Multidimensional Matching: Hukou status in the

Marriage Market

Danyan Zha

November 30, 2018

Abstract

Following P. Chiappori et al. (2012), I develop a two-dimensional matching model

on the marriage market in China, where individuals differ in a continuous attribute

(e.g. socioeconomic status) and a discrete attribute (hukou status). Surplus gain

from marriage depends on the discrete characteristic with the rationale that an urban

hukou is much more valuable than a rural hukou, it’s harder for a husband to move

to the wife’s place and a limited quota of hukou change is available upon marriage.

I first derive some general properties of the stable matching, then characterize the

closed-form solutions by specifying a quadratic surplus function. Using China 1990 1%

sample census and using educational attainment as a proxy for socioeconomic status,

model predictions are validated. There are fewer across-hukou type marriages and even

fewer rural husband-urban wife match. Matching is assortative on education within

each marriage type. Urban husbands with rural wives have on average fewer schooling

years than those with urban wives. Moreover, rural wives with rural husbands have on

average more schooling years than those with urban husbands.

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1 Introduction

Marriage market has attracted economists’ attention since Becker’s seminal work in mar-

riage since 1970s.(Becker, 1973). Theoretical work on matching models under Transferrable

Utility model contributes furthermore to economists’ tool set in marriage market research.

However, most attention is put in one dimensional matching model because of its tractabil-

ity, namely, positive assortativeness with supermodular surplus function.

China started to implement the Hukou system in 1958, which is a normal household

registration system that records where each households lives. However, a special feature is

that households are classified as agricultural or non-agricultural type that specify different

opportunities, including work and education. For example, children with agriculture hukou

can’t go to public primary or junior high schools in the urban township of the same county.

Urban residents also enjoy additional food stamps in 1980s. Moreover, before 1984, there is

also a strict migration ban that rural residents are not allowed to live or let alone work in

urban areas. People can change their hukou status in theory, but the quota is very limited

before 1984. Since 1984, there are heterogeneous hukou reforms across different districts that

gradually allow rural residents working outside their own village. People first start to move

to the township in the same county and as time goes by, some also move to the city in the

same province or to a different province.

In this paper, following (P. Chiappori et al., 2012), I develop a two dimensional matching

model on marriage market in China. Each agent has a continuous trait (namely, Socioeco-

nomic status) and a discrete trait (hukou status) that can be agriculture or non-agriculture.

In the model, two possibilities of surplus functions are analyzed. The first is assuming high

surplus as long as one partner has urban hukou. This has the rationale that hukou status

can be changed to the spouse’s status upon marriage. However, this is not compatible with

the social norm in China that it’s pretty hard for a husband to move to a wife’s place after

mariage. Hence, an alternative surplus function is analyzed, which allows only high surplus

for a couple with urban husband. Moreover, a limited quota of hukou change is introduced

to mimic practice. Predictions are derived from the model and I use China 1990 1% sample

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census data to check how predictions work.

A brief literature review is given in section 2, and models are demonstrated in section

3, while section 4 shows the empirical results from data. Section 5 concludes and states

possible future work.

2 Literature Review

This paper is closely related to (P. Chiappori et al., 2012), where they build a two-dimensional

model with smoking as the discrete characteristic. The theoretical analysis of matching on

Transferrable Utility dates back to (Shapley & Shubik, 1971) and (Becker, 1973). In one-

dimensional matching model, with supermodularity of surplus function, positive assortative-

ness is achieved, making it very tractable. There is also a close connection between matching

models, transportation problems and hedonic models. However, multidimensional matching

model is much more complicated.

From a more applied perspective, there is a large literature on how the family registra-

tion system in China hukou affects urban/rural development and migrant workers’ welfare.

However, how it affects marriage market in equilibrium is less researched. There is even less

research considering the different surplus generated by different couple types. Empirically,

research has found marriage patterns respond to surplus change. (Nie & Xing, 2011) found

that urban husband-rural wife couple share jumps up after 1998 when it was announced that

new-born babies could freely choose their hukou status from either father or mother, while

it was followed after mother automatically before.

More broadly, studying hukou system in China also allows us to link labor market with

marriage market, and studies how migration decision is made. (Dupuy et al., 2014) builds a

marriage matching model of two locations, each with a distinct labor market and marriage

market, to analyse people’s decisions of migration to work and migration to wed. Empiri-

cally, there are also many papers that document how marriage market influences individuals’

migration choice. (Edlund, 2005) argues that the attractiveness of high-income men in urban

areas contributes to the empirical stylized fact that young women outnumber young men in

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urban area though a high skilled labor market in urban area may predict the opposite with

the assumption that there are more skilled men than women. (Weiss et al., 2013) shows that

young women migrate from mainland China to Hong Kong for better marriage prospect

after the lifting of migration ban which causes more women emigration from Hong Kong.

Using Danish data, (Gautier et al., 2010) argues that cities serve as a role of providing dense

marriage market, which influences singles and couples’ location choices.

3 The Model

a The basic framework

a.1 Populations and surplus

This part is borrowed heavily from (P. Chiappori et al., 2012). There are two populations:

men’s and women’s. They are of equal size and are normalized to one. Both men and

women differ in two dimensions. First, they are characterized by a continuous attribute:

their socioeconomic status which is a summary proxy for income, education, prestige and

so on. For simplicity, I assume that this index is uniformly distributed over the interval

[0, 1], although more general settings could be considered. Second, agents differ in terms of

hukou status, a discrete dimension. A woman (man) is thus formally characterized by a pair

(x,X) ( (y, Y )) where x, y ∈ [0, 1] is the individual’s continuous socioeconomic index, and

X, Y ∈ {A,N} defines the individual’s hukou status. For the time being, we assume that

hukou is independent of socioeconomic status.

I then consider the workhorse model used on marriage market: a frictionless matching

model with transferable utility (TU) as in (Becker, 1973) and Shapley-Shubik(1971). The

key assumption in this model is that for any couple, there exists a marital surplus that the

couple can decide how to divide upon marriage. In this model, the marital surplus depends

on both the socioeconomic status and hukou status of each partner. Moreover, following the

hukou change rule, let’s assume that as long as one of the partner has non-agriculture hukou,

the other could change his/her hukou to non-agriculture. Thus, the surplus function Σ has

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the form:

Σ((x,X), (y, Y )) =

f(x, y), if either of the spouse has a non-agriculture hukou

λf(x, y), if X = Y = A

where x and y denote the wife’s and husband’s respective socioeconomic indices, and where

the function f is strictly increasing and supermodular, and satisfies f(0, 0) = 0. Here, λ < 1

represents the decrease in surplus generated by a couple both with agriculture hukou and the

assumption that agriculture hukou enjoys fewer benefits than non agriculture hukou in the

analysis period of time in China. Note that a mixed couple (agriculture and non-agriculture)

is the same as that of a couple both with non-agriculture hukou, but strictly larger than an

agriculture hukou pair.

a.2 Stable matching

A matching is defined as a measure µ on the set ([0, 1] × {A,N})2 and four functions

(uA(x), uN(x), uA(y), uN(y) that captures the utilities of individuals of different types. The

only constraint on the measure µ is that its marginal should be equal to the initial male

and female distributions. A matching is stable if it satisfies individual rationality and no

blocking pair condition. Individual rationality requires that no matched individual would

be better off remaining single. No blocking pair condition requires that no two individuals

would prefer being matched together to their current situation. Hence stability would require

for any (x,X), (y, Y ) we have that

uX(x) + vY (y) ≥

λf(x, y), if X = Y = A

f(x, y), otherwise

where the equality is satisfied on the support of the matching measure µ, i.e. where we

observe a positive probability of matching for that couple type.

Existence of a stable matching is guaranteed by the property of a TU model. Stability

in a TU framework is equivalent to the maximization of aggregate surplus over all possible

assignments; therefore the problem boils down to the existence of a solution to a simple

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maximization problem, for which one can readily check that the standard conditions are

satisfied.

As for pureness, as defined in (P. Chiappori et al., 2012), a matching is pure if almost

all women with same attributes (x,X) are matched with probability one to exactly one type

of agent (y, Y ) = ρ(x,X) and the same applies to men. In one-dimensional case where we

have positive assortative matching with supermodularity assumption, the stable matching is

one-to-one and hence is pure. However, in the current model, we would need the ”twisted”

condition((P.-A. Chiappori et al., 2010)), i.e., for almost all (x0, X), the partial derivative of

the surplus Σ((x,X), (y, Y )) with respect to x0 at two different points (x0, X), (y1, Y1) and

(x0, X), (y2, Y2) are equal to each other if and only if ((y1, Y1) = (y2, Y2). This property is

hard to be true in the current setting: if a woman with index x0 has an agriculture hukou

and marries with an urban man with index y1, the partial of the surplus with respect to x

is ∂f(x0,y1)∂x

. If she is mated with a rural man with index y2, the partial is λ∂f(x0,y2)∂x

. We may

still have:∂f(x0, y1)

∂x= λ

∂f(x0, y2)

∂x

with y2 > y1 since λ < 1. Therefore, the stable matching may not be pure in the current

setting.

a.3 The Symmetric Case

Let me start with a particular case when men and women are symmetric. That is to say:

(1) the surplus function is symmetric f(x, y) = f(y, x), and (2) characteristic distributions

are exactly the same for men and women. Let αM , αF indicate the share of agents with

agriculture hukou among men and women in the society. In the symmetric case, we have

αM = αF = α. Unlike P. Chiappori et al. (2012), in which the stable matching is the same

regardless of α as long as αM = αW , in my setting, the stable matching would depend on

whether α ≤ 0.5 or α > 0.5.

When α ≤ 0.5, the stable matching can easily be solved and characterized:

Proposition 1. Under the symmetry assumptions (1) and (2) above, and when α ≤ 0.5,

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there exists a unique stable matching, which is completely assortative, namely:

• In each couple, agents have the same SES.

• All agents with non-agriculture hukou are married with agents with same SES and

agriculture hukou. The remained agents with non-agriculture hukou are married to

each other with same SES. Agriculture - agriculture couple does not exist. In particular,

when α = 0.5, we only have cross type marriage, i.e., urban man matched with rural

woman and rural man matched with urban woman.

Apparently, if α ≤ 0.5, the stable matching is not pure. In the case α < 0.5, an agent

with non-agirculture hukou is matched with an agent with non-agriculture hukou and same

SES with probability 1−2α1−α , and is matched with an agent with agriculture hukou and same

SES with probability α1−α .

Intuitively, when α ≤ 0.5, there exist more agents with non-agriculture hukou, therefore,

agriculture-agriculture type matching could be fully avoided to maximize surplus. Figure 1

gives an example of the matching pattern. Area with same color indicates a match in the

equilibrium.

Lastly, let’s compute the corresponding utilities. Without loss of generality, let’s assume

f(0, 0) = 0. In the case when α < 0.5, assume Ms. x (with non-agriculture hukou) marries

Mr. y (also with non-agriculture hukou) at the stable match; note that x = y by the previous

proposition. Let uN(x) (resp. vN(y)) denote her (his) utility. Then, by stability:

uN(x) = maxsf(x, s)− vN(s)

where the maximum is reached for s = y. By the envelope theorem:

u′N(x) =∂

∂xf(x, y)

where the right-hand side derivative is taken at the point (x, x). It follows that

uN(x) =

∫ x

0

∂

∂xf(s, s)ds+K

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where K is a constant. Symmetrically,

vN(y) =

∫ x

0

∂

∂yf(s, s)ds+K ′

and by individual rationality,

uN(0) = K ≥ 0

vN(0) = K ′ ≥ 0

uN(0) + vN(0) = f(0, 0) = 0

Hence K = K ′ = 0, uN(x) = vN(x) = 12f(x, x)

Moreover, we know that

uA(x) + vN(x) = f(x, x)

uN(x) + vA(x) = f(x, x)

Hence uN(x) = vN(x) = 12f(x, x)

When α > 0.5, there exist fewer agents with non-agriculture hukou, therefore, it’s impossible

for every couple to enjoy the benefit of urban hukou. In this case, the stable matching would

depend on the magnitude of λ and the surplus function. However, there exists general

prediction as stated in the following proposition:

Proposition 2. When α > 0.5, under the symmetry assumptions (1) and (2) above, there

exists a unique stable matching, and

• All agents with non-agriculture hukou are married with the agents with agriculture

hukou

• There exists a threshold x, all women with x > x and agriculture hukou are married

with the top men with non-agriculture hukou. The symmetric argument applies to men.

• The smaller λ is, the lower the threshold x is.

• In the extreme cases: when λ = 0, x = 1− 1−αα

; when λ = 1, x = 1.

Figure 2 gives an example of the matching pattern when λ is in between of the two

extreme cases.

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b The general case: preliminary results

In a general case without the symmetry assumption, let us keep the assumption that hukou

status and SES characteristics are independent. From previous analysis, we know that if

αM ≤ 0.5, αW < 0.5, every couple would enjoy the benefit of urban hukou. The stable

matching is completely assortative. For the discussion below, assume αM > 0.5, αW >

0.5 which (1) introduces more interesting stable matching and (2) is more consistent with

empirical facts. Also denote µ as the matching function, pN(x) be the probability that

an urban woman with SES x marries a rural man (then 1 − pN(x) is the probability that

she marries an urban man), similarly, denote pA(x), qN(y) and qA(y) as the probability of

marrying a rural partner for a rural woman, an urban man and a rural man. For the stable

matching, there are some properties which are true regardless of the distributions and the

specific function format of the surplus as long as it’s supermodular.

Result One: Assortative matching within each marriage type In the stable matching,

given the hukou types of men and women, matching is positive assortative on SES.

Proposition 3. Consider two matched couples, (x,X), (y, Y ) and (x′, X), (y′, Y ) with iden-

tical hukou status. For almost all couples, x ≥ x′ if and only if y ≥ y′

Proof. This is implied by supermodularity of surplus function f(x, y) directly.

Result Two: No Randomization for Agents with non-agriculture hukou Individ-

uals with non-agriculture hukou never marry with agents also with non-agriculture hukou if

hukou status is independent of SES statuses.

Proposition 4. Assume hukou status is independent of SES, then ∀x, y, pN(x) = 1, qN(y) =

1

Proof. If there is an open set of x that pN(x) < 1. Taking these individuals, they are married

with non-agriculture agents with similar SES (due to positive assortativeness within each

marriage type) with some positive probability, correspondingly, there will be some other

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agriculture individuals who are married with each other with similar SES with some positive

probability. We can rearrange these two couples and enjoy a higher social surplus because

1 + 1 > 1 + λ.

Different from (P. Chiappori et al., 2012), properties of stable matching outcomes under

situation with different agriculture population shares among males and females are pretty

similar to the symmetric case.

c Model Variation

To provide practical implications, I need to adopt two features of the context into the model.

They are:

• An urban husband – rural wife couple generates more surplus than a rural husband –

urban wife couple.

• There is a limited quota of hukou change from agriculture to non-agriculture upon

marriage.

These two features are rationalized below.

c.1 Surplus

Basic set-up is same as before except surplus function is changed to:

Σ = f(x, y) if Y = N

Σ = λf(x, y) if Y = A

In China, the norm after marriage is that the wife moves to the husband’s place, hence it’s

harder for a rural man to move to urban district after getting married with an urban wife. In

another word, the total surplus generated by this couple match should be smaller than a rural

woman matched with an urban husband, i.e., Σ((x,A), (y,N)) > Σ((x,N), (y, A)). Let me

further assume Σ((x,A), (y,N)) = Σ((x,N), (y,N)), Σ((x,N), (y, A)) = Σ((x,A), (y, A)).

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All men are indifferent between marrying a rural wife or an urban wife with the same SES.

All women prefer a partner with non-agriculture hukou status than agriculture hukou status

while the SES is the same.

The second feature states that quota of changing hukou status from agriculture to non-

agriculture is limited. Denote the absolute level of the quota as β, this indicates that

β << αM . Before 1990s in China, there is a policy that states that the number of people

that can change hukou should not exceed 0.1% of the urban population. This rationalizes a

very small β.

c.2 Stable matching

No Quota constraint Let me add the two features one by one. Let’s assume there is no

quota constraint, with the surplus function above, the main result is as following:

Proposition 5. In stable matching outcome, there exists threshold x̃ between 0 and 1, such

that:

• ∀x ≥ x̃, pA(x) = pN(x) = 0

• ∀x < x̃, pA(x) ∈ (0, 1), pN(x) ∈ (0, 1)

• qA(y) ∈ (0, 1), qN(y) ∈ (0, 1)

When urban men are very attractive as shown in the surplus function, the very top urban

men are married with both urban and rural women with very top SES. Note also that urban

woman and rural woman plays no different roles in the surplus function. For the other

urban men and all rural men, each woman is matched with an urban man and a rural man

with positive probability. While the rural man should have a higher SES characteristic than

the urban man if there exists any woman that is matched with both of them with positive

probability.

With Quota Constraint Assume there is an absolute quota of changing hukou from agri-

culture to non-agriculture, β, which is a relatively small number. The result is as following:

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Proposition 6. If αM + β ≥ αF , in stable matching outcome, there exists thresholds x̃1, x̃2,

x̃1 > x̃2, such that:

• ∀x ≥ x̃1, pA(x) = 0, ∀x ≥ x̃2, pN(x) = 0

• ∀x̃1 ≥ x ≥ x̃2, pA(x) = 1, pN(x) = 0

• ∀x ≤ x̃2, pA(x) = 1, pN(x) ∈ (0, 1)

If αM + β < αF , in stable matching outcome, there exists thresholds x̃1, x̃2, x̃′2, x̃1 >

x̃2, x̃1 > x̃′2, such that:

• ∀x ≥ x̃1, pA(x) = pN(x) = 0

• ∀x̃1 ≥ x ≥ x̃2, pA(x) = 1; ∀x̃1 ≥ x ≥ x̃′2, pN(x) = 0

• ∀x ≤ x̃2, pA(x) ∈ (0, 1); ∀x ≤ x̃2, pN(x) = 0

With limited quota, the very top rural women are matched with urban men. Among

remained population, we are put in a similar situation as ”smoking” case in (P. Chiappori

et al., 2012), only urban-urban match induces more surplus, as in the nonsmoker-nonsmoker

case. Hence conclusion there can be applied here. If αM + β < αF , after exhausting the

quota, we are left with more urban men than urban women. Hence we know that there is

no randomization for all high SES and possible randomization for low SES. In the remained

population, there is no urban husband-rural wife couple since pA(x) = 1. If αM + β ≥ αF ,

after exhausting the quota, we are left with more urban women than urban men. There is no

randomization for all high SES and rural women with low SES. There is randomization on

rural men with low SES. Urban husband-rural wife doesn’t exist in this remaining population

since pN(x) = 0

d A quadratic example

d.1 Specification

As stated above, the exact form of the stable matching depends on the distributions and

surplus function. Let me now consider a simple example with the following assumptions:

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(i) independence of hukou status and SES characteristics (ii)the percentage of agriculture

population among males (αM) is less than the percentage of agriculture population among

females (αF ), i.e., αM < αF (iii) the absolute number of hukou change quota is β < αF −αM(iv) The distribution of SES characteristics for males and females, F (x) andG(y), are uniform

distribution between [0, 1] (v)the gain generated from marriage is quadratic, specifically, I

assume that:

f(x, y) = xy

as used in previous papers.

d.2 Formal characterization of the stable matching

I now show that it is possible in this simple setting to completely solve the matching model

in closed form. Regarding hukou status, at most four marriage types may appear: N-N, A-A,

N-A, A-N. Within each category, supermodularity implies that matching will be assortative;

i.e., men with higher SES will marry wives with higher SES.

Marital patterns My main result is the following:

Proposition 7. There exists a unique stable matching, and there exists three numbers, x̃1, x̃2,

p, x̃1 > x̃2, all between 0 and 1,

x̃1 = 1− β

αF

such that the unique stable matching has the following properties:

• All agents marry.

• For all x > x̃1, a rural woman with SES x is matched with probability 1 to an urban

husband with SES:

y =1

1− αMx− αM

1− αMan urban woman with SES x is also matched with probability 1 to an urban husband

with SES:

y =1

1− αMx− αM

1− αM

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i.e. All top women marry ”down” since 11−αM

< 1

• For all x ∈ (x̃2, x̃1), a rural woman with SES x is matched with probability 1 to a rural

husband with SES:

y =αFαM

x− αF − αM − βαM

For all x ∈ (x̃2, x̃1), an urban woman with SES x is matched with probability 1 to an

urban husband with SES:

y =1− αF1− αM

x+αF − αM − β

1− αM

i.e., rural women marry ”up” since αF

αM> 1 while urban women marry ”down” since

1−αF

1−αM< 1

• For all x ∈ [0, x̃2), a rural woman with SES x is matched:

– with probability p to an urban husband with SES:

y = (αF + λ(1− αF ))x

– with probability 1− p to a rural husband with SES:

y = (αF + λ(1− αF ))x

For all x ∈ [0, x̃′2), an urban woman with SES x is matched with probability 1 to an

urban husband with

y = (αF + λ(1− αF )

λ)x

i.e. Rural women marry ”down” and are indifferent between a rural man and an urban

man since there is no more quota of hukou change available. Urban women marry ”up”

since αF+λ(1−αF )λ

> 1

Interpretation At the top SES distribution, all women marry ”down” to an urban hus-

band because of a limited supply of urban husbands with high SES. After exhausting the

quota, individuals with same hukou status and high SES marry to each other while rural

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women marry ”up” because the very top is married to an urban husband and urban women

marry ”down” because the non-availability of very top urban husband. For women with

low SES, the urban woman is married with an urban man to take full advantage of the

urban-urban couple surplus and they marry ”up” because the outside option for an urban

man is to marry a rural wife. While the rural women are indifferent between marrying a

rural husband or an urban husband with same SES.

Surplus allocation A by-product of the result is the intra-household allocation of re-

sources implied by equilibrium conditions. In the example above, they are exactly pinned

down.

Comparative Statics Comparative statics properties can be studied using the closed

form solutions.

• A large β, namely, more quota of hukou change, decreases the first threshold x̃1, benefits

rural females and urban males, while it hurts urban females and rural males because

it increases competition.

• A large λ, by reducing the welfare cost of having an agriculture hukou, increases the

threshold x̃2, x̃′2, and benefits rural males and rural females, but also urban males.

However, it hurts urban females by reducing their comparative advantage.

• Increasing of αM benefits rural females and hurts all other agents.

• Increasing of αF hurts rural females but benefits all others.

e Model Predictions

In practice, the frictionless process described in the model is never observed. Actual match-

ing involves multidimensional characteristics, some of which may even not be observed by

econometricians. It may be furthermore affected by random shocks or search cost. For all

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these reasons, observed matching patterns are largely stochastic. However, previous anal-

ysis could suggest some specific patterns, more specifically, we may expect the following

regularities to hold:

Prediction 1. Rural husband-urban wife couple should be less frequent than rural wife-urban

husband couple.

Prediction 2. Among couples with same marriage type, matching should be assortative on

SES.

Prediction 3. In observed data, urban husbands married with a rural wife should have a

lower SES than those married with an urban wife.

Prediction 4. When two urban men with the same low SES marry respectively a rural

woman and an urban woman, the urban woman should on average be of lower SES than the

rural woman.

When two urban men with the same high SES marry respectively a rural woman and an

urban woman, the difference of the two wives should be insignificant.

Next session is devoted to testing these predictions.

4 Empirical Application

a Data

Estimations are based on China 1990 1% sample census, a household type survey conducted

by China National Bureau of Statistics in 1990. Hukou status, education, marriage status

are available in this dataset. Though there is no information on historical hukou status,

but given hukou change quota is relatively small and all change is from agriculture to non-

agriculture upon marriage, all predictions are still valid with this twist. Furthermore, I

restrict my sample to women between 18 and 33, men between 20 and 35.

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b A first look at the Data

In the sample, we have 50.61% male; 75.27% of male population have agriculture hukou while

79.38% of female population have agriculture hukou. This confirms the previous assumption

αF > αM . The main characteristics of the data are described in table 1-3.

Table 1 gives the summary statistics of married and single individuals. Males are more

educated than females on average. Urban population are also more educated than rural

population. Table 2 breaks couples into four marriage types and present their summary

statistics. Table 3 reports the observed matching pattern. As we can see from the results,

cross-hukou type marriage appears much less in data than random matching may suggest.

Urban husbands with rural wives have on average fewer schooling years than those with

urban wives. Urban wives with rural husbands also have on average fewer schooling years

than those with urban wives.

c Result

Table 4 presents evidence of sorting by education within each marriage type, In this table,

own education is regressed on spouse’s education controlling own age and city fixed effect.

Not surprisingly, there is clear evidence of positive assortative-ness by education. In table 5,

own education is regressed on spouse education, spouse’s hukou status controlling own age

and city fixed effect.

5 Conclusion

A two-dimensional matching model where one dimension of the characteristics is hukou status

is applied to analyze the marriage market in China in the early 1990s when there is limited

quota to transfer hukou status upon marriage. I show that with reasonable assumptions

on the surplus structure, the model predicts that there are fewer cross-hukou type marriage

and predicts the selection into the four marriage types. In future work, this model can be

extended to investigate how marital trends have changed in recent periods of China when

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the value difference between urban hukou and rural hukou decreases.

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Figure 1: Symmetric Case with αM = αW < 0.5

Rural Urban

Women

Rural Urban

Men

Figure 2: Symmetric Case with αM = αW > 0.5

Rural Urban

Women

Rural Urban

Men

19

Figure 3: Asymmetric Case with αM > 0.5 and unlimited Quotas

Rural Urban

Women

Rural Urban

Men

Figure 4: Asymmetric Case with αW − β > αM > 0.5 and Limited Quotas

Rural Urban

Women

Rural Urban

Men

20

Table 1

Summary Statistics: Married versus Singles

Means (Standard Deviations)

Husband’s age 20-35, Wife’s age 18-33

Census 1990

Married

Men Women

Age 28.42 26.55

(4.17) (3.84)

Education 8.39 6.75

(3.04) (3.74)

Rural 0.77 0.81

(0.42) (0.39)

N 1,228,450 1,230,784

Singles

(never married)

Men Women

Age 23.09 20.28

(3.39) (2.18)

Education 8.01 7.50

(3.38) (3.27)

Rural 0.71 0.76

(0.45) (0.43)

N 563,295 523,414

21

Table 2: Summary Statistics: Married Couples by Hukou Status

Means (Standard Deviations)

Husband’s age 20-35, Wife’s age 18-33

Census 1990

Rural-Rural Urban-Urban

Husband Wife Husband Wife

Age 27.65 25.96 29.28 27.59

(4.02) (3.68) (3.40) (3.31)

Education 7.73 5.89 10.86 10.49

(2.85) (3.40) (2.32) (2.27)

N 813,073 170,310

Urban-Rural Rural-Urban

Husband Wife Husband Wife

Age 28.74 26.63 28.11 26.28

(3.74) (3.52) (3.94) (3.75)

Education 9.35 8.33 8.79 8.45

(2.57) (3.01) (2.81) (3.08)

N 28,320 6,941

22

Table 3

Husband’s age 20-35, Wife’s age 18-33

Census 1990

A. Observed Matching

Rural Wife Urban Wife

Rural Husband 79.82% 0.68%

813,073 6,941

Urban Husband 2.78% 16.72%

28,320 170,310

B. Random Matching

Rural Wife Urban Wife

Rural Husband 59.25% 15.75%

Urban Husband 19.75% 5.25%

23

Table 4

Sorting by education

Husband’s age 20-35, Wife’s age 18-33

Census 1990

Rural-Rural Urban-Urban

Wife’s

Education

Husband’s

Education

Wife’s

Education

Husband’s

Education

Spouse’s Education 0.369 0.287 0.432 0.458

(0.004) (0..005) (0.009) (0.009)

N 813,073 813,073 170,310 170,310

Rˆ2 0.16 0.16 0.22 0.22

Urban Husband

& Rural Wife

Rural Husband

& Urban Wife

Wife’s

Education

Husband’s

Education

Wife’s

Education

Husband’s

Education

Spouse’s Education 0.288 0.215 0.417 0.344

(0.010) (.008) (0..015) (0.013)

N 28,320 28,320 6,941 6,941

Rˆ2 0.07 0.07 0.16 0.16

Note: All regressions include: own age and city fixed effect.

Robust errors in parentheses.

24

Table 5

Regression of Education by hukou status on spouse’s education

and hukou status

Husband’s age 20-35, Wife’s age 18-33

Census 1990

Wife’s Education Husband’s Education

Rural Urban Rural Urban

Spouse’s Education 0.368 0.431 0.287 0.403

(0.004) (0.008) (0.005) (0.008)

Rural Spouse -1.69 -0.944 -0.356 -0.661

(0.054) (0.051) (0.047) (0.038)

N 841,393 177,251 820,014 198,630

Rˆ2 0.17 0.24 0.16 0.22

Note: All regressions include: own age and city fixed effect.

Robust errors in parentheses.

25

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