Emission Certificates by byrnetown73

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									                             Emission Certificates

                                        Konrad Hoppe


                                    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

                                              1
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



                                                 2
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
 2    certificates for free. The monopolist solves the problem

                                                                      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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