tre relationsrip between investment and demand witr ...

GIUSEPPE TRAVAGLINI

TRE RELATIONSRIP BETWEEN INVESTMENT AND DEMAND

WITR UNCERTAINTY AND IRREVERSIBLE CAPITAL

Estratto da: GIORNALE DEGLI ECONOMISTI E ANNALI DI ECONOMIA

Ottobre-Dicembre 1996

THE RELATIONSHIP BETWEEN INVESTMENT

ANO OEMANO WITH UNCERTAINTY

ANO IRREVERSIBLE CAPITAL*

INTRODUCTION

The existing literature on economie fluctuations has already discussed most of the questions involved in the relationship between investment and uncertainty. Recently a novel approach has been developed considering the investment opportunity as a financial option, which gives the holder the right to buy the asset by a future date for a certain price. Usually, this option value literature deals with the effects of uncertainty caused by price signals, such as the variability of price of output or the volatility of costs of inputs (Mc­ Donald-Siegel, 1986; Pindyck, 1991; Ingersoll-Ross, 1992; Dixit, 1992; Berto­ la-Caballero, 1994. For a complete survey Dixit-Pindyck, 1993; Travaglini, 1996). The main point is that in presence of uncertainty and irreversible choices the firrns can have incentive to deIay the investment, waiting for new information about the expected profitability, even if they are risk neutral. In this approach, however, the role of demand has received scant attention. Actually, there are a small number of researches that deal explicitly with these issues. The main two articles are Cuckierman (1980) and Bemanke (1983). From their analysis emerges an explanation for aggregate investment spending, which suggests that variations in uncertainty pIays a dominant causal role. By the way, one of the major limits of these models is that they are not based on the current financial valuation techniques or, altemativeIy, they do not make use of stochastic dynamic programming as in the modem literature on irreversible investment.

. I am most gratcful to ENRICO SALTARI for many insightful conversatìons. The author thanks GIORGIO RODANO and an anonimous referee for the useful comments and suggestìons. Financìal support from MINISTERO PER L'UNIVERSITÀ E LA RICERCA SCIENTIFICA E TECNOLOGICA (fund 40% 1995) is gratcfully acknowledged. Thc usual dis­ claimcrs apply.

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GIUSEPPE TRAVAGLI~,a

Starting from these views, our aim is to provide a more up-to-date for­ mulation of that intuition, dealing with the group of various "accelerator" modeis of fixed investment. More precisely, this paper reinterprets the flexi­ ble accelerator in a model with irreversible investment and continous-time uncertainty. The first innovative aspect, respect to the approach of option value theory, is that uncertainty on investment is caused by quantity signaIs, that is by stochastic fluctuations in demando It wilI be shown that if the in­ vestment is irreversible, the firm chooses a smaller capitaI stock when the ir­ reversibility coinstraint is not binding. Moreover, this level is an inverse function of the total cost determined by the option value of investment. The second presumption, is that the accelerator mechanism can be extended to take account of uncertainty effects on investment decisions. This view is il­ lustrated, deriving a flexible accelerator coefficient which is a non increasing function of the variance of demando The way the firms behave over time is as follows. Since demand curve fluctuates randomly, the firm formulates at the current time an investment programme in such a way to avoid the possibility of excess capacity. This implies that the desired capitaI stock can be smaller than the case of reversibility. Unlike previous contributions (Nickell, 1974; Nickell, 1978), to bring out this result we do not assume to have a fixed coefficients technology. Produc­ tion function cmùd be perfectly flexible but uncertainty and failures of sec­ ondary markets can coinstrain the allocations of productive factors. Further, adjustment costs are not incorporated into the optimal planning process, al­ thought irreversible assumption can be viewed as an infinite adjustment cost of decumulation. Our claim is that expectations and uncertainty influence negativeIy investment plans. This is tme in any option value model, but not in models with irreversibility assumption and uncertain demand 1 . The con­ sideration of these features offers new insights to understand the interreIa­ tion between transitory shocks to the aggregate demand and the permanent change to the capitaI stock. Moreover, it may help to understand why some­ times in presence of increasing aggregate demand the firms tend not to ad­ just their own productive capacity. The paper is organized as follows. The next section briefly sUlTeys the literature on the accelerator principle. The third one introduces and solves our model. The results are discussed in the Iast two sections, in which we I To say, Nìckell concludes that the higher will be the demand growth rate. «the shorter will be the period during which investment is zero, given a constant period of decline» (p. Il, 1974). This result is derived under the assumption that at a point in time demand suddenly starts to drop, «after which the original upward trend is ex­ pected to continue» (p. 13, 1974).

THE RELATIONSHIP BETWEEN INVESTMENT Ac'lD DEMAc\lD WlTH UNCERTAINTY

551

further develop some considerations about the linkages among aggregate de­ mand, investment spending and the rate of growth.

1.

THE ACCELERATOR PRINCIPLE

In its crudest form the acceleration principle states that an increase in future level of aggregate demand (D) will induce an increase in current net aggregate investment (I11). In this case we can write (1)

where v is the desired capital-output rate. In this macroeconomic version, the coefficient v is a fLxed value: the fluctuations of the aggregate demand imply proportional and instantaneous variations of capital stock. However, this cor­ respondence between investment and demand appears too restrictive; in­ deed, it implies that firms should use uniquely quantities and not prices as signals in making their investment decisions. Moreover, it implies that the expectations are always satisfied ex post, and that consequently the choices of firms are independent on uncertainty about the future movements of ag­ gregate demando Finally, the expression (1) suggests that the expansion path of the investment is always reversible, that is a default of demand can always be followed by a proportional decumulation of capital stock. An in all, these assumptions tend to make this formulation of the accelerator principle too vague to be usefully employed in the study of the macroeconomic fluctua­ tions which links investment to aggregate demando Some of these limits can be overcame if the accelerator coefficient v is regarded as a variable rather than a f:ixed parameter. The neo-classical ap­ proach of investment, proposed by Jorgenson (1963) can be employed to this purpose. In this setting it is assumed that the firm maximizes its net present value on an infinite horizon, and that the rate of capital accumulation is de­ termined by the investment gross function J(t) = (dKldt) + oK(t), where (dKJdt) .k(t) is the net investment and o is the depreciation rate of the capi­ tal stock in each periodo Figure 1 shows that if firm needs to expand its out­ put, say, of a quantity .1Y = Yi - Y o' it must enhance proportionally the bulk of the inputs employed in production (capitaI and labour).

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GIUSEPPE TRAVAGLINI

Yo

Fig. 1

Clearly, since the curve that links isoquants is aline (isocline), the corre­ sponding technology is a production function homogeneous of degree one. As a consequence, the slope of isoquants is independent from the scale of production, depending exclusively on the optimai proportion of inputs. Since the expectations of the firm are seIf-fulfilled, the future level of isoquant that meets demand is known with certainty at the current time. In these circum­ stances, once calcuiated the optimal amount Xi of an input required to pro­ duce a given quantity Y, the physical amount of other inputs can be obtained using the optimal proportion p X/Y. From this property we can derive a sim­ pIe form of accelerator mechanism. To say, the rate of growth of capitaI fac­ tor can be written as L1K = p.dY. This equation has the same meaning of the expression (1); the main difference is that the former is a macroeconomic re­ Iation while the Iatter is achieved through a micro founded procedure. In the case of the Jorgenson's dynamic approach the optimal solution of the neo­ classical model is found when the marginal product of capitaI F K is equated to the real user cost (cip). F = q (8 + r) = !!..­ K p p

(2)

where q is the cost of capitaI, r the interest rate and p the price of output. Now, take the Cobb-Dougias production function Y Ka and rewrite the expression (2) as K* =

ap

q (8 + r)

Y = K D (y, r) = p(r) Y

(3)

THE RELATIONSHIP BETWEEN INVESTMENT AND DEMAND WITH UNCERTAINTY

553

From the relation of gross investment we can obtain the level of net in­ vestment In = I (t) - 8K (t) :K, and substituting in (3)

r

ap Y(t) = v(r) Y(t) q (8 + r)

(4)

The equation (4) is an alternative form of the accelerator principle. As concerns the macroeconomic specification, the coefficient v is no longer a fixed value, but it is a function of prices that enter into its formulation. For this reason the expression (4) is called flexible accelerator mechanism. It should be noted that a more general form of the flexible accelerator can be derived assuming that the firm incurres an adjustment co~t C whose magnitude varies positively with the speed of expansion In == K (Eisner Strotz, 1963; Lucas, 1967a, 1967b; Gould, 1968). Traditionally, the C func­ tion is supposed to be strictly convex (C' (K) >, C' (tè) > O, O per K > O). Un­ der this assumption, if the user cost is defined as [q (8 + r) - C' (K)] then the present value is maximized by the policy which maximizes pF [K (t), L (t)]

wL (t) - [q (8 + r) - C' (K)] K (t)

in each periodo Thus the investment rate is an inverse function of the rental price, and a decreasing function of the marginaI adjustment costs which vary with the net investment. It is apparent that the current capital stock is now related to the level of previous and future periods because of adjustment costs. Further, this model implies that the value of the accelerator coefficient v(.) varies over time with the dynamics of net investment. Usually, the theoretical discussion on the capability of the accelerator principle to explain the relationship among demand, production and invest­ ment do not go further. Nothing is said about the prospects that the expecta­ tions of firrns may be unfulfilled ex posto Clearly, if this is the case the flow of net investment would be caused not only by the expected level of demand, but also by the risk that that path of demand should be differenL Conse­ quently, the net investment becomes a function of past and future level of the expected demando and each point on the expansion path would be, now, af­ fected byall the influences previously assumed away. Starting from this point, in the rest of the paper we wìll try to amend the traditional versions of accelerator, in order to show how uncertainty can alterate the pattern of net investment forecasted by the alternative forrns of accelerator principle. This result will allow to understand why the invest­ ment curve appears so unstable, and why sometimes thedecisions of firrns and investors are so sensitive to the expected future pattern of the economic activity.

l j I

554

2.

GIUSEPPE TRAVAGLINI

INSTABILITY OF AGGREGATE DEMAND AND THE EFFECTS OF UNCERTAINTY, WITH IRREVERSIBLE CAPITAL

In this section we discuss the role of uncertainty on investment deci­ sions, getting into perspective the fluctuations of real variables, namely the volatility of demand curve. To facilitate the investigation we assume that markets are characterized by priee rigidity. This assumption is formulated for four different reasons. First of all, the "stylised facts" often show that shocks whieh affect economie activities may modify the Ievel of aggregate demand and production, rather than the Ievel of prices. Rence, it seems that demand shocks play an important role in investment decisions. As second point, this assumption makes it possible to rescue the originaI keynesian in­ tuition that investment responds to quantity signais. Further, this aspect is remained up till now unexplored by the option value theory. Finally, the as­ sumption of stieky prices permits to mark out the effects of uncertainty caused by fluctuations in aggregate demando The generaI strategy is to take the vector of prices as given, so there is no reason to expect any particular covarìation between prices and investment in response to demand shocks. I consider a representative firm with a production function given by Y t = G [Kt , L t ], with GL > O, GLL < O e GK > O, GKK < O, where L is the numbers of workers employed, and K the stock of capital in each periodo The main as­ sumption is that capitaI is irreversible. The assumption of irreversibility is a non-negativity constraint on the rate of decumulation of capital. FormallYI it may be written as dK ~ O. The producer acts as a monopolist. Re chooses the priee while the leve! of output per period is determined by the demand func­ tion he faces. The interest is in the effects of fluctuations of demand on in­ vestment decision in that economy at the current time. Let's assume that de­ mand grows at an average rate a, but future values are lognormally distrib­ uted with a variance s2 which grows linearly with the time horizon. With these assumptions the stochastic dynamies of demand can be described by the geometric Brownian motion (5)

dD = aDdt + sDdz

where dD describes the dynamies of demand, while dz is the increment of a Wiener process dz

=

,

e (t)Jd~ with e (t) - N (0,1). It implies that dz - N (O, dt).

Every investment decision is taken at the beginning of each period, thus the outcome is unknown since the future value of demand is always uncertain. If capitaI is irreversibIe, the uncertainty about the future pattern of demand can affect the expansion path of real investment. Indeed, if expectations of firm on future demand are unfulfilled the new capitaI equipment can not be

THE RELATIONSHIP BETWEEN INVESTMENT AND DEMAND WITH UNCERTAINTY

555

"undone", and the producer bears the entire cost of investment in any states of world. This view is very different from the traditional outlook of accelerator principle, discussed previously. There, it has been assumed that a proportional and immediate co-movement between demand and investment exists. But, what happens when the choice of inputs is influenced by unexpected fluctua­ tions on demand curve? In other words, in presence of uncertainty and irre­ versibility, is the intertemporal cost minimization modified by the fact that the position of the isoquant changes continuously and erratically over time?

2.1 The Optimal Amount of Inputs: the Case of Certainty and Reversibility We first derive the demand for capital and Iabour in the case of certainty and reversibility. This optimal solution will be referred to as benchmark. To discuss the issues it is hepful to describe the process of investment decisions in terrn of intertemporal cost-minimizing choice. This is an unusual view to analyse the intertemporal choices of firrns, but it is useful to deal with the in­ vestment strategies of producers, and to derive a modified accelerator mech­ anism. According to the Jorgenson model, we assume that the production func­ tion is homogeneous of degree one, and takes form Yt = G [K1, L t ]. K is capi­ taI and L is Iabour. The gross investment in each period is given by I (t) = (dK/dt) + oK (t), and the labour employed on production process is subject to the dynamic accumulation constraint dL/dt Le Note that this equation de­ scribes the cumulative labour tumover processo Indeed, dL t > O when firms hire, and dL t < Owhen the firrn fires. Further, let's assume that both the cost q. of capital, and the cost w of labour are fixed, and the price of output is P = 1 For simmetry with capitaI, we assume that remuneration of labour is treated on a stock basis. Indeed, it could be argued that at the current time the firrn must evaluate the present value of the investment programme, taking into account the global cost of the productive factors. After this, both the terrns r and (r + o) convert the stock prices to the flow prices. Hence rw measures the cash outlay per unit of labour. The minimization problem is 00

Ct = 111in L"K,

f[wit

o

+q

CKt

+ oK)] exp (-rt) dt

(6)

with the constraints Ko and Lo given. We characterize the solution using the calculus of variations with constraints. More precisely, we form a La­ grangian integrand function J as follows

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GIUSEPPE TRAVAGLINI

(7)

where is the Lagrange multiplier. From (7) we get the solution =>

GL

r\IV

GK

q(8+r)

(8)

that is, in each period, the technical rate of substitution must be equaI to the economic rate of substitution. Since the prices are assumed to be fixed, the marginai productivity of labour and capitai are constant, whereby the dy­ namic nature of the problem appears to be effectively statico Thus, this result is coherent with the Jorgenson solution: the optimal value of firm is reaIized when the marginaI revenue meets the marginaI cost.

2.2 The Cost-Minimizing Process with Uncertainty and Irreversibility Up to this point few macroeconomic insights have been gained to make clear the limits of traditional accelerator principle. We therefore need to go further, wondering what happens if the capitai is irreversible, dK?:.O, and if firm faces a flow of demand dD which evolves according to the diffusion process (5). The problem of finding a cost-minimizing way to produce a giv­ en level of output may be written as:

max [- wdLt - qdKt - q8Kf dt) + Et (dV) + dK?O

(9)

This is a problem of stochastic dynamic optimization written in the form of the Bellman equation. Now, the future level of production is a function of the state variable D which fluctuates in a unpredictable manner. In this set­ ting we assume Yt = D( as a coinstraint. Thus there is no reason to expect the isoquant is optimal, it is just imposed exogenously to the firmo This approach to the accelerator relationship has been, originally, proposed by Grossman (1972). If totaI production is coinstrained by the demand curve at the going price level, firms will see output as predetermined when making investment decisions. In this case output would enter the investment function. Conse­ quently, the present vaIue of the firm is a function of Dc Further, absent ad­ justment costs, the other instrument of maximization is the level L. But the factor L is reversible (it can be choosen at any point in time), and hence it is not a state variable of the problem. Given the demand and the~price rigidity, the minimization of the present

THE RELATIONSHIP BETWEEN INVESTMENT AND DEMAND WITH UNCERTAINTY

557

value of costs is equivalent here to the vaIue maximization of the firmo The generaI functional form of the present value is written as V (Kt' Dt) Our first task is to characterize the equilibrium of this economy, proceeding in two steps. The first one, is to solve the expected value of the discounted value function at future time applying Ito' s lemma. The second one, is to calculate the desired stock of capitaI in each periodo Written in this way the value function is positive, and can be solved maximizing the right-hand side.

Result 1. In presence of both uncertainty on demand and irreversible capitaI, the desired capitaI stock KD is smaller than the desired leve! in tbe case of reversibility, in any period of time. The bulk of the input K is deter­ mined by the optimal condition V K = é > c, where é is the user cost of capitalo The total cost is a non decreasing function of uncertainty, proxied by the variance of demand S2. Proof. The first step is to resolve the expectation lemma

(dV) applying Ito's

(10)

Using this expression substitute in (9) for Et (dV), and derive with re­ spect to dK to obtain the following first order condition (11)

This condition is familiar. It states that the value of investment is a de­ creasing function of the cost of capitaI, q. Now return to (9) and repiace V K' This gives

Further, calculate the total derivatives of the value function with respect to the two optimal choices K and L*. This aIIows to get tbe optimal path of both capitaI accumulation and labour demando This statement is an applica­ tion of the envelope theorem. In this case, I get two non-homogeneous sec­ ond order differential equations (13a)

558

GICSEPPE

TRAVAGLI~I

(Bb) where to simplify the notation we have written V K '" (/J and VL == il. These ex­ pressions are the valuation functions stemmed from the Bellman equation (9), describing the necessary conditions that have to be satisfied on any opti­ mal path. To solve the homogeneous part of the two equations I try the func­ tions (/J CD) == ADI! and il (D) := BDII, for (Ba) and (13b) respectively. Sustituting and solving obtain and

(14)

The labour demand is given by2 2

Simple substitutions provide the two fundamental characteristic equations

Far each quadratic equation the two roots are respectively

81, O2

l

=2 1 2

a

±r(~ 52

~J

2r

r-

_l2 f3Jf3z=-- a ± [ (-"$2

2, 52

t

r

$2

with 01 > l and 02 < O

with

f3J > l and f3z < O

(a.l)

(a.2)

Thus the solution of the homogeneous part of the two equations is a linear com­ bination of the two solutions. To it we add any particular solution. Therefore the gen­ eral solution is et> = Al DO, + AlDo, - -~'"~._'"-"

r

(b.l)

(b.2) Yet, these two equations can be simplified. Actually, when the demand goes to zero, D O, the value of finn goes to zero V (O) O, and VK et> O, that is the mar­ ginaI value of investment tends to zero. Thus, since 02 < Oit must be A 2 = O in order to respect that condition. The solution (a. l) is reduced to (continues)

THE RELATIONSHIP BETWEEN INVESTMENT ANO OEMANO WITH UNCERTAINTY

r

w

=>

559

(15)

The value of the multiplier is thus the marginal present cost of an addi­ tional unit of Iabour. Whether the irreversibility coinstraint never binds, the option value of capitaI is zero and the solution is coherent with the Jorgen­ son's solution with certainty and reversibility. Indeed, substituting for IL in the particular solution of capitaI it gets GL/wr == GK/q(o + r), which states that the marginaI productivity condition holds at any single point in time. Clear­ Iy, this result is equal to the conditon (8) and hence the probIem is a se­ quence of static problems. A more complex argument is called far when the capitaI is irreversible. Now, capitaI accumulation is affected by the volatility of demand, and the in­ stalled capitaI stock depends on the whole past history of Dt' To make the value function (13a) easily solvabIe we need three conditions. When invest­ ment is irreversibIe, the Cobb-Douglas production function is essential to ob­ tain a closed solution. Thus assume Yt == KaLl- a. The marginal productivity of capitaI can be written as G K == aJ(a-1Ll-a == aY/K. Now replace G K and À in (c). Then, assume the current rate of profit is positive. Now, there is an opportu­ nity cost to delay investment. Thus, at some high enough demand D* the op­ portunity cost to renounce the yields is too high to keep the option alives. At this point the firm makes the investment to produce a quantity of output equal to demand, that is y* D*. FinallYI the function must satisfy two boundary conditions, namely the value matching and the smooth pastìng con­ dition. The former, is applied to the first order condition V K q, and it matches the marginaI values of the unknown function tP (D t ) to those of the known cost function. Basically, it gives a system of two equations

AY I1>1 + hY = c

{ eAyO- 1 + h = O (follaws)

Cc) The first term is the option value to invest, that is the worth to delay the invest­ ment maintaining the opportunity alives. It is caused by the irreversibility of capitaI input. ConverselYI the second term reflects the fundamental solution of the optimiza­ don problem. With respect to condition (b.2) the complete solution requires that the two homogeneous solutions vanish. This happens because labour input is perfectly reversible. and consequently the option value to hire new workers always goes to ze­ ro. This property substantìally simplifies the present model.

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GIUSEPPE TRAVAGLINI

where h wra/KGu and c =q (6 + r)/p is the Jorgenson's user cost. The solu­ tion of this system provides the following optimal condition

O hY* =--c=c A

O-l

(16)

O.E.D. To understand this condition, consider an increase in capital stock at time t. The left-hand side gives the present value of marginal benefits FK = hY, and the right-hand side gives the marginai total user cost 8 of doing so. Under the assumption of irreversibility the right-hand side depends on future demand pattern. This feature is expressed by the coefficient m == 0/(0 - 1) > 1, since O> 1. It implies that the present marginaI user cost 8 is bigger than the Jorgenson's user cost, 8 > c, that is the opportunity cost m to delay invest­ ment creates a gap between the marginaI revenue and the direct marginal user cost c. Fig. 2 illustrates this result. The curve MRKR identifies the value of reversibie capital with decreasing marginal productivity. KDy is the desired capital stock corresponding to the user cost c. On the other hand, in our ex­ ampie the optimal allocation is met when VK 8> c. It means the irreversible capital must be characterized by an higher marginal productivity, for any level of stock, to make worthwhile the amount KDy' This feature is represent­ ed by the curve MRKI, that is the marginaI productivity of irreversible capi­ taI. The difference between (8 > c) provides a misure of option value to in­ vesto To evaluate the effect of uncertainty on the totai marginal user cost 8 differentiate totally the expression lP with respect to s, taking into account the only positive root 01, It gives (17)

From the study of quadratic equation lP it easy to shown that alP/ao > O. Further, alP/as (J"O (O - 1) > O, and consequently to respect condition (17) it must be a()/as < O. In other words, the value of the positive root 8 decreases when s raises, and therefore m == 8/(8 1) increases.

THE RELATIONSillP BETWEEN INVESTMENT AND DEMAND WITH UNCERTAlNTY

561

q

q l - - - > t - -.........

--_ _ _--MRKI

q

--_ _ _ _ _ MRKR

K

Fig. 2

It hints that the gap between è and c becomes larger as uncertainty about the future path of demand accrues. To conclude, under these assump­ tions the stock of capitaI in any period of time is a decreasing function of de­ mand shocks. 2.3 The Accelerator Mechanism with Uncertainty and Irreversibility

Now, I tUffi to investigate the accelerator principle to show how the flow of net investment is sensitive to the dynamic fluctuations of the demand curve. Result 2. In presence of uncertainty on demand and irreversible capitaI, the accelerator coefficient is equal to v = [(O - 1/0) (wra/cGL )]. It is a function of the structural parameters of the economy, and it is inversely related with the option value of investment. Proof. Take the efficient condition (16) *

O

h Y =--c=c A

O 1

and substitute for h

awrlKGL and y* = D*. It gets

=> (D, r, é)

=

(~cGwr) D* L

The coefficient (awrlèG L ) is the flexible accelerator, expressed as a func­ tion of prices, technology and of the uncertainty s. Since the capitaI input K

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GIUSEPPE TRAVAGLINI

Ìs also a functìon of the current and expected future demand it can be de­ rived an accelerator relation (18)

O.E.D. This model delivers the familiar accelerator mechanism between Ìnvest­ ment and demand, but it is augmented by the factor (l/m) to take account of uncertainty. lt shows that not only the stock, but also the net investment is smaller than the case of reversibility because (e - 1)/ e < 1. Further, since the derivative ae/as < 0, the value of the positive root given by expression (al) converges to one as the limit of S2 tends to infinity. Thus, as uncertainty on future demand accrues the net investment tends to zero because (e -1)/ econ­ verges to zero, too. The remarkable aspect of this result is that the demand growth rate represents just part of the information set taken into account for an investment decision. lndeed, if the demand path over time is character­ ized by an high degree of uncertainty about the future states of world, the net investment may not raise at the current time. lntuitively, it happens be­ cause the investment choices of firms are affected not only by the expected value of profit, but also by the volatility of demand that can sometimes pro­ vide negative profits. This is exactly the meaning of equation (5).

2.4 Relationship between Investment Growth and Demand Fluctuations These features bring us to the relevant issue whether the partial equilib­ rium model discussed above can offer insights to understand the interrela­ tion between investment and demand in the actual dynamie economie processo The main aspect of the mode! is that aggregate demand is thefundamen­ tal force of the growth processo The stochastic dynamics is introduced by the process (5), while irreversibility is considered an essential element in capital accumulation decisions. However, these assumptions make it possible to generate a different causation relation of the growth processo First of all, the rate of output growth is drive n by investment decisions of firrns. This result contrasts with the canonical model of growth, Le. SoIow mode!, in which capitaI accumulation is determined by the saving decisions oE households. Moreover, the fluctuating growth rate of demand constraines the invest­ ment behaviour oE firrns and hence the rate of output growth. This is a key Eeature because, differently from the Ramsey model, demand does not move automatically with product. This appealing result emerges from the bct that the investment function (18) depends on the option value to invest measured

THE RELATIONSHIP BETWEEN INVESTMENT AND DEMAND WITH UNCERTAINTY

563

by the multiplier m. Thus, even if demand increases because of a higher vari­ ance, the investment may decrease because the higher variability can change in the opposite direction the investment decisions of firms and hence the capitaI stock. We want stress that this result do not require any a priori as­ sumptions on the existence of any acce]eration relation. This is uniquely the result of the dynamie minimization process in presence of uncertainty and irreversible capitalo In fact, the idea whieh comes from equation (18) is that the rate of in­ vestment is a decreasing function of uncertainty, and the willingness to rise the bulk of capital may depend not onIy on the growth rate of expected de­ mand, but also on the unstable pattern describes by the effective demando This is a key property of the model. Since demand fluctuates continuousIy over time because of transitory shocks, the firms prefer to reduce their own capital stock since it is irreversible. This choice modifies permanently the productive capacity of the economy and may Ieave the aggregate demand unsatisfied. As a consequence, investment choiees may be inefficient over time not only for consumers, but also for firms. Over many intervaIs of time, output could increase without any investment if the capitaI stock in the pre­ vious periods exceeds the desired one. This is to say that the optimal alloca­ tion of the firms in one period can come to nothing by the unpredictable fluctuations of demand in future time. For exampIe, starting from a tempo­ rary equilibrium K* and in presence of negative shocks, capitaI stock cannot be undone because of irreversibility; thus iL remains abundant in the next periods. In turn, whether the shock is positive the capitaI equipment may not adjust completely (because investment is a sort of commitment), and it will stay behind the buIk necessary to cover the actual demando Consequent­ Iy, the reduced incentives to invest today tends to Iower the growth rate of output, and tends to modify the trend of demando This spill-over effect can get worse the firm's expectations and eventually reduces further the rate of investmenL Note that these insights may be useful to reinterpret the role of aggre­ gate demand in the traditional growth model. We focus ed, in fact, on a non­ monetary model but we obtain different results from the intertemporal neo­ classical mode!, and the, more up-to-date, REC approach. This happens be­ cause differently from those models the source of disturbances affects direct­ ly demand, and the propagation mechanism is the investment function; thus investment and consumption can show divergent dynamies. Moreover, short­ run fluctuations of demand can be associated with shift in Iong-run econom­ ic perspectives. This finding suggests that transitory but unpredictable fluc­ tuations of future consumption pattern can alterate permanently the strate­ gie behaviour of firms at the current time and give rise to the potential for a

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cumulatively unstable processo This fact can contribute to generate variable trends of the long-run behaviour of aggregate economie variables, and to ex­ plain the observed variations in growth trends of real product, consumption and investment (Stock - Watson, 1988; King et alt., 1991). Thus, the tradi­ tional dichotomy between growth and stabilization policies misses an impor­ tant connection between the two policy goals. This fact bring us to the last point. In the last fifteen years the macroeco­ nomie debate has related the two phenomena of business cycle and econom­ ie growth. The RBC literature has shown that temporary technological shocks can modify permanently the trend of the economie path. Other theo­ rists have pointed out some mechanisms through whieh numerous determi­ nants, namely human capitaI, learning-by-doing, research and development spending, may affect an economie growth path (Aghion-Saint Paul, 1993). AlI these works emphasize that long-run rate of growth is an "endogenous" out­ come of the economie system. Overal!, the criticaI aspect of these contribu­ tions is that no research analyses in dept the linkages between the invest­ ment decisions and the economie growth path. Starting from these features, we proposed a tentative partial equilibrium model in whieh investment spending is driven by aggregate demando We treated investment as an in­ dipendent mechanism, sensitive to the transitory shock of aggregate de­ mando This casual relation can imply a disequilibrium process which acts as an endogenous stimulus for further movements of output. CONCLUSIONS

The main scope of this paper has been to focus on the investment deci­ sions of firms in presence of a demand curve whieh fluctuates continuously over time. The features discussed here illustrate that the pattern of net in­ vestment can be caused by uncertainty on unexpected demand shocks. This insight is important because the modern literature on economie fluctuations has provided a large class of models which focus on output and welfare ef­ fects from movements in demand, but with no role for dynamics. This view makes it possible to underestimate the key role of investment choices in the process of capitaI accumulation, and to misunderstand the complete set of issues whieh impinges upon the growth path of economie activity. The an­ swer to this view must be that firms may find more convenient to postpone investment decisions, and let a part of demand to be unsatisfied. This is caused by the fact that in a properly dynamie framework uncertainty about the future unknown pattern of demand affects the state of confidence and expectations of producers at any time. Of course, this aspect explains why the marginaI productivity condition

THE RELATlONSHIP BETWEEN INVESTMENT AND DEMAND WITH UNCERTAlNTY

565

of capital input and the associated accelerator mechanism appear as a func­ tion of variance. Each firm takes as given the diffusion process of demando Further, aH firrns have a complete inforrnation set available at any time t. For these reasons the expectations are rational, but in a different way from the usual one. Indeed, in the present model the positive root is a function of both the drift a and its variance S2. It means that to include both average values and variances in the expectation function may help to better predict the future value of an investment project. To say differently, it should be very inefficient to use only the inforrnation included in the mathematical expectation a, and not to utilize the remaining information set contained in the second moment of the process (5). Thus the behaviour of firrns may be the cause of the inef­ fectiveness of any macroeconomic policy. Indeed, if the aim is to stimmate economie activity, expectations and beliefs about the choiees of the regulator, could be more important than interest rate policy or fiscal policy. This means that credibility should be viewed as the major goal of any stabilization policy. Certainly, our model is sensitive to the hypothesis tllat characterize its structure. To say, uncertainty is exogenously given, and the investigation is conducted on the hypothesis of representative firm and partial equilibrium. For this reason the possible interrelation between investment and growth has been discussed only on an heuristic ground. The further step would be in searching for a rule in which investment decisions and consumption deci­ sions are both coinstrained by uncertainty and irreversibility. But, some as­ sumptions are necessary to gain insights about the relationship which links investment and demand in the case of irreversibility. Obtaining such insights can help to reduce the volatility of investment schedule which sometimes makes the cyclical fluctuations so unstable.

GIUSEPPE TRAVAGLINI

UNIVERSITY OF URBINO

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