The multiproduct firm: Demand relationships and decision-making

The Multiproduct Firm: Demand Relationships and Decision-Making

STEPHEN HILL Department of Applied Economics, University of Wales Institute of Science and Technology, UK

If a firm produrea more than one product, the price of one may affect the demand for others. Examination of this relationship reveals that for the profit maxhnizing firm the price differential between products is determined by differences in costs, differences in own price and cross price elasticities and the relative revenue from each product. The introduction of advertising results in an optimal advertising budget for each product that is the weighted s u m of own and cross advertising elasticities divided by the average cost per advertisement.

1. INTRODUCTION

The purpose of this paper is to examine the conse- quences for the multiproduct firm of selling goods which are demand related. The major external variables for the firm producing any good are the decisions made by the producers of complementary or substitute goods. Where these other goods are produced by the same firm, then the decision problem is not only one of reacting to the external decision environment, but also of determining the optimal price differentials for these goods produced internally. It is to this second aspect that this paper is addressed.

To assist the exposition some simplifying assump- tions are made about the nature of cost and revenue functions. The basic result obtained is that for the profit maximizing firm the optimal price differential between two products is determined by differences in costs, the own price elasticities, the cross price elasticities and the relative importance of the two goods in terms of total revenue to the firm.

2. THE GENERAL CASE

The basic problem to be considered is that for the firm producing more than one good, the price of one may affect the demand for the others. The goods may be substitutes, in which case increasing the price of one increases demand for another (e.g. the firm producing both proprietary and retail brands), or complements, where increasing the price of one reduces demand for the other (such as integrated products and spare parts).

Assume a firm produces two goods X and Y Let P,, Q,, P,, 0, be the price and quantity of the two goods, respectively. The nature of the demand relationships between the two can be found by determining the marginal revenue of each.'

For the firm, total revenue (TR) will be

TR = PxQx + Pyay (1)

If the goods are demand related, the demand for one depends on the price of both, i.e.

Qy = dP,, P,) (3) The nature of the demand relationship is indicated by the partial derivatives. If dQ,/dP, > O , X and Y are substitutes and by implication dQy/dPx > 0. If these partial derivatives are negative, the two goods are demand complements.

The marginal revenue (MR) for each product can be found by differentiating the total revenue function.

The first two terms are the marginal revenue di- rectly associated with each product, and show that if the demand curve for the product slopes down marginal revenue is less than price (since dP,/dQx < 0). The third term refers to the change in the total revenue associated with each product when sales of the other product change, i.e. Qy(dPy/aQx) shows the effect on the total revenue from Y when an extra unit of X is sold. The sign of this cross marginal revenue term depends on whether the goods are demand substitutes or complements. If

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90 MANAGERIAL AND DECISION ECONOMICS, VOL. 3, NO. 2, 1982 OHeyden & Son Ltd, 1982

the goods are substitutes, aP,laQ, is negative so that increased sales of X leads to reduced revenue from Y.

The marginal revenue from each product can be expressed in elasticity terms.

Let

Then qx is the price elasticity of demand for X, and qxy is the cross price elasticity of demand for X with respect to the price of Y.

Now

(7)

Thus the marginal revenue of X is determined by the price of X, the elasticity of demand for X and the ratio of total revenue from Y to the total revenue from X times the reciprocal of the cross elasticity of demand. Then if Y is a substitute for X, the greater the degree of substitution (larger qxy) the smaller the marginal revenue from X. This makes intuitive sense since the price of X must be lowered to sell an extra unit, which also leads to lower sales of Y. Similarly, the larger the revenue from Y compared to X, the greater this cross elasticity effect. If the two goods are complements then the reduction in the price of X necessary to increase sales leads to increased sales of Y, therefore the marginal revenue of X is higher because of the addition to revenue from Y. Similarly

3. DETERMINING THE OPTIMAL PRICE DIFFERENTIAL

Assume that the average costs of the two goods are constant at C, and C,, respectively. Then the total

cost (TC) for the firm will be

TC = C,Q, + CyQy In the short run, at least the firm will operate with a capacity constraint. Assuming that the two goods require similar production facilities, this capacity constraint can be written as Z 3 Q, + Q,. The problem for the firm is then to maximize profit subject to this capacity constraint. The Lagrangean function L is formed by adding the constraint in implicit form multiplied by the artificial variable h to the profit function TR-TC, i.e.

=p,Q, +pyQy -C,Q, - C,Q, + A(Z- Q, - Q,) = P,f(P,, p,) + P,g(P,, f',) - C,Q, - C,Qy +A(Z- Q, - Q,) (10)

The Kuhn-Tucker conditions for maximization are

Taking the first two and eliminating A

P, + Q,(%) + Q, (2) - cX

Utilizing the results in terms of elasticities

or

=c,-c, (11)

Then the optimal price differential is related to the

STEPHEN HILL is lecturer in Managerial Economics in the Department of Applied Economics, UWIST. He graduated from University College of Wales, Aberystwyth and then studied at University College, Cardiff obtaining an MSc. Econ. Recent publications include articles on managerial decision-making and the economics of worksharing. Joint author with Julian Gough of fundamentals of Managerial Economics (Macmillan, 1979).

Address: Department of Applied Economics, UWIST, King Edward VII Avenue, Cardiff, CFI 3NU, UK.

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OHeyden & Son Ltd, 1982

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MANAGERIAL AND DECISION ECONOMICS, VOL. 3, NO. 2, 1982 91

difference in average costs, the respective elasticities of demand and the ratio of total revenues times the respective inverse cross elasticities of demand. The profit maximizing price differential will be higher the greater the difference in costs or elasticities. If the two products are not demand related, the cross elasticities of demand are zero and the optimal price differential equation becomes

P, 1+- - P , 1+- =cx-c, (12) ( t,) ( :,) where the price differential depends only on the differences in costs and demand elasticities.

This simplified equation has been used to analyse two situations:

1.

2.

Price discrimination, in which case C, = C, and the price difference depends only on the difference in elasticities. The equation shows that the higher price will be charged in the less elastic market.

Retail and proprietary brand pricing, where the difference in price can be related to differences in costs and elasticities, with the presumption that the retail brand entails both lower costs and a higher demand elasticity.’ However, the use of Eqn (12) ignores the mutual demand inter- dependence. The two goods share the same characteristics and are obvious demand substitutes. Then the optimal price differential between them can only be determined by explicit reference to the extent by which the price of one affects the demand for the other.

Equation (11) as a more general form is useful for analysing a number of other situations. The motor car manufacturer producing a number of models will be aware that the price charged for one has a significant effect on the demand for the others. In the demand complements case the pricing of spare parts is an obvious application. If the demand for spare parts is less elastic than for the original good and the cross elasticity of demand is low, the manufacturer can safely charge a premium price for the spare .part without disturbing the market for the original good. Note that if the sale of the original good creates a captive market for the component and the component market is important relative to the original good market (in revenue terms) then the optimal strategy will be to charge a low price for initial good and a high price for the component.

It can be shown (Appendix) that the cross price elasticities of demand for two goods bear a relationship to each other that depends upon the own price elasticities of demand. For a given cross elasticity of demand for X with respect to the price of Y, then the less elastic the demand for Y the smaller the effect of a change in price of X on the demand for Y (i.e. the less elastic the cross elasticity of demand

for Y). Thus, for given value of the other elasticities, the more the demand for X is affected by the price of X the more that demand will also be affected by the price of Y.

In the proprietary-retail brand situation, the demand for the proprietary brand is expected to be less elastic than the demand for the retail brand, depending on the success of the manufacturers brand-image creation policies.’ Then, for a given cross elasticity of demand for the retail brand, the less elastic the demand for the properietary brand the less the cross elasticity of demand for the proprietary brand and the smaller the cross marginal revenue effect of an increase in the price of the proprietary brand. Then a successful brand creation policy not only reduces the elasticity of demand, but also reduces the problem of demand inter- dependence.

4. ADVERTISING

If the price of one good affects the demand for the other, then it is likely that advertising one of the goods will have an affect on the demand for the other. Let A,, A, be the quantity of advertising on each good. Then

in all cases and

if the goods are substitutes. Now

The marginal revenue of the last unit of advertising on each good can be found by differentiation.

These results can again be expressed in terms of elasticities. Define the advertising elasticity of demand E by

A concept analogous to the cross price elasticity of demand is needed. Define the cross elasticity of demand for X with respect to advertising on Y (E,,)

92 MANAGERIAL AND DECISION ECONOMICS, VOL. 3, NO. 2, 1982 OHeyden & Son Ltd, 1982

as

, and E,,=-- Ax (18) aQ, Ay Exy =--

aA, Q, aAx Qy Now

EYX Ex +- PY Q Y pxQx AX AX

=-

Then the marginal revenue of an extra unit of advertising on X increases with the elasticity of advertising on X and the cross advertising elasticity of Y with respect to X. If the two goods are complements, the cross advertising elasticity is posi- tive and the marginal revenue from advertising on X will be larger the more advertising increases the demand for Y. Conversely, if the two goods are substitutes, increased avertising on X will reduce demand for Y (E,,, <0) and marginal revenue will be lower the greater this cross elasticity effect.

The expression for the marginal revenue of advertising on each good can be written as

and

The marginal revenue of advertising on X is the weighted sum of the elasticities of X and Y with respect to advertising on X, using as weights the total revenues from X and Y. Then the larger the revenue from Y, the greater the cross elasticity effect. For substitutes the marginal revenue from X is reduced according to the size of the total revenue from Y and the size of cross advertising elasticity, while for complements marginal revenue is increased by the same factors.

5. INTEGRATING PRICE AND ADVERTISING: THE GENERAL MODEL

The purpose of this section is to synthesize the results obtained so far into an explanation of profit maximizing price and advertising decision-making for the dual product firm with demand related products.

The simplifying assumption of constant average product costs can be dropped in favour of the more general production cost function.

TC = C(Q,) + E(Q,>

However, we shall assume that average advertising costs are constant for each product, so that total

advertising costs are a,Ax + %A,

where a,, a, are the average advertising costs on X and Y, respectively.

Using the general demand functions (Eqns 13 and 14), incorporating the new cost functions to- gether with the capacity constraint Z 5 Q, + Q,, the new Lagrangean function can be formed

n=p,Q, + ~ , Q Y - C ( Q , ) - ~ ( Q y ) - ~ ~ , -a,A,+A(Z-Q,-Q,) (21)

This is maximized by finding the partial derivatives with respect to the variables Ox, Q,, A,, A, and A, and setting these equal to zero.

an -= p, + Q, (2) aQ,

+Qy cz) - A = O (22)

an a Q Y

-- - p, + QY(%)

+Q, (::) - A = O (23)

an = P, (2) + P, (2) aAX

- 4 = O

- a , = O

an - = Z - Q, - Q, ah

Q,(an/aQ,)=O, Q,(drI/aQ,>=O, A(dn/aA)=O, Ay(an/aAy) = 0, A,(an/dA,) = 0 and Q,, Q,, A,, A 30 . Eliminating A from Eqns (22) and (23) and expressing in elasticity terms gives

P, 1+-+=- -P, 1+-+-- ( rlx P,Q, rlxy ( rly pya, pxQx rlyx 1 This is the equation for the optimal price differential for the two products, interpreted in Section 3 above with the difference that the price differential is now related to the difference in marginal rather than average costs. This equation is a more general form, and the similarity to Eqn (11) occurs because average costs were then assumed constant and therefore equal to marginal costs.

Equations (24) and (25) can also be expressed in elasticity terms, using the results of the previous section to obtain

1 - l ~ x Q , ~ x + ~ y Q y ~ y x ~ = ~ A,

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i.e. PxQx&, + pyQy&yx A, =

Then the optimal quantity of advertising is the weighted sum of advertising elasticity and cross advertising elasticity, divided by the average cost per advertisement. Other things constant, for com- plement goods the larger the cross advertising elasticity the larger the optimal advertising. For substitute goods, the larger the cross elasticity or the greater the total revenue of the other good, the smaller the advertising quantity. In either case the optimal advertising quantity will fall as the advertising cost increases.

Applying this result to the proprietary-retail brand situation, the more that advertising of the proprietary brand reduces demand for the retail brand then the lower the optimal proprietary brand advertising. In the original goods-components situation, the more that advertising of the original good increases the demand for components then the greater the original good advertising. Similarly the more important the component good market in terms of total revenue the greater the advertising on the original good. Then for example, razor manufacturers relying on the lucrative replacement

blade market will find it profitable to sustain advertising on the original razors by more than the razor market alone would justify.

6. CONCLUSION

If a firm sells more than one good, and if these goods are demand related, then the marginal revenue of either output or advertising will be in- fluenced by how that affects the demand for the other, as measured by the relevant cross elasticity. In the case of complements, marginal revenue will be increased above that due to the single product and decreased in the case of substitutes.

Thus the determination of optimal price differentials and advertising budgets for demand related goods requires explicit consideration of the way in which decisions taken in one market affect the other. The analysis also demonstrates the crucial role of the relative revenue of each market in the determination of price and advertising in the other. If the cross elasticities and revenue of the other market are large enough, the price and advertising decisions of a particular market may bear little relation to the demand characteristics of that market.

REFERENCES

1. J. Gough and S. Hill, Fundamentals of Managerial 3. S. Nickell and D. Metcalf, Economic Journal 88 (350),

2. D. Morriss and J. Nightingale, Managerial and Decision Economics, p. 183. Macmillan (1979).

Economics 1, 132-137 (1980).

254-268 (1971).

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APPENDIX. THE RELATION BETWEEN CROSS ELASTICITIES

Manipulation of the elasticity definitions is sufficient to show that cross price elasticities have a relationship to each other that depends on own price elasticities.

Recall the cross price elasticity definition (Eqn 6)

Multiply throughout by (P,./P,)(dP,/aP,)

Then the cross price elasticity of demand for X with

respect to the price of Y equals the elasticity of demand for X times the elasticity of the price of X with respect to the price of Y.

Now multiply throughout by ( Q,lQ,)(~Q,laQ,>

P, aP, Q, aQ,

P, aQ, Q, dPx

?Ixy =qxEaP,Q,aQ, =q,---- a, aP, P,

or

n x y q y x = rlxrly

For the advertising elasticities, similar manipulation reveals that

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The multiproduct firm: Demand relationships and decision-making