Green Product Innovation
in the Context of Habit Formation1
Christos Constantatos
Department of Economics
University of Macedonia
Markus Herrmann
Department of Economics, CREATE,
Université Laval
and
Eftichios Sartzetakis
Department of Economics
University of Macedonia
October 13, 2015
1 Please
address all correspondence to Markus Herrmann, Département d’économique, Pavillon
J.-A.- DeSève, 1025, avenue des Sciences-Humaines, Université Laval, Québec, Canada, G1V 0A6.
E-mail: markus.herrmann@ecn.ulaval.ca.
Abstract
We examine the introduction by a single firm, of a new product-version with superior environmental characteristics in a market that has been previously served by a competitive
industry producing a brown product-type. Due to brand-name and advertising, consumers,
while unable to distinguish the brown-type producers among themselves, they are able to
distinguish the new product from the traditional one. Consumers are environment-conscious,
their consciousness being expressed as a reduction in their willingness to pay for any unit
of a given product-type according to the total damage generated by the production of that
type. If the green product has a product-differentiation advantage, habit formation prevents
it from instantly gaining its corresponding market share. We set a continuous-time dynamicmodel in continuous time in order to study the interaction of these two effects.
Keywords: Habit formation; Environmental consciousness, pollution, habit formation, dynamic analysis.
JEL classification:
Working document
October, 21 2015
1
1.1
Model framework
Firms
Consider a product X, initially sold by a competitive industry, the total amount produced
at time t being q1 (t), where t < t0 . The environmental quality of the product is assumed to
be in direct relation to the industry’s emissions,
e1 (t) = δ1 q1 (t) − v1 (t),
where δ1 is a pollution parameter and v1 is the industry’s total abatement. The cost of
abatement is quadratic in the amount of abated units of the pollutant,
C1 (v1 (t)) = (k1 /2)v1 (t)2 ,
where k1 ≥ 1 is a cost parameter related to that industry.
The introduction of a new product which is a substitute to product X, confers
monopoly power to an innovating firm, denoted 2. Innovation occurs at t = t0 . The patent
length is exogenous to the firm and given by T . We assume that firm 2’s emissions and cost
of abatement function have the same form as that of industry 1. Hence,
e2 (t) = δ2 q2 (t) − v2 (t),
C2 (v2 (t)) = (k2 /2)v2 (t)2 ,
where the innovating firm 2 may have a comparative advantage in polluting less, δ2 ≤ δ1 , a
comparative advantage in abatement, k2 ≤ k1 , or both. Assume for now that parameters δ2
and k2 are exogenous to the innovating firm.
1
The total environmental damage from the flow of pollution is
D(E) = E 2 ,
(1)
where E = e1 + e2 represents the aggregate net emissions related to product type 1 and 2.1
1.2
Consumers
We assume that a representative consumer derives utility from pure consumption at each
instant of time t given by the utility function,2
1
e=
U
α
e i qi −
2
i=1,2
X
!
X
qi2 + 2γq1 q2
+M
(2)
i=1,2
where α
ei > 0, γ ∈ [0, 1) measures the degree of substitutability between two types of
products, 1 and 2, and M represents the amount of the numeraire good. We assume for now
that γ is exogenous.3
Consumers are myopic in the sense that they only care about current consumption.
Furthermore, they have a preference for the type of good they are used to consume and
may be aware of the flow of emissions related to their consumption of product type i = 1, 2.
Loyalty to product type i is assumed to be captured via Ψ(Xi (t)), where Xi represent
cumulative sales of product type i, discounted at rate ρ. We will thus assume that the
consumer weights past sales and we define:
Z
t
Xi (t) =
eρ(τ −t) q(τ )dτ
(3)
−∞
1
The model could be extended to account for stock pollution.
In what follows, we abstract from the time argument when this should not cause any confusion.
3
We may later want to address a possible discrete choice by the innovating firm between : γ1 > γ2 , where
γ1 can be chosen without incurring any fixed cost and γ2 can be chosen when incurring a fixed cost of F .
The fixed cost can be interpreted as an advertisement cost, which allows to decrease the substitutability
between goods, and hence to increase the dissimilarity between goods.
2
2
Differentiating (3) with respect to time by using Leibniz’s rule, we have
Z
t
Ẋi (t) = q(t) − ρ
eρ(τ −t) q(τ )dτ
−∞
= q(t) − ρXi (t),
(4)
and, as a consequence, consumer loyalty is to be interpreted as a renewable resource.4
Environmental awareness is modelled as in Constantatos et al. (xyz) via a decreasing
willingness to pay for consuming the potentially polluting product type i, α − φei , where φ
is a preference parameter. Accounting for consumer loyalty and environmental awareness,
we adapt Constantatos et al. and write:
α
ei = α + Ψ(Xi ) − φei .
(5)
Substituting (5) into (2) and making use of the definition of emissions of product type i, ei ,
we obtain:
!
1
U=
(α + Ψ(Xi ) + φvi ) qi −
2
i=1,2
X
X
(1 + 2φδi ) qi2 + 2γq1 q2
+ M.
(6)
i=1,2
Assuming prices pi and that both goods are consumed in equilibrium (M sufficiently high),
we obtain the system of inverse demand functions
Pi (qi , qj , vi , Xi ) = α + Ψ(Xi ) + φvi − (1 + 2φδi ) qi − γqj ,
i, j = 1, 2, i 6= j,
(7)
where the amount of abatement for product type i, vi , affecting environmental quality, is
exogenous from the point of view of the consumer, as are the cumulative sales Xi in the
short run.
4
If the flow of production were to be constant, q(τ ) = q̄, we would necessarily observe a steady-state level
of sales at t given by X(t) = q̄/ρi . Furthermore, it can then be shown that Ẋi (t) = 0.
3
1.3
Equilibrium on the competitive fringe market
Given the competitive nature in the fringe market selling product type 1, we can derive the
equilibrium quantity sold for this product by assuming that industry 1 takes the price of its
output as exogenously given. The maximization problem is given by
max p1 q1 − C1 (v1 ) − τ e1 ,
q1 ,e1
(8)
subject to v1 = δ1 q1 − e1 and where τ is the environmental tax levied on the pollution flow.
Assuming an interior solution, we find the first order conditions
p1 =
and
−
∂C1 ∂v1
∂v1 ∂q1
∂C1 ∂v1
= τ,
∂v1 ∂e1
(9)
(10)
which state that (i) the market price must equalize the marginal cost of production, given
by the increase in the abatement cost, and (ii) the marginal abatement cost must equalize
the environmental tax. Substituting for the respective derivative of the abatement cost, we
get
p1 = k1 (δ1 q1 − e1 )δ1
k1 (δ1 q1 − e1 ) = τ.
Combining both equations yields: p1 = δ1 τ , i.e. the price of product 1 equals the marginal
environmental cost caused by production of type 1. Furthermore, given that v1 = δ1 q1 − e1 ,
we can characterize the abatement level as: v1 = τ /k1 . It follows that the higher the
environmental tax or the lower the cost parameter of abatement, the higher the equilibrium
level of abatement of the competitive fringe.
Making use of the inverse demand function (7) for i = 1 as well as the equilibrium
abatement level of the competitive fringe, v1 = τ /k1 , we can calculate the output of product
4
type 1,
q1 (q2 , X1 ) =
α + Ψ(X1 ) + φ(τ /k1 ) − δ1 τ − γq2
1 + 2φδ1
(11)
The aforementioned maximization program assumes that industry 1 is myopic in the sense
that it does not account for intertemporal, aggregate sales X1 . Under this assumption, an
industry accounting for consumer loyalty would face an arbitrage consisting in selling more
now, with the benefit of increasing consumers’ willingness to pay later. However, selling
more would be prohibitive for the individual firm, as it implies higher marginal production
costs to that firm only, while the increased loyalty will benefit the overall industry.5
1.4
The innovating firm’s problem
The innovating firm accounts for industry 1 as a competitive fringe on the current market, as
well as for the evolution of its intertemporal sales. It follows that the maximization problem
of firm 2 is:
Z
T
max
q2 ,v2
e−rt ? [P2 (q2 , q1 (q2 , X1 ), v2 , X2 ) q2 − C2 (v2 ) − τ (δ2 q2 − v2 )] dt
(12)
t0
subject to
Ẋ1 = q1 − ρX1 ,
X1 (t0 ) = X1,0 , given,
(13)
Ẋ2 = q2 − ρX2 ,
X2 (t0 ) = 0,
(14)
where (q2 (t), v2 (t)) are the control variables and (X1 (t), X2 (t)) the state variables of the
problem.
5
This problem of a competitive fringe is thus similar to that of an resource extracting industry having
open access to a common resource pool, like, e.g., the high sees fisheries.
5
We write the present-value Hamiltonian function
H = e−rt [P2 (q2 , q1 (q2 , X1 ), v2 , X2 ) q2 − C2 (v2 ) − τ (δ2 q2 − v2 )]
+λ1 (t)[q1 (q2 , X1 ) − ρX1 ] + λ2 (t)[q2 − ρX2 ],
(15)
where λi is the shadow value (in present-value terms) associated to intertemporal, aggregate
sales Xi , i = 1, 2. Necessary conditions for optimality, assuming an interior solution, imply:
0 =
0 =
λ̇1 =
λ̇2 =
∂H
∂P2 ∂P2 ∂q1
∂q1
−rt
P2 (.) +
=e
+
q2 − τ δ2 + λ1
+ λ2
∂q2
∂q2
∂q1 ∂q2
∂q2
∂H
∂C2
−rt ∂P2
=e
q2 −
+τ
∂v2
∂v2
∂v2
∂q1
∂H
−rt ∂P2 ∂q1
= −e
q 2 − λ1
+ ρλ1
−
∂X1
∂q1 ∂X1
∂X1
∂H
∂P2
−
= −e−rt
q2 + ρλ2
∂X2
∂X2
(16)
(17)
(18)
(19)
as well as transversality conditions
λ1 (T ) = 0,
(20)
λ2 (T ) = 0.
(21)
From equations (20) and (21) we conclude that the innovating firm behaves like a myopic firm
once its patent has expired at T . This is because we ignore any bequest function associated
to cumulative sales X1 (T ) and X2 (T ). We thus implicitly assume that once the patent
expires, the innovating firm becomes a price taker as entry of new firms selling product type
2 will occur.
We conjecture (to be shown) that the shadow value associated to firm 1’s intertemporal sales, λ1 is negative (λ1 ≤ 0), while firm 2 attributes a positive value to its own,
intertemporal sales (λ2 ≥ 0).
In what follows, we interpret the aforementioned necessary conditions for optimality.
6
Rearranging equation (16) yields

<0
<0
z}|{ z}|{
 ∂P2 ∂P2 ∂q1 
∂q1 rt
rt

P2 (.) + 
 ∂q2 + ∂q1 ∂q2  q2 = τ δ2 − λ1 ∂q2 e − λ2 e .
|{z} | {z }

<0
>0
The left-hand side of the equation above represents the marginal revenue of producing product type 2. It accounts for the classical, negative direct effect on price of increasing output
(∂P2 /∂q2 ) and an indirect, positive effect on price, because increasing output q2 lowers the
competitive industry’s output ((∂P2 /∂q1 )(∂q1 /∂q2 )). The right-hand side represents the full
marginal cost of production. It comprises the tax cost as well as the implicit value of consumer loyalty for both product types, in current-value terms (multiplied by ert ). Given our
∂q1 rt
conjecture that λ1 ≤ 0, λ2 ≥ 0 and ∂q1 /∂q2 < 0, we have λ1 ∂q
e + λ2 ert > 0 and, as a
2
consequence, the full marginal cost is lower than the tax cost. This is because increasing
sales of product type 2 by the innovating firm increase its own consumer loyalty and shrinks
that of its competitor.
Rearranging equation (17), we get:
∂P2
∂C2
=τ+
q2 ,
∂v2
∂v2
which states that the marginal cost of abatement equals the marginal benefit of abatement,
which comprises the avoided tax payment and the increased willingness-to-pay due to the
higher, perceived quality.
Considering the dynamic efficiency conditions, we obtain for X1 (see (18)), we have
λ̇1 = −e
−rt
∂P2
∂q1
q 2 + λ1
+ρλ1 .
∂q1
∂X1
|{z}
|
{z
}
<0
7
>0
With respect to the innovator’s intertemporal sales, X2 (see (19)), we get
λ̇2 = −e−rt
∂P2
+ ρλ2 .
∂X2
Research questions
• How does the private optimum compare to the social optimum?
• Economic instruments: Is there a need for subsidizing innovation? Taxing the brown
product to disengage loyalty?
• How does the entrance of a new firm affect the socially optimal tax level?
• How does consumer awareness substitute for adjusting the tax level? How is consumer
loyalty related to this?
8
2
Preliminary simulations
Preliminary simulations have been carried out in order to give an analytical characterization
of the profit-maximizing choices of the monopolist with respect to the quantity produced
(q2 ) and the quantity of pollutant abated (v2 ). Graphical representations of the evolution
the incumbent industry’s decisions (q1 , v1 ), of the state variables (X1 , X2 ) and corresponding
shadow values (λ1 , λ2 ) are also shown.
Baseline parameters are given in Table 1. Note that the functional form Ψ(Xi ) = ψXi
has been retained for now.
benchmark price
differentiation
abatement valuation
habit valuation
habit depreciation
pollution coefficient
cost parameter
discount rate
patent length
unit tax
initial state
α1
α2
γ
φ
ψ
ρ
δ1
δ2
k1
k2
r
T
τ
X1 (0)
X2 (0)
5
5
1
0.3
0.1
0.35
0.7
0.7
5
5
0.03
75
1.25
1
0
Table 1: Baseline parameter values
The baseline case is shown in Figures 1 and 2. In this case, only the brand name differentiates the two products. Cumulative sales at time 0 are given respectively by X1 (0) = 1
and X2 (0) = 0. As the simulation results show, the monopolist (firm 2) choses a decreasing pattern of output and abatement decisions (q2 and v2 ), while the representative firm
of the competitive fringe market abates at a constant rate and increases production over
time (overall emissions are increasing for firm 1 and decreasing for firm 2). This results in
important habit formation for the competitive fringe firm as compared to the innovating
9
firm, an effect that feeds back into the production decision of the innovating firm (see the
left panel of Figure 2). The implicit values of habit formation as perceived by the innovating
firmhave the awaited sign and converge to 0 as the patent ends. Indeed, the innovating firm
attributes a positive value to the habit formation with respect to its own product (λ2 > 0),
while a negative value is associated to habit formation in relation to the product sold by the
competitive fringe (λ1 < 0).
Dynamic comparative exercises
We have run dynamic comparative exercises in relation to the pollution coefficient of
the innovating firm (δ2 ) and its production cost parameter (k2 ).
Turnpike property: I’ve chosen a relative long time horizon (T = 75) in order to show
the “turnpike property”, which implies that variables converge to a stable level (close to
their steady state value, which we still have to calculate analytically! To be done, probably
with the help of Mathematica.) before they diverge and satisfy transversality conditions
λ1 (T ) = λ2 (T ) = 0.
Interpretation: To be done, but things look fine. A decrease in δ2 (from 0.7 to 0.5)
characterizes a less polluting green good. The sales related to the green good, q2 , go up,
while q1 decreases. As a result, abatement for the green firm increases, emissions tend to
rise over the whole planning horizon and so does habit formation X2 (t) (see Figure 6). The
innovating firm perceives a higher shadow value λ2 associated to X2 , and interestingly also
tends to attribute a higher shadow cost to the competitive rival’s habit formation (X1 ); see
Figure 7.
10
3.5
2
1.8
3
1.6
Control variables q2, v2, q1
2.5
1.4
1.2
q1
2
q2
e1
1
v1
e2
v2
1.5
0.8
0.6
1
0.4
0.5
0.2
0
5
10
15
Time periods
20
25
0
5
10
15
20
25
Figure 1: Control and decision variables of firms 1 and 2
11
9
0.4
8
X1
lambda1
X2
lambda2
0.3
7
0.2
Shadow values for X1, X2
State variables X1, X2
6
5
4
3
0.1
0
−0.1
2
−0.2
1
0
5
10
15
Time periods
20
25
−0.3
5
10
15
Time periods
Figure 2: State and co-state variables
12
20
25
3.5
D e c i s i on v ar i a b l e s ,
( q 1( t ) , q 2( t ) )
3
2.5
q 1( t ) | δ 2
q 2( t ) | δ 2
q 1( t ) | δ 2
q 2( t ) | δ 2
q 1( t ) | δ 2
q 2( t ) | δ 2
2
=
=
=
=
=
=
0.5
0.5
0.6
0.6
0.7
0.7
1.5
1
0.5
10
20
30
40
T im e
50
60
70
Figure 3: Varying δ2 and its impact on decision variables q1 (t), q2 (t)
0.42
0.4
D e c i s i on v ar i a b l e s ,
( v 1( t ) , v 2( t ) )
0.38
0.36
v 1( t ) | δ 2
v 2( t ) | δ 2
v 1( t ) | δ 2
v 2( t ) | δ 2
v 1( t ) | δ 2
v 2( t ) | δ 2
0.34
0.32
=
=
=
=
=
=
0.5
0.5
0.6
0.6
0.7
0.7
0.3
0.28
0.26
0.24
10
20
30
40
T im e
50
60
70
Figure 4: Varying δ2 and its impact on decision variables v1 (t), v2 (t)
13
2
1.8
1.6
D e c i s i on v ar i a b l e s ,
( e 1( t ) , e 2( t ) )
1.4
1.2
e 1( t ) | δ 2
e 2( t ) | δ 2
e 1( t ) | δ 2
e 2( t ) | δ 2
e 1( t ) | δ 2
e 2( t ) | δ 2
1
0.8
=
=
=
=
=
=
0.5
0.5
0.6
0.6
0.7
0.7
0.6
0.4
0.2
0
10
20
30
40
T im e
50
60
70
Figure 5: Varying δ2 and its impact on emissions e1 (t), e2 (t)
9
8
7
S t at e v a r i ab l e s ,
( X 1( t ) , X 2( t ) )
6
X 1( t ) | δ 2
X 2( t ) | δ 2
X 1( t ) | δ 2
X 2( t ) | δ 2
X 1( t ) | δ 2
X 2( t ) | δ 2
5
4
=
=
=
=
=
=
0.5
0.5
0.6
0.6
0.7
0.7
3
2
1
0
10
20
30
40
T im e
50
60
70
Figure 6: Varying δ2 and its impact on state variables X1 (t), X2 (t)
14
0.8
0.6
C os t at e v ar i ab l e s ,
( λ 1( t ) , λ 2( t ) )
0.4
0.2
λ 1( t ) | δ 2
λ 2( t ) | δ 2
λ 1( t ) | δ 2
λ 2( t ) | δ 2
λ 1( t ) | δ 2
λ 2( t ) | δ 2
0
=
=
=
=
=
=
0.5
0.5
0.6
0.6
0.7
0.7
−0.2
−0.4
−0.6
−0.8
10
20
30
40
T im e
50
60
70
Figure 7: Varying δ2 and its impact on costate variables λ1 (t), λ2 (t)
15
3.5
D e c i s i on v a r i ab l e s ,
( q 1( t ) , q 2( t ) )
3
2.5
q 1( t ) | k 2
q 2( t ) | k 2
q 1( t ) | k 2
q 2( t ) | k 2
q 1( t ) | k 2
q 2( t ) | k 2
2
=
=
=
=
=
=
3
3
4
4
5
5
1.5
1
0.5
10
20
30
40
T im e
50
60
70
Figure 8: Varying k2 and its impact on decision variables q1 (t), q2 (t)
0.6
0.55
D e c i s i on v ar i a b l e s ,
( v 1( t ) , v 2( t ) )
0.5
v 1( t ) | k 2
v 2( t ) | k 2
v 1( t ) | k 2
v 2( t ) | k 2
v 1( t ) | k 2
v 2( t ) | k 2
0.45
0.4
=
=
=
=
=
=
3
3
4
4
5
5
0.35
0.3
0.25
10
20
30
40
T im e
50
60
70
Figure 9: Varying k2 and its impact on decision variables v1 (t), v2 (t)
16
2
D e c i s i on v ar i a b l e s ,
( e 1( t ) , e 2( t ) )
1.5
1
e 1( t ) | k 2
e 2( t ) | k 2
e 1( t ) | k 2
e 2( t ) | k 2
e 1( t ) | k 2
e 2( t ) | k 2
0.5
=
=
=
=
=
=
3
3
4
4
5
5
0
−0.5
10
20
30
40
T im e
50
60
70
Figure 10: Varying k2 and its impact on emissions e1 (t), e2 (t)
9
8
7
S t at e v a r i ab l e s ,
( X 1( t ) , X 2( t ) )
6
X 1( t ) | k 2
X 2( t ) | k 2
X 1( t ) | k 2
X 2( t ) | k 2
X 1( t ) | k 2
X 2( t ) | k 2
5
4
=
=
=
=
=
=
3
3
4
4
5
5
3
2
1
0
10
20
30
40
T im e
50
60
70
Figure 11: Varying k2 and its impact on state variables X1 (t), X2 (t)
17
0.4
0.3
C os t at e v ar i ab l e s ,
( λ 1( t ) , λ 2( t ) )
0.2
λ 1( t ) | k 2
λ 2( t ) | k 2
λ 1( t ) | k 2
λ 2( t ) | k 2
λ 1( t ) | k 2
λ 2( t ) | k 2
0.1
0
=
=
=
=
=
=
3
3
4
4
5
5
−0.1
−0.2
−0.3
10
20
30
40
T im e
50
60
70
Figure 12: Varying k2 and its impact on costate variables λ1 (t), λ2 (t)
18