Further generalizations of the cake-eating problem under uncertainty

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  • DISCUSSION

    FURTHER GENERAL IZAT IONS OF THE CAKE-EAT ING

    PROBLEM UNDER UNCERTAINTY

    1. INTRODUCTION

    HoteUing [4] and Gale [3] have studied the following problem. A community possesses a non-replenishable resource, say a coal deposit or, simply, a 'cake'. The cake is of known size. At what (variable) rate should the cake be con- sumed over time?

    Employing as criterion the familiar integral of discounted instantaneous utilities, with an infinite horizon and with the utility function increasing and concave in the rate of consumption, it can be shown that it is optimal to con- sume the cake at a rate which declines through time, the manner of decline being determined by the rate of discount and by the curvature of the utility function.

    Later, the analysis was extended to accommodate uncertainty about the size of the cake (Kemp [5] ). In particular, it was shown that if the criterion is taken to be the integral of expected discounted utilities then the optimal rate of consumption is not necessarily monotonic in time.

    Suppose now that it is believed that at some unknown future time the com- munity's coal resources will be supplemented by the discovery of oil, that is, that a second cake, again of unknown size, will become available. What, in these circumstances, is the optimal consumption plan?

    In the present note it will be shown that the analysis of Kemp [5] can be extended to accommodate a sequence of cakes, each of unknown size and uncertain delivery date. A subsidiary objective is to show that the analysis can be made to accommodate cakes which are not consumed instantaneously. The emphasis throughout will be on problem formulation rather than on the solution of particular cases.

    2. A SEQUENCE OF CAKES

    In the problem studied by Kemp [5] one seeks a planned consumption path which maximizes the integral of discounted expected utilities; that is, one seeks to maximize

    Theory and Decision 8 (1977) 363-367. All Rights Reserved. Copyright 9 1977 by D. Reiclel Publishing Company, Dordrecht.Holland.

  • 364 DISCUSSION

    "2. J(**)(c) =- J e-Ptu(ct)II(Ct)dt + J e-ptu(O)(1 - II(Ct))dt

    O 0

    where p is the non-negative rate of discount, u(ct) is a strictly concave utility function of the rate of consumption c t planned for time t, C t = ~ crdr is the cumulative consumption planned for time t, and II(Ct) is the subjective prob- ability that the cake is larger than C r Without loss, the utility function can be so scaled that u(0) = 0. The task then is to find

    (1) max fl**)(c) = max J e-ptu(ct) II (Ct)dt

    s.t. Ct = ct, ct >~ 0 and Co = 0

    With suitable additional restrictions on the utility function, (1) reduces to a simple problem in the classical calculus of variations.

    2.1. Let us now take an easy half.step forward by supposing that a second cake of unknown size is expected with certainty at time T. The task then is to find

    (2) T

    max:( )-max { f e 0

    + .~ e-Ptu(ct)II(2)(C~ 1) + C(2))dt } T

    s.t. f P~') t < T

    ct= [6,(1)+C(2) tl>T

    where C~ 0 is the cumulative consumption of the ith cake (i = l, 2), II0)((7(1)) is the probability that something of the first cake remains at time t, t < T, that is, the probability that the first cake is larger than C O), and H(2)(C O)

  • DISCUSSION 365

    + 4 2) is the probability that something of at least one cake remains at time t, t >t T. Evidently problem (2) can be extended to accommodate the pos. sibility that any finite number of cakes will become available at known times.

    2.2. We can now dispense with the assumption that the delivery date of the second cake is known. Suppose instead that the delivery date is known up to a subjective probability distribution. The task then is to find

    (3) max S w(T)j(T) dT c

    0

    where 6o(T)dT is the probability that the second cake will arrive during the interval [T, T + dT]. Substituting for j(73, (3) becomes

    f max co(T) e-mu(ct) II(O(C}O)dt r

    0 0

    + fe-~ C}2))dtJdT T

    or, integrating by parts,

    (4) f (c)n( )(c} ))a( )a ' max e-~ t 1 ~ t t c

    0

    0

    where I2(t) ~1 - f0 t r is the probability that the delivery date is later than time t.

    Again the problem can be generalized by allowing for more than two cakes. One might suppose that the expected time of arrival of each of rn cakes is distributed independently of the expected time of arrival of each of the remaining m - 1 cakes. Or one might suppose that the interval between the arrival of the/th cake and the arrival of the ( /+ 1)th cake is distributed independently of the time of arrival of the ]th cake.

  • 366 DISCUSSION

    It remains true of this more general problem that the optimal rate of con- sumption is not necessarily monotonic in time.

    In a special case, any residue of the first cake vanishes upon the arrival of the second cake (coal mines are rendered absolete by a 'break-through' in the harnessing of solar energy), so that II (2) is a function of Ct (2) only. In a second special case, the two cakes are of known sizes, ~0) and ~(2), with only the date of delivery of the second cake uncertain. Then (4) simplifies, for

    l1 i fC O) I> C (1)

    1 if Ct(0 + C~ 2) < C 0) + C (2)

    + el2)) = 0 if c~ 1) + c (2) i> ~0) + ~(2)

    (o f course, it is never optimal to set C~ 0 + C~ 2) > ~(2); hence II(2)(Ct 0) + Ct (2)) = 1 always.) Combining the two special cases one arrives at a very special case closely related to one studied by Dasgupta and Heal [2], the chief difference being that in [2] the second cake is delivered as a steady flow, so many slices per time interval. Neither the above very special case nor that studied by Dasgupta and Heal is of much interest in its own right since uncertainty about the size of the cake is ignored. This does not mean that they are not useful stepping-stones on the way to more realistic models.

    3. CAKES WHICH ARE NOT CONSUMED INSTANTANEOUSLY

    In all cake-eating problems studied to date it has been assumed that in the act of consumption the cake is destroyed instantaneously. Among resources, one thinks of coal, oil and natural gas but not of gold or iron ore.

    Suppose alternatively that any slice of the cake is enjoyed over some finite period of time 0 and then, in reduced form, returned to the cake (re-cycled). Let ct be the proportion of the slice which returns, with 0 < ot < 1. Then we can consider the following non-linear control problem with a fixed (or time- dependent) delay:

    max J(**)(c) = max [ e-atu(ct)I I(C,)dt {c) {c} o

    d - -

  • DISCUSSION 367

    s.t. Ct = ct - ~ ct ~ 0 and Co = 0

    There now are available methods of handling problems of this kind. (See, for example, Blatt [ 1] and Teo and Moore [6] .) It remains true that the optimal

    path of consumption is not necessarily monotonic.

    School of Economics University of New South Wales

    MURRAY C. KEMP

    REFERENCES

    [ 1 ] Blatt, J. M., 'Control Systems with Time Lags', School of Mathematics, University of New South Wales 1975.

    [2] Dasgupta, P. and G. Heal, 'The Optimal Depletion of Exhaustible Resources', Review of Economic Studies, Symposium, 1974.

    [3] Gale, D., 'On Optimal Development in a Multi-Sector Model', Review of Economic Studies 34 (January 1967), 1-18.

    [4] Hotelling, H., 'The Economics of Exhaustible Resources', Journal of Political Economy 39 (April 1931), 137-175.

    [5] Kemp, M. C., 'How to Eat a Cake of Unknown Size', Chapter 23 in M. C. Kemp, Three Topics in the Theory of International Trade (Amsterdam: North-HoUand 1976).

    [6] Teo, K. L. and E. J. Moore, 'Necessary Conditions for Optimality for Control Problems with Time Delays Appearing in Both State and Control Variables', School of Mathematics, University of New South Wales 1975.