House price model based on maginal analysis of house market

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 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  ( z1 , z2 , , zn ) to represent one house’s attributes, then, hedonic house price model is p  f ( z )  f ( z1 , z2 , , zn ) 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 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 Electronic copy available at: https://ssrn.com/abstract=3966052 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 p  Mc0  c0 (§2) Traditional economics’ house price model didn’t consider house attribute, but in house market, house attribute does strongly influence house price We aim to build a basic theoretical model, then, we set some assumptions, and assume house market is perfectively competitive and reached long run equilibirum We found, in perfectively competitive market reached long run equilibrium, under certain assumptions such as houses have a homogeneous attribute, quality, house price will be (1) p q q( z ) c0  c0 q0 q ( z0 ) where p is one house’s price, q  q( z ) is this house’s quality, q0  q ( z0 ) is the Electronic copy available at: https://ssrn.com/abstract=3966052 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  ( z1 , z2 , , zn ) are one house’s own attributes, z0  z10 , z20 , , zn0 are attributes of house on city periphery The deduction of formula (1) is arranged in Section Formula (1) looks new, but, the key logic of formula (1) is quite simple, price equals marginal cost, one basic idea of present economics Formula (1) can be changed into p c0  Here, p  pq is the price of house quality, c0  Mc0  Mcq is the q q0 q0 Mq0 q marginal cost to provide house qualtiy (see §4) Then, formula (1) becomes pq  Mcq The above equation means, price of house quality pq equals marginal cost of house quality Mcq That price equals marginal cost is one basic idea of present economics (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 q( z ) c0 q ( z0 ) This paper’s house price model, formula (1) p Traditional economics’ house price model p  Mc0  c0 p  f ( z) Hedonic house price model Explaining Variable Model Form Estimated Model R-square e0.043d c0 e0.043d0 0.754 c0 p  c0 0.453 z p  e3.1680.014d 0.019 z , z0 , c0 p Electronic copy available at: https://ssrn.com/abstract=3966052 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 Rsquare 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 ( z0 ) are given, let c0  k , formula (1) becomes p  q c0  c0 q  k gq( z1 , z2 , , zn )  f ( z1 , z2 , , zn ) ， q0 q0 q0 here, f ( z1 , z2 , , zn ) is also a hedonic house price model (see §6) And, traditional economics’ house price model is also the special situation of formula (1), when houses are homogeneous in attribute When houses are homogeneous in attribute, z  z0 , then, q  q( z )  q( z0 )  q0 , 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 provides literature review Section provides a marginal analysis of house market, and builds the model Section provides the more theoretical expression of the model Section shows the model’s difference from hedonic house price model Section tests the model Section explains real world’s house price by the model Section concludes Electronic copy available at: https://ssrn.com/abstract=3966052 Literature Review Traditional economics assumes commodity is homogeneous, and argues, at long run competitive equilibrium, price equals the minimum long run average cost and equals long run marginal cost ( Mc0 ), which means p  LACmin  Mc0 , such as Samuelson et al (2005, 2010), Pindyck et al (2013), Varian (2014), Mankiw (2016) And, the idea that price equals marginal cost can also be traced back to David Ricardo(1817), Stuart Mill (1848), Alfred Marshall (1890), etc., even Karl Marx (1867) The assumption, commodity is homogeneous, implies that commodity is the same 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 As analyzed in section 3, on city periphery, there are often plenty of land to build house, then, the 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 Electronic copy available at: https://ssrn.com/abstract=3966052 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 (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 Electronic copy available at: https://ssrn.com/abstract=3966052 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 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 z  ( z1 , z2 , , zn ) Where z is the vector of house attributes, z1 , z2 , , zn are one house’s all attributes Electronic copy available at: https://ssrn.com/abstract=3966052 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  ( z1 , z2 , , zn ) 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 z0  z10 , z20 , , zn0 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 z0  z10 , z20 , , zn0 In one city, at given time, the city periphery will be on given location or given place, then, z0  z10 , z20 , , zn0 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 c0 to represent the production cost of house on city periphery Then, when an extra unit of house is produced, the production cost of the extra unit of house will be c0 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 z0  z10 , z20 , , zn0 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 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 Electronic copy available at: https://ssrn.com/abstract=3966052 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 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 Mc  c0 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 Mr  p0 According to traditional economics, in perfectively competitive market reached long run competitive equilibrium, marginal revenue ( Mr ) will equal marginal cost Electronic copy available at: https://ssrn.com/abstract=3966052 ( Mc ), then we get p0  Mr  Mc  c0 This means, house price on city periphery ( p0 ) will equal the production cost of house on city periphery ( c0 ) p0  c0 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 p0  c0 Since as set in above, the attributes of house on city periphery is z0  z10 , z20 , , zn0 Then, we can find, in perfectively competitive house market reached long run competitive equilibrium, the price of house on city periphery will be p0  c0 , the attributes of house on city periphery is z0  z10 , z20 , , zn0 Note that, in a given city, at given time, c0 and z0 are given (see footnote in page 12) 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 z0  z10 , z20 , , zn0 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 10 Electronic copy available at: https://ssrn.com/abstract=3966052 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 z0  z10 , z20 , , zn0 And, c0 , z0 are given Suppose in the city, house A’s attributes are z  ( z1 , z2 , , zn ) , 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 A will be higher than p0  c0 , 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 11 Electronic copy available at: https://ssrn.com/abstract=3966052 q  q( z )  q( z1 , z2 ,, , zn ) where q is one house’s quality, z  z1 , z2 ,, , zn 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 z0  z10 , z20 , , zn0 and price p0  c0 The house with attributes z0  z10 , z20 , , zn0 will have a quality q0  q( z0 )  q( z10 , z20 , , zn0 ) Then, in perfectively competitive house market reached long run competitive equilibrium, the extra unit of house will have quality q0  q( z10 , z20 , , zn0 ) and price p0  c0 Suppose there is one house in the same city, its attributes are z  z1 , z2 ,, , zn and quality is q  q( z1 , z2 ,, , zn ) , and this hosue’s price is p Similar to Olsen(1969)’s idea, we assume that, in perfective competitive market reached long run competitive equilibrium, houses will have the same quality/price, then, we can get q q0 q0  = , p p0 c0 then we get p (1) q q( z ) c0  c0 q0 q ( z0 ) where p is one house’s price, q is this house’s quality, q0 is the quality of extra unit of house and q0 is also the quality of house on city periphery, c0 is the production cost of the extra unit of hosue and c0 is also the production cost of house 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 This paper assumes house’s attribute is decided by house’s location In a given city at given time, the 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 12 Electronic copy available at: https://ssrn.com/abstract=3966052 After a test in §6 and analysis in §7, we will find, formula (1) does exist in real world’s house market, and can explain real world’s house price better than other models, such as hedonic house price model Totally, formula (1) considers three factors in house price decision, the three factors are: one house’s own attributes ( z ), attributes of house on city periphery ( z0 ), and production cost of house on city periphery ( c0 ) Formula (1) represents this paper’s three factors house price model 3.3 The Price Model When House Has One Ordinal Attribute Formula (1) is the theoretical model for house price when house has one cardinal attribute, house’s quality We created the concept of house quality, to discover the basic law of house price decision We find the basic law of house price decision, which is formula (1) But, in real world’s market, house’s attribute is often ordinal, not cardinal, such as, house’s distance to city center is only an ordinal attribute Formula (1) can help us to understand the house price decision when house has one ordinal attribute Based on formula (1), we can find, when house has one ordinal attribute, there will exist the following three qualitative relationships in house market Relationship Given production cost of house on city peripehry ( c0 ) and the attribute of house on city periphery ( z0 ), house price ( p )will be positively related with house’s attribute ( z ) Relationship Given house’s attribute ( z ) and the production cost of house on city peripehry ( c0 ), house price ( p )will be negatively related with the attribute of house on city periphery ( z0 ) Relationship Given house’s attribute ( z ) and the attribute of house on city periphery ( z0 ), house price ( p ) will be positively related with production cost of house on city periphery ( c0 ) The above three relationships represents this paper’s house price model when house has one ordinal attribute We argue, when house has several ordinal attributes, the basic law of house price 13 Electronic copy available at: https://ssrn.com/abstract=3966052 decision will be similar to house price decision when house has one ordinal attribute In real world’s house market, house’s attribute is often ordinal, then, the above three relaitonships can be more frequently used to explain real world’s house price A More Theoretical Expression of the House Price Model As already mentioned, in perfectively competitive house market reached long run competitive equilibrium, house’s marginal cost ( Mc0 ) will be the production cost of house on city periphery ( c0 ), then, Mc0  c0 In this paper, the quality of the extra unit of house is called marginal quality In house market, house’s marginal quality ( Mq0 ) will be the quality of house on city periphery ( q0 ) , then, Mq0  q0 Since Mc0  c0 and Mq0  q0 , then formula (1) can also be expressed by the following model p (2) q Mc0 Mq0 where Mq0 is house’s marginal quality at equilibirum, Mc0 is house’s marginal cost at equilibrium Formula (2) is a more theoretical expression of this paper’s house price model Further analysis shows, formula (2) is also a general price model for various commodities in competitive market, when commodity’s quality or attribute is considered in price decision.1 This means, this paper’s house price model is not an isolated model, but a special use of a more general model in house market We asssume house and house’s quaity are dividable and additive We set that, H is the quantity of house added, c is the production cost added because of H , q is the house quality added because of H As analyzed in above, c0  Mc0 = the marginal cost to produce house, q0  Mq0  q H is the marginal quality in The proving of (2) is similar to the proving of formula (1) The only difference is, here, Mq0 , Mc0 c is H p, q, are price, quality, marginal quality, marginal cost of commodity The two conditions for us to get formula (2) are: (a) commodities have the same quality/price, (b) marginal revenue equals marginal cost More analysis is in the author’s working paper “A two factors price model based on deeper research of consumer choice” 14 Electronic copy available at: https://ssrn.com/abstract=3966052 producing house Then c0 Mc0 c / H c c     Mcq This means, =Mcq in q0 Mq0 q / H q q0 fact is the marginal cost to provide house quallity Formula (1) can be changed into analyzed above , p c0 p   pq is the price of house quality As q q0 q c0 =Mcq is the marginal cost to provide house quallity Then, q0 formula (1) becomes pq  Mcq The above equation means, the price of house quality equals the marginal cost to provide house quality This means, the key logic of formula (1) is quite simple, at quality or attribute level, price equals marginal cost That price equals marginal cost is one baisc idea of present economics’ price theory This means, formula (1) is only a use of present economics’ price theory, but at quality or attribute level Rosen (1969) already addressed pq  Mcq , but Rosen didn’t develop a model like formula (1) based on pq  Mcq As analyzed in this paper, formula (1) is the basic theoretical model for urban house price, and well explains real world’s house prices The Difference from Hedonic House Price Model, etc We can find, this paper’s house price model, formula (1), considers all the three factors that influence house price, then is the more general house price model Hedonic house price model and traditional economics’ house price model considers only one factor, and can be seen as the special situation of formula (1) Hedonic house price model argues house’s price will be decided by house’s attributes According to hedonic house price model, house price will be p  f ( z) Where p is one house’s price, z  z1 , z2 , , zn are this house’s attributes Hedonic house price model only considers z (one house’s own attributes) in this house’s price decision While this paper’s house price model, formula (1), considers 15 Electronic copy available at: https://ssrn.com/abstract=3966052 z , z0 , c0 in house price decision After the following analysis, we can find, hedonic house price model is a special situation of formula (1) when z0 (attribut of house on city periphery) and c0 (production cost of house on city periphery) are given In formula (1), q  q( z1 , z2 , , zn ) , z1 , z2 , , zn are one house’s attributes In formula (1), when production cost of house on city periphery ( c0 ) and attribute of house on city periphery ( z0 ) are given, c0 and q0  q ( z0 ) are given, let c0  k , then k is also q0 given Then, formula (1) will become p c q c0  q  k gq( z1 , z2 , , zn ) q0 q0 Let k gq( z1 , z2 , , zn )  f ( z1 , z2 , , zn ) Then, formula (1) becomes p  f ( z1 , z2 , , zn ) The model is also a hedonic house price model The above analysis means, hedonic house price model is only a special situaiton of this paper’s house price model The special situation is that, attribute of house on city periphery ( z0 ) and production cost of house on city periphery ( c0 ) are given When attribute and production cost of house on city periphery are given? The answer is, in the same city at given time, attribute and production cost of house on city periphery are given Because, in different cities, the attribute of house on city periphery and the production cost of house on city periphery will be different, and, in the same city, as time elapses, the cost to produce house on city periphery and the attribute of house on city periphery will change.(Note that, in the same city, in different year, city peripehry might be at different place, since as time elapses city might become larger or smaller.) Then, according to formula (1), we can find, hedonic house price only fits to explain the house price in the same city at given time, but cannot explain the house price in many cities or the house price during many years 16 Electronic copy available at: https://ssrn.com/abstract=3966052 The empirical analysis in§6.3 shows that, hedonic house price model doesn’t fit to explain the house prices in many cities, and, by evidence from real world’s market, we can find that, hedonic house price model cannot explain house price during many years Such as, in Beijing City of China, during 2004-2020, the average house price went up from 700 dollars per square meter to 8000 dollars per square meter, more than 10 times It’s obvious that, this rise of house price during 2004-2020 in Beijing cannot be mainly explained by the rise of house’s attribute And, as analyzed by many researchers, hedonic house price model can explain house price in the same city at given time.(see Kain and Quigley (1970), Wabe (1971), Evans (1973), Paul Cheshire, et al (1995), Wen Hai-zhen (2005), etc.) Formula (1) can explain house price in many cities and can explaing house price during many years Such as, the empirical analysis in§6.3 shows that, formula (1) can explain house price in many cities And, the analysis in §7 shows that, formula (1) can explain house price during many years Traditional economics’ house price model will be p  c0 (see §2) Traditional economics’ house price model considers only one factor ( c0 ) and ignored the other two ( z , z0 ) Traditional economics’ house price model also can be seen as a special suituation of this paper’s model, when houses are homogeneous in attribute When houses are homogeneous in attribute, then z  z0 , then, q  q( z )  q( z0 )  q0 , then, formula (1) will become p q c0  1gc0  c0 q0 which is just traditional economics’ house price model Test the Model 6.1 Test the House Price Model when House Has One Ordinal Attribute Relationship 1,2,3 in §3.3 represents the house price model when house has one ordinal attribute Here, we will test Relationship 1,2,3, by data of 31 major cities in China The detailed test is arranged in Appendix A.1 Since the three relationships are only qualitative, then, we will test the three 17 Electronic copy available at: https://ssrn.com/abstract=3966052 relationships by figure and by correlation coefficient Relationship is the new relationship discovered by this paper, then, we will focus on Relationship Relationship shows that, one house’s price is negatively related with the attribute of house on city periphery (given one house’s own attribute and given the production cost of house on city peripehry) In the test, house attribute is house’s distance to city center (d), house price is per square meter price (In major cities of China, the distance to city center is the most important attribute that influence per square meter house price) After test, we find, the correlation coefficient between house price and attribute of house on city periphery is -0.749, and the significance is less than 0.01 This means, in real world’s house market, given house’s attribute and given production cost of house on city periphery, house price is significantly negatively related with attribute of house on city periphery Figure 1: House Price and Attribute of House on City Periphery House Price Attribute of House on City Periphery (Kilometers) From the above Figure 1, we can find that, house price is obviously negatively related with the attribute of house on city periphery (Here, one house’s own attribute and the production cost of house on city periphery are given) From the test, we also found that, given production cost and attribute of house on city periphery, one house’s price is positively related with this house’s own attribute And, from the test, we also found that, given one house’s own attribute and given the attribute of house on city periphery, one house’s price is positively related with the production cost of house on city periphery Figure below shows that, house price is obviously positively related with the production cost of house on city periphery 18 Electronic copy available at: https://ssrn.com/abstract=3966052 Figure 2: House Price and Production Cost of House on City Periphery House Price (1000/m2) Production Cost of House on City Periphery (1000/m2) From the test, we find, the above Relationship 1, 2,3 in §3.3 exist in real world’s market This also implies that, formula (1) is a successful theoretical model From the test, we find, in real world’s house market, house price in the city is strongly influenced by each of the three factors: house’s own attribute, attribute of house on city periphery, and production cost of house on city periphery 6.2 Test the House Price Model when House Has One Cardinal Attribute We tested formula (1) by 120 communities’ house price in 31 cities in china After hypothesis test, we find, formula (1) does exist in real world’s house market The detailed hypothesis test is in Appdendix A.2 6.3 Comparision of the Three Different House Price Models Base on above empirical data of 31 cities’ house market, we provide a comparison of the three different house price models For detail, see Appendix A.3 Based on above data and formula (1), we got the following house price model The model represents this paper’s house price model (3) p q q (d ) e 0.043d c0  c0  0.043d0 c0 q0 q (d ) e 0.043d Where p is one house’s price, q  e is this house’s qualtiy, d is this house’s 0.043d0 distance to city center; q  e is qualtiy of house on city periphery, d is city periphery’s distance to city center The R-square of the model is 0.754, which is high enough, since here, we considers only one attribute of house in house price decision, and house often has several important attributes that influence house price 19 Electronic copy available at: https://ssrn.com/abstract=3966052 Based on above data, we also got a hodenic house price model p  e3.1680.014d the R-square of the model to explain house price is 0.019 Why the R-square of hedonic house price model is quite low? The reason is, as analyzed in section 5, hedonic house price model fits to explain house price in the same city at given time, but here, the data of house price is from 31 cities, then, hedonic house price model loses its power As analyzed in §2, traditional economics’ house price model will be p  Mc0  c0 Where c0 is the production cost of house on city periphery Based on above data of house price, we find, the R-square of traditional economics’ house price model is 0.453 As shown in the following table, we can find, formula (3) has the highest R-square (0.754).This means, this paper’s house price model explains real world house price better than other models Here, the R-square (0.754) is high enough, since only one attribute of house is considered in house price decision, while house has several attributes that strongly influence house price Model Name Model Form Explaining Variable Estimated Model R-square z , z0 , c0 e0.043d p  0.043d0 c0 e 0.754 q( z ) c0 q ( z0 ) This paper’s house price model p Traditional economics’ house price model p  Mc0  c0 c0 p  f ( z) z Hedonic house price model p  c0 (Estimation is not needed) 0.453 p  e3.1680.014d 0.019 Explaining Real World’s House Price In this section, we can find, this paper’s house price model explains real world’s house price much better than other models In real world’s house market, there are three important price phenomena The three price phenomena are very common, though some people might not notice them all The first phenomenon is that, in the same city, house on better location will has a higher per square meter price Such as, in big cities, the location nearer to city center is 20 Electronic copy available at: https://ssrn.com/abstract=3966052 always the better location, and, in big cities, house nearer to city center always has a higher per square meter price Such as, as illustrated in the following Figure 3, in Beijing City, house nearer to city center often has a higher per square meter price Figure 3: House Price and House Location (Beijing City) House Price (1000/m2) 180 160 140 120 100 80 60 40 20 Distance to City Center (Kilometers) 10 11 12 13 14 15 16 17 18 The second phenomenon is that, as the production cost of house goes up, house price in the city will go up Such as, in Beijing City, during 2004-2021, as the production cost of house on city periphery went up from 300 dollars per square meter to 5000 dollars per square meter, the average house price in the city went up from 700 dollars per square meter to 8000 dollars per square meter The third phenomenon is that, as the city becomes larger, house price in city center will become higher Such as, as illustrated in the following figure 4, in 31 cities, the larger city often has a higher city center house price (Here, we use the distance between city center and city periphery to represent city’s size, and, we use the relative house price instead of house price to reduce the noise brought by house price level) Figure 4: City Center House Price and City Size City Center House Price 6.00 5.00 4.00 3.00 2.00 1.00 0.00 10 20 30 40 Size of City (Kilometers) We argue, a successful house price model should be able to explain the above three house price phenomena, since the above three house price phenomena are quite 21 Electronic copy available at: https://ssrn.com/abstract=3966052 common in big cities around the world We can find, neither hedonic house price model nor traditional economics’ house price model can explain all the above three important house price phenomena Hedonic house price model can explain the first houses price phenomenon (house on better location has higher price), but cannot explain the second and the third house price phenomenon Traditional economics’ house price model can explain the second house price phenomena (house price will go up as production cost goes up), but cannot explain the first and the third house price phenomenon As mentioned in formula (1), this paper’s house price model is p q c0 q0 where p is one house’s price, q is this house’s quality, q0 is the quality of house on city periphery, c0 is the production cost of house on city periphery We can find that, this paper’s house price model can explain all the above three house price phenomena The above house price model shows that, given q0 and c0 , one house’s price p is positively related with this house’s quality q And, according to the definition of house quality in this paper, house on better location will have higher quality q Then, according to above house price model, in the same city at given time, house on better location will have higher quality q , then will have a higher p ( As already analyzed, in the same city at given time, q0 and c0 are given) Here, we can find that, this paper’s house price model can explain the first house price phenomenon, house on better location will have a higher price The above house price model shows that, given q and q0 , one house’s price p is positively related with c0 (the production cost of house on city periphery)，if the production cost of house on city periphery ( c0 ) goes up, in the city, all house’s price will go up We can find that, this paper’s house price model can explain the second house price phenomenon in real world’s market The above house price model shows that, given q and c0 , house’s price p is 22 Electronic copy available at: https://ssrn.com/abstract=3966052 negatively related with q0 (quality of house on city periphery), if q0 goes down, p will go up When one city becomes larger and larger, compared with house in city center, house on city periphery will have a relatively lower and lower quality ( q0 ).1 Then, according to above house price model, (suppose the production cost of house on city periphery c0 keeps unchanged), as the city becomes larger and larger, house price in city center ( p ) will become higher and higher We can find that, this paper’s house price model can explain the third house price phenomenon, house price in city center will become higher as city becomes larger The second house price phenomenon and the third house price phenomenon in fact are the house price phenomenon during many years, since only during many years, house’s production cost and city size might change dramatically Then, the above analysis in fact implies, this paper’s house price model can explain house price during many years Conclusions This paper developed a house price model, in perfectively competitive house market reached long run competitive equilibrium The house price model is p= q c0 q0 where p is one house’s price, q is this house’s quality, q0 is the quality of house on city periphery, c0 is the production cost of house on city periphery The house price model provides better explanation to real world’s house price than other models, and can be the basic theoretical model for urban house price Reference Adam Smith (1776) 1976 An Inquiry into the Nature and Causes of the Wealth of 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Time Preference and the Valuation of Some Aspects of Environment in the London Metropolitan Region.” Applied Economics 3(4):247-255 Steven C Bourassa, Eva Cantoni and Martin Hoesli 2007.“Spatial Dependence, Housing Submarkets, and House Price Prediction,” The Journal of Real Estate Finance and Economics, 35(2),143-160 Wen Hai-zhen, Jia Sheng-hua, Guo Xiao-yu 2005 “Hedonic price analysis of urban housing: An empirical research on Hangzhou,China.” Journal of Zhejiang University Science (Science in Engineering) 6A(8): 907-914 William, Alonso 1964 Location and Land Use Cambridge MA: Harvard University Press 24 Electronic copy available at: https://ssrn.com/abstract=3966052 ... (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... Hedonic house price model and traditional economics’ house price model considers only one factor, and can be seen as the special situation of formula (1) Hedonic house price model argues house? ??s... zn ) The model is also a hedonic house price model The above analysis means, hedonic house price model is only a special situaiton of this paper’s house price model The special situation is that,