OMEGA. The Int. JI of Mgmt Sci., Vol. 7. No. I. pp. 80-81 0306-0483/79/0201-0080502.00/0 ¢) Pergamon'Press Ltd, 1979, Printed in Great Britain

A Decision Model with Stochastic Cost and Demand Curves

THE STANDARD mict'oeconomic textbooks consider pricing decisions for profit maximization when the sell- ing price-market demand relationship and the produc- tion volume-production cost relationship are given deterministically; and in recent research efforts (e.g. 1"2] and 1"3]), the probability distribution of a product's profit is derived from stochastic estimates of market demand, market price and production cost. In the first model 'price' is the decision variable and the 'world' is completely deterministic: in the second model the stochastic nature of the 'world' is explicitly recognized but the effects of managerial decisions on cost and sales figures are completely ignored. This paper attempts to combine the favorable characteristics of both models and considers the realistic situation in which the firm has to set a selling price for the product: therefore, 'price' is not a stochastic variable but a decision vari- able. At the same time, the price-demand and volume- cost relationships are known only in stochastic terms. The problem is then to determine the selling price to optimize the firm's objective.

LINEAR DEMAND AND COST CURVES

Assume that, through a combination of statistical analysis of historical data and experts' forecasts, the stochastic relationship between unit selling price P and the planning period's market demand D for a given product is given by the regressive linear function

D = ao + alP + • (1)

and that the total cost T of manufacturing D units dur- ing the planning period is estimated to be

Tffi b o + blD + fl, (2)

where a and fl are normally distributed error terms with mean O and standard deviation 0% and op respect- ively. Assuming that the plant can always manufacture sufficient units to meet the actual demand in the cur- rent period, the period's profit Z from the product is then

Z = D P - T

ffi (ao + a l p + ~t)P - bo

- b l (ao + a t e + ot) - fl (3)

Z is clearly stochastic, with mean and standard devi- ation as

E(Z) ffi - a ~ P 2 + (ao + atbt)P - (bo + a0bt) (4)

and b2~r2 o~)1/2 a(Z) = (/'2a~ + ~ ~ + (5)

If the objective is to maximize expected profit, then by solving d[E(Z)]/dP ffi O, it can be easily verified that to set the selling price at

PI ffi (ao - albl)/2at (6)

will maximize E(Z). However, it is now well recognized that, under condition of uncertainty, a decision maker should consider the trnde-off between the expected retu-m ]/nd ffie-ns'~-(0i: tlie return's standard deviation)

resulting from an action. One common way to effect this trade-off explicitly is to maximize the utility func- tion

U(Z) = E(Z) - ko(Z), (7)

where k is the mean-standard deviation trade-off factor. The value of P[~ that maximizes U(Z) cannot be expressed in a convenient algebraic form, but lower and upper bounds of P~ can be easily derived. Figure 1 sketches the curves E(Z)' •(Z) and a straight line kPtT~. From (7), [dE(Z)/dP]r~, = [d~Z)/dP'le~,; i.e. P* is the P-value where E(Z)'s gradient equates ¢(Z)'s gradient, and Fig. 1 illustrates clearly that P~ < P[ always holds. The point

Pt = (ao + atbl - k¢)/2at (8)

is the point where E(Z)'s gradient equates the gradient of the fine kP¢~, and Fig. 1 also illustrates that P~ > PI always holds. Hence P[~ are bounded as

P ~ > P * > Pt (9)

and an accurate value of P[ can be found numerically between the bounds by using standard search tech- niques (see, e.g. [4]).

As a numerical example, let k = 1, and

D f l O O O - P + o l , a ~ = S x l0 3 "]

T = 5000+ 100D+fl , ~r~ 16 x 10°J ~ (10)

From the foregoing, it follows that P~ = 550, P~ = 514.6, a n d / ~ that maximizes U(Z) is somewhere in between. Using the Fibonacci Search subroutine in the IMSL library I l l , P~ is found to be $515.5, with corresponding E(Z) = 196,310 and a(Z) = 37,346. Given the simplicity of (7)' the same answers can also be obtained in about ten minutes using a calculator.

80

NON-LINEAR DEMAND AND COST CURVES

Although it is well recognized that the price-demand and volume-cost relationships are often non-linear, the linear relationships considered above are really quite adequate for most cases. This is because the accuracy of the relevant results depends only on t h e a ccuracy of the functional relationships in the neighborhood of p~. Thus, if necessary, one can obtain a crude initial estimate of P~ using linear functions that approximate the non-linear relationships over a wide range of P values; after that, refined linear functions closely fitted to the non-linear relationships in the neighborhood of P~ can be used to obtain more accurate estimates of P~ and other relevant values.

Nevertheless, if one insists on using non-linear demand and cost functions, the approach of the preced- ing section can be easily extended to handle the higher degree of polynomials of the demand-cost functions, even though the algebraic manipulations will become more tedious. For example, for the case of quadratic demand and cost curves

D = ao + a~P + a2 P2 + a = Do + • and j~ (11)

T = bo + bt D + b2D 2 + fl

Memoranda 81

0 / \

FIG. 1.

where ~t and fl are normally distributed error terms, one can derive the mean and variance of the profit Z = DP - T as

E(Z) = (P - bl) Do - bo - b2D~ - b2tT 2 !

and / (12) Var(Z) 2 4 = 2b,a~ + ~ + [2a2b2p 2

- ( l -2alb2)p + [bl + 2aob,)]2tT~

Given (12), the value of ~ that maximizes E ( Z ) - kcr(Z) can be easily determined by standard search techniques.

CONCLUSION

In a typical pricing, capacity planning or product selection problem, the management has to make a number of decisions. The effects of these decisions are only known in a stochastic sense, and the decisions should be made such that they are consistent with the firm's objective on making profit and avoiding risk. The model presented in this paper attempts to incorporate realistically all the above three components, whereas earlier models (see e.g. [2] and ['3] have often over- looked (i) the stochasticity of the environment, (ii) the difference between the decision variables and the sto- chastic environmental factors, or (iii) the risk-aversion component in the firm's objective. Although the pre-

sented model considers only one decision variable, we should point out that the addition of other relevant variables (e.g. advertising expenses in the demand func- tion and subcontracting in the cost function causes no conceptual difficulty. Practical solution of these models are straightforward with standard search procedures.

REFERENCES

I. International Mathematical and Statistical Libraries 0975) IMSL Library l, Reference Manual, Edition 4 (Fortran IV) S/370/360.

2. Ko'rrAs J, LAU AHL & LAU HS 0978) A general approach to stochastic management planning models: an overview. Acctg Rev. 52, 389-401.

3. LIAo M (1975) Model sampling: a stochastic CVP analysis. Acctg Rev. 49, 780-790.

School of Business Administration 8001 Natural Bridge Road University o f Missouri St Louis Missouri 63121 USA

John F Kottas Hon-Shiang Lau

(September 1978)