level of exchange rate E is continuous at E. As a consequence, a is continuous following a rise in π. 15

17 and real balances are independent or complements ( U cm ), then a π is negative and hence λ jumps down to its new equilibrium value λ 1. If consumption and real balances are substitutes ( U cm < ), then the response of λ is ambiguous depending on the substitution degree of consumption and real balances (the sign of Ucm r Ucc ). Furthermore, from equations (13c) and (21b) we have: a > π = aλ ( λ1 λ ) ( π1 π ) aπ = aπ kλ ( kλ h11aλ ) <. (22) Obviously, if U cm, then a rise in the rate of devaluation leads to a surplus in the current account. If U cm <, then an increase in the rate of devaluation results in ambiguous responses in the current account depending on the substitution degree of consumption and real balances. These results stand in sharp contrast with Calvo (1981). Substituting equation (21a) into equations (19a)-(19c) and using the result of equation (21b), we have the exact paths of k, q, and a. As the sign of U cm will affect the jump direction and magnitude of λ and hence the evolution of the economy, we only focus on the case of U in order to save space. cm In Figure 3 the initial equilibrium is the same as that in Figure 2, meaning that k & = intersects q& = ( λ ) at point Q in the upper panel and point Q lies at the locus a& = ( λ, q = 1, π ) in the lower panel. In response to an unanticipated permanent increase in π, in the upper panel q& = ( λ ) shifts leftwards to q& = ( λ 1 ) as a result of a decrease in λ while k & = remains intact; 13 the new equilibrium is at Q 1, where the capital stock decreases from k to k in the long run. The result 13 Under the situation where U, λ π < is true from equation (21b). Analogous with footnote 9, it is clear cm from equations (1a) and (1b) that k π q& = = H H λ π < λ k and k λ k& = =. 16

18 of a decrease in the long-run capital stock is opposite to that of Petrucci (23). At the same time, a& = ( λ, q = 1, π ) shifts leftwards to a& = ( λ 1, q = 1, π1) in the lower panel, 14 and in association with k the new steady state is at Q 1 whereby the steady-state value of a increases from a to a. Before we proceed with the analysis, one point worth noting is that E is continuous even if π rises. 15 Accordingly, the stock of financial assets a is continuous in response to an increase in π. As a consequence, in the lower panel the stable arm ZZ should link the initial steady state Q with the new steady state Q 1 since k is also predetermined, otherwise the economy is divergent. Based on this information, in the upper panel of Figure 3 at the instant of a rise in the rate of devaluation, the market value of capital must at once decrease from 1 to q so as to place the economy exactly at Q on the stable arm SS. At the same time, in the lower panel the economy stays put at point Q, because k and a are predetermined. Thereafter, q will increase and k will decrease along the SS curve towards its long-run equilibrium Q 1 in the upper panel, while in the lower panel a continues to increase in association with a decrease in k along the ZZ locus towards its long-run equilibrium Q. 1 During the transitional dynamics, the economy experiences 14 Since π and λ affect equation (1c), from equation (1c) we have: k π a& = = J λ J k λ π Jπ J k = [ J λ ( λ π ) Jπ ] J k. Substituting the result of λ π from equation (21b) into the above equation, using the result of a π and a λ from equation (13c), and manipulating the resulting equation, we obtain: k π a& = = Jπ kλ ( r J k h11) r ( kλ h11aλ ) J k. By using J π > if U cm from footnote 5, k λ h11 aλ < from footnote 8, J k >, h 11 >, and k λ >, we have k π. a& = < 15 Please see footnote

19 capital decumulation and a continuing current-account surplus following a rise in the rate of devaluation. Before ending our discussion, one point should be addressed. Endogenous leisure-labor discretion, investment with adjustment costs, and perfect capital mobility are the key factors to obtain the contrasting results. First, if we abstract endogenous leisure-labor discretion from the model, then the dynamic system will reduce to equation (1a) and ~ q& = qr fk ( k) = H ( k, q), (1b') ~ a & = f ( k) r ( a m( λ, π )) c( λ, π ) C( I( q)) = J ( k, q, a, λ, π ), (1c') ~ ~ ~ ~ ~ ~ where H k = fkk >, H q = H q >, J k = f k >, J q = J q <, J a = J a >, J λ = ( cλ ~ r m λ ) >, and J π = J π >. According to equations (1a) and (1b'), the economy stays put at the initial stationary equilibrium Q in the upper panel of Figure 2 and Figure 3 regardless of the shock of a devaluation or that of the rate of devaluation. At the same time, with an unchanged capital stock k, the economy always remains intact at its initial steady state Q in the lower panel of Figure 3 when the rate of devaluation is increased, while the stock of financial assets decreases discretely so as to take the economy at once to a point exactly at the a& = ( λ 1, q = 1, π ) locus in the lower panel of Figure 2 if a devaluation is undertaken by the authorities. Second, if adjustment costs are absent in the model, then equation (4c) reduces to: q = λ. (4c') From equations (4c'), (4d), (4e), and (4f), the non-arbitrage condition between holding foreign bonds and holding real balances is the same as equation (5), and that between holding foreign bonds and holding capital is: 18

20 r = f k ( k, l). (23) Equations (5) and (23) imply that foreign bonds, real balances, and capital are perfect substitutes. Accordingly, we define a ~ b m k as the stock of total real wealth and hence equation (8) can be revised as: ~ a & = f ( k, l) r a~ c r ( m k). (24) Furthermore, with the feature of & λ = and λ = λ at all times, the economy only has one dynamic equation [equation (24)]. Since equation (24) exhibits an instability property ( a ~ & a ~ = r > ), then a once-and-for-all change in λ will lead the current account to remain in equilibrium following a devaluation shock or the shock of the rate of devaluation, and hence there are no transitional dynamics. Third, if we do not consider the role of perfect capital mobility, then the perfect foresight equilibrium of this small open economy with capital immobility is described by equations (3), (4a), (4b), (4e), (6a), and q& = qu ( c, m) λ f ( k, l) qπ, (6b') m k m& = f ( k, l) c C( I), (8'') where q q λ, and λ is not a constant over time. As the system contains four differential equations, we cannot present any graphical analysis. In the steady state we can easily infer that a devaluation shock has no effects on the long-run endogenous variables and the shock of the rate of devaluation leads to Brock (1974)-like conclusions, 16 i.e., a rise in the rate of devaluation leads to a 16 The steady-state conditions for this case can be condensed to the following: q ˆ = 1, V ( ˆ) l = U ( c ˆ, m ˆ ) f ( ˆ, lˆ c l k ), U m ( cˆ, mˆ ) = U c ( cˆ, mˆ )( ρ π ), f ˆ, k ( k lˆ) = ρ, and f ( k ˆ, ˆ) l = cˆ, where a hat over a variable denotes its stationary value. 19

21 decrease in real balances and an ambiguous response in capital, labor employment, and consumption depending on U cm. Obviously, Calvo s results are robust. 4. Concluding Remarks This paper introduces endogenous leisure-labor discretion, capital formation with adjustment costs, and perfect capital mobility into the Calvo (1981) framework and investigates the effects of devaluation and the rate of devaluation on capital formation and the current account. It is found that a stepwise devaluation leads to capital accumulation and a current-account deficit in the long run and during the adjustment process, whereas a rise in the rate of devaluation results in an ambiguous impact on capital formation and the current account depending on the relationship between consumption and real money balances in the utility function. If consumption and real balances are additively separable or complements, then a rise in the rate of devaluation definitely generates capital decumulation and a current-account surplus in the steady state as well as on the transition path. 2

22 q UU q = 1 Q k & = k q& = ( λ ) SS k a a Q ZZ k ( λ,,π ) a & = q k Figure 1

23 q q Q q =1 Q Q 1 k & = SS q& = ( λ ) q& = ( λ 1 ) k k k a a Q a Q ( λ, = 1 ) a& = q, π a k Q 1 k ZZ a& = λ 1, q = 1, π ( ) k Figure 2

24 q q =1 Q 1 Q k & = q Q q& = ( λ 1 ) SS q& = ( λ ) k k k a a Q 1 a Q ( λ, = 1 ) a& = 1 q, π1 ( λ, = 1 ) a& = q π, ZZ k k k Figure 3

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