making a utility model for CO2 compensating travel demand

I'm trying to make a model which is build around the idea of carbon offsetting fuel (the consumer pays an extra fee per litre fuel for the compensation of the emitted CO2). The goal here is to make a model which takes the following into account:

• it maximizes utility
• it has a budget constraint
• it has an demand for kilometres driven (which gives positive utility) --> travel demand
• these kilometres driven however also emit CO2 which the consumer is aware of and dislikes (the consumer experiences negative utility from emitting CO2
• there is a demand for compensation (the consumer doesn't recieve negative utility from driving if it compensates. However the consumer has to pay more for the same km driven).
• optional: the consumer can also consider taking public transportation (which is consider as an non polluting means of travel, however this is less comfortable than driving in your own car.

i've already tried different options, but i can't seem to get my demands for driving and compensated driving to make completely sense. I hope there is someone here who can help me with this problem

• Can you show us what you have tried? May 12 '20 at 16:36
• \begin{equation} U=\alpha_0q_0+\alpha_1\sqrt{q_1}+\alpha_2\sqrt{q_2}+\alpha_3e^{e_2q_1E_m(1-q_c)} \end{equation} \begin{equation} Y=q_0+p_1q_1+p_2q_2+p_cq_1q_cE_m \end{equation} \begin{equation*} \alpha_1, \alpha_2, e_2, E_m>0 \end{equation*} \begin{equation} \alpha_3<0 \end{equation} alpha1,2,3,e$_2$ = function parameters\\ E$_m$ = pollution per km\\ q0= other consumption\\ q1= km driven\\ q2 = km public transportation\\ qc = factor of km driven which are compensated\\ --> here my qc demand didnt make sense. Probably because i assumed it was a factor 0<qc<1
– M H
May 13 '20 at 8:47
• \begin{equation} U=\alpha\sqrt{N}+\beta E_mN(1-\frac{C}{N}) \end{equation} \begin{equation} Y=p_dV+(p_d+p_c)*C \end{equation} \begin{equation} \alpha >0, \beta<0 \end{equation} \begin{equation} N=D+C \end{equation} N = Kilometres driven\\ D = Dirty km (without CO2 compensation)\\ C = Clean km (CO2 compensated)\\ pd = price dirty km\\ pd + pc = price compensated km \\
– M H
May 13 '20 at 8:57
• \begin{equation} U=N^\alpha E^\beta \end{equation} \begin{equation} N=D^\gamma C^\delta \end{equation} \begin{equation} U=(D^\gamma C^\delta)^\alpha*D^\beta \end{equation} \begin{equation} Y=p_dD+(p_d + p_c)C \end{equation} E=environmental damages.\\ this model gives me sensible demands. However i don't think this cobb-douglass function really caries the right idea behind the model
– M H
May 13 '20 at 9:05