imported input price and the current account in an optimizing model without capital mobility

Journal of Economic Dynamics and Control 15 (1991) 91-101. North-Holland

Imported input price and the current account in an optimizing model without capital mobility

Partha Sen* London School of Economics, London WC2A 2AE, UK Unicersity of Illinois, Urbana-Champaign, IL, USA

This paper analyses the effect of an oil-price increase in an economy which cannot borrow in the international capital markets. Current-account surpluses and deficits are matched by reserve changes at the central bank. It is shown in an optimizing setup that an unanticipated oil-price increase worsens the current account and lowers the real interest rate contrary to models which assume perfect capital markets.

1. Introduction

The world saw two dramatic increases in the price of oil in the 1970s and an almost equally dramatic collapse in the 1980s. This led to a flurry of activity among the open-economy macroeconomists to model the effect of such price changes. The literature, which is too large to catalogue here, certainly enhanced our knowledge of such supply-side shocks [see, e.g., Buiter (19781, Findlay and Rodriquez (1977), Obstfeld (19801, Fender (198511.

The earlier models were all ‘ad hoc’ in the sense that the behavioural equations did not have explicit microeconomic underpinnings. More recently there has been a burgeoning literature which is based on optimizing models [see, e.g., Bruno and Sachs (1985), Sachs (19821, Svensson (19841, Matsuyama (19871, Nielson (1988)].

This literature based on optimizing models assumes the existence of perfect capital markets and typically abstract from monetary considerations. This makes the analysis of the developing-countries experience very difficult in that a vast majority of them do not have access to the world capital markets. It is also true that most of these countries (almost one hundred of

*I am grateful to an anonymous referee for comments.

0165-1889/91/$03.500 1991-Elsevier Science Publishers B.V. (North-Holland)

J.E.D.C.-D

92 P. Sen, Imported input price and the current account

them) to this day operate fixed exchange-rate systems. This paper seeks to model the behaviour of an oil-importing country based on an optimizing model operating a fixed exchange-rate system and without capital mobility.

The perfect capital market models [e.g., Svensson (19841, Sachs (1982), and Nielsen (1988)] predict that a permanent increase in the price of oil would cause a current-account surplus under reasonable assumptions [see Matsuyama (19871, where this may not happen]. Now the experience of the (non-oil) less developed countries was exactly the opposite [for a survey see Khan and Knight (198311. Also, for a small open economy the rate of interest in this model is given exogenously (and assumed equal to the rate of time preference), so an analysis of endogenous domestic interest rate change is ruled out by assumption [again Matsuyama (19871, who models the economy as being populated by agents with finite lives, is an exception]. There is some evidence, at least for the first oil-price increase, that the real interest rate fell in most countries. Our model predicts both a current-account deficit and a fall in the interest rate following an unanticipated increase in the price of oil.

The paper is organized as follows. In section 2 we set out the model and its solution. The next section looks at the effect of the oil-price increase. Finally, section 4 contains the conclusions. All the mathematical derivations are to be found in an appendix.

2. The model

The oil-importing small economy is assumed to produce one good whose price is fixed in the world market. It also takes the price of oil as given. The economy cannot borrow or lend in the international capital markets. Any current-account imbalance is financed by equal and opposite changes in the official settlements balance because the monetary authorities of this economy maintain a fixed exchange rate. The model incorporates the two features of a large number of small oil-importing countries alluded to in the Introduction viz., borrowing constraints and fixed exchange rates.

There is a representative household and a representative firm. All agents are price takers and have perfect foresight.

2.1. The household

The representative household maximizes its lifetime utility which depends .on its consumption of the final good and its holding of money balances. The latter is controversial, but the results here are not sensitive to this assumption [e.g., if it is replaced by a cash-in-advance constraint which is nothing but a unitary velocity case of the more general case; see Calvo (1986) for a

P. Sen, Imported input price and the current account 93

discussion],

maxjm[U(C) + V(m)]exp( -bt) dt, 0

(I)

where C is the consumption of the final good, m is the level of real money balances, b is the discount rate. We assume additive separability of U and V, and wherever no ambiguity arises we will supress the time subscript for C and m. The instantaneous utility functions U and V are strictly concave in their respective arguments. Labor is inelastically supplied (and normalized equal to one).

The households flow budget constraint is given by

h=w+ra-c-im, (2)

where

a=m+K (3)

is the stock of wealth, w is the real wage rate, K is the capital stock (see below), and r is the real interest rate (a dot over a variable denotes its time derivative). The opportunity cost of holding real money balances should be the nominal interest rate but we assume that the world rate of inflation is zero (which by the purchasing power parity assumption translates into a zero domestic inflation rate under a fixed exchange-rate system).

The first-order conditions for this problem are

U(C) =A,

V(m) =Ar.

(4)

(5)

The co-state variable A evolves according to

i=h(b-r). (6)

In addition, eqs. (2) and (3) and the following transversality conditions need to be satisfied,

limexp( -6t)A(t) 20, t-m

(7)

limexp(-Gt)A(t)a(t) =O. t-m

(8)

Note that these two together with (2) gives us the usual intertemporal budget constraint.


2.2. The firm

The representative firm maximizes its discounted sum of profits by choosing its inputs oil (N), labor (L), and the stock of capital (KI to produce the (one) final good. Ideally, we would like to introduce costs of adjustment and make investment the choice variable [as, e.g., in Sen and Turnovsky (19891, Matsuyama (19881, Murphy (198811, but in the present case that renders the model analytically intractable. So we follow Obstfeld (1989) and assume that capital goods can be bought and sold instantaneously in the world market (for foreign exchange). Note that this implies that the stock of foreign exchange reserves can change discontinuously. It is, however, not possible for households or firms to borrow in the world capital markets.

The firm’s intertemporal maximization thus becomes one of static opti- mization, i.e., in each period max F(K, N, L) - K - qN - wL, choosing K, L, and N, where F is a constant returns-to-scale production function with positive but diminishing marginal productivity for L, K, and N, and q is the price of the imported input (oil),

F,(K, N, L) = y, (9)

F,(K,N,L) =q, (10)

FL( K, N, L) = w. (11)

The conditions (9), (lo), and (11) equate the marginal productivity of each factor with its price.

In the absence of overwhelming empirical support for an alternative we follow other writers [e.g., Svensson (1984)] and assume all factors of production are cooperant (or Edgeworth-complementary), i.e., all cross-partial derivatives of F are positive. In the earlier literature there was a lot of discussion about the complementarity versus substitutability (in the Edge- worth sense) of oil with other factors of production [again see Svensson (1984) for a discussion].

2.3. The government

The government or the monetary authorities maintain a fixed exchange-rate regime so that the current-account surplus, which is also the balance-of-pay- ments surplus, is the increase in the money supply (except that following the arrival of new information the money stock may jump through a discrete sale or purchase of capital abroad),

ti=F-qN-C-k. (12)


We restrict our attention to the case where the monetary authorities have enough foreign exchange reserves to maintain a hxed exchange-rate system. An interesting extension of the analysis presented here would be an expected future breakdown of a fixed-rate system due to an expected increase in the price of oil.

2.4. Macroeconomic equilibrium

We collect here all the optimality conditions imposing on them the requirement of macroeconomic equilibrium (specifically L = l),

u’(C) =A,

V(a -K) =hF,(K,N),

F,(KTN) =q,

(13)

(14)

(15)

h=hb-V(a-K), (16)

k=F(K,N)-qN-C. (17)

It is possible to solve eqs. (12) to (15) for C, K, and N in terms of the two dynamic variables (i.e., A and a) and the exogenous variable q (details are given in the appendix),

C=C(h), C’ < 0, (18a)

K=K(A,a,q), K,>O, K,>O, K,<O, (I@)

N=N(A,a,q), N,>O, N,>O, N,<O. (18~)

Note that these are all partial effects and most of the derivatives work through the cooperancy assumption.

A rise in A lowers consumption and lowers the demand for money and hence given real wealth increases K which via cooperancy increases N. An increase in a (other things constant) lowers the marginal utility of money and given a requires an increase in K to lower the interest rate. Similarly a rise in q, other things equal, reduces the demand for N (own-price effect is always negative) which through Edgeworth-complementarity reduces K.

2.5. Dynamics

Substituting the values of C, K, and N from (18) in (16) and (17) we obtain the two dynamic equations of the system. Linearizing these around the steady


Fig. I

state we find that ci = 0 locus slopes downwards in a-A space. h = 0 could slope either upwards or downwards and the unstable arm is sloping upwards. In either case the equilibrium is a saddlepoint with the saddlepath sloping downwards.

2.6. The long-run equilibrium

The long-run equilibrium is obtained by setting the time derivatives of A and a equal to zero in eqs. (12)-(17). These can be solved for the values of -- - - C, A, N, K, and a, where an overbar denotes a steady-state value. Using eq. (3) gives us the value of FE.

U(C) =A, --

l”(Z-K) =hF,(K,N),

-- F,(K,N) =q,

V’(a-K) =Ab,

-- F(K,N) -qm=c.

(194

( 19b)

(19c)

( 194

3. Effect of an increase in the price N

3.1. An unanticipated permanent increase

Suppose now starting from a position of long-run equilibrium the price of oil increases unexpectedly and this increase is expected to be permanent. In


Fig. 2

reality, of course, there is always doubt as to whether a change in price is permanent or temporary, but for expositional purposes the route we follow is the one chosen by many in the literature.

Let us first examine the long-run multipliers of the price change. These are reported below (see appendix for details):

dc/dq < 0, dh/dq > 0,

dp/dq < 0, dK/dq < 0, diE/dq < 0, (20)

dZ/dq < 0.

The own-price effect is always negative, so the use of the imported input falls. Across steady states value-added falls and hence consumption falls. The marginal utility of consumption, however, rises and this implies that the marginal utility of money must rise, which causes real balances to decline [remember L”(E)/h = bl. The new steady-state stock of capital is lower. To see this, it must be recalled that across steady states the marginal product of capital must be equal to the discount rate. The fall in N (through cooperancy) lowers the marginal product of capital. Finally, we note that since the stock of capital and money are lower, the new level of wealth (5) is definitely lower.

As shown in fig. 2, the new long-run equilibrium (E,) is to the north-west of the original one (E,). It is not possible a priori to tell what happens to the new stable arm. As drawn in the diagram, it lies above the old one although one cannot rule out the possibility of it lying below the old one.


a

Fig. 3

The impact effect of the increase in q is thus unclear since it depends crucially on which way A goes. But one important empirical observation is predicted by the model - that an increase in q leads the oil-importing economy to run current-account deficits. It can be shown by solving for m in terms of A and a,

m =m(h,a), m,<O, l>m,>O.

Therefore L+Z = m,/i + m,ci < 0, since ,i > 0 and h < 0 along the adjustment path.

Another observed behaviour of actual economies viz. the decline in the real rate of interest [see Svensson (198411 also follows since along the adjustment path

/i/h=b-F,>O.

So FK is below its long-run value of b.

3.2. A temporary increase in q

Suppose we have a temporary increase in q which was previously unanticipated, i.e., at time 0, q rises to a new higher value and it is known with certainty that at time T it will go back to its original value. In this case, as


shown in fig. 3,’ while the oil-price increase is in effect, the qualitative behaviour of the economy is similar to the increase in price which is permanent, e.g., the economy runs down its wealth and is running trade deficits. Also on impact FK falls. When the oil-price increase is reversed, deficits and wealth decumulation are replaced by surpluses and saving and the real rate of interest rises to a level above its long-run value.2

Notice the sharp contrast in the literature in the case of perfect world capital markets between a permanent and a temporary rise in price of oil is missing here.

An anticipated future increase in 4 has ambiguous effects on the current account which depends crucially on whether the new stable arm is above or below the old one.

We see that an increase in the price of oil which is permanent causes a current-account deficit which is in contrast to the result obtained by Sachs (1982) Nielsen (1988), and Svensson (1984). In our model, the steady-state stock of capital falls like in Butlin (1985) who also assumes a fixed labour supply. For a temporary increase in the price of oil the current accounts are similar to those obtained by Svensson (1984). For an immediately imple- mented increase in the price of oil the real rate of interest falls (assuming the economy starts from a long-run equilibrium).

4. Conclusions

In this paper we have modelled a small open economy without access to the world capital market. Current-account surpluses or deficits were financed by changes in foreign exchange reserves by maintaining a fixed exchange rate.

The model predicts that an oil-price increase which was previously unanticipated (irrespective of whether it is permanent or temporary) lowers the real interest rate on impact and leads to current-account deficits along the adjustment path. Anticipated future increases may lead to surpluses or deficits.

The model needs to be extended in many directions to make it more realistic. We have assumed zero borrowing opportunities, but in actual fact some opportunities do exist (otherwise there would be no LDC debt problem!). The capital-accumulation equation should include costs of adjustment as in Murphy (1988) and the issue of cross-partial derivatives (in production) needs to be examined more carefully. The neo-classical labour market structure should be replaced by a richer one.

‘The economy must now follow an unstable trajectory which hits the old saddlepath SOSO leading to E, exactly at the time the oil-price increase is reversed.

‘Note that this is true irrespective of the position of the new saddlepath (i.e., in fig. 3 it does not matter whether the new saddlepath is S,S, leading to E, or S,S, leading to E,).


Appendix

The derivatives in eq. (18) are as follows:

CA = I/V,

K, = (-W,&/A,

K, = V”FNN/A,

K, - -hF&A,

N, = bFdA ,

N2 = ( -V”FKN)/A,

N3=(V”+AFKK)/A,

where

A = (AF,& - hFKKFNN - V”FNN) < 0.

The dynamics of the system can be expressed as (by linearizing around the steady state)

Now

a,, = b + V’K, 2 0,

a,* = I”‘( 1 - K,) > 0,

azl = bK, - C, > 0,

a ,,=bK,>O.

The determinant of A = - C,(l - KJV” < 0, so the eigenvalues are of opposite signs. The long-run comparative statics results are given below:

dE/dq = - FKN/D < 0,

diii/dq = FJD < 0,


where D = FKKFNN - F& = 0,

dc/dq = FK d@iq -N < 0,

dh/dq = (l/U”) dC/dq > 0,

dFi/dq = (b/V”) dh/dq < 0,

dii/dq = diZ/dq + di?/dq < 0.

References

Bruno, M., and J. Sachs, 1985, The economics of worldwide stagflation (Blackwell, Oxford). Buiter, W.H., 1978, Short-run and long-run effects of external disturbances under a floating

exchange rate, Economica 45, 251-272. Butlin, M.W., 1985, Imported inputs in an optimising model of a small open economy, Journal of

International Economics 18, 101-131. Calvo, G., 1986, Temporary policies: Predetermined exchange rates, Journal of Political Econ-

omy 94, 1319-1329. Fender, J., 1985, Oil in a dynamic two good model, Oxford Economic Papers 37, 249-263. Findlay, R. and C.A. Rodriquez, 1977, Intermediate imports and macroeconomic policy under

flexible exchange rates, Canadian Journal of Economics 10, 208-217. Khan, M.S. and M.D. Knight, 1983, Determinants of current account balances of non-oil

developing countries in the 1970’s: An empirical analysis, International Monetary Fund Staff Papers 30, 819-842.

Matsuyama, K., 1987, Current account dynamics in a finite horizon model, Journal of Interna- tional Economics 23, 299-313.

Murphy, R.G., 1988, Sector specific capital and real exchange rate dynamics, Journal of Economic Dynamics and Control 12, 7-12.

Nielson, S.B., 1988, Terms of trade, the current account and imports of intermediate and investment goods, Mimeo. (University of Copenhagen).

Obstfeld, M., 1980, Intermediate imports, the terms of trade and the dynamics of the exchange rate and the current account, Journal of International Economics 10, 461-480.

Obstfeld, M., 1989, Fiscal deficits and relative prices in a growing world economy, Journal of Monetary Economics 23, 461-484.

Sachs, J.D., 1982, Energy and growth under flexible exchange rates: A simulation study, in: J. Bhandari and B. Putnam, eds., The international transmission of international disturbances under flexible exchange rates (M.I.T. Press, Cambridge, MA).

Sen, P. and S.J. Turnovsky, 1989, Deterioration of the terms of trade and capital accumulation: A re-examination of the Laursen Metzler effect, Journal of International Economics 26, 227-250.

Svensson, L.E.O., 1984, Oil prices, welfare, and the trade balance, Quarterly Journal of Economics 99, 649-672.

imported input price and the current account in an optimizing model without capital mobility

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