Decision-making in a Multiproduct Firm: A Reply

STEPHEN HILL Deparbent of Eeonamicsand Banking, UW", cprdi&, UK

In the earlier paper (Hill, 1982) it was demon- strated that for the dual-product firm

STR SPX SPY SQ, SQ, SQ,

MR, =-= P, + Q, * -+ Qy * -

Difficulties arise because of the assumption that

6PX 1 -- SQ, -saJsp,

As Brunner describes, in the two-variable case this is true only in restrictive circumstances, since (using the original notation);

This only equals

1 GQJSP,

if either SQJSP, or SQy/SP, = 0, i.e. if at least one of the cross-elasticities is zero. Similar problems arise with the inversion of SQJSP,, which is only possible if one of the own price elasticities is zero. Thus the elasticity specification of the earlier paper cannot be used without qualification. Now in economic terms it is certainly possible that one product may be demand-related to the other with- out the converse also being true, but this certainly robs the model of its intended generality in familiar elasticity terms. However, the inappropriateness of the normal elasticity formulation does not eliminate the economic significance of the model.

Three solutions can be envisaged. The first is to avoid the inverse function problem by redefining the solutions in terms of some kind of 'inverse elasticities'. Let

Then Eqn (8) from the earlier paper can be re- placed by

So the substitution is not the familiar own- and cross-price elasticities of demand but some sort of quantity-price elasticities. These 'inverse elas- ticities' (hillasticities perhaps!) can be interpreted by considering the constituent partial derivatives. The derivative of the price of x with respect to the quantity x shows how the price of x will change in response to a change in the demand for x, whilst the cross-partial derivative shows the effect of a change in the demand for x on the price of y. The question of whether price or quantity is the decision variable is thus reawakened, with antecedents all the way back to Marshall. The economist habitually de- scribes the linear demand curve as of the form D = a - bP, and then plots price on the vertical axis corresponding to the mathematician's function P = a/b-Dlb. Given the definitions of inverse elas- ticities the results of the earlier paper can be re- worked without loss of generality.

The second solution is to avoid the elasticities formulation altogether and rework in terms of par- tial derivatives. Recalling the first-order conditions for maximization:

Spy c, SPx SQ, SQ,

P, + Q, - + 0, * - -

=Py+Qy *-+Q, SPY a-- Spx cy SQY SQY

then Q SP Q SPY p 1 + - - " . 2 + 2 . - P, SQ, P, SQ,

=Cx-Cy -P , 1+--.- Qy SPY +-.-) Q, SP, ( Py SQ, P, SQ,

This does not have the neatness of the former solution, but tells more or less the same story, since the optimal price differential is related to the differ- ence in average costs and the way that each demand affects both own price and the price of the other product. The mathematics may have changed, but the economic logic is basically the same.

The third and most interesting solution is to reconsider the decision process involved. In a world of uncertainty, a sensible approach would be to determine prices and then see how many could be sold. So price is the decision variable, with quan- tities determined by the market. Translating this

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60 MANAGERIAL AND DECISION ECONOMICS, VOL. 5, NO. 1, 1984 0 Wiley Heyden Ltd, 1984

into mathematical terms, it may make more sense to be concerned with the marginal effect of a change in price, rather than quantity.

Recall that TR = PxQ, + Pyay

Then

-- SQ, SQY dip, SPX SPX

- Q, + P x - +PY * - STR

- +x. P 2+=2 SQ P Q P .- SQ, Qx(l Q, SP, P,Q, Qy SP,

= Qx( 1 + rlx + TR, rlyx

-

where qx is the normal price elasticity of demand for x, TR,, TRY refer to the total revenues from each product and qyx is the normal cross-elasticity of demand for y in response to a change in the price of x. So the marginal effect of a change in the price of x depends upon the sales of x, the elasticity of demand for x, the relative importance of the two markets and the cross-elasticity of demand for y. The higher the cross-elasticity of demand for y, and the more important the market for y compared with x, the more the effect of increasing the price of x on total revenue will be offset by gains in the revenue from y.

This approach can be fruitfully applied to the profit-maximizing dual-product firm. Assume that average costs for each product are constant at C: and C,. Then

T = P,Qx + pyay - GQ, - GQy Differentiating with respect to P, and P,, and

setting equal to zero:

Now

PY SQY P, 8Qx Q, SP, Q Y SPY

9 rly=---, rl*=---

P, SQy 9 rlyx--.-

Q, Spx - P SQ,

Qx Spy .-

rlxy = y Substituting;

CxQx GQy r lyx -- r lx - - - r lyx=o

PY Q Y Q, + Qxrlx +- PX p, PX

CyQy CxQx rly--rlxy = o pxQx Qy +Qyrly +- rlxy --

PY PY PY

Multiply the first equation by P,, and the second by P,:

pxQx + pxQXqx + (py - G)Qyrlyx - C,Q,rlx = 0

Pyay + pyQyrly + (P, - C,)Q, . rlxy - C,Qyrlx = 0

Gathering terms:

P,Q, + (Px - Cx)Qxrlx + (py - Cy)Qyrlyx = 0 Py Qy + (Py - G)Qyqy + (Px - CJQxqxy = 0

Now let T, be the pricehost margin on x ( P x - C,), similarly for ny. Then

P,Q, + TXQxqx + TyQyqyx = 0 PyQy + ~ Y Q y v y + ~xQxqxy = O

These equations can be used to eliminate Q,, Q,, since

(px + n;rlx)Qx = -(Tyrlyx)Qy and

so that -(.nxrlxy)Qx = (py + ~ ~ r l ~ > Q ,

- ( P X + .rr,rlx) - - - (Txrlq)

=YrlYx PY + T Y r l Y

(PY + . r r , r l Y P X + .srxrlx> = ~ x T y r l x y r l y x

p y p x + Txr lxPy + P x T y r l y

+ .rr,Tx ( r l x r l y - rlxyrlyx) = 0

Now let qxqy - qxyqyx = J (since this is the Jaco- bian determinant), and recall that T is the price/cost margin for each product. Then

PYPX(1 + r l x + rly + J) - p y c x r l x - p x C , q y

PY(PX(1 + rl, + rly +J> - Grl, - CXJ) - (PXCY + PYQJ + c x q = 0

= PxCyqy + P,c,J - CXC,J

which implies that

Now let

MANAGERIAL AND DECISION ECONOMICS, VOL. 5, NO. 1, 1984 61

or the profit margin per unit sold, then

CY - 1 + r x q x

py rlY +r rJ --1+-

So the ratio of c a t s to price for one product is determined by the elasticity of demand for that product, the rate of profit and the elasticity of demand for the other product and the difference between the product of own- and cross-elasticities of demand (since J = qxqy - 7)x,,qyx). Note that for the single-product case this equation reduces to

1 - _ G-l+- PY TY

and the more elastic the demand, the closer costs are to price.

The results on advertising in the demand-related situation remain the same, so that the optimal ad- vertising budget is still

PxQx&, + pyQy&yx A, = a,

where E ~ , syx, are the normal advertising and cross- advertising elasticities. Brunner’s point about (TRy/TR,). (l/qxy) not disappearing when qxy equals zero is correct but not longer relevant. In the ‘inverse elasticities’ formulation what is required is that &y should equal zero to get the unrelated

products solution, which remains as

since the difficulties about inverse partial deriva- tives disappear with +xy.

The requirement to transform local into global optima should have been made explicit in the ear- lier formulation, rather than the implicit assumption that the objective function is concave. Since the constraint was a linear inequality it satisfied the convexity requirement of a global optimum.

Given the difficulties of manipulating partial de- rivatives, the appendix about the relationship of cross- to ordinary elasticities must necessarily fall. However, this was no more than an interesting aside. The conclusion remains valid that the deter- mination of optimal price differentials and advertis- ing budgets for demand-related products requires explicit consideration of the way in which decisions taken in one market affect the other. I am grateful to Brunner for bringing these mathematical difficul- ties to my attention.

Acknowledgement

I am very grateful to David Ellis, W S T Mathematics Depart- ment, for unravelling the mathematical complexities. Any re- maining erors are, of course, my sole responsibility.

REFERENCE

S. Hill (1982) Managerial and Decision Economics 3, 90-94.

62 MANAGERIAL AND DECISION ECONOMICS, VOL. 5, NO. 1, 1984