Incentivised duopoly: profit maximisation or sales maximisation? BAM 2002 British Academy of Management Annual Conference Hosted by Middlesex University Held in Novotel, Hammersmith, London Malcolm Brady Dublin City University Business School Glasnevin Dublin 9 Ireland Tel. +353 1 7005188 Fax +353 1 7005446 Email malcolm.brady@dcu.ie Web site: http://webpages.dcu.ie/~bradym Abstract This paper examines and compares strict profit maximisation policy and strict sales maximisation policy in a duopoly where managers are incentivised by allowing them maximise a linear combination of sales and profits as an objective function. The paper examines the behaviour of the duopoly under combinations of strict profit maximisation and strict sales maximisation and examines the conditions under which a strict sales maximisation policy may outperform a strict profit maximisation policy. The paper uses a game theoretic framework to compare strategies under different cost structures. The paper uses analytical and computer simulation methods to examine firm behaviour. 1
Introduction Fershtman and Judd (1987) have proposed a model of duopoly in which owners recognise that managers are motivated by sales as well as by profits and therefore set managers the task of maximising an objective function that is a linear combination of sales and profits, rather than the traditional task of maximising profits. Their paper suggests that at equilibrium in a nondifferentiated duopoly where firms have identical costs and where uncertainty exists owners will choose a policy other than strict profit maximisation ie. owners will select an objective function where the incentive parameter is less than one; they suggest that 'profit maximising contracts generally do not arise in equilibrium'. Other authors have also suggested that firms pursue objectives other than pure profit maximisation: Baumol suggests that firms may pursue sales growth maximisation (Baumol, 1962) or sales revenue maximisation (Koutsoyiannis, 1979: ch.15); Marris (1963) suggests that firms set out to maximise a balanced rate of sales growth; Jensen and Meckling (1976) and Fama (1980) suggest that managers are motivated to increase their perquisites; Cyert and March (1992:7) suggest that profit maximisation is either only one among many goals of business firms or not a goal at all. Fershtman and Judd s incentive factor α typically lies on a continuum between one and zero and represents the weighting assigned to profits in the objective function; one minus the incentive factor (1- α) represents the weighting applied to sales. Their paper seeks to determine the optimum incentive factor for owners to choose. Fershtman and Judd assume an inverse linear demand function (p = a bq) for the duopoly and a linear variable cost function with no fixed cost (C = cq) for each firm. From Fershtman and Judd s model we can show that, where costs for both firms are identical, owners will select an α with value less than one when a > c (ie. always); owners therefore will allow managers include sales in their objective function. Where costs are not identical owners will select an incentive parameter less than one whenever a > 3c 1-2c 2 which will generally be the case when costs of the two firms are close. Should costs be greatly different then this inequality may no longer hold and one of the owners may select an incentive factor that is equal to or greater than one ie. an incentive function that is strictly profit maximising or even penalises for sales. This paper considers the behaviour of the duopoly for incentive factors at the two extreme points on the continuum between 0 and 1. This restriction allows three types of duopoly behaviour: both factors set at one implying that both firms adopt a strict profit maximising approach, both factors set at zero implying that both firms adopt a strict sales maximising approach, and one factor set to one and one factor set to zero implying one firm adopts a strict profit maximising approach and the other firm adopts a strict sales maximising approach. These three combinations of factors form the basis of the analysis that follows. Duopoly with one firm acting as strict profit maximiser and the other as strict sales maximiser Let us assume firm one is a strict profit maximiser and firm two is a strict sales maximiser. Let us also assume that variable cost is a linear function of quantity and there are no fixed costs. The objective functions given to managers by their owners, and which managers are expected to maximise, are therefore as follows: 2
and O 1 = (p - c 1 ) q 1 O 2 = p q 2 where subscript one refers to firm one and subscript two refers to firm two. These are special cases of the generalised Fershtman and Judd incentive objective function: O = απ+ (1 - α)r where α is equal to one for the strict profit maximiser and α is equal to zero for the strict sales maximiser. These two positions may be viewed as two extremities of a continuum on α between zero and one. Interdependence of the two firms is, as usual, created by means of the demand function: ie. p = a - b Q p = a - b (q 1 + q 2 ) where price is an inverse linear function of quantity. Following the usual Cournot equilibrium analysis approach we obtain the partial derivative of the objective function with respect to quantity and set the result to zero to obtain a maximum. This gives the following reaction functions: q 1 = {1/2b} {a - bq 2 - c 1 } and q 2 = {1/2b} {a - bq 1 } Solving for q 1 and q 2 by substitution gives: q 1 = {1/3b} {a - 2c 1 } and q 2 = {1/3b} {a + c 1 } We note that q 2 is always greater than q 1, given positive costs; ie. the strict sales maximiser always produces a greater quantity than the strict profit maximiser. Price at the Cournot equilibrium point is determined to be p = {1/3} {a + c 1 } Now let us examine profits of the two firms: Π 1 1,0 = (p - c 1 ) q 1 = {1/9b} {a - 2c 1 } {a - 2c 1 } Π 2 1,0 = (p - c 2 ) q 2 3
= {1/9b} {a + c 1-3c 2 } {a + c 1 } where Π 1 1,0 represents profits of firm one when firm one chooses strict profit maximisation (α= 1) and firm two chooses strict sales maximisation (α = 0), and similarly for firm two. Profits of the sales maximiser will be greater than profits of the profit maximiser when Π 2 1,0 > Π 1 1,0, ie. when {a + c 1-3c 2 } {a + c 1 } > {a - 2c 1 } {a - 2c 1 } ie. when a (2c 1 - c 2 ) > (c 1 2 + c 1 c 2 ) When costs of both firms are identical, ie. c 1 = c 2 = c, this condition reduces to a > 2c. We also note from the reaction function above that positive q 1 requires a > 2c 1. This implies that in the case of identical costs, strict sales maximisation yields greater profits than strict profit maximisation for all feasible equilibria (ie. where both quantities are positive). The intuitive argument is as follows: profit for both the profit maximiser and the sales maximiser is quantity multiplied by price less cost. If we refer to price less unit cost as unit margin, profit is equal to quantity multiplied by unit margin. Where costs are the same, unit margin will be the same for both firms; this implies that the firm that sells the greater quantity - the sales maximiser - will make the greater profit. This is so because in a duopoly price results from the combined action of both firms, not the action of one firm alone. Comparison with a duopoly where both firms are strict profit maximisers When both firms act as strict profit maximisers both sets of owners set their managers the traditional objective functions: and O 1 = (p-c 1 ) q 1 O 2 = (p-c 2 ) q 2 Analysis similar to that of the previous section yields the following results for quantity, price and profit for firm two: q 2 = {1/3b} {a + c 1-2c 2 } p = {1/3} {a + c 1 + c 2 } Π 2 1,1 ={1/9b} {a + c 1-2c 2 } 2 with analogous expressions for firm one. As expected, price is higher and combined quantity lower than in the previous case. Now let us examine the conditions under which profit for firm two, when it acts as a strict sales maximiser and firm two is a strict profit maximiser, would be greater than the profit it would earn if it too acted as a strict profit maximiser. This occurs when Π 2 1,0 > Π 2 1,1 ie. when 4
{a + c 1-3c 2 } {a + c 1 } > {a + c 1-2c 2 } 2 Expanding brackets and gathering terms reduces this to a > (4c 1 2 - c 1 c 2 ) / c 2 When costs are equal this further reduces to a > 3c. Clearly a = 3c is a change point: below this point, ie. when a < 3c, it is in the best interests of the firm to act as a strict profit maximiser; above this point, ie. when a > 3c, it may be in the interest of the firm to act as a strict sales maximiser. Comparison with a duopoly where both firms act as strict sales maximisers Let us now consider the case where both firms adopt a strict sales maximisation strategy. In this case the objective functions for both firms are similar: and O 1 = pq 1 O 2 = pq 2 Following the same analysis procedure as above we find expressions for quantity, price and profit for firm one: q 1 = q 2 = a/ 3b p = a/3 Π 1 0,0 = {1/9b} {a (a - 3c 1 )} with an analogous expression for profit for firm two. As one would expect, quantities produced are larger for both firms and price is lower than in the previous case. Profit is less than the profit that the firm would make if both firms were profit maximisers when Π 1 0,0 < Π 1 1,1 ie. when a (a - 3c 1 ) < {a + c 2-2c 1 } 2 When costs are identical this condition becomes a > -c, which is always true for positive cost. Profit when both firms are sales maximisers is therefore always less than profit when both firms are profit maximisers, where costs are identical. In similar fashion we can show that profit for a sales maximiser, the other firm also being a sales maximiser, is greater than the profit it would get if it was a profit maximiser, the other firm continuing to be a sales maximiser, when Π 1 0,0 > Π 1 1,0 ie. when a (a - 3c 1 ) > {a - 2c 1 } {a - 2c 1 } which reduces to a > 4c 1. Note that this condition does not require identical costs. 5
A game theoretic view of incentive duopoly We can now put these results together in the form of a payoff matrix. To do this we use some numbers as values for the parameters of our model. Figure 1 shows results for a duopoly where costs are identical and where a < 3c: profits have been determined for the two firms where parameters a, b, and c have values 25, 0.0001, and 9 respectively and where a profit tax rate of 40% applies; profit figures shown are in thousands. A unique Nash equilibrium exists at the profit maximisation/ profit maximisation position (171,171). Managers have a straightforward decision to make here: both firms will choose strict profit maximisation. Firm Two Firm One 171, 171 33, 159 159, 33-56, -56 Figure 1. Payoff matrix for duopoly where a < 3c The payoff matrix for a duopoly where 3c < a < 4c is shown in figure 2; parameter values are the same as for figure 1 except that c is set at 8 (making a > 3c). This payoff matrix is of the hawk-dove archetype with two Nash equilibria (Osborne and Rubinstein, 1994:17) occurring at the positions: sales maximiser/ profit maximiser (198, 54) and profit maximiser/ sales maximiser (54, 198). This is also referred to as the 'chicken' game: each firm is better off being a sales maximiser provided the other firm is not. If both firms select a sales maximiser strategy both are worse off. Firm Two Firm One 193, 193 54, 198 198, 54 17, 17 Figure 2. Payoff matrix for duopoly where 3c < a < 4c A third situation to examine is that where a > 4c; a payoff matrix for this is shown in figure 3. In computing these figures c is set at 6 for both firms (making a > 4c). Here a single Nash equilibrium exists where both firms adopt a sales maximisation strategy. However, this is a 6
Prisoner s Dilemma type game as an alternative exists where both firms would be better off, but this alternative is not a Nash equilibrium. Firm Two Firm One 241, 241 113, 269 269, 113 117, 117 Figure 3. Payoff matrix for duopoly where a > 4c Conclusion We know from Fershtman and Judd's work that owners will select an incentive function other than strict profit maximisation in many cases, and always where costs are identical ie. when a > c. This research examined the case where owners are restricted to strict policies ie. strict profit maximisation or strict sales maximisation. We found that when one firm adopts a strategy of strict sales maximisation and the other one of strict profit maximisation the strict sales maximiser produces better profits than the strict profit maximiser for a > 2c (which represents all feasible equilibria). We also found that the profits of this sales maximiser are better than its profits would be if both firms were profit maximisers when a > 3c, but worse when a < 3c. Where a > -c, ie. always, a duopoly where both firms adopt a sales maximisation strategy yields worse results for both firms than if both firms adopted strict profit maximisation. However, for a > 4c, the sales maximiser, where both firms are sales maximisers, will perform better than it would have as a profit maximiser, where the other firm was a sales maximiser. A broad overall conclusion is that where both firms have identical and relatively high costs (a < 3c) both firms will adopt strict profit maximisation policies. Where both firms have identical and mid-range costs (3c < a < 4c) one of the firms will adopt a sales maximisation strategy and equilibrium considerations will force the other to adopt a profit maximisation strategy, even though it will underperform its rival. For identical and relatively low costs (a < 4c) a sub-optimal equilibrium will exist. Trends in profit figures under different strategies as costs change are shown in figure 4. As expected, where the duopoly contains a profit maximiser and a sales maximiser, profit levels for the sales maximiser always exceed those of the profit maximiser. To the left of the line a = 4c (ie. for a > 4c) a sales maximisation strategy may be in the best interests of managers and owners: if the other firm recognises the Nash equilibrium and chooses sales maximisation then profits are made; and if the other firm selects profit maximisation then even larger profits are made. To the right of the line a = 4c (ie. for a < 4c) sales maximisation is risky because if 7
the other firm also chooses sales maximisation then both firms will make poor profits or even incur losses. To the right of the line a = 3c (ie. for a < 3c) then profit maximisation is clearly the best strategic choice for managers or owners: if the other firm recognises the Nash equilibrium and also chooses profit maximisation then both firms make profits; if the other firm chooses sales maximisation then the firm will still make some, albeit low, profit. An incentive still exists for managers or owners to adopt a sales maximisation strategy when 3c < a < 4c: if managers are measured not only on the absolute position achieved but also on the relative position then the hawk-dove Nash equilibria positions are more favourable to the (winning) manager as this manager significantly outperforms his or her rival ie. the difference in levels of firm profits is large as shown in figure 4. To a manager being significantly ahead of your rival in market share and profit may be more important than the absolute value of profit. Focal point considerations may apply in determining which of the two firms will choose sales maximisation and which will choose profit maximisation in order to maintain the Nash equilibrium: for example, if managers of one of the firms had a reputation for sales maximisation this may be lead the other firm to choose profit maximisation. 500 400 300 Profit 200 100 0-100 0 2 4 6 8 10 12 14 P1 1,1 P1 1,0 P1 0,1 P1 0,0-200 -300-400 Sales maximisation a = 4c a = 3c Profit maximisation Cost Figure 4: Profit vs. cost for four duopoly situations The above model can be situated within duopoly theory as shown in figure 5. The reaction curves for the standard Cournot case where both firms set themselves profit maximisation as an objective are shown as lines A and B and their intersection point w is the Cournot Nash equilibrium. The duopoly situation where one firm sets itself profit maximisation as an objective function and the other sets itself sales revenue maximisation as its objective function is represented by the reaction curves B and C; the intersection point x represents a Nash equilibrium for this situation. The reaction curve C which represents the sales maximiser is parallel to reaction function A (the reaction curve it would have if it was a profit maximiser) but shifted to the right. This shift to the right occurs because costs are not taken into account 8
in the sales maximisers reaction function and so the curve intercepts both axes at higher values. Whether or not sales maximisation is advantageous to the firm depends on the isoprofit curve on which the intersection point lies. Should C intersect with B between w and x then the sales maximiser will make more profit than it would have done as a profit maximiser. Point x is the point at which an isoprofit curve is tangent to B at intersection point x and represents the maximum profit that the sales maximiser can make. This point x is the von Stackleberg equilibrium point but was reached by making a conjecture different to that made by von Stackleberg. The shift to the right can continue until point y and the firm will be better off than it would be as a profit maximiser. A shift to the right beyond point y will cause the firm to make less profit than it would as a profit maximiser. The sales maximiser reaction curve that yields an equilibrium at the von Stackleberg point can be determined using a procedure similar to that of Koutsoyiannis (1979:235). The profit function for firm one and the reaction curve for firm two are: Π 2 = (a bq 2 bq 1 c 2 )q 2 and q 1 = {1/2b} {a bq 2 c 1 } Substituting the reaction function expression for q 1 into the profit function we get: Π 2 = (a bq 2 ½(a bq 2 c 1 ) c 2 )q 2 Differentiating this with respect to q 2, setting to the result to zero to find the maximum point, and solving gives: q 2 * = {1/2b} {a + c 1 2c 2 } Inserting this expression into the reaction curve for q 1 gives: q 1 * = {1/4b} {a 3c 1 + 2c 2 } The point (q 1 *, q 2 * ) is the von Stackleberg equilibrium. The reaction curve of the sales maximiser is: q 2 = {1/2b} {a bq 1 } Substituting into this the expressions for q 1 * and q 2 * and simplifying gives: a = 6c 1 c 2 which reduces to a = 5c in the case of identical costs. Therefore, for identical costs, when a = 5c the reaction curve of the sales maximiser will intersect the reaction curve of the profit maximiser at the von Stackleberg point. Where a < 5c the equilibrium point will lie to the right of the von Stackleberg point and where a > 5c it will lie to the left. Figure 5 also shows how both firms would be worse off if both choose sales maximisation as an objective. In that case both reaction curves shift out to the right and the Nash equilibrium occurs at point z; isoprofit curves for both firms are lower than at this point than at the Cournot equilibrium point. 9
q 1 a/b C (a-c)/b A a/2b D (a-c)/2b B w z x y (a-c)/2b a/2b (a-c)/b a/b q 2 Figure 5. Reaction curves The payoff matrices discussed above have been constructed using strategies where the incentive factor α has a value 1, for strict profit maximisation, or zero, for strict sales maximisation. These can be viewed as similar to pure strategies in game theory terms. Selection by an owner of an incentive factor value between zero and one is similar to the concept of a mixed strategy in game theory terms. A mixed strategy in game theory represents the probability of an actor selecting one or other of the pure strategies and the payoffs of the mixed strategy are represented as expected utilities. In our duopoly model we are choosing a linear combination of the two pure strategies as the strategy to be implemented by a firm and the payoffs are actual not expected; this is a similar although not identical concept but may provide an avenue for further research. Bibliography Baumol, William (1962). On the theory of the expansion of the firm, American Economic Review, v.52, n.5, pp.1078-87. Cyert, Richard and James March (1992). A behavioural theory of the firm, Blackwell Business (2 nd edition, first published 1963). Fama, Eugene (1980). Agency problems and the theory of the firm, Journal of Political Economy, v.88, pp.288-307. Fershtman, Chaim and Kenneth Judd (1987). Equilibrium incentives in oligopoly, American Economic Review, v.77, n.5, pp.927-940. Jensen, Michael and William Meckling (1976). Theory of the firm: managerial behaviour, agency costs and ownership structure, Journal of Financial Economics, v.3, pp.305-360. Koutsoyiannis, A (1979). Modern microeconomics, Macmillan. 10
Marris, Robin (1963). A model of the managerial enterprise, Quarterly Journal of Economics, v.77, n.2, pp.185-209. Osborne, Martin J. and Ariel Rubinstein (1994). A course in game theory, MIT Press. 11