M.-C. CHANG
298
performance. A product with a large θ means that it gen-
erates less environmental harm, i.e., a “green” product,
where θ (0, ). A firm’s cost function is described as
c(q,
), with c(0,
) = 0, c
(q,
) > 0, c
(q,
) > 0, cq(q,
)
> 0, and cqq(q,
) 0, where the subscripts stand for par-
tial derivatives, pollution abatement costs are increas-
ingly costly, and marginal production costs are non-de-
creasing. A firm’s cost function in our model also satis-
fies the traditional hypothesis in the environmental eco-
nomics literature by Palmer et al. [6], i.e., cq
(q,
) > 0.
Polluting emissions damage the global environment
and personal health due to the ingestion of polluted air,
water, and food. We denote the social cost D(e) as the
global environmental damage, and denote the private
cost d(e) as the personal health damage. The consumers
are a continuous distribution over [0, 1]. A consumer of
type s has a maximized willingn ess to pay for the product
to be s, and each consumer purchases at most one unit of
the product at price p. Since the global environmental
damage is the same for each buyer and each non-buyer, it
does not affect our analytic result. A buyer’s net utility is
s d(e) D(e) p, while a non-buyer’s net utility is
D(e). We assume that d(0) = 0, d' (e) > 0, and d''(e) 0.
3. Game without Environmental Regulation
The game without environmental regulation is a two-
stage game. At stage 1, the firm determines the product’s
environmental performance. At stage 2, the firm sets the
price. We use backward induction to obtain a sub-game
perfect Nash equilibrium (SPNE).
sU is assigned as the marginal consumer who is indif-
ferent to buy produ cts or not. The superscript “U” sta nds
for the case of a game without environmental regulation.
sU is solved through the Eq uation (1) as follow:
1sdf sp
0. (1)
From Equation (1), we have sU = sU(p, θ), where sU
[0, 1], and from Equation (1), we obtain the relationship
between sU and p by the Implicit Function Theorem as:
1
1
U
p
sdf
0
,
. (2)
The result of Equation (2) tells us when price (p) in-
creases, it induces the critical point sU to shift to right and
it approaches 1. In other words, when price increases, it
makes the demand quantity (1 sU) decrease. Hence,
consumer behavior satisfies the demand law.
The demand function that firm faces is qU = 1 sU,
and the firm’s profit function is:
,, ,
UU
ppqpcqp
. (3)
Deriving Equation (3) with respect to parameter p and
let it be zero, we have the result in Stage 2 as:
0
UU
qU
pqs c. (4)
From Equation (1), we also obtain the relationship
between sU and θ by the Implicit Function Theorem as:
10
U
s
, (5)
where 22
011dfdfqd f
0
,
and 2
1q
qcdf
2
. The sign of U
is de-
cided by the sign of 1. When the product’s marginal
environmental damage is large enough, i.e.,
2
q
dqc f
, it induces 1 < 0 and 0
U
s
. This
implies that given one unit of emission with large health
damage to consumers, an increase in a product’s envi-
ronmental performance induces the product’s demand
quantity to increase.
We next examine the relationship between price (p)
and the product’s environmental performance (θ). The
comparative static result in Equation (4) is:
22
2
0,
for.
U
q
q
pdfqdfq
sc
dqc f
(6)
This tells us that the clearer the product is, the higher
the price will be when the product’s marginal environ-
mental damage is large enough. There are two negative
effects in a consumer’s utility: a large marginal environ-
mental damage of the product and the high price. How-
ever, the product’s marginal environmental damage can
be mitigated by increasing its environmental perform-
ance. Hence, when the product’s marginal environmental
damage is large, the consumers are willing to spend a lot
more to purchase the product with high environmental
performance. Some studies on marketing research pro-
vide various evidence to support our finding such as
Cairncross [7], and Cason and Gangadharan [8]. They
concluded that some consumers are willing to pay a
higher price on biodegradable and 3R (Reduce, Reuse
and Recycle) products. We propose this as:
Proposition 1. In a game without environmental regu-
lation, when the product’s marginal environmental da-
mage is large, the consumers are willing to pay a high
price to purchase the product with a high environmental
performance.
We solve the equilibrium solution at stage 1. Recall
the firm’s profit function in Equation (3). Derive Equa-
tion (3) with respect to parameter θ and let it be zero. We
obtain the optimal product’s environmental performance
θU that maximizes the firm’s profit. Substitute θU into
Equation (4) and Equation (3), and the equilibrium solu-
tions in the game without environmental regulation are
{θU, pU,
U, eU}.
4. Game with Environmental Regulation
We now introduce an environmental regulation into the
game. At stage 0, the regulator sets the emission standard
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