So thanks a lot for presenting your work. I think you have a rather complex setting, and succeeded very well in putting this complex setting into somewhat nice looking and instructive equations into a nice model. Let us go through this model in several steps. And let us start with the demand for filters, which is, I think, from a model setting here, the beginning of the model. We have, you have polluting firms. And these polluting firms, they minimise their costs, which consists of the price I pay for filters, and the price they pay for the remaining emissions. That is, they have to buy permits for the remaining emissions. If you look at this, this is a very nice setting.
Firms will minimise their costs. And the nice thing here is that you have a decreasing productivity of filters, that is, the first filter will remove much emissions, the second filter less, and so on. So firms, depending on the price of filters and the price of permits, will find an interior solution, an interior optimum, how many filters to demand. And by the market clearing condition for the permit market, which is this one here, you link emissions and the demand for filters to the total number of permits applied by the regulator, which is close to your main question, where you ask, how many permits should the regulators supply?
Now let’s move to the clean tech market, which is a core element of your model. And this clean tech market is modelled very nicely. Because you have old firms and old technology. That is, these are firms that have built filters for decades and have a certain technology to do so. And this technology is freely available. On the other hand you have one big firm, or a potentially big firm, and this big firm has a new development. It can produce filters at much lower cost. And as this is a new development, this firm holds a patent and can exploit this patent to make its own profit.
The nice thing about this model is that you do not only presume that there’s market power, but you explain why there’s market power. The second nice thing about this part of the model is that, if you look at these equations, and if you look at the marginal costs
they imply this looks something like this: You have the new development, the new technology, which has constant marginal costs, and then you have the old technology, which start where the marginal costs start, at the same level, but then increase if you increase production. That is, whenever a price for filters occurs on the market that is larger than the marginal cost of the big firm, the old technology will still be used on this market. So the new firm, the innovator, cannot throw out all the old firms just by charging a too low price. To do this this firm would make a profit of zero, which is clearly not optimal.
So if you want to solve this model, I think the way to do this is to start with this equation, as you said and figure out how many filters the polluting firms will demand. Then go to the clean tech market. Figure out, for a given price, how many filters will be supplied by the old technology. And then the difference between demand and the old filters is what the big firm, the innovator, will have to produce. And then you can ask, what is the best price that the innovator would like to see? Because this is a firm that has market power it will optimise over its price.
And finally, once you know how many filters will be used, and what is a price for permits and a price for filters, you can figure out by looking at social welfare, how many permits the regulator should supply. Which is the core question you are going to analyse. Just two remarks on notation. First of all, I think this should be a small xi here. And I would ask you to use different indices in the sector and this sector, not always i, which can be somewhat confusing to the reader. But overall I think that’s a very nice start, a very nice model, and I’m looking forward to the final results.