Question

In a region that must reduce emissions, three polluters currently emit 30 units of emissions together. The three firms have the following marginal control cost functions that describe how marginal costs vary with the amount of emissions each firm reduces.

Suppose this region needs to reduce emissions by 18 units and plans to do it using a form of cap-and-trade after grandfathering total allowances.
(a) How many emission allowances will the control authority grandfather (i.e., give for free)?
(b) Suppose the authority grandfathers an equal number of allowances to each firm. In other words,
each firm should reduce the equal amount of emission out of 18 total reduced units. What is the
total control cost each firm needs to pay for this equal emission reduction?
(c) Assuming no market power and that demand equals supply, the companies can trade their
emission allowances in a market for the emission permit. What price would be paid for those
allowances?
(d) How many allowances would each firm be expected to buy or sell?
(e) If the control authority decided to use an emission tax rather than cap-and-trade, what tax rate
would achieve the 18-unit reduction cost-effectively? Why?

Firms emission reduction	Firm 1 Marginal cost	Firm 2 Marginal cost	Firm 3 Marginal cost
1	$1	$1	$2
2	$1.5	$2	$3
3	$2	$3	$4
4	$2.5	$4	$5
5	$3	$5	$6
6	$3.5	$6	$7
7	$4	$7	$8
8	$4.5	$8	$9
9	$5	$9	$10
10	$5.5	$10	$11

Answer 1

Answer to a :-

Total MC of firm 1 = 32.5

Total MC of firm 2 = 55

Total MC of firm 3 = 65

Total cost of all the firms = 32.5+55+65= 152.5

% share of each firm in total cost =

Firm 1 = 32.5/152.5*100=21.01%

Firm 2 =36.06

Firm 3 = 42.6%

Total emission by all the firms = 30

Expected reduction = 18 units

Permissible reduction = 30-18= 12 units

Quota grandfathering for each firm :-

Firm 1 = 21.01% of 18 = 4 units approximately

Similarly

Firm 2 = 6 units

Firm 3 = 8 units

Thus allowing to emit firm 1 3 units , firm 2 to emit 4 units and firm 3 to emit 5 units each

Answer to b :-

​​​​​​Suppose each one is allowed equal allowances that is 12/3= 4 each ..

Thus emission reduction will be also equal that is 18/3= 6 each

Thus control cost of each firm =

Firm 1 = 6*0.5=3 $

Firm 2 = 6*1=6$

Firm 3= 6*1 =6$

Answer to c :-

If demand equals to supply the new equilibrium will be set on 5 units of quantity.

Thus each firm has an allowance of emitting 4 units , when they trade the allowance to get permit each one has to pay an additional cost for 1 unit (5 -4)

Firm 1 = 0.5*1=0.50$

Firm 2 :-1 *1=1$

However for firm 3 there is an additional cost in the first unit itself thus it will sell for $5 to get a permission of $5

(5-5)=0

Thus firm 3 will not bear any additional cost

Answer to d :-

Each firm is expected to sell 4 allowances each

Answer to e :-

New price at 4 units = $11.5

Initial price at 5 units = 3+5+6=$14

change in price =14-11.5=2.5

% tax rate = 2.5/14*100=17.86%