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Stochastics and Statistics

Optimal pricing and advertising in a durable-good duopoly

Anand Krishnamoorthy a,*, Ashutosh Prasad b, Suresh P. Sethi b

a College of Business Administration, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816-1400, United Statesb School of Management, The University of Texas at Dallas, Richardson, TX 75083-0688, United States

a r t i c l e i n f o

Article history:

Received 31 March 2008

Accepted 6 January 2009Available online 10 January 2009

Keywords:

Control

Dynamic programming

Game theory

Marketing

Differential games

a b s t r a c t

This paper analyzes dynamic advertising and pricing policies in a durable-good duopoly. The proposed

infinite-horizon model, while general enough to capture dynamic price and advertising interactions in

a competitive setting, also permits closed-form solutions. We use differential game theory to analyze

two different demand specifications – linear demand and isoelastic demand – for symmetric and asym-

metric competitors. We find that the optimal price is constant and does not vary with cumulative sales,

while the optimal advertising is decreasing with cumulative sales. Comparative statics for the results are

presented.

2009 Elsevier B.V. All rights reserved.

1. Introduction

Decisions on advertising and pricing are inherently dynamic. Advertising effects are both immediate and continue to persist after the

advertisement is withdrawn, due to the memory of the advertisement and the state dependence in buying behavior. Thus, the omission of consideration of future effects results in under-advertising. Pricing dynamics are also quite common and include skimming and penetration

pricing, which are long-run strategies, and price promotions, which are temporary changes in price. Furthermore, pricing and advertising

can interact in their dynamic effects. We consider these dynamic aspects in determining optimal pricing and advertising decisions in this

paper.

Well-known models of advertising effects on sales in the economics and management literature include those by Vidale and Wolfe

(1957), Nerlove and Arrow (1962), and Sethi (1983). Some of these models were descriptive to begin with, but using optimal control theory,

it is possible to derive their profit-maximizing dynamic advertising policies. The models have also been extended to competitive settings.

Some papers that deal with dynamic advertising decisions are Bass et al. (2005), Deal (1979), Erickson (1985, 2008), He et al. (2007, 2008),

Naik et al. (2008), Nair and Narasimhan (2006), Sethi (1973, 1983), Sorger (1989), and Wang and Wu (2001) . In contrast, far fewer models

feature both price and advertising decisions, owing mainly to the fact that introducing price competition in extant models of advertising

competition renders the analysis intractable. This paper specifically addresses this gap in the literature by presenting and solving a dy-

namic duopoly model with inter-related pricing and advertising decisions.

The proposed model is a differential game extension of a recent model by Sethi et al. (2008; SPH, hereafter) that examined advertising

and price decisions by a monopolist firm in a durable-good market. The SPH model’s dynamics is specified as

_ xðt Þ ¼ quðt ÞDð pðt ÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 xðt Þ

p ; xð0Þ ¼ x0 2 ½0;1; ð1Þ

where x(t ) is the cumulative sales (as a fraction of market potential) at time t , D( p(t )), with D0 ( p(t )) < 0, captures the impact of price p(t ) on

the rate of change of cumulative sales, u(t ) is the advertising effort at time t , and q is the effectiveness of advertising. A useful feature of the

SPH model is that it permits closed-form solutions for both price and advertising, a feature that is otherwise lacking in the literature and that

we are able to partially retain in this competitive extension. The square-root term ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xðt Þ

p captures decreasing returns and market satu-

ration similar to the Vidale and Wolfe (1957) formulation, and it also captures an element of word-of-mouth interaction as noted by Sethi

(1983) and Sorger (1989) in the following expansion, for small values of x: ffiffiffiffiffiffiffiffiffiffiffi1 x

p ð1 xÞ þ xð1 xÞ: ð2Þ

0377-2217/$ - see front matter 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2009.01.003

* Corresponding author. Tel.: +1 407 823 1330; fax:+1 407 823 3891.

E-mail address: [email protected] (A. Krishnamoorthy).

European Journal of Operational Research 200 (2010) 486–497

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The modeling of durable goods dynamics is important in economics and management (e.g., Mansfield, 1961; Bass, 1969). A durable good is

one that once purchased by the customer does not need to be repurchased for a lengthy period of time. Examples are cars, televisions, wash-

ing machines, and microwave ovens. In contrast, consumables and perishable goods, such as grocery items, need to be repeatedly repur-

chased. The nature of durable goods means that the market potential depletes with sales and therefore over time, and eventually,

saturation is reached. Thus, the dynamic decisions must also take into account that sales obtained in the present are lost in the future.

Whereas the early durable goods models were descriptive in nature, it did not take long for modelers to posit the effect of decision vari-

ables such as price and advertising on these models and attempt to find normative guidelines. For example, the Bass (1969) model was

extended to include pricing decisions by Robinson and Lakhani (1975). A review of such models is provided by Mahajan et al. (1990). A

more recent study of optimal pricing policies for a monopolist is the paper by Krishnan et al. (1999), based on the following proposed

extension of the Bass (1969) model:

_ xðt Þ ¼ ða þ bxðt ÞÞð1 xðt ÞÞ 1 b_ pðt Þ pðt Þ

; ð3Þ

where p(t ) is the priceat time t , _ p(t ) the change in price at time t , and a, b, and b are model parameters. They find that either a monotonically-

declining or an increasing–decreasing pricing pattern is optimal.

It should be noted that these pricing prescriptions do not take into account the impact of competition. With a few exceptions, the dura-

ble-goods diffusion models, such as the Bass (1969) model, apply to category-level sales and do not account for within-category, brand-

level competition. In contrast, there are advertising models for perishable goods that emphasize market share competition but in which

the category sales remains constant over time because the market does not deplete (e.g., Prasad and Sethi, 2004). In this paper, we make

a contribution by providing optimal pricing and advertising policies in a durable good category in the presence of competition.

Krishnan et al. (2000) propose a brand-level diffusion model to analyze the impact of a late entrant on the diffusion of different brands

of a new consumer durable and that of the category as a whole. They argue that in categories where the primary question is ‘‘whether or

not to buy the category” rather than ‘‘whether or not to buy the brand”, potential adopters of a brand should come from the remainder of

the market (i.e., the unfulfilled market potential). Their model is given by

_ xiðt Þ ¼ ðai þ bi xðt ÞÞð1 xðt ÞÞ; ð4Þwhere _ xiðt Þ is the adoption rate of brand i, xðt Þ ¼

Pi xiðt Þ is the cumulative adoption of the category, and ai and bi are diffusion coefficients.

Notice from Eq. (4) that the market pool of potential adopters of brand i is 1 x(t ), i.e., the proportion of consumers who have not yet bought

any brand in the category, and not 1 xi(t ). The adoption rate depends on attracting these remaining potential customers, similar to the Bass

(1969) model and to the model we propose. However, in our case, price and advertising also influence the adoption rate.

Teng and Thompson (1984) incorporate price and advertising in a new-product oligopoly model but limit their analysis to the case of

price leadership (i.e., there is only one price, that of the largest firm) and resort to numerical analysis to show that the optimal price and

advertising patterns are high initially and then decrease over time. In contrast, under our specification, we are able to solve the differential

game explicitly and obtain that the price is constant and advertising should decrease over time.

There also exist other dynamic models of pricing and advertising. In one of the earliest models of price and advertising in a dynamic

duopoly, Thepot (1983) uses Nerlove–Arrow-type dynamics to obtain the open-loop pricing and advertising decisions under exogenous,

exponential demand growth. Gaugusch (1984) models a duopolyin which one firm chooses its price and does not advertise, while the otherchooses its advertising effort under a fixed price. He finds that the first firm increases its price while the rival decreases its advertising rate.

Dockner and Feichtinger (1986) derive the optimal price and advertising decisions of firms operating in a sticky-price oligopoly. For trac-

tability, they analyze a duopoly and find that the optimal price and advertising should decrease over time if the actual demand is lower

than that specified by the dynamic sales equation.

Chintagunta et al. (1993) analyze a Nerlove–Arrow model of price and advertising in a duopoly in which the total market expands exog-

enously over time. Using numerical analysis, they find that in equilibrium, the advertising and pricing decisions follow the Dorfman–Stei-

ner rule. Mesak and Clark (1998) derive the optimal pricing and advertising policies for a new-product monopolist and, as in Chintagunta

et al. (1993), find that the advertising-sales and advertising-price relationships are of Dorfman–Steiner-type.

It is worth noting that the aforementioned dynamic models of price and advertising are not applicable to durable goods markets (i.e.,

markets in which the market potential depletes over time). In this paper, we analyze a model of a durable-good duopoly, and are able to

derive explicit solutions for the optimal pricing and advertising policies.

The rest of the paper is organized as follows. The next section presents the model and the related assumptions. Section 3 presents the

analysis and the results for the case of linear demand. Section 4 presents the analysis and the results for the case of isoelastic demand, and

Section 5 presents a discussion of the results and managerial implications. Section 6 concludes with a summary and directions for futureresearch.

2. Model

We start by listing the notation in Table 1.

Denote the cumulative sales of firm i, i 2 {1,2}, at time t by si(t ). The rate of change of cumulative units sold, which is the instantaneous

sales, is denoted _siðt Þ, and is given by

_siðt Þ ¼ dsiðt Þdt

¼ qiuiðt ÞDið piðt ÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT siðt Þ s jðt Þ

q ; i; j 2 f1;2g; i – j; ð5Þ

where si(t ) + s j(t ) is the cumulative sales of the category at time t , T is the market potential, subscript j refers to the rival firm, ui(t ) denotes the

advertising effort of firm i at time t , qi is the effectiveness of firm i’s advertising, and Di( pi(t )) is the demand function for firm i, specified as a

function of own price, pi(t ), at time t . Thus, this is a competitive extension of the SPH model. This model has the desirable property that thesales rate goes to zero as the market depletes. Consistent with the literature on durable-goods diffusion models, the potential adopters of

A. Krishnamoorthy et al. / European Journal of Operational Research 200 (2010) 486–497 487



firm i come from the remainder of the market, i.e., from consumers who have not yet purchased from the product category. As in the liter-

ature, the sales can be divided by the market potential T to normalize it to 1.

Each firm chooses its advertising and price to maximize its discounted infinite-horizon profit, given by

maxuiðt Þ; piðt Þ

J i ¼Z 1

0

er it ðð pi miÞ_siðt Þ C ðuiðt ÞÞÞdt ; ð6Þ

where r i is the discount rate of firm i, m i is the marginal cost of production of firm i, and C (ui(t )) is the cost of firm i’s advertising.

Firm i’s total advertising expense is specified as

C ðuiðt ÞÞ ¼ c i2uiðt Þ2; ð7Þ

where we refer to c i/2 as the unit cost of advertising, for convenience. This specification is common in the literature, where the cost of adver-

tising is assumed to be convex and, more specifically, quadratic (e.g., Sethi, 1983; Sorger, 1989). It captures the diminishing returns to adver-

tising. Alternatively, one can use linear advertising costs and have advertising appear as a square-root in the state equations.

The differential game between the two firms can therefore be summarized as follows:

maxu1ðt Þ; p1ðt Þ

J 1 ¼R 10 er 1t ð p1 m1Þ_s1ðt Þ c 1

2 u1ðt Þ2

dt ;

maxu2ðt Þ; p2ðt Þ

J 2 ¼R 10 er 2t ð p2 m2Þ_s2ðt Þ c 2

2 u2ðt Þ2

dt ;

ð8Þ

s:t: _s1ðt Þ ¼ q1u1ðt Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT s1ðt Þ s2ðt Þ

p D1ð p1ðt ÞÞ;

_s2ðt Þ ¼ q2u2ðt Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT s1ðt Þ s2ðt Þ

p D2ð p2ðt ÞÞ:

ð9Þ

In this paper, we adopt the feedback solution concept for differential games. This better reflects the competitive dynamics of the two rivalsover time since feedback equilibria are subgame perfect. In addition, several papers provide evidence that a feedback solution fits the data

better than its open-loop counterpart (e.g., Chintagunta and Vilcassim, 1992).

Next, we perform a detailed analysis of the model. To obtain the optimal policies, we solve the differential game given by (8) and (9) to

obtain the feedback Nash equilibriumstrategies. For expositional convenience, we will suppress the time-dependence of the state and con-

trol variables when no confusion arises.

The Hamilton–Jacobi–Bellman (HJB) equation for firm i is

r iV i ¼ maxui; pi

ð pi miÞ qiui

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT si s j

p Dið piÞ

c i2u2i þ

@ V i@ si

qiui


q Dið piÞ

þ@ V i

@ s jq ju j


q D jð p jÞ

;

8>><>>: ð10Þ

where V i = V i(si,s j) is the value function of firm i.

Writing the first-order conditions for pi and ui from the HJB equation in (10), we get

qiui


q Dið piÞ þ ð pi miÞ qiui


q D0

ið piÞ

þ @ V i@ si

qiui


q D0

ið piÞ

¼ 0; ð11Þ

ð pi miÞ qi


q Dið piÞ

c iui þqi


q Dið piÞ

@ V i@ si

¼ 0: ð12Þ

We presently assume that the solutions are in the interior and later show that this is true.

To determine the optimal pricing and advertising strategies of the two firms, we need to specify the demand function. We start with the

linear demand specification.

3. Linear demand specification

We now consider the following linear demand specification:

Dið piðt ÞÞ ¼ ai bi piðt Þ; ð13Þ

where ai is the demand intercept and bi represents price sensitivity. The linear demand function is one of the most commonly used in theliterature (e.g., Petruzzi and Dada, 1999).

Table 1

Notation.

si(t ) Cumulative sales of firm i at time t

T Market potential

ui(t ) Advertising effort of firm i at time t

pi(t ) Price of firm i at time t

c i Coefficient associated with the advertising cost of firm i

qi Effectiveness of advertising of firm i

mi

Marginal cost of production of firm i

ai Demand intercept of firm i (linear demand)

bi Price sensitivity of firm i (linear demand)

gi Price elasticity of firm i (isoelastic demand)

r i Discount rate of firm i

V i(si, s j) Value function of firm i when its cumulative sales is si and its rival’s is s j

488 A. Krishnamoorthy et al. / European Journal of Operational Research 200 (2010) 486–497



Substituting (13) and simultaneously solving the two first-order conditions in (11) and (12) yields the optimal price and advertising

policies, denoted pi ðsi; s jÞ and u

i ðsi; s jÞ, respectively, which are given in Proposition 1.

Proposition 1. The optimal feedback pricing and advertising strategies of firm i are given by

pi ðsi; s jÞ ¼

1

2

ai

bi

þ mi @ V i@ si

; ð14Þ

ui ðsi; s j

Þ ¼ qi

4c ibi

ai

þ b

i

@ V i

@ si mi

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT

si

s jq : ð

15

ÞThe sales trajectories corresponding to the equilibrium in (14) and (15) can be obtained by substituting the optimal strategies in (14) and

(15) and solving the two state equations in (9). Substituting the optimal solutions from (14) and (15) into the HJB equation in (10) and sim-

plifying yields

V iðsi; s jÞ ¼ 1

32c ic jr ib2

i b jc jb jq

2

i ai þ bi

@ V i@ si

mi

4

þ 4c ib2

i q2

j

@ V i@ s j

a j þ b j@ V j@ s j

m j

3 !

ðT si s jÞ: ð16Þ

To prove that the pair of strategies in (14) and (15) forms a feedback equilibrium, we need to show that there exist two continuously-

differentiable functions V i(si, s j), i, j 2 {1,2}, i– j, which satisfy the partial differential equations in (16) and the boundary condition that

limt ?1 V i(si(t ),s j(t )) = 0.

We propose the following form for the value function V i(si, s j):

V iðsi; s j

Þ ¼ ki

ðT

si

s jÞ )

@ V i

@ si ¼ @ V i

@ s j ¼ ki:

ð17

ÞWith this, and Proposition 1, we can conclude that the optimal advertising is decreasing with cumulative category sales and, therefore,

over time. Note that each firm’s advertising is positive and increasing in the unfulfilled market potential. In other words, firms choose

high advertising levels not only if their cumulative brand sales are low, but also when the rival’s cumulative brand sales are low. This

is because a low cumulative sales level of either firm means there is more of the unfulfilled market potential to tap into. We can also

conclude, given the linear value function and Proposition 1, that p i ðsi; s jÞ > 0 if the condition

ai

biþ mi þ ki > 0 holds, which is clearly true

since the value function should be positive. The optimal price is independent of the cumulative sales level and thus constant over

time.

To explore the solution fully, we need further insight into the constant ki, since it directly affects the optimal decisions and the value

function. Equating the coefficients of T si s j in (16), we have

ki ¼c jb jq

2i ðai biðki þ miÞÞ4 4c ib

2

i q2 j kiða j b jðk j þ m jÞÞ3

32c ic jr ib2

i b j; i; j 2 f1;2g; i – j: ð18Þ

We first consider the case of symmetric firms.

3.1. Symmetric firms

Eq. (18) represents the system of simultaneous quartic equations that has to be solved to obtain the Nash equilibrium of the differential

game. For the case of symmetric firms, i.e., c i = c j = c , r i = r j = r , qi = q j = q, mi = m j = m, ai = a j = a, and bi = b j = b, a symmetric solution will be

obtained with ki = k j = k. Rewriting Eq. (18), we now have the following quartic equation to solve for k:

k ¼ q2ða 5bðk þ mÞÞða bðk þ mÞÞ3

32cr b2 : ð19Þ

Since the value function is positive, k > 0 on the left-hand side of Eq. (19). For the right-hand side to be positive, we require either that

k > ab m or k < a

5b m. The former can be ruled out because for demand to be non-negative, we require a b piP 0, or pi 6 a/b. Moreover,

we know from the solution for price in (14) that pi ¼ 1

2ab þ m þ k

. Therefore, k > a

b m is infeasible.

Remark 1. The solution must also satisfy the boundary condition that limt ?1V i(si(t ),s j(t )) = 0. We find that k < a

5b

m satisfies this,

whereas for k > ab

m, it is not satisfied.

With the solution for k, the optimal price and advertising solutions, "i, can be rewritten as

pi ðsi; s jÞ ¼

1

2

ab þ m þ k

; ð20Þ

ui ðsi; s jÞ ¼

q4c b

ða bðk þ mÞÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT si s j

q : ð21Þ

It is evident that the solution for the optimal price in (20) is a constant.

The comparative statics for the parameters on the variables of interest are given in Table 2. Since we have assumed symmetry, u2, p

2,

and V 2 have the same comparative statics as u1, p

1, and V 1, respectively.

Remark 2. For the comparative statics of k w.r.t. the model parameters, we have @ k@ c < 0, @ k@ r < 0, @ k@ q > 0, @ k@ m < 0, @ k@ a > 0, and @ k

@ b < 0.1

1 All the proofs can be found in the Appendix.


http://-/?-

http://-/?-



In summary, the comparative statics of k w.r.t. the model parameters are in the expected directions (i.e.,k decreases with the unit cost of

advertising, the discount rate, the marginal cost of production, and the price sensitivity of demand, and increases with the effectiveness of

advertising and the baseline demand). Since the value functions are directly proportional to k, the comparative statics for k carry through to

that for the value functions of the two symmetric firms.

Next, consider the comparative statics of price w.r.t. the model parameters.

Remark 3. For the comparative statics of p w.r.t. the model parameters, we have @ p@ c < 0, @ p@ r < 0, @ p@ q > 0, @ p@ m > 0, @ p@ a > 0, and @ p

@ b < 0.

We find that, for every parameter except m, the comparative statics of price are the same as those of k. For the marginal cost m, we find

that the optimal price is increasing in m. These signs are in the expected directions.

Finally, consider the comparative statics of advertising (taking s1 and s2 as given) w.r.t. the model parameters.

Remark 4. For the comparative statics of u w.r.t. the model parameters, we have @ u@ c < 0, @ u@ r > 0, @ u@ q > 0, @ u@ m < 0, @ u@ a > 0, and @ u

@ b < 0.

In other words, the optimal advertising intensity decreases with the unit cost of advertising, the marginal cost of production, and the

price sensitivity of demand, and increases with the effectiveness of advertising and the baseline demand. One would expect that due to the

carryover effect of advertising, firms that value the future more would advertise at higher levels, but, interestingly, we find that lower dis-

count rates lead to lower levels of advertising.

We next turn our attention to the case of asymmetric competitors.

3.2. Asymmetric firms

We now examine the duopolistic competition between asymmetric firms trying to maximize their discounted profits in an infinite-hori-

zon setting. To solve the differential game, one has to solve the following set of simultaneous equations:

k1 ¼ 1

32c 1c 2r 1b2

1b2

c 2b2q2

1ða1 b1ðk1 þ m1ÞÞ4 4c 1b

2

1q2

2k1ða2 b2ðk2 þ m2ÞÞ3

; ð22Þ

k2 ¼ 1

32c 1c 2r 2b1b22c 1b

1q

2

2ða2

b2ðk2 þ

m2ÞÞ4

4c 2b

2

2q

2

1

k2ða1

b1ðk1 þ

m1ÞÞ3 : ð

23Þ

Given the solutions for k1 and k2, we see that the optimal price and advertising levels are indeed positive, as assumed previously. The optimal

price and advertising policies can now be rewritten as

pi ðsi; s jÞ ¼

1

2

ai

bi

þ mi þ ki

; ð24Þ

ui ðsi; s jÞ ¼

qi

4c ibi

ðai biðki þ miÞÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT si s j

q : ð25Þ

We use Mathematica to numerically solve the system of quartic equations given by (22) and (23) for a range of parameter values.

The comparative statics for the parameters on the variables of interest in the asymmetric case are presented in Table 3.

From Table 3, one can see that ki is increasing in c j, m j, and b j, and decreasing in r j, q j, and a j. The results for the own parameters are

similar to those in the symmetric case (i.e., ki is decreasing in c i, r i, m i, and bi, and increasing in qi and ai).

For the comparative statics of price w.r.t. the model parameters, we find that pi is decreasing in c i, r i, and bi, and increasing in qi, mi, and

ai. This is consistent with our findings in the symmetric case. For the cross parameters, we find that p i decreases with r j, q j, and a j, andincreases with c j, m j, and b j (see Fig. 1).

For the comparative statics of advertising (taking s1 and s2 as given) w.r.t. the model parameters, we find that, consistent with our find-

ings in the symmetric case, ui is decreasing in c i, mi, and bi, and increasing in r i, qi, and ai. For the cross parameters, we find that ui increases

with r j, q j, and a j, and decreases with c j, m j, and b j (see Fig. 2).

Table 2

Comparative statics for the symmetric case with linear demand.

Variables c r q m a b

k ; ; " ; " ;

pi ; ; " " " ;

ui ; " " ; " ;

V i ; ; " ; " ;

" increase; ; decrease.

Table 3

Comparative statics for the asymmetric case with linear demand.

Variables c i c j r i r j qi q j mi m j ai a j bi b j

ki ; " ; ; " ; ; " " ; ; "

pi ; " ; ; " ; " " " ; ; "

ui ; ; " " " " ; ; " " ; ;

V i ; " ; ; " ; ; " " ; ; "





This concludes our study of the linear specification for the demand function Di( pi(t )). In the next section, we analyze a nonlinear

specification.

4. Isoelastic demand specification

We now consider the isoelastic demand function:

Dið piðt ÞÞ ¼ piðt Þgi ; ð26Þso called because the elasticity of demand is gi, a constant. It is assumed that the demand is elastic, i.e., gi > 1, so that the optimal price is

finite.

Substituting (26) and simultaneously solving the two first-order conditions in (11) and (12) yields the optimal price and advertising

policies given in Proposition 2. We assume @ V i@ si

< 0 so that pi ðsi; s jÞ > 0 and later show this is indeed true.

Proposition 2. The optimal feedback pricing and advertising strategies of firm i are given by:

pi ðsi; s jÞ ¼

gi

gi 1

mi @ V i

@ si

; ð27Þ

ui ðsi; s jÞ ¼

qi

c igi

gi

gi 1

mi @ V i

@ si

giþ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT si s j

q : ð28Þ

In Proposition 2, note that, as with linear demand, each firm’s advertising is increasing in the unfulfilled market potential. In other

words, firms choose high advertising levels when there is more of the unfulfilled market potential to tap into.

The sales trajectories corresponding to the equilibrium in (27) and (28) can be obtained by substituting the optimal price and adver-

tising decisions in (27) and (28) and solving the two state equations in (9). Substituting the optimal solutions from (27) and (28) into

the HJB equation in (10) and simplifying, we have

V i

ðsi; s j

Þ ¼

1

2r i

q2i mi @ V i

@ si

2gi

gi1

mi @ V i

@ si

2gi

c iðgi 1Þ2

þ

2q2 j

@ V i@ s j

g jg j1

m j @ V j

@ s j

2g jþ1

c jg j

0B@

1CAð

T

si

s j

Þ:

ð29

Þ

10.2c = ,

2[0,0.4]c ∈

0.10 0.15 0.20 0.25 0.30 0.35 0.40c2

1.25

1.30

1.35

1.40

1.45

p1, p2

p2

p1

11 ρ = ,

2[0,2] ρ ∈

0.5 1.0 1.5 2.0 ρ 2

1.15

1.20

1.25

1.30

1.35

1.40

p1, p2

p2

p1

10.2m = ,

2[0,0.4]m ∈

0.10 0.15 0.20 0.25 0.30 0.35 0.40m2

1.24

1.26

1.28

1.30

1.32

p1, p2

p2

p1

10.5 β = ,

2 [0,1] β ∈

0.4 0.6 0.8 1.0 β 2

1.5

2.0

p1 , p2

p2

p1

Fig. 1. Comparative statics of p1 and p2 for asymmetric firms with linear demand. (Unless otherwise stated, theparameter values in Figs. 1 and2 are: c 1 = 0.2, c 2 = 0.2, r 1 = 0.1,

r 2 = 0.1,q1 = 1, q2 = 1, m1 = 0.2, m2 = 0.2,a1 = 1, a2 = 1, b1 = 0.5,b2 = 0.5, and T s1 s2 = 1. Due tospace restrictions, the graphs for {r 1 = 0.1, r 2 2 [0,0.2]} and {a1 = 1, a2 2 [0,2]}

are not presented here.)


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We try the following form for the value function V i(si, s j):

V iðsi; s jÞ ¼ kiðT si s jÞ ) @ V i@ si

¼ @ V i@ s j

¼ ki: ð30Þ

Equating the coefficients of T si s j in (29), we have

ki ¼ 1

2r i

q2i ðmi þ kiÞ2 gi

gi1

ðki þ miÞ

2gi

c iðgi 1Þ2 2q2

j k jg j

g j1

ðk j þ m jÞ

2g jþ1

c jg j

0B@

1CA; i; j 2 f1;2g; i– j: ð31Þ

As before, the analysis of this equation is discussed for symmetric and asymmetric firms. We first consider the case of symmetric firms.

4.1. Symmetric firms

Eq. (31) represents the system of simultaneous equations that need to be solved to obtain the Nash equilibriumof the differential game.

For symmetric firms, c i = c j = c , r i = r j = r , qi = q j = q, mi = m j = m, gi = g j = g, and ki = k j = k. Eq. (31) now becomes

k ¼q2ðm þ kÞðm þ ð3 2gÞkÞ gðmþkÞ

g1

2g

2cr

ðg

1

Þ2

: ð32Þ

Given the solution for k, the optimal price and advertising solutions can be rewritten as

10.2c = ,

2[0,0.4]c ∈

0.10 0.15 0.20 0.25 0.30 0.35 0.40c2

1.5

2.0

2.5

u1, u2

u2

u1

11 ρ = ,

2[0,2] ρ ∈

1.0 1.5 2.0 ρ 2

1.0

1.5

2.0

u1 , u2

u2

u1

10.2m = ,

2[0,0.4]m ∈

0.10 0.15 0.20 0.25 0.30 0.35 0.40m2

1.3

1.4

1.5

u1, u2

u2

u1

10.5 β = ,

2[0,1] β ∈

0.4 0.6 0.8 1.0 β 2

1.5

2.0

2.5

u1, u2

u2

u1

Fig. 2. Comparative statics of u1 and u2 (given s1 and s2) for asymmetric firms with linear demand.

Table 4

Comparative statics for the asymmetric case with isoelastic demand.

Variables c i c j r i r j qi q j mi m j gi g j

ki ; " ; ; " ; ; " ; ;

pi ; " ; ; " ; " " ; ;

ui

; ; " " " " ; ; ; "

V i ; " ; ; " ; ; " ; ;





pi ðsi; s jÞ ¼

gðm þ kÞg 1

; ð33Þ

ui ðsi; s jÞ ¼

qc g

gðm þ kÞg 1

gþ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT si s j

q : ð34Þ

We use Mathematica to numerically solve Eq. (32) for k.

As before, the solution for the optimal advertising is decreasing with cumulative sales and the optimal price is a constant.

Finally, we consider the case of asymmetric firms.

4.2. Asymmetric firms

To solve the differential game for asymmetric firms, one needs to solve the following set of simultaneous equations:

k1 1

2r 1

q21ðm1 þ k1Þ2 g1ðm1þk1Þ

g11

2g1

c 1ðg1 1Þ2 2q2

2k1

g2ðm2þk2Þg21

2g2þ1

c 2g2

0B@

1CA ¼ 0; ð35Þ

k2 1

2r 2

q22ðm2 þ k2Þ2 g2ðm2þk2Þ

g21

2g2

c 2ðg2 1Þ2

2q21k2

g1ðm1þk1Þg11

2g1þ1

c 1g1

0B@

1CA ¼ 0: ð36Þ

The solutions are complicated, and numerical analysis is used to obtain the set of positive-real roots that satisfy the system of equations in

(35) and (36). Given the solutions for k1 and k2, the optimal price and advertising solutions can be rewritten as

pi ðsi; s jÞ ¼

gi

gi 1

ðmi þ kiÞ; ð37Þ

ui ðsi; s jÞ ¼

qi

c igi

gi

gi 1

ðmi þ kiÞ

giþ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT si s j

q : ð38Þ

The comparative statics for the parameters on the variables of interest in the asymmetric case are presented in Table 4.

From Table 4, one can see that ki is decreasing in c i, r i, mi, and gi, and increasing in qi. For the cross parameters, we find that ki decreases

with r j, q j, and g j, and increases with c j and m j. Since the value function of firm i is directly proportional to ki, the comparative statics for ki

carry through to that for the value functions of the respective firms.

10.2c = ,

2 [0,0.4]c ∈

0.10 0.15 0.20 0.25 0.30 0.35 0.40c2

12

14

16

18

20

22

p1, p2

p2

p1

11 ρ = ,

2[0,2] ρ ∈

1.0 1.5 2.0 ρ 2

5

10

15

20

25

30

p1, p2

p2

p1

10.2m = ,

2[0,0.4]m ∈

0.10 0.15 0.20 0.25 0.30 0.35 0.40m2

13.0

13.5

p

1, p

2

p2

p1

11.25η = ,

2 [1.01,2]η ∈

1.4 1.6 1.8 2.0η 2

10

20

30

40

p1, p2

p2

p1

Fig. 3. Comparative statics of p1 and p2 for asymmetric firms with isoelastic demand. (Unless otherwise stated, the parameter values in Figs. 3 and 4 are: c 1 = 0.2, c 2 = 0.2,r 1 = 0.1, r 2 = 0.1, q1 = 1, q2 = 1, m1 = 0.2, m2 = 0.2, g1 = 1.25, g2 = 1.25, and T s1 s2 = 1. Due to space restrictions, the graphs for {r 1 = 0.1, r 2 2 [0,0.2]} are not presented here.)




The comparative statics of price w.r.t. the model parameters are presented in Fig. 3. We find that pi is decreasing in c i, r i, and gi, and

increasing in qi and mi. For the cross parameters, pi decreases with r j, q j, and g j, and increases with c j and m j.

Fig. 4 presents the comparative statics of advertising (taking s1 and s2 as given) w.r.t. the model parameters. We find that ui is decreasingin c i, mi, and gi, and increasing in r i andqi. For the cross parameters, we find that ui increases with r j, q j, and g j, and decreases with c j and m j.

In summary, the comparative statics on the common parameters (i.e., c i, c j, r i, r j, qi, q j, mi, and m j) are in the same direction in both the

linear and the isoleastic demand specifications, suggesting that the results are robust across different demand specifications.

5. Discussion

Proposition 1 deals with the linear demand specification and it provides the expressions for the optimal price and the optimal adver-

tising. It finds that the optimal price is constant and that the optimal advertising decreases with cumulative sales. We now discuss the rel-

evance of these results.

The result that the optimal price is constant may be implied by sales dynamics models, e.g., Bass (1969), that do not model price as a

control variable but still fit sales data well; possible explanations for this are that price is constant or correlated with time (see Bass et al.,

1994). Bayus (1992) estimates the empirical price trends of three consumer durables – console TVs, CD players, and telephones – and finds

that their prices have declined over time. However, price time series in durables goods is complicated by technology and cost factors. For

example, Dolan and Simon (1996, p. 293) describe price trends for personal computers from 1987 to 1992, showing it remained flat in theeducation segment, declined in the home and business segments, and increased in the scientific segment. The price declines were attrib-

uted to cost decreases and intense competition and the price increases to added technological value.

Proposition 1 also states that the optimal advertising level decreases with cumulative category sales. This is because a high cumulative

sales level of either firm means there is less of the unfulfilled market potential to tap into. This result is consistent with the observation that

firms begin to drastically reduce their advertising efforts in the decline stage of the product life cycle ( Ferrell and Hartline, 2008, p. 286).

Comparing the results in the linear-demand case in Tables 2 and 3, one can see that the comparative statics results for the own-param-

eters in the asymmetric case are the same as those in the symmetric case, i.e., (a) the value-function coefficient of a firm is decreasing in its

unit cost of advertising, discount rate, marginal cost of production, and price sensitivity, and increasing in its advertising effectiveness and

demand intercept, (b) the optimal price of a firm is decreasing in its unit cost of advertising, discount rate, and price sensitivity, and

increasing in its advertising effectiveness, marginal cost of production, and demand intercept, and (c) the optimal advertising level (taking

sales as given) of a firm is decreasing in its unit cost of advertising, marginal cost of production, and price sensitivity, and increasing in its

discount rate, advertising effectiveness, and demand intercept.

For the cross parameters, we find that (a) the value-function coefficient of a firm is decreasing in the rival firm’s discount rate, adver-

tising effectiveness, and demand intercept, and increasing in the rival’s unit cost of advertising, marginal cost of production, and price sen-sitivity, (b) the optimal price of a firm is decreasing in the rival firm’s discount rate, advertising effectiveness, and demand intercept, and

10.2c = ,

2[0,0.4]c ∈

0.10 0.15 0.20 0.25 0.30 0.35 0.40c2

2.0

2.5

3.0

3.5

4.0

u1, u2

u2

u1

11 ρ = ,

2[0,2] ρ ∈

1.0 1.5 2.0 ρ 2

1.0

1.5

2.0

2.5

3.0

u1 , u2

u2

u1

10.2m = ,

2[0,0.4]m ∈

0.10 0.15 0.20 0.25 0.30 0.35 0.40m2

2.09

2.10

2.11

2.12

2.13

2.14

u1, u2

u2

u1

11.25η = ,

2[1.01,2]η ∈

1.4 1.6 1.8 2.0η 2

2.0

2.5

3.0

3.5

4.0

4.5

u1, u2

u2

u1

Fig. 4. Comparative statics of u1 and u2 (given s1 and s2) for asymmetric firms with isoelastic demand.




increasing in the rival’s unit cost of advertising, marginal cost of production, and price sensitivity, and (c) the optimal advertising level (tak-

ing sales as given) of a firm is decreasing in the rival firm’s unit cost of advertising, marginal cost of production, and price sensitivity, and

increasing in the rival’s discount rate, advertising effectiveness, and demand intercept.

Whereas Proposition 1 deals with a linear demand function, Proposition 2 examines an isoelastic demand function. The optimal price

and advertising paths, however, are not qualitatively affected. Therefore, the empirical support needed for Propositions 1 and 2 is identical.

Qualitatively, Propositions 1 and 2 state the same thing – that the optimal price is a constant and that the optimal advertising declines

as the market potential depletes. This is a good thing because it suggests that the results are at least somewhat robust to specification

changes. A remaining practical issue for the firm is how to estimate the parameters of the model and decide which specification to use.

Although we do not examine econometric issues here, a few points are in order. As Chintagunta and Jain (1995) point out, when firms make

their decisions strategically, the levels of marketing mix variables like price and advertising are endogenously determined, i.e., the first-

order conditions for Nash equilibria in price and advertising are functions of the state variable (market share or, as in our case, sales).

As a result, the estimation of the model parameters should specify a system of simultaneous equations that consists of both the response

functions and the equilibrium conditions. Ignoring this endogeneity problem will lead to inconsistent parameter estimates and, therefore,

incorrect decision-making by managers. More recently, researchers have begun using Kalman filter to estimate the parameters of dynamic

response models (e.g., Naik et al., 1998). Since the state equation in our model is nonlinear in sales, one has to use extended Kalman filter to

estimate the model parameters (see Naik et al., 2008).

Looking at the results in Table 4, one can see the following comparative statics results for the own-parameters in the isoelastic case: (a)

The value-function coefficient of a firm is decreasing in its unit cost of advertising, discount rate, marginal cost of production, and price

elasticity, and increasing in its advertising effectiveness, (b) the optimal price of a firm is decreasing in its unit cost of advertising, discount

rate, and price elasticity, and increasing in its advertising effectiveness and marginal cost of production, and (c) the optimal advertising

level (taking sales as given) of a firm is decreasing in its unit cost of advertising, marginal cost of production, and price elasticity, and

increasing in its discount rate and advertising effectiveness.

For the cross parameters, we find that (a) the value-function coefficient of a firm is decreasing in the rival firm’s discount rate, adver-

tising effectiveness, and price elasticity, and increasing in the rival’s unit cost of advertising and marginal cost of production, (b) the opti-

mal price of a firm is decreasing in the rival firm’s discount rate, advertising effectiveness, and price elasticity, and increasing in the rival’s

unit cost of advertising and marginal cost of production, and (c) the optimal advertising level (taking sales as given) of a firm is decreasing

in the rival firm’s unit cost of advertising and marginal cost of production, and increasing in the rival’s discount rate, advertising effective-

ness, and price elasticity.

Comparing the results in the linear and the isoelastic demand cases, we see that the comparative statics on the common parameters

(unit cost of advertising, discount rate, advertising effectiveness, and marginal cost of production) are in the same direction in both demand

specifications, suggesting that the results are robust across the two demand specifications.

In the literature review, we discussed how our results compare with those of extant dynamic models of price and advertising, with and

without competition. Consider, for example, the monopoly model of Sethi et al. (2008). Consistent with that model, wefind that, in the case

of linear demand, the value-function coefficient and the optimal price decrease with the discount rate and the price sensitivity of demand,

and increase with the effectiveness of advertising. The optimal advertising intensity increases with the discount rate and the effectiveness

of advertising, and decreases with the price sensitivity of demand. In addition to the three parameters, we also derive the comparative stat-

ics for two additional parameters (the marginal cost of advertising and the baseline demand) that are not in the SPH model. While the SPHmodel considers the case of a monopolist, our analysis also presents the comparative statics with respect to the competitor’s parameters.

The comparative statics result that the optimal advertising ui ðsi; s jÞ is increasing in the discount rate r is not obvious. In fact, some non-

durable-goods models of advertising dynamics suggest that the optimal advertising should decrease if the discount rate increases (e.g., Bass

et al., 2005). To understand the intuition behind our result, we first note that advertising has twoeffects: it increases the current sales and it

also affects future sales via the state variable. The latter is sometimes called the carryover effect of advertising. In a nondurable-goods mod-

el, the carryover effect is positive, i.e., both current and future sales increase with advertising. By definition, as the discount rate increases,

the contribution of future sales to the firm’s objective decreases. Thus, the effectiveness of advertising is reduced because the carryover

effect has become less important. This leads to the conclusion in the literature that advertising should decrease when the discount rate

increases.

In contrast, in a durable-goods model, the size of the market is fixed. Thus, if the current sales increases, the future sales must decrease.

Another way to state the two effects of advertising for the durable goods case is that advertising leaves the total sales unaffected but it

increases the allocation of sales to the near term versus the future. When the discount rate increases, it is profitable to have a greater allo-

cation of sales to the near term. Therefore, the optimal advertising is increasing in the discount rate. This intuition has not been pointed out

in previous studies.Next, we discuss the practical meaning of the value-function coefficient ki. The value function for firm i at time t , denoted V i(si(t ), s j (t )), is

the value of its total net discounted future profit stream, assuming that each firm makes optimal decisions starting from the sales (si (t ), s j

(t )) at time t . We know from the optimal control theory that@ V iðsi;s j Þ

@ si, evaluated at the optimal (si (t ), s j (t )), is called the shadowprice at time t ,

and it is the marginal improvement of the value function if a small increase is applied to the starting sales for firm i at time t (Sethi and

Thompson, 2000, p. 35). In our case, @ V i@ si

¼ ki is the shadow price and it is constant and negative. It is negative because in a durable goods

setting, due to a fixed sales potential, an increase of the initial sales level reduces the potential for further earnings due to market

saturation.

Since each firm’s profit in the symmetric case (for given si and s j) is directly proportional to k, the two firmswould want k tobe as high as

possible. The analysis shows that k is increasing in q and a and decreasing in c and r . Therefore, one of the ways for firms to increase their

profit would be to increase the effectiveness of their advertising, i.e., increase q. To increase the effectiveness of advertising, firms could

create more alternative advertisements and use pre-testing to select the best ad copy ( Gross, 1972).

In our analysis, we also noted the impact of the value-function coefficient k on the optimal pricing and advertising decisions and profit

of the firm. Our analysis can, therefore, help managers determine the profit-maximizing levels of advertising and price. Once k is deter-

mined, the application of the formulae to practice is straightforward. The approximate value of k can be obtained offline, and the mo-ment-to-moment decisions can be made using the value of k from the paper.




We acknowledge that the predictions of the model are subject to the model’s specification. In particular, we assume that the interaction

between price and advertising is multiplicative. Future research could investigate how our results change when the interaction between

price and advertising is more complicated than that specified in this paper. One could also formulate a Stackelberg game to examine how

the results change when there is sequential entry instead of simultaneous entry (e.g., see Fruchter and Messinger, 2003). It would also be

interesting to extend the current model to study price and advertising competition in a three-or-more-firm oligopoly along the lines of

Fruchter (1999) and Naik et al. (2008), and the impact of uncertainty on the optimal price and advertising decisions, as in Prasad and Sethi

(2004).

6. Conclusions

This paper analyzes a model of advertising and price competition in a dynamic durable-good duopoly. The previous literature on opti-

mal price and advertising decisions in this setting is very limited. Theoretically, we extend a recent monopoly model by Sethi et al. (2008),

which incorporates important elements such as advertising and price interaction, market saturation, and feedback solutions, by including

competition and without losing the advantage of analytical tractability. Using differential game theory, we obtain the optimal advertising

and pricing decisions for two different demand specifications and present the comparative statics for symmetric and asymmetric compet-

itors. An important feature of the proposed model is that, while it is realistic enough to capture price and advertising in a competitive set-

ting, it allows for explicit feedback solutions.

The analysis reveals that the optimal advertising effort should decrease over time as more of the market potential is captured, and that

the optimal price is stationary. Normative results based on the analysis of the model for symmetric and asymmetric competitors suggest

that when the demand is linear in price, each firm’s optimal price and profit should increase with its advertising effectiveness and base

demand, and decrease with its unit cost of advertising, discount rate, and price sensitivity. These should also increase with the rival’s unit

cost of advertising, marginal cost of production, and price sensitivity, and decrease with the rival’s discount rate, the effectiveness of the

competitor’s advertising, and its base demand. Interestingly, the optimal advertising effort moves in the same direction for both compet-

itors (i.e., decreases with the unit cost of advertising, marginal cost of production, and price sensitivity, and increases with the discount

rate, advertising effectiveness, and base demand).

For isoelastic demand, we find that each firm’s optimal price and profit should increase with its advertising effectiveness, and decrease

with its unit cost of advertising and discount rate. These should also decrease with the rival’s discount rate, advertising effectiveness, and

price elasticity, and increase with the rival’s unit cost of advertising and marginal cost of production. As with linear demand, the optimal

advertising effort moves in the same direction for both competitors (i.e., decreases with the unit cost of advertising and marginal cost of

production, and increases with the discount rate and advertising effectiveness).

The current study leaves open avenues for future research. A promising avenue for future research would be to study how the results

change when there is sequential entry instead of simultaneous entry. Such models would be formulated as Stackelberg games (e.g., see

Fruchter and Messinger, 2003; He et al., 2007, 2008, 2009). Specifically, one could obtain and analyze feedback Stackelberg equilibria in

a vertical supply chain dealing with durable goods along the lines of He et al. (2009), which considers non-durables. Another fruitful exten-

sion is to study price and advertising competition in an oligopoly along the lines of Fruchter (1999) and Naik et al. (2008). Although duop-

oly models are representative of many real-world markets, modeling markets characterized by three or more firms might yield additional

insights.

Proof of Remark 2

Denote Eq. (19) as f ð; kÞ ¼ k q2ðabð5kþmÞÞðabðkþmÞÞ3

32cr b2 ¼ 0. The implicit function theorem yields, for any parameter v, @ k@ v ¼ @ f =@ v

@ f =@ k. We have

@ k

@ c ¼ q2ða bð5k þ mÞÞða bðk þ mÞÞ3

4c bð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ < 0;

@ k

@ r ¼ q2ða bð5k þ mÞÞða bðk þ mÞÞ3

4r bð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ< 0;

@ k

@ q ¼ qða bð5k þ mÞÞða bðk þ mÞÞ3

2bð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ > 0;

@ k

@ m ¼ q

2

ða b

ð4k

þ m

ÞÞða b

ðk þ

mÞÞ

2

8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2 < 0;

@ k

@ a ¼ q2ða bð4k þ mÞÞða bðk þ mÞÞ2

bð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ> 0; and

@ k

@ b ¼ q2ða2 2kab b2ðk þ mÞð5k þ mÞÞða bðk þ mÞÞ2

2b2ð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ

:

Note that a2 2kab b2(k + m)(5k + m) can be written as (a b(5k + m))(a + b(3k + m) ) + 2b2k(5k + m), which is positive since k < a5b

m.

Therefore, @ k@ b < 0.

Proof of Remark 3

For the comparative statics of price, note from (20) that for any parameter v, except for m, the sign of @ p@ v is the same as that of

@ k@ v. Therefore, we have

@ p@ c < 0,

@ p@ r < 0,

@ p@ q > 0,

@ p@ a > 0, and

@ p@ b < 0. For m, we have

@ p@ m ¼

12 1 þ

@ k@ m

¼

8cr bþq2ðabðkþmÞÞ3

2ð8cr bþq2ð2abð5kþ2mÞÞðabðkþmÞÞ2Þ > 0.




Proof of Remark 4

From (21), we have

@ u

@ c ¼ qð16cr b þ q2ð3a bð5k þ 3mÞÞða bðk þ mÞÞ2Þða bðk þ mÞÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT s1 s2

p

8c 2bð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ < 0;

@ u

@ r ¼ q3ða bð5k þ mÞÞða bðk þ mÞÞ4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT s1 s2

p

8cr bð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ> 0;

@ u

@ q ¼ ð8cr b þ q2ða bmÞða bðk þ mÞÞ2Þða bðk þ mÞÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT s1 s2

p

4c bð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ > 0;

@ u

@ m ¼ qða bðk þ mÞÞð8cr b þ q2ða bðk þ mÞÞ3Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT s1 s2

p

2c ð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ< 0;

@ u

@ a ¼ qða bðk þ mÞÞð8cr b þ q2ða bðk þ mÞÞ3Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT s1 s2

p

2c bð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ> 0; and

@ u

@ b ¼ qða bðk þ mÞÞð8cr bðaþ bðk þ mÞÞ þq2ðaþ bmÞða bðk þ mÞÞ3Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT s1 s2

p

4c b2ð8cr b þ q2ð2a bð5k þ 2mÞÞða bðk þ mÞÞ2Þ< 0:

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