House price model based on maginal analysis of house market

Hedonic house price model only considers one house’s own attribute in price decision, but as shown in Section 6, in real world’s house market, house attribute on city periphery and hous

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House Price Model Based on Marginal Analysis of House

Market

(Hay Nong Tsinghua University; Asian Economics Theory Research Center)

The paper built the basic theoretical model for urban house price, based on a marginal analysis of house market The contributions are the three (1) We discovered,

in real world’s house market, there are three factors that strongly influence house price, and we show the reason (2) We built a three factors house price model, and tested the model by empirical data The model explains real world’s house price better than present models (3) The new model is the more general model, present house price models, such as hedonic house price model, are the special situation of the new model

1 Introduction

The value of house constitutes the major part of wealth Such as, in China, the value of house constitutes three quarters of total household wealth, and, in the United States, the value of house constitutes one third of total household wealth (CGB, et al, 2018) Then, how house’s value or price is decided, is important both in theory and in practice But, till now, a basic theoretical model for urban house price is still needed Hedonic house price model is widely used by researchers to explain house price, such as Kain and Quigley (1970), Wabe (1971), Evans (1973), Paul Cheshire, et al (1995), Wen Hai-zhen (2005) Hedonic house price model argues house price is decided

by house’s objective attributes or characteristics, such as house’s size, house’s distance

to city center, etc We use z( , , , )z z1 2 z n to represent one house’s attributes, then, hedonic house price model is p f z( ) f z z( , , , )1 2 z n

Hedonic house price model only considers one house’s own attribute in price

decision, but as shown in Section 6, in real world’s house market, house attribute on

city periphery and house production cost on city periphery also strongly influence

house price (see Figure 1, Figure 2 in Section 6)

Why attribute and production cost of house on city periphery have strong influence

on house price? The reason can be found by marginal analysis of house market In house

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market of one city, when an extra unit of house is built, the extra unit of house will often

be built on city periphery (see §3) As known by economists, the extra unit of commodity has strong influence on market price (Ricardo 1817, Marshall 1890, Samuelson 2005, etc.) Such as, Samuelson (2005), Mankiw (2016) etc argued, in perfectively competitive market reached long run competitive equilibrium, the cost of extra unit of commodity (the marginal cost) decides commodity’s market price And, it’s obvious that, when attribute is considered in price decision, the attribute of the extra unit of commodity also will influence commodity’s market price Such as, as the newly produced computer (the extra unit of computer provided to market) has better attribute such as higher CPU speed, the old computer’s price will often drop down

Then, totally, when attribute is considered in house price decision, in the city, there are three factors that strongly influence house price The three factors are house’s own attribute, house attribute on city periphery (the extra unit house’s attribute), house production cost on city periphery (the extra unit house’s production cost) Hedonic price model only considers one factor and ignores the other two, then has obvious limitations Traditional economics assumes commodity is homogeneous, and argues, at long run competitive equilibrium, commodity’s price equals the long run marginal cost (Mc0) (see literature review in §2) In house market, the extra unit of house produced

is often on city periphery, then, house’s marginal cost (Mc0) is just the production cost

of house on city periphery (c0), then, traditional economics’ house price model will be

(1) 0 0

( )( )

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quality of house on city periphery, c0 is the production cost of the house on city periphery Here, house’s quality is a function of house’s attributes z( ,z z1 2, ,z n)are one house’s own attributes,

p Mc

The above equation means, price of house quality p equals marginal cost of house q

quality Mc That price equals marginal cost is one basic idea of present economics q

(see Samuelson et al 2005, Varian 2014, Mankiw 2016) This means, formula (1) is

only a use of present economics, but at quality or attribute level

We tested formula (1) by empirical data, and found, formula (1) does exist in real world’s market In the test, the significance is lower than 0.01

Formula (1) considers all the three factors (z ,z0 ,c0 ) in house price decision Hedonic house price model only considers house’ own attribute z, and ignores z0, c0 Traditional economics’ house price model only considers cost c0, and ignored z, z0 Then, formula (1) has obvious advantages We find, formula (1) explains real world’s house price better than other two models (see §6, 7)

By data of 31 major cities in China, we provided a comparison of the three models

in explaining real world’s house price (§6.3) The result of comparison is in table below

Model Name Model Form Explaining

Variable Estimated Model R-square This paper’s house price

model, formula (1) 0 0

( )( )

d d

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In the comparison, we use per square meter house price (p), and we use house’s

distance to city center (d) to represent house’s attribute, since in major cities of China,

house’s distance to city center is the most important attribute that influences per square meter house price (see §6.3)

From the comparison, we find, formula (1) has the highest R-square This means, formula (1) explains real world’s house price better than other models Here, the R-square of formula (1), 0.754, is high enough, since we only consider one attribute

(distance to city center d) of house, and house has several attributes that influence price

Here, hedonic price model’s R-square is the lowest, only 0.019 The reason is, as analyzed in §5, hedonic house price model fits to explain house price in the same city, but here, the house price is from 31 cities, not from the same city

We find, hedonic house price model can be seen as the special situation of formula (1) when c0, z0 are given When c0, z0 are given, c0 and q0 q z( )0 are given, let 0

q q z q z q , then formula (1) becomes p1gc0 c0

This paper’s contributions are the following three (1) We discovered, in real world’s house market, there are three factors that strongly influence house price, and

we show the reason (2) We built a three factors house price model, and tested the model

by empirical data The model explains real world’s house price better than present models (3) The new model is the more general model, while present house price models, such as hedonic house price model, are the special situation of the new model

The remainder of the paper is organized as follows Section 2 provides literature review Section 3 provides a marginal analysis of house market, and builds the model Section 4 provides the more theoretical expression of the model Section 5 shows the model’s difference from hedonic house price model Section 6 tests the model Section

7 explains real world’s house price by the model Section 8 concludes

in attribute, such as each computer has the same CPU speed By assuming commodity

is homogeneous, traditional economics excludes attribute and only considers cost in price decision But in real world’s market, commodity is often heterogeneous in attribute and attribute does strongly influence price Such as, in real world’s computer market, computer may have different CPU speeds, and the computer with higher CPU speed will often have higher price Then, to explain real world’s price, attribute should

be considered

In big cities, when house is built in the city, mostly, the land near city center is better land and will be used first, after the land near city center is used up, the land farer

from city center will be used Then, in big city, when an extra unit of house is built, the

extra unit of house will be built on city periphery In traditional economics, marginal

cost is defined as the total cost increased when an extra unit of commodity is produced

Then, according to the definition of marginal cost, in house market, the marginal cost

of house ( Mc0 ) will be the production cost of house on city periphery ( c0) Then, traditional economics’ house price model will be pMc0 c0

It’s obvious that, traditional economics’ house price model (house price equals house’s marginal cost) cannot well explain house price in real world’s market In real world’s competitive house market, in big cities, house price on city periphery might equal house’s marginal cost (house’s production cost on city periphery).1 But, house

supply of house is easily to be increased Then, in competitive house market, on city periphery, if house price is higher than production cost, more houses will be built, until house price equals production cost

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price in city center will often be much higher than house’s marginal cost (house’s production cost on city periphery) This means, house price in city center can’t be explained only by house’s marginal cost

Hedonic price model is widely used by researchers to explain house price, such as Kain and Quigley (1970), Wabe (1971), Evans (1973), Paul Cheshire, et al (1995), Wen Hai-zhen (2005) Hedonic house price model argues house’s price will be decided by house’s attributes The limitation of hedonic house price model is that, hedonic house price model only considers house’s own attribute, but as shown in Section 6 (Figure 1, 2), attribute of house on city periphery and production cost of house on city periphery also strongly influence house price Then, hedonic price model in fact ignored the other two important factors in house price decision, then, hedonic price model has obvious limitations and cannot be the basic theoretical model for urban house price

Besides above two major schools of house price model, some other house price models were also built, such as James M Poterba, et al (1991), Steven C Bourassa et

al (2001), Peter Abelson et al (2005), etc But these models also cannot theoretically explain real world’s house price decision

Till now, we still need a basic theoretical model for urban house price

Olsen (1969)argued, in house market, different house has a homogeneous attribute, house’s quality or house’s housing service, and, at long run competitive equilibrium, houses will have the same quality/price

Olsen (1969)’s idea is a good abstraction of the real world’s house price

phenomenon that better house has higher price Such as, in real world’s house market,

in big cities, the house nearer to city center is often the better house (the house with better transportation, shorter distance to office, better medical service, etc.), and, the house nearer to city center often has higher per square meter house price

That better commodity (commodity with better attribute or higher quality) has

higher price is also widely existing in many other markets Such as, in iron ore market,

Then, on city periphery, house’s tends to equal house’s production cost While as analyzed in the following, house’s production cost on city periphery represents house’s marginal cost Then, house price on city peripehry tends to equal house’s marginal cost.

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the iron ore that contains more iron element often has higher price

This paper assumes better house has higher price, then, Olsen (1969)’s idea will

be used in this paper, because Olsen (1969)’s idea is a good abstraction of the price

phenomenon that better house has higher price

3 A Marginal Analysis of House Market, and House Price Model

Since we aim to build a basic theoretical model, then, in this paper, we always assume house market is perfectively competitive and reached long run competitive equilibrium We always assume house’s attribute is positive

3.1 A Marginal Analysis of House Market

Marginal analysis can be widely used to explain consumer choice, producer choice and price decision Marginal analysis was used by many economists, such as Menger (1871), Jevons (1871), Walras (1899), Ricardo (1817), Mill (1848), Marshall (1890), Clark (1899), etc Here, we provide a marginal analysis of house market In this marginal analysis, house attribute is considered

As already mentioned, in big cities, when house is built in the city, mostly, the land near city center is better land and will be used first, after the land near city center is used up, the land farer from city center will be used Then, in big city, when an extra

unit of house is built, the extra unit of house will be built on city periphery

The Production Cost and the Attribute of the Extra Unit of House

House is the commodity that has several attributes, such as house’s distance to city center, house’s size, etc House’s these attributes can be divided into two categories The first category of attributes are attributes that are directly decided by house’s location, such as house distance to city center, etc The second category of attributes are attributes of house itself, such as house’s size, house’s decoration, etc The second category of attributes are not directly decided by house’s location, and are often called structural attributes

In this paper, one house’s all attributes are represented by

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In this paper, for simplification, we assume houses are the same in the second category of attributes This also means, we assume houses are the same except houses are on different locations Under this assumption, in the same city, once the location of house is given, house’s attributes z( ,z z1 2, ,z n) are given

As mentioned above, in house market, when an extra unit of house is built, the extra unit of house will be built on city periphery We use

0 1 , 2 , , n

z z z z to

represent the attributes of house on city periphery Then, above on above analysis we

can find, in house market, when an extra unit of house is built, the extra unit of house will have attributes

0 1 , 2 , , n

z z z z In one city, at given time, the city

periphery will be on given location or given place, then,

Then, above marginal analysis shows, in house market of one city, when an extra unit of house is built, the production cost of the extra unit of house is c0, the attributes

of the extra unit of house are

0 1 , 2 , , n

The Three Factors that Influence House Price

Traditional economics argues, the production cost of extra unit of commodity (the marginal cost) will influence commodity’s price (Samuelson 2005, Mankiw 2016, etc.) This argument was verified by empirical data of real world’s house market As shown

by figure 2 in Section 6, in 31 major cities of China, as the the production cost of house

on city periphery (the production cost of extra unit of house) goes up, house price in the city often goes up

Here, a new question arises Will the attribute of extra unit of commodity also

influence commodity’s market price?

If we observe the real world’s prices, we can find, the answer is yes Such as, in computer market, the newly produced computers can be seen as the extra units of computer provided to the market And, in real world’s computer market, as the newly

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produced computer has better attribute, such as higher CPU speed, the old computer’s price will often drop down

This paper found, in house market, the attribute of the extra unit of house strongly

influence house price As illustrated by figure 1 in section 6, in 31 major cities of China,

house price is strongly influenced by the attribute of house on city periphery, while the

attribute of house on city periphery represents the attribute of extra unit of house

As mentioned above, the attribute of house on city periphery (the attribute of the extra unit of house) influences house price, the production cost of house on city periphery (the production cost of the extra unit of house) influences house price And, it’s obvious that, one house’s own attributes will influence this house’s price Then,

totally, there are three factors that influence one house’s price The three factors are:

one house’s own attribute z , the attribute of house on city periphery z0 , the production cost of house on city periphery c0

Since in house market, there are three factors that influence house price, then, house price model should consider all the three factors in house price decision In the following, we will discover how house price is decided by the three factors

The Price and the Attribute of the Extra Unit of House

Marginal cost is the total cost increased when an extra unit of commodity is

produced Then, according to the definition of marginal cost, in house market, the

marginal cost ( Mc ) will be the production cost of house on city periphery ( c0), then

0

Mcc

Marginal revenue is the total revenue increased when an extra unit of commodity

is produced And as mentioned above, in house market, the extra unit of house will be

on city periphery Since house market is assumed to be perfectively competitive, then, according to the definition of marginal revenue, in house market, the marginal revenue (Mr ) will equal the price of house on city periphery ( p0) then

0

Mr p

According to traditional economics, in perfectively competitive market reached long run competitive equilibrium, marginal revenue (Mr) will equal marginal cost

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( Mc ), then we get

p MrMcc

This means, house price on city periphery ( p0) will equal the production cost of

house on city periphery ( c0)

p c

It’s easy to understand why on city periphery house price will equal house production cost. On city periphery, since there are often plenty of land to build house, then, the supply of house on city periphery is easily increased If the price of house on city periphery is higher than production cost, there will exist profit, then, more houses will be built, then house price will drop down, until price of house equals production cost of house

The above marginal analysis shows, in perfectively competitive house market reached long run competitive equilibrium, the price of house on city periphery will be

As analyzed, the house on city periphery represents the extra unit of house in house market Then, the above analysis also means, in perfectively competitive house market reached long run competitive equilibrium, when extra unit of house is built, the extra unit of house will have a price p0 c0 , and, the extra unit of house have attribute

0 1 , 2 , , n

3.2 The Price Model When House Has One Cardinal Attribute

Traditional economics argues, in perfectively competitive market at long run

competitive equilibrium, commodity’s price is decided by the production cost of extra

unit of commodity (Samuelson 2005, Mankiw 2016, etc., even Marshall 1890)

Traditional economics’ this argument was based on the assumption that commodity is homogeneous, such as each unit of computer is homogeneous But, in

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house market, houses are heterogeneous in attribute, and house attribute does influence house price We argue, in house market, for one house with given attribute, this house’s

price is decided by both the production cost of the extra unit of house and the

attribute of the extra unit of house

As analyzed above, in perfectively competitive house market reached long run competitive equilibrium, when extra unit of house is built, the extra unit of house will have a price p0 c0, and, the extra unit of house have attribute

0 1 , 2 , , n

And, c0, z0 are given

Suppose in the city, house A’s attributes are z( ,z z1 2, ,z n) , zz0 This means, house A has better attributes than the extra unit of house (the house produced

on city periphery) It’s easy for us to know that, house A’s price p Awill be higher than

p c , since house A has better attributes Why house A will have a higher price? The reason lies in consumer’s choice behavior Here, if house A’s price is the same as or even lower than the extra unit of house’s price, all consumer will choose house A, then, the house A’s price will go up (The price of the extra unit of house is assumed given) But the above analysis only tells us whether house A’s price is higher or lower than p0 c0, since the above analysis is only qualitative The important question is,

quantitatively, how house A’s price is decided?

To answer this question, a deep analysis on consumer choice is required, which might need 20 pages Fortunately, Olsen (1969)’s idea can help us to answer this question quickly

Olsen (1969)argued, in house market, different house has a homogeneous attribute, house’s quality or house’s housing service, and, at long run competitive equilibrium, houses will have the same quality/price Olsen (1969)’s idea is a good abstraction of consumer’s influence on house price, better house (house with better attributes) will have higher price

Similar to Olsen (1969)’s idea, we assume each house has a homogeneous and cardinal attribute, quality q And, house’s quality q is function of house’s attributes Then

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1 2( ) ( , ,, , )n

q q z q z z z

where q is one house’s quality, zz z1, 2,, ,z n are this house’s all attributes

As analyzed above, in perfectively competitive house market reached long run competitive equilibrium, the extra unit of house will have attributes

q q z q z z z Then, in perfectively competitive house market reached

long run competitive equilibrium, the extra unit of house will have quality

=

q q q

p p c , then we get

(1) 0 0

( )( )

on city periphery Here, the house market is assumed to be perfectively competitive and reached long run equilibrium In a given city, at given time, z0, q0, c0 can be seen as given.1

city periphery will be on given location, then, house’s attributes are given, then, the quality of house is given In a given city at given time, the production cost is also given

If we give up that assumption that house’s attributes are decided by location, in given city at given time, z0, q0, c0 are also given In a given city, at given time, when house market reached long run competitive equilibrium, house produced on city periphery will have the best match of house attributes and production cost under given technology and factor prices (labor cost, land cost etc.) Here, the best match of house attributes and production cost can be seen as the solution of the optimum problem of house production, then can be seen as given, though attributes of house and production cost of house might be related And, house’s attributes are given means house quality is given For more analysis, please contact the author.