# Question on Finding the Correct Emission Tax $t$

Let there be two companies $$U_{1}$$ and $$U_{2}$$ where, initially $$U_{1}$$ produces and sells $$x$$ units at $$p=18$$. Production costs are $$C_{U_{1}}(x)=\frac{1}{6}x^3$$ and in the process $$a$$ units of greenhouse gas is emitted, where $$x=20a$$.

I then calculated the profit-maximizing output $$x^{*}=6$$, while the units of GHG emitted is $$a^{*}=0,3$$ and the profit is $$G_{U_{1}}(x^{*})=72$$.

Next assume that the second company $$U_{2}$$ starts producing. It produces $$y$$ units of a different good, so that $$U_{1}$$ profits from it. Namely $$G_{1}(x)=18x-\frac{1}{6}x^3+4y$$, while $$U_{1}$$'s production has a negative effect on the profit of $$U_{2}$$, namely $$G_{U_{2}}(y)=24y-y^2-200a$$

I then determined the profit-maximizing output of $$U_{2}$$ which is $$y^{*}_{U_{2}}=12$$ as well as the profit $$G_{2}(y^{*})=144-200\frac{6}{20}=84$$

Now my problem: How high would an emission tax $$t$$ have to be, so that $$U_{1}$$ selects the most efficient number of units to produce, meaning that the effects on $$U_{2}$$ are completely internalized.

Normally, if $$U_{1}$$ and $$U_{2}$$ were producing the same goods I would simply look at maximizing $$G_{1}+G_{2}$$, however, $$U_{1}$$ and $$U_{2}$$ are producing different goods. Surely, in order for $$U_{1}$$ to internalize the negative effects it has on $$U_{2}$$, it needs to completely halt its production.

What am I supposed to do?

• Welcome to the site. You have indicated that the emissions from $U_1$'s production affect $U_2$'s profit. But what is the process or mechanism by which $U_2$'s production or output affects $U_1$'s profit? In other words, what lies beyond the formula $G_{1}(x)=18x-\frac{1}{6}x^3+4y$? – Adam Bailey Jan 27 at 22:08
• @AdamBailey I was only given the formula and it said the $U_{1}$ profits from the $U_{2}$'s production of $y$ units. – MinaThuma Jan 27 at 22:17