Various cities across China are restricting new car purchases in an effort to curb pollution and traffic using either auctions or lotteries. Which of these two methods works better?

In efforts to contain rapidly-escalating congestion and pollution, city governments across China are experimenting with numerous methods to pull cars off the road. Some of these policies, such as restricting certain plate numbers from driving on certain days, reduce usage of existing cars. But recently, cities have started restricting car purchases. Two main models have emerged: lotteries in which licenses are handed out free (or for a small fee) to random winners and auctions in which the highest bidders get the licenses. There is a growing list of cities that use these models. Beijing, Hangzhou, and Guiyang use lotteries while Shanghai and Tianjin use auctions. Guangzhou split the difference and issues half of available licenses through an auction and the other half through a lottery.

These competing approaches raise the question: which is a better method of allocating scarce licenses? Issuing an equal number under either method results in the same number of cars on the road and reduces congestion and pollution equally. So does it make any difference? It does, but let’s back up for a moment. We first have to define what we mean by “better”.

To understand how economists define “better” let’s go back to my childhood. My mother baked excellent apple pies and whenever she did so, I always wanted as large a slice as possible. To get a large slice two things mattered. The first was the size of the pie. The bigger the better so I preferred my mother use a nine- rather than eight-inch pie pan. Once the pie was ready, I had to share it with my brothers and parents. So I also cared about how the pie was divided. Economists think about “better” in the same way: how big the pie is and how its gets divided up. The former is called “efficiency” and the latter “equity”.

Let’s apply these two principles to auto restrictions. If an outcome is efficient, the total benefit is as large as possible (just like the pie being as large as possible). For this to be the case it must be that anyone who could gain from trading does so. The auction method is efficient because it exhausts all possible trades. Suppose there are 10,000 licenses being auctioned. The 10,000 people willing to pay the most will get them. Once the auction ends, there are no useful trades left. Since the 10,000 people who value the licenses most already have them, there is no one else willing to pay enough for them to give them up. If there were they would have outbid them in the auction in the first place.

The lottery outcome is very different. In a lottery anyone who desires a license even a little can throw their hat in the ring and, if they are lucky, win one. Is the benefit here as large as possible or, put differently, are all trades exhausted? No. There are some people not lucky enough to win but who would be willing to pay more for a license than some who were lucky. There is a price (above the winner’s value and below the loser’s) at which they would be willing to trade. This would make both of them better off and increase efficiency.

Why don’t these trades happen? Because lottery rules disallow transferal of licenses once won. That the lottery does not satisfy all gainful trades is made painfully obvious by black markets for licenses. In Beijing licenses in illegal secondary markets have sold for as much as $33,000. Those lucky enough to win Beijing’s lottery–odds are projected to be one in 150 this year–receive an extremely valuable asset. For many of them this asset is more valuable to others and they would like to realize these higher returns even at the risk of prosecution.

Many people regard a lottery system as fairer because everyone has an equal chance of winning. This is an issue of equity or how we divide the pie. Economists have little definitive to say about equity because it requires weighting different people’s well-beings. How should we weight the well-being of lottery winners relative to losers? One must take a stand on this to judge the lottery’s equity. Most economists would defer from doing so but would say that if you regard the lottery as fairer then you must be willing to sacrifice some efficiency by using it.

For those who prefer the lottery there is a way to have the best of both worlds–keep the lottery but allow for a legal secondary market. Those who win the lottery could decide whether to keep their license or sell it to someone who values it more. If we did this, after the dust settles the same people should have the licenses as if we had auctioned them. This surprising result is called the Coase Theorem after Nobel Prize winner Ronald Coase. The intuition is the following. Suppose we award 10,000 licenses through a lottery. If some of the winners are not among the 10,000 people who value the licenses the most then they will be willing to sell them to the people who do. It may take several trades but the licenses should eventually end up in the hands of those who value them most. But this is exactly what the auction would yield and is equally efficient. The only difference from the auction is that the lottery winners will be richer and the government poorer–a difference of equity.

The main criticism of an auction is that mostly wealthy people will be able to drive. This is a valid concern but a lottery with no secondary market is a poor antidote to this–it arbitrarily chooses winners (some of whom will be wealthy yet pay nothing) and does not put the licenses in the hands of those who value them most. Better to auction off the licenses and have the government use the proceeds to help poorer households or invest it in public transit infrastructure to provide better options for the carless. If we are concerned that the government may not accomplish this redistribution, then keep the lottery but allow for a legal secondary market. There is no reason to force the lucky lottery winners to drive if they would rather spend their winnings on something they prefer more.