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Optimal Pricing and Advertising in a Durable-Good Duopoly*
Anand Krishnamoorthya, Ashutosh Prasadb, and Suresh P. Sethib a College of Business Administration, University of Central Florida, Orlando, FL 32816-1400
b School of Management, The University of Texas at Dallas, Richardson, TX 75083-0688
European Journal of Operational Research, forthcoming
February 2009
*Corresponding author. Address: College of Business Administration, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816-1400. Telephone: +1 407 823-1330; Fax: +1 407 823-3891; E-mail: [email protected].
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Optimal Pricing and Advertising in a Durable-Good Duopoly
Abstract
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 asymmetric 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.
Keywords: Control; Dynamic programming; Game theory; Marketing; Differential games
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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immediate and continue to persist after the advertisement is withdrawn, due to the memory of the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 dynamic 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
0( ) ( ) ( ( )) 1 ( ), (0) [0,1],x t u t D p t x t x xρ= − = ∈ (1)
where x(t) is the cumulative sales (as a fraction of market potential) at time t, D(p(t)), with
'( ( )) 0D p t < , 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 ρ 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 1 ( )x t− captures decreasing returns and market saturation
3
similar to the Vidale-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:
1 (1 ) (1 )x x x x− ≈ − + − . (2)
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,
washing machines, and microwave ovens. In contrast, consumables and perishable goods, such
as grocery items, need to be repeatedly repurchased. 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 variables 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:
( )( ) ( ( ))(1 ( ))(1 )( )
p tx t a bx t x tp t
β= + − − , (3)
where p(t) is the price at time t, ( )p t the change in price at time t, and a, b, and β 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 durable-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
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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
( ) ( ( ))(1 ( ))i i ix t a b x t x t= + − , (4)
where ( )ix t is the adoption rate of brand i, ( ) ( )iix t x t=∑ is the cumulative adoption of the
category, and ai and bi are diffusion coefficients. Notice from equation (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 ( )ix 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 duopoly in which one firm chooses its
price and does not advertise, while the other chooses 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 tractability, they analyze a duopoly and find that the optimal price
5
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 exogenously over time. Using numerical analysis,
they find that in equilibrium, the advertising and pricing decisions follow the Dorfman-Steiner
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 future research.
2. Model
We start by listing the notation in Table 1.
<Insert Table 1 about here>
Denote the cumulative sales of firm i, {1,2}i∈ , at time t by si(t). The rate of change of
cumulative units sold, which is the instantaneous sales, is denoted ( )is t , and is given by
( )( ) ( ) ( ( )) ( ) ( ),ii i i i i i j
ds ts t u t D p t T s t s tdt
ρ= = − − , {1,2}, i j i j∈ ≠ , (5)
where ( ) ( )i js t s 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, ρi is the
effectiveness of firm i’s advertising, and ( ( ))i iD p 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 the sales rate goes to zero as the market depletes.
6
Consistent with the literature on durable-goods diffusion models, the potential adopters of 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 literature, 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
( )( ), ( )
0
max ( ) ( ) ( ( )) ,i
i i
r ti i i i iu t p t
J e p m s t C u t dt∞
−= − −∫ (6)
where ir is the discount rate of firm i, im is the marginal cost of production of firm i, and
( ( ))iC u t is the cost of firm i’s advertising.
Firm i’s total advertising expense is specified as
2( ( )) ( )2
ii i
cC u t u t= , (7)
where we refer to / 2ic as the unit cost of advertising, for convenience. This specification is
common in the literature, where the cost of advertising is assumed to be convex and, more
specifically, quadratic (e.g., Sethi, 1983; Sorger, 1989). It captures the diminishing returns to
advertising. 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:
1
1 1
2
2 2
211 1 1 1 1( ), ( )
0
222 2 2 2 2( ), ( )
0
max ( ) ( ) ( ) ,2
max ( ) ( ) ( ) ,2
r t
u t p t
r t
u t p t
cJ e p m s t u t dt
cJ e p m s t u t dt
∞−
∞−
= − −
= − −
∫
∫ (8)
1 1 1 1 2 1 1
2 2 2 1 2 2 2
s.t. ( ) ( ) ( ) ( ) ( ( )),
( ) ( ) ( ) ( ) ( ( )).
s t u t T s t s t D p t
s t u t T s t s t D p t
ρ
ρ
= − −
= − − (9)
In this paper, we adopt the feedback solution concept for differential games. This better
reflects the competitive dynamics of the two rivals over 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).
7
Next, we perform a detailed analysis of the model. To obtain the optimal policies, we
solve the differential game given by (8-9) to obtain the feedback Nash equilibrium strategies. For
expositional convenience, we will suppress the time-dependence of the state and control
variables when no confusion arises.
The Hamilton-Jacobi-Bellman (HJB) equation for firm i is
( ) ( )
( )
2
,
( ) ( ) ( )2
max ( ) ,i i
i ii i i i i j i i i i i i j i i
ii i u p i
j j i j j jj
c Vp m u T s s D p u u T s s D p
srV
Vu T s s D p
s
ρ ρ
ρ
∂ − − − − + − − ∂= ∂ + − − ∂
(10)
where ( , )i i i jV V s s= is the value function of firm i.
Writing the first-order conditions for pi and ui from the HJB equation in (10), we get
( ) ( )' '( ) ( ) ( ) ( ) 0,ii i i j i i i i i i i j i i i i i j i i
i
Vu T s s D p p m u T s s D p u T s s D p
sρ ρ ρ
∂− − + − − − + − − =
∂ (11)
( )( ) ( ) ( ) 0.ii i i i j i i i i i i j i i
i
Vp m T s s D p c u T s s D p
sρ ρ
∂− − − − + − − =
∂ (12)
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:
( ( )) ( )i i i i iD p t p tα β= − , (13)
where αi is the demand intercept and βi represents price sensitivity. The linear demand function is
one of the most commonly used in the literature (e.g., Petruzzi and Dada, 1999).
Substituting (13) and simultaneously solving the two first-order conditions in (11-12)
yields the optimal price and advertising policies, denoted *( , )i i jp s s and *( , )i i ju s s , respectively,
which are given in Proposition 1.
Proposition 1: The optimal feedback pricing and advertising strategies of firm i are given by
* 1( , ) ,2
i ii i j i
i i
Vp s s m
sαβ
∂= + − ∂
(14)
8
2
* ( , ) .4
i ii i j i i i i j
i i i
Vu s s m T s s
c sρ
α ββ ∂
= + − − − ∂ (15)
The sales trajectories corresponding to the equilibrium in (14-15) can be obtained by
substituting the optimal strategies in (14-15) and solving the two state equations in (9).
Substituting the optimal solutions from (14-15) into the HJB equation in (10) and simplifying
yields
( )34
2 2 22
1( , ) 432
.ji ii i j j j i i i i i i j j j j i j
i j ji j i i j
VV VV s s c m c m T s s
s s sc c rβ ρ α β β ρ α β
β β
∂ ∂ ∂ = + − + + − − − ∂ ∂ ∂
(16)
To prove that the pair of strategies in (14-15) forms a feedback equilibrium, we need to
show that there exist two continuously-differentiable functions ( , )i i jV s s , , {1,2}, i j i j∈ ≠ ,
which satisfy the partial differential equations in (16) and the boundary condition that
lim ( ( ), ( )) 0i i jtV s t s t
→∞= .
We propose the following form for the value function ( , )i i jV s s :
( , ) ( ) .i ii i j i i j i
i j
V VV s s k T s s ks s
∂ ∂= − − ⇒ = = −
∂ ∂ (17)
With 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 *( , ) 0i i jp s s > if the condition 0ii i
im k
αβ
+ + >
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 ik , since it directly
affects the optimal decisions and the value function. Equating the coefficients of i jT s s− − in
(16), we have
9
( ) ( )342 2 2
2
( ) 4 ( ), , {1, 2}, .
32j j i i i i i i i j i j j j j
ii j i i j
c k m c k k mk i j i j
c c r
β ρ α β β ρ α β
β β
− + − − += ∈ ≠ (18)
We first consider the case of symmetric firms.
3.1. Symmetric Firms
Equation (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.,
i jc c c= = , i jr r r= = , i jρ ρ ρ= = , i jm m m= = , i jα α α= = , and i jβ β β= = , a symmetric
solution will be obtained with i jk k k= = . Rewriting equation (18), we now have the following
quartic equation to solve for k: 2 3
2
( 5 ( ))( ( ))32
k m k mkcr
ρ α β α ββ
− + − += . (19)
Since the value function is positive, k > 0 on the left-hand side of equation (19). For the right-
hand side to be positive, we require either that k mαβ
> − or 5
k mαβ
< − . The former can be
ruled out because for demand to be non-negative, we require 0ipα β− ≥ , or /ip α β≤ .
Moreover, we know from the solution for price in (14) that * 1 ( )2ip m kα
β= + + .
Therefore, k mαβ
> − is infeasible.
Remark 1: The solution must also satisfy the boundary condition that lim ( ( ), ( )) 0i i jtV s t s t
→∞= . We
find that 5
k mαβ
< − satisfies this, whereas for k mαβ
> − , it is not satisfied.
With the solution for k, the optimal price and advertising solutions, ∀ i, can be rewritten as
* 1( , ) ( ),2i i jp s s m kα
β= + + (20)
( )2* ( , ) ( ) .4i i j i ju s s k m T s s
cρ α ββ
= − + − − (21)
10
It is evident that the solution for the optimal price in (20) is a constant.
<Insert Table 2 about here>
The comparative statics for the parameters on the variables of interest are given in Table
2. Since we have assumed symmetry, *2u , *
2p , and 2V have the same comparative statics as *1u ,
*1p , and 1V , respectively.
Remark 2: For the comparative statics of k w.r.t. the model parameters, we have 0kc∂
<∂
,
0kr∂
<∂
, 0kρ∂
>∂
, 0km∂
<∂
, 0kα∂
>∂
, and 0kβ∂
<∂
.1
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 0pc∂
<∂
,
0pr∂
<∂
, 0pρ∂
>∂
, 0pm∂
>∂
, 0pα∂
>∂
, and 0pβ∂
<∂
.
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.
1 All the proofs can be found in the Appendix.
11
Remark 4: For the comparative statics of u w.r.t. the model parameters, we have 0uc∂
<∂
,
0ur∂
>∂
, 0uρ∂
>∂
, 0um∂
<∂
, 0uα∂
>∂
, and 0uβ∂
<∂
.
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 discount 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-horizon setting. To solve the differential game, one has to solve
the following set of simultaneous equations:
( )( ) ( )( )( )4 32 2 21 2 2 1 1 1 1 1 1 1 2 1 2 2 2 22
1 2 1 1 2
1 432
k c k m c k k mc c r
β ρ α β β ρ α ββ β
= − + − − + , (22)
( )( ) ( )( )( )4 32 2 22 1 1 2 2 2 2 2 2 2 1 2 1 1 1 12
1 2 2 1 2
1 432
k c k m c k k mc c r
β ρ α β β ρ α ββ β
= − + − − + .
(23)
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
* 1( , ) ( ),2
ii i j i i
ip s s m k
αβ
= + + (24)
( )( )2* ( , ) .4
ii i j i i i i i j
i iu s s k m T s s
cρ
α ββ
= − + − − (25)
We use Mathematica to numerically solve the system of quartic equations given by (22-
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.
<Insert Table 3 about here>
12
From Table 3, one can see that ki is increasing in cj, mj, and βj, and decreasing in rj, ρj,
and αj. The results for the own parameters are similar to those in the symmetric case (i.e., ki is
decreasing in ci, ri, mi, and βi, and increasing in ρi and αi).
<Insert Figure 1 about here>
For the comparative statics of price w.r.t. the model parameters, we find that pi is
decreasing in ci, ri, and βi, and increasing in ρi, mi, and αi. This is consistent with our findings in
the symmetric case. For the cross parameters, we find that pi decreases with rj, ρj, and αj, and
increases with cj, mj, and βj (see Figure 1).
<Insert Figure 2 about here>
For the comparative statics of advertising (taking s1 and s2 as given) w.r.t. the model
parameters, we find that, consistent with our findings in the symmetric case, ui is decreasing in
ci, mi, and βi, and increasing in ri, ρi, and αi. For the cross parameters, we find that ui increases
with rj, ρj, and αj, and decreases with cj, mj, and βj (see Figure 2).
This concludes our study of the linear specification for the demand function ( ( ))i iD p t . In
the next section, we analyze a nonlinear specification.
4. Isoelastic Demand Specification
We now consider the isoelastic demand function:
( ( )) ( ) ii i iD p t p t η−= , (26)
so called because the elasticity of demand is iη , a constant. It is assumed that the demand is
elastic, i.e., 1iη > , so that the optimal price is finite.
Substituting (26) and simultaneously solving the two first-order conditions in (11-12)
yields the optimal price and advertising policies given in Proposition 2. We assume 0i
i
Vs
∂<
∂ so
that *( , ) 0i i jp s s > and later show this is indeed true.
Proposition 2: The optimal feedback pricing and advertising strategies of firm i are given by:
* ( , ) ,1
i ii i j i
i i
Vp s s m
sη
η ∂
= − − ∂ (27)
13
1
* ( , ) .1
i
i i ii i j i i j
i i i i
Vu s s m T s s
c s
ηρ ηη η
− + ∂
= − − − − ∂ (28)
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 (27-28) can be obtained by
substituting the optimal price and advertising decisions in (27-28) and solving the two state
equations in (9). Substituting the optimal solutions from (27-28) into the HJB equation in (10)
and simplifying, we have
( )( )
2 12222
2
2111( , )
2 1.
jij jii i i
j ji i ij j ji i i
i i j i ji j ji i
VVV V mm ms ss s
V s s T s sr cc
ηη ηη ρρηη
ηη
− +− ∂∂ ∂ ∂ − − − ∂ − ∂∂ − ∂ = + − − −
(29)
We try the following form for the value function ( , )i i jV s s :
( , ) ( ) .i ii i j i i j i
i j
V VV s s k T s s ks s
∂ ∂= − − ⇒ = = −
∂ ∂ (30)
Equating the coefficients of i jT s s− − in (29), we have
( ) ( )
( )
( )2 12
2 22
2
2111 , , {1, 2},
2 1.
jiji
j j j ji i i i iji
ii j ji i
k k mm k k mk i j i j
r cc
ηη ηη ρρηη
ηη
− +− + + + −− = − ∈ ≠ −
(31)
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
Equation (31) represents the system of simultaneous equations that need to be solved to obtain
the Nash equilibrium of the differential game. For symmetric firms, i jc c c= = , i jr r r= = ,
i jρ ρ ρ= = , i jm m m= = , i jη η η= = , and i jk k k= = . Equation (31) now becomes
14
( )
( )
22
2
( )( ) (3 2 )1
2 1
m km k m kk
cr
ηηρ ηη
η
− +
+ + − − =−
. (32)
Given the solution for k, the optimal price and advertising solutions can be rewritten as
* ( )( , ) ,1i i j
m kp s s ηη
+=
− (33)
1* ( )( , ) .
1i i j i jm ku s s T s s
c
ηηρ
η η
− + +
= − − − (34)
We use Mathematica to numerically solve equation (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:
( ) ( )
( )
( )1 22 2 12 1 1 1 2 2 22 2
1 1 1 2 11 2
1 21 2 21 1
21 11 0,
2 1
m k m km k k
kr cc
η ηη η
ρ ρη η
ηη
− − + + + + − − − − =
−
(35)
( ) ( )
( )
( )2 12 2 12 2 2 2 1 1 12 2
2 2 2 1 22 1
2 22 1 12 2
21 11 0.
2 1
m k m km k k
kr cc
η ηη η
ρ ρη η
ηη
− − + + + + − − − − =
−
(36)
The solutions are complicated, and numerical analysis is used to obtain the set of
positive-real roots that satisfy the system of equations in (35-36). Given the solutions for k1 and
k2, the optimal price and advertising solutions can be rewritten as
( )* ( , ) ,1
ii i j i i
i
p s s m kη
η
= + − (37)
15
( )1
* ( , ) .1
i
i ii i j i i i j
i i i
u s s m k T s sc
ηρ ηη η
− +
= + − − − (38)
The comparative statics for the parameters on the variables of interest in the asymmetric
case are presented in Table 4.
<Insert Table 4 about here>
From Table 4, one can see that ki is decreasing in ci, ri, mi, and ηi, and increasing in ρi.
For the cross parameters, we find that ki decreases with rj, ρj, and ηj, and increases with cj and mj.
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.
<Insert Figure 3 about here>
The comparative statics of price w.r.t. the model parameters are presented in Figure 3.
We find that pi is decreasing in ci, ri, and ηi, and increasing in ρi and mi. For the cross parameters,
pi decreases with rj, ρj, and ηj, and increases with cj and mj.
<Insert Figure 4 about here>
Figure 4 presents the comparative statics of advertising (taking s1 and s2 as given) w.r.t.
the model parameters. We find that ui is decreasing in ci, mi, and ηi, and increasing in ri and ρi.
For the cross parameters, we find that ui increases with rj, ρj, and ηj, and decreases with cj and mj.
In summary, the comparative statics on the common parameters (i.e., ci, cj, ri, rj, ρi, ρj, mi,
and mj) 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 advertising. It finds that the optimal price is constant and that the
optimal advertising decreases with cumulative sales. We now discuss the relevance 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
16
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 the education segment, declined in the home and business segments,
and increased in the scientific segment. The price declines were attributed 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-parameters 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, advertising effectiveness, and demand intercept, and
increasing in the rival’s unit cost of advertising, marginal cost of production, and price
sensitivity, (b) the optimal price of a firm is decreasing in the rival firm’s discount rate,
advertising effectiveness, and demand intercept, and increasing in the rival’s unit cost of
advertising, marginal cost of production, and price sensitivity, and (c) the optimal advertising
level (taking 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.
17
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, advertising effectiveness, and price elasticity, and
increasing in the rival’s unit cost of advertising and marginal cost of production, (b) the optimal
18
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 effectiveness, 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, we find 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 statics for two additional parameters (the marginal cost of advertising and
the baseline demand) that are not in the SPH model. While the SPH model 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 *( , )i i ju s s is increasing in the
discount rate r is not obvious. In fact, some nondurable-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 two
effects: 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 model, 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
19
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 allocation 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 ( ( ), ( ))i i jV s t s t , is the value of its total net discounted future
profit stream, assuming that each firm makes optimal decisions starting from the sales
( ( ), ( ))i js t s t at time t . We know from the optimal control theory that ( , )i i j
i
V s ss
∂
∂, evaluated at
the optimal ( ( ), ( ))i js t s t , is called the shadow price 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, ii
i
Vk
s∂
= −∂
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 sj) is directly proportional
to k, the two firms would want k to be as high as possible. The analysis shows that k is increasing
in ρ and α 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 ρ. 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 determined,
20
the application of the formulae to practice is straightforward. The approximate value of k can be
obtained offline, and the moment-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 optimal 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 competitors. An important feature of the proposed model is that,
while it is realistic enough to capture price and advertising in a competitive setting, 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
21
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 competitors
(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; He et al., 2008; He et al., 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
extension is to study price and advertising competition in an oligopoly along the lines of Fruchter
(1999) and Naik et al. (2008). Although duopoly models are representative of many real-world
markets, modeling markets characterized by three or more firms might yield additional insights.
22
Appendix
Proof of Remark 2
Denote equation (19) as 2 3
2
( (5 ))( ( ))(., ) 032
k m k mf k kcr
ρ α β α ββ
− + − += − = . The implicit
function theorem yields, for any parameter χ, //
k ff k
χχ∂ ∂ ∂
= −∂ ∂ ∂
. We have
( )2 3
2 2
( (5 ))( ( )) 04 8 (2 (5 2 ))( ( ))
k k m k mc c cr k m k m
ρ α β α ββ β ρ α β α β
∂ − + − += − <
∂ + − + − +,
( )2 3
2 2
( (5 ))( ( )) 04 8 (2 (5 2 ))( ( ))
k k m k mr r cr k m k m
ρ α β α ββ β ρ α β α β
∂ − + − += − <
∂ + − + − +,
( )3
2 2
( (5 ))( ( )) 02 8 (2 (5 2 ))( ( ))
k k m k mcr k m k m
ρ α β α βρ β β ρ α β α β∂ − + − +
= >∂ + − + − +
,
2 2
2 2
( (4 ))( ( )) 08 (2 (5 2 ))( ( ))
k k m k mm cr k m k m
ρ α β α ββ ρ α β α β
∂ − + − += − <
∂ + − + − +,
( )2 2
2 2
( (4 ))( ( )) 08 (2 (5 2 ))( ( ))
k k m k mcr k m k m
ρ α β α βα β β ρ α β α β∂ − + − +
= >∂ + − + − +
, and
( )( )
2 2 2 2
2 2 2
2 ( )(5 ) ( ( ))
2 8 (2 (5 2 ))( ( ))
k k m k m k mkcr k m k m
ρ α αβ β α β
β β β ρ α β α β
− − + + − +∂= −
∂ + − + − +. Note that
2 22 ( )(5 )k k m k mα αβ β− − + + can be written as 2( (5 ))( (3 )) 2 (5 )k m k m k k mα β α β β− + + + + + ,
which is positive since 5
k mαβ
< − . Therefore, 0kβ∂
<∂
.
Proof of Remark 3
For the comparative statics of price, note from (20) that for any parameter χ, except for m, the
sign of /p χ∂ ∂ is the same as that of /k χ∂ ∂ . Therefore, we have 0pc∂
<∂
, 0pr∂
<∂
, 0pρ∂
>∂
,
0pα∂
>∂
, and 0pβ∂
<∂
. For m, we have
23
2 3
2 2
1 8 ( ( ))(1 ) 02 2(8 (2 (5 2 ))( ( )) )
p k cr k mm m cr k m k m
β ρ α ββ ρ α β α β
∂ ∂ + − += + = >
∂ ∂ + − + − +.
Proof of Remark 4
From (21), we have
( )( )
2 2 21 2
2 2 2
16 (3 (5 3 ))( ( )) ( ( ))0
8 8 (2 (5 2 ))( ( ))
cr k m k m k m T s suc c cr k m k m
ρ β ρ α β α β α β
β β ρ α β α β
+ − + − + − + − −∂= − <
∂ + − + − +,
( )3 4
1 22 2
( (5 ))( ( ))0
8 8 (2 (5 2 ))( ( ))k m k m T s su
r cr cr k m k mρ α β α ββ β ρ α β α β
− + − + − −∂= >
∂ + − + − +,
( )( )
2 2 21 2
2 2
8 ( )( ( )) ( ( ))0
4 8 (2 (5 2 ))( ( ))
cr m k m k m T s suc cr k m k m
β ρ α β α β α β
ρ β β ρ α β α β
+ − − + − + − −∂= >
∂ + − + − +,
( )( )
2 31 2
2 2
( ( )) 8 ( ( ))0
2 8 (2 (5 2 ))( ( ))
k m cr k m T s sum c cr k m k m
ρ α β β ρ α β
β ρ α β α β
− + + − + − −∂= − <
∂ + − + − +,
( )( )
2 31 2
2 2
( ( )) 8 ( ( ))0
2 8 (2 (5 2 ))( ( ))
k m cr k m T s suc cr k m k m
ρ α β β ρ α β
α β β ρ α β α β
− + + − + − −∂= >
∂ + − + − +, and
( )( )
2 31 2
2 2 2
( ( )) 8 ( ( )) ( )( ( ))0
4 8 (2 (5 2 ))( ( ))
k m cr k m m k m T s suc cr k m k m
ρ α β β α β ρ α β α β
β β β ρ α β α β
− + + + + + − + − −∂= − <
∂ + − + − +.
24
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27
Table 1: Notation
( )is t Cumulative sales of firm i at time t
T Market potential
( )iu t Advertising effort of firm i at time t
( )ip t Price of firm i at time t
ic Coefficient associated with the advertising cost of firm i
iρ Effectiveness of advertising of firm i
im Marginal cost of production of firm i
iα Demand intercept of firm i (linear demand)
iβ Price sensitivity of firm i (linear demand)
iη Price elasticity of firm i (isoelastic demand)
ir Discount rate of firm i
( , )i i jV s s Value function of firm i when its cumulative sales is is and its rival’s is js
Table 2: Comparative statics for the symmetric case with linear demand
Variables c r ρ m α β k ↓ ↓ ↑ ↓ ↑ ↓
*ip ↓ ↓ ↑ ↑ ↑ ↓ *iu ↓ ↑ ↑ ↓ ↑ ↓
iV ↓ ↓ ↑ ↓ ↑ ↓
Legend: ↑ increase; ↓ decrease.
28
Table 3: Comparative statics for the asymmetric case with linear demand
Variables ic jc ir jr iρ jρ im jm iα jα iβ jβ
ik ↓ ↑ ↓ ↓ ↑ ↓ ↓ ↑ ↑ ↓ ↓ ↑ *ip ↓ ↑ ↓ ↓ ↑ ↓ ↑ ↑ ↑ ↓ ↓ ↑ *iu ↓ ↓ ↑ ↑ ↑ ↑ ↓ ↓ ↑ ↑ ↓ ↓
iV ↓ ↑ ↓ ↓ ↑ ↓ ↓ ↑ ↑ ↓ ↓ ↑
Legend: ↑ increase; ↓ decrease.
Table 4: Comparative statics for the asymmetric case with isoelastic demand
Variables ic jc ir jr iρ jρ im jm iη jη
ik ↓ ↑ ↓ ↓ ↑ ↓ ↓ ↑ ↓ ↓ *ip ↓ ↑ ↓ ↓ ↑ ↓ ↑ ↑ ↓ ↓ *iu ↓ ↓ ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↑
iV ↓ ↑ ↓ ↓ ↑ ↓ ↓ ↑ ↓ ↓
Legend: ↑ increase; ↓ decrease.
29
Figure 1: Comparative statics of p1 and p2 for asymmetric firms with linear demand 2
1 0.2c = , 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
1.45p1, p2
p2
p1
1 1ρ = , 2 [0,2]ρ ∈
0.5 1.0 1.5 2.0r2
1.15
1.20
1.25
1.30
1.35
1.40
p1, p2
p2
p1
1 0.2m = , 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
1.32p1, p2
p2
p1
1 0.5β = , 2 [0,1]β ∈
0.4 0.6 0.8 1.0b2
1.5
2.0
p1, p2
p2
p1
2 Unless otherwise stated, the parameter values in Figures 1-2 are: 1 0.2c = , 2 0.2c = , 1 0.1r = , 2 0.1r = , 1 1ρ = ,
2 1ρ = , 1 0.2m = , 2 0.2m = , 1 1α = , 2 1α = , 1 0.5β = , 2 0.5β = , and 1 2 1T s s− − = . Due to space restrictions, the graphs for { 1 0.1r = , 2 [0,0.2]r ∈ } and { 1 1α = , 2 [0,2]α ∈ } are not presented here.
30
Figure 2: Comparative statics of u1 and u2 (given s1 and s2) for asymmetric firms with linear demand
1 0.2c = , 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
1 1ρ = , 2 [0,2]ρ ∈
1.0 1.5 2.0r2
1.0
1.5
2.0
u1, u2
u2
u1
1 0.2m = , 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
1 0.5β = , 2 [0,1]β ∈
0.4 0.6 0.8 1.0b2
1.5
2.0
2.5
u1, u2
u2
u1
31
Figure 3: Comparative statics of p1 and p2 for asymmetric firms with isoelastic demand3
1 0.2c = , 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
1 1ρ = , 2 [0,2]ρ ∈
1.0 1.5 2.0r2
5
10
15
20
25
30
p1, p2
p2
p1
1 0.2m = , 2 [0,0.4]m ∈
0.10 0.15 0.20 0.25 0.30 0.35 0.40m2
13.0
13.5
p1, p2
p2
p1
1 1.25η = , 2 [1.01,2]η ∈
1.4 1.6 1.8 2.0h2
10
20
30
40
p1, p2
p2
p1
3 Unless otherwise stated, the parameter values in Figures 3 and 4 are: 1 0.2c = , 2 0.2c = , 1 0.1r = , 2 0.1r = ,
1 1ρ = , 2 1ρ = , 1 0.2m = , 2 0.2m = , 1 1.25η = , 2 1.25η = , and 1 2 1T s s− − = . Due to space restrictions, the graphs for { 1 0.1r = , 2 [0,0.2]r ∈ } are not presented here.
32
Figure 4: Comparative statics of u1 and u2 (given s1 and s2) for asymmetric firms with isoelastic demand
1 0.2c = , 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
1 1ρ = , 2 [0,2]ρ ∈
1.0 1.5 2.0r2
1.0
1.5
2.0
2.5
3.0
u1, u2
u2
u1
1 0.2m = , 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
1 1.25η = , 2 [1.01,2]η ∈
1.4 1.6 1.8 2.0h2
2.0
2.5
3.0
3.5
4.0
4.5
u1, u2
u2
u1
