Real asset returns, inflation and activity in a small, open, Cash-in-Advance economy

Journal of International Money and Finance 32 (2013) 234–250

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Real asset returns, inflation and activity in a small, open, Cash-in-Advance economy Arman Mansoorian a, *, Mohammed Mohsin b a b

Department of Economics, York University, 4700 Keele Street North York, Ontario, M3J 1P3 Canada Department of Economics, University of Tennessee, Knoxville, TN 37996, USA

a b s t r a c t JEL classifications: F31 F32 E4 Keywords: Real asset returns Inflation Economic activity Open economy Cash-in-Advance

The effects of inflation are worked out for a small open economy with Cash-in-Advance (CIA) constraints on bond purchases. If all transactions are subject to CIA constraints, an increase in the inflation rate will reduce savings, bringing about a current account deficit, while the capital stock will be unaffected. If investment is not subject to CIA constraints, an increase in the inflation rate will encourage investment and reduce savings, bringing about a current account deficit. Numerical evaluation of the model gives rise to falls in real interest rates that are in line with recent empirical findings. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction There is considerable empirical evidence that the nominal rates of return on assets do not generally keep up with inflation. This negative correlation between inflation and real asset returns was originally thoroughly investigated by Fama (1981); and it has been investigated by several authors since then.1 Barnes et al. (1999), in fact, conclude that this phenomenon extends to other assets; for most countries that they consider, inflation has a negative impact on the real rate of return on a variety of assets including bonds. When regressing the level or the changes in the safe nominal interest rates on the level or the changes in the inflation rate they find that for most countries the coefficient on the inflation

* Corresponding author. E-mail address: [email protected] (A. Mansoorian). 1 See, for example, Fama and Gibbons (1984), Geske and Roll (1983), Wahlroos and Berglund (1986), Serletis (1993), Thornton (1993), Rapach (2003) and Rapach and Wohar (2005). 0261-5606/$ – see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jimonfin.2012.04.014

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rate is positive, but significantly smaller than unity, indicating a significant fall in the real rates of return resulting from an increase in the inflation rate.2 Fama explained this negative relationship between asset returns and inflation by arguing that inflation reduces expected economic activity, which in turn reduces the return on equities. One important shortcoming of this explanation is that it implies that inflation will reduce real activity, which is not consistent with the empirical literature that has been the focus of the New Keynesian literature.3 The open economy implications of the negative relationship between inflation and real asset returns have also not been studied in the literature. Indeed, Barnes et al. (p. 740) note that “many monetary models imply that inflation cannot affect the steady state real rate of return on assets.” In this paper we propose a simple workable model which is tractable enough that we could use it to study some basic open economy implication of the negative relationship between inflation and real asset returns. We assume that real bond returns fall when inflation increases because bonds are purchased with money. The assumption that bonds are purchased with money was central to development of the traditional IS-LM-BP model. In the textbook treatment of the IS-LM-BP model when domestic interest rates are above (below) foreign rates, there is a huge capital inflow (outflow), and balance of payments surplus (deficit). This implied an excess demand for (supply of) the domestic currency on the foreign exchange market, which rose (lowered) the value of the domestic currency. Moreover, empirically, for small open economies changes in domestic interest rates do lead to substantial movements in the price of the country’s currency on the foreign exchange markets, which are mainly induced by the movements of short term capital. From this, one could infer that international transactions involving assets are financed with money. We consider a small open economy model in which bond purchases are subject to Cash-in-Advance (CIA) constraints.4,5 We show that in this setting, following an increase in inflation, the adjustments of consumption, investment and net foreign assets exhibit Keynesian features, even though prices are fully flexible. The New Keynesian literature (in the spirit of Woodford (2003)) has made considerable efforts to obtain such dynamics using price setting behaviour in imperfectly competitive environments.6 With CIA constraints on bond purchases, the real rate of return on bonds is not constant. In that setting, inflation, brought about by the rate of devaluation of the domestic currency, acts as a tax on bond purchases. Hence, the real rate of return on bonds depends on the rate of devaluation. Hence, in order to have a well defined steady state equilibrium, in which the real rate of return on bonds is equal to the rate of time preference, we make the rate of time preference endogenous. As in Uzawa (1968),

2 See their Table 3, where they report the results of the regressions of changes in the money market rate on changes in inflation for different countries. The estimated coefficients on the changes in the inflation rate are between
0.0383, for India, and 0.1182, for New Zealand. But for most countries this coefficient is positive and very small: 0.0127 for Australia, 0.0532 for Canada, 0.0186 for France, 0.0055 for Germany, 0.1095 for Italy, 0.0434 for Japan, 0.0235 for the U.K., and 0.0515 for the U.S. 3 Authors such as Marshall (1992), Bakshi and Chen (1996) and Stulz (1986) have tried to explain this phenomenon in terms of Lucas (1978) type asset pricing models. In these models a permanent increase in the inflation rate does not affect steady state real interest rates. 4 We speculate that the literature has not given enough attention to this channel through which monetary policy impinges upon the open economy because it is believed that “One difficulty with this (i.e., the CIA) approach is that the introduction of the various constraints, embodying the role played by money in transactions, can very quickly become intractable” (Turnovsky, 1997, p. 20). See also Turnovsky (2000, p. 264). Similarly, Blanchard and Fischer (1989, p. 155) state that “Models based explicitly on (CIA) constraints.can quickly become analytically cumbersome.” The analysis in the present paper is very tractable because we adopt a continuous time setting. The reader is referred to Calvo (1987), Calvo and Vegh (1994, 1995) and Edwards and Vegh (1997) for other discussions of open economy issues with CIA constraints on consumption. 5 The small open economy literature has considered several channels though which monetary policy can affect the economy. Obstfeld (1981a,b) combines the money-in-utility model of Sidrauski (1967) with Uzawa (1968) preferences, where the rate of time preference is an increasing function of instantaneous utility. Mansoorian (1996) considers the importance of habits, combining the habit formation model of Ryder and Heal (1973) with money-in-utility. There, the rate of time preference is fixed, but instantaneous utility depends on habits, which are modelled as a weighted sum of past levels of instantaneous utilities. Shi and Epstein (1993) combine endogenous time preference with habit formation, making the rate of time preference an increasing function of habits. Mansoorian and Mohsin (2006) consider a model with a fixed rate of time preference and no habits, but with endogenous labour and a Cash-in-Advance (CIA) constraint on consumption expenditures. 6 See, for example, Roberts (1995) and Rudd and Whelan (2007).

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the rate of time preference is assumed to be an increasing function of instantaneous utility. In that case, when there is a change in the rate of devaluation the representative agent’s rate of time preference will adjust in order to set the inflation-adjusted real rate of interest to the rate of time preference in the steady state.7,8 We show that for reasonable functional forms and parameter values a one percentage point increase in the inflation rate would result in a 0.88 percentage point fall in the steady state rate of time preference (and, hence, in the real rate of interest), which is very much in line with the empirical findings of Barnes et al. (1999), reported in footnote 2 above. We will consider three different settings. First, we will consider a small open economy with a fixed output in which there are CIA constraints on bond purchases as well as on consumption. This setting is studied first as it will bring out clearly the mechanisms at work. Second, we will consider a small open economy with production when there are CIA constraints on consumption, investment and bond purchases. Third, we will consider a small open economy with production and investment, in which there are CIA constraints on consumption and bond purchases, but not on investment. As in Obstfeld (1981b), Calvo (1987) and Mansoorian and Mohsin (2006), we assume that the central bank targets the rate of change of the exchange rate (not the rate of growth of money per se).9 This precludes complicated, yet no so crucial, off steady state effects, similar to those analyzed by Fischer (1979). First, consider the model with a fixed output. An increase in the rate of devaluation of the domestic currency will act as an increase in a tax on bond purchases. This will reduce the return on savings, which will increase consumption, bringing about a current account deterioration.10 On impact, there is no effect on his real money holdings: the money received from the sale of bonds is equal to the money required to finance extra consumption goods purchased. Over time, as the country’s net foreign asset position deteriorates consumption and real money holdings decline until they reach their new steady state levels. Second, consider the model with capital, and CIA constraints on consumption, investment and bond purchases. In that case, the rate of devaluation will act as a tax not only on bond purchases, but also on capital. As a result, an increase in the rate of devaluation will reduce the return on bonds and capital by the same amount, leaving the steady state capital stock unaffected. Nevertheless, the dynamics of consumption, net foreign assets and real money holdings will be qualitatively the same as in the case with fixed production discussed above. Third, consider the model with investment, and CIA constraints on consumption and bond purchases, but not on investment. In that case, the increased inflation, in line with the traditional Keynesian models, boosts both investment and consumption. In that case, an increase in the inflation rate reduces the return on bonds, but not on investment, which encourages capital accumulation.11

7 Uzawa preferences have been relatively popular in the open economy literature; see, e.g., Obstfeld (1981a,b, 1990), Devereux and Shi (1991), and Mendoza (1991). Schmitt-Grohé and Uribe (2003) use the assumption of endogenous time preference as a remedy for some technical problems encountered when performing real business cycle type analysis for open economies. The reader is, nevertheless, referred to Obstfeld and Rogoff (1996, pages 723–725) for the controversies surrounding Uzawa preferences. 8 Notice, when both the real rate of return on internationally traded bonds and the rate of time preference are exogenous, for the small open economy to have a well defined steady state they must be set equal to each other (see, for example, Sen and Turnovsky (1989a,b)). 9 This assumption is consistent with the assumptions in the literature concerned with the time consistency of monetary policy (e.g., Kydland and Prescott (1977), Backus and Driffill (1985), and Walsh (1995)), where it is also assumed that the central bank targets the inflation rate (not the rate of growth of money per se). Recently, Mishkin (2000) also argued that the central banks of most developed as well as emerging countries do indeed target the inflation rate rather than the rate of growth of money. 10 Recent work by Ahmed and Rogers (2000), Schmidt (2006), Betts and Devereux (2001), and Holman and Neumann (2002) provide evidence in favour of the proposition that after a monetary easing, which increases inflation, consumption increases. 11 The empirical literature seems to have concluded that an increase in the inflation rate will encourage growth and capital accumulation at low inflation rates, which is consistent with the third model studied in this paper. The empirical literature has, nevertheless, concluded that for high level of inflation, an increase in the inflation rate appears to harm growth and investment. Sarel (1996) estimated this crucial threshold level of inflation to be 8 per cent, which is in line with the recent estimate of 9 percent by Espinoza et al. (2010). On the other hand, Khan and Senhadji (2001) estimated this threshold to be between 1 and 3 percent for industrialized countries and between 7 and 11 percent for developing countries. The reader is also referred to Ahmed and Rogers (2000), Ericsson et al. (2001) and Rapach (2003) for strong evidence in favour of an increase in the inflation rate leading to an increase the long run capital stock for moderate inflation levels.

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Also, the fall in the real rate of return on bonds reduces savings. With the increase in investment and fall in savings, the country runs a current account deficit.12 In sum, in this paper we propose a simple workable model that generates a reasonable negative relationship between inflation and real asset returns that is in line with the empirical findings of Barnes et al. (1999), is consistent with the assumption central to the development of the traditional IS-LM-BP model, and is tractable that we could use it to study some basic open economy implication of the negative relationship between inflation and real asset returns. The predictions of all three models regarding the behaviour of consumption after an inflation shock are consistent with some recent empirical findings (see footnote 10). The predictions of the third model regarding the behaviour of capital after an inflation shock are consistent with the experiences of countries with relatively low inflation rates (see footnote 11), while the prediction of the second model regarding the capital stock are consistent with the superneutrality results implied by Classical models (e.g., Sidrauski, 1967). Finally, our results regarding the effects inflation on the capital stock are in sharp contrast to Stockman’s (1981) results for a closed economy without bonds. Stockman showed that an increase in the inflation rate would lead to a fall in steady state capital if there were CIA constraints on consumption as well as on investment, whereas the steady state capital would be unaffected if there were CIA constraints on consumption only. The reason was that in a closed economy without bonds, the rate of return on savings is equal to the return on capital. When investment is subject to a CIA constraint, an increase in the inflation rate reduces the return on savings, which reduces steady state capital. On the other hand, when there are no CIA constraints on investment then changes in the inflation rate do not affect the return on savings. The paper is organized as follows. The model without capital is presented in Section 2. The model with capital is presented in Section 3. Some concluding remarks are made in Section 4. 2. The model without capital 2.1. The model and the perfect foresight path The model is that of a small open economy, with perfect capital mobility, flexible prices, and no uncertainty, with a representative household who has perfect foresight. The foreign currency price of the single good in the model is fixed at P*. The economy is small and takes P* as given, which is set equal to 1. The domestic currency price of this good is P ¼ EP*, where E is the exchange rate (the price of foreign currency in terms of domestic currency). The rate of inflation is equal to the rate of depreciation of the domestic currency (E_ t =Et ), which is denoted by εt . The representative agent has Uzawa preferences, where the rate of time preference is an increasing function of instantaneous utility

Z

N

Uðct Þe
Qt e
rt dt

(1)

0

_ ¼ qðUðc ÞÞ
r where Q t t 0

(2) 00

with U 0 ð,Þ > 0, q ð,Þ > 0 and q ð,Þ > 0 for all values of consumption c. Notice, the rate of time prefZ t qðcs Þds.13 erence is qðct Þ, while r is the real interest rate. Also,
Qt
rt ¼
0

12 These results are similar to the effects of taxes on international borrowing and lending (see, for example, Shi (1994)). A tax on international borrowing and lending will also leave the return on capital unaffected while reducing the return on bonds. The dynamics of the model will be qualitatively very similar to the case discussed above. Nevertheless, there are very important dissimilarities between our model with no CIA constraint on investment and a model with taxes on international borrowing. The dissimilarities arise from that fact that in the present model an increase in the inflation rate also taxes consumption expenditures. 13 This re-formulation of Uzawa preferences is due to Epstein (1987) and Obstfeld (1990), and it considerably simplifies the solution method. Our solution to this problem follows closely Obstfeld’s (1990, pp. 56–58) method.

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The agent is endowed with y units of the good at any time t. He also receives monetary transfers with real values of st from the government. There are two kinds of assets in the model, money balances and internationally traded bonds. These bonds have a fixed price of unity in terms of the foreign currency and also in terms of goods, as P* ¼ 1. Each bond pays r units of the foreign currency (or goods, again, as P* ¼ 1) at any time. The real assets of the representative agent are

at ¼ mt þ bt ;

(3)

where bt is his bond holdings, and mt his real domestic money balances. His flow budget constraint is14

a_ t ¼ y þ rbt þ st
ct
εt mt ;

(4)

and he should satisfy the intertemporal solvency condition

Lim e
rt at  0:

(5)

t/N

As explained in the Introduction, in the traditional IS-LM-BP models it was assumed that all transactions, even those involving bonds, are subject to CIA constraints. Hence, the representative household also faces the CIA constraint15

mt ¼ ct þ b_ t ;

(6)

where b_ t are his bond purchases at time t. This constraint can be rewritten as

b_ t ¼ mt
ct :

(7)

His problem will then be to maximize (1) subject to (2)
(5), (7), and the initial conditions a0 ; b0 , taking the paths of fst g, and fεt g as exogenously given to him by the government. Using (3) to eliminate mt , the current value Hamiltonian for his problem can be written as

H ¼ Uðct Þe
Qt þ Lt ½rbt þ y þ st
ct
εt ðat
bt Þ þ Jt ½at
bt
ct  þ Ft ½qðUðct ÞÞ
r where Lt ; Jt and Ft are the shadow prices associated with assets a, bonds b and time preference Q: The optimality conditions for this problem are:

Hc ¼ 00U 0 ðct Þe
Qt ¼ Lt þ Jt
Ft q U 0 ðct Þ;

(8)

L_t ¼ rLt
Ha 0 L_t ¼ Lt ðr þ εt Þ
Jt ;

(9)

0

J_ t ¼ rJt
Hb 0J_ t ¼ ðr þ 1ÞJt
ðr þ εt ÞLt

(10)

F_ t ¼ rFt
HQt 0F_ t ¼ rFt þ Uðct Þe
Qt ;

(11)

and the transversality conditions. Now, following Obstfeld (1990, pp. 56–58), define

lt ¼ Lt eQt 0l_ t ¼ L_t eQt þ Lt eQt ½qðUðct Þ
r

(12)

14 _ t =Pt and bond purchases Notice, the representative agent spends his savings y þ rbt þ st
ct on real money accumulation M _ t =Pt ¼ y þ rbt þ st
ct . Substituting in this expression a_ t ¼ m _ t þ b_ t (from Eq. (3)) and the fact that b_ t ; that is, b_ t þ M _ t =Pt ¼ m _ t þ εt mt , we obtain Eq. (4). M 15 Notice, here, money is used to buy bonds and consumption goods, and in the second model we will be discussing below, it will be used to purchase capital as well. Bonds and capital, being vehicles for transferring purchasing power over time, will affect utility only indirectly by influencing current and future consumption, and therefore do not enter the utility function.

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jt ¼ Jt eQt 0j_ t ¼ J_ t eQt þ Jt eQt ½qðUðct Þ
r;

(13)

ft ¼ Ft eQt 0f_ t ¼ F_ t eQt þ Ft eQt ½qðUðct Þ
r:

(14)

Substituting from (12)–(14) into (8)
(11) we can re-write the optimality conditions as: 0

U 0 ðct Þ ¼ lt þ jt
fq U 0 ðct Þ;

(15)

l_ t ¼ lt ðεt þ qÞ
jt ;

(16)

j_ t ¼ jt ð1 þ qÞ
ðr þ εt Þlt ;

(17)

f_ t ¼ qft þ Uðct Þ:

(18)

Turning to the government side of the model, in this paper we abstract completely from government expenditures on goods and services, and concentrate exclusively on monetary policies. The central bank chooses the real monetary transfers st in order to achieve a fixed rate of devaluation of the domestic currency ε, subject to its flow constraint, which says that it should finance its expenditures st with seigniorage:

st ¼ M_ t =Pt ¼ m_ t þ mt εt ;

(19)

Mt being the nominal money supply. Notice in writing (19) we have assumed for simplicity that the central bank does not have any bonds or foreign reserves, and the government does not have any debt. We are, therefore, assuming that we have a perfectly flexible exchange rate system; and the central bank controls the rate of depreciation of the domestic currency ε by choosing the transfers st appropriately. Hence, the money stock mt is demand determined: for a given rate of devaluation the central bank stands ready to supply any amount of real balances that are demanded. Along a perfect foresight path the allocations are such that the representative agent’s expectations about fst g and fεt g coincide with the actual paths of these variables. To solve for the perfect foresight path first combine the government budget constraint (19) and the household’s flow budget constraint (4) to obtain

b_ t ¼ rbt þ y
ct ;

(20)

according to which the current account balance b_ t is equal to national income rbt þ y less absorption ct . The perfect foresight path is the solution to the dynamic system described by (15)–(18), and (20). Notice, without on-going technological growth all real variables and the co-state variables will have a growth rate of zero in the steady state; and we can use standard techniques to solve for the saddlepath. To solve for the saddlepath first linearize (15) around the steady state to obtain

      0 ðc
cÞ ¼ G l
l þ j
j
q U 0 f
f ;

(21) 00

0

where overbars denote steady state values and G ¼ 1=½U 00 þ fðU 02 q þ U 00 q Þ: Next, linearize (16)
(18) and (20), and use (21) to derive the following system of differential equations:

2_3 2 A11 l 6 j_ 7 6 6 7 ¼ 6 A21 4 f_ 5 4 A31 _b A41

A12 A22 A32 A42

A13 A23 A33 A43

32 3 0 l
l 6 7 0 7 76 j
j 7; 0 54 f
f 5 A44 b
b

(22)

0

0

where the exact expressions for Aij are as follows: A11 ¼ q þ ε þ Glq U 0 , A12 ¼ Glq U 0
1, 02 0 0 02 A21 ¼ Gjq U 0
r
ε, A22 ¼ Gjq U 0 þ 1 þ q, A23 ¼
Gjq U 02 , A13 ¼
Glq U 02 , 0 0 0 0 0 0 0 0 0 0 A31 ¼ A32 ¼ Gðfq U þ U Þ, A33 ¼ q
Gq U ðfq U þ U Þ, A41 ¼ A42 ¼
G, A43 ¼ Gq U and A44 ¼ r:

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The system of differential equations (22) has one predetermined variable, b. Therefore, for saddlepath stability the coefficient matrix in (22) should have one negative and three positive eigenvalues. Let x denote the negative eigenvalue, and ½a1 ; a2 ; a3 ; 1 its corresponding eigenvector, where the exact expressions for ai (i ¼ 1, 2, 3) are reported in Appendix I. The saddle path corresponding to the system of differential equations (22) will then be given by the following equations:

bt
b ¼ ðb0
bÞext ;

(23)

lt
l ¼ a1 ðb0
bÞext ;

(24)

jt
j ¼ a2 ðb0
bÞext ;

(25)

ft
f ¼ a3 ðb0
bÞext :

(26)

Eqs. (23)–(26) describe the perfect foresight path. 2.2. The effects of exchange rate policies Now consider the effects of a permanent unanticipated increase in the rate of devaluation of the domestic currency at time 0. To this end, first consider the steady state effects. The steady state is described by (7), (15)–(18), and (20) with l_ ¼ j_ ¼ f_ ¼ b_ ¼ 0: Differentiating these equations totally we obtain the steady state effects on b, c and m:

db
q < 0; ¼ 0 dε r q U 0 ð1 þ 2q þ εÞ

(27)

dc dm
q < 0: ¼ ¼ 0 dε dε q U 0 ð1 þ 2q þ εÞ

(28)

From Eqs. (21) and (23)–(26) one can readily derive the adjustments of c to the new steady state. In order to obtain the adjustment of real money holdings m combine Eqs. (7) and (20) to obtain

mt ¼ rbt þ y:

(29)

After linearizing this equation around the steady state and using Eq. (23), we obtain the adjustment of m along the perfect foresight path:

mt
m ¼ rðb0
bÞext :

(30)

The adjustments of c, b and m are depicted in Figs. 1–3, respectively. An increase in the rate of devaluation acts as an increase in a tax on bond purchases, which reduces the return on savings.16 Hence, on impact, there is an increase in consumption, a fall in savings, and a current account deterioration. In order to increase his consumption the representative agent sells some of his bond holdings. On impact, therefore, there is no effect on his real money holdings (see Eq. (29)): the money received from the sale of bonds is equal to the money required to finance the extra consumption goods

16 Notice, in the absence of labour-leisure choice the tax on consumption brought about by inflation will not have a substitution effect. In fact, in the absence of CIA constraint on bond purchases (and with no capital) the increased inflation will have no effect on the equilibrium. Hence, by abstracting completely from labour-leisure choice, we are able to isolate completely the mechanism by which the CIA constraint on bond purchases will impinge on the economy.

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241

Fig. 1. The adjustment of c with fixed output.

Fig. 2. The adjustment of b with fixed output.

Fig. 3. The adjustment of m with fixed output.

purchased. Over time, as the country’s net foreign asset position deteriorates consumption and real money holdings decline until they reach their new steady state levels. It will be instructive to contrast these results with Obstfeld’s (1981a,b), who also uses Uzawa preferences, but introduces money into the model as an argument in instantaneous utility. In Obstfeld’s model there is a unique level of utility that must be maintained in the steady state. This is determined by the equality of the rate of time preference with the real rate of interest, which in his

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model is unaffected by exchange rate policies. Thus, an increase in the rate of devaluation, by increasing steady state inflation, will result in a fall in steady state real money holdings and a rise in steady state consumption. In order to increase steady state consumption, the representative agent must accumulate assets along the adjustment path. Thus, in the short run after the increase in the rate of devaluation there will be a sharp fall in consumption and real money holdings, leading to a current account surplus. After that, both consumption and real money holdings increase along the adjustment path to the new steady state equilibrium. Hence, our results, obtained with CIA constraints, are in sharp contrast to Obstfeld’s. 3. The model with capital Until now the discussion has been confined to an endowment economy. In this section, we will extend the model to incorporate production. We will first consider the case in which there are CIA constraints on all transactions involving consumption, bonds and investment; we then consider the case in which there are CIA constraints in consumption and bond purchases only. 3.1. The case with CIA on consumption, investment and bond purchases Assume that output is produced with a neo-classical production function, yt ¼ f ðkt Þ, where kt is the capital stock at time t, f 0 ð,Þ > 0 and f 00 ð,Þ < 0. Further, assuming no depreciation, to have a meaningful model for investment, we assume

k_ t ¼ It
JðIt Þ;

(31)

where JðIt Þ are the adjustment costs associated with It ; it represents the amount of investment goods that are lost in the installation process. Hence, if It is the amount spent on investment, then JðIt Þ is the part of these expenditures that is wasted in the installation process. The function JðIt Þ is assumed to be a non-negative convex function, with J 0  0 and J 00 > 0: By choice of units we may set Jð0Þ ¼ 0 and J 0 ð0Þ ¼ 0. Assuming for simplicity that all domestic capital are held by the representative agent, his total assets which in this subsection can be accumulated with real money holdings are now

at ¼ mt þ bt þ kt :

(32)

Substituting for y into (4), and using (32), we obtain the representative agent’s flow budget constraint17

    a_ t ¼ f kt
J It þ rbt þ st
ct
εt mt :

(33)

When all purchases (i.e., consumption, investment and bond purchases) are subject to CIA constraints, the CIA constraint facing the representative agent will be given by

mt ¼ ct þ b_ t þ It ;

(34)

where It denotes Investment expenditures at time t. One can re-write this equation as

b_ t ¼ mt
ct
It :

(35)

The problem of the representative agent is then to maximize (1) subject to (2), (5), (31)–(33) and (35) and the initial conditions, a0 ; b0 and k0 . Using (32) to eliminate mt , the Hamiltonian for his problem can be written as:

17 _ t =Pt Notice, the representative agent spends his savings f ðkt Þ þ rbt þ st
ct on investment It, real money accumulation M _ t =Pt
εt mt , we obtain Eq. (33). _t ¼ M and bond purchases b_ t . Using (31), (32) and the fact that m

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243

H ¼ Uðct Þe
Qt þ Lt ½rbt þ f ðkt Þ
JðIt Þ þ st
ct
εt ðat
bt
kt Þ þ Jt ½at
bt
kt
ct
It  þ Ft ½qðUðct ÞÞ
r þ Qt ½It
JðIt Þ where Q is the shadow price of capital. The optimality conditions for this problem are:

Hc ¼ 00U 0 ðct Þe
Qt ¼ Lt þ Jt
Ft q U 0 ðct Þ;

(36)

HI ¼ 00
Lt J 0 þ Jt ¼ Qt ½1
J 0 ;

(37)

L_t ¼ rLt
Ha 0 L_t ¼ Lt ðr þ εt Þ
Jt ;

(38)

J_ t ¼ rJt
Hb 0J_ t ¼ ðr þ 1ÞJt
ðr þ εt ÞLt ;

(39)

Q_ t ¼ rQt
Hk 0Q_ t ¼ rQt þ Jt
ðf 0 ðkÞ þ εt ÞLt ;

(40)

F_ t ¼ rFt
HQt 0F_ t ¼ rFt þ Uðct Þe
Qt ;

(41)

0

and the transversality conditions. Now we define

qt ¼ Qt eQt 0q_ t ¼ Q_ t eQt þ Qt eQt ½qðUðct Þ
r:

(42)

Substituting from (12)–(14) and (42) into the optimality conditions (36)–(41), we obtain: 0 U 0 ðct Þ ¼ lt þ jt
fq U 0 ðct Þ;

(43)

jt
lt J 0 ¼ qt ½1
J 0 ;

(44)

l_ t ¼ lt ðεt þ qÞ
jt ;

(45)

j_ t ¼ jt ð1 þ qÞ
ðr þ εt Þlt ;

(46)

f_ t ¼ qft þ Uðct Þ;

(47)

q_ t ¼ qt q þ jt
lt ðf 0 ðkt Þ þ εt Þ:

(48)

Using (44)–(46) in (48), and then noting that in a steady state j_ t ¼ q_ t ¼ J 0 ð0Þ ¼ 0, it follows that the steady state capital stock, k, is determined by

f 0 ðkÞ ¼ r:

(49)

With r fixed, the steady state capital stock is unaffected by an increase in the devaluation rate. The reason is that, with CIA constraints on both investment and bond purchases, changes in the rate of devaluation act as an increase in a tax on both capital and bond holdings by exactly the same amount.18 Our numerical evaluation of the model reveal that for the reasonable functional forms and parameter

18 This result for the open economy is in contrast to Stockman’s result for a closed economy. In Stockman’s model, when there was a CIA constraint on all transactions, an increase in the rate of growth of money reduced the steady state capital stock, because in his model the increase in the rate of steady state inflation rate taxed holdings of capital, reducing savings and thus investment.

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values a one percentage point increase in the inflation rate reduces the steady state real interest rate, as described by the steady state rate of time preference, by 0.88 percentage points.19 With the steady state capital stock k unaffected by the increase in the depreciation rate, we are left with the country’s net foreign assets b as the only predetermined variable upon which the saddlepath of the model will depend, giving rise, most likely, to a monotonic adjustment of the variables. This monotonicity, together with the fact that the steady state level of k is unaffected by the increase in ε, implies that k, and therefore output, will be left unaffected by the increase in ε. It can be readily shown that the adjustments of c, b and m will then be qualitatively the same as in the case with fixed output in Figs. 1–3.

3.2. The case with CIA on consumption, and bond purchases only When only consumption and bond purchases are subject to CIA constraints the model becomes slightly more complicated. Hence, this subsection will be divided into two parts. We first present the model, and then we discuss the effects of an increase in the inflation rate. 3.2.1. The model When only consumption and bond purchases are subject to CIA constraints a very important distinction should be made between assets the accumulation of which requires real balances versus assets the accumulation of which does not require real balances. In this sense, we should make a distinction between bonds and money on the one hand and capital on the other. Hence, now we revert to constraint (3), which states that assets the accumulation of which requires real balances are now only bonds and real money holdings. Also, we re-write the flow budget constraint of the representative agent accordingly:

a_ t ¼ f ðkt Þ
It þ rbt þ st
ct
εt mt :

(50)

Notice, the representative agent spends his savings f ðkt Þ þ rbt þ st
ct on investment It, real money _ t =Pt
εt mt , we obtain _ t =Pt and bond purchases b_ t . Using (3) and the fact that m _t ¼ M accumulation M Eq. (50). Comparing this explanation for (50) and the explanation for constraint (33) in footnote 14 reveals that they are equivalent, except that now we have made a fundamental distinction between bonds and real money holdings (assets the accumulation of which requires money), and capital (asset the accumulation of which does not require money). Notice, the right hand side of Eq. (50) is the total additional value of real money balances that the representative agent wants to carry with him/her; and the left hand side of the equation says he/she can carry this extra real money holdings either in the form of cash or in bonds. The problem of the representative agent is then to maximize (1) subject to (2), (3), (5), (7), (31) and (50). Using (3) to eliminate mt , the Hamiltonian for his problem can be written as:

H ¼ Uðct Þe
Qt þ Lt ½rbt þ f ðkt Þ
I þ st
ct
εt ðat
bt Þ þ Jt ½at
bt
ct  þ Ft ½qðUðct ÞÞ
r þ Qt ½It
JðIt Þ where L; J; and F are shadow prices associated with the respective state variables. The optimality conditions for this problem will be given by equations (43), (45)–(48) and

HI ¼ 00
lt þ qt ½1
J 0  ¼ 0:

(51)

19 These results are available upon request. To evaluate the model numerically, we use the functional forms and parameter values used in Appendix II.

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245

Again, to solve for the perfect foresight path first combine the government budget constraint (19) with the household’s flow budget constraint (50) to obtain

b_ t ¼ rbt þ f ðkt Þ
ct
It ;

(52)

according to which the current account balance b_ t is equal to national income rbt þ f ðkt Þ less absorption ct þ It . This completes the discussion of the model. We now turn to the effects of an increase in the inflation rate in this model. 3.2.2. The effects of an increase in the inflation rate Now consider the effects of a permanent unanticipated increase in ε at time 0. First consider the steady state effects. Differentiating equations (31), (35), (43), (45)–(48), (51) and (52) at the steady state, with l_ ¼ j_ ¼ f_ ¼ b_ ¼ k_ ¼ 0 we can derive the steady state effects of an increase in ε on c, k and b: 0

q U 0 dc dk ¼ 00 > 0; dε f dε

(53)

dc
q < 0; ¼ 0 dε ð1 þ 2q þ εÞq U 0

(54)

  db 1 dc dk <0: ¼
f0 dε r dε dε

(55)

The intuitions for these results are as follows. Because bond purchases are subject to CIA constraints while investment is not, an increase in the inflation rate reduces the real return on bonds relative to capital. Hence, there is an increase in investment, increasing the steady state capital stock (Eq. (53)). Also, the reduction in the real rate of return on bonds makes it less attractive to save. Hence, along the adjustment path savings is low, which reduces the steady state level of consumption (Eq. (54)). With savings low and investment high, along the adjustment path the country runs a current account deficit; and its steady state net foreign asset position deteriorates. Finally, consider the adjustment of real money holdings. From Eq. (3), the steady state real money holdings are equal to the steady state consumption level, which falls after the increase in the inflation rate. Also, substituting for b_ t from (3) into (52), we obtain the equation for real money holdings: mt ¼ rbt þ f ðkt Þ
It : Hence, on impact, with b and k predetermined and I increasing, m falls. The detailed discussion of the saddlepath, and the adjustment of the economy to its new steady state are presented in Appendix II. 4. Conclusions In this paper we have constructed the model of a small open economy assuming that there are CIA constraints on bond purchases. With this assumption inflation acts as a tax on bond purchases, reducing the real return on bonds, which is consistent with several empirical studies. This assumption was also central to development of the traditional IS-LM-BP model. We started with an economy with a fixed output, when both consumption and bond purchases were subject to CIA constraints. We showed that an increase in the rate of devaluation reduces the return on bonds, which reduces savings, increases consumption, and brings about a current account deterioration. Then we extended the model to incorporate production and capital. We showed that when investment as well as consumption and bond purchases are subject to CIA constraints, an increase in the rate of devaluation reduces the return on bonds and capital by the same amount, leaving the steady state capital stock unaffected. Then, the dynamics of consumption, real money holdings, and the country’s net foreign asset position are very similar to the case with a fixed output.

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On the other hand, when there are CIA constraints on consumption and bond purchases, and not on investment, the model exhibited Keynesian type adjustments with regards to investment and consumption. In that case an increase in the inflation rate reduces the return on bonds, but not on investment, which encourages capital accumulation. Further, the fall in the return on bonds discourages savings. Hence, on impact, there is an increase in consumption; and then consumption falls along the adjustment period until it reaches its lower steady state level. With the increase in investment and fall in savings, the country runs a current account deficit. Our results regarding the effects of inflation on the capital stock are the reverse of Stockman’s results for a closed economy without bonds. In Stockman’s model an increase in the inflation rate would lead to a fall in steady state capital if there were CIA constraints on consumption as well as on investment, whereas the steady state capital would be unaffected if there were CIA constraints on consumption only. Acknowledgements We would like to thank a co-editor and the referees for their very constructive comments. All the remaining errors are ours alone. Appendix I In this Appendix we report the exact expressions for the elements of the eigenvectors corresponding to the negative eigenvalues of the coefficient matrices in (22) which were denoted by a1 ; a2 and a3 :

a1 ¼

ðA44
xÞ½A23 A12
A13 ðA22
xÞ ; A21 ðA12 A43
A13 A42 Þ þ A23 fðA11
xÞA42
A12 A41 g þ ðA22
xÞfðA13 A41
A43 ðA11
xÞg

a2 ¼

ðA44
xÞ½A21 A13
A23 ðA11
xÞ ; A21 ðA12 A43
A13 A42 Þ þ A23 fðA11
xÞA42
A12 A41 g þ ðA22
xÞfðA13 A41
A43 ðA11
xÞg

a3 ¼

ðA44
xÞ½ðA11
xÞðA22
xÞ
A12 A21  : A21 ðA12 A43
A13 A42 Þ þ A23 fðA11
xÞA42
A12 A41 g þ ðA22
xÞfðA13 A41
A43 ðA11
xÞg

Appendix II The perfect foresight path corresponding to the model with CIA constraints on consumption and bond purchases, but not in investment, is the solution to the dynamic system described by equations (31), (35), (43), (45)–(48), (51) and (52). First, linearize Eqs. (43) and (51) in order to obtain

  0 ðc
cÞ ¼ G1 l
l þ ðj
jÞ
q U 0 ðf
fÞ ;

(56)

ðI
IÞ ¼ G2 ½ðq
qÞ
ðl
lÞ;

(57)

00

0

where G1 ¼ 1=½U 00 þ fðU 02 q þ U 00 q Þ and G2 ¼ 1=½lJ 00 ðIÞ. Now linearize equations (31), (45)–(48) and (52) substituting out ct
c and It
I from them using (56) and (57) to obtain the following differential equation system:

2_3 2 B11 l 6 j_ 7 6 B21 6 7 6 6 f_ 7 6 6 7 ¼ 6 B31 6 q_ 7 6 B41 6 7 6 4_5 4 B51 k B61 b_

B12 B22 B32 B42 0 B62

B13 B23 B33 B43 0 B63

0 0 0 B44 B54 B64

0 0 0 B45 0 B65

3 32 0 l
l 7 6 0 7 76 j
j 7 6f
f7 0 7 7; 76 7 6 0 7 76 q
q 7 0 54 k
k 5 B66 b
b

(58)

A. Mansoorian, M. Mohsin / Journal of International Money and Finance 32 (2013) 234–250 0

0

247 02

where the coefficients Bij are as follows: B11 ¼ q þ ε þ G1 lq U 0 , B12 ¼ G1 lq U 0
1, B13 ¼
G1 lq U 02 , 0 0 02 0 B21 ¼ G1 jq U 0
r
ε, B22 ¼ G1 jq U 0 þ 1 þ q, B23 ¼
G1 jq U 02 , B31 ¼ B32 ¼ G1 ðfq U 0 þ U 0 Þ, 0 0 0 0 02 B33 ¼ q
G1 q U 0 ðfq U 0 þ U 0 Þ, B41 ¼ G1 qq U 0
f 0 ðkÞ, B42 ¼ G1 qq U 0 ; B43 ¼
G1 qq U 02 ; B44 ¼ q, 0 B45 ¼
lf 00 ðkÞ, B51 ¼
G2 ; B54 ¼ G2 , B61 ¼
G1 þ G2 ; B62 ¼
G1 , B63 ¼ G1 q U 0 ; B64 ¼
G2 , B65 ¼ f 0 ðkÞ and B66 ¼ r: The system of differential equations (58) has two predetermined variable, k and b. Hence, for saddlepath stability the coefficient matrix should have two negative and four positive eigenvalues. As the coefficient matrix in (58) is not block recursive, closed form solutions for the solution of the saddlepath are not available. Hence, from this point on, the model should be evaluated numerically, with specific functional forms for preferences and the production function. Following the real business cycle literature, we assume that UðcÞ ¼ ½ðcÞ1
s
1=½1
s, qðUðcÞÞ ¼ blnð1 þ cÞ, and FðkÞ ¼ ka . Following Cooley and Prescott (1995, pp. 20–22) and Mendoza (1991, p. 804) we set s ¼ 2; b ¼ 0:11; a ¼ 0:32, b ¼ 0:04 and r ¼ 0:04. With these functional forms and parameter values one can show that the model exhibits saddlepath stability. We can, thus, derive the numerical solution for the saddlepath, and work out the effects of an increase in the rate of devaluation of the domestic currency. Assume that initially the inflation rate is 4% and the central bank increases it to 6%. We calculate the levels of all the variables at the steady states equilibria corresponding to both inflation rates. The steady state levels of consumption and real money holdings fall by 2% while the steady state level of capital increases by 2.6%. At the same time, the steady state net foreign assets per GDP fall from
21.3563 to
21.4616. The dynamic adjustments of capital, consumption, net foreign assets and real money holdings are presented in Figs. 4–7, respectively. As explained in the main text, the increase in the inflation rate encourages capital accumulation (Fig. 4), because it reduces the real rate of return on bonds, while leaving the return on capital unaffected. The fall in the real rate of return on bonds also discourages savings. Hence, on impact, there is an increase in consumption; and then over time consumption falls, until it reaches its new lower steady state level (Fig. 5). With the fall in savings and increase in investment the country runs a current account deficit, and its net foreign asset position deteriorates over time (Fig. 6). Finally, with regards to the adjustment of real money holdings (Fig. 7) first note, as shown in the main text, mt ¼ rbt þ f ðkt Þ
It . The sharp increase in investment after the increase in the inflation rate results in a sharp fall in the real money holdings. Over time, as capital adjusts towards its higher steady state level, investment falls and output increases, while the deterioration in the country’s net foreign asset position reduces the interest on bond holdings. With rbt falling over time on the one hand, and f ðkt Þ
It increasing over time on the other, the real money holdings may adjust non-monotonically, as shown in Fig. 7.

Fig. 4. The adjustment of k.

Fig. 5. The adjustment of c.

Fig. 6. The adjustment of b.

Fig. 7. The adjustment of m.

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249

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