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A common argument in the economics of transportation is that publicly provided roads free of charge will result in congestion. Users only internalize the average costs of using the road instead of the marginal social cost which includes the reduction in travel speed for other drivers using the road.

The standard economic remedy for this situation is a Pigouvian tax equal to the difference in marginal private cost and marginal social cost.

Such congestion taxing schemes are however in general politically infeasible. Only few examples worldwide of congestion taxing exits. Examples are The Stockholm congestion tax and Singapore's Electronic Road Pricing (ERP) system, which was first introduced in 1975.

However, the Swedish congestion tax may be deducted from taxable income for both private individuals and businesses. Private individuals may deduct the congestion tax for business journeys, and for traveling between the home and workplace according to the usual rules of car cost deduction (1:80 SEK/km), that is if the distance is at least 5 kilometers (3.1 mi) and the time saved by traveling by own car compared to public transport is at least 2 hours per day.

My question is what is the purpose of implementing a tax only to refund drivers at least partially what they pay in taxes? Why want the drivers simply realize this and act as if there was no tax?

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Pigouvian taxes are actually supposed include some tax returns because they are supposed to be revenue-neutral.

The point is that the tax still distorts the incentives to purchase 'dirty' good. Unless the refund perfectly mimics the consumption tax, the tax still will have desired effect. What you care about substitution effects (distorting private decision in a way that people purchase less dirty good). The intention of Pigouvian tax is not to have income effect (e.g. to make people's budget constraint more biding).

For example, if two goods are substitutes (let's say driving car $x$ or going via bus $y$), and you charge extra tax $(1+t)$ on top of the price of the good you will still provide people incentive to switch from consuming car rides even if you given them some tax rebate $g$ to their income $m$, provided that the tax rebate is not exactly proportional to consumption of the car drives or even higher i.e. $ g = \tau x ∧\tau<t. $

Consider following illustration. Let us first suppose that $g$ is just general lump-sum rebate, which actually is not that far from the Swedish system for consumers that only gives you flat rebate 1:80 SEK/Km for distance to your work not 1:80 SEK/km travelled total (according to the Wikipedia description and English translation the linked source). Thus if we assume your distance to work is fixed (i.e. you cant move home) $g$ would be just flat tax rebate. Even if it is not 100% realistic to assume you cannot move it is good starting point to understand intuition.

Under the above assumptions Swedes would face the following problem:

$$U(x,y) \quad \text{ s.t.} (1+t)p_1x+p_2y=m +g$$

Dividing first two focs of the problem will give us marginal rate of substitution given by:

$$\frac{U'_y}{U'_x}=\frac{p_2}{(1+t)p_1}$$

This tells us that the flat tax rebate is irrelevant, the tax still has the desired effect of making consumption of good $x$ less desirable and it works as designed.

Now consider alternative case where the tax deduction actually depends on distance traveled not just distance to work, but tax deduction is not equal to the tax rate. The Swedish congestion taxes vary between 15 SEK and 45 SEK according to the government website. So the tax deductibility seems to be much smaller than tax rate. So even if you would be deducted based on amount of travel by car rather than distance to work, i.e. case where $g=\tau x$, this would change the problem as follows:

$$U(x,y) \quad \text{ s.t.} (1+t)p_1x+p_2y=m +\tau x$$

Dividing first two focs of the problem will give us marginal rate of substitution given by:

$$\frac{U'_y}{U'_x}=\frac{p_2}{(1+t -\tau)p_1}$$

Now you can see that as long as $t>\tau$ the Pigouvian tax will still have some effect. Now one might ask why Swedish government would not just directly implement tax rate $t_2= t-\tau$, since that would be the de facto 'true' tax here, but the tax would still have an effect.

Lastly, there are actually empirical studies of Swedish congestion tax that showed this tax did reduced congestion (Börjesson & Kristoffersson; 2018). The authors state:

In 2016, the congestion charges in Sweden celebrated their tenth anniversary. They have been effective in reducing congestion in metropolitan areas and they are socially beneficial in both Stockholm and Gothenburg

So in the end the theory is confirmed also by empirical evidence.

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    – 1muflon1
    Commented Jan 18, 2022 at 16:20
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My guess is that not all who pay the congestion charge will be refunded. Namely the ones traveling short distances (more able to use other modes) and the ones not commuting to work (more able to travel during other times). This can be enough to get to the free-flow state of optimal speed.

This is likely made to deal with the shortcomings of a congestion charge that does not take into account distances people drive in the congestion hours and differences in value of time.

However, I believe that this system might make living and commuting from outside the ring more attractive as commuting is fast and practically doesn't cost congestion charge for these commuters. So in the long run, you will still have congestion and might have to lower the refund maybe even until there is no refund anymore.

Many transport economists ignore the long run elasticity of travel demand that includes how relocation decisions changes city planning decisions. Most transport economists define the long run with a maximum of 10 years, and do not consider relocation effects.

Ideally (from a theoretical economic perspective), a better solution would be a kilometer charge that depends on space, car model and time.

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