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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?

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  • $\begingroup$ 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 $G_{1}(x)=18x-\frac{1}{6}x^3+4y$? $\endgroup$ – Adam Bailey Jan 27 at 22:08
  • $\begingroup$ @AdamBailey I was only given the formula and it said the $U_{1}$ profits from the $U_{2}$'s production of $y$ units. $\endgroup$ – MinaThuma Jan 27 at 22:17

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