Control of buffer stocks modeled by functional differential equations

Pergamon Nonlinenr Analysis, 7’hmy, Methods dr Applicotiom, Vol. 30, No. 2, pp. 1087-1097. 1997

Proc. 2nd World Congress of Nonlinear Analysts B 1997 Elsevier Science Ltd

PII: SO362-546X(97)00364-7

Printed in Great Britain. All tights reserved 0362-546X/97 $17.00 + 0.00

Control of Buffer Stocks Modeled by Functional Different ial Equations

Robert Underwood t and Graham Davis $

t Department of Mathematical and Computer Sciences,Colorado School of Mines, Golden, CO, 80401

$ Division of Economics and Business,Colorado School of Mines, Golden, CO, 80401

keywords: buffer stocks, control, functional differential equations.

1. INTRODUCTION

Commodity price risk is of concern to policy-makers in developed and developing nations (e.g., Claessens and Duncan in [2]). From this has come an interest in commodity price control via an international buffer stock mechanism, not so much to control the direction of prices as to damp the extreme fluctuations, thereby raising economic welfare. Early models of buffer stocks assumed that the price innovations of the commodity to be managed were stochastic, either as in a random walk, which reflects permanent price innovations, or as temporary harvest shocks. In both cases buffer stock smoothing of prices is doomed to failure due to eventual stock-out, with the timing of this stock-out a function of the resources available to hoard inventory. It should be noted that buffer stocks are a natural feature of commodity markets in that stock is withheld from the market in the hopes of making money when the price moves upward.

Of concern is that the random walk hypothesis is implausible from an economic standpoint, and random supply shocks do not readily apply to the production of minerals, even t.hough their price innovations have much the same statistical characteristics as agricultural commodities [3]. Thus, the anticipated failure of buffer stock schemes may be entirely an artifact of an erroneous model for the price process, at least with reference to minerals. This paper reexamines the efficacy of buffer stocks for mineral price management given a continuous cobweb model of mineral markets. In doing so, we extend the work of Mackey [7] and [8] and Lichtenberg and Ujihara (61 who replicate mineral price movements using cobweb models, and without having to assume some exogenous stochastic process which is inconsistent with the structure of mineral supply. The resulting problem corresponds to a null controllability problem which is related to steering the price’s state to its equilibrium value.

Our goal in this paper is to introduce the reader to a model and control mechanism for the movement of commodity prices using buffer stocks to modify these movements. It is our intent to show that this viewpoint is a reasonable characterization of this process. However, at this stage these models and control policies have not under gone the extensive statistical analysis needed to accurately model commodity price movements. This analysis will be the subject of future work.

Let us begin our discussion by reviewing a classic dynamic model for the change in prices caused by the interaction of supply and demand of goods. We will assume that demand is governed by:

D(t) = CY - @P(t) (1)

where (Y > 0 and 0 > 0, and supply by

s(t) = -7 + cw(t - h) + u(t) (2)

1087

1088 Second World Congress of Nonlinear Analysts

where 6 > 0. Later we will use u(t) as a control mechanism for the process. The strategies for u will be used to emulate a speculative marketing board which we will refer to as a buffer stock manager. The form of the control will be designed in accordance with this economic structure. For the m.oment we will consider the uncontrolled system with u(t) s 0.

The most elemental model will be described first. In [l], the operation of the model is described as: “ given P(t - 1) in the previous period, the quantity supplied to the market is S(t) in the current market and P(t) is set to clear the supply, i.e. P(t) and the amount bought and sold X(t) are given by:

X(t) = D(t) = s(t).” (3)

It can be easily seen from (l),(2), and (3) that the solution to the above system is given by

P(t) = (--s/p)” -t l -;,,oJn (a + r),

where n = 1,2, + . . . Clearly, the system is stable when S//3 < 1. In this case we will refer to the steady state solution as pa, where

P a+7

“=p+s’ (5)

2. THE WALRASIAN MODEL

In this setting we assume that the dynamic model for price movement is described by:

dP(t) - = -k(S(t) - D(t)) dt

which becomes dW - = -k VP(t) + SP(t - h) - (a + 7) + u(t)]

dt (7)

when (1) and (2) are substituted into (6). The parameter k in (6) is the speed of response for the system. The use of (6) as a price adjustment mechanism mimics the actual relationship between stock changes and price changes in mineral commodity markets.

For u(t) E 0, the system (7) is stable when the characteristic equation

X + k/3 + kbemxh = 0, (8)

has roots in the left half plane. If we let X = p+ iv, then (8) becomes

P = -kp - k&+cos (A),

u = kSe-ph sin (vh).

In order for p 5 0, we must have -kp - k6e-ph cos vh < 0. In this case,

(9)

e-ph cos (uh) 2 -$

If S/p < 1, there can be no solution with /J > 0. Therefore, if PO/6 > 1, then p < 0. Thus, az with the cobweb model when S/p < 1, we have stability. However, if a/p 1 1 it is still possible to have stability with p < 0. This is easiest to see for the case 0 = 0. If we set v = 0, then (9) must satisfy ephp = -k6. It is clear from the graph of ephp that there is a solution where p < 0 if 6 is not too large. In [5] Hayes has looked at this situation in detail. The region of stability mapped out in Figure ?? summarizes his results. To locate this region, set p = 0 in (9). This implies that

Second World Congress of Nonlinear Analysts 1089

Figure 1: Stability Regions

-P vh = arccos -, 6

and therefore, substituting into the second part of (9) we find that

-P arccos - 6

= kbh sin(arccos $)

= k6h 41 - (p/S)2

Thus, arccos ($)

k6h = dG-@pp’

When the solution is stable, it follows that the steady state solution is

(10)

as it was for the simple cobweb model.

3. SPECULATIVE STOCKHOLDING AS A DESTABILIZING MECHANISM

In pioneering work, Mackey in [7] and (81 examines a similar market system. Mackey is interested in the variations in commodity prices, and in [7] the particular role of speculative storage in causing these fluctuations. In his work, speculative producers or a marketing board are assumed to withhold all of current production for a predetermined period of time A(P) where 0 5 A(P) 5 Amax. Typically the storage delay, A, is either a monotone decreasing or a unimodal function of current market price, but the delay does not respond to price innovations should the predetermined storage plan be loss-making. While Mackey brings to light the possibility of having variable time delays in his model, his analysis assumes that storage delay is constant. This is reasonable since under his assumptions A’(P) does not affect the local stability conditions. Under his storage strategy, the supply given by (2) is modified to become

s(t) = (-7 + 6P(t - h)) - S(t) + S(t - h - A), (11) = S(t-h-A),

= -7+6P(t-h-A),


where S is the supply when no modification has been made and S(t) is the supply when Mackey’s storage policy is implemented. From our viewpoint depicted by (2), we see that he has taken the control to be

u(t) = -S(t) + S(t - h - A) = 6(P(t - h - A) - P(t - h)). 02) From (11) and (12), we see that this storage strategy is equivalent to increasing the fixed produc-

tion lag h by a constant factor A. In light of Figure 1, we can understand Mackey’s central concern. When /3/S < 1, (i.e. S//3 > l), a system operating below the curve (10) which is stable can be transformed into an unstable structure by letting u be chosen by (12). Mackey concludes that such storage policies “may either destabilize a previously stable market, or exacerbate an unstable market situation by increasing the amplitude and period of oscillations in commodity prices.” This indict- ment against speculative storage leads naturally to the question of compensating storage and reiease activities within Mackey’s unstable system by a global buffer stock scheme. In the next section we will offer this viewpoint as an alternative market strategy.

4. A BUFFER STOCK CONTROL STRATEGY

Suppose we wish to reduce the variation of a commodity price such as copper. For instance, a country such as Zambia may be concerned about large fluctuations in prices and may seek to damp these fluctuations. We will explore the use of a buffer stock to achieve this end. A buffer stock is formed by buying the product, storing the product, and then releasing the stock at some later time.

Let the buffer stock level, Q(t) be defined by

J t Q(t) = - u(s)&3 (13)

to where to is the initial time for the model and u(t) is our control whereby we remove or add stock to the market.

I f we ignore funding restrictions on the buffer stocks and focus just on the issue of controlling the price about its equilibrium value, we could set

u(t) = s(p, - P(t - h))

when P(t) = p,. The result would be that P’(t) = 0 from that moment on and P(t) would stay at the desired price P(t) = pm. Unfortunately, n(t) may become negative making such a strategy impractical.

Observe that if we set u(t) = ~f(-7+6P(t-h)) for all t ( re ar g dl ess of P(t) value) and substitute u into (7), we have a new value S = (1 - f)c5 for 6 so that

which indicates we have improved the stability. However, the buffer stock must eventually be released which makes this strategy impossible to implement.

Similarly, we could set u(t) = g (P(t) -pm). In th is case, the coefficient for the P(t) term in (7) changes from -kp to -kp( 1 + g). The increase in this coefficient should improve the dampening in the controlled solution. Unfortunately, this control could lead to the addition of stock when no such stock is available.

However, we can combine these two strategies as shown below:

-fS(t) = -f(-7 + 6P(t - h)) if P(t) < p,, u(t) = 9 VW - POJ if P(t) 2 p,, and Q(t) > 0.

0 if P(t) 2 p, and Q(t) < 0 (14)


Consider the following example. Let h = 1, a = 2, p = 1, 7 = 0,6 = 3, and L = 1. Further assume that P(t) = .2 for -1 2 t 2 0. When left uncontrolled the price is unstable as shown in Figure 2 .

Figure 2: Price without any Control

I f we use the control described by (14) with f = .75 and g = 3, we obtain the result displayed in Figure 3. Observe that this strategy does seem to bound the solution but the oscillations are still large. Also, the buffer stock grows quite large.

the (w&s)

Figure 3: Controlled Price

A draw back of these strategies is that they require an estimate of p,. This is a value which practitioners may have trouble estimating. As will be seen later, this concern can be circumvented by using a simple moving average constructed from the prior year. Finally, we have seen in the above example a situation where it is possible to bound prices. Mackey’s concern about instability is due to his specific strategy. Our strategy appears much more hopeful. In Section 6, we will see that a similar strategy may applied to the copper market.

5. SUPPLY AND DEMAND MODELS WHERE PRODUCTS ARE SUBSTITUTABLE

The model in Section 4, can be modified to include cross-price effects from close substitutes such as copper and aluminum. In the model that follows let the subscript , i = 1,2, denote products 1 and 2.

The model we will use to describe the interaction is shown below:

a(t) = [a? + @2(t)l - la: - P:P2ww

&PI = -71 + &pl(t - h) + UlV)

Dz(t) = [ai + c&w)1 - lo,0 - P:pl(t)lp2w

S2(4 = -72 + 62P2@ - h) + U2(&

(15)

where 4, /3!, yi, and 6i are all assumed to be nonnegatitive.


In this model, the demand of product l’s good is affected by the price of product 2’s good and vice versa. The supply equation remains as before. First we will describe the Walrasian model for two products. We will assume that

[ g]=[ h ISI W - DI WI k2[S2W - D2Wl ]* (1’3

We will use the following notation:

and

(17)

The Walrasian model (16) with this notation becomes,

+ W~W)~zW [ *IsfPl(W’d1)] + [ h;;;dd;] _ [ M;;]

P’(t) = -AoP(t) - AiP(t - h) + c + d4(t)P&) + Bu(t) (18)

To appreciate some of the differences which the two dimensional interaction has introduced, we will first consider the two dimensional version of cobweb model. This viewpoint is equivalent to setting P’(t) E 0 in (18). To further simplify the model. Consider the case where pi = /3; = 0 and q(t) = uq(t) = 0 for all t. With the above notation, the two dimensional cobweb model (18) can now be expressed as

P(t) = A,‘[-AIP(t - h) + b]. 09)

The eigenvalues for A,‘Al are

(20)

Since all the coefficients are nonnegative, we see that in this case the largest eigenvalue increases as ai and CY~ increase away from zero. In this situation, the oscillations will grow and might become undamped if (Y: and CY$ are sufficiently large. Thus, the interaction between the two products is injurious to the stability of the prices.

There is another situation that is straight forward to analyze. Suppose only product 2 affects product l’s demand. That is o’& = 0 and /3: = 0. Then the equation for PI becomes


pi(t) = -h{ @ -a:9<t,14w + be@ - h) (21) -171 + a;1 - +*(t) + w(t) 1

In the case 0: > 0 and Pz(t) > 0 as it should, the coefficient fl - &Pz(t)] 5 py. Again the dampening is lessened.

The variation of parameters formula (22) b e ow makes it clear what we can and cannot achieve 1 with a buffer control. Let

9(t) = exp-tP?(t - to) - Pi 11 P2(7W).

Then,

PI(t) = 4(t) {S(O) + Jt q+(s) (-71 + 61q(s - h) to

-a’: - a;P*(s, + Ul(S,) ds}

= 4(t) {S(O) + 1; 4% (SOJ(4 - Q? - 4P2(4 + w) d..}

(22)

where Se,1 (t) is the current supply of copper with no control removing stock from this supply. Suppose s(t) < Pl,. The best we can do is to remove all of the stock leaving us with

e(t) = 4(t) {Pi(O) + 1; 9%) (-4 - 4w) as} .

Since 9(t) > 0, we have no hope of further control through 7~2 and P2.

6. AN APPLICATION TO THE COPPER AND ALUMINUM MARKETS

To gain insight into how the control mechanisms might work under real market conditions, we looked at the copper and aluminum markets. First we will examine copper market and then the two markets together where they are allowed to interact.

The observed weekly movement in prices from February 29, 1984 through December 27, 1989 is shown in Figure 4.

Figure 4: Copper Prices

First we will consider the model for copper alone. We assumed that the model for these prices is described by (7) where h = 1 i.e. 1 year, k = .268, (Y = 8762,p = 21.5, y - 5620, and 6 = 24.7. These


time in years

Figure 5: Actual and Modeled Prices Over a Two Year Span

parameters have been calculated from world demand and supply data by estimating elasticities for copper. A comparison of the real data with model using these parameters is shown in Figure 5.

We also fit the data statistically using a least squares approach in a limited sense. For instance, we assumed that the lag was exactly one year. The model shown below provides a preliminary view of the efficacy of the functional differential equation described by (7). For the graph show in Figure 6, p = 4.66, and 6 = 43.72 width the other values the same as when the elasticities are estimated directly. To obtain a more accurate model is beyond the scope of this paper. Our concern here is

Figure 6: Actual and Modeled Prices Over a Two Year Span

to see whether the functional differential equation (7) is a reasonable economic characterization, and we feel that these graphs indicate this to be the case. Observe that t5/fl = 1.15 > 1 in the first case and 9.38 in the second. Therefore, there may be an eigenvalue of the system with the real part the right half plane. Examination of the graph in Figure 1 shows that indeed one of the eigenvalues does


not have its real part in the right half plane. Instabilities may occur. As the model now stands is too roughly estimated to closely track the actual prices. However,

from the point of the economics it is an adequate characterization.

70

64

62.

Equilibrium Solution

0.5 1 1.5 2

time in years

Figure 7: Controlled Prices of Copper

Now we describe our experience with the copper and aluminum market. In the simulation of copper and aluminum prices that follows, we used the parameters below. As before we assume that the lag is h = 1. For copper, the parameters are kl = .268,c~p = 8697.85, a; = -1.283,/?? = 23~3; = .03,71 = -5620, and 61 = 24.7. For aluminum the parameters are k2 = .12,aq = 13864.2,~~: = -1.283,/&j = 44.54,& = .Ol, 72 = -9085, and 62 = 54.3. Our current economic analysis indicated that the interaction between copper and aluminum was weak.

To control the prices, we used the strategy,

i

-f(Pl)S(t) = -f(-7 + 6P(t - h)) if P(t) 5 .995poo u(t) = 0 if .995p, < P(t) < ~~1.04

!7ww(t)) if ~~1.04 5 P(t)

where Q(t) is given by (13) and f and g are defined as follows:

(23)

.OOl if .99poo < R(t) I .995p,,

.005 if .98&, < PI(t) 5 .99pW f= .015 if .98poo < R(t) I .985Pca

.03 if .975p, < PI(t) 5 .98p,, and

.05 if R(t) 5 .975P,,

(24)

0 if n 5 7.5; .125 if 7.5 < R 2 10 .15 if 10 < R < 15,

9= .1875 if 15 < Sz 5 20, (25) .2125 if 20 < n 5 30, .2375 if 30 < R 5 40, and .25 if 40 < R.

As can be seen in Figure 7, this control is moderately successful. The key to the success of the control is to feed the stock gently back into the market. I f the stock is fed back too quickly, the control can chatter, that is rapidly oscillate between taking stock out and then returning stock soon thereafter. Indeed we can think of our control action as a sliding mode control, and consequently the chatter is


to be expected. Observe that this control has been designed to have a band width where no control action takes place. This coasting feature helps to alleviate some (but by no means all) of the chatter. The risk run by this approach is that size of the buffer may get too large and may not decrease to zero. In our simulation, we can see from Figure 8 that we were able to avoid the problem. However, this level of control was achieved because we assumed that the model used was the correct one.

in 1OOOtons 15,

20 .O 60 80 100

time in weeks

Figure 8: Buffer Stocks for Copper

The graph of aluminum prices in our two dimensional model are shown in Figure 9.

Figure 9: Aluminum Prices while Copper is Controlled

As mentioned before, it may be difficult to know the actual equilibrium of the system. An alternative design is to control the system about a moving average of previous prices. In the graphs below we show the results when the moving average was made over the prior 52 weeks. While this

Figure 10: Copper Prices when Controlled using a Moving Average

control keeps the prices damped, the buffer stock stays nonzero for a longer period of time. For


storage cost reasons It, is desirable to have as small a stock as possible and to have it go to zero as soon as possible.

7. CONCLUSIONS

From the experimental evidence presented here, we can see that in our setting it is possible to control prices through buffer stocks. Whether it can be successfully, implemented is another question. Several issues remain unresolved. First, can we adequately model the economic system governing the movement of prices. If one looks at our policy carefully, it becomes clear that one aspect of it is to buy when the price is low and to sell when the price is high. In one case we use the equilibrium directly, while in the other we use a moving average surrogate. Furthermore, the actual parameters in the control strategy play a significant role in the success of the the control policy. In the real world, the time of price movements would require that we obtain the parameter estimates from models of the process. Ultimately, the controller should recursively update its estimates in a manner analogous to Kalman filters. Lastly, now that we have a better understanding of reasonable controller designs, we need to examine the problem from an optimal control viewpoint.

References

[l] Allen, R.G.D. Mathematical Economics, MacMillan and CO Ltd, London (1956).

[2] Claessens, Stijn and Ronald C. Duncan, Managing Commodity Price Risk in Developing Countries, The Interna- tional Bank for Reconstruction and Development, The Johns Hopkins University Press, Baltimore, Md, (1993).

[3] De&on, Angus and Guy Laroque, On the Behavior of Commodity Prices, Review of Economic studies, Vol. 59, ~~1-23 (1992).

[4] Ghosh, S., C.L. Gilbert, A.J. Hughes Hallet, Optimal Stabilizaiion of the Copper Market, Resources Policy, Butterworth and Co (Publishers) Ltd (1982) 201-214.

[5] Hayes, N.D. Roots of the Transcendental Equation Associated with a Certain Difference-Differential Equation, J. Lond. Math, Sot., 25, (1950), 226-232.

[6] Lichtenberg, A. J., A. Ujihara Appbxtion of Nonlinear Mapping Theory lo Commodity Price Fluctuations, Journal of Economic Dynamics and Control 13 (1989) 225-246.

[7] Mackey, M., Commodity Price Fluctuations: Price Dependent Delays and Nonlinearities as Ezplanatory Factors. Journal of Economic Theory, 48, (1989) 497-509.

[S] Mackey, M. Cotwumer Memory and Price Fluctuations in Commodity Markets: An Integrodifferential Model

Journal of Dynamics and Differential Equations, Vol. 1, No. 3, (1989).

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Control of buffer stocks modeled by functional differential equations