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The Black Market for Beijing License Plates∗
Øystein Daljord†Guillaume Pouliot‡Junji Xiao§Mandy Hu¶
May 4, 2021
Abstract
Black markets can reduce the effects of distortionary regulations by reallocating scarce
resources toward consumers who value them most. The illegal nature of black markets,
however, creates transaction costs that reduce the gains from trade. We take a partial
identification approach to infer gains from trade and transaction costs in the black mar-
ket for Beijing car license plates, which emerged following their recent rationing. We find
that at least 11% of emitted license plates are illegally traded. The estimated transaction
costs suggest severe market frictions: between 61% and 82% of the realized gains from
trade are lost to transaction costs.
Keywords: informal economy/underground economy, optimal transport, partial identi-
fication, semiparametric and nonparametric methods
JEL Codes: E26, D450, P230, C140
∗Warm thanks to Shanjun Li for comments and data. Thanks to John Barrios, St´ephane Bonhomme, Dan
Black, Jean-Pierre Dub´e, Alfred Galichon, Chris Hansen, Ryan Kellogg, Carl Mela, Luxi Shen, Avner Strulov-
Shlain, Baohong Sun, Chad Syverson, and Thomas Wollmann for comments. Thanks to Charles Ahlstrom,
Xinyao Kong, Cheryl Liu, Chun Pong Lau, and Omkar Katta for research assistance.
The R package implementing the methods developed herein, as well as pedagogical documentents for in-
structors, are available on the corresponding author’s website.
†Chicago Booth School of Business, University of Chicago, 5807 South Woodlawn Avenue, Chicago,
IL 60637, USA. E-mail: Oystein.Daljord@chicagobooth.edu. Web: faculty.chicagobooth.edu/oystein.
daljord
‡Harris School of Public Policy, University of Chicago, 1155 E 60th St, Chicago, IL 60637 USA. E-mail:
guillaumepouliot@uchicago.edu. Web: sites.google.com/site/guillaumeallairepouliot. Corresponding
author.
§Department of Economics, Lingnan University. 8 Castle Peak Rd, Tuen Mun, Hong Kong. E-mail:
junji.xiao@gmail.com. Web: https://jjxiao.weebly.com/
¶CUHK, Department of Marketing, Room 1105, 11/F, Cheng Yu Tung Building, 12 Chak Cheung
Street Shatin, N.T., Hong Kong. E-mail: mandyhu@baf.cuhk.edu.hk. Web: bschool.cuhk.edu.hk/staff/
hu-mandy-mantian
arXiv:2105.00517v1 [econ.GN] 2 May 2021
1 Introduction
The informal economy, the trade in goods and services that goes undetected in official statis-
tics, makes up an estimated one-sixth of the GDP in the world economy (Schneider, Buehn
and Montenegro,2010). An important segment of the informal economy is black markets,
where goods and services are traded illegally. Black markets emerge in response to restric-
tions on trade that create gains from trade. Such restrictions include prohibition, taxation,
and rationing. Even though black markets are usually considered undesirable per se, it has
been suggested that they can improve welfare by reducing the impact of regulatory distor-
tions (Davidson, Martin and Wilson,2007). Nevertheless, the illicit nature of black markets
creates transaction costs, such as potential legal liabilities, search costs, or the difficulties of
enforcing any kind of contract, which reduce the said gains from trade. The question of their
welfare impact is therefore case-specific and largely empirical. It is furthermore challenging.
Indeed, due to their being illegal, black market transactions are typically unobserved, and
little is known about how such markets perform.
In this paper, we ask how much of the gains from trade are lost to transaction costs in a black
market, which is informative about unintended effects of trade restrictions and the efficacy
of enforcement.
We address this question in the context of the black market for license plates in Beijing. In
2011, the Beijing government restricted driving within the city limits to all but those who
have Beijing license plates in an effort to tackle increasing pollution and congestion. The
license plates were rationed and the quota was allocated by lottery. While Shanghai has
used auctions to allocate license plates since the 1990s, the auction format has found lim-
ited public support. Li, S. (2018) reports survey evidence which shows that, even though a
majority of Beijing residents recognized the need for rationing, less than 10% preferred an
auction, and about 40% preferred a hybrid lottery and auction mechanism.1At the time
the Beijing lottery was introduced, there were growing political concerns over the increasing
auction prices in Shanghai, which by 2010 had reached the level of the price of a typical new
car. Having considered an auction format politically unviable, Beijing chose a lottery out of
1See footnote 13 of Li, S. (2018).
1
fairness concerns (Huang and Wen,2019).
Allocating license plates by lottery creates gains from trade: some lottery winners may prefer
to trade their license plates rather than use the plates themselves. In Section 2, we report
anecdotal evidence of a black market for license plates that emerged soon after the intro-
duction of the lottery. News reports note that the market largely took place online, while
car dealerships seem to have acted as occasional intermediaries. The market was supplied
by both corrupt officials and lottery winners. Enforcement of the non-transferability was
seemingly lax, according to anecdotal evidence. Both rentals, without formal ownership,
and purchases, with or without formal ownership, were offered in the market. The reported
transaction prices are of a similar magnitude to the Shanghai auction prices.
We find evidence of black market trade in comprehensive car sales data. We argue that if
the license plates were allocated to a random subset of the population of car buyers, and
there was no black market trade, then fewer cars would be sold, but the sales distribution
would not shift following the rationing. If instead black market trade allocated license plates
towards wealthier households who empirically purchase more expensive models (Li, S.,2018;
Xiao, Zhou and Hu,2017), the average prices would likely shift upwards. We use a standard
difference-in-differences approach to document such an upward shift in the average Beijing
car prices following the rationing. We find no such price shifts in comparable nearby cities,
such as Tianjin and Shijiazhuang, that did not ration license plates.
We do find evidence of a clear upward shift in the average prices in Tianjin after it intro-
duced a rationing mechanism, which distributed about 40% of the license plates by auction
and the rest by lottery. By design, an auction allocates licenses to a non-random subset of
the population of car buyers just as a black market would reallocate license plates towards
those who are willing to pay, and leads to similar shifts in the sales distribution. We do not
find that the price jump in Beijing was caused by car dealerships adapting their pricing to
the rationing. Instead, we find evidence that vertical price restraints, which were widespread
in the industry at the time, precluded car dealerships from responding to the rationing by
adapting their prices.
2
In addition to being of intrinsic interest, the volume of trade on the black market is essential
to assessing the gains from trade. Though the difference-in-differences estimates are consis-
tent with the existence of black market trade, they carry little information about the volume
of trade. Our main empirical challenge is that the black market transactions are unobserved.
To overcome this challenge, we develop a novel, transparent, and intuitive empirical approach
based on optimal transport methods to identify a lower bound for the gains from trade. We
exploit the fact that a lottery, which randomly samples car buyers from the population, will
generate a different sales distribution than a black market that allocates licenses to a selected
subset of the population. We show how a distance between the two sales distributions is
informative about the volume of black market trade. To address the potential issue of time
trends in the sales distributions, we develop an analogue to difference-in-differences for distri-
butions which uses shifts in the sales distribution in the nearby city of Tianjin, where there
was no rationing at the time, to control for common trends. We find that at least 11% of the
quota is traded on the black market.
In the final step, we combine information from a variety of sources, including news reports, to
infer transaction costs in a market equilibrium model. To further bound the volume of trade,
we combine estimates of the willingness-to-pay for Beijing license plates from the recent lit-
erature (Li, S.,2018) with anecdotal evidence. We thereby produce, in Table 8, decreasingly
nested estimates of identified intervals for the volume of trade and share of transaction costs
that correspond to increasingly nested sets of assumptions. By a Coase theorem type argu-
ment, we infer transaction costs as the wedge between demand and supply that is necessary
to rationalize the estimated volume of trade.
Though some of the anecdotal evidence points to lenient enforcement of the non-transferability
of the license plates (see Section 2), our estimates suggest otherwise. Firstly, we find that
sizable gains from trade are left unrealized. While the market would realize RMB 18.8 billion
in gains from trade in the absence of transaction costs, we find that the realized net gains
from trade lie in a plausible range from RMB 1.3 billion to RMB 5.3 billion.2Secondly, we
find that a plausible range from 61% to 82% of the realized gross gains from trade in license
2At the current exchange rate, dividing by seven gives a rough estimate of the dollar equivalents.
3
plates are lost to transaction costs.
Our paper ties in different literatures. One is an older, and mostly theoretical, literature
on the economic impact of rationing on welfare and incentives for illegal trade (Tobin, J.,
1952;Dr`eze, J. H.,1975;Stahl and Alexeev,1985;Dye and Antle,1986). Our paper also
complements a literature within the field of public finance that estimates the size of the in-
formal economy. Various indirect measures have been proposed, from monitoring excessive
electricity consumption to currency velocity; see Schneider, Buehn and Montenegro (2010)
for an overview. The interest in this literature however tends to be in the scale of tax evasion
in the macroeconomy. Instead, our interest is in measuring the performance of a particular
black market, which calls for a more tailored empirical approach.
Finally, the market we study is of interest in itself. Our paper complements a small literature
on welfare effects of the recent rationing of car sales in larger Chinese cities (Li, S.,2018;
Xiao, Zhou and Hu,2017;Tan, Xiao and Zhou,2019;Huang and Wen,2019). Importantly,
this literature has assumed away the existence of a black market. Our results show that black
market trade plausibly ranges from 11% to 27% of the quota, and as such cannot be ignored
in a thorough economic analysis of car sales in China.
We give an overview of the lottery rationing mechanism and the black market in Section
2. We describe the data in Section 3and report a standard event study in Section 4. The
optimal transport methods are laid out in Section 5along with the estimation results. We
develop and evaluate the market equilibrium model in Section 6, including the estimated
gains from trade and transaction costs. We conclude with a brief discussion in Section 7.
2 The lottery and the black market for license plates
In late December 2010, the Beijing government announced that, effective January 2011, car
license plates would be required to drive without restrictions within the city limits.3A quota
of non-transferable license plates, which was set to about 40% of the previous year’s sales,
was allocated by lottery. The lottery application process was simple: the pecuniary costs
3The first public suggestion of a plan to ration car licenses was issued on December 13, 2010. See Section
Gfor details about the lottery implementation as well as exact restrictions and eligibility criteria.
4
were low and the application could be completed quickly online. Each applicant had the
same probability of winning a license plate, and there were limits on the number of applica-
tions each household can submit. Immediately following the introduction of the lottery, the
number of newly registered cars in Beijing was reduced to the level of the quota.
Allocating the license plates by lottery creates gains from trade: rationed prospective car
buyers may find license plate holders or corrupt officials who are willing to sell theirs. News
outlets, including government controlled ones, reported of a black market for license plates
that emerged soon after the introduction of the lottery. Table 1gives results from a search
in Chinese news reports that include mentions of black market transaction prices. The most
frequent mentions of yearly rental prices for a license plate are between RMB 6,000 and
RMB 12,000 annually. To provide context, the average monthly wage in Beijing in 2011 was
around RMB 5,000. Given an interest rate of about 6% at the time, the twenty-year net
present value (NPV) of the rental income stream of a license plate ranges from RMB 73,000
to RMB 145,000.
The news reports of purchase prices range from RMB 16,000 to RMB 650,000. Note that the
latter price is for a so-called Jing-A plate, which carries particular prestige, is predominantly
used by party officials and certain institutions, and is not available via the lottery. According
to a New York Times article, these seem to be rare transactions of exceptionally high value
(Guo,2016). The majority of reported purchase prices, without ownership, are between RMB
70,000 to RMB 120,000. The upper end of that range is on par with the Ford Focus, which
was the best-selling car at the time (South China Morning Post,2014).
Reuters and New York Times reported a number of online sites that match license plate
holders with buyers (Shen,2014;Guo,2016). Car dealerships seem to act as intermediaries.
In 2014, Reuters quoted a sales representative who claims that he can provide prospective
buyers with license plates (Shen,2014).4Such reports suggest that the non-transferability of
license plates is leniently enforced.
Perhaps more surprisingly, The Atlantic reported that some license holders engage in fake
4Wang Shaoyong, sales manager at a Peugeot dealer in Beijing, said his shop provides car buyers with
license permits from a partner firm that “has many car plates registered in its name.”
5
Table 1: Black market transaction prices in 2011 and 2012 in Chinese news reports
Article Date Newspaper Rental Price
(RMB 1,000/mo)
Purchase Price∗
(RMB 1,000)
Jul 23, 2011 China Times 0.5-2 20 (N); 40-90 (N)
Sep 18, 2011 Beijing Times 1†
Oct 8, 2011 China National Radio 0.5-1
Oct 11, 2011 China Economic Weekly 1 80-100 (U)
Dec 19, 2011 Beijing News 85 (U); 130 (U); 75 (U)
Feb 18, 2012 Rule of Law Weekend 650‡(Y) ; 260 (Y); 50 (Y).
Aug 29, 2012 Beijing Youth Daily 0.5-1 150 (Y); 30-40 (N)
Sep 10, 2012 Beijing Times 0.8 40 (N)
Dec 15, 2012 Workers Daily 0.5 200+ (Y); 170 (U); 16 (N); 30-50 (N)
Dec 19, 2012 Guangming Daily 30 (U)
Dec 26, 2012 ChinaNews.com 80 (U); 160 (U); 200 (U)
∗The letters in the bracket in the column of purchase prices indicate the ownership status: Y = with
ownership, N = without ownership, U = ownership unclear.
†First year is free, and then the price is RMB 1,000 per month.
‡Jing-A is a license plate that is used predominantly by governments and institutions. These license plates
command a premium in the market. These are not available through the lottery.
Sources (by row): Wang (2011); Beijing Times (2011); China Radio Network (2011); Beijing News (2011);
Rule of Law Weekend (2012); Xiaohong (2012); China Industry Network (2012); Guangming Daily (2012);
China News Network (2012)
marriages with buyers in order to legally transfer them a license (Carlson,2017). The quoted
price for such a transaction is RMB 90,000.
The market is reportedly supplied by both license plate holders and corrupt officials. Ac-
cording to South China Morning Post, officials involved with drafting the lottery rules have
experienced unusual luck in the lottery and there have been accusations of the lottery being
rigged (South China Morning Post,2012).5The New York Times reported that the head of
the Beijing Department of Transportation was sentenced to life in prison in 2015 for selling
Jing-A license plates, which suggests stricter enforcement. This sentence, however, appears to
be part of general secretary Xi Jinping’s campaign against corruption amongst party officials,
and may not be representative of the overall policy.
In sum, we find anecdotal evidence of the existence of a material market for license plates.
With respect to enforcement, the evidence is mixed, leaning towards lenient. Though a given
5The Beijing News reported that a record 1.26 million residents competed for fewer than 20,000 plates
in a single month. “Liu Xuemei” was so lucky to have won two plates on that month. Cynical car-plate
hunters wondered if they should change their name to stand a better chance in the next contest. Eagle-eyed
internet users soon discovered Liu Xuemei was the name of the director of the vehicle and driver management
department of the Ministry of Public Security. Liu Xuemei, in her 30s, was in charge of drafting rules for
vehicle permits.
6
Table 2: Rationing Mechanisms
Beijing Tianjin Shijiazhuang
2015 Per Capita GDP (USD) 18731 13355 4576
Rationing Mechanism Lottery Auction and Lottery No Rationing
Announcement Date Dec 23, 2010 Dec 15, 2013 n/a
Implementation Date Jan 1, 2011 Dec 16, 2013 n/a
Average Quota (per mo) 10,000 8,000 n/a
Quota Allocation
(electric:lottery:auction) 0:1:0 1:5:4 n/a
Vehicles without local plate restricted restricted n/a
Table 3: Summary Statistics for MSRP (RMB)
Count Mean Std. Dev. Min. Max.
5,065,356 150,356 100,919 20,800 1,305,000
range of transaction prices is consistently reported across outlets, we find little indication of
either the volume of black market trade or of the magnitude of the transaction costs. The
question of the volume of the Beijing black market for licenses seems to remain an open one.
We next turn to the car sales data with which we estimate the size of the market.
3 Data
We obtained vehicle registration data from Webinsight Technology and Information Corpora-
tion, a Chinese marketing research company. The vehicle registration data have observations
at the city-month level on aggregate registration of all vehicle models available from January
2010 to December 2015 in 35 cities in China. The vehicle models are identified by their
unique codes as catalogued by the Motor Vehicles’ Type and Model Designation, published
by the National Standard of People’s Republic of China, and various car characteristics are
provided.
The vehicle quota data are collected from each city’s official publications.6The data in-
clude the number of applicants, the quota, the average value of winning bids (in the auction
rationing mechanism), and the share of each type of rationing mechanism if multiple mecha-
nisms are applicable, see Table 2. Table 3aggregates the variables across cities and years.
6The official publications of information on allocation mechanisms are from Shanghai International Com-
modity Auction Co. Ltd., Beijing Passenger Vehicle Quota Administration Office, and Tianjin Information
System of Passenger Vehicle Quota Administration.
7
4 Difference-in-differences analysis
Figure 1displays the average car prices in Beijing, Tianjin, and Shijiazhuang on a monthly
level before and after the introduction of the Beijing lottery. Tianjin is the fourth largest city
in China and is about 130 kilometers away from Beijing. With a GDP of about two-thirds
of Beijing’s GDP, Tianjin experienced the fastest economic growth of the major Chinese
cities in the period covered by the data. Shijiazhuang, 270 kilometers away from Beijing,
is the capital and the largest city of North China’s Hebei Province. Although it has about
a quarter of the GDP of Beijing and about a tenth of the population, Shijiazhuang has a
similar pre-trend to Tianjin and Beijing. Shijiazhuang had not introduced rationing by 2016,
while Tianjin introduced a hybrid auction/lottery format in January 2014. The average price
in Beijing is seen to increase by 24% following the introduction of the lottery. The average
prices in Tianjin and Shijiazhuang increased by about 3% in the same period. Assuming
common trends across the cities, we get a difference-in-differences estimate of the effect of
the rationing on the average car prices of about 21%. See Section Cfor the corresponding
difference-in-differences regressions.
Figure 1: Monthly average car price (RMB 1,000) for Beijing, Shijiazhuang, and Tianjin.
Beijing introduced the lottery in January 2011, and Tianjin introduced the hybrid
auction and lottery mechanism in January 2014.
8
The price jump in Beijing could be a supply side price response to the rationing. However,
when we compare the prices of cars that were sold in 2010 with the prices of the same car
models in 2011, we find very little changes in prices. While, as detailed below, we do not
have matched data for the transaction price of each sold car, only its MSRP, we do have
access to proprietary data from a market research firm detailing the changes in price of each
car model before and after the lottery. These are presented in Figure 2.
Figure 2: Distribution of price changes between 2010 and 2011. Nearly all of the density is
near 0, suggesting that prices did not change much between 2010 and 2011.
The lack of supply side price responses is explained by the vertical restraints that were
used in the industry at the time. In 2014 and 2015, following a large scale price investigation,
the National Development and Reform Commission fined car manufacturers, which included
Audi, Chrysler, Mercedes-Benz, Dongfeng-Nissan and BMW, a total of more than RMB
2.5 billion for practicing resale price maintenance for cars and spare parts (see Cai,2016).
While measurement errors in prices, i.e., differences between the measured manufacturers’
suggested retail price (MSRP) and the actual purchase price, is a common concern in studies
of demand for cars in the United States, it is not a threatening empirical issue in the case
at hand. Such measurement errors are less of an issue in the Chinese data since under
resale price maintenance, the MSRPs are enforceable minimum prices. Indeed, transaction
prices are unavailable for large data sets but will typically not deviate from MSRP by a
9
substantial margin. Li, S. (2018) concurs and his analysis likewise relies on this assumptions.
As he explains, “there could be potential pitfalls in using MSRPs when they are different
from the transaction prices due to promotions,” however “different from the promotion-heavy
environment in the U.S., China’s auto market has infrequent promotions from manufactures
or dealers and retail prices are often very close to or the same as MSRPs, a phenomenon
commonly seen for luxury products in China.” See Xiao, Zhou and Hu (2017) and references
therein for further discussion.
The bottom line, as we see it, is that not only are the price differences between MSRPs and
transaction prices small, they are effectively constant for any given model. This makes it
very unlikely to induce over-counting of sales.
Even more to the point, we may directly verify the robustness of the results. We have access
to a proprietary data set that is considerably smaller but is less susceptible to measurement
error since it contains transaction prices. In particular, the total number of households that
were randomly selected to collect the proprietary data in Beijing over the years 2010 and
2011 is 3,718, whereas the number of households present the Beijing MSRP data during the
same time frame is 974,573.
Using this proprietary data set, we are able to match a subset of the models in our MSRP
dataset with the transaction price, and we analyze this subset using MSRPs and transaction
prices separately.
We find that the absolute difference of the results is quite small, suggesting that measurement
error that arising from using MSRPs instead of transaction prices is not a threat to the validity
of the analysis. A longer discussion of this direct robustness check can be found in Section I
of the Appendix.
Though car dealerships were effectively not allowed to change their prices, Feenstra (1988)
suggests that car dealerships may upgrade the quality of their product lines in response to
the quota. The gain from an increase in the intensive margin can partially offset the loss from
the extensive margin following the rationing. The increase in the average price in Beijing
10
could therefore be a strategic price response by the manufacturers to the market contraction.
If so, the average prices for the models that were sold following the rationing, but not before,
would be higher than the models that were sold both before and after the introduction of
the lottery. Figure 3shows barely noticeable changes in the distribution of the prices of the
models offered before the lottery (pre) and the models that were only offered after the lottery
(post). The post-lottery distribution seems to shift slightly to the left, if anything. In sum,
we find little evidence that the price jump was caused by supply side responses.
Figure 3: Distribution of Beijing car prices in 2010 (pre-lottery) and new car prices in 2011
(post-lottery). There is little change in the car prices before the lottery and the
car prices of new models after the lottery, irrespective of the number of cars sold
at each price. The parameters of the superimposed Gaussian curves are the mean
and variance of log MSRP for the respective set of data.
There were other policies in place that may have affected car sales. The Beijing subway
system expanded rapidly from 39 stations in 2002 to more than 300 in 2014. In particular, a
number of new lines that expanded the network from 228 kilometers to 336 kilometers were
opened in December 2010, just before the rationing went into effect. Public transportation
appeals disproportionately to lower income households who may subsequently select out of
the car market, which would lead to a change in demand that is unrelated to the rationing
(Xie, L.,2016). Since the subway expansion in 2010 coincided with the rationing, it is
11
hard to directly test this alternative account in the data. There were, however, three other
major subway expansions: in December 2012 (70 kilometers), May 2013 (57 kilometers), and
December 2014 (62 kilometers). Figure 1shows no apparent shift in the Beijing prices at any
one of these expansions. The lack of responses at these later expansions suggests the effect
of the expansion in December 2010 was also modest.
The Beijing price jump may instead be explained by black market trade. If the Beijing lottery
randomly selected car buyers from the population of car buyers, if there were no illicit trade
in license plates, and if income and preferences did not change right after the lottery, then
the sales distribution would not shift with the rationing. The price jump is instead consistent
with a black market that reallocates the license plates to wealthier households who buy more
expensive cars.
Figure 1shows a similar, if smaller, price jump in Tianjin at the beginning of 2014 when
Tianjin introduced license plate rationing. Tianjin used a lottery/auction hybrid mechanism
that allocates 50% of the license plates by lottery, 40% by auction, and reserves the remainder
for electric cars. By design, the auction, like the black market, allocates license plates to those
who have the highest willingness to pay. The Beijing price jump is consistent with a share
of the Beijing license plates being allocated in part by a lottery and in part by a price-based
mechanism.
Though the data show price jumps following the rationing in both Beijing and Tianjin, it
is not obvious that price-based allocation mechanisms, be it a black market or an auction,
would shift sales towards more expensive cars. On the one hand, the price of a license
plate serves as a tax on a car purchase that leads a given buyer to substitute towards less
expensive cars. On the other hand, price-based allocation mechanisms select wealthier buyers
that purchase more expensive cars. The net effect could be that these mechanisms shift
sales towards less expensive cars. Tan, Xiao and Zhou (2019) and Xiao, Zhou and Hu
(2017), however, document increasing expenditures in Chinese cities that use lottery/auction
hybrids. In Shanghai, which has used auctions to allocate license plates since 1996, there is a
strong positive correlation between auction prices and car expenditures, so the net effect has
12
empirically proved to be positive.7We will remark, however, that our estimation methods
do not require the sales to shift towards more expensive cars.
5 Inferring bounds on the volume of trade using optimal trans-
port
The main empirical challenge is that we do not observe transactions in the black market
directly. Though a 21% increase in the average price may seem substantial, one cannot infer
the volume of trade from the first moments of the sales distributions. We instead wish to infer
the volume of trade from the observed displacements of the sales distributions. Figure 4shows
overlaid histograms of car prices in Beijing in 2010 and 2011. The sales distribution clearly
shifts to the right after the introduction of the lottery. This displacement is consistent with a
black market that reallocates license plates to wealthier households, but, under assumptions
we specify below, is inconsistent with a lottery that randomly selects car buyers from the
population. Our strategy is to infer a lower bound for the volume of black market trade by
quantifying the displacement of the car sales distributions in Figure 4using optimal transport
theory and methods.
7The average Shanghai auction price in 2010 was RMB 39,000 and increased to RMB 81,000 in 2015, while
the average car price increased from RMB 173,000 to RMB 215,000 over the same period.
13
Figure 4: Histograms of Beijing car sales in 2010 (pre-lottery) and 2011 (post-lottery). Un-
like Figure 3, this plot considers the number of cars sold at each MSRP value.
Optimal transport has a long history in economics and operations research (see Kantorovitch,
1958 for an early treatment). Furthermore, optimal transport has recently witnessed renewed
interest in economics and applied econometrics, including the analysis of identification of dy-
namic discrete-choice models (Chiong, Galichon and Shum,2016), in vector quantile regres-
sion (Carlier, Chernozhukov and Galichon,2016), in empirical matching models (Galichon,
Kominers and Weber,2018), and in latent variables (Arellano and Bonhomme,2019). To
the best of our knowledge, this is the first principled use of optimal transport for reduced
form analysis in economics, notwithstanding matching applications. The R package diftrans,
implementing the estimators developed in the methodological section, is available on the
corresponding author’s website.
We develop two estimators. First, the before-and-after estimator uses displacement, i.e., the
shift in the sales distribution following the rationing in Beijing, in order to estimate the vol-
ume of trade. Second, the difference-in-transports estimator uses a difference between the
displacement in Beijing and the corresponding displacement in Tianjin. Similar to a standard
difference-in-differences estimator, this difference in displacements controls for common eco-
nomic trends. As with standard regression methods, we must specify a cost function which, in
14
the case of optimal transport, is the distance between points in the support of the compared
probability mass functions.
In the following, we refer to a buyer as a household that purchases a car with an illegally
purchased or rented license plate. We refer to a seller as a household that won a license plate
and then decided to sell or rent it to a buyer. We refer to the transaction between a buyer
and a seller as a trade.
In the following, it is useful to adopt a potential outcomes notation. Let Ppre(r, b) and
Ppost(r, b) be the potential pre-lottery and post-lottery population sales distributions in Bei-
jing, respectively. The index r∈ {0,1}is one when there is rationing post-lottery, and zero
otherwise. The index b∈ {0,1}is one when there is a black market post-lottery, and zero
otherwise. Distinguishing between the rationing and the presence of a black market in the
potential outcomes model will allow for a precise comparison between the assumptions needed
for both estimators.
5.1 Identification Assumptions and Optimal Transport Theory
We first provide three assumptions that are sufficient to identify a lower bound for the volume
of trade. The first rules out anticipatory sales effects of the rationing; the pre-lottery sales
distribution does not depend on whether there is rationing and/or a black market in the
post-lottery period.
Assumption 1 No anticipation:
Ppre(r, b) = Ppre
for r∈ {0,1}and b∈ {0,1}.
Assumption 1seems plausible given that there was only a week between the announcement
of the rationing and its implementation. It is possible that some anticipation occurred in
December 2010, the month preceding the inauguration of the lottery. For robustness, we
produce the analysis with and without December 2010 (see Section H.1). We find that the
results are quite robust; both of our estimates change by at most one percent when we omit
December 2010.
15
Next, we assume away time trends that are unrelated to the lottery. Specifically, we assume
that the pre-lottery sales distribution is equal to the potential post-lottery sales distribution
when there is no rationing and no black market.
Assumption 2 No time trends:
Ppre =Ppost(0,0).
Assumption 2rules out trends in car preferences such as, say, increasing demand for SUVs.
It implies that a household that wins the lottery would purchase a car, in 2011 in a world
without rationing, according to the same price distribution as it would in 2010. It also
precludes car dealerships from changing the car prices following the lottery. This assumption
is consistent with vertical restraints that regulated retail prices in the car market at the time
as discussed in Section 2. It is possible to do away with this assumption (see Section 5.3).
The next assumption concerns the effect of rationing on car preferences.
Assumption 3 No general equilibrium effects:
Ppost(0,0) = Ppost (1,0).
Assumption 3is an analogue to the common stable unit treatment value assumption, also
called no interference assumption, in the program evaluation literature. It forces the potential
post-lottery sales distribution, without rationing and with no black market, to equal the
potential sales distribution post-lottery, with rationing and with no black market. An example
of general equilibrium effects that violates Assumption 3would be if rationing license plates,
which leads to fewer cars on the roads, increased the willingness-to-pay for a car. We however
believe that such general equilibrium effects are negligible, since the rationing controls the
flow of new cars to Beijing (on the order of 260,000 per year), and not the stock (on the order
of five million at the time). The assumption also rules out a direct effect of the rationing
on the car preferences. This assumption is common in the literature on rationing, but it is
untestable, and it has been contested in the past. One example is Tobin, J. (1952), which
entertains the idea that rationing can change tastes over time.
16
“Experience under rationing may alter the consumer’s scale of preferences. He
may learn to like pattern of expenditures into which rationing forces him, or to
dislike it even more intensely than if it had not been forced upon him.” (p. 548).
Shen, Hu and Dang (2019) explores the hypothesis that lottery winners experienced a sense
of luck that shifted their preferences towards more expensive car models, which violates As-
sumption 3. Neither Shen, Hu and Dang (2019)’s hypothesis nor Assumption 3seems to be
testable in our data, and thus must be assessed according to their plausibility.
A potential threat to the validity of Assumption 3is that the rationing can mechanically in-
duce a difference in the observed distributions Ppre and Ppost(1,0) by bringing about a change
in their composition. Specifically, it is possible that the distribution of cars purchased with
a new license and the distribution of cars purchased with an existing license –i.e., one which
was previously associated with another car– are different. Because only the former purchases
are being rationed, the observed distribution may change due to the fact that its composition
has changed. As detailed in Section 5.2 and in Section Dof the Appendix, the magnitude of
this effect may be assessed, and it is not a threat to our analysis.
Another potential issue deserving discussion is the possibility that households planning to
purchase multiple cars, upon winning the lottery, purchase a more expensive car than they
otherwise would have, knowing that they will in all likelihood only get to buy one. We argue
that such an effect does not threaten our analysis. First, the many conditions such buyers
must satisfy suggest that they are few and far apart; we speak of prospective car buyers
who had at least two fewer cars than they wanted to purchase in the short- or medium-term,
were substituting at least one less car for a substantially more expensive car, but were not
willing to go purchase it on the black market. Second, and more importantly, our estimator
is specifically robust to this concern. Indeed, upon inspection of the transport map corre-
sponding to the solution of the optimal transport estimator presented in Section 5.2, one finds
that approximately 99% of the trades that account for the estimated total are for more than
RMB 100,000 (which, for instance, is almost half of the 95th percentile household income in
China).8This means we would only have as confounders the unlikely households who are
8This approximation requires a unique and interpretable solution ˆγ, as defined below. This obtains by
17
spending an additional amount of at least RMB 100,000 on the intensive margin because they
are constrained on the extensive margin, and are still not willing to transact on the black
market. We consider those to be very unlikely households.
If license plates are reallocated by black market trade, then Ppost(1,1) may not equal Ppost(1,0).
If a buyer purchases a different car than the seller would have purchased, had he not traded
his license, then we can think of that trade as shifting or transporting mass from Ppost(1,0)
to Ppost(1,1). The smallest number of such trades required to account for the shift in dis-
tributions gives a lower bound estimate of share of black market trade. It is a lower bound
since, for instance, a trade where the buyer purchases the same car as the seller would have
purchased, had he not sold his license, goes undetected. Since all license plates can be traded
without shifting the sales distribution, the upper bound for the volume of trade is 100% of
the quota. The counterfactual distribution Ppost(1,0) is identified since Assumptions 1,2, and
3imply that
Ppre =Ppost(1,0).(1)
We could have given condition (1) directly as an assumption. By distinguishing between
Assumptions 2and 3, and only relaxing Assumption 2in Section 5.3 where we develop
a difference-in-transports estimator, we get a clear comparison of sufficient conditions for
identification.
Under these assumptions, the minimum amount of mass we need to transport between the
(observable) distributions Ppre and Ppost(1,1) estimates a lower bound for the volume of
trade. For ease of exposition, we drop the potential outcome arguments from Ppost(1,1)
and, abusing notation, let Pj(x) for j∈ {pre, post}and x∈ X denote the probability mass
functions in the rest of this section, where Xis the support of car prices. If we observed the
population distributions Ppre and Ppost directly, we could compute the desired lower bound
using a ground cost Ci,j +λ|xi−xj|, for all i, j, for small λ, as it picks out a solution with small trades. We
used λ= 0.01.
18
via the following oracle problem,
OT (Ppre ,Ppost) = minimize
γ∈ΓZc(x0, x1)γ(x0, x1)dx0dx1
subject to
Zγ(x0, x1)dx1=Ppre(x0),for all x0∈ X ,
Zγ(x0, x1)dx0=Ppost(x1),for all x1∈ X .
(2)
The optimal transport cost has two components. The first is c(x0, x1), the cost of transport
between point x0in Ppre and point x1in Ppost. The second is γ(x0, x1), the amount of mass
that is transported between point x0in Ppre and point x1in Ppost. The optimization is over
γ∈Γ, where Γ is the set of all bivariate probability distributions. The constraints ensure that
the marginal distributions exactly equate the observed distributions Ppre(x) and Ppost (x). We
choose the cost specification
c(x0, x1) = (|x1−x0|>0).(3)
This function assigns the same cost for transport between any pair of points x0,x1when
x06=x1, and zero cost when x0=x1. The objective function of the optimal transport problem
therefore exactly represents a volume of trade that transforms the pre-lottery distribution
into the post-lottery distribution. The optimal transport cost OT for the problem in (2) can
be interpreted as the smallest volume of trade that is consistent with the data. Our estimand
is then
s(Ppre,Ppost ) = OT (Ppr e,Ppost ).(4)
An example may be instructive. Suppose cars are sold either at a price p1or a price p2, so
X={1,2}. Pre-lottery, eight cars are sold: six at p1and two at p2, such that the population
distribution is Ppre =1
8[6,2]. Post-lottery, the quota is set to four cars. One is sold at p1and
19
three at p2, such that the population distribution is Ppost =1
4[1,3]. It is immediately clear
that at least two license plates must be traded to rationalize the shift in the sales distribution.
One solution γthat satisfies the constraints in (2) is
γ=1
4
1 2
0 1
.
The diagonal terms imply no transport and hence no cost. We can therefore calculate the
optimal transport by summing all off-diagonal terms of γ. This gives OT (Ppr e,Ppost ) = 50%
as a lower bound on the share of illegal trade. This example assumed away sampling variation
in the empirical distributions. We account for sampling variation in Section 5.2 below.
There may be more than one γthat solves (2). Our parameter of interest is however not
γ, but OT (Ppre ,Ppost), which is clearly unique. Multiplicity of γsolutions therefore has no
implications for the identification of a lower bound for the volume of trade.
It is tempting to visualise, in Figure 4, the displacement as the area of the two histograms
that is non-overlapping. This turns out to be correct; it represents the total variation dis-
tance between the two distributions, which obtains as the dual representation of the optimal
transport distance when the cost is as specified in (3).
The estimand in (4) does not require that buyers purchase more expensive cars than the
sellers would have purchased if they could not trade, i.e., that the sales distribution shifts to
the right; it only requires that they purchase cars of different prices so that the trades are
observed. Any displacement of the sales distributions is counted as trade.
5.2 Before-and-after estimator
We need to account for the fact that we have access to the sample distributions, and not the
population distributions. The sample pre-lottery distribution is ˆ
Ppre,npre , where npre is the
number of observations, and ˆ
Ppost,npost and npost are defined analogously for the post-lottery
20
period.9We can approximate the population problem in (2) with its sample analog,
OT (ˆ
Ppre,npre ,ˆ
Ppost,npost ) = minimize
γ∈ΓX
i,j
γi,j Ci,j
subject to
X
i
γi,j =ˆ
Ppre,npre (j),for all j,
X
j
γi,j =ˆ
Ppost,npost (i),for all i,
γi,j ≥0,for all i, j,
(5)
where C=11T−diag(1). The observed sample distributions ˆ
Ppre,npre and ˆ
Ppost,npost are
plotted in Figure 5. However, we cannot directly apply the program in (5) to ˆ
Ppre,npre and
ˆ
Ppost,npost because sampling uncertainty will inflate the transport estimate. Even in the null
case of no trade in license plates, i.e., if both ˆ
Ppre,npre and ˆ
Ppost,npost were composed of draws
from the same distribution Ppre, sampling variation would lead to differences in the realized
distributions.
Figure 5: Exact empirical distributions of Beijing car sales in 2010 (pre-lottery) and 2011
(post-lottery). These are labeled ˆ
Ppre,npre and ˆ
Ppost,npost , respectively.
9The pre-lottery period is the year 2010, and the post-lottery period is limited to be the year 2011 unless
otherwise specified.
21
An effective and transparent approach to smooth out sampling variation while maintaining
interpretability is to ignore small moves: we can attribute zero cost to transport mass up to
some small distance d≥0, where the distance is measured in RMB. We want to choose d
large enough to not confound sampling variation with trades but small enough to detect as
many trades as possible. An approximate but immediate tool for visualizing the smoothing
implied by ignoring differences of up to dRMB is the histogram of car prices evaluated at
different values of das a bin width, see Figure 6.
To select an appropriate tuning parameter dso as to smooth out the sampling variation, we
use as an approximation the variation in transport cost between empirical distributions with
observed sample sizes npre and npost, where both empirical distributions are drawn from the
population distribution of car buyers, Ppre. We select dsuch that the transport cost in the
placebo problem is less than the precision level at which we report the transport cost. We
report the percentage cost up to its first decimal, and thus require a placebo cost of less
than 0.05 percent. That way, sampling variation can be safely ignored when interpreting
point estimates. We rely on the mean placebo transport cost, but inspection of the quantiles
reveals a rule of thumb relying on the later would produce a similar estimate.
22
Figure 6: Smoothed empirical distributions of Beijing car sales in 2010 (pre-lottery) and
2011 (post-lottery) at candidate values of d. The width dof the bins in each of
these plots is different, approximating the different levels of smoothing applied to
Figure 5.
23
To implement this smoothing criterion, we replace the cost matrix Cwith a thresholded
version
Ci,j (d) = (|xi−xj|> d),
where xiis the ith entry of X. We then solve the sample optimal transport problem
OT (ˆ
Ppre,npre ,ˆ
Ppost,npost ) = minimize
γ∈ΓX
i,j
γi,j Ci,j (d)
subject to
X
i
γi,j =ˆ
Ppre,npre (j),for all j,
X
j
γi,j =ˆ
Ppost,npost (i),for all i,
γi,j ≥0,for all i, j.
(6)
The placebo transport cost is the expected transport cost when both the pre-lottery and post-
lottery empirical distributions are drawn from the same population of car buyers. Because
the population distributions are not observed, we use the empirical distribution as a plug-in,
and look at the average transport cost when drawing an empirical distribution from the same
plug-in distribution. Specifically,
ˆsplacebo(d) = Eˆ
Ppre [OTd(˜
Pnpre ,˜
Pnpost )].
This simple and transparent approach is in line with our intuitive notion of sampling un-
certainty. We select a dsuch that in expectation, or with prescribed probability, only a
controlled quantity of detected trades will in fact be attributable to sampling variation. This
controlled quantity may be chosen to be negligible in terms of the degree of precision of
the reported output. Trades corresponding to changes in price of less than the threshold d
RMB will be ignored, which will attenuate the lower bound. That is again in line with our
philosophy of trading off a tighter but less reliable bound for a more conservative approach
delivering a credible lower bound on the volume of trade.
We next probe the sensitivity of the estimates over a range of values of d’s. Figure 7plots
24
Figure 7: Real and placebo costs as a function of d. The placebo cost in Beijing at each dis
the mean of 500 simulated values of OTd(˜
Ppre,npre ,˜
Ppre,npost ), while the real cost
is OTd= (ˆ
Ppre,ˆ
Ppost).The vertical line at d= 25,000 is the smallest bandwidth
for which the placebo cost is less than 0.05%.
ˆs(d) = OTd(ˆ
Ppre,npre ,ˆ
Ppost,npost ) and ˆsplacebo (d) against d. While ˆs(d) estimates the lower
bound for the market share, ˆsplacebo (d) estimates the sampling uncertainty as a function of
d.
According to our rule of thumb, the results in Table 4suggest using d= 25,000. The placebo
transport cost ˆsplacebo(25,000) is negligible at the one decimal accuracy level. We therefore
choose ˆs(25,000) = 9.8% as our preferred before-and-after estimate.
Another criterion for selecting a d, which was introduced in the discussion of Assumption 3
and is detailed in Section Dof the Appendix, is that it smooths out the displacement due to
the changes in the proportion of first-time and returning buyers. As detailed in the Appendix,
we find that d= 10,000, the estimated effect is about one percent, and for d= 20,000, the
estimated effect is less than 0.1 percent.
Table 4: Beijing 2010 placebo transport costs
d: 10,000 15,000 20,000 25,000 30,000 50,000 70,000 90,000
ˆs(d) : 15.0% 13.6% 12.1% 9.8% 8.6% 5.4% 4.1% 3.0%
ˆsplacebo (d) : 0.08% 0.07% 0.06% 0.04% 0.04% 0.007% 0.004% 0.004%
25
5.3 Difference-in-transports estimator
The results in the previous section relied on Assumption 2, which precludes time trends in
the sales distributions. This assumption may be challenged. For instance, income growth
or changes in car preferences could lead the sales distributions to shift for reasons that
are unrelated to black market trade. In Section 4, we used the sales data from Tianjin to
control for common trends using difference-in-differences regressions. We draw on the same
idea for our second estimator. We want to use the observable displacement of the Tianjin
sales distribution to account for external factors (changes in income, preferences, etc.) that
would have shifted the sales distribution in Beijing, had it not introduced rationing. We
call this analogue to the standard difference-in-differences estimator a difference-in-transports
estimator. Figure 8shows that the sales distribution in Tianjin, where there was no rationing,
may have shifted slightly to the right. The displacement in Tianjin is clearly less pronounced
than the corresponding displacement in Beijing, which is seen in Figure 4.
Figure 8: Histograms of Tianjin car sales in 2010 (pre-lottery) and 2011 (post-lottery).
The target quantity is the displacement between the observed, post-lottery Beijing sales
distribution, with rationing and a black market, and the counterfactual, post-lottery Beijing
sales distribution, with rationing and no black market,
OT (PBeij ing,post (1,0),PBeij ing,post (1,1)).(7)
26
Its oracle difference-in-transports proxy is given by the transport cost of the Beijing sales
distributions before and after the lottery with rationing and a black market, net of the
transport cost that would have been incurred if there was rationing, but no black market,
OT (PBeij ing,pre,PB eijing,post(1,1)) −O T (PBeijing,pre ,PBeijing,post(1,0)) (8)
By the triangle inequality of Proposition 55 in Peyr´e, G. (2018), the population quantity sdit
with C=C(0) produces a lower bound on the quantity of interest,
OT (PBeij ing,pre,PB eijing,post(1,1)) −O T (PBeijing,pre ,PBeijing,post(1,0))
≤OT (PBeij ing,post (1,1),PBeij ing,post (1,0)).
(9)
This quantity depends on the unobservable OT (PBeij ing,pre,PB eijing,post(1,0)). In contrast to
the before-and-after case, we now relax Assumption 2, thereby allowing the sales distribution
in Beijing to shift between the pre- and post-lottery period for reasons other than the lottery,
i.e., PBeijing,post(0,0)) 6=Ppre. We instead use shifts in the Tianjin sales distribution to
control for external factors. We replace Assumption 2with the following parallel trends-like
assumption.
Assumption 4 Equal displacement:
OT (PBeij ing,pre,PB eijing,post(0,0)) = O T (PTianj in,pre(0,0),PT ianj in,post(0,0)).
This assumption requires the observed displacement in Tianjin to equal the corresponding,
counterfactual displacement in Beijing were there to be no rationing and no black market
trade. Assumptions 3and 4together imply
OT (PBeij ing,pre,PB eijing,post(1,0)) = O T (PTianj in,pre,PT ianj in,post(0,0)).(10)
27
We can therefore write the estimand (8) in terms of optimal transports between observable
distributions
sdit =OT (PBeij ing,pre,PB eijing,post(1,1)) −O T (PTianj in,pre(0,0),PT ianj in,post(0,0)),(11)
which is the optimal transport analogue of the traditional difference-in-differences estimand.
Although (11) is well estimated by
OTd(ˆ
PBeijing,pre ,ˆ
PBeijing,post(1,1)) −OTd(ˆ
PT ianjin,pre (0,0),ˆ
PT ianjin,post (0,0))
for a small but non-zero value of d, we would like our difference-in-transports estimate to give
a lower bound on OTd(ˆ
PBeijing,post(1,1),ˆ
PBeijing,post(1,0)), the sample version of the target
quantity, and thus guarantee that the desirable population inequality (9) always holds in
sample. We show below that the estimator
ˆsdit =OT2d(ˆ
PBeijing,pre ,ˆ
PBeijing,post(1,1)) −OTd(ˆ
PT ianjin,pre (0,0),ˆ
PT ianjin,post (0,0)) (12)
delivers this property. Under Assumption 4, and for dsufficiently large to smooth out sam-
pling variation, ˆsdit(d) is an accurate estimate of
OT2d(PBeij ing,pre,PB eijing,post(1,1)) −OTd(PBeij ing,pre ,PBeij ing,post (1,0)),(13)
and lower bounds the sample analogue of the population quantity of interest. This can be
shown using Theorem 1. The intuition is that a standard triangle inequality obtains for the
smoothed optimal transport, but only if we oversmooth the less-than-or-equal side of the
inequality.
Theorem 1 Let OTd(·,·)be the discrete optimal transport problem described in (6)with
C=C(d), and let ˆ
Pa,ˆ
Pband ˆ
Pcbe three probability mass functions. Then, the following
inequality holds
OT2d(ˆ
Pa,ˆ
Pb)−OTd(ˆ
Pc,ˆ
Pb)≤OTd(ˆ
Pa,ˆ
Pc).(14)
A proof is given in Section Aof the Appendix. Theorem 1guarantees that
28
Table 5: Beijing 2011 placebo transport costs
d: 4,000 7,000 10,000 15,000 20,000 25,000 30,000 35,000
ˆs∗
dit(d) : 9.89% 11.87% 10.15% 6.83% 4.99% 4.66% 4.43% 3.40%
ˆsplacebo (d) : 0.16% 0.05% 0.03% 0.03% 0.03% 0.03% 0.03% 0.02%
OT2d(ˆ
PBeijing,pre ,ˆ
PBeijing,post(1,1)) −OTd(ˆ
PBeijing,pre ,ˆ
PBeijing,post(1,0)) (15)
is a lower bound for the sample analog of (7) for all d, and a lower bound for (7), the volume
of trade, for any sufficiently large tuning parameter d.
The most informative lower bound is then given as
ˆs∗
dit = ˆsdit(d∗) = max
d≥dOT2d(PBeij ing,post ,PBeij ing,pre)−OTd(PT ianjin,post ,PT ianjin,pre ),
where dis a data-dependent minimum value, and may be thought of as the smallest value of
dfor which equal displacement holds in sample and such that noise is filtered out from the
sample analog of the estimand (7).10
Remark that an additional virtue of the 2dover-correction in the difference-in-transports
estimator (motivated above by the in-sample triangle inequality of Theorem 1) is that the
subtracted transport cost itself having truncation only at level d, the difference likewise over-
corrects for the sampling variation, thus producing a lower bound estimate that requires
milder sampling variation smoothing.
We may likewise assess the plausibility of the equal displacement assumption by inspecting
trends when both Beijing and Tianjin are in a comparable regime. We inspect displacements
in Beijing and Tianjin for 2014 to 2015. In those years, Tianjin had a lottery and an auction
while Beijing had a lottery and a black market, with both of them seeing similar average
price jumps after the introduction of their own rationing scheme, suggesting they might be
comparable. This is confirmed empirically in Figure 9, where we see almost equal displace-
10The required value for the smoothing parameter dmay be characterized by the following bounding
argument,
OT2d(ˆ
PBeijing ,post(1,1),ˆ
PBeijing ,pre)−OTd(ˆ
PT ianjing,post (1,0),ˆ
PT ianjing,pr e)
=OT2d(ˆ
PBeijing ,post(1,1),ˆ
PBeijing ,pre)−OTd(ˆ
PBeijing ,post(1,0),ˆ
PBeijing ,pre)
≤OTd(ˆ
PBeijing ,post(1,1),ˆ
PBeijing ,post(1,0))
≤OT (PBeij ing,post (1,1),PBeijing ,post(1,0)),
where the first equality holds (approximately) for dfor which the equal displacement holds (approximately)
in sample, the first inequality holds by the triangle inequality of Theorem 1, and the last inequality holds for
dlarge enough, as given by Table 5.
29
ment for d≥2,000. While it is more natural to compare trends or displacements when
both cities are without rationing since this is the counterfactual we want to account for when
differencing transports, we do not observe the Beijing price distribution before 2010, making
such a comparison unavailable to us. Observe, from Figure 9, that non-trivial parallel trends
are captured and differenced out for dup to at least 15,000.
Figure 9: Post-Trends. The Beijing placebo is given by the optimal trans-
port OTd(PB eijing ,2014,PB eijing ,2015), and the Tianjin placebo is given by
the optimal transport OTd(PT ianj in,2014,PT ianj in,2015). The difference is
OTd(PBeij ing,2014 ,PBeij ing,2015 ) - OTd(PT ianj in,2014,PT ianj in,2015).
The above evidence suggests that noise is filtered out, up to our sensitivity criteria, for
d≥7,000, and that equal displacement holds in sample for d≥2,000; we thus pick d=
max{2,000,7,000}. For the final step, we choose the most informative admissible d≥d.
The solid yellow line in Figure 10 plots ˆsdit(d), the difference between the dotted green line
(Tianjin) and the dashed blue (Beijing) line, for a range of dvalues.
30
Figure 10: Choosing the most informative admissible d. The Beijing cost is given by the op-
timal transport OTd(PB eijing ,2010,PB eijing ,2011), and the Tianjin cost is given by
the optimal transport OTd(PT ianj in,2010,PT ianj in,2011). The difference uses twice
the bandwidth to compute Beijing’s cost: OT2d(PB eijing,2010,PB eijing,2011)−
OTd(PT ianj in,2010,PT ianjin,2011 ); the maximum difference occurs when d=
7,000, when the transport cost is roughly 11.9%.
The difference-in-transports estimate is seen to be most informative at d∗= 7,000, which
gives ˆsdit(d∗) = 11.9%. The cost attributable to the change in the composition of the distri-
bution, detailed and calculated in Section Dof the Appendix, is less than 0.9%, and we give
11% as a robust estimate of the volume of the black market.
6 Inferring transaction costs and prices
We infer the unobserved transaction costs and prices from a market equilibrium model that
combines our estimated volume of trade with empirical results from the literature. We de-
rive our model from a common version of the Coase Theorem: if the initial allocation leaves
gains from trade, as is expected when license plates are allocated by lottery, households will
transact until the gains from trade are exhausted. A market price will form that equates
the number of lottery winners willing to sell their license plates to the number of prospective
buyers. Transaction costs, which may be both pecuniary and non-pecuniary, are frictions
that reduce the volume of trade. We bound the transaction costs by treating these as a tax
on trade necessary to rationalize the estimated volume of black market trade.
31
One likely important transaction cost component is potential legal liabilities. For instance, a
new car must be registered in the name of the legal owner of the license plate, and therefore
the formal owner, not the driver, will be liable for traffic accidents. The fact that trade in
license plates is illegal may also give rise to non-pecuniary transaction costs, such as feelings
of culpability. Search costs are another component: buyers and sellers need to meet in the
market. The reported existence of online market places for license plates, however, suggests
that search costs are small.11
Potential enforcement, and perhaps moral hazard of the kind associated with leaving the
formal car ownership in someone else’s name, are likely the key transaction cost components.
While there may be further transaction cost components, we do not attempt to distinguish
between these. The transaction costs in our model are thus a simple quantitative measure of
the frictions in the black market.
We derive a demand and supply curve using the willingness-to-pay for license plates estimated
in Li, S. (2018). The estimates are derived by combining a BLP random coefficients discrete
choice model, using car registration like we do, with micro moments from household survey
data. These estimates are represented by a function v(n) with range RMB [0,280000].12
We take Li, S. (2018)’s estimated willingness-to-pay for a license plate v(n) to be known
without sampling variation. It is useful in the following to write v(n) in terms of a cumu-
lative distribution function F(v), which returns the share of households in the market with
a willingness-to-pay less than v, i.e. F(v) = 1
NPN
n=1 (v(n)≤v). Since v(n) is strictly
decreasing, we can recover Ffrom vas
v−1(v)
N= 1 −F(v).
We assume that only prospective car buyers enter the lottery.
Assumption 5 The lottery draws q winners with equal probability from the population of N
11Bhave and Budish (2018) notes that online market places for ticket resale decreases transaction costs and
increases the volume of resold tickets in the secondary market.
12Shanjun Li has generously shared his estimates with us. While Li, S. (2018)’s Figure 3 shows positive
willingness-to-pay for quantities far beyond the unrationed, pre-lottery market equilibrium, we require that
the marginal willingness-to-pay for a license plate at the unrationed market equilibrium is zero, i.e., v(N) = 0.
Our results are, however, not very sensitive to this modification.
32
prospective car buyers, whose willingness-to-pay for a license plate is distributed according to
the cumulative distribution function F.
It follows immediately from Assumption 5that both the qsellers’ (winners) and the N−q
buyers’ willingness-to-pay are distributed according to the cdf F. We set q= 260,000, the
quota in 2011, and the market size to N= 700,000, the total sales of new cars in 2010 before
the lottery. Assumption 5rules out speculators, which we define to be those who enter the
lottery with no intention of buying a car if they win a license plate. An influx of speculators
is consistent with the sharp decline in the lottery odds, which went from 10% in the first
auction in January 2011, to a monthly average of 4% for 2011, and dropped further to a
monthly average of 2% in 2012. We relax Assumption 5and allow for speculators in Section
Fof the Appendix.
We next make an assumption about how prices and transaction costs form.
Assumption 6 Each trade generates transaction costs of 2t, which are equally borne by the
buyer and the seller. The transaction price p equates demand to supply given the transaction
costs.
A buyer is willing to buy a license plate if v > p +t, and a seller is willing to sell a license
plate if v≤p−t. There are N−qrationed prospective buyers that demand license plates,
while there are qlottery winners that supply licenses. This gives the demand and supply
functions
D(p, t)=(N−q)(1 −F(p+t)),
S(p, t) = qF (p−t).
(16)
Figure 11 plots the demand and supply curves derived from Li, S. (2018)’s estimates. Suppose
first that there are no transaction costs. Then demand equals supply at price pnotc and the
quantity is qnotc. The volume of trade that occurs without transaction costs qnotc is less than
the quota qsince lottery winners with valuation in excess of the market clearing transaction
price prefer not to trade. Adding Assumptions 5and 6brings the upper bound of the volume
of trade down from 100% of the fraction to supper =qnotc
q=N−q
N= 62% of the quota. The
estimate of supper implies that though our lower bound estimate is ˆs= 11% of the quota, this
market can support no more than 11%
62% = 18% of trades in equilibrium.
33
Figure 11: Demand and supply derived from v(n). The figure is drawn to scale with quan-
tity in 10,000 and price in RMB 1,000.
The transaction costs tin Figure 11 are seen to drive a wedge between the supply and demand
that lowers the volume of illegal trade to sq. The total transaction costs, summed over buyers
and sellers, that rationalize the trade are hence 2tsq.
Theorem 2 Suppose that Fis known, continuous, and strictly increasing. Suppose that the
market clears at transaction price p and transaction cost t for a known volume of sq black
market trades. Then the transaction prices and costs are identified.
We can alternatively dispense with the homogeneous transaction cost assumption and allow
buyers and sellers to bear different transaction costs, i.e., tbuyer 6=tseller. Though we cannot
jointly identify the buyer- and seller-specific transaction costs along with the transaction
price, a pair ˜pbuyer =p+tbuyer and ˜pseller =p−tseller is identified. This interpretation
does not affect the identification of the total transaction costs, which are sq(˜pbuyer −˜pseller ).
Since these interpretations are observationally equivalent, we cannot infer which side of the
market bears the majority of the transaction costs. Hence, we maintain the homogeneous
transaction cost interpretation. The willingness-to-pay F, which we derived from Li, S.
(2018)’s estimates, satisfies the continuity and monotonicity restrictions.
The implied transaction costs and prices at ˆsare given in Table 6, along with 95% confidence
intervals.13 We show in Section Ethat ˆ
tis an upper bound for the transaction cost at ˆs, and
13We use subsampling to produce draws of ˆsfor inference on the share of trade in Table 6.
34
Table 6: Implied Transaction Prices, Costs, and Net Gains.
Estimates at ˆs= 11% 95% CI
ˆp105 [91,121]
ˆ
t100 [83,118]
total transaction costs 5.7 [3.8,7.3]
net gains from trade 1.3 [0.5,2.5]
Estimates of transaction prices and costs are in RMB 1,000. Gains
from trade and transaction costs are in RMB billon.
that the implied ˜pbuyer and ˜pseller at ˆsare upper and lower bound estimates, respectively. In
the model, trades take place between a selection of buyers with particularly high valuations
and sellers with particularly low valuations. This pattern is consistent with the literature on
the resale market of tickets where trades are observed (Bhave and Budish,2018;Leslie and
Sorensen,2013) and where search costs are relatively small.
6.1 Transaction costs and net gains from trade
We compare estimates of the total transaction costs and net gains from black market trade
to two benchmarks: one where non-transferability is strictly enforced and one where there
are no transaction costs. The latter can be interpreted as the case with no enforcement. We
consider transaction prices to be transfers between sellers and buyers that do not affect gross
gains from trade (the area between the demand and the supply curve up to bsq). Table 6
shows that the gross gains from trade is RMB 7.0 billion at ˆs= 11%, but transaction costs
sum up to RMB 5.7 billion. The net gains from trade are the gross gains from trade minus
the transaction costs. The lower bound estimate for the net gains from trade is RMB 1.3
billion, while the upper bound, for the case with no transaction costs at a black market share
qnotc = 62% and price pnotc = RMB 59,000, is RMB 18.8 billion.
6.2 Further bounds on transaction costs
Under the assumptions of the equilibrium model, the data are consistent with a volume of
trade that ranges from 11% to 62 % of the quota. This range implies transaction costs from
zero to RMB 5.7 billion and transaction prices from RMB 59,000 to RMB 105,000. We can
tighten the bounds with information on the transaction prices if we are willing to use the
news reports from Section 2as information about transaction prices. Table 7shows how
35
Table 7: Transaction Prices and Costs
p t s Net Gains from Trade Share Transaction Costs
59 0 62% 18.8 0%
57 21 49% 12.8 30%
64 42 37% 8.2 49%
73 59 27% 5.3 61%
105 100 11% 1.3 82%
The calculations are conditional on sranging from the estimated lower bound
ˆsto the upper bound supper that the market can support, and assume no
speculators. Prices and costs are in RMB 1,000. Gains from trade are in RMB
billion.
the implied transaction costs and the volume of trade vary conditional on a range of known
transaction prices. We can therefore tighten the bounds further if we are willing to use the
news reports from Section 2as informative about transaction prices.
Though we found a wide range of reported transaction prices, most are above RMB 70,000.
Li, S. (2018) estimates that the (counterfactual) market clearing auction price is about RMB
75,000 (see its Figure 3). It appears that rentals is the most common sales format. The most
frequently reported rental prices range is from RMB 500 to 1,000 per month. A twenty-year
NPV of RMB 6,000 in annual rental income is RMB 73,000, which seems conservative, yet
plausible.
Assumption 7 RMB 73,000 is a lower bound for the transaction price.
Table 7shows that Assumption 7tightens the upper bound for sthat is consistent with the
data from 62% to 27%, and shifts the lower bound for the share of transaction costs from
zero to 61%. We distinguish the realized gross gains from trade from the potential gains from
trade, which are those that could be realized in the black market if there were no transaction
costs. In our framework, a black market without transaction costs would theoretically be as
efficient as an auction. Since the potential gains from trade are RMB 18.8 billion, the share
of the realized net gains from trade lies in the range from 7% to 28%.
Table 7allows readers who prefer a more conservative lower bound on the transaction prices
to trade weaker assumptions against wider bounds for the quantities of interest.
Table 8summarizes the assumptions and the corresponding inferred bounds for the market
36
Table 8: Transaction prices and transaction costs in RMB 1,000.
Data Assumptions s p t Share transaction
costs
car sales 1,2,3[14,100]% R+R+[0,100]%
car sales 1,3,4[11,100]% R+R+[0,100]%
car sales, F 1,3,4,5,6[11,62]% [59,105] [0,100] [0,82]%
car sales, F,
news reports 1,3,4,5,6,7[11,27]% [73,105] [59,100] [61,82]%
1. No anticipation
2. No time trends
3. No general equilibrium
4. Equal displacement
5. Market participants
6. Market equilibrium
7. Lowest plausible transaction price
performance measures p,s,t, and the share of gains from trade lost to transaction costs.
We started with inferring a lower bound for the volume of trade using comprehensive car
sales data on millions of car sales under weak assumptions that are common in the program
evaluation literature. Under these assumptions, the car sales data, which are indirectly related
to our quantities of interest, gave an informative lower bound for sbut did not deliver useful
bounds on transaction prices and transaction costs. We then imposed increasingly strong
assumptions on black market transactions (Assumptions 5,6, and 7). These assumptions are
seen to be powerful in tightening the bounds, in particular Assumption 7. We note that this
assumption, which serves like a highly informative prior on a variable that is directly related
to our quantities of interest, is supported by only a handful of news reports.
7 Discussion
It is puzzling that the Beijing government seems to allow a substantial volume of black
market trade. The government can close the black market if they wish by allowing for
trade. One reason may be that the scale of trade makes the costs of strict enforcement
prohibitive. Another reason may be that some trade takes place between family members,
e.g., a daughter wins a license and lets her parents register a car in her name. Such trades
may be less politically expedient to enforce.
Lax enforcement may also strike a balance between allocative efficiency and equity concerns
37
in a society where only 8% of the population approves of auctions. Black market trade
may be viewed as a second-best solution that ameliorates the worst misallocations resulting
from the lottery allocations. However, more transparent mechanisms are available. Other
Chinese cities that later introduced rationing (Guangzhou, Tianjin, Hangzhou, Shenzhen,
and Shijiazhuang) chose a hybrid lottery/auction mechanism which offer a different balance
between efficiency and equity. Huang and Wen (2019) estimates that the hybrid mechanism
in Guangzhou, which allocates about 50% of the license plates by lottery and the other half
by auction, preserves 83% efficiency. Our results give that the Beijing black market preserves
at most 30% efficiency, down to as little as 10% efficiency.
However, a relatively large black market need not mean that enforcement is lenient. Indeed,
some of the evidence suggests that enforcement is strong. The high willingness-to-pay for
license plates in the market creates strong incentives for trade. If interest is in an equitable
allocation, then the size of the transaction costs (up to 82% of the gains from trade) suggests
that enforcement is in fact quite effective in precluding illegal trade of high private value.14
If interest is instead in an efficient black market, the size of the transaction costs suggests
relaxing enforcement.
Our hope is that the methodological developments detailed herein will find applications and
extensions beyond the specific subject matter of this article. In that respect, because it
accounts for trends and self-corrects for sampling variation –i.e., is less sensitive to tuning– we
suggest the difference-in-transports estimator as preferable to the before-and-after estimator,
when a match is available.
Certainly, the case at hand can be considered at a higher level of generality, where the key
observables are sample distributions before and after the time of treatment. While we aimed
to count the number of subjects with nonzero treatment effects, a different choice of ground
cost could be tailored to produce different estimator.
More sophisticated reduced form applications of optimal transport methods are being con-
sidered. For instance, optimal transport presents itself as a natural means with which to
14We use enforcement here to mean not only active enforcement but also any deterrent effect of codifying
the license plates as non-transferable.
38
compare distributions in bunching type estimation problems (Pukelis,2020), and to assess
covariate balance in program evaluation problems (see Gracie,2020,Rosenbaum,1989, and
Rosenbaum,2012). More general methodology could consider accommodating covariates,
instruments, and specialized inference. Relatedly, Gunsilius (2020) uses optimal transport
as the appropriate notion of distance to extend the synthetic control method to the case
in which experimental units correspond to entire distributions, and Arkhangelsky (2019)
leverages optimal transport theory for identification in specific clustered panel data setups.
8 Summary
Black markets may serve as an alternative allocation mechanism to auctions in places where
auctions are either practically or politically infeasible. We adopted a partial identification
approach in the spirit of Manski, C. (2003) to analyze the performance of the black market
for Beijing license plates. We estimated different bounds for the volume of trade, the gains
from trade, and the transaction costs corresponding to different and increasingly strong as-
sumptions. We found that the black market plausibly reallocates between 11% and 37% of
the rationed license plates and conservatively estimated the net gains from trade in the black
market to be between RMB 1.3 billion and RMB 5.3 billion. Yet, the black market realizes
between 7% and 28% of the potential gains from trade, 61% to 82% of which are lost to
transaction costs. The size of the transaction costs suggests that enforcement is strong and
that the black market realizes modest efficiency gains compared to hybrid lottery/auction
allocation mechanisms that are used in other large Chinese cities.
39
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43
Appendix: The Black Market for Beijing License Plates
For Online Publication
Øystein Daljord Guillaume Pouliot Mandy Hu Junji Xiao
May 4, 2021
A Proofs
A.1 Theorem 1
The proof follows immediately from the argument of Proposition 55 in Peyr´e, G. (2018), and
the analogous triangle inequality for the pseudo-distance C(d),
(|xi−xj|>2d)−(|xj−xk|> d)≤(|xi−xk|> d),
where xlis the lth entry of X.
A.2 Theorem 2
For any pair of transaction costs and prices, the valuation of the marginal seller is vseller =p−t
and the valuation of the marginal buyer is vbuyer =p+t. Together, we get
p=1
2(vseller +vbuyer )
t=1
2(vbuyer −vseller )
(1)
Equating demand to supply using (16) at the estimated lower bound of trades sq, the marginal
valuations are uniquely recovered by inverting F−1(s) = vseller and F−1(1 −sq
N−q) = vbuyer .
It follows immediately that pand tare uniquely determined given sq.
1
arXiv:2105.00517v1 [econ.GN] 2 May 2021
B Additional Figures
Figure 1: Placebo costs as a function of d. The curves represent the mean, standard devia-
tion, and the quantiles of 500 simulated values of OTd(˜
Ppre,npre ,˜
Ppre,npost ) for each
value of din Beijing. The vertical line at d= 25,000 is the smallest bandwidth
for which the placebo cost is less than 0.05%.
C Difference-in-differences regression
After aggregating different car models by city, time period (pre- and post-lottery), and price,
we run the following difference-in-differences specifications in logs for three control groups:
Tianjin, Shijiazhuang, and Tianjin and Shijiazhuang combined. The regression model is
pj,c,t =α0+α1Beijingj,c,t +α2postj,c,t +α3B eijingj,c,t ×postj,c,t +j,c,t (2)
where pj,c,t is log of the jth price observation in city cin month tand where Beijingj,c,t is
an indicator that is one for price observations from Beijing and zero for cities in the relevant
control group. The indicator postj,c,t is one in all months after the introduction of the lottery
in Beijing, and zero otherwise. Table 1shows that the estimated price jumps, ˆα3, are similar,
and around 21% across the specifications.
2
Table 1: Diff-in-diff regression results for different control groups
Tianjin Shijiazhuang Both
Beijing 0.266 0.233 0.255
(0.002) (0.002) (0.001)
post 0.035 0.027 0.032
(0.002) (0.003) (0.001)
Beijing ×post 0.208 0.215 0.210
(0.002) (0.003) (0.002)
Constant 11.493 11.527 11.505
(0.001) (0.002) (0.001)
Observations 1,411,305 1,200,418 1,637,150
R20.074 0.059 0.076
D Transport costs due to changes in the composition of sold
car prices in Beijing
From Yang et al. (2014) and the Beijing Municipal Commission of Transport cited therein, we
know the number of licenses emitted on a monthly basis in Beijing in 2010. We also observe,
as detailed in Section 3, the monthly sales for each model and price. Therefore, we can
estimate for each month of 2010 the proportion φf,t of first-time buyers and the proportion
φr,t returning buyers. Specifically, we estimate
φf,t =ρ·Lt
Piqi,t
, φr,t = 1 −φf,t,
where tindexes the time period, qi,t is the quantity of cars sold at price piin month t, and
Ltis the number of licenses given out in period t. The coefficient ρis the proportion of new
cars, as opposed to used cars, purchased with a license. We take this proportion to be ρ= 0.5
(Yang et al.,2014).
The monthly estimates are convenient here because the underlying population distributions
of preferences of first purchases, f, and returning purchases, r, are plausibly stable, as is ρ,
while we get identifying variability from (φf,t, φr,t) across t.
We may then solve for our best guesses ˆ
f(ρ), ˆr(ρ) for the distributions, as a function of ρ, by
3
solving
min
f,r∈ΛX
i,t
(φf,t fi+φr,tri−pi,t)2.
Given ˆ
f(ρ), ˆr(ρ), we can compute
ˆppre := θf,pre ˆ
f+θr,pre ˆr
as well as
ˆppost := θf ,post ˆ
f+θr,post ˆr,
where θf,pre and θr,pre are, respectively, the fraction of car purchases in 2010 for which a
license is used for the first time and the fraction for which the costumer used a returning
license, i.e., one that has been previously used to purchase another car. The scalars θf,post
and θr,post are defined analogously for 2011.
Consequently, we can compute before-and-after estimate for a chosen smoothing parameter
d,
OTd(ˆ
f(0.5),ˆr(0.5)).
Table 2: Estimated cost due to change in composition
d OTd(ˆ
f, ˆr)
4,000 2.4%
8,000 1.8%
10,000 1.1%
20,000 0.04%
The correction applies to the before-and-after estimator for tuning dand to the difference-in-
transports estimator for the value of the tuning coefficient on the transport cost in Beijing. For
instance, for the difference-in-transports with tuning d= 4,000 and smoothing 2d= 8,000
on the Beijing transport cost, the appropriate correction according to Table 2is 1.8%.
E Comparative statics in transaction costs and prices
We derive the comparative statics for the transaction costs and price estimators. We drop the
hat-notation for expositional convenience. Taking derivatives of the equilibrium conditions
in (1), we get
2∂p(s)
∂s =1
f(vseller )−q
N−q
1
f(vbuyer )
4
and
2∂t(s)
∂s =−q
N−q
1
f(vbuyer )−1
f(vseller ).
It is immediately clear that the implied transaction costs decrease with the volume of trade
s. The same is not necessarily true for the transaction price, which may increase or decrease,
depending on the shape of f. However, both ˜pbuyer =p+tand ˜pseller =p−tare monotonic
in s. These statics show that, conditional on a lower bound estimate s, t and ˜pbuyer =p+t
are upper bound estimates, and ˜pseller =p−tis a lower bound estimate.
F Extension to speculators
We now relax Assumption 5and allow for speculators. Define speculators as having zero
willingness-to-pay for a license plate if non-transferability is strictly enforced, i.e. a speculator
will never purchase a car if she wins a license. Speculators have two effects in the market:
they crowd out car sellers on the supply side (shift the supply curve down) and they increase
the number of car buyers on the demand side (shift the demand curve out). Suppose that
a share zof the license plates are allocated to speculators, and suppose that a speculator’s
reservation price is zero. Then the demand and supply are
D(p, t)=(N−q(1 −z))(1 −F(p+t)),
S(p, t) = zq +(s−z)
sqF (p−t).
(3)
Speculators have a supply curve that is flat from zero to zq, and increasing thereafter to q.
As zgoes to s, the supply curve becomes vertical at q. On the demand side, the demand
curve shifts out towards v(n) as zgoes to one and all prospective car buyers must turn to
the black market for license plates.
Table 3: Estimates of transaction prices and costs at ˆs= 11% and z= 0.11%. Prices and
costs are in RMB 1,000. Gains from trade are in RMB billion.
Estimate 95% CI
ˆp110 [98,125]
ˆ
t105 [98,125]
Total transaction costs 6.0 [4.0,7.9]
Net gains from trade 1.1 [0.4,2.1]
In Table 3, we report results assuming that all 11% illegal trades are by speculators. Since
the supply curve is now flat from zero to sq, all speculators sell at ˜pseller = 0. We see that
now ˆp=ˆ
t. At the same time, the demand curve shifts out to v(n). The outward shift in the
demand curve dominates the downward shift in the supply curve: both transaction prices
5
and costs increase, from 105 to 110 and from 100 to 105, respectively. The net gains from
trade go down by 15%, largely due to further taxation of buyers.
One limit case of interest is when speculators completely crowd out prospective car buyers
and there are no transaction costs, i.e., when z=qand t= 0. Then our model gives the same
allocation as Li (2018)’s counterfactual auction market, where the supply curve is vertical at
qand demand is v(n). It follows that Li (2018)’s analysis applies.1
G Allocation of Vehicle License Plates
G.1 Beijing
According to the Beijing Municipal People’s Government data, the number of motor vehicles
in Beijing has grown by 81.8% from 2.58 million in 2005 to 4.69 million in November 2010.
To control the number of motor vehicles, the city of Beijing has introduced a set of rules in
2010 (henceforth, ‘the rules’).
There have been several revisions to the rules after they were first introduced in 2010. The
most recent draft update to the rules was released in June 2020 (Beijing Municipal Commis-
sion of Transport,2020c). The updated rules are expected to be implemented in 2021. In the
rest of this section, we focus on the initial version of the rules that was released in December
24, 2010 by the Beijing Municipal Commission of Transport (2010a,b).
G.1.1 Application Procedure
To win a license plate, individuals need to participate in a lottery if they want to buy a new
or second-hand car, receive a car as a gift, or change a license plate that belongs to another
city in China to a Beijing license plate .
Individuals can participate in the lottery through the following procedure:
1. Start an application to obtain an application code.
2. After the application is approved, the application code will be validated.
3. The validated code can be used to participate in the lottery.
1except for the congestion and pollution externalities, which we have ignored.
6
Applications can be made either in person or online. There are 16 government offices that
accept applications made in person. There are six main steps to apply online:
1. Log in to the application website, create an account and choose the appropriate appli-
cation category.
2. Read and accept the rules.
3. Fill in personal information, such as name, identification number, email address, contact
information, etc.
4. Choose the license category.
5. Fill in more personal information, such as date of birth, vehicle type, etc.
6. Check and confirm the information.
G.1.2 Lottery Mechanism
The lottery for individuals is held on the 26 of each month with the first taking place on
January 2011. The lottery policy was officially announced and implemented on December 24,
2010, but had been publicly hinted at as early as December 13, 2010 (Yang et al.,2014). The
non-winning validated codes will remain valid in the same year, and they are automatically
allocated to the next lottery. These codes will remain valid until December 31 of the same
year in which they are generated. Applicants who win the lottery must complete the vehicle
registration process within six months. Otherwise, they will forfeit their rights to a license
plate.
Licenses for alternative fuel vehicles were not included in the initial set of rules. Licenses
for these vehicles were introduced in the revised set of rules that was effective in January 1,
2014 (Beijing Municipal Commission of Transport,2013). Based on this revision of the rules,
alternative fuel vehicles were referred to as the electric vehicles that are listed in the product
catalog. The quota for the total number of license plates for vehicles was 240,000 in 2011.
According to the bureau, the lottery is entirely random and was designed by data scientists
to ensure fairness.
For some statistics related to the license plates lottery in Beijing, please refer to Section
G.1.5. For details on the revisions to the winning rates in the lottery, please refer to Section
G.1.6.
7
G.1.3 License Plates for Individuals
Individuals who do not have a vehicle can participate in the lottery if they fulfill one of the
following requirements:
1. They hold a Beijing domicile.
2. They belong to the army or police force in Beijing.
3. They are citizens from Hong Kong, Macau or Taiwan, or they are from other parts of
the world who hold a valid Beijing residence permit and have resided in Beijing for one
year.
4. They are a Chinese citizen who do not have a Beijing domicile but have a Beijing work
and stay permit.
5. They are allowed to live in Beijing and have paid social security and income tax for five
consecutive years.
G.1.4 Transfer of License Plates
The license plates are non-transferable. If a vehicle is transferred to another person or the
vehicle is reported as a scrapped vehicle, the license plate still belongs to the owner. On the
other hand, if the vehicle is sent to another person as a gift, then the original owner of the
vehicle cannot keep the license plate. For property transfers due to divorce or inheritance,
judicial order will be followed instead of the rules.
G.1.5 Application Statistics
Table 4shows the details of the first vehicle license plate lottery for individuals (Beijing Mu-
nicipal Commission of Transport,2011a,b,2020a,b).
Table 4: Summary Statistics for the Beijing License Lottery in January 2011
Number of Valid Application Codes Quota Winning Rate
187,240 17,600 9.39%
8
G.1.6 Winning Rate
In light of the surge in the number of applications for vehicle license plates, the Beijing Mu-
nicipal Commission of Transport had implemented several revisions to the lottery mechanism
to increase the winning rate for individuals who were unable to win a license plate after many
lotteries. In this section, we focus on the Municipal’s first attempt in modifying the winning
rate (Beijing Municipal Commission of Transport,2013).
In the first attempt to modify the winning rate, the winning rate is higher if an individual
has participated in the lottery for 24 times without winning. Depending on the license type,
the winning rate is going to be a multiple of the base winning rate of a certain lottery. This
is illustrated by Table 5.
Table 5: Revised Winning Rate in 2013
Number of Lottery Losses Winning Rate
with C5 License∗Other License
1 – 24 2r r
25 – 36 3r2r
37 – 48 4r3r
49 – 60 5r4r
61 – 72 6r5r
.
.
..
.
..
.
.
rrefers to the base winning rate in a particular month
∗C5 licenses are designated to small passenger cars for people with disabilities
The 2017 revision (Beijing Municipal Commission of Transport,2017) to the winning rate
uses a similar method as in Beijing Municipal Commission of Transport (2013) with a few
differences. Namely, the winning rate will be modified once an individual loses the lottery
six times instead of 24. Subsequently, the winning rate will be modified after 12 attempts as
in Table 5.
Any contracts or prepayments on vehicle sales that have been reported to the bureau before
the implementation of the rules were exempt from the requirements of the rules.
G.2 Tianjin
The rationing rules were introduced in Tianjin in order to control the number of vehicles,
alleviate the problem of traffic congestion and to improve the air quality of the city.
The measures to control vehicle purchase and restrict the traffic in Tianjin were first an-
nounced in a press conference by the Tianjin Municipal People’s Government at 7 p.m. of
9
December 15, 2013 (The State Council of the People’s Republic of China,2013). The con-
trols and restrictions were effective five hours later on December 16, 2013 at midnight. The
Tianjin Municipal People’s Government suspended any new vehicle registration and vehicle
transfer in Tianjin between December 16, 2013 and January 15, 2014 in order to ensure a
smooth transition and preparation for the new rules.
After the rules were announced in the press conference, some 4S stores (stores that sell
vehicles in China) had over 70 consumers queued up to purchase vehicles, and the vehicles
of some major brands were all sold out before midnight of December 16, 2013 according to a
news report by the National Business Daily (2013). According the same news report, Toyota
sold around 2,000 vehicles in five hours following the press conference, which equals to 20%
of the annual vehicle sales in Tianjin.
The first draft of the rules (henceforth, ‘the rules’) was effective for one year. In January 1,
2015, the control rules were updated and remained effective for the next five years. The most
recent update to the rules was released in 2019 and implemented on January 1, 2020. This
set of rules would remain effective for the next five years. In the rest of this section, we focus
on the initial draft of rules by the Tianjin Municipal Transportation Commission (2013) that
was released in December 15, 2013.
G.2.1 Application Procedure
In general, the application procedure is as follows:
1. Start an application to obtain an application code. The allocation mechanism has to
be stated in the application.
2. After the application is approved, the application code will be validated.
3. The validated code can be used to participate in an auction or a lottery.
The applications can be made in person or online. There are 14 government offices that
accept the applications that are made in person. There are five main steps in making the
online application for individuals:
1. Log in to the application website, create an account and choose the appropriate appli-
cation category.
2. Read and accept the rules.
10
3. Fill in the personal information, such as name, identification number, email address,
etc., and choose to participate in the lottery or the auction.
4. Fill in the contact information such as the mobile phone number. An SMS will be sent
to the phone number to confirm that the phone number is correct.
5. The application code and application form will be generated by the system.
G.2.2 Allocation Mechanism
Vehicle license plates are allocated every 12-month cycle. The number of quotas available in
each cycle is 100,000. The license plates are allocated monthly. Any unused quota in a cycle
cannot be reallocated to the next cycle. The quotas are allocated according to the ratio of
1:5:4 for different vehicle types as shown in Table 6.
Table 6: Allocation of License Plates in Tianjin per cycle
Vehicle type Allocation mechanism Number
Energy-efficient vehicles Lottery 10,000
Normal vehicles Lottery 50,000
Normal vehicles Auction 40,000
Among the available license plates in each cycle, 88% of the them are allocated to individuals.
The rest are allocated to organizations.
Each code can be used in either an auction or a lottery. All applications made after the 9th
of the month will be allocated to the next month. Applicants who wish to withdraw from the
application should do so before the 20th of the month. Once an individual wins the auction
or lottery, they need to complete the application and use the winning code within six months.
Otherwise, the code will be deactivated.
G.2.3 License Plates for Individuals
Individuals can only have one validated code and can apply for an application code if they
fulfill the following requirements:
1. They reside in Tianjin and fulfill one of the requirements:
(a) They hold a Tianjin domicile.
(b) They belong to the army or police force in Tianjin.
11
(c) They hold a valid personal identification from Hong Kong, Macau or Taiwan, or
other parts of the world, and have lived in Tianjin for at least 9 months each year
for the past two years.
(d) They do not hold a Tianjin domicile but they have a Tianjin residence permit and
have paid for social security for at least 24 months.
2. They hold a valid driving license.
3. They do not have a vehicle registered under their name or the vehicle or they do not
hold a license plate that has been cancelled.
4. They do not have a valid vehicle license plate or do not have a license plate that is
renewable.
5. They do not have an overdue winning code or did not give up any wining code in the
past two years via the car lottery.
G.2.4 Lottery
The exact number of license plates available in the lottery is announced by the 9th of each
month. The lottery is held on the 26th of each month or the next working day if the 26th
is not a working day. Like the Beijing lottery, the lotteries are organized every month for
individuals. For some statistics for the license plates lottery in Tianjin, please refer to Section
G.2.6.
G.2.5 Auction
Auctions are organized via the internet and details for each auction is released by the 18th
of each month.
To participate in an auction, applicants need to pay a security deposit of RMB 2,000 per
application code. The reservation price for each license plate is RMB 10,000. There is
no maximum bid in the auction. The auctions for individuals and organizations are held
separately.
In each auction, the highest bid wins. If two bidders submit the same bid, the bidder who
submits the bid earlier wins. Results for the auctions will be announced within five working
days after the auction day. If a winning bidder does not pay for the bid, the security deposit
will be forfeited.
12
Any unallocated quota for vehicle licenses in a certain month will be reallocated to the next
month.
Some statistics for the license plates auction in Tianjin are given in Section G.2.6.
G.2.6 Application Statistics
Table 7summarizes the average price, lowest price and number of license plates available in
the auctions for vehicle license plates in the February 2014 (the first auction) and June 2020
(the latest month). Note that the highest bid price for the auctions is not provided by the
auction house. The 2014 auction data is obtained from the Tianjin Municipal Transporta-
tion Commission (2016), while the 2020 auction data is obtained from the Tianjin Small Ve-
hicle License Plates Auction System (2016).
Table 7: Vehicle License Plate Auctions in Tianjin in February 2014 and June 2020
Average Price
(CNY)
Lowest Price
(CNY)
Number of License
Plates
February 2014 16,340 10,000 3,203
June 2020 22,022 20,500 3,059
As shown in Table 7, the average price for individuals in June 2020 is higher than the
corresponding average price in February 2014. In addition, the reservation price for the
individual license plates are no longer hit in June 2020.
On the other hand, the lottery for vehicle license plates are also competitive, but they are
a little less competitive than in Beijing. In the first Tianjin vehicle lottery in February 26,
2014, there was a total of 151,139 validated application codes but only 4,450 normal vehicles
and 910 renewable vehicles (Tianjin Municipal Transportation Commission,2014). Table 8
summarizes the details of the lottery in June 2020 (Tianjin Municipal Transportation Com-
mission,2020).
Table 8: Vehicle License Plate Lottery in Tianjin in June 2020
Category Number of valid License plates Winning
application codes allocated rate
Individuals
Renewable vehicles 3,406 3,406 100.00%
Normal vehicles 809,957 8,747 1.08%
Organizations
Renewable vehicles 21 21 100.00%
Normal vehicles 9,996 500 5.00%
13
Although no official data are available by the Tianjin Municipal Transportation Commission
on the breakdown between individuals and organizations in the 2014 lottery data for Tianjin,
the first lottery for license plates in Tianjin is much less competitive than the most recent
one; the number of valid application codes in June 2020 is 544.78% more than the number
of valid application codes in February 2014.
H Robustness to Marginal Data
The Beijing lottery began in January 2011. It is possible that anticipation occurred as early as
mid-December 2010 (Yang et al.,2014). This motivates us to drop the month of December
from the control year 2010 and repeat the analysis. Lottery winners do not immediately
purchase a vehicle upon obtaining their license, and distributional differences may arise due
to the sales only “getting started” with rationed license when there is the quota-regulated
flow of them but no stock. For that reason, we also consider the analysis when, in addition
to omitting December 2010, we omit January and February 2011.
Our results are largely unchanged, suggesting that anticipation of or response to the restric-
tions does not drive our results. To summarize, when omitting only December 2010, the
difference-in-transports estimate smoothing both differenced transport costs by dis 13.5%,
and the conservative difference-in-transports estimate smoothing the Beijing transport cost
by 2dis 10.9%. When omitting December 2010, January 2011 and February 2011, the
difference-in-transports estimate smoothing both differenced transport costs by dis 14.2%,
and the difference-in-transports estimate smoothing the Beijing transport cost by 2dis 11.6%.
Section H.1 replicates the figures and tables of the main text when omitting December 2010.
Section H.2 replicates the figures and tables when omitting December 2010, January 2011
and February 2011. Section H.3 collects the estimates from the main and the two alternative
specifications.
14
H.1 Excluding December 2010
Figure 2: Monthly average swtprice (RMB 1,000) for Beijing, Shijiazhuang, and Tianjin.
This figure is similar to Figure 1.
Figure 3: Distribution of Beijing car prices in 2010 (pre-lottery) and new car prices in 2011
(post-lottery). This figure is similar to Figure 3.
15
Figure 4: Smoothed empirical distributions of Beijing car sales in 2010 (pre-lottery) and
2011 (post-lottery). This figure is similar to Figure 4.
Figure 5: Exact empirical distributions of Beijing car sales in 2010 (pre-lottery) and 2011
(post-lottery). This figure is similar to Figure 5.
16
Figure 6: Smoothed empirical distributions of Beijing car sales in 2010 (pre-lottery) and
2011 (post-lottery) at candidate values of d. This figure is similar to Figure 6.
17
Figure 7: Real and placebo costs as a function of d. This figure is similar to Figure 7.
Figure 8: Placebo costs as a function of d. This figure is similar to Figure 1in Appendix.
18
Figure 9: Smoothed empirical distributions of Tianjin car sales in 2010 (pre-lottery) and
2011 (post-lottery). This figure is similar to Figure 8.
Figure 10: Post-Trends. This figure is similar to Figure 9.
19
Figure 11: Choosing the most informative admissible d. This figure is similar to Figure 10.
Table 9: Diff-in-diff regression results for different control
groups. This table is similar to Table 1.
Tianjin Shijiazhuang Both
Beijing 0.273 0.235 0.260
(0.002) (0.002) (0.001)
post 0.026 0.014 0.022
(0.002) (0.003) (0.002)
Beijing ×post 0.201 0.213 0.205
(0.002) (0.003) (0.002)
Constant 11.502 11.540 11.515
(0.001) (0.002) (0.001)
Observations 1,275,829 1,078,516 1,486,251
R20.078 0.061 0.079
20
H.2 Excluding December 2010, January 2011, and February 2011
Figure 12: Monthly average swtprice (RMB 1,000) for Beijing, Shijiazhuang, and Tianjin.
This figure is similar to Figure 1.
Figure 13: Distribution of Beijing car prices in 2010 (pre-lottery) and new car prices in
2011 (post-lottery). This figure is similar to Figure 3.
21
Figure 14: Smoothed empirical distributions of Beijing car sales in 2010 (pre-lottery) and
2011 (post-lottery). This figure is similar to Figure 4.
Figure 15: Exact empirical distributions of Beijing car sales in 2010 (pre-lottery) and 2011
(post-lottery). This figure is similar to Figure 5.
22
Figure 16: Smoothed empirical distributions of Beijing car sales in 2010 (pre-lottery) and
2011 (post-lottery) at candidate values of d. This figure is similar to Figure 6.
23
Figure 17: Real and placebo costs as a function of d. This figure is similar to Figure 7.
Figure 18: Placebo costs as a function of d. This figure is similar to Figure 1in Appendix.
24
Figure 19: Smoothed empirical distributions of Tianjin car sales in 2010 (pre-lottery) and
2011 (post-lottery). This figure is similar to Figure 8.
Figure 20: Post-Trends. This figure is similar to Figure 9.
25
Figure 21: Choosing the most informative admissible d. This figure is similar to Figure 10.
Table 10: Diff-in-diff regression results for different con-
trol groups. This table is similar to Table 1.
Tianjin Shijiazhuang Both
Beijing 0.273 0.235 0.260
(0.002) (0.002) (0.001)
post 0.025 0.023 0.024
(0.002) (0.003) (0.002)
Beijing ×post 0.212 0.214 0.213
(0.002) (0.003) (0.002)
Constant 11.502 11.540 11.515
(0.001) (0.002) (0.001)
Observations 1,220,450 1,033,006 1,412,183
R20.079 0.060 0.080
26
H.3 Summary of Results
Table 11: Results
Original Section H.1 Section H.2
Diff-in-diff (t-stat) 21% (103.992) 20.5% (97.853) 21.3% (98.590)
Before-and-after (d) 15% (10,000) 14% (10,000) 14.6% (10,000)
Diff-in-transports (d) 14.3% (7,000) 13.5% (7,000) 14.2% (7,000)
Conservative diff-in-transports (d) 11.9% (7,000) 10.9% (7,000) 11.6% (7,000)
The diff-in-diff estimate uses both Tianjin and Shijiazhuang as the control group. The other
estimates are reported with their corresponding bandwidths. The differences-in-transport
reports the transport cost when using dfor both Beijing and Tianjin. The conservative
analogue uses 2das the bandwidth for Beijing calculations. Section H.1 excludes December
2010 from the analysis. Section H.2 excludes December 2010, January 2011, and February
2011 from the analysis.
27
I Robustness to the use of Manufacturer suggested retail price
as a proxy for transaction prices
Figure 22: Optimal transport cost using MSRPs and transaction prices, and their difference,
for matched data, as well as optimal transport cost using MSRPs for full data.
Focusing our attention on the sales of cars for which we have both the MSRP and transaction
price, we compute the transport cost between 2010 and 2011 for bandwidths between 0 and
40,000. As the above figure shows, the transport cost using the transaction prices is very
similar to the transport cost using the corresponding MSRPs. The absolute difference of
these transport costs does not exceed 5% beyond a bandwidth of 3000, and for bandwidths
between 4,000 and 33,000, the absolute difference is less than 3%. Thus, on the subset of data
for which we have both MSRPs and transaction prices, our results are robust to the choice
of MSRP and transaction price. In particular, the before-and-after estimator for d= 20,000
is 27.5% when using transaction prices and 27.4% when using MSRPs.
Note that because the effective sampling scheme of the proprietary data is different from
that of the non-rationed purchases or lottery –corrupted or not– data, the transport cost is
different when restricted to the car models which both are in the proprietary data and could
be matched to entries in our main data set.
28
Indeed, this is readily explained. The proprietary data with information on transaction prices
comes from a survey of a small number of people that are supposedly selected at random:
1799 people in Beijing during 2010 and 1919 people in Beijing during 2011. Since very few
individuals are sampled, there are fewer distinct models for which we have transaction price
data: 401 car models in 2010 and 233 car models in 2011. From this small sample of models,
we successfully match a yet slightly smaller subsample: 374 car models in 2010 and 229 car
models in 2011.
We may assess the impact of this sampling in practice. Assuming that successful matches
are random selections, we can simulate this process and create a subsample of MSRPs. If
we randomly sample subsets of the size of the proprietary data and then sample uniformly
at random a further subset of matched models of size equal to that of the matched sample,
we obtain transport costs to the full population distribution of the same order of magnitude
as that of the matched sample in MSRP. The sample selection explains the difference in
MSRP transport cost for this matched subsample, and the approximate equivalence for this
matched subsample of the MSRP and transaction price transport costs serves as very desirable
robustness verification.
29
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31