You can pay a subsidy with a tax, but it won't eliminate the deadweight loss

The government's new plan for incentivising a switch to electric vehicles (EVs) is interesting, as reported in the New Zealand Herald last week:

The Government is signalling its intention to slash the price of imported electric and hybrid vehicles by up to $8000 in a bid to make greener cars cheaper for Kiwis.

But it is also planning to slap a new fee of up to $3000 on the import of vehicles with the highest greenhouse gas emissions.

The Government has today opened a six-week consultation period before it introduces new legislation in Parliament later this year...

The Government is proposing discounts of up to $8000 for zero-emission new imported vehicles, such as electric vehicles (EVs).

That number would be $6800 for plug-in hybrid electric vehicle (PHEVs) and $4800 for hybrids.

The level of the discount depends on the total net emissions of the vehicle...

A used Mazda Axela, which is one of New Zealand's most popular imported vehicles, would cost $7200 after an $800 discount.

But a new Land Rover Sports V8 would be slapped with a $3000 high-emissions fee.

A $22,000 Toyota Hiace would cost an extra $1400 after the fee was applied.

Genter said the policy would be cost neutral – meaning the money gained through the fees from higher emitting vehicles would offset the subsidies provided to the lower emission cars.

A specific excise tax on the sale of a good, such as high-emission vehicles, will raise revenue for the government, but it also creates a deadweight loss - there is some economic welfare from the market for those vehicles that is lost, because fewer of them are being traded. This is illustrated in the diagram below. If the market were left alone, it would operate with a price of P0, and Q0 high-emission vehicles would be traded. When the excise tax is imposed, we represent that with the new curve S+tax. The price the consumer pays for a high-emissions vehicle increases to PC, but the effective price for the seller decreases to PP (which is the consumer's price PC, minus the amount of the tax paid to the government). The quantity of high-emissions vehicles decreases to QT.

However, now think about economic welfare. Consumer surplus is the difference between the amount that consumers are willing to pay (shown by the demand curve), and the amount they actually pay (the price). In the diagram, at the equilibrium price and quantity, consumer surplus is the triangle AEP0. Producer surplus is the difference between the amount the sellers receive (the price), and their costs (shown by the supply curve). In the diagram, at the equilibrium price and quantity, consumer surplus is the triangle P0ED.  Total welfare is the sum of the two areas (consumer surplus and producer surplus), and is equal to the triangle AED.

Once the tax is imposed, the consumer surplus decreases to ABPC, while the producer surplus decreases to the area PPCD. The government gains the area of tax revenue, which is the rectangle PCBCPP (this rectangle is the per-unit amount of the tax, multiplied by the quantity of taxed vehicles). Total welfare is the sum of all three areas (consumer surplus, producer surplus, and government revenue), or ABCD. Notice that total welfare with the tax is lower than it is without the tax, by the area BEC. That is the deadweight loss of the tax - lost economic welfare as a result of the tax reducing the quantity of high-emissions vehicles traded.

So, we lose economic welfare in the market that is taxed. Does that mean that we gain welfare in the market that is subsidised? Actually, it doesn't. The diagram below shows the effect of a subsidy on the market for EVs. If the market were left alone, it would operate with a price of PA, and QA EVs would be traded. When the subsidy is introduced (and assuming it is paid to the importers of EVs), we represent that with the new curve S-subsidy. The price the consumer pays for a high-emissions vehicle decreases to PE, but the effective price for the seller increases to PF (which is the consumer's price PG, plus the amount of the subsidy paid to the seller by the government). The quantity of EVs increases to QS.

Now consider the areas of economic welfare. Without the subsidy, consumer surplus is the area FGPA, and producer surplus is the area PAGH. So, total welfare without the subsidy is the area FGH. With the subsidy, the consumer surplus increases to the area FJPG, while the producer surplus increases to the area PFKH. The government subsidy is the rectangle PFKJPG (this rectangle is the per-unit amount of the subsidy, multiplied by the quantity of subsidised vehicles). The subsidy is negative welfare - it reduces total welfare, because the government could instead use that subsidy money to pay for schools, roads, etc. So, it has an opportunity cost (it is not free money). Total welfare with the subsidy is the sum of consumer and producer surplus, minus the area of the subsidy. This is tricky because all the areas overlap, but if you work it out you'll find that total welfare is now FGH-GKJ. So, total welfare with the subsidy is lower than without the subsidy, by the area GKJ - the subsidy also creates a deadweight loss.

Now, combining the two markets, it is clear that the government could use the revenue that it raises from the tax on the high-emissions vehicle market, to pay for the subsidy on the EV market. However, that only pays the subsidy - it does nothing about the deadweight loss in either market. [*] So, while the policy may be cost neutral from a government fiscal standpoint, it clearly isn't cost neutral for society as a whole.

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[*] Now, you could argue (rightly) that the high-emissions vehicle market has a negative externality, and so too many high-emissions vehicles are traded relative to the welfare-maximising quantity. So, a tax on that market would actually increase total welfare (once you factor in the externality). However, that still leaves the deadweight loss in the EV market.

You could also argue that the EV market has a positive externality, since EV use reduces the number of high-emissions vehicles, and so too few EVs are traded relative to the welfare-maximising quantity. So, a subsidy on that market would actually increase total welfare (once you factor in the externality).

However, you couldn't argue that both of those things are true, since you would be double-counting the externality. Either there is a negative externality of high-emissions vehicles, or a positive externality of EVs, but there can't simultaneously be welfare increases for both of those things.