The real question is, "Can CO2 be given more value by using it to make plastics than the value of the energy used to power the process?"
This is a much harder problem. A good start might be a thermodynamic analysis.
Ethylene monomers have an enthalpy of formation from pure CO2 and H2O of 12.9 kWh/kg, plus losses due to process inefficiency. It's only 7.3 kWh/kg if you have a "free" source of hydrogen to provide the remaining energy, but typically industrial hydrogen comes from fossil fuels.
Industrial electricity prices in the US and China are around 10c(=0.66 Yuan)/kWh, so that much electricity goes for about $1.30. But ethylene prices are only around $1.07/kg,[1] so from this admittedly simplistic analysis it would seem the answer is no, it's not worth it. This is before considering the extra energy required to concentrate CO2 from dilute form in the atmosphere or an exhaust flue.
Someone else on this thread said it best -- the tree wins the technology contest. Self replicating, solar powered, soil building, carbon sequestering, water/air cleaning, etc. Oh yeah, and trees are literally[2] what makes it rain.
[1] http://asia.nikkei.com/Markets/Commodities/Ethylene-prices-u...
[2] By transpiring coastal rainfall, trees "kick it back up" and transport water to the inner parts of the continent like an irrigation system. Hence why deforestation leads to desertification. http://www.nature.com/nature/journal/v496/n7445/abs/nature11...
In all seriousness, they gather the solar energy themselves, absorb the CO2 themselves, after a small initial investment you can walk away and the result is plenty of useful materials (including bioplastics)!
They absorb CO2 and other particulate pollution, provide shade which has measurable reductions in cooling costs, divert storm water and improve property values.
Also solar has it's own carbon footprint, and not as small as people think.
The raw materials for these panels are mined, transported, the factories in China that make those panels use likely coal as their primary energy source, then the panels are transported again and again and again, and finally installed and maintained by people who likely don't show up to work in an EV.
You need to be very careful abstracting the true carbon footprint of anything and sadly too many people seem to ignore this fact.
Just because the item you buy and it's use is environmentally friendly it doesn't mean that the supply chain is.
In this case you can easily end up releasing more carbon by building the solar panels and all the machinery needed for sequestering the CO2 than you can actually effectively sequester within the effectively lifespan of your machinery.
You also need to take into account in the cost since each dollar (or any other unit of currency) has a carbon footprint on it's own based on the economy it was generated in.
There is a good reason why some things are not effective, viable or just plain stupid, otherwise you end up planning your freaking solar roadways or worse building them.
The guy that commutes in a coal rolling SUV pays more so he stops and buys an EV pickup instead.
The coal burning power station that he charges it with pays more so they get replaced with gas or solar or wind.
The company that ships the panels chooses between different shipping methods based on price, which includes a cost of carbon.
And so on. The magic of free markets (I'll end this sentence early since we're not doing politics this week.
I see the core problem as. We already have great gasoline infrastructure. It's very energy dense. We just need to learn how to make green gasoline.
The toughest part, however, is that the process would release methane gas into the air. So it's unclear if it's really greenhouse-gas neutral.
In theory, though, if you could trap the methane and then sequester the hydrocarbons you made, you'd have a way to pull CO2 out of the atmosphere.
The reason we use mostly oil to synthesize plastic is that someone else already put in the energy to convert CO2: the plants.
Even though you can't avoid the thermodynamics here and cheat the energy difference, of course there could be more efficient, cheaper and cleaner ways to go from CO2 to plastic, and those would be interesting in terms of dealing with climate change.