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Durable goods, habits, time preference, and exchange rates
North American Journal of Economics and Finance 14 (2003) 115–130
Durable goods, habits, time preference, and exchange rates Arman Mansoorian a , Simon Neaime b,∗ a
b
Department of Economics, York University, Toronto, Canada Department of Economics, Institute of Financial Economics, American University of Beirut, 850 Third Avenue, 18th Floor, New York, NY 10022-6297, USA
Received 27 June 2001; received in revised form 24 July 2002; accepted 23 October 2002
Abstract We work out the role of durable goods regarding the effects of exchange-rate policies in the money-in-utility model. If for whatever reason the stock of durables is above its steady-state level, then along the adjustment path consumption expenditures would be rising and the current account would be in surplus, but real money holdings would be constant. Without habits, money is super-neutral. Habits break super-neutrality. With habits and durable goods, the adjustment of consumption expenditures and the current account after a change in the rate of depreciation will likely be non-monotonic, while the adjustment of real balances will be monotonic. With Uzawa preferences and durable goods, the adjustment of real money balances will also be monotonic, while the adjustment of consumption and the current account will be non-monotonic. © 2002 Elsevier Science Inc. All rights reserved. JEL classification: F31; F32; F41 Keywords: Durable goods; Money; Habits; Exchange-rate policies; Time preference
1. Introduction The literature concerned with the effects of exchange-rate policies has paid little attention to the important role of durable goods. This is despite recent findings in the asset-pricing literature (e.g., Ferson & Constantinides, 1991; Heaton, 1995), suggesting that durable goods play an important role in improving the empirical performance of consumption-based asset-pricing models. The purpose of the present paper is to work out some implications of ∗
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durable goods for exchange-rate policies under three popular specifications of intertemporal utility: time-separable preferences, the habit-persistence model, and Uzawa preferences. Obstfeld (1981a, 1981b) was the first to discuss some policy issues for a small, open economy in an optimizing framework under the assumption that instantaneous utility is a function of consumption and real money holdings, as in Sidrauski (1967), and that the rate of time preference is an increasing function of instantaneous utility (as in Uzawa, 1968). In his (1981b) paper, he considers the effects for such an economy of a central bank target rate of money growth. The results in Obstfeld (1981a) reveal that the analysis is considerably simplified if it is assumed that the central bank targets the rate of devaluation of the domestic currency.1 This assumption is also made by Calvo (1981) and Djajic (1982). The analysis in the present paper will be confined to this case. Obstfeld (1981b) shows that in that setting, with perfectly flexible prices, a once-and-for-all devaluation will not have any effect. On the other hand, an increase in the rate of devaluation will lead to a sharp fall in consumption and real money holdings in the short run, and hence a current-account surplus. Over time, both consumption and real money holdings increase along the adjustment path to the new steady-state equilibrium. The reason is as follows. The equality of the rate of time preference and the world interest rate dictates the level of instantaneous utility which must be maintained in the steady state. The increase in the rate of devaluation of the domestic currency, therefore, requires substitution of consumption for real money holdings along a given steady-state indifference curve. Hence, in order to be able to finance his higher steady-state consumption, the representative agent must accumulate bonds along the adjustment path. Therefore, on impact, the increase in the rate of devaluation leads to a reduction in consumption and a current-account surplus. By the homotheticity of preferences, it follows that on impact real money holdings also fall. Along the adjustment path, both consumption and real money holdings rise over time. Ryder and Heal (1973) provide an alternative formulation of intertemporally dependent preferences to the Uzawa function. In their formulation, the rate if time preference is fixed, but instantaneous utility depends on both current full consumption and the habitual standard of living, represented by a weighted sum of past levels of full consumption.2 In a recent paper, Mansoorian (1996) uses this habit-persistence model to show that the effects of an increase in the rate of devaluation are very similar to Obstfeld’s (1981a), if preferences exhibit adjacent substitutability, but differ sharply if preferences exhibit adjacent complementarity.3 In the present paper, we discuss the importance of durable goods for exchange-rate policies. We start with a model with time-separable preferences (i.e., a model with no habits and a fixed rate of time preference) in which consumption goods exhibit durability. We 1 With fully flexible prices, this is equivalent to the central bank targeting the inflation rate, which according to Mishkin (2000) is the policy followed by many developed and emerging economies. The assumption that the central bank targets the inflation rate is also consistent with the assumptions made in the literature on the time consistency of monetary policies (e.g., Backus & Driffill, 1985; Kydland & Prescott, 1977; Walsh, 1995). 2 Full consumption is the utility derived from current consumption and current real money holdings. 3 With adjacent substitutability, an increase in current consumption will decrease the marginal utility of consumption in the near future relative to the marginal utility in the distant future. Similarly, with adjacent complementarity, an increase in current consumption will increase the marginal utility of consumption in the near future relative to the marginal utility in the distant future.
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show that an increase in the rate of devaluation of the domestic currency is super-neutral. It has no effect on consumption expenditures, the stock of durable goods, or the country’s net foreign asset position. Nevertheless, the holdings of real balances fall. These are the Sidrauski super-neutrality results extended to an open-economy setting with durable goods. They arise because preferences are time-separable. The focus of this paper is not only the issue of super-neutrality, as in the literature following Sidrauski, but also the effects of consumer durables on the dynamics of open-economy models. If the stock of durable goods is above its long-run level, the representative agent adjusts the stock by substantially reducing current purchases. Hence, along the adjustment path consumption expenditures are below their long-run level, and there is a current-account surplus. Nevertheless, because preferences are time-separable, and the rate of time preference is equal to the interest rate, real money holdings are constant over time. If one relaxes the time-separable preference structure by incorporating habits, then the results change fundamentally. Since the fall in real money holdings reduces welfare, the presence of habits requires a corresponding fall in the steady-state habitual standard of living. This requires a fall in the steady-state stock of durable goods, a corresponding fall in consumption expenditures, and a fall in net foreign asset holdings, as well as a fall in real money holdings. Thus, habits break super-neutrality even though there is complete absence of nominal rigidities. We follow the empirical literature and assume that with habits, preferences exhibit adjacent complementarity (as opposed to adjacent substitutability).4 As shown in Mansoorian (1996), with habits but non-durable goods and adjacent complementarity of preferences, the increase in the rate of devaluation causes the representative agent to maintain the habitual standard of living by reducing savings. Hence, after the increase in the rate of devaluation there is an increase in consumption and a current-account deficit, while real money holdings fall. Along the adjustment path, consumption and real money holdings are declining, and the country runs a current-account deficit. Durability in consumption goods changes the dynamics of the model fundamentally. In that case, it is not only the steady-state stock of habits, but the stock of durable goods which would likely have to adjust downwards in response to an increase in the depreciation rate. To reduce the stock of durables, the representative agent reduces his current purchases, which tends to lead to a current-account surplus. Hence, the habits and durability effects have competing influences on the adjustment of consumption and the current account. Adjustment of these variables will, therefore, be non-monotonic. On the other hand, in the case of durable goods and no habits, real money holdings do not change over time when the stock of durables is adjusted downwards. It follows, that the simultaneous presence of habits and durability makes adjustment of real money holdings along the adjustment path monotonic, and qualitatively the same as in the model with habits and non-durability. Another means by which one can break super-neutrality in the absence of nominal rigidities is through Uzawa preferences, as shown by Obstfeld’s contributions discussed 4 Constantinides (1990) requires a high degree of adjacent complementarity in order to solve the Mehra and Prescott (1985) equity-premium puzzle with the habit-formation model. Backus, Gregory, and Telmer (1993) require a high degree of adjacent complementarity in order to account for excessively high variation in the expected return on the forward relative to spot exchange rates.
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above.5 Uzawa preferences are another popular specification of preferences for open economies.6 The presence of durable goods again fundamentally changes the dynamics under Uzawa preferences, where an increase in the rate of devaluation requires substitution of consumption for real money holdings along the given indifference curve in the steady state. This implies an increase in the steady-state stock of durable goods. To increase this stock, the representative agent increases consumption expenditures above their steady-state levels. This adjustment competes with the dynamics of the Obstfeld model. Hence, the response of consumption and the current account is again non-monotonic. Since money balances do not exhibit durability, their adjustment is monotonic. The paper is organized as follows. The model with durable consumption goods no habits and a fixed rate of time preference is presented in Section 2. The effects of exchange-rate policies are discussed in Section 3. Some concluding remarks are made in Section 4.
2. The model with durable goods and time-separable preferences 2.1. The model The model corresponds most closely to Obstfeld’s (1981a). The foreign-currency price of the single good in the model is fixed at P∗ , which the small economy takes as given. The domestic-currency price of this good is P = EP∗ , where E is the exchange rate (the price of foreign currency in terms of domestic currency). With flexible goods prices, the inflation ˙ rate is equal to the rate of depreciation of the domestic currency (E/E), which is denoted by ε. The objective function of the representative agent is ∞ e−θt ω(ct + st , mt ) dt, (1) 0
where θ is the rate of time preference, ω(·) is a homothetic utility function, which depends on the current consumption of the services of consumer durables (ct +st ) and on current money holdings (mt ).7 The instantaneous utility derived by the representative agent at time t from consumption of the services of durable goods and from real money holdings is measured by ω(ct + st , mt ). We refer to it as full consumption. 5 Uzawa showed that Sidrauski’s super-neutrality result depended on the assumption of a fixed rate of time preference. In a closed economy in the steady state, the rate of time preference should be equal to the marginal productivity of capital. This dictates a unique level of capital (and, hence, of consumption) for the steady state. This result will not hold if the rate of time preference is a function of instantaneous utility. 6 In addition to Obstfeld’s classic contributions, see, for example, Devereux and Shi (1991) and Mendoza (1991). 7 The money-in-utility model has had a long history in monetary economics, starting from Sidrauski. One recent prominent paper that employs such a specification is by Obstfeld and Rogoff (1995). An alternative specification would be the cash-in-advance model, which is used by, for example, Calvo (1987). In that setting mt = ct ; and it can be shown that, as in Stockman’s (1981) original work, changes in the rate of growth of money will not have any effect on the steady-state equilibrium.
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In Eq. (1), ct is the amount of consumer durables purchased at time t, while st is the stock of durables inherited from the past. These durable goods depreciate at the rate of δ. Thus, t eδ(τ −t) cτ dτ, (2) st = −∞
and the evolution of st is given by s˙t = ct − δst .
(3)
Output is fixed at y units of the good at any time t. The representative agent also receives monetary transfers with real values of Tt from the government. There are two kinds of assets in the model, money balances and internationally traded bonds, whose foreign currency price is fixed and whose rate of return is fixed at r. The real assets of the representative agent are at = bt + mt ,
(4)
where bt is his bond holdings. His flow budget constraint is8 a˙ t = y + rat + Tt − ct − (r + εt )mt .
(5)
Finally, the intertemporal solvency condition, lim e−rt at ≥ 0,
t→∞
(6)
prevents the agent from borrowing without bound. The representative agent’s problem is to maximize (1), subject to (3), (4)–(6), and the initial conditions (i.e., s0 and a0 ), taking the paths of {Tt }, and {ε t } as given. Along a perfect foresight path, the agent’s expectations about {Tt } and {εt } coincide with the actual paths of these variables. In this paper we abstract completely from government expenditures on goods and services, and concentrate exclusively on exchange-rate policies.9 The government chooses the real monetary transfers, Tt , in order to satisfy its flow constraint, which says that it finances its transfers (Tt ) from the interest on its international bond holdings and from seigniorage ˙ t, Tt = rRt + εt mt + m
(7)
where Rt is the central bank’s foreign reserves at time t, held in the form of internationally traded bonds. As stated in Section 1, we assume that the central bank is targeting the rate of depreciation of domestic currency (i.e., the inflation rate) and not the rate of growth of money per se. This is the assumption made by Obstfeld (1981b), and is consistent with the assumptions 8 According to this constraint the amount of assets the representative agent accumulates should be equal to his net income (rat + y + Tt ) minus his total expenditures (ct + (r + εt )mt ). 9 Obstfeld (1981a) allows for government purchases of goods and services. He, however, assumes that such government expenditures are unproductive, or, alternatively, that the utility of the representative agent is strongly separable between private consumption and public goods. We could make this assumption without changing any of our results.
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made in the literature on the time consistency of monetary policies.10 As may be clear from Eq. (7), the central bank can target the inflation rate either by intervening on the foreign exchange market (by adjusting R, through purchases and sales of bonds), or by adjusting the transfers T.11 Clearly, if the bank targets {ε t } by adjusting T, then we would have a flexible exchange rate system (with R fixed). Finally, we integrate the budget constraints of the representative agent (5) and the central bank (7) in order to obtain the dynamics of the country’s net foreign assets f (≡ b + R): f˙ = rf + y − c,
(8)
according to which the country’s current account f˙ is equal to total income rf + y less absorption c. 2.2. Discussion The optimization problem of the representative agent and all optimality conditions are derived in the Appendix. Here we highlight some of the important features of the optimality conditions. First, note that for a steady state to exist in a small open economy with a fixed rate of time preference and access to unlimited borrowing or lending at a fixed rate of interest, r, we must assume that the rate of time preference is equal to the rate of interest, θ = r. If θ > r, the rate of return required by the representative agent (i.e., θ ) is larger than the market rate of return (i.e., r). In that case, the representative agent will want a consumption profile that will be increasing over time indefinitely. On the other hand, if θ < r, then the rate of return required by the representative agent is smaller than the market rate of interest. In that case, the representative agent will want a consumption profile that will be declining over time indefinitely. None of these is consistent with the existence of a steady state, that is, a long run equilibrium in which all variables are constant over time. The assumption that θ = r is a standard assumption in the literature,12 which follows formally from Eq. (A.4) in the Appendix. The assumption that θ = r has important implications for the dynamics of the model. With this assumption the rate of return required by the representative household is equal to the market rate of return. As a result, the representative agent wants a level of full consumption ω(ct + st , mt ) that is constant over time. This then implies that the representative agent will want a level of total services of durable goods ct + st and of real money holdings mt that are also constant over time. This is shown formally by Eqs. (A.11), (A.13) and (A.14) in the Appendix. The result that along the optimum path ct +st is constant over time, in turn, has important implications for the dynamics of ct and st . To see this, suppose for whatever reason that the current stock of durables, say, s0 is larger than its steady state value s¯ . In that case, to converge to s¯ , s must be declining over time along the adjustment path. Hence, in order 10 Also see Fischer (1979) on the complexities involved in discussing the adjustment of the economy after a change in the rate of growth of money in Sidrauski’s (1967) model when the inflation rate is endogenous. 11 We assume that money is a completely unbacked asset, that is, the central bank is not obliged to convert money into any other asset on demand. As a result, the central bank can adjust the transfers T freely in order to control the rate of devaluation ε. 12 See, for example, Sen and Turnovsky (1989a, 1989b).
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Fig. 1. Adjustments of m, c and f when s0 < s¯ .
to maintain the services of durable goods enjoyed by the household ct + st at a constant level, it must be the case that current purchases of durable goods ct must be increasing over time along the adjustment path. Thus, in such a case, the level of c along the adjustment path will be below its steady-state level, c. ¯ Intuitively, if the current stock of durables, s0 , is larger than its steady-state value, then the representative agent must run down the stock by reducing his current purchases, ct , below their steady-state level c. ¯ With consumption so low (and savings high) along the adjustment path, the country will be running a current-account surplus (i.e., we will have f˙ > 0, and f will be increasing over time). These hypothetical paths for m, c and f are shown in Fig. 1.
3. The effects of exchange-rate policies In this section, we consider the effects of exchange-rate policies. Without nominal rigidities, a once-and-for-all devaluation has no effect on real variables. Hence, we confine the analysis to exchange-rate policies involving an increase in the rate of devaluation, ε. We assume that the economy has a fully flexible exchange-rate system. Hence, an increase in the rate of devaluation is brought about by an increase in the rate of growth of money.13 We first study the effects with time-separable preferences (i.e., with durable goods and no habits and a fixed rate of time preference). Then we review the results for the case in which consumption goods are non-durable, but there are habits. After that, we study the effects with habits and with durable consumption goods. Finally, we study the case with Uzawa preferences in which there are no habits, but in which the rate of time preference is an increasing function of full consumption ω(ct + st , mt ). 3.1. Time-separable preferences and durable goods In the Appendix we show that the combination of time-separable preferences and habits implies that an increase in the rate of devaluation has no effect on the goods market (i.e., on the steady-state levels of consumption, c, the stock of durable goods, s, or the country’s net foreign asset position, f ), but it reduces the level of real money holdings, m. This is 13 Notice m = M/(EP∗ ), where M is the nominal balances, E is the exchange rate and P∗ is the price level in terms of the foreign good. In the steady state m is constant. This means that, with P∗ fixed, if the rate of devaluation, ε, increases, then the steady-state rate of growth of nominal balances must also increase by the same amount.
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the Sidrauski monetary super-neutrality result extended to an open economy with durable consumption goods. Sidrauski showed that in a closed economy with time-separable preferences, and without durable goods, there is a dichotomy between the goods sector and the monetary sector, which implies that changes in the inflation rate have no effect on consumption. The increase in the inflation rate nevertheless reduces real money holdings, m, because it increases the cost of holding real money balances.14 In a small, open economy with time-separable preferences and durable goods, Sidrauski’s results still hold. An increase in the rate of devaluation increases the inflation rate. This does not affect the goods market, but reduces real money holdings. To see how this super-neutrality result can be broken with habits, notice that the fall in real money holdings reduces the steady-state level of full consumption, ω. If we proceed by introducing habits, then steady-state habits must fall as ε rises. This, then, implies that steady-state c and s will likely fall. As s adjusts downward, the dynamics of m, c, and f depicted in Fig. 1 interact with the dynamics of the model with habits and non-durable goods. Before we discuss this interaction, it will be instructive to discuss the dynamics of the model with habits and non-durable goods. 3.2. Habits and non-durable goods The model with habits and non-durable goods is discussed in Mansoorian (1996). There, ω(·) is a function of ct and mt . Instantaneous utility, U(·), is then a function of full consumption, ω(·), and the habitual standard of living, ht , inherited from the past. Hence, the preferences of the representative agent are given by ∞ e−θt U (ω(ct , mt ), ht ) dt, 0
where ht is a weighted sum of past levels of ω, with exponentially declining weights given to more distant values of ω. Hence, t ht = ρ e−ρt eρτ ω(cτ , mτ ) dτ, (9) −∞
from which it follows that the evolution of ht is given by h˙ t = ρ[ωt − ht ].
(10)
As mentioned in Section 1, we confine the analysis to preferences with adjacent complementarity, as it is this property which has made the habit-formation model popular in the asset-pricing literature. With adjacent complementarity, the representative agent tries 14 More precisely, Sidrauski considers the effects of an increase in the rate of growth of money, and shows that steady-state consumption and capital stock are unaffected, while steady-state real money holdings fall. Fischer shows that if the central bank controls the rate of growth of money, then there will be some off-steady-state effects as the inflation rate will have to continuously adjust to clear the money market. Notice first, that the steady-state inflation rate is equal to the steady-state rate of growth of money in Sidrauski’s model. Second, if the central bank controls the inflation rate instead of the rate of growth of money, then there will be no off-steady-state effects such as those discussed by Fischer.
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Fig. 2. Adjustments of m, c and f with habits and non-durables.
to maintain his habitual standards of living in response to a shock with relatively small adjustments of current ω. An increase in the rate of depreciation of the domestic currency leads to a fall in the steady-state habitual standards of living, as long as the nominal interest rate is positive. The reason is that with money-in-utility, instantaneous utility is maximized when it is satiated in real money holdings. If initial nominal interest rates are positive, then the equilibrium is away from this optimum. Hence, if ε increases further, then instantaneous utility falls. With adjacent complementarity, the representative agent tries to maintain the relatively high habitual standards of living he has inherited from the past. Hence, although the immediate effect of the increase in the depreciation rate is to reduce real money holdings, they stay above their new steady-state level. Immediately after the increase in the depreciation rate consumption increases, savings fall, and the country runs a current-account deficit. Along the adjustment path to the new steady state, mt and ct decline and the country continues to run a current-account deficit. The adjustment paths of mt , ct , and ft with habits and non-durable goods are shown in Fig. 2. 3.3. Habits and durable goods Hence, habits capture the fact that an increase in the rate of devaluation reduces welfare. The steady-state level of habits falls with an increase in the rate of devaluation. This, then, breaks the super-neutrality result discussed above. The results for the two cases discussed above allow us to explain the results for the model with habits and durable goods intuitively. Preferences are now given by ∞ e−θt U (ω(ct + st , mt ), ht ) dt, 0
where st is the stock of durable goods inherited from the past, and is given by Eq. (2), while the stock of habits ht , is a weighted sum of past levels of ω(ct + st , mt ): t ht = ρ e−ρt eρτ ω(cτ + sτ , mτ ) dτ. (11) −∞
The evolution of ht is, thus, given by h˙ t = ρ[ωt − ht ].
(12)
As explained above, an increase in ε increases the cost of holding money and thus leads to a fall in welfare. Hence, the steady-state habitual standards of living fall. Real money
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holdings fall. The steady-state levels of consumption expenditures and durables are likely to fall as well.15 It therefore follows from Eq. (8), that the steady-state stock of net foreign assets falls. As discussed above, both the habits and durability effects tend to lead to a fall in m when ε rises. Without habits, m falls to its new steady-state level instantly, and it stays there forever. With habits, m also falls immediately after the increase in ε. However, now it stays above its new long-run level, as the representative agent tries to maintain the habitual standards he has inherited from the past. After this, m falls further along the adjustment path until it reaches its new steady-state level. It is, thus, clear that with the simultaneous presence of habits and durable goods, adjustment of m will be qualitatively the same as in the model with habits and non-durables. However, the habits and durability effects will have competing influences on the behavior of c and f. To reduce the steady-state stock of durables, the representative agent reduces his consumption expenditures below their steady-state level, which tends to bring about a current-account surplus. On the other hand, as discussed above, the representative agent wants to maintain the relatively high habitual standards he has inherited from the past. This habit-persistence effect tends to increase consumption, which tends to bring about a current-account surplus. Backus, Kehoe, and Kydland (1994) document evidence from various countries suggesting that the trade balance is negatively correlated with current and future movements in the terms of trade, but positively correlated with past movements. In a recent paper, Mansoorian (1998) uses the habits-and-durability model without money to show that Backus et al.’s findings indicate that habit effects should be dominant in the short run and durability effects in the long run. In the present model, an increase in ε leads to an immediate increase in consumption expenditures and a current-account deficit. However, when the durability effects become dominant, consumption starts to fall and the current account starts to improve. This is the J-curve phenomenon in a monetary setting.16,17 The adjustments of c and f for this case are shown in Fig. 3. On the other hand, Heaton’s empirical findings on the behavior of asset prices suggest that the durability effects will be dominant in the short run; but eventually the habit effects will dominate. If this is the case, then immediately after the increase in ε, there will be a fall in consumption expenditures and a current-account surplus. However, when the habit effects 15 The effect on the steady state levels of c and s is ambiguous for the following reason. When ε increases, the agent substitutes c¯ + s¯ for m ¯ (substitution effect). On the other hand, when habits fall, steady-state ω also falls (Eq. (11)). This tends to reduce both c¯ + s¯ and m ¯ (habit effects). Hence, there are two competing influences on c + s and its steady state level cannot be signed. Nevertheless, for the rest of the paper we assume that s¯ falls when ε increases. The reasons are explained in footnote 16. 16 Notice, following from footnote 15, if s¯ increases with ε, we do not have the non-monotonic adjustment of c or f, as both the habits and durability effects have similar influences on adjustment. However, the empirical results reported by Backus et al. suggest that adjustment of the current account is non-monotonic. 17 This is an alternative to the textbook explanation of the J-curve (see, e.g., Krugman & Obstfeld, 2000, pp. 466–468), which relies on institutional rigidities that fix the volumes of exports and imports in the short run. The textbook treatment also assumes that prices are fixed in the short run, and the real depreciation is originally brought about by a nominal devaluation of the domestic currency. Unlike the textbook treatment, the explanation for the J-curve presented in this paper is embedded in an intertemporal utility-maximizing framework.
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Fig. 3. Adjustments of c and f with habits dominant in the short run and durability in the long run.
become dominant, consumption expenditures will start to fall and the current account will start to improve. The adjustments of c and f for this case are shown in Fig. 4. 3.4. Uzawa preferences Finally, consider the model with Uzawa preferences, in which there are no habits, but where the rate of time preference is an increasing function of instantaneous utility t ∞ U (ct , mt ) exp − θv dv , where θv = θ(U (cv , mv )). (13) 0
0
Obstfeld (1981a) considers the effects of exchange-rate policies with Uzawa preferences when consumption goods are non-durable. As the small, open economy takes the world real interest rate as given, the equality of the rate of time preference and the world interest rate dictates the level of instantaneous utility which must be maintained in the steady state. An increase in the rate of devaluation of the domestic currency would, therefore, by increasing the cost of holding real balances, require substitution of consumption for real money holdings along the same indifference curve in the steady state. Hence, in order to be able to finance his higher steady-state consumption, the representative agent accumulates bonds along the adjustment path. Therefore, on impact, the increase in the rate of devaluation leads to a huge reduction in consumption and a current-account surplus. By the homotheticity
Fig. 4. Adjustments of c and f with durability dominant in the short run and habits in the long run.
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of preferences, it follows that on impact real money holdings also fall by a huge amount. Along the adjustment path, both consumption and real money holdings rise over time. With consumer durables, these results change substantially. With durables, ct should be replaced with ct + st in Eq. (13). As money does not exhibit durability, its dynamics are as in Obstfeld (1981a). On the other hand, there are two competing effects on the dynamics of consumption and the current account. First, there is the effect without durable goods discussed above, which from now on we refer to as the “time-preference” effect. Second, there is the durability effect. As may be clear from new Eq. (3), the steady-state stock of durables has to increase. As we have seen already, this tends to increase consumption expenditures above their steady-state level. Hence, again, the dynamics of the current account are likely to be non-monotonic. If the durability effects are dominant in the short run and the time preference effects in the long run, then immediately after the increase in the rate of devaluation there will be an increase in consumption expenditures and a current-account deficit. But when the time preference effects start to dominate, then consumption expenditures start to fall and the current account turns into a surplus. This, again is the J-curve phenomenon in a monetary setting. That we should have such dynamics with Uzawa preferences and durable goods also follows from our previous analysis. As shown in Mansoorian (1996), the dynamics of the model with habits and adjacent substitutability of preferences are qualitatively similar to the dynamics of the model with Uzawa preferences.18 Hence, the dynamics of the model with Uzawa preferences and durable goods should be qualitatively the same as the habit-persistence model with durable goods, when habits are combined with adjacent substitutability of preferences, and when the steady-state stock of durables increases with the rate of devaluation. Hence, although Uzawa preferences cause the steady-state levels of c and f to rise, adjustment of c and f when durability effects are dominant in the long run are as in Fig. 3, and adjustments when durability effects are dominant in the short run are as in Fig. 4.
4. Conclusion This paper studies the effects of exchange-rate policies in the presence of durable goods. The setting is that of a small, open economy in which instantaneous utility depends on real money holdings. Three popular specifications of intertemporal utility are considered: time-separable preferences, the habit-persistence model, and Uzawa preferences. It is shown that with time-separable preferences, changes in the devaluation rate have no effect on the stock of durable goods, on consumption expenditures, and on the country’s net foreign assets. This is the Sidrauski super-neutrality result extended to an open economy with durable goods. Although money is super-neutral with time-separable preferences, an increase in the devaluation rate reduces real money holdings. With money-in-utility, this reduces welfare. Hence, extending the model to incorporate habits breaks super-neutrality. With habits, the 18
See footnote 3 for an explanation of the difference between adjacent complementarity and adjacent substitutability.
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fall in welfare resulting from the fall in real money holdings is likely to lead to a fall in the steady-state stock of durable goods. The habit effect and the durability effect have competing influences on the adjustment of consumption expenditures and the country’s net foreign asset holdings. These variables are likely to adjust non-monotonically along the path to the new steady state. Real money holdings, however, adjust monotonically, because they do not exhibit durability. With Uzawa preferences, the model does not have habits, but the rate of time preference is an increasing function of instantaneous utility. With such preferences, an increase in the rate of devaluation requires substitution of consumption for real money holdings in the steady state. The presence of durable goods would, in addition, require an increase in the steady-state stock of durables. The time-preference and the durability effects tend to have competing influences on consumption expenditures and the current account, making adjustment of these variables non-monotonic. On the other hand, as real balances do not exhibit durability, their adjustment is monotonic, and depends only on the time-preference effect.
Acknowledgements The authors would like to thank the editor of the journal and two anonymous referees for valuable comments and suggestions on an earlier draft. Financial support from the Social Sciences and Humanities Research Council of Canada (grants 410-97-1212 and 410-01-1018), and the University Research Board of the American University of Beirut are also gratefully acknowledged.
Appendix In this appendix, we derive formally the perfect-foresight path, and the effects of exchangerate policies for the case with durable goods and time-separable preferences. For this purpose, we first solve the representative agent’s problem, and then consider the country’s external adjustment along the equilibrium path, in light of the assumptions we have made about central bank policies. The problem of the representative agent is then to maximize lifetime utility (1) subject to constraints (3)–(6). The Hamiltonian for this problem is H = ω(ct + st , mt ) + φt [ct − δst ] + µt [rat + y + Tt − ct − (r + ε)mt ], where φ is the shadow price of durable goods, while µ is the shadow price of assets. The optimality conditions for this problem are Hc ≡ ω1 + φ − µ = 0,
(A.1)
Hm ≡ ω2 − µ(r + ε) = 0,
(A.2)
˙ −Hs + θ φ = −ω1 + φδ + θ φ = φ,
(A.3)
˙ −Ha + θ µ ≡ −rµ + θ µ = µ,
(A.4)
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and the transversality conditions lim e−θt φt st = 0,
(A.5)
lim e−θt µt at = 0,
(A.6)
t→∞
and t→∞
Clearly, for (A.4) to converge, we will need r = θ,
(A.7)
which means that along the adjustment path, µ is always at its steady-state level µ. ¯ In order to derive the optimum path of the economy, first linearize (A.1) and (A.2) around the steady state, using the fact that with r = θ along the adjustment path dµ = 0, to obtain ω22 ¯ (ct − c) ¯ = −(st − s¯ ) − (φt − φ) (A.8) ω11 ω22 − ω12 ω21 and (mt − m) ¯ =
ω21 ¯ (φt − φ), ω11 ω22 − ω12 ω21
(A.9)
where over-bars denote steady-state values. Next, linearize (3) and (A.3) around the steady state, and use (A.7)–(A.9) in order to obtain s˙t st − s¯ a11 a12 = , (A.10) 0 a22 φt − φ¯ φ˙ t where a11 = −(1 + δ), a12 = −[ω22 /(ω11 ω22 − ω12 ω21 )], and a22 = 1 + r + δ. The system of differential equations in (A.10) has one positive and one negative eigenvalue. Hence, as s is a state variable, while φ is a jump variable, the differential equation system (A.10) exhibits saddlepath stability. The negative eigenvalue of the system is −(1 + δ); and the stable path is given by the following equations: st − s¯ = (s0 − s¯ ) e−(1+δ)t ,
(A.11)
φt − φ¯ = 0.
(A.12)
Using (A.11) and (A.12) in (A.8) and (A.9) we obtain the adjustments of c and m along the optimum path ct − c¯ = −(st − s¯ ) = −(s0 − s¯ ) e−(1+δ)
(A.13)
mt − m ¯ = 0.
(A.14)
and Next, consider the dynamics of the country’s net foreign assets f (≡ b + R): f˙ = rf + y − c.
(A.15)
Linearizing this expression around the steady state, using (A.13), we obtain f˙t = r(ft − f¯) + [s0 − s¯ ] e−(1+δ)t .
(A.16)
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The solution to (A.16) is ft − f¯ = −
1 1 −(1+δ)t ¯ + (f0 − f ) + (s0 − s¯ ) e (s0 − s¯ ) ert . 1+δ+r 1+δ+r (A.17)
Clearly, for (A.17) to converge the coefficient on ert must be 0. Hence, (f0 − f¯) +
1 (s0 − s¯ ) = 0 1+δ+r
(A.18)
is an important condition which must hold for saddlepath stability of the model. This condition determines, for the initial values of the predetermined variables f0 and s0 , how the steady-state values of these variables must be related for saddlepath stability. Next we show that with time-separable preferences, and with habits, an increase in the rate of devaluation will have no effect on the goods market (i.e., on the steady-state levels of consumption, c, the stock of durable goods, s, or the country’s net foreign asset position, f), but it will reduce the level of real money holdings, m. For this purpose, consider first the steady state of the economy described above. This is characterized by (A.1), (A.2), and by Eqs. (3), (A.3) and (A.15) with s˙ = φ˙ = f˙ = 0. There are five equations in six unknowns: ¯ and µ. c, ¯ m, ¯ s¯ , f¯, φ, ¯ The sixth equation is (A.18). These equations are all independent of the level of the exchange rate, E. Hence, as in Obstfeld, a once-and-for-all devaluation does not have any effect on this economy, because all prices are fully flexible. Next, note (3), (A.15) and (A.18) implicitly determine the steady-state levels of s, f and c. As these equations are independent of ε, it follows that changes in the rate of devaluation have no effect on c, ¯ s¯ or f¯. This is the Sidrauski monetary super-neutrality extended to an open-economy setting with durable consumption goods. It arises, essentially, because preferences are time-separable. To see how this super-neutrality result could be broken with habits, notice that by differentiating (A.1)–(A.3) at the steady state, using the fact that dc¯ = d¯s = df¯ = 0, we obtain dm −µ/(r + ε) = < 0. dε [1 + 1/(δ + r)](ω12 − ω22 )/(r + ε) Hence, steady state falls as ε rises, because steady state c and s are unaffected while m falls.
References Backus, D., & Driffill, J. (1985). Inflation and reputation. American Economic Review, 75(3), 530–538. Backus, D., Gregory, A. W., & Telmer, C. I. (1993). Accounting for forward rates in markets for foreign currency. Journal of Finance, 48, 1887–1908. Backus, D., Kehoe, P., & Kydland, F. (1994). Dynamics of the trade balance and the terms of trade: The J-curve? American Economic Review, 84, 84–103. Calvo, G. A. (1981). Devaluation: Levels versus rates. Journal of International Economics, 11, 165–172. Calvo, G. A. (1987). Balance of payment crisis in a cash-in-advance economy. Journal of Money Credit and Banking, 19, 19–32.
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Constantinides, G. M. (1990). Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy, 98, 519–543. Devereux, M., & Shi, S. (1991). Capital accumulation and the current account in a two country model. Journal of International Economics, 30, 1–25. Djajic, S. (1982). Balance of payments dynamics and exchange rate management. Journal of International Money and Finance, 1, 179–192. Ferson, W. E., & Constantinides, G. M. (1991). Habit formation and durability in aggregate consumption: Empirical tests. Journal of Financial Economics, 29, 199–240. Fischer, S. (1979). Capital accumulation on the transition path in a monetary optimizing model. Econometrica, 47, 1433–1439. Heaton, J. (1995). An empirical investigation of asset pricing with temporally dependent preference specifications. Econometrica, 63, 681–717. Kydland, F., & Prescott, E. (1977). Rules rather than discretion: The inconsistency of optimal plans. Journal of Political Economy, 85, 473–491. Krugman, P., & Obstfeld, M. (2000). International economics: Theory and policy (5th ed.). New York: Harper Collins Publishers. Mansoorian, A. (1996). On the macroeconomic policy implications of habit persistence. Journal of Money Credit and Banking, 28, 119–129. Mansoorian, A. (1998). Habits and durability in consumption, and the dynamics of the current account. Journal of International Economics, 44, 69–82. Mehra, R., & Prescott, E. (1985, March). The equity premium: A puzzle. Journal of Monetary Economics, 15, 145–161. Mendoza, E. (1991). Real business cycles in a small open economy. American Economic Review, 81, 797–818. Mishkin, F. S. (2000). Inflation targeting in emerging-market countries. American Economic Review Papers and Proceedings, 90, 105–109. Obstfeld, M. (1981a). Capital mobility and devaluation in an optimizing model with rational expectations. American Economic Review Papers and Proceedings, 71(2), 217–221. Obstfeld, M. (1981b). Macroeconomic policy, exchange rate dynamics, and optimal asset accumulation. Journal of Political Economy, 89(6), 1142–1161. Obstfeld, M., & Rogoff, K. (1995). Exchange rate dynamics redux. Journal of Political Economy, 103, 624–660. Ryder, H. E., & Heal, G. M. (1973). Optimal growth with intertemporally dependent preferences. Review of Economic Studies, 40, 1–31. Sen, P., & Turnovsky, S. J. (1989a). Deterioration of the terms of trade and capital accumulation: A re-examination of the Laursen–Metzler effect. Journal of International Economics, 26, 227–250. Sen, P., & Turnovsky, S. J. (1989b). Tariffs, capital accumulation and the current account in a small open economy. International Economic Review, 30, 811–831. Sidrauski, M. (1967). Rational choice and patterns of growth in a monetary economy. American Economic Review Papers and Proceedings, 30, 534–544. Stockman, A. C. (1981). Anticipated inflation and the capital stock in a cash-in-advance economy. Journal of Monetary Economics, 8, 387–393. Uzawa, H. (1968). Time preference, the consumption function, and optimal asset holdings. In J. N. Wolfe (Ed.), Value, capital and growth: Papers in honour of Sir John Hicks. Chicago, IL: Aldine Publishing Company. Walsh, C. (1995). Optimal contracts for central bankers. American Economic Review, 85(1), 150–167.
Durable goods, habits, time preference, and exchange rates Arman Mansoorian a , Simon Neaime b,∗ a
b
Department of Economics, York University, Toronto, Canada Department of Economics, Institute of Financial Economics, American University of Beirut, 850 Third Avenue, 18th Floor, New York, NY 10022-6297, USA
Received 27 June 2001; received in revised form 24 July 2002; accepted 23 October 2002
Abstract We work out the role of durable goods regarding the effects of exchange-rate policies in the money-in-utility model. If for whatever reason the stock of durables is above its steady-state level, then along the adjustment path consumption expenditures would be rising and the current account would be in surplus, but real money holdings would be constant. Without habits, money is super-neutral. Habits break super-neutrality. With habits and durable goods, the adjustment of consumption expenditures and the current account after a change in the rate of depreciation will likely be non-monotonic, while the adjustment of real balances will be monotonic. With Uzawa preferences and durable goods, the adjustment of real money balances will also be monotonic, while the adjustment of consumption and the current account will be non-monotonic. © 2002 Elsevier Science Inc. All rights reserved. JEL classification: F31; F32; F41 Keywords: Durable goods; Money; Habits; Exchange-rate policies; Time preference
1. Introduction The literature concerned with the effects of exchange-rate policies has paid little attention to the important role of durable goods. This is despite recent findings in the asset-pricing literature (e.g., Ferson & Constantinides, 1991; Heaton, 1995), suggesting that durable goods play an important role in improving the empirical performance of consumption-based asset-pricing models. The purpose of the present paper is to work out some implications of ∗
Corresponding author. Tel.: +1-9613-829944; fax: +1-9611-744484. E-mail address: [email protected] (S. Neaime).
1062-9408/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved. PII: S 1 0 6 2 - 9 4 0 8 ( 0 2 ) 0 0 1 1 5 - 8
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durable goods for exchange-rate policies under three popular specifications of intertemporal utility: time-separable preferences, the habit-persistence model, and Uzawa preferences. Obstfeld (1981a, 1981b) was the first to discuss some policy issues for a small, open economy in an optimizing framework under the assumption that instantaneous utility is a function of consumption and real money holdings, as in Sidrauski (1967), and that the rate of time preference is an increasing function of instantaneous utility (as in Uzawa, 1968). In his (1981b) paper, he considers the effects for such an economy of a central bank target rate of money growth. The results in Obstfeld (1981a) reveal that the analysis is considerably simplified if it is assumed that the central bank targets the rate of devaluation of the domestic currency.1 This assumption is also made by Calvo (1981) and Djajic (1982). The analysis in the present paper will be confined to this case. Obstfeld (1981b) shows that in that setting, with perfectly flexible prices, a once-and-for-all devaluation will not have any effect. On the other hand, an increase in the rate of devaluation will lead to a sharp fall in consumption and real money holdings in the short run, and hence a current-account surplus. Over time, both consumption and real money holdings increase along the adjustment path to the new steady-state equilibrium. The reason is as follows. The equality of the rate of time preference and the world interest rate dictates the level of instantaneous utility which must be maintained in the steady state. The increase in the rate of devaluation of the domestic currency, therefore, requires substitution of consumption for real money holdings along a given steady-state indifference curve. Hence, in order to be able to finance his higher steady-state consumption, the representative agent must accumulate bonds along the adjustment path. Therefore, on impact, the increase in the rate of devaluation leads to a reduction in consumption and a current-account surplus. By the homotheticity of preferences, it follows that on impact real money holdings also fall. Along the adjustment path, both consumption and real money holdings rise over time. Ryder and Heal (1973) provide an alternative formulation of intertemporally dependent preferences to the Uzawa function. In their formulation, the rate if time preference is fixed, but instantaneous utility depends on both current full consumption and the habitual standard of living, represented by a weighted sum of past levels of full consumption.2 In a recent paper, Mansoorian (1996) uses this habit-persistence model to show that the effects of an increase in the rate of devaluation are very similar to Obstfeld’s (1981a), if preferences exhibit adjacent substitutability, but differ sharply if preferences exhibit adjacent complementarity.3 In the present paper, we discuss the importance of durable goods for exchange-rate policies. We start with a model with time-separable preferences (i.e., a model with no habits and a fixed rate of time preference) in which consumption goods exhibit durability. We 1 With fully flexible prices, this is equivalent to the central bank targeting the inflation rate, which according to Mishkin (2000) is the policy followed by many developed and emerging economies. The assumption that the central bank targets the inflation rate is also consistent with the assumptions made in the literature on the time consistency of monetary policies (e.g., Backus & Driffill, 1985; Kydland & Prescott, 1977; Walsh, 1995). 2 Full consumption is the utility derived from current consumption and current real money holdings. 3 With adjacent substitutability, an increase in current consumption will decrease the marginal utility of consumption in the near future relative to the marginal utility in the distant future. Similarly, with adjacent complementarity, an increase in current consumption will increase the marginal utility of consumption in the near future relative to the marginal utility in the distant future.
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show that an increase in the rate of devaluation of the domestic currency is super-neutral. It has no effect on consumption expenditures, the stock of durable goods, or the country’s net foreign asset position. Nevertheless, the holdings of real balances fall. These are the Sidrauski super-neutrality results extended to an open-economy setting with durable goods. They arise because preferences are time-separable. The focus of this paper is not only the issue of super-neutrality, as in the literature following Sidrauski, but also the effects of consumer durables on the dynamics of open-economy models. If the stock of durable goods is above its long-run level, the representative agent adjusts the stock by substantially reducing current purchases. Hence, along the adjustment path consumption expenditures are below their long-run level, and there is a current-account surplus. Nevertheless, because preferences are time-separable, and the rate of time preference is equal to the interest rate, real money holdings are constant over time. If one relaxes the time-separable preference structure by incorporating habits, then the results change fundamentally. Since the fall in real money holdings reduces welfare, the presence of habits requires a corresponding fall in the steady-state habitual standard of living. This requires a fall in the steady-state stock of durable goods, a corresponding fall in consumption expenditures, and a fall in net foreign asset holdings, as well as a fall in real money holdings. Thus, habits break super-neutrality even though there is complete absence of nominal rigidities. We follow the empirical literature and assume that with habits, preferences exhibit adjacent complementarity (as opposed to adjacent substitutability).4 As shown in Mansoorian (1996), with habits but non-durable goods and adjacent complementarity of preferences, the increase in the rate of devaluation causes the representative agent to maintain the habitual standard of living by reducing savings. Hence, after the increase in the rate of devaluation there is an increase in consumption and a current-account deficit, while real money holdings fall. Along the adjustment path, consumption and real money holdings are declining, and the country runs a current-account deficit. Durability in consumption goods changes the dynamics of the model fundamentally. In that case, it is not only the steady-state stock of habits, but the stock of durable goods which would likely have to adjust downwards in response to an increase in the depreciation rate. To reduce the stock of durables, the representative agent reduces his current purchases, which tends to lead to a current-account surplus. Hence, the habits and durability effects have competing influences on the adjustment of consumption and the current account. Adjustment of these variables will, therefore, be non-monotonic. On the other hand, in the case of durable goods and no habits, real money holdings do not change over time when the stock of durables is adjusted downwards. It follows, that the simultaneous presence of habits and durability makes adjustment of real money holdings along the adjustment path monotonic, and qualitatively the same as in the model with habits and non-durability. Another means by which one can break super-neutrality in the absence of nominal rigidities is through Uzawa preferences, as shown by Obstfeld’s contributions discussed 4 Constantinides (1990) requires a high degree of adjacent complementarity in order to solve the Mehra and Prescott (1985) equity-premium puzzle with the habit-formation model. Backus, Gregory, and Telmer (1993) require a high degree of adjacent complementarity in order to account for excessively high variation in the expected return on the forward relative to spot exchange rates.
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above.5 Uzawa preferences are another popular specification of preferences for open economies.6 The presence of durable goods again fundamentally changes the dynamics under Uzawa preferences, where an increase in the rate of devaluation requires substitution of consumption for real money holdings along the given indifference curve in the steady state. This implies an increase in the steady-state stock of durable goods. To increase this stock, the representative agent increases consumption expenditures above their steady-state levels. This adjustment competes with the dynamics of the Obstfeld model. Hence, the response of consumption and the current account is again non-monotonic. Since money balances do not exhibit durability, their adjustment is monotonic. The paper is organized as follows. The model with durable consumption goods no habits and a fixed rate of time preference is presented in Section 2. The effects of exchange-rate policies are discussed in Section 3. Some concluding remarks are made in Section 4.
2. The model with durable goods and time-separable preferences 2.1. The model The model corresponds most closely to Obstfeld’s (1981a). The foreign-currency price of the single good in the model is fixed at P∗ , which the small economy takes as given. The domestic-currency price of this good is P = EP∗ , where E is the exchange rate (the price of foreign currency in terms of domestic currency). With flexible goods prices, the inflation ˙ rate is equal to the rate of depreciation of the domestic currency (E/E), which is denoted by ε. The objective function of the representative agent is ∞ e−θt ω(ct + st , mt ) dt, (1) 0
where θ is the rate of time preference, ω(·) is a homothetic utility function, which depends on the current consumption of the services of consumer durables (ct +st ) and on current money holdings (mt ).7 The instantaneous utility derived by the representative agent at time t from consumption of the services of durable goods and from real money holdings is measured by ω(ct + st , mt ). We refer to it as full consumption. 5 Uzawa showed that Sidrauski’s super-neutrality result depended on the assumption of a fixed rate of time preference. In a closed economy in the steady state, the rate of time preference should be equal to the marginal productivity of capital. This dictates a unique level of capital (and, hence, of consumption) for the steady state. This result will not hold if the rate of time preference is a function of instantaneous utility. 6 In addition to Obstfeld’s classic contributions, see, for example, Devereux and Shi (1991) and Mendoza (1991). 7 The money-in-utility model has had a long history in monetary economics, starting from Sidrauski. One recent prominent paper that employs such a specification is by Obstfeld and Rogoff (1995). An alternative specification would be the cash-in-advance model, which is used by, for example, Calvo (1987). In that setting mt = ct ; and it can be shown that, as in Stockman’s (1981) original work, changes in the rate of growth of money will not have any effect on the steady-state equilibrium.
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In Eq. (1), ct is the amount of consumer durables purchased at time t, while st is the stock of durables inherited from the past. These durable goods depreciate at the rate of δ. Thus, t eδ(τ −t) cτ dτ, (2) st = −∞
and the evolution of st is given by s˙t = ct − δst .
(3)
Output is fixed at y units of the good at any time t. The representative agent also receives monetary transfers with real values of Tt from the government. There are two kinds of assets in the model, money balances and internationally traded bonds, whose foreign currency price is fixed and whose rate of return is fixed at r. The real assets of the representative agent are at = bt + mt ,
(4)
where bt is his bond holdings. His flow budget constraint is8 a˙ t = y + rat + Tt − ct − (r + εt )mt .
(5)
Finally, the intertemporal solvency condition, lim e−rt at ≥ 0,
t→∞
(6)
prevents the agent from borrowing without bound. The representative agent’s problem is to maximize (1), subject to (3), (4)–(6), and the initial conditions (i.e., s0 and a0 ), taking the paths of {Tt }, and {ε t } as given. Along a perfect foresight path, the agent’s expectations about {Tt } and {εt } coincide with the actual paths of these variables. In this paper we abstract completely from government expenditures on goods and services, and concentrate exclusively on exchange-rate policies.9 The government chooses the real monetary transfers, Tt , in order to satisfy its flow constraint, which says that it finances its transfers (Tt ) from the interest on its international bond holdings and from seigniorage ˙ t, Tt = rRt + εt mt + m
(7)
where Rt is the central bank’s foreign reserves at time t, held in the form of internationally traded bonds. As stated in Section 1, we assume that the central bank is targeting the rate of depreciation of domestic currency (i.e., the inflation rate) and not the rate of growth of money per se. This is the assumption made by Obstfeld (1981b), and is consistent with the assumptions 8 According to this constraint the amount of assets the representative agent accumulates should be equal to his net income (rat + y + Tt ) minus his total expenditures (ct + (r + εt )mt ). 9 Obstfeld (1981a) allows for government purchases of goods and services. He, however, assumes that such government expenditures are unproductive, or, alternatively, that the utility of the representative agent is strongly separable between private consumption and public goods. We could make this assumption without changing any of our results.
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made in the literature on the time consistency of monetary policies.10 As may be clear from Eq. (7), the central bank can target the inflation rate either by intervening on the foreign exchange market (by adjusting R, through purchases and sales of bonds), or by adjusting the transfers T.11 Clearly, if the bank targets {ε t } by adjusting T, then we would have a flexible exchange rate system (with R fixed). Finally, we integrate the budget constraints of the representative agent (5) and the central bank (7) in order to obtain the dynamics of the country’s net foreign assets f (≡ b + R): f˙ = rf + y − c,
(8)
according to which the country’s current account f˙ is equal to total income rf + y less absorption c. 2.2. Discussion The optimization problem of the representative agent and all optimality conditions are derived in the Appendix. Here we highlight some of the important features of the optimality conditions. First, note that for a steady state to exist in a small open economy with a fixed rate of time preference and access to unlimited borrowing or lending at a fixed rate of interest, r, we must assume that the rate of time preference is equal to the rate of interest, θ = r. If θ > r, the rate of return required by the representative agent (i.e., θ ) is larger than the market rate of return (i.e., r). In that case, the representative agent will want a consumption profile that will be increasing over time indefinitely. On the other hand, if θ < r, then the rate of return required by the representative agent is smaller than the market rate of interest. In that case, the representative agent will want a consumption profile that will be declining over time indefinitely. None of these is consistent with the existence of a steady state, that is, a long run equilibrium in which all variables are constant over time. The assumption that θ = r is a standard assumption in the literature,12 which follows formally from Eq. (A.4) in the Appendix. The assumption that θ = r has important implications for the dynamics of the model. With this assumption the rate of return required by the representative household is equal to the market rate of return. As a result, the representative agent wants a level of full consumption ω(ct + st , mt ) that is constant over time. This then implies that the representative agent will want a level of total services of durable goods ct + st and of real money holdings mt that are also constant over time. This is shown formally by Eqs. (A.11), (A.13) and (A.14) in the Appendix. The result that along the optimum path ct +st is constant over time, in turn, has important implications for the dynamics of ct and st . To see this, suppose for whatever reason that the current stock of durables, say, s0 is larger than its steady state value s¯ . In that case, to converge to s¯ , s must be declining over time along the adjustment path. Hence, in order 10 Also see Fischer (1979) on the complexities involved in discussing the adjustment of the economy after a change in the rate of growth of money in Sidrauski’s (1967) model when the inflation rate is endogenous. 11 We assume that money is a completely unbacked asset, that is, the central bank is not obliged to convert money into any other asset on demand. As a result, the central bank can adjust the transfers T freely in order to control the rate of devaluation ε. 12 See, for example, Sen and Turnovsky (1989a, 1989b).
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Fig. 1. Adjustments of m, c and f when s0 < s¯ .
to maintain the services of durable goods enjoyed by the household ct + st at a constant level, it must be the case that current purchases of durable goods ct must be increasing over time along the adjustment path. Thus, in such a case, the level of c along the adjustment path will be below its steady-state level, c. ¯ Intuitively, if the current stock of durables, s0 , is larger than its steady-state value, then the representative agent must run down the stock by reducing his current purchases, ct , below their steady-state level c. ¯ With consumption so low (and savings high) along the adjustment path, the country will be running a current-account surplus (i.e., we will have f˙ > 0, and f will be increasing over time). These hypothetical paths for m, c and f are shown in Fig. 1.
3. The effects of exchange-rate policies In this section, we consider the effects of exchange-rate policies. Without nominal rigidities, a once-and-for-all devaluation has no effect on real variables. Hence, we confine the analysis to exchange-rate policies involving an increase in the rate of devaluation, ε. We assume that the economy has a fully flexible exchange-rate system. Hence, an increase in the rate of devaluation is brought about by an increase in the rate of growth of money.13 We first study the effects with time-separable preferences (i.e., with durable goods and no habits and a fixed rate of time preference). Then we review the results for the case in which consumption goods are non-durable, but there are habits. After that, we study the effects with habits and with durable consumption goods. Finally, we study the case with Uzawa preferences in which there are no habits, but in which the rate of time preference is an increasing function of full consumption ω(ct + st , mt ). 3.1. Time-separable preferences and durable goods In the Appendix we show that the combination of time-separable preferences and habits implies that an increase in the rate of devaluation has no effect on the goods market (i.e., on the steady-state levels of consumption, c, the stock of durable goods, s, or the country’s net foreign asset position, f ), but it reduces the level of real money holdings, m. This is 13 Notice m = M/(EP∗ ), where M is the nominal balances, E is the exchange rate and P∗ is the price level in terms of the foreign good. In the steady state m is constant. This means that, with P∗ fixed, if the rate of devaluation, ε, increases, then the steady-state rate of growth of nominal balances must also increase by the same amount.
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the Sidrauski monetary super-neutrality result extended to an open economy with durable consumption goods. Sidrauski showed that in a closed economy with time-separable preferences, and without durable goods, there is a dichotomy between the goods sector and the monetary sector, which implies that changes in the inflation rate have no effect on consumption. The increase in the inflation rate nevertheless reduces real money holdings, m, because it increases the cost of holding real money balances.14 In a small, open economy with time-separable preferences and durable goods, Sidrauski’s results still hold. An increase in the rate of devaluation increases the inflation rate. This does not affect the goods market, but reduces real money holdings. To see how this super-neutrality result can be broken with habits, notice that the fall in real money holdings reduces the steady-state level of full consumption, ω. If we proceed by introducing habits, then steady-state habits must fall as ε rises. This, then, implies that steady-state c and s will likely fall. As s adjusts downward, the dynamics of m, c, and f depicted in Fig. 1 interact with the dynamics of the model with habits and non-durable goods. Before we discuss this interaction, it will be instructive to discuss the dynamics of the model with habits and non-durable goods. 3.2. Habits and non-durable goods The model with habits and non-durable goods is discussed in Mansoorian (1996). There, ω(·) is a function of ct and mt . Instantaneous utility, U(·), is then a function of full consumption, ω(·), and the habitual standard of living, ht , inherited from the past. Hence, the preferences of the representative agent are given by ∞ e−θt U (ω(ct , mt ), ht ) dt, 0
where ht is a weighted sum of past levels of ω, with exponentially declining weights given to more distant values of ω. Hence, t ht = ρ e−ρt eρτ ω(cτ , mτ ) dτ, (9) −∞
from which it follows that the evolution of ht is given by h˙ t = ρ[ωt − ht ].
(10)
As mentioned in Section 1, we confine the analysis to preferences with adjacent complementarity, as it is this property which has made the habit-formation model popular in the asset-pricing literature. With adjacent complementarity, the representative agent tries 14 More precisely, Sidrauski considers the effects of an increase in the rate of growth of money, and shows that steady-state consumption and capital stock are unaffected, while steady-state real money holdings fall. Fischer shows that if the central bank controls the rate of growth of money, then there will be some off-steady-state effects as the inflation rate will have to continuously adjust to clear the money market. Notice first, that the steady-state inflation rate is equal to the steady-state rate of growth of money in Sidrauski’s model. Second, if the central bank controls the inflation rate instead of the rate of growth of money, then there will be no off-steady-state effects such as those discussed by Fischer.
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Fig. 2. Adjustments of m, c and f with habits and non-durables.
to maintain his habitual standards of living in response to a shock with relatively small adjustments of current ω. An increase in the rate of depreciation of the domestic currency leads to a fall in the steady-state habitual standards of living, as long as the nominal interest rate is positive. The reason is that with money-in-utility, instantaneous utility is maximized when it is satiated in real money holdings. If initial nominal interest rates are positive, then the equilibrium is away from this optimum. Hence, if ε increases further, then instantaneous utility falls. With adjacent complementarity, the representative agent tries to maintain the relatively high habitual standards of living he has inherited from the past. Hence, although the immediate effect of the increase in the depreciation rate is to reduce real money holdings, they stay above their new steady-state level. Immediately after the increase in the depreciation rate consumption increases, savings fall, and the country runs a current-account deficit. Along the adjustment path to the new steady state, mt and ct decline and the country continues to run a current-account deficit. The adjustment paths of mt , ct , and ft with habits and non-durable goods are shown in Fig. 2. 3.3. Habits and durable goods Hence, habits capture the fact that an increase in the rate of devaluation reduces welfare. The steady-state level of habits falls with an increase in the rate of devaluation. This, then, breaks the super-neutrality result discussed above. The results for the two cases discussed above allow us to explain the results for the model with habits and durable goods intuitively. Preferences are now given by ∞ e−θt U (ω(ct + st , mt ), ht ) dt, 0
where st is the stock of durable goods inherited from the past, and is given by Eq. (2), while the stock of habits ht , is a weighted sum of past levels of ω(ct + st , mt ): t ht = ρ e−ρt eρτ ω(cτ + sτ , mτ ) dτ. (11) −∞
The evolution of ht is, thus, given by h˙ t = ρ[ωt − ht ].
(12)
As explained above, an increase in ε increases the cost of holding money and thus leads to a fall in welfare. Hence, the steady-state habitual standards of living fall. Real money
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holdings fall. The steady-state levels of consumption expenditures and durables are likely to fall as well.15 It therefore follows from Eq. (8), that the steady-state stock of net foreign assets falls. As discussed above, both the habits and durability effects tend to lead to a fall in m when ε rises. Without habits, m falls to its new steady-state level instantly, and it stays there forever. With habits, m also falls immediately after the increase in ε. However, now it stays above its new long-run level, as the representative agent tries to maintain the habitual standards he has inherited from the past. After this, m falls further along the adjustment path until it reaches its new steady-state level. It is, thus, clear that with the simultaneous presence of habits and durable goods, adjustment of m will be qualitatively the same as in the model with habits and non-durables. However, the habits and durability effects will have competing influences on the behavior of c and f. To reduce the steady-state stock of durables, the representative agent reduces his consumption expenditures below their steady-state level, which tends to bring about a current-account surplus. On the other hand, as discussed above, the representative agent wants to maintain the relatively high habitual standards he has inherited from the past. This habit-persistence effect tends to increase consumption, which tends to bring about a current-account surplus. Backus, Kehoe, and Kydland (1994) document evidence from various countries suggesting that the trade balance is negatively correlated with current and future movements in the terms of trade, but positively correlated with past movements. In a recent paper, Mansoorian (1998) uses the habits-and-durability model without money to show that Backus et al.’s findings indicate that habit effects should be dominant in the short run and durability effects in the long run. In the present model, an increase in ε leads to an immediate increase in consumption expenditures and a current-account deficit. However, when the durability effects become dominant, consumption starts to fall and the current account starts to improve. This is the J-curve phenomenon in a monetary setting.16,17 The adjustments of c and f for this case are shown in Fig. 3. On the other hand, Heaton’s empirical findings on the behavior of asset prices suggest that the durability effects will be dominant in the short run; but eventually the habit effects will dominate. If this is the case, then immediately after the increase in ε, there will be a fall in consumption expenditures and a current-account surplus. However, when the habit effects 15 The effect on the steady state levels of c and s is ambiguous for the following reason. When ε increases, the agent substitutes c¯ + s¯ for m ¯ (substitution effect). On the other hand, when habits fall, steady-state ω also falls (Eq. (11)). This tends to reduce both c¯ + s¯ and m ¯ (habit effects). Hence, there are two competing influences on c + s and its steady state level cannot be signed. Nevertheless, for the rest of the paper we assume that s¯ falls when ε increases. The reasons are explained in footnote 16. 16 Notice, following from footnote 15, if s¯ increases with ε, we do not have the non-monotonic adjustment of c or f, as both the habits and durability effects have similar influences on adjustment. However, the empirical results reported by Backus et al. suggest that adjustment of the current account is non-monotonic. 17 This is an alternative to the textbook explanation of the J-curve (see, e.g., Krugman & Obstfeld, 2000, pp. 466–468), which relies on institutional rigidities that fix the volumes of exports and imports in the short run. The textbook treatment also assumes that prices are fixed in the short run, and the real depreciation is originally brought about by a nominal devaluation of the domestic currency. Unlike the textbook treatment, the explanation for the J-curve presented in this paper is embedded in an intertemporal utility-maximizing framework.
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Fig. 3. Adjustments of c and f with habits dominant in the short run and durability in the long run.
become dominant, consumption expenditures will start to fall and the current account will start to improve. The adjustments of c and f for this case are shown in Fig. 4. 3.4. Uzawa preferences Finally, consider the model with Uzawa preferences, in which there are no habits, but where the rate of time preference is an increasing function of instantaneous utility t ∞ U (ct , mt ) exp − θv dv , where θv = θ(U (cv , mv )). (13) 0
0
Obstfeld (1981a) considers the effects of exchange-rate policies with Uzawa preferences when consumption goods are non-durable. As the small, open economy takes the world real interest rate as given, the equality of the rate of time preference and the world interest rate dictates the level of instantaneous utility which must be maintained in the steady state. An increase in the rate of devaluation of the domestic currency would, therefore, by increasing the cost of holding real balances, require substitution of consumption for real money holdings along the same indifference curve in the steady state. Hence, in order to be able to finance his higher steady-state consumption, the representative agent accumulates bonds along the adjustment path. Therefore, on impact, the increase in the rate of devaluation leads to a huge reduction in consumption and a current-account surplus. By the homotheticity
Fig. 4. Adjustments of c and f with durability dominant in the short run and habits in the long run.
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of preferences, it follows that on impact real money holdings also fall by a huge amount. Along the adjustment path, both consumption and real money holdings rise over time. With consumer durables, these results change substantially. With durables, ct should be replaced with ct + st in Eq. (13). As money does not exhibit durability, its dynamics are as in Obstfeld (1981a). On the other hand, there are two competing effects on the dynamics of consumption and the current account. First, there is the effect without durable goods discussed above, which from now on we refer to as the “time-preference” effect. Second, there is the durability effect. As may be clear from new Eq. (3), the steady-state stock of durables has to increase. As we have seen already, this tends to increase consumption expenditures above their steady-state level. Hence, again, the dynamics of the current account are likely to be non-monotonic. If the durability effects are dominant in the short run and the time preference effects in the long run, then immediately after the increase in the rate of devaluation there will be an increase in consumption expenditures and a current-account deficit. But when the time preference effects start to dominate, then consumption expenditures start to fall and the current account turns into a surplus. This, again is the J-curve phenomenon in a monetary setting. That we should have such dynamics with Uzawa preferences and durable goods also follows from our previous analysis. As shown in Mansoorian (1996), the dynamics of the model with habits and adjacent substitutability of preferences are qualitatively similar to the dynamics of the model with Uzawa preferences.18 Hence, the dynamics of the model with Uzawa preferences and durable goods should be qualitatively the same as the habit-persistence model with durable goods, when habits are combined with adjacent substitutability of preferences, and when the steady-state stock of durables increases with the rate of devaluation. Hence, although Uzawa preferences cause the steady-state levels of c and f to rise, adjustment of c and f when durability effects are dominant in the long run are as in Fig. 3, and adjustments when durability effects are dominant in the short run are as in Fig. 4.
4. Conclusion This paper studies the effects of exchange-rate policies in the presence of durable goods. The setting is that of a small, open economy in which instantaneous utility depends on real money holdings. Three popular specifications of intertemporal utility are considered: time-separable preferences, the habit-persistence model, and Uzawa preferences. It is shown that with time-separable preferences, changes in the devaluation rate have no effect on the stock of durable goods, on consumption expenditures, and on the country’s net foreign assets. This is the Sidrauski super-neutrality result extended to an open economy with durable goods. Although money is super-neutral with time-separable preferences, an increase in the devaluation rate reduces real money holdings. With money-in-utility, this reduces welfare. Hence, extending the model to incorporate habits breaks super-neutrality. With habits, the 18
See footnote 3 for an explanation of the difference between adjacent complementarity and adjacent substitutability.
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fall in welfare resulting from the fall in real money holdings is likely to lead to a fall in the steady-state stock of durable goods. The habit effect and the durability effect have competing influences on the adjustment of consumption expenditures and the country’s net foreign asset holdings. These variables are likely to adjust non-monotonically along the path to the new steady state. Real money holdings, however, adjust monotonically, because they do not exhibit durability. With Uzawa preferences, the model does not have habits, but the rate of time preference is an increasing function of instantaneous utility. With such preferences, an increase in the rate of devaluation requires substitution of consumption for real money holdings in the steady state. The presence of durable goods would, in addition, require an increase in the steady-state stock of durables. The time-preference and the durability effects tend to have competing influences on consumption expenditures and the current account, making adjustment of these variables non-monotonic. On the other hand, as real balances do not exhibit durability, their adjustment is monotonic, and depends only on the time-preference effect.
Acknowledgements The authors would like to thank the editor of the journal and two anonymous referees for valuable comments and suggestions on an earlier draft. Financial support from the Social Sciences and Humanities Research Council of Canada (grants 410-97-1212 and 410-01-1018), and the University Research Board of the American University of Beirut are also gratefully acknowledged.
Appendix In this appendix, we derive formally the perfect-foresight path, and the effects of exchangerate policies for the case with durable goods and time-separable preferences. For this purpose, we first solve the representative agent’s problem, and then consider the country’s external adjustment along the equilibrium path, in light of the assumptions we have made about central bank policies. The problem of the representative agent is then to maximize lifetime utility (1) subject to constraints (3)–(6). The Hamiltonian for this problem is H = ω(ct + st , mt ) + φt [ct − δst ] + µt [rat + y + Tt − ct − (r + ε)mt ], where φ is the shadow price of durable goods, while µ is the shadow price of assets. The optimality conditions for this problem are Hc ≡ ω1 + φ − µ = 0,
(A.1)
Hm ≡ ω2 − µ(r + ε) = 0,
(A.2)
˙ −Hs + θ φ = −ω1 + φδ + θ φ = φ,
(A.3)
˙ −Ha + θ µ ≡ −rµ + θ µ = µ,
(A.4)
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and the transversality conditions lim e−θt φt st = 0,
(A.5)
lim e−θt µt at = 0,
(A.6)
t→∞
and t→∞
Clearly, for (A.4) to converge, we will need r = θ,
(A.7)
which means that along the adjustment path, µ is always at its steady-state level µ. ¯ In order to derive the optimum path of the economy, first linearize (A.1) and (A.2) around the steady state, using the fact that with r = θ along the adjustment path dµ = 0, to obtain ω22 ¯ (ct − c) ¯ = −(st − s¯ ) − (φt − φ) (A.8) ω11 ω22 − ω12 ω21 and (mt − m) ¯ =
ω21 ¯ (φt − φ), ω11 ω22 − ω12 ω21
(A.9)
where over-bars denote steady-state values. Next, linearize (3) and (A.3) around the steady state, and use (A.7)–(A.9) in order to obtain s˙t st − s¯ a11 a12 = , (A.10) 0 a22 φt − φ¯ φ˙ t where a11 = −(1 + δ), a12 = −[ω22 /(ω11 ω22 − ω12 ω21 )], and a22 = 1 + r + δ. The system of differential equations in (A.10) has one positive and one negative eigenvalue. Hence, as s is a state variable, while φ is a jump variable, the differential equation system (A.10) exhibits saddlepath stability. The negative eigenvalue of the system is −(1 + δ); and the stable path is given by the following equations: st − s¯ = (s0 − s¯ ) e−(1+δ)t ,
(A.11)
φt − φ¯ = 0.
(A.12)
Using (A.11) and (A.12) in (A.8) and (A.9) we obtain the adjustments of c and m along the optimum path ct − c¯ = −(st − s¯ ) = −(s0 − s¯ ) e−(1+δ)
(A.13)
mt − m ¯ = 0.
(A.14)
and Next, consider the dynamics of the country’s net foreign assets f (≡ b + R): f˙ = rf + y − c.
(A.15)
Linearizing this expression around the steady state, using (A.13), we obtain f˙t = r(ft − f¯) + [s0 − s¯ ] e−(1+δ)t .
(A.16)
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The solution to (A.16) is ft − f¯ = −
1 1 −(1+δ)t ¯ + (f0 − f ) + (s0 − s¯ ) e (s0 − s¯ ) ert . 1+δ+r 1+δ+r (A.17)
Clearly, for (A.17) to converge the coefficient on ert must be 0. Hence, (f0 − f¯) +
1 (s0 − s¯ ) = 0 1+δ+r
(A.18)
is an important condition which must hold for saddlepath stability of the model. This condition determines, for the initial values of the predetermined variables f0 and s0 , how the steady-state values of these variables must be related for saddlepath stability. Next we show that with time-separable preferences, and with habits, an increase in the rate of devaluation will have no effect on the goods market (i.e., on the steady-state levels of consumption, c, the stock of durable goods, s, or the country’s net foreign asset position, f), but it will reduce the level of real money holdings, m. For this purpose, consider first the steady state of the economy described above. This is characterized by (A.1), (A.2), and by Eqs. (3), (A.3) and (A.15) with s˙ = φ˙ = f˙ = 0. There are five equations in six unknowns: ¯ and µ. c, ¯ m, ¯ s¯ , f¯, φ, ¯ The sixth equation is (A.18). These equations are all independent of the level of the exchange rate, E. Hence, as in Obstfeld, a once-and-for-all devaluation does not have any effect on this economy, because all prices are fully flexible. Next, note (3), (A.15) and (A.18) implicitly determine the steady-state levels of s, f and c. As these equations are independent of ε, it follows that changes in the rate of devaluation have no effect on c, ¯ s¯ or f¯. This is the Sidrauski monetary super-neutrality extended to an open-economy setting with durable consumption goods. It arises, essentially, because preferences are time-separable. To see how this super-neutrality result could be broken with habits, notice that by differentiating (A.1)–(A.3) at the steady state, using the fact that dc¯ = d¯s = df¯ = 0, we obtain dm −µ/(r + ε) = < 0. dε [1 + 1/(δ + r)](ω12 − ω22 )/(r + ε) Hence, steady state falls as ε rises, because steady state c and s are unaffected while m falls.
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