Deterministic and Stochastic Models in Evaluation of Land Value

Copyright © IFAC Modelling and Control of National Economies, Beijing, PRC, 1992

DETERMINISTIC AND STOCHASTIC MODELS IN EV ALVATION OF LAND V ALVEI Baoqun Yin, Xiaoxian Yang and Qiugui Chen Department of Mathematics, ChiNJ University of Science and Technology, Hefei, 230026, PRC

The purpose of this paper is to study, as simply as pOSSible, the problems of optimal maintain investment programme of a building on the land and evaluation of land value. First, we assume that the dynamic model for the change of building quality is deterministic bilinear model. By using the maximum principle introduced by rontrajgin, we discuss these problems. ~econdly, we assume that it is Ito stochastic differential equation, and by using the method of stochastic optimal control, we lead optimal maintain investment programme for general quality effectual function of the building on the land, when the house-owner obtains the greatest expectant profit in one maintain period, under the constant continuous discount rate.

~bstract.

Keywords. Management system; optimal control; stochastic control; evaluation of land value; optimal maintain investment programme.

INTRuDUCTIUN

all results have clearly nothing to do with building quality. Therefore, we assume the dynamic model satisfied by building quality ~ x( t) is a bilinear one:

In the management system of civic economy, it naturely appears the transactions of conveyance, mortyage, rent, sale and so on, of the house on the land. Consequentl~ how to do scientific evaluation for land value looks more and more important. Generally speaking, what is called land value is computed out with mean profit that the house-owner obtains, after constructing all kinds of building on the land.

x(t ) =- et x ( t ) + x ( t ) u ( t ) , then we obtain the results in keeping with reality. Yang\1990)introduced a fixed tiny economic management model of regular, uncontinuous maintain investment, under the constant continuous discount rate, not only optimal maintain investment programme was obtained, but problem of evaluation of land val- . ue was solved. This paper discusses the tiny stochastic economic management model of evaluation of land value. ~ince the model introduced by Yang(1990)is actually the first order movement equation of the stochastic model given by this paper, we can naturely derive same result.

results(Yuji, etc.,19b2; Yang,1990) of evaluation for land value were given from assumption that the dynamic model for the change of building quality is a linear model of exponential attenuation:

~arly

i \ t ) =- C1X ( t ) TU ( t ) . But the drawback of linear model is that

l~roject supported by the National Natural

DETEHMINI~TIC

~cience Foundation of China.

51

BILIN~AR

MUDEL

Let x(t) be quality of a building on the

(

land at time t. and 0. be the rate of exponential attenuation with time t. In the time interval (t. t+tJ. the house-owner of the building does regular inspection and maintain to preserve original quality of

~(t) =- '}H R -Pt -J-lt>[-o.tU(tl} ox =- (t)e =-k( t) e- Pt +}\( t

)[a.-~g( m.) ~(t-k1:)

.=1

l\.(T)=O

(6)

the building or to add certain modernization equipments.

~uppose

Then. the solution of (6) is obtained

quality Xlt) is

an attenuational model: (1 ) where

N

exp[-(Pto.)S+ L:: g(m,,)JS~(V-ln:)clvJtcls ( 7)

(2 )

1<"1

()

Beside s . we have

Ci>O> mj(>O, X(O)=X ••

itt)=-ax(tJ+x(tJU(t)

forthermore. supposing that after renting his house on the land. the profit that

{

the house-owner gains in the time inter-

x(O)=xo

(8)

val [0. TJ is and it has a solution:

where H(t)x(t) is rent. exp(-pt) is the decreasing weight function. and

The optimal maintain investment programme

P>O is

mt should be satisfied by the following condition:

the discount rate. it shows that his income expressed by the function rt\t)x(t)N

2::mk~(t-k~)

is not isoplethic on the time

~~1

T is a time period of rent; N=[T/~l. here [.] is maximum integer to unsurpass 1>. 1>;

or

~

is a maintain investment interval. mk is investment fund of the house-owner at kth time maintain or adding equipments; g(m~) is quality effectual function in kth time

Integrating o n both sides of (10) with

maintain. and assume it satisfies the fo-

respect to t

from 0 to T. we obtain

llowing conditions: ( 11 ) g

=.

( 0)· 0

dg-(m) ... 0

dm

"7

,

Where A(k1:) and x\k~). k=1 .2 •••.• N. may

Now let's try to find optimal maintain in-

be computed from (7) and (9). (11) is the

vestment programme m!. k=1 .2 •••.• N. when

equation satisfied by the optimal mainta-

the house-owner obtains maximum profit.

in investment programme m:. Rccording to (11 J . the corresponding optimal ma i ntain

According to the

~ontrajgin's

investment pro g ramme m~ can be computed.

maximum pri-

nciple(Suresh. and Qorald, 1981 ) . we have

For example. let

g(m~)=~~ • R(t)=RQe~t.

where H. is a constant. then we have

1']:>0 and o..-t-f':>'l •

h \ t ) [ - el.X ( t l+x ( t ) u ( t )] N

where u(t)=~g(m~)£(t-k.). The adjoint k=l

(12 )

equation is We also obtain by the integral mean

52

J~

theorem

=

max T•• ltT.·1IIj, ... (,;l• •• .

Where C(X) is expenses of the building when construction quality is x. and V is land value when t=o.

Again making use of (11), we have

k=1 ,2, ••• ,N

If J' is positive. the house-owner will not generally give up the land; if J- is negative. generally no people will buy the land. 80 in the market of so-called complete compete, usually only when J'=O, normal transactions will be done.

( 14)

After arrangement we obtain

k=1 ,2, ••• ,N

Under the normal transactions. that is. when J =0. the evaluation of land value V should be

( 1 5)

( 1 5) is a precise result. Where at exists objectivly, but it can't be computed. In order to save investment fund, we assume ~=O, that is, we replace mk by ;;

2

mt= [c~exp(~.afrT\)J • k=1,2, ...• N

e- pt clt - C. ('lCT.) efT, }

Assume C(x)=cx. then C(x~)=cx~, where c is a constant. Be s ides. x(t) must satisfy the following condition:

(16)

where

Ht {

k=1 ,2 ••••• N

( 17)

T~~QO.

) =- c;(Xlt)

x(To)=xT.'

l'

of:.. f3Triif S(t-T.-h)XCtl, 1<=1

'i.'.,,-t<+oo

(22)

its solution is

Clearly Ck~O. ~o long as we properly choose ~. we may successi 'fely compute m",. k=1. 2 •••.• N. from (16). As

(21 )

8uppose R(t)=koexp(~t), where R. is a constant, 1'» and then have

°

the limit form of (15) is

(24) and and

k=1.2 ..... N

( 19)

Owing to the investment programme mf: m~x According to the previous results. the house-owner obtains maximum profit with the increase of time. At this time, there exists optimal maintain time peri od and

(mk)=m', and m~ exists, the series is convergence in (22 ) . we have

T:

optimal building quality x~ (that is building quality when the house-owner obtains maximum profit). Then he carries on optimal maintain investment programme m;. Consequently, the total profit is

8ubstituting \24) and \25) into (21), elds

53

yi-

(27 )

Making use of the integral mean theorem, the optimal maintain period is obtained T*"=_1_[1 C(cHP-!J)_G ]

011

n

Now we assume that the dynamic model of building quality x(t) which is a stochastic process defined in the probability space ~, F, ~ ) . is Ito differential equation

(28)

r.

R.

::>TOCHaSTIC MODEL

00

where o
+00.

lo a 1

::iince the

life of the building is always finite, and maintain frequency can't be infinite, 00

~~,rm;

"-1 using

dx ( t

is a finite value. To prolong the

period as far as possible, we may take 8~= O , at this time, have maximum period T' __1_

'-l'J

1 C(c;(.-tP-!J) 11 R.,

)=- Cl x l t ) eft t

1l (t )dt +
(35)

where

Here the goal introducing the $ -function is to continue discrete maintain. l'[XCOl=J.J= 1; the mean of initial quality is defined

(29)

and

as E(xo)=x o . Assume that

Therefore, we can obtain

a BTOwn movement process, and Etdw)=o,

Substituting (28) and (32) into (27) yields

E[c1w(t)c1W(T)l= S (t-l:); dw(t) and x. is mutually indepedent; 0 is a constant, that is strengh of interference.

where 0'::

er•.:: M••

R.

'

is

Supposing that after renting his house on the land, the mean profit that the house-owner gains in the time interval [0, TJ is

Let Br.=O, we obtain

rrIl= _1_ 1 . "((HP-I) 10 ,,'"

w(t)

tt~O)

The detailed calculation of t30) may look up appendix . Then we have

('34)

where H(t)x(t) is rent, 7"" 0 is the constant continuous discount rate; T is a period of rent; N=[T/r]; ~ is a maintain investment interval; lilt? is investment

°

54

fund of the house-owner at kth time maintain or adding equipments; g(m~) is quality effectual function in kth time maintain, and assume it satisfies the following conditions:

(4), so we have from (9)

k=l,2, ••• ,N

g (O)=O·, when

m:>

0

,

dg(m) 0 elm ~O, dgtrn) dm~ <

As T--.too, stationary optimal maintain investment programme is obtained

Now let's try to find optimal maintain investment programme mk , k=1 ,2, .•. ,N, when the house-owner obtains maximum profit. At this time, (35) has a solution(~ksendal,

m'

~

HO¥ever, with the development of cities and the increase of population, modernization and public equipments must be filled up, so the rent increases with time, and we can take the rate of rent as

[...BfulLJ2.

(46)

(i"tr-~

above determinstic bilinear model, we have similar results for evaluation of land value. Since there exists optimal maintain period while optimal building quali ty is E(xi.l (that is building quality when the house-owner obtains maximum mean profit), he carries on optimal maintain investment programme m~. The total mean profi t is

AS

T!,

(40) ~

=1. .,.

It's economic significance lies in that after deciding maintain time interval only according to rent coefficient, the house-owner can decide optimal investment fund of kth time maintain or adding equipments without considering the renting time.

1985) :

where Ro is a proportion constant, the rate of exponential growing.

(45)

(38)

is

Under the assumption of (40), substituting (39) into (37), if also naming

•

m~x:

J =T.. XT•• ~

{E J1'T.OO [R.(t)JC{t)-~m..b(t-T.-kr)]ertdt 00

":1.2.··· -E[C(Xr.l]e-l"r. }-ECv)

Where E[c(x)]=c(Etx)] is mean expenses of the building when construction quality is x, and E(v) is mean land value when t=o.

since the first term in the right of t41) only has something to do with initial condition xO ' while each term in the second is mks function, we have

_mICe-O:t} t R..2<..- ['- e-(c.lTt-~) 1J CI--+"Y-1')

(47)

Generally speaking, only when J*=O, normal transactions will be done. Atthis time_, evaluation of land value E(v) should be

(42) (48)

then, this yields

When quality effectual function g(m)=prm ' we have from (46) For example, specificly taking quality effectual function as g(m k

)=,8.rrt\ '

mf=£[R(T.+"d]2. ...

4-

(49)

a+Y-1

Thus, this gives k=l ,2, •.• ,N

(44)

ECv)= max {EJ+ooRLt!X(t1e-i'tat- pl ::;~~..

where ,8 is a positive proportion constant. Apparently, (44) satisfies the condition

55

T.

4-

f1;=1 [R.tT.-tktj)~ l"'_~)a (ct1'

~ he

Besides, we have

economic significance of the above

results lies in that the science basis of value should be put forward when the house-owner does the transactions of conveyance, mortyage, rent, sale and so on, of the house on the land. REFERENCES where supposing that l'"- 2~::>- O. ;3ubsti tuting (51) into (50), yields

E( v)= l"flcl.)( T..XT.

value with optimal control theory. Transactions of the ;3ociety of Instru-

{R(To) E.(xT.)e- rr• ;- 2.. [~]2. o.-t- r-~ 4 cH r-Yj

e-(r-2~)t

[

YUji,I., etc.,(,982).Bvaluation of land

1-e-(r-l~n

J -TT. e

-(1:

1

-C[E():T.)]e· j

ment & Control Engineers, Vol. le, No. 3, 306-308. Yang,X,X,',99 ) . Maximum principle appli-

(52)

cation in evaluation of land value. Journal of China University of ~cience

Taking the first partial derivatives of

{.1

and Technology, Vol. 2 J , No.4, 41 7 --

with respect to T and E(x T.) in (52),

422.

we obtain that optimal mean building qua-

~uresh,~,S"

lity should be satisfied by the following

and L.T.Gorala(1981). Optimal

Control Theory Applications to Manage-

necessary condition:

ment dCience, Boston, Martinus Nijhoff Publishing. ~ksendal.B.(1985).

~tochastic

Differential

Equation Introduction with Applications,

~rinted

in Germany, 38-45.

A.PPBNDIX If the solutions of (53) and ,54) are T~, E(x~

The third term in the right-hand side of

), respectively, land value should be

(30) is obtained from the first partial dervative of F with respect to T, where (55)

And further, assume C(x)=cx, where c is a constant. Note 1];>-0, R,t)=R.e1Jt , according to (54), optimal maintain period is obtained

T" = 1-1 o YJ TI

r c.(~t-r-I'jl ] l

R,

(56)

Substituting (56) into (53), optimal mean quality of the building is obtained

(57)

At last, land value should be

(58)

CONCLU;3ION

56