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A game-theoretic approach to the division of profits from economic land development
oml !kicace and IJ
18 (19
The seminal work by AJonso ory of urban land use has in Fujita (6986, 1988)]. The been extended by many aurathem central tool of analysis is the : maximum rent each participant in the market can pay at each location. As a result of the different possible uses of a location, there is a whole family of bid rents for any given location. The equilibrium rent is the ma bid rents, which is geographically ret?resented as the upper en a rent curves. In most literature on land use theo.ry, the normal post perfectly compe?itive land market. Namely, it is assumed t many renters-that no one can influence the land ~rke bg his in&vi for ~YKSQ action. Although ?I sizable contiguo”p~plot orf iand is nmggy &=.Wii.%SGfi production activities, this as t of land use is generaZy _“VU”IIIIY to the mfinitely divisible case iu we want to model a situation in z&k% the are
This paper is part of the author’s Ph.D. d niversity of Pennsylvani Ima&TX. Smith, IL S Foundation under gratefully acknowledgx!. 2/88/%3.500 1988, Elsevier Skien
X Asami, Division of profits from economic bnd dedopwmt
234
developers can extract a certain amount of the economic profit associated with land use.’ It is generaliy the case that the developer has av r&s which could alternative ensembles of land development requirements. In rat words, there is gener andowners with whom the devel collection f each landowner’s parcel the rent negotiations, the locatio over rents. of land may influence his bargaining power. It is also of interest to see the in the land market. resuhs of competition among &Wh~iS ere there are o the land mar&et, and analyzes the p&ii shares which developers and landowners might reasonably expect to obtain. To do so, we focus on a simple land-development situation in which there are m de&OF= a& n landowners Each landowner owns nnp _-_-_e llnit -_I_ nf __ land and esh developer Can earn a positive profit, k, from some economic use of k contiguous units of land (where 1 s k sn). ence, the bargaining process for land can be regarded as a game situ n in which each profitable coalition must include both (at least) one developer and (at least) k contiguous landowners Within this game-theoretic fkameworlc,‘itis appropriate to ask what profit share each participant might reasonably expect to obtain. In view of the it is of particular interest to identify those participants who btain positive shares, and those who cannot. Our approach to oblem is to consider the effectiveness of each participant’s ‘defection in those *profitable coalitions’ to which he may belong. In this context, our main sbiective is to show that for the class of land-development games, the set of rticipants who can effectively demand positive profit shares is precisely e set of ‘essential’ participants, i.e., those who must long to f~~fvymaxiid&
p&&&
-9:i:-, ~GQ c~msea_
sq
i&u&
&y&w=
iandowrz: s depending on the numbers of developers and iandowners and l~~~R_I~zh +f’+ relative locations. More generally, those results that in dOpGS
obtain a significant share of the profits resulting from land -3: if and only if the number of developers is less than or equai to .a ;_9*> .-r--Z__ mm nun3 r of development patterns available, each of which the economic rent available to markets will tend significantly less than total landthese resuits also su st that when obtainable by par& of l&, the the k&ions of their sites. In particular, t zo
p”3 ._c.
owners.Thepaper shows that the developer shares positive our
el.
profit from land
for some land u To deve!op these results,
discussion.
ers to divide a game, there are m develo
and n landowners? t D={&...,i&} and rs and landowners, re landowner owns one umt of land a line segment [Qn). landowners are indexed by the position of their plots of la the jth landowner, called Ij; owns all land in the interval f j - 1,j) for each oining land units ila r &zD requires k units (isksfl) of order to gain a profit of k units (i.e., one unit of profit r unit of !and) f?*om the development of land. It is assumed that this de4 develop one set of k adjoining land units) is the o obtain a positive profit. Each developer di ED must pay a land rent, /Ij, to landowner IjE L, if a, chooses to develop lis land unit. Given any prevailing collection of rents, i)the developer &s associated profit, xi’,is given by i-
(P
1,.
. . , pm),
By assuming that the opportunity cost of land is no landowner !j’s profit, yj, is given by yj=pj =0
if land j is develo otherwise.
‘This analysis is motivated by Binmore’s (1885) t
P&km.
I! Asumi, Divisionof profitsfiom economicknd da&m
236
pas
on of the arran
in C is ‘relatively smafls, then
ent of land units belonging to the landowners in C. of sets of k contiguous land units in C which are
DP={{l,+, ,..., zS+~}cL:s=o,...,n-k}.
(3)
.--2 E $ “&.&ch eicme_t of z%pconsists of Eandow-nersOf k coatiguo__ urn_..of iaurU* includes one developer and one element of DIP, then this coalition e at ieast a profit of k The counting function, #( .), mentioned above is then defined by the maximum number of mutually disjoint elements of DP which are contained in a coalition. More formally, let lc be the icator function defined by lc= 1 if C contains an element of DP, =O
if 85dolss not contain an element of DP.
(4)
by adding one developer to C the coalition. can develop at
ous units of land. To derive the maximum number ces to count the maximum number of partition, c, of C, 1, tion of C into n subsets (which can
vision ofprofitsfro
characteristic functio
237
for anvH
If the number of developers in C (i.e., !C n
number of tic)= NC)
disjoint development se, u(C) is equal to the
This section is devoted to an informal discussion of results reported in the sholuld conslult the 3naenaBilr fnr 3__ +srs*a paper. Interested read_ --_WLY _~~~----~ _-a __ treatment. Consider an example, LDG(1,3,2). In f$is game, cdy cue &v&wr se&a to develop two contiguous land units, while there are three land units available. There are two development patte s possible, i.e., {IL,I,} and {1,,1,}. Given the framework of the game above, a coalition Cc {d,, 1J2,Z3] can make a positive profit by the development only if C includes one developer and at least {l,,l,} or (I, The maximum joint profit of all the players is 2. A coalition which can e this maximal profit is called maxi~%zz!!~ prsfitable coalition.Notice in this example that dl and I2 are contai in every maximally profitable coalition. r words, d1 and & are indispensable players for a coalition to make a aximal profit (of 2). Such players are c essential (in the set of 438 An essential r, being a player neces y Pr coalitions, can reasonably expect a positive shaz of profit in such a situation. Such a positi can effectively threaten
that for any land-develop denote the set of non-negative integers and the set of ively.
e plotit
4, wtik
only
two
YI Asami, Divisionofprojiis from economic land dew
in (ii) and (iii) is of particular interest, since even if
nt [Alonso (1964), Fujita (1986, 1988)], it is an excess demand for 1 (to introduce perfect it is concluded that rent represents the economic profit achievable at each location. In the example above, ot make any positive profit and yet it is. t the rent of land unit 2 is 5 say. The model above, thus,
an introduction of explicit negotiation pCdUi~S thin which it ma
A number of procedures (including both a simultaplayers and a sequential negotiation and each landowner) are possible and
assumptions of this model can be made I varihons in land-development profits
sible gods, such as housing, are anal in Gerber (1985), ax2 (1985). The opcr and lanciownersis the spatid rz!ztioa of (1985). Their models, however, do oat awxn
is assumed that: Cl (normality). u(4) = 0. sitivity). v(H) 4 8. C.3 (monotonicity). VA, The game 6 =(H, v) is designated as a profit-division game Cl-C.3 are satisfied. e class of profit-division games is de it-tuple $=($i: iEH)E is designated as an allocution of the iffg
be the class of profitabilecc) tim able coalitionsof G, res
and the sub&w of
If for any member, i EH, of a given coalitiq r E2H, we desi to leave coalition C as his defiction threat iz C, thy;: the ‘effectiveness’gf the defection threat in the game G is defined as r’oiimw Definition1. For any ie
i is said to be
(i) eflective in (ii) iw,@ctlee in G iff V 05
%e dlwatisn is a weaker con rationality is not
in Mi
1
-
L)e (ii) If E $0, is.,
c
&=~L+~(k[n/kJ/jL+~)=k[n/k]=v(Du L).
LtEDUL 12thskkr, for example, a gme G=(H,u) with H=(l,2) and u(H)=o({l))=S d(2))= _*^..n c && p&&e _ ra$aL& tc ~_~ i’s 8 ?ILmx3reofthisgamtis~pty.Inthisgame,it~ is just a littk larger than phyer 2%.But the admissibk allocation gives rise to (@,O)), @=,.dl profit is &ractcd by player 1. If the core is nonempty, however, a se? of bdly acimmble allocatiollsalway!3cmtains the core. see Theorem 1.
rom (i) and (ii), the core CGis non-empty. Finally, applying
cororrtwy 1.
Q.E.
eorems 1 and 2 to this land-develo
v G =L
, if m=l, then &=A&
Alonso, W, 1964, Location and land use (Harvard University Press, Cambridge, MA). Asami, Y., 1986, Admissible allocations with an application to a land-development game Working Paper Series in Regional Science and Transportation, no. 107 (University 01 Pennsylvania, Philadelphia, PA). Baumol, W.J., J.C. Panzar and R.D. Will@ 1982, Contestable markets and the theory o industrial structure (Harcourt-Brac+Jovanovich, New York). A.E. Roth, ed., &me-theoretic models o cpe\367-1114 0-J --. --*ouma! of CJdm Economics 18,364-378. Fuji&a,M, 1986, Urban land use theory, in: J. Jaskold Gabszewicz J.-F. Thisse, M. Fujita ant theory (Harwood Academic Pub&hers, Chur) 73-149. FUj
__I. a Lo * “_
~
~,j
p~suri~~
u~~~~.sgi
~~~
iamb*agq
RX, 1985, Existence and description of housing market equilibrium, Regional S&ma and Urban Economics 15,383-401. 1953, Some theorems on n-person games, Ph.D. dissertation (Department o Matiatics, Princeton University, Princeton, NJ). Kaneko, M., 1982, The central assignment game and the assignment markets, Journal o Mathematical Economies 10,205-232. Kaneko, M., 1983, Housing markets with indivisibiities, Journal of Urban Economics 13,22-50. Kats, A., Y. Tauman, 1985, Coalition production economies with divisible and indivisible inpuu Asymptotiz results, Journal of Mathematical Economics 14,19-42 Laing, J.D., B. Slot&k, 1986, On transfers of wealth and markets for votes in the provision CI public goods Mimeo. (Department of Decision Sciences, Wharton School, University a Pennsylvania, Philadelphia, PA). Lute, R.D. and H. Raiffa, 1957, Games and decisicns (Wiley, New York). Moulin, H., 1981, Game theory for the social sciences (New York University Press, New York).
Gerber,
18 (19
The seminal work by AJonso ory of urban land use has in Fujita (6986, 1988)]. The been extended by many aurathem central tool of analysis is the : maximum rent each participant in the market can pay at each location. As a result of the different possible uses of a location, there is a whole family of bid rents for any given location. The equilibrium rent is the ma bid rents, which is geographically ret?resented as the upper en a rent curves. In most literature on land use theo.ry, the normal post perfectly compe?itive land market. Namely, it is assumed t many renters-that no one can influence the land ~rke bg his in&vi for ~YKSQ action. Although ?I sizable contiguo”p~plot orf iand is nmggy &=.Wii.%SGfi production activities, this as t of land use is generaZy _“VU”IIIIY to the mfinitely divisible case iu we want to model a situation in z&k% the are
This paper is part of the author’s Ph.D. d niversity of Pennsylvani Ima&TX. Smith, IL S Foundation under gratefully acknowledgx!. 2/88/%3.500 1988, Elsevier Skien
X Asami, Division of profits from economic bnd dedopwmt
234
developers can extract a certain amount of the economic profit associated with land use.’ It is generaliy the case that the developer has av r&s which could alternative ensembles of land development requirements. In rat words, there is gener andowners with whom the devel collection f each landowner’s parcel the rent negotiations, the locatio over rents. of land may influence his bargaining power. It is also of interest to see the in the land market. resuhs of competition among &Wh~iS ere there are o the land mar&et, and analyzes the p&ii shares which developers and landowners might reasonably expect to obtain. To do so, we focus on a simple land-development situation in which there are m de&OF= a& n landowners Each landowner owns nnp _-_-_e llnit -_I_ nf __ land and esh developer Can earn a positive profit, k, from some economic use of k contiguous units of land (where 1 s k sn). ence, the bargaining process for land can be regarded as a game situ n in which each profitable coalition must include both (at least) one developer and (at least) k contiguous landowners Within this game-theoretic fkameworlc,‘itis appropriate to ask what profit share each participant might reasonably expect to obtain. In view of the it is of particular interest to identify those participants who btain positive shares, and those who cannot. Our approach to oblem is to consider the effectiveness of each participant’s ‘defection in those *profitable coalitions’ to which he may belong. In this context, our main sbiective is to show that for the class of land-development games, the set of rticipants who can effectively demand positive profit shares is precisely e set of ‘essential’ participants, i.e., those who must long to f~~fvymaxiid&
p&&&
-9:i:-, ~GQ c~msea_
sq
i&u&
&y&w=
iandowrz: s depending on the numbers of developers and iandowners and l~~~R_I~zh +f’+ relative locations. More generally, those results that in dOpGS
obtain a significant share of the profits resulting from land -3: if and only if the number of developers is less than or equai to .a ;_9*> .-r--Z__ mm nun3 r of development patterns available, each of which the economic rent available to markets will tend significantly less than total landthese resuits also su st that when obtainable by par& of l&, the the k&ions of their sites. In particular, t zo
p”3 ._c.
owners.Thepaper shows that the developer shares positive our
el.
profit from land
for some land u To deve!op these results,
discussion.
ers to divide a game, there are m develo
and n landowners? t D={&...,i&} and rs and landowners, re landowner owns one umt of land a line segment [Qn). landowners are indexed by the position of their plots of la the jth landowner, called Ij; owns all land in the interval f j - 1,j) for each oining land units ila r &zD requires k units (isksfl) of order to gain a profit of k units (i.e., one unit of profit r unit of !and) f?*om the development of land. It is assumed that this de4 develop one set of k adjoining land units) is the o obtain a positive profit. Each developer di ED must pay a land rent, /Ij, to landowner IjE L, if a, chooses to develop lis land unit. Given any prevailing collection of rents, i)the developer &s associated profit, xi’,is given by i-
(P
1,.
. . , pm),
By assuming that the opportunity cost of land is no landowner !j’s profit, yj, is given by yj=pj =0
if land j is develo otherwise.
‘This analysis is motivated by Binmore’s (1885) t
P&km.
I! Asumi, Divisionof profitsfiom economicknd da&m
236
pas
on of the arran
in C is ‘relatively smafls, then
ent of land units belonging to the landowners in C. of sets of k contiguous land units in C which are
DP={{l,+, ,..., zS+~}cL:s=o,...,n-k}.
(3)
.--2 E $ “&.&ch eicme_t of z%pconsists of Eandow-nersOf k coatiguo__ urn_..of iaurU* includes one developer and one element of DIP, then this coalition e at ieast a profit of k The counting function, #( .), mentioned above is then defined by the maximum number of mutually disjoint elements of DP which are contained in a coalition. More formally, let lc be the icator function defined by lc= 1 if C contains an element of DP, =O
if 85dolss not contain an element of DP.
(4)
by adding one developer to C the coalition. can develop at
ous units of land. To derive the maximum number ces to count the maximum number of partition, c, of C, 1, tion of C into n subsets (which can
vision ofprofitsfro
characteristic functio
237
for anvH
If the number of developers in C (i.e., !C n
number of tic)= NC)
disjoint development se, u(C) is equal to the
This section is devoted to an informal discussion of results reported in the sholuld conslult the 3naenaBilr fnr 3__ +srs*a paper. Interested read_ --_WLY _~~~----~ _-a __ treatment. Consider an example, LDG(1,3,2). In f$is game, cdy cue &v&wr se&a to develop two contiguous land units, while there are three land units available. There are two development patte s possible, i.e., {IL,I,} and {1,,1,}. Given the framework of the game above, a coalition Cc {d,, 1J2,Z3] can make a positive profit by the development only if C includes one developer and at least {l,,l,} or (I, The maximum joint profit of all the players is 2. A coalition which can e this maximal profit is called maxi~%zz!!~ prsfitable coalition.Notice in this example that dl and I2 are contai in every maximally profitable coalition. r words, d1 and & are indispensable players for a coalition to make a aximal profit (of 2). Such players are c essential (in the set of 438 An essential r, being a player neces y Pr coalitions, can reasonably expect a positive shaz of profit in such a situation. Such a positi can effectively threaten
that for any land-develop denote the set of non-negative integers and the set of ively.
e plotit
4, wtik
only
two
YI Asami, Divisionofprojiis from economic land dew
in (ii) and (iii) is of particular interest, since even if
nt [Alonso (1964), Fujita (1986, 1988)], it is an excess demand for 1 (to introduce perfect it is concluded that rent represents the economic profit achievable at each location. In the example above, ot make any positive profit and yet it is. t the rent of land unit 2 is 5 say. The model above, thus,
an introduction of explicit negotiation pCdUi~S thin which it ma
A number of procedures (including both a simultaplayers and a sequential negotiation and each landowner) are possible and
assumptions of this model can be made I varihons in land-development profits
sible gods, such as housing, are anal in Gerber (1985), ax2 (1985). The opcr and lanciownersis the spatid rz!ztioa of (1985). Their models, however, do oat awxn
is assumed that: Cl (normality). u(4) = 0. sitivity). v(H) 4 8. C.3 (monotonicity). VA, The game 6 =(H, v) is designated as a profit-division game Cl-C.3 are satisfied. e class of profit-division games is de it-tuple $=($i: iEH)E is designated as an allocution of the iffg
be the class of profitabilecc) tim able coalitionsof G, res
and the sub&w of
If for any member, i EH, of a given coalitiq r E2H, we desi to leave coalition C as his defiction threat iz C, thy;: the ‘effectiveness’gf the defection threat in the game G is defined as r’oiimw Definition1. For any ie
i is said to be
(i) eflective in (ii) iw,@ctlee in G iff V 05
%e dlwatisn is a weaker con rationality is not
in Mi
1
-
L)e (ii) If E $0, is.,
c
&=~L+~(k[n/kJ/jL+~)=k[n/k]=v(Du L).
LtEDUL 12thskkr, for example, a gme G=(H,u) with H=(l,2) and u(H)=o({l))=S d(2))= _*^..n c && p&&e _ ra$aL& tc ~_~ i’s 8 ?ILmx3reofthisgamtis~pty.Inthisgame,it~ is just a littk larger than phyer 2%.But the admissibk allocation gives rise to (@,O)), @=,.dl profit is &ractcd by player 1. If the core is nonempty, however, a se? of bdly acimmble allocatiollsalway!3cmtains the core. see Theorem 1.
rom (i) and (ii), the core CGis non-empty. Finally, applying
cororrtwy 1.
Q.E.
eorems 1 and 2 to this land-develo
v G =L
, if m=l, then &=A&
Alonso, W, 1964, Location and land use (Harvard University Press, Cambridge, MA). Asami, Y., 1986, Admissible allocations with an application to a land-development game Working Paper Series in Regional Science and Transportation, no. 107 (University 01 Pennsylvania, Philadelphia, PA). Baumol, W.J., J.C. Panzar and R.D. Will@ 1982, Contestable markets and the theory o industrial structure (Harcourt-Brac+Jovanovich, New York). A.E. Roth, ed., &me-theoretic models o cpe\367-1114 0-J --. --*ouma! of CJdm Economics 18,364-378. Fuji&a,M, 1986, Urban land use theory, in: J. Jaskold Gabszewicz J.-F. Thisse, M. Fujita ant theory (Harwood Academic Pub&hers, Chur) 73-149. FUj
__I. a Lo * “_
~
~,j
p~suri~~
u~~~~.sgi
~~~
iamb*agq
RX, 1985, Existence and description of housing market equilibrium, Regional S&ma and Urban Economics 15,383-401. 1953, Some theorems on n-person games, Ph.D. dissertation (Department o Matiatics, Princeton University, Princeton, NJ). Kaneko, M., 1982, The central assignment game and the assignment markets, Journal o Mathematical Economies 10,205-232. Kaneko, M., 1983, Housing markets with indivisibiities, Journal of Urban Economics 13,22-50. Kats, A., Y. Tauman, 1985, Coalition production economies with divisible and indivisible inpuu Asymptotiz results, Journal of Mathematical Economics 14,19-42 Laing, J.D., B. Slot&k, 1986, On transfers of wealth and markets for votes in the provision CI public goods Mimeo. (Department of Decision Sciences, Wharton School, University a Pennsylvania, Philadelphia, PA). Lute, R.D. and H. Raiffa, 1957, Games and decisicns (Wiley, New York). Moulin, H., 1981, Game theory for the social sciences (New York University Press, New York).
Gerber,