# Emission Certificates by byrnetown73

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```									                             Emission Certiﬁcates

December 9, 2009

Let us ﬁrst consider the situation with n participants without certiﬁcates. I’ll denote
the quantity produced by every single ﬁrm with q and the aggregated demand with D.
Futhermore I’ll assume a linear demand function

D =a−p

A lecturer told me one day never to trust in linear functions concerning markets, but linear
functions are enough for my purpose to show you some interesting impacts of emission
certiﬁcates. Hence we assume a linear cost function without ﬁx costs as C(q) = cq.

In the case of the polypoly the market equilibrium can be found via

p∗ ∈ arg max{qp − C(q)} ⇔ p∗ = c
q

Hence we have a equlibrium demand of D∗ = a − c and every single ﬁrm has proﬁts of

D∗
πi = (p∗ − c)      =0
n
=0

Let us now introduce certiﬁcates with prize z and assume that every ﬁrm has to buy one
certiﬁcate for every unit of output. Hence the cost function changes a bit to C(q) = (c+z)q.
Every ﬁrm solves the problem

p∗ ∈ arg max{qp − C(q)} ⇔ p∗ = c + z
q

∗
Therefore we obtain a equilibrium demand in the case of certiﬁcates of Dz = a − (c + z)
and every ﬁrm obtains proﬁts of
∗
Dz
πi = (p∗ − (c + z))      =0
n
=0

Our ﬁrst result is, that the proﬁts don’t change in a situation with certiﬁcates compared

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to one without those emission rights.

Let us now analyze a situation in which the certiﬁcates will be emitted via grandfathering.
Every ﬁrm gets an speciﬁed amount of those certiﬁcates for free. Let us assume that
every ﬁrm gets as many emission rights as they would need for their production before the
new law is introduced. Let us further assume that there is an exchange market for these
certiﬁcates where they can buy or sell rights they don’t need respectively need to have. I
will denote the amount of certiﬁcates a ﬁrm owns with m. We need to improve our cost
function a bit because the rights have opportunity cost character:

C(q) = (c + z)q − z · m

The optimal output is not aﬀected by this change, hence we still obtain an equilibrium
price of p∗ = (c + z) and a demand of D∗ = a − (c + z). But the proﬁts change a bit.
D∗        a−c                                a−c
Every ﬁrm produced without certiﬁcates q =           n    =     n    units of output, hence m =    n

D∗ ∗
So the proﬁt can be calculated via the following deliberation. The revenues are                n p      =
a−(c+z)                                                        a−(c+z)
n    (c+z)   and the cost of production are   C ∗ (q)   =      n    (c+z)    hence on the production
side they don’t realize a proﬁt. But we realized a change in the amount of units produced.
They make a proﬁt with selling the certiﬁcates:

D∗           a − c a − (c + z)                  z2
πi = z m −          =z         −                         =      >0
n              n        n                       n

Isn’t that surprising?! There were many debates hold that especially ﬁrms in polypoly
markets don’t have a chance to shoulder the increased costs and have to laid oﬀ employees
when the government introduces emission certiﬁcates, but we have seen that they have
positive proﬁts and therefore larger proﬁts when certiﬁcates would be grandfathered. And
in the case of bought certiﬁcates they face the same proﬁts compared to the case with no
emission rights.

Now I want to consider the case of a monopoly. Because he will get the whole market,
I’ll improve the notation a bit. The demand function will be x = a − p ⇔ p = a − x and
C(x) = cx The problem of a monopolist is

x∗ ∈ arg max{(a − x)x − cx} ⇔ a − 2x∗ = c
x
a−c
⇔ x∗ =
2
∗    a−c        a+c
⇒ p =a−     ⇔ p∗ =
2          2

Therefore we can calculate the proﬁt with

a−c     a+c                 1
πM =               −c             = (a − c)2
2       2                  4

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In the ﬁrst case of emission certiﬁcates he has to buy one for every unit x at a prize of z.
Therefore he has to solve

x∗ ∈ arg max{(a − x)x − (c + z)x} ⇔ a − 2x∗ = c + z
x
a − (c + z)
⇔ x∗ =
2
∗     a − (c + z)        a+c+z
⇒ p =a−              ⇔ p∗ =
2               2

and the proﬁt in this situation is

M         a+c+z                a − (c + z)  1
πz =             − (c + z)                  = (−a + c + z)2
2                       2       4

now we can determine the diﬀerence between those two situation via comparative statics:

M
π > πz
1           1
⇔ (a − c)2 >   (−a + c + z)2
4           4
⇔ z < 2(a − c)     ,z = 0

this equality is fulﬁlled for every positive x∗ as you can see:

x∗ > 0
a − (c + z)
⇔             > 0
2
⇔ z < (a − c) < 2(a − c)

And now let’s check the situation with grandfathering. The monopolist will get m = x∗ =
a−c

a − (c + z)
x∗ ∈ arg max{(a − x)x − (c + z)x + z · m} ⇔ x∗ =
x                                               2
a+c+z
⇒ p∗ =
2

therefore we can calculate the proﬁt with

M                   a − (c + z)   a − (c + z)           a − (c + z)    a−c
πz       =    a−                               − (c + z)             +z
2             2                     2          2
1
=     ((a − c)2 + z 2 )
4
z2
and this is   4    larger than in the case of no emission rights.

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