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Investment in fixed capital and competitive industry dynamics.

1. Introduction

It has often been observed that real-world economies stagger through time constrained by the remnants of past mistaken investment decisions. This paper explores a model in which producers in a single competitive industry purchase a fixed capital good which, once acquired and installed, is specific to that industry. As such, the model is designed as a modest but necessary step towards understanding the possible dynamic implications of asset specificity.(1) By integrating production and investment decisions, the model may also be seen as being complementary to those cobweb models which explore production dynamics with given fixed factors of production.(2)

The model involves the simplest possible assumptions about technology, depreciation, costs, market demand, and market clearing. Our adjustment rules are in harmony with Marshall's claim:

When therefore the amount produced (in a unit of time) is such that the demand price is greater than the supply price, then sellers receive more than is sufficient to make it worth their while to bring goods to the market to that amount; and there is at work an active force tending to increase the amount brought forward for sale. On the other hand, when the amount produced is such that the demand price is less than the supply price, sellers receive less than is sufficient to make it worth their while to bring goods to the market on that scale; so that those who were just on the margin of doubt as to whether to go on producing are decided not to do so, and there is an active force at work tending to diminish the amount brought forward for sale (1920, p. 345).(3)

Our analysis - which is presented in the spirit of the claim of Baumol and Benhabib that the most promising role of dynamic models is that of 'enriching the list of recognized possible outcomes' (1989, p. 80) - shows that the industry can, indeed, display a rich diversity of possible behaviours. Our focus throughout is on comparative dynamics. Following an examination of the implications for the behaviour of market price of the speeds with which production and investment respond to deviations of market price from the appropriate supply price, we explore, in particular, the importance of the rate of depreciation in the fixed capital good: a change in this rate can have repercussions on the behaviour of market price via a change in the absolute level of full cost, via a change in the composition of costs, and, for an industry-specific asset, via a change in the physical carry-over from one period to the next. More generally, a recurrent theme is to identify factors which can impart stability to the behaviour of the industry. In addition to the durability and specificity of physical capital goods, crucial considerations are capacity (under-)utilisation decisions and the cautiousness of producers when expanding their capacities.

2. Technology, costs, and market demand

The technology involves a well-defined production period and fixed input-output coefficients: to produce one unit of the product requires one unit of machine services, one unit of labour services, and one unit of raw materials. Machines have a delivery lag (or gestation lag) of one production period. They depreciate at a rate 0 [less than] [Delta] [less than or equal to] 1 per production period irrespective of usage and, once installed, are assets specific to the industry, their use involving no opportunity cost. In contrast, producers can immediately acquire raw materials and hire labour services on contracts which last for the ensuing production period.

At date t, which constitutes the beginning of the [t.sup.th] production period, each producer has a stock of machines and, constrained by that, decides on the output to produce during the ensuing period for sale at date t + 1. Industry output(4) produced during the [t.sup.th] period for sale at date t + 1 is denoted by [q.sub.t+1]. The cost of an input dose comprising one unit of raw materials and one unit of labour services is v, where v is incurred at the beginning of the production period. At date t, each producer also decides on the number of machines to order for delivery at date t + 1. The aggregate gross investment in machines decided upon at date t, denoted by [I.sub.t], determines the aggregate number of machines available for use at the beginning of the next period, denoted by [k.sub.t+1], that is, [k.sub.t+1] = (1 - [Delta])[k.sub.t] + [I.sub.t]. The acquisition cost of a machine, denoted by m, is incurred at the time of delivery, where m includes any installation cost. Producers can finance the acquisition of machines, raw materials, or labour services by borrowing at a given market rate of interest, where i denotes the rate of interest corresponding to the duration of the production period.

Whereas v, m, and i are given to the industry and remain constant over time, the price of the product is endogenous. At each date price adjusts instantaneously to clear the market for the perishable commodity, the stationary market demand curve being linear with unitary slope. The equilibrium market price at date t is given by

[Mathematical Expression Omitted] (1)

Note the discontinuity: price tends to a as quantity tends to zero but, if the industry ceases production, price is deemed to be zero, there being no price for participants to observe.(5)

A static analysis would identify a long-period equilibrium involving a stationary price, [p.sup.*]; a stationary capacity fully utilised in each period, [q.sup.*] = [k.sup.*]; stationary replacement investment, [I.sup.*] = [Delta][k.sup.*]; and zero pure profits for producers. Zero pure profits requires that a replacement machine have a zero net present value. At the delivery date, a new machine costs m and has a value in terms of the discounted quasi-rents from its future use given by

([p.sup.*] - v(1 + i))(1/1 + i + 1 - [Delta]/[(1 + i).sup.2] + [(1 - [Delta]).sup.2]/[(1 + i).sup.3] = . . .) = [p.sup.*] - v(1 + i)/1 - [Delta] [summation of] [(1 - [Delta]/1 + i).sup.j] = [p.sup.*] - v(1 + i)/i + [Delta] (2)

so that the stationary price is [p.sup.*] = v(1 + i) + m(i + [Delta]). We define c [equivalent to] v(1 + i) + m(i + [Delta]) to represent the full per unit cost of the product. The corresponding output would be [q.sup.*] = [k.sup.*] = a - [p.sup.*] = a - c, where, by assumption, a [greater than] c. The stationary state is shown in Fig. 1.

The conventional terminology for m(i + [Delta]) would be the 'user cost' of a machine.(6) We employ this term, but it needs to be kept in mind that, once machines are installed, their use does not impose any opportunity cost. It is a standard claim that an institutional arrangement where producers themselves purchase machines is equivalent to one where there is a rental market for machine services with a rental cost equal to the user cost. In our context, however, this equivalence does not hold, at least outside a stationary state. Although producers in our industry can order machines at a given price - either because they are (collectively) sufficiently insignificant purchasers or because of Bertrand-like competition between producers of machines - the fact that, once acquired and installed, they become assets specific to our industry implies that the number of machines carried over from the current period sets a lower limit to the stock of machines available for use (at zero opportunity cost) during the next period.(7) It should be acknowledged that fixed capital assets might be specific to an industry but that their ownership or use might still be transferable between producers within the industry. However, given our assumption of uniform producers facing fixed input - output coefficients, such intra-industry transactions would, to use Walras' phrase, be 'theoretically without rational motive' (1954, p. 269).

The stationary state constitutes an industry equilibrium in the sense that, if it were established, there would be no endogenous forces leading to change. But is there any reason to believe that this equilibrium would be achieved? It is worth establishing a crucial, if elementary, point immediately. Suppose that each producer, endowed with perfect information about market conditions, were to expect price [p.sup.*] to prevail forever in the future. Would this ensure the attainment of that price? The problem is precisely that, for a price expectation of [p.sup.*], the net present value of a new machine is zero and any non-negative investment would maximise a producer's net worth. Consequently, in the absence of a forward market, there is no mechanism which would ensure that producers, acting independently, would undertake precisely the necessary aggregate investment, [I.sup.*] = [Delta][k.sup.*], and thereby produce the necessary aggregate output, [q.sup.*]. Thus the notion of an equilibrium price as one which pre-reconciles plans is not helpful in the present context. We need to postulate specific dynamic processes for the industry.

3. Short-period dynamics

At date t, producers decide on the outputs to produce over the [t.sup.th] period for sale at date t + 1 at a price which is uncertain at the time the decisions are taken. They have inherited stocks of machines, which include any newly-delivered machines ordered at date t - 1. Since market conditions will, in general, have changed from when the investment decisions were taken, there is no reason why producers would automatically use all their machines over the ensuing period. Being committed to incur the sunk costs of the inherited machines, they would base their production decisions on the effective variable cost v(1 + i). Interpreting v(1 + i) as the Marshallian short-period supply price, we assume that producers follow a rule whereby, subject to the capacity constraint, they increase output if the current (or going) price exceeds this supply price and reduce output if the going price is less than this supply price.(8) Specifically

[q.sub.t+1] - [q.sub.t]/[q.sub.t] = [Mu]([p.sub.t] - v(1 + i)) subject to 0 [less than or equal to] [q.sub.t+1] [less than or equal to] [k.sub.t] (3)

or, equivalently

[q.sub.t+1] = (1 - [Mu]v(1 + i))[q.sub.t] + [Mu][p.sub.t][q.sub.t] subject to 0 [less than or equal to] [q.sub.t+1] [less than or equal to] [k.sub.t] (4)

where [Mu] [greater than] 0 is the adjustment speed for production. A feature of this adjustment hypothesis is that a cessation of industry production is permanent: this follows from the fact that [q.sub.t] = 0 directly implies [q.sub.t+1] = 0 and it does not depend in any way on our deeming that a zero output gives rise to a zero price.(9) For future reference, note also that, since price tends to a as quantity tends to 0, [Phi] [equivalent to] a - v(1 + i) constitutes an upper-bound on the quasi-rent per unit, so that, from (3), [Phi][Mu] is an upper-bound on the rate of change of production, these upper-bounds being 'strict' since a zero output implies a zero price.

It is instructive to seek to disentangle the different forces operating within the market by initially isolating the implications of (3) through examining the short-period dynamics. For this purpose, we postulate a constant stock of machines, [Mathematical Expression Omitted], maintained period by period by replacement investment, [Mathematical Expression Omitted]. Our short-period is, of course, an analytical construct, not a calendar period of time.(10)

For establishing analytical results, it is convenient initially to suppose that the capacity constraint is not binding. In this case, there are two fixed points: output [Mathematical Expression Omitted] with price [Mathematical Expression Omitted] and a zero output with a zero price. Assuming further that the non-negativity constraint on price is not binding, a - [q.sup.t] can be substituted for [p.sub.t] in (3). A transformation then yields the logistic mapping

[x.sub.t+1] = (1 + [Mu][Phi])[x.sub.t](1 - [x.sub.t]) where [x.sub.t] [equivalent to] [Mu][q.sub.t]/(1 + [Mu][Phi]) (5)

where, for the moment, we disregard the non-negativity constraint on x. We review briefly the well-known properties of this mapping in terms of [Mathematical Expression Omitted], this parameter being of central economic importance.(11) There are two fixed points for (5): the origin x = 0, which is repelling since [Mu] [greater than] 0, and [Mathematical Expression Omitted], which corresponds to [Mathematical Expression Omitted]. If [Mu] [less than or equal to] 2/[Phi], the fixed point [Mathematical Expression Omitted] is an attractor. As [Mu] passes through 2/[Phi] this fixed point loses its stability and gives rise to period-doubling up to the accumulation point at approximately 2.57/[Phi]. For 2.57/[Phi] [less than] [Mu] [less than] 3/[Phi], the orbit may be attracted to a cycle of any integer periodicity or be chaotic. The maximum value of x is [M.sub.x]([Mu], [Phi]) = (1 + [Mu][Phi])/4. For [Mu] [less than] 3/[Phi], so that [M.sub.x]([Mu], [Phi]) [less than] 1, any seed [x.sub.0] in [0,1 ] gives rise to a trajectory which remains in [0,1]. For [Mu] = 3/[Phi], so that [M.sub.x]([Mu], [Phi]) = 1, the attractor is tangent to the unit square, this being the classic case of an attractor which is chaotic but not strange.(12) In contrast, for [Mu] [greater than] 3/[Phi], so that [M.sub.x]([Mu], [Phi]) [greater than] 1, almost every seed gives rise to a trajectory which escapes from [0, 1], that is, there is some t such that [x.sup.t] [greater than] 1.(13) The latter would imply [x.sub.t+1] [less than] 0 for the unconstrained logistic (5). In contrast, for our model, where quantity cannot be negative, escape translates into industry closure.

Consider now the possible impact of the constraint that price be nonnegative. It follows immediately from eq. (4) that l/v(1 + i) is a critical adjustment speed. For [Mu] [less than] 1/v(1 + i), the non-negativity of price is sufficient to ensure that the industry continues to operate, that is, [p.sub.t] [greater than or equal to] 0 implies [q.sub.t+1] [greater than] 0. In contrast, for [Mu] [greater than or equal to] /v(1 + i), the first occurrence of a zero price is sufficient (though not necessary) to bring about industry closure, that is, [p.sub.t] = 0 implies [q.sub.t+1] = 0. The impact of the non-negativity of price is illustrated in Fig. 2, a bifurcation diagram(14) for [Mu] based on parameters a = 8, v = 1.75, and i = 0.1, and for a capacity constraint which is not binding. These values are such that [Phi] = 6.075, the critical values for (5) being 2/[Phi] [approximately equal to] 0.329, 2.57/[Phi] [approximately equal to] 0.423, and 3/[Phi] [approximately equal to] 0.494. For [Mu] [less than] 0.483 - which is derived from [M.sub.q]([Mu], [Phi]) = [M.sub.x]([Mu], [Phi])(1 + [Mu][Phi])/[Mu] [less than] a - the non-negativity of price has no impact on the attractor and behaviour corresponds to that of (5): that is, for [Mu] [less than or equal to] 0.329 the fixed point is attracting, 0.329 [less than] [Mu] [less than or equal to] 0.423 constitutes the period-doubling region, and 0.423 [less than] [Mu] [less than] 0.483 is a chaotic region. Over the range 0.483 [less than] u [less than] 1/v(1 + i) [approximately equal to] 0.519, the industry continues to operate with the periodic occurrence of a zero price. In contrast, the first occurrence of a zero price causes closure for [Mu] [greater than or equal to] 0.519. Note that the explicit incorporation of the nonnegativity of price extends the range of adjustment speeds for which the industry is active: (5) would imply almost certain industry closure for [Mu] [greater than or equal to] 3/[Phi] [approximately equal to] 0.494.

Consider now the impact of the capacity constraint. A critical stock of machines is that stock the full use of which would lead to a zero quasi-rent per unit, namely, [Mathematical Expression Omitted], as shown in Fig. 1. If [Mathematical Expression Omitted], the short-period equilibrium involves an output equal to [Mathematical Expression Omitted] with the corresponding price being [Mathematical Expression Omitted]. This fixed point is necessarily an attractor: at any output strictly less than [Mathematical Expression Omitted], quasi-rent per unit would be strictly positive resulting in an increase in output. Sooner or later, the capacity constraint would be binding. In contrast, if [Mathematical Expression Omitted], the relevant short-period fixed point involves [Mathematical Expression Omitted] with [Mathematical Expression Omitted]. If the capacity constraint does not impinge on the attractor, the latter may exhibit a cycle of any integer periodicity or be chaotic or involve industry closure. If the attractor does include the capacity constraint being binding, this being a first-order system, the behaviour of market price must be periodic. Figure 3 is a bifurcation diagram for [Mathematical Expression Omitted] based on parameters [Mu] = 0.55, a = 8, v = 1.75, and i = 0.1, so that [Mathematical Expression Omitted]. For these particular parameters, industry closure would occur in the absence of a capacity constraint. However, given a capacity constraint, closure only occurs for [Mathematical Expression Omitted] (since [q.sub.t] = 7.89 implies [q.sub.t+1] = 0 even though [p.sub.t] [greater than] 0).(15)

An obvious objection to analysing the short-period dynamics is that producers would not continue automatically to maintain their stocks of machines irrespective of the behaviour of prices. Indeed, the system may be attracted to a fixed point where quasi-rent per unit is zero: producers simply could not afford to continue to undertake replacement investment. Clearly postulating that they do so is an analytical device designed to isolate the dynamics of the production adjustment process. But it also serves to alert us at an early stage to the possibility of a tension between short-period and long-period forces.

4. Long-period dynamics

Producers base investment decisions on the per unit cost, c, interpreted here as a Marshallian long-period supply price. Specifically, the rate of change in the aggregate stock of machines is proportional to the difference between the current (or going) price and the long-period supply price

[k.sub.t+1] - [k.sub.t]/[k.sub.t] = [Sigma]([p.sub.t] - c) subject to [k.sub.t+1] - [k.sub.t]/[k.sub.t] [greater than or equal to] - [Delta] (6)

where [Sigma] [greater than] 0 denotes the adjustment speed for investment and where the lower bound, equivalent to [k.sub.t+1] [greater than or equal to] (1 - [Delta])[k.sub.t], reflects the inability of producers to sell second-hand machines.(16) Note that, since price tends to a as quantity tends to 0, [Phi] [equivalent to] a - c constitutes an upper bound on what we may term pure profit per unit, so that [Sigma][Phi] is an upper-bound on the rate of change in the aggregate stock of machines. The corresponding aggregate gross investment in machines at date t is given by

[I.sub.t] = max [[Delta][k.sub.t] + [Sigma]([p.sub.t] - c)[k.sub.t]; 0] (7)

Replacement investment [Delta][k.sub.t] is thus adjusted (upwards or downwards) in the light of the difference between the current price and the long-period supply price. It should be noted that, as for the production adjustment hypothesis, there are other plausible formalisations of the Marshallian story.

The dynamic process (6) is essentially of second-order: [k.sub.t+1] depends on [p.sub.t] which depends on [q.sub.t] which in turn depends on (is constrained by) [k.sub.t-1]. If, for the moment, it is assumed that producers always use their available capacities and if the non-negativity constraint on price and the lower-bound [k.sub.t+1] [greater than or equal to] (1 - [Delta])[k.sub.t] are disregarded, applying an appropriate transformation to (6) yields a logistic-delay

[y.sub.t+1] = (1 + [Sigma][Phi])[y.sub.t](1 - [y.sub.t-1]) where [y.sub.t] [equivalent to] [Sigma][k.sub.t]/(1 + [Sigma][Phi])

In contrast to the flip-bifurcation route to chaos of the logistic, the logistic-delay exhibits the Hopf-bifurcation route to chaos: as [Sigma] passes through 1/[Phi], the fixed point involving [k.sup.*] = a - c loses its stability and gives rise via a Hopf bifurcation to an attracting invariant circle involving a distorted period-six cycle approached through an oscillatory trajectory.(17) Following the emergence of a period-seven window at around 1.15/[Phi], the attractor switches back and forth between being quasi-periodic, periodic, or strange until [Sigma] reaches 1.271/[Phi]. The latter value - being the value at which the invariant circle is tangent to the boundaries of the unit square - is critical with respect to the likelihood of escape: for [Sigma] [less than] 1.271/[Phi] there are finitely many regions of seeds ([y.sub.0], [y.sub.1]) which give rise to trajectories which escape from the unit square, whereas for [Sigma] [greater than or equal to] 1.271/[Phi] there are infinitely many regions which do so. If [Sigma] [greater than or equal to] 1.38/[Phi] all trajectories eventually escape.(18)

The derivation of (8), though instructive, involved a lot of 'ifs': in our context, depending on the parameters, behaviour may be constrained by the non-negativity of price and/or by the lower bound [k.sub.t+1] [greater than or equal to] (1 - [Delta])[k.sub.t]. Indeed, escape for (8) translates into the latter constraint being binding. Furthermore, imposing full-capacity utilisation is only an analytical device to isolate the pure or unadulterated investment dynamics.

Before examining the long-period dynamics by means of simulations, we must consider the issue of viability. The model comprising (3) and (6) does not take into account the possibility of bankruptcy. In other words, certain parameter configurations give rise to long-period attractors where on average producers' revenues do not cover their outlays; yet the industry does not close. Notwithstanding the ability of producers to borrow or lend at the going rate of interest, such behaviour would not be viable over the long-term. To incorporate an explicit bankruptcy condition into the formal model itself would be either complex or arbitrary (or both).(19) However, it is appropriate to incorporate some viability requirement into the generation of the bifurcation diagrams. To stipulate that for long-period behaviour to be sustainable the average of ([p.sub.t] - c) must be non-negative would, in fact, be a weak requirement. This is because at any date, if capacity was not fully used over the previous period, [p.sub.t] - c exceeds the profit per unit actually realised over that period.(20) Accordingly we impose the stronger requirement - which we call the 'break-even condition' - that

[summation over t] ([p.sub.t][q.sub.t] = v[q.sub.t+1] - m[I.sub.t]) [greater than or equal to] 0

where [p.sub.t][q.sub.t] - v[q.sub.t+1] - m[I.sub.t] is simply the difference between revenue and outlays at date t (and has no profit connotation) and where the summation is taken over the horizon over which the iterations are plotted.(21) Whereas it must be acknowledged that it is a rather crude condition which takes no account of the time profile of the differences between revenues and outlays, taking the latter into account would have conveyed a spurious air of precision to what are only intended to be indications of possible behaviours. For our purposes, the force of the break-even condition is simply that, if it is violated, the industry is deemed to close, this being represented by a zero price in the bifurcation diagram.

The full system - which comprises the coupled interaction of (3) and (6) - is particularly complex. Its behaviour depends inter alia on the relative magnitudes of [Mu] and [Sigma]. We have seen that for the short-period dynamics, if ***, period-doubling first occurs as the adjustment velocity [Mu] increases through 2/[Phi]; whereas, for the separate behaviour of (6), a Hopf bifurcation occurs as the adjustment velocity [Sigma] increases through 1/[Phi]. The latter carries over to the full system: even if 1/[Phi] exceeds 2/[Phi], the fixed point involving [q.sup.*] = [k.sup.*] and [p.sup.*] = c is attracting for [Sigma] [less than] 1/[Phi] whatever the value of [Mu]. This is because of the manner in which (3) and (6) are coupled: the stock adjustment process is, in a natural sense, the primary force.

Figures 4(a) and 4(b) are bifurcation diagrams for the adjustment speed based on a = 8, v = 1, m = 2, i = 0.1, and [Delta] = 0.5. Whereas Fig. 4(a) assumes automatic full-capacity utilisation, that is, it reflects the unadulterated investment dynamics implied by (6), Fig. 4(b) is for the full system (3) and (6) on the assumption that [Mu] = [Sigma]. The Hopf bifurcation occurs as the adjustment speed passes through 1/[Phi] = 0.175. In Fig. 4(a), industry closure - through violation of the break-even condition(22) - occurs for [Sigma] [greater than] 0.188. In contrast, for the full system, there is a visible period-six cycle for [Sigma] = [Mu] between 0.183 and 0.233 and closure, also due to the violation of the break-even condition, does not first occur until [Sigma] = [Mu] [greater than] 0.32. The comparison shows starkly that the full system is viable over a significantly wider range of adjustment speeds than is the unadulterated investment dynamics: capacity (under-)utilisation decisions can, indeed, have a powerful impact.

There is no a priori reason why the adjustment speeds [Sigma] and [Mu] should be equal. Accordingly, Fig. 5 shows a bifurcation diagram for the production adjustment speed over the range 0.15 [less than or equal to] [Mu] [less than or equal to] 0.55 for a given investment adjustment speed of [Sigma] = 0.25 [greater than] 1/[Phi], based on parameters a = 8, v = 2, m = 2, i = 0.1, and [Delta] = 0.5. As [Mu] increases through 0.36 an exact period-six cycle gives way to a distorted period-six cycle; as it increases through 0.46, the attractor appears decidedly noisy. It might be that producers would be more cautious with respect to investment decisions than with respect to production decisions; and this might be manifested in [Sigma] [less than] [Mu]. However, in the absence of any compelling a priori reason to expect one particular adjustment coefficient to be greater than the other, we assume henceforth for our simulations that [Delta] = [Mu].

Before exploring further the system comprising (3) and (6), we should consider briefly whether the proposition that capacity (under-)utilisation decisions can have a powerful stabilising impact is robust with respect to alternative adjustment hypotheses. Consider, for example, the production adjustment rule. An alternative formalisation of Marshall's story would be

[q.sub.t+1] - [q.sub.t]/[q.sub.t] = [Lambda] ([p.sub.t] - v(1 + i))/v(1 + i) subject to 0 [less than or equal to] [q.sub.t+1] [less than or equal to] [k.sub.t]

where [[p.sub.t] - v(1 + i)]/v(1 + i) could be interpreted as a rate of quasi-rent. Since v and i are here assumed constant, this would be equivalent to (3) for [Lambda] = [Mu]v(1 + i). Another possibility would be

[q.sub.t+1] - [q.sub.t]/[q.sub.t] = [Gamma] [([p.sub.t] - v/v) - i] subject to 0 [less than or equal to] [q.sub.t+1] [less than or equal to] [k.sub.t]

where ([p.sub.t] - v)/v constitutes a rate in the classical sense of a rate of return on an outlay made at the outset of the period. This would be equivalent to (3) for [Gamma] = [Mu]v.

The transformation of (3) into a logistic and (6) into a logistic-delay does depend on taking the proportionate rates of change of output and of investment. We would regard this as plausible. However, it would also be consistent with Marshall's story to postulate that the absolute rates of change of production and of investment are proportional to the differences between the going price and the appropriate supply price. An interesting feature of this case is that the interaction of the production and investment processes can give rise to complex dynamic behaviour even though their separate behaviours - each being linear - could not. For present purposes, it should be noted that the full system would be viable over a wider range of parameters than would be the unadulterated investment dynamics. Currie and Kubin (1997) confirm the stabilising force of capacity (under-)utilisation decisions for a seemingly quite different production hypothesis, namely, one where producers normally use all their machines and only under-use them (without necessarily ceasing production) if current price falls below some threshold level.(23)

5. User cost, depreciation, and asset specificity

Consider now the significance of user cost m(i + [Delta]) for the system comprising (3) and (6). Figure 6 is a bifurcation diagram for user cost between 0.01 and 2 based on a = 8, v = 2.5, i = 0.1, [Delta] = 0.5, and [Mu] = [Sigma] = 0.3, where changes in user cost should for the moment be thought of as arising from a change in the purchase price of a machine.(24) The impact on the range of variation of price of a reduction in user cost is not monotonic since it embodies two effects: a fall in user cost not only reduces the full cost c but also reduces the proportion of costs which, from the perspective of production decisions, are sunk. That the absolute magnitudes of costs matter follows from our earlier demonstration of the importance of [Phi] and [Phi] for the separate behaviours of (3) and (6), respectively. That the composition of costs matters for the interaction of (3) and (6) can be seen from Fig. 1. Ceteris paribus the lower the proportion of costs which are sunk, the closer v(1 + i) is to c. If the going price is between v(1 + i) and c then the stock of machines decreases whereas (the inherited stock of machines permitting) output increases and moves further away from [q.sup.*](= [k.sup.*]). The closer v(1 + i) is to c, the less the likelihood of this: the lower the proportion of costs which are sunk, the less the tension between short-period and long-period forces.(25)

Consider now the significance of the depreciation rate. A change in [Delta] not only alters user cost, thereby having two effects, but also has a third impact via the change in the proportion of machines carried over from one period to the next. In order to isolate this latter effect, assume that a change in [Delta] is accompanied by a compensating change which keeps user cost constant at 1.5, the other parameters being a = 8, v = 2.5, i = 0.1, and [Mu] = [Sigma] = 0.3. Figure 7(a) is a bifurcation diagram for the depreciation rate between 0.01 and 0.55: for [Delta] below 0.53, the constraint [k.sub.t+1] [greater than or equal to] (1 - [Delta])[k.sub.t] impacts on the attractor. Figure 7(b), where the dotted line represents [k.sub.t+41] = (1 - [Delta])[k.sub.t], shows the attractor(26) for the stock of machines at [Delta] = 0.418. The exact period-six cycle emerges as [Delta] decreases through 0.378 and, as [Delta] tends to 0 (as the slope of the dotted line tends to 1), the amplitude of the cycle tends to zero. Thus, by increasing the lower-bound on the rate of change in the aggregate stock of machines, an increase in the mean lifetime of a machine exercises a stabilising influence by reducing the range over which market price varies on the attractor.

But what is the significance of asset specificity? In fact, it is a key factor in that it enables the depreciation rate to have its third impact on this industry. If machines were not specific to this industry once installed but could be rented at a given price equal to the user cost m(i + [Delta]), where rental contracts were arranged one production period in advance, the only amendment to the model would be that the constraint in (6) would become simply [Delta]k/k [greater than or equal to] - 1. Provided that the two effects via costs of a change in [Delta] are nullified by taking a compensating change which keeps user cost constant, the impact of asset specificity can be identified by comparing the behaviour of market price at the actual value of [Delta] with what the behaviour would be at [Delta] = 1. For the parameters underlying Fig. 7(a), asset specificity as such has no impact for [Delta] above 0.53. For [Delta] below 0.53, its impact is greater the lower is [Delta], that is, the longer the mean lifetime of a machine.(27)

The claim that decreases in [Delta] are likely to be stabilising does need to be qualified. Note well that the model is not designed to accommodate [Delta] = 0; in that case, increases in producers' stocks of machines would be strictly irreversible. Moreover, even with [Delta] being strictly positive, for certain parameter configurations the operation of processes (3) and (6) (leaving aside the break-even condition) results in the capital stock exploding. For example, for a = 8, v = 2, m = 2, and [Delta] = i = 0.1, the stock of machines explodes as the adjustment velocity increases above about 0.35. Consider Fig. 8, where, for present purposes, we have suspended the break-even condition. For [Mu] = [Sigma] = 0.353 - just above 2/[Phi] = 0.345 - the production adjustment process prevails over the capital stock adjustment process in terms of price determination: a period-two cycle emerges, with the range between the maximum and minimum prices thereafter increasing with the adjustment speed. Beyond 0.354 - the value at which the break-even constraint would have been first violated had it not been suspended in generating Fig. 8 - there is an explosion in investment. Such an explosion is due to an asymmetry: whereas there is a well-defined lower bound on [Delta]k/k, there is no analogous physical upper limit on [Delta]k/k. The result is a ratchet-like effect, albeit not a strictly monotonic one: net investment at the high price overwhelms the limited disinvestment via depredation at the low price.(28) Clearly this could not be sustained for long: industry closure would follow inexorably from the bankruptcy of producers if they tried to persist in their behaviour.

6. Caution

It is plausible that cautious producers, recognising that it may take some time to reduce their stocks of machines through depreciation, might well limit their ordering of new machines when price is relatively high. This might be manifested in the magnitude of the adjustment velocity, [Sigma]. An alternative manifestation of cautiousness would be an upper bound on [Delta]k/k.(29) Pursuing the latter, consider the implications of imposing a 'flexibility constraint', as invoked by Day (1994) and by Huang (1995).(30) The adjustment process for the stock of machines becomes

[k.sub.t+1] - [k.sub.t]/[k.sub.t] = [Sigma] ([p.sub.t] - c) subject to - [Delta] [less than or equal to] [k.sub.t+1] - [k.sub.t]/[k.sub.t] [less than or equal to] [Beta] (9)

so that

[I.sub.t] = [Delta][k.sub.t] + [Sigma]([p.sub.t] - c)[k.sub.t] subject to 0 [less than or equal to] [I.sub.t] [less than or equal to] ([Delta] + [Beta])[k.sub.t] (10)

where [Beta] [greater than] 0 is the flexibility coefficient. Interpreting the flexibility constraint as the manifestation of a rule of thumb employed by producers who recognise that it may take some time to reduce machine stocks through depreciation, it seems plausible that [Beta] = [Delta]. Figure 9(a), in which the upper and lower dotted lines represent [k.sub.t] = (1 + [Beta])[k.sub.t-1] and [k.sub.t] = (1 - [Delta])[k.sub.t-1] respectively, shows the attractor for the aggregate stock of machines for [Mu] = [Sigma] = 0.4, a = 8, v = 2, m = 2, i = 0.1, and for [Delta] = [Beta] = 0.1. The flexibility constraint prevents what otherwise would be an explosion in the stock of machines by establishing a trapping set.(31) Figure 9(b) shows the histogram for price; the mean price is 2.42, compared to a full cost, and unstable fixed-point for price, of 2.4. Figure 9(c) shows the trajectory of price for the last 100 dates of the horizon. It should be noted that imposing [Beta] = [Delta] is not sufficient to preclude the possibility of an explosion whatever the other parameters: for example, investment would explode for [Mu] = [Sigma] [greater than] 1.94, given the other parameters on which Fig. 9 is based. Nevertheless, the system involving (3) and (9) with [Beta] = [Delta] is viable over a significantly wider range of parameters than the system comprising (3) and (6).

7. Stochastic forces

A recurrent issue with dynamic models is whether individuals would learn from their experiences. Where the behaviour of market price is chaotic or involves cycles of long periodicity, there is no particular presumption that they would learn. However, it is highly unlikely that producers would persist in behaving in ways which gave rise, say, to a regular period-six cycle. The first individual to recognise this cycle would be able to make a killing in the market without this requiring any understanding of the process generating the cycle. If enough producers realised this, industry behaviour would itself change, rendering what they had learned redundant. Seeking to incorporate processes of learning - which to be meaningful would certainly require accommodating differences between producers - is beyond the scope of this paper.

It is instructive, however, to consider very briefly the implications of allowing for forces which, from the perspective of this industry, are stochastic. Specifically suppose that the market demand curve is subject to random shifts. For purposes of comparison, we take the same parameters as those on which Figs 9(a), 9(b), and 9(c) are based, except that the intercept on the demand curve is now a random variable uniformly distributed between 7.5 and 8.5. Figure 10(a) shows the new attractor for the stock of machines (where no dotted lines have been added). Figure 10(b) is a histogram for price. Note that the uniform shocks are translated into non-uniform price distributions: the model is operating in a non-trivial way in the sense that the assumptions concerning the underlying stochastics are not simply translated into price distributions with the same form as the shocks. The introduction of the stochastic forces results in a relatively modest increase in the mean of price from 2.42 to 2.47 but in a marked increase in the variance of price from 0.08 to 0.35, the increase in the difference between maximum and minimum prices from 1.22 to 4.1 being a significant multiple of the range over which the demand intercept varies.(32) Moreover, whatever producers might have made of the time path in Fig. 9(c), the one in Fig. 10(c) does not exhibit a readily discernible pattern.(33) And since the average rate of pure profit - whether one takes the average going rate or the average realised rate - is strictly positive, they would not be compelled to change their behaviour. More generally, the significance of incorporating stochastic forces is that doing so extends the ranges of behavioural parameters which might be sustainable in the sense of giving rise to price dynamics which might reasonably be consistent with producers maintaining their behaviour.

8. Conclusions

Successive generations of economists have, at least until relatively recently, been disposed to present reassuring pictures of Marshallian processes of market adjustment. We would not wish to reinforce any impression that Marshall himself would have subscribed to such portrayals. He insisted:

But in real life such [market] oscillations are seldom as rhythmical as those of a stone hanging freely from a string; the comparison would be more exact if the string were supposed to hang in the troubled waters of a mill-race, whose stream was at one time allowed to flow freely, and at another partially cut off. . . . If the person holding the string swings his hand with movements partly rhythmical and partly arbitrary, the illustration will not outrun the difficulties of some very real and practical problems of value (1920, p. 288).

Whilst, of course, Marshall could not have anticipated the discoveries of modem non-linear dynamics, it is striking that, in presenting his vision of the behaviour of markets, he drew analogies with fluid dynamics and turbulence, these now being known to be natural sources of complex and chaotic dynamics.(34)

Our industry can, indeed, exhibit turbulent behaviour. The possibility of erratic behaviour cannot be attributed to contrived assumptions: at every stage we have deliberately invoked the simplest possible assumptions about technology, depreciation, costs, and market demand. Nor can the possibility of erratic behaviour be dismissed on the grounds, say, that the adjustment speeds needed for such behaviour are too high to be plausible: for any given adjustment coefficients [Mu] and [Delta] - however small - there are values for a, v, c, i, and [Delta], and thereby for [Phi] and [Phi], which would give rise to such behaviour.(35) If it is important to seek to identify the sorts of factors which can give rise to complex market behaviour, it is equally important to understand the sorts of factors which can impose bounds on behaviour and which can impart stability to the system. Our model has highlighted, in particular, the significance of capacity (under-)utilisation decisions, of the cautiousness of producers when expanding their capacities, and of the durability and specificity of physical capital goods. Examining a single industry in isolation and with just one type of fixed capital good is, of course, only a first step in understanding the possible behaviours of real-world economies which, constrained by the remnants of previous mistaken investment decisions, are staggering through time.

ACKNOWLEDGEMENTS

The authors are grateful for the valuable comments of Paul Madden, Stan Metcalfe, Ian Steedman, and the editor and two anonymous referees of Oxford Economic Papers.

1 In recent years, asset specificity has received considerable attention from the new institutional economists, albeit from a different perspective from our own. See Williamson (1985) or Pitelis (1993).

2 See, for example, Holmes and Manning (1988), Hommes (1991, 1994, 1997), and Huang (1995). For a model of investment, see Day (1994). See also Currie and Kubin (1995) on the implications of non-linearities for partial analysis: their cautionary remarks are relevant to this analysis.

3 The adjustment hypotheses are also consistent with Walras' conception of output adjustment under free competition (1954, p. 225).

4 Our analysis in terms of industry aggregates may be interpreted as being based on a representative producer. To consider explicitly a representative producer and then aggregate over producers would complicate the exposition needlessly. The dynamics are sufficiently complex to warrant not attempting to accommodate differences between producers in this paper.

5 The rationale for the assumed discontinuity is explained below.

6 See Jorgenson (1963) for the derivation of user cost in a continuous time framework.

7 The fact that on installation machines become fixtures means that, were there to be a rental market, there would be possibilities for opportunistic behaviour at the time of contract renewal. See Williamson (1985) or Pitelis (1993).

8 Given that the postulated technology involves fixed input-output coefficients, it would not be plausible to assume that producers seek to maximise expected profits on the basis of subjectively certain price expectations (whether naive, adaptive, or rational).

9 It would have made no substantive difference to our analysis if we had assumed instead that price is undefined when output is zero. We could have modified eq. (3) to allow for the possibility that, if output is zero, potential entrants might perceive an opportunity for profits. However, any such amendment would be ad hoc. As Howrey and Quandt (1968, p. 351) acknowledged, there are often complications with 'behavior at the origin'.

10 For a discussion of what Marshall himself intended by his time periods, see Currie and Steedman (1990, p. 218).

11 Baumol and Benhabib (1989) and Kelsey (1988) provide excellent introductions to chaotic economic models and examine, in particular, the properties of the logistic mapping.

12 At the accumulation point for the period-doubling, the attractor is strange but not chaotic. See Ott (1993, p.205).

13 The set of seeds for which this would not happen constitutes a Cantor set. See Devaney (1989, p. 38).

14 For all our simulations, we take seeds which are based on an initial price of 1.5 times the (static) equilibrium price, where the latter, depending on context, is either the short-period or the long-period equilibrium. For all bifurcation diagrams, we disregard the first 300 iterations and plot the subsequent 700 iterations.

15 Note the existence of a period-three cycle from 7.54 to 7.72, confirming that the Li-Yorke conditions hold. On the Li-Yorke conditions, see Day (1994, ch. 7). See also Schmidt (1995) for an extended treatment of the properties of a constrained logistic.

16 As observed in Section 2, maximisation on the basis of naive expectations would not be meaningful.

17 For an expository treatment of different types of bifurcations, see Lorenz (1989, ch. 3).

18 Currie and Kubin (1997) explore in some depth the investment dynamics with full capacity utilisation for a similar model for which [Delta] = 1. For detailed treatments of the remarkable properties of the logistic-delay, see Aronson et al. (1982) and Pounder and Rogers (1980). It should be noted that the logistic delay may have multiple attractors; see Fick et al. (1991).

19 See Day (1994) for a model in which bankruptcy is formally incorporated. This fits naturally into Day's model since producers finance investment from profits. See also Day et al. (1974) on the role of working capital in a model in which producers acquire a durable capital good with no delivery lag.

20 We have in mind defining the realised (pure) profit per unit as [p.sub.t] - v(1 + i) - m(i + [Delta])([k.sub.t]/[q.sub.t]). It should be acknowledged, however, that, for [Delta] [less than] 1, the appropriate definition of pure profit is problematic outside a stationary state, particularly where the use of inherited machines imposes no opportunity cost.

21 Since the time horizon is long - namely, 700 periods for our bifurcation diagrams - the arbitrariness which results from what price and quantity happen to be at the beginning of that horizon and from ignoring, say, the terminal value of machines at the end of that horizon is minimal.

22 Presenting a bifurcation diagram without imposing the break-even condition would have been misleading for the unadulterated investment dynamics implied by (6): for the latter, for [Delta] [less than] 1, the industry could never close however high the adjustment velocity (and however high the losses sustained by producers).

23 It should be noted that a limiting (and implausible) case of this hypothesis would be expected profit maximisation on the basis of subjectively certain naive expectations: where producers would fully use the available capacity for [p.sub.t] [greater than] v(1 + i) and close down for [p.sub.t] [less than] v(1 + i).

24 The model is not designed to accommodate zero user costs (or zero variable costs). In checking for the break-even constraint, we used the purchase cost of a machine as implied, at the user cost involved, by the assumed values for i and [Delta].

25 The significance of sunk costs is explored in Currie and Kubin (1997).

26 For all simulations of attractors, we disregard the first 2,000 iterations and plot the subsequent 10,000 iterations.

27 The impact can be identified in Fig. 7(a) by comparing behaviour at the actual value of [Delta] with behaviour for [Delta] above 0.53.

28 The significance of our assumption that price is deemed to be zero with zero production should be noted. If, instead, we were to assume that zero production gave rise to a (shadow) price of a then, for any [Delta] [less than] 1, industry closure - in the sense of there being from the operation of (3) and (6) some T for which [q.sub.t] = 0 for t [greater than or equal to] T - would (leaving aside the break-even condition) be accompanied by an exponential explosion in the (unused) capital stock. It should be stressed that our assumption of a discontinuity is not artificial but corresponds with reality: if output is zero, there is no price to be observed.

29 In our model, there is already a well-defined upper bound on changes in production implied by the (albeit changing) capacity constraint.

30 See also Weddepohl (1995) for an analysis of the implications of bounds on the rate of price change in a two-good exchange economy.

31 For detailed treatments of trapping sets using the first return map for a first-order difference equation system, see Day (1994) and Huang (1995).

32 In fact, if the range over which the demand intercept is stochastic is increased sufficiently, the possibility of an explosion in the capital stock is reinstated.

33 On the implications of noise for the behaviour of a logistic, see Kelsey (1988). For a formal treatment of the question of whether, in the presence of noise, rational producers could detect patterns in forecasting errors, see Hommes (1997).

34 It is interesting that Walras also invoked an analogy with turbulence by likening the market to 'a lake agitated by the wind, where the water is incessantly seeking its level without ever reaching it' and by insisting that 'just as a lake is, at times, stirred to its very depths by a storm, so also the market is sometimes thrown into violent confusion by crises, which are sudden and general disturbances of equilibrium' (1954, pp.380-1).

35 As Day (1994, pp. 224-5) observes in his analysis of price tatonnements, for any adjustment coefficient there exist perfectly reasonable demand and supply systems which do not converge.

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