PLEASE SOLVE ALL QUESTIONS

$($a$)$ Find the area as a function of $q,A_c(q),$ below the marginal cost curve, $MC(q)=c+dq,$ where $c>0$

and $d>0.A_c(q)$ is the shaded area shown in the figure below. The area under the marginal cost

curve, $A_c(q),$ represents the total cost to the company to produce $q$ units of oil.

Hint: divide the area under the curve $MC(q)$ into the area of a triangle and the area of a rectangle

and add them together. The width of the rectangle is $q$ and the height of the rectangle is $c.$

$($b$)$ Find the area as a function of $q,A_r(q),$ below the marginal revenue curve, $MR(q)=a-bq,$ for

qin$[0,\frac{a}{b}].$ Here $a>0$ and $b>0.$ The area under the marginal revenue curve, $A_r(q),$ represents

the total revenue the company gains by producing $q$ units of oil.

Hint: again divide the area under the curve $MR(q)$ into the area of a triangle and the area of a

rectangle and add them together. The width of the rectangle is $q$ and the height is $a-bq.$

$($c$)$ Assume that $a>c$ and qin$[0,\frac{a}{b}].$ The total profit the oil company makes as a function of quantity

produced, $q,$ is given by $P(q)=A_r(q)-A_c(q).$ Find the quantity $q^{**}$ that the company should

produce in order the maximize its profit. Make sure to justify why the quantity you found is indeed

the global maximum.

$($d$)$ What is the total profit that the company makes when it is producing the quantity which maximizes

its profit? In other words, what is $P(q^{**})?$

$($e$)$ Another method of finding the quantity $q^{**}$ that the company should produce in order the maximize

its profit is to look for where $MR(q)$ and $MC(q)$ intersect. Why should this intersection point be

where profit is maximized?

In the scenario above, the oil company will typically produce the quantity of oil that maximizes its profit, with

no consideration to the climate emissions produced in the extraction of oil. One method that governments

use to reduce the climate emissions involved in the extraction of oil is to impose a carbon tax on the fuel

used to extract oil. In BC$,$ the carbon tax is currently $$\frac{65}{?}$ tonne. This essentially increases the marginal

cost of producing some quantity $q$ of oil by some fixed positive amount $t.$

$3.$

$(************)$ Assume the carbon tax is applied so that the Marginal Cost is now $M{C}_t(q)=c+t+dq$

and that $a>(c+t).$

$($a$)$ What is the quantity produced by the oil company that maximizes profits now? For the other parts

of this question, call this quantity tilde$(q).$ The profit function is now $P_1(q)=A_r(q)-A_{ct}(q),$ where

$A_{ct}(q)$ is the area under the new marginal cost curve, $M{C}_t(q).$ Assume again that qin$[0,\frac{a}{b}]$ and

make sure to justify why the quantity you found is indeed the global maximum.

$($b$)$ What is the total profit that the company makes when it is producing the quantity which maximizes

its profit now? In other words, what is $P_1(tilde(q))?$

$($c$)$ Is the new quantity where profit is maximized, tilde$(q),$ less than the quantity that maximized profit with

no carbon tax, $q^{**}?$

$($d$)$ How big does the carbon tax, $t,$ need to be so that the new quantity produced where profit is

maximized is half of the old quantity produced? In other words, how big does $t$ need to be in order

for tilde$(q)=\frac{q^{**}}{2}?$