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, 20 f() = , NE, p2(1 - 1/(2m))2, p2/4., (i/yi = 0), (j/yi < 0, i = j). /yi = j/yi < 0.

j,,.

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" ",.,,,,.

, ,.

., " ",., (, ).

,.

3.2, :

ui = ui(x, y) (i I), fj = fj(y, x) 0 (j J).

- :

ui (x,y) max (32) (x,y) "congestion tendency" - "" f(.).

ui(x, y) i (I i = i0), fj(y, x) 0 (j J), (33) k k (xk - wi ) yj (k K) (34) i i j,, :

4 (p, x, y) () (32)(34),,.

k yj,.. ui,y(x, y) 0 (i I), f k (x, y) 0 (r = j), k r,yj j, (ui,yk(x, y) = 0 (i I, k = k), f k(x, y) = 0 (r, s J : s = j, k K)) r,ys j (x, ) - (x, y) : u(x, y) = u(x, ),, (x, y) k j, yj < j.

k k k yj (),,,.

xk i.

, (x int(X)), : - (x, ).

,, :.

-.

,, x , :,., : ().

3.2 ( ).,, u1 = u1(x1, x2) u2 = u2(x1, x2), 1 1 2 (x1, x1 ) 1 (x2). , u1(x1, x2)/ 1 2 x2 < 0,,, u1(x1, x2)/x2 > 0, 1 1 2 40 u1(x1, x2)/x2 = 0, x2 40. wi = 20.

1 2 1 2 : (A) : w1 = 0, w2 = 40, (B).

( ),. (), (, -) " ": -. A B,,, - (..2 ).). A.

B,, (..2 ).) x2 x1 1 wB wB x1 x1 x1 xwA wA 2 1 2 ) ). 2:

. -,,,.

B, A .

-,,,, " ".

,,,.

3.3 ( : ) : 3, 1 ( ) 2 : 1- 2-. j = 1, 2 j-, aj yj = gj(aj, y-j)(, y1 = g1(a1, y2), (-y1) = a1, y2,,, ) j(a, y) := pjyj -p3aj. x1, x2, x3 u(x1, x2, x3) x3.

,, w3 a1 + a2 + x3, (w1, w2) = 0, :

0 x, 0 x3 w3, p1x1 + p2x2 + p3x3 p3w3 + 1 + 2. (35),.235.

- (p,x,,y) x 0, a, x3 < w3 ( x, a ).

. L(x, x3, ) := ui(x1, x2, x3) - (px + p3x3 - p3w3 - 1 - 2).

,, : ( uk := u(x)/xk, u3 := u(x)/x3):

uk(x)/ur(x) = pk/pr (k, r = 1, 2, 3). (36) (), :

ak 1/k (y) = pk/p3 = uk/u3 (k = 1, 2). (37),. -,.. :

ui(x1, x2, x3) max (38) x,a,y xk yk = gk(ak, yk) (k = 1, 2), (39) a1 + a2 + x3 w3, 0 x, 0 a. (40), (p, x,, ) (0 (x, a)),,,,, - :

ak ar ar uk(x, x3)/u3(x, x3) = 1/k (k, ) - r (r, )/r (r, ) ((k, r) = (1, 2), (k, r) = (2, 1)). (41) :

yk 1) : r (r, k) = 0, (41) (x,, ) (36),(37) (x,, y) ( ), x = x.

(37) (41)).

- 1, .

2),,,, : x = x.

"",, (x, y, ) y1 y2 (2, y1) > 0, : 1 (1, y2) = 0 (,, ).

, u(.).,,, :

da1 > 0, da2 = 0, dx3 = -da1 < 0. (40).

a 1 dy1 = 1 (1, y2)da1 > 0.

,, y1 ady2 = 2 (2, y1)1 (1, y2)da1 > 0.

( ) u, du = u1(x)dy1 + u2(x)dy2 + u3(x)dx3 = (42) ya1 a(u11 + u22 1 - u3)da1 > 0. (43), (37), :

da1,., (x, ) -.

"".

: -,..

y1 < 1,,.

3.4 (. 234). 2, : , .

Y := {(y1, y2) I y1 + y2 1}, R+|. ui = ui(x1, x2, x2 ) (i = 1, 2),, i i -i, ( ) x2 ) (i = 1, 2),. -i, (1/2).,, : ,, ( )., ., - x,,, x, ( 2).

, :

, (, ) -.

3.3,.

I., :

1. " " () x., x, (, ).

2. " " p,,,.,,. (, ).

- 1 2.,.,.

3. ().

1 2,, ",,. 3,,, (, );,..

3.5 ( 3.3) d1, d2 1 2.,,, dk k a. ( a1 := y1):

(p1 + d1)g1(a1, y2) - p3a1 max a1 0.

a, (36), a (p1 + d1)/p3 = 1/1(, ) = u1(x)/u3(x),., (41),,, (x,, ) (x,, ), ( p3 = 1) :

yd1/p3 := 2 (, )/2a (, ),., : p = p,,, (p, x, ).,,.

, :,,. ( ).

II.

.

,.,,.,, " " ""., "",, -,.

3.3.1 ("" ). 1) (.. ),. 2) ( ) ( ),,.

-, ( ). .

, :

1..

2. (, ).

"".

3.6 ( 3.3) 3.3.2 3.3 -.

. 1- :,.

R.Coase,, 1, y1 p2. y2, p1. a ( a1 := y1):

2 (p1 + p1)y1 - p2y1 - p3a1 max y1 = g1(a1, y1), 0 < a1.

2 y1,a1,y, : 2 = 1.,, y p1 = d1 = 2 /2a p3.,, 2,.

2,.,, 1-, :,.

3.7 ( 3.2) -,. " ": A B - (..2 )).

,,. (3.2),,,,, B,, A,,..

,,.

,,,,,,.,.

4.1 4.1.1, ; xk (i-), i k k ( yj + wk ) j k xk yj + wk.

i j,,. (. non-rivalness).

:,, -,.. - ;

, -.

,. (, ),. k i- xk i( yj + wk, xk ), xk i j -i -i ( i), i . k k i(.) = yj + wk i(.) = yj + wk - xk. j j j=i j.

4.1.2 k, I, xk i k : xk = yj + wk.i j. (non-excludability),,.,,,, (,, ), .

(, ), ( ).,,,.24 ( ) ,.

.,.

, ( ) ( ) ;, -., /,,.,, ( ).

( : )...

Kpriv, Kpub .

,,.,, k xk = yj (k Kpub) ( i j )., - (x, ):

(x, ) P [ I u(x) = max u(x) : (44) x,y ui(xi) ui(xi), i I \ {} ; (45) fj(yj) 0, j J ; (46) k k xk yj + wi, k Kpriv ; (47) i iI j iI k xk = yj, k Kpub, i I ]. (48) i j,.

-, -,,.,, ui.

( := 1):

k k L(x, y,, , ) := iui(.) + jfj(.) + k( yj + wi - xk) + i i j kKpriv j i i k + k( yj - xk). (49) kKpub j (, ) (, ),. :

1 2 1 uk (x, )/uk (x, ) = fk ()/fk () (j J, k1 Kpub, k2 Kpriv) (50) i i j j i 1 2 1 uk (x, )/uk (x, ) = fk ()/fk () (i I, j J, k1, k2 Kpriv) (51) i i j j 25.,.

Samuelson, P.A. (1954) "The Pure Theory of Public Expenditure," Review of Economics and Statistics, 350-356.

, (),, f(y) = g(y2,..., yl) - y1,. (44) :

(x, ) P [ I u(x) = max u(x) : (52) x,y ui(xi) ui(xi), i I \ {} xi 0, x1 = y1 g(y2,..., yl), i I (53) i xk yk, k = 1 ]. (54) i i 4.2 () (52),, q = (qi)iI I Rn, : qi = p1. i x1.

i l { ui(x1, x2,..., xl) max | xi 0, qix1 + pkxk i. } (55) i i i i i xi k={ py max | y1 g(y2,..., yl) }. (56) y 4.2.1 () 26, (53) (54), (x, y,p, q), l (qix1 + pkxk) = p, qi = p1, (x, y) i i k=2 i i (56) (55) (p, q).

: () : y = x1 (i).

i .

4.2.1 (52) u(.) g(.) ( ), [u1(xi) > 0 k Kpriv : uk(xi) > 0 i I, xi 0], grad(g()) = 0. :

i i (I) ( x 0) -.

(II) - (x, ) (p, q) (1,..., n), (x,, p, q) .

.Lindahl, E. (1919) "Positive Losung, Die Gerechtigkeit der Besteuerung".

(, ;. 1 2). 1 2);

,,, (x1,..., x1 ) i n ,.

(I), ( ) -,.. (x, y, p, q) :

u1(x)/uk(x) = qi/pk, p1/pk = -1/k(y). (57) i i,, (55) (56). i, qk = p1, k.

,,, 1 2 1 2 1 uk (y1, x)/uk (y1, x) = pk /pk = k (y)/k (y). (58) i i,,, , (49), (x, y), (52)., (x, y) (52), -.

(II),.., (57) (58)., p :=, (52), (49), qi := p1(u1/uk)/( u1/uk) (59) i i j j jI qi = p1, k ,., II, l i := qix1 + pkxk, i k=2 i, ( (59)).,,, x, (55) (56). () x, -,, (x,, p, q) .

[[[]]]],,,. (, tatonnement,.)., "" ;,.,., - -,.,,,.

1. t (p(t), q(t)).,, xk(t) (k = 2, l) x1(t).

i i > 0, :

pk(t)/ t = ( xk(t) - yk(t)) (k = 2,..., l), i i qi(t)/ t = (x1(t) - y1(t)), p1(t) := qi(t).

i i,,,.

;, (, qi/p1 xi ),., x1(t) i,,, ( m).

, (. free-rider effect)., -,,.

4.3 ( ), :,.,. - " ".,,.

i- ti 0;

i.

: p1y1 = ti.

i (,, u1 > 0), i.,,, t-i, ti.

:

ui(x1, x2,..., xl) max, (60) i i i ti,xi l (ti, xi) 0, ti + pkxk i, x1 = y1 = (t1 + tj)/p1, i i k=2 j=i 4.3.1,, (p, t, x, y),, (i) (56), (60), (ii), x1 = y1 (i), (iii) l i ( ) pk(yk - xk) = 0.

k=2 i i,, (t, x, y) ( t = 0, x 0) -. t > 0, x1 = y :

L/t = u1((t1 + tj)/p1, x)/p1 - = 0, (61) j=i L/xk = uk(y1, x) - pk = 0 (k = 2, l). (62) .

, (xi 0, ti = 0) ( i 0 ti 0) :

L/ti = u1((t1 + tj)/p1, x)/p1 - + i = 0, (63) i j=i L/xk = uk(y1, x) - pk = 0 (k = 2, l). (64) i i. :

u1(y1, x)/uk(y1, x) = p1/pk = -1/k(y) (i : ti > 0). (65) i i (58):

. (51) x, (x, y) = (x, ) : 1) m = 1, 2) ( u1(y1, xi) = 0 (i = 1)), 3) i,.

, :

4.3.1 (52),, (p, t, x, y), : (y1, x) 0,, u1(y1, xi) 0 (i I), i ti > 0, u1(y1, xi) > 0 i = i1, i = i2. (I) i -, (II) (),, -.

. (I) (. ), ti > 0 (, ). (II) :

1 dy1 > 0 dxk = dyk = -dy1u1 /uk < 0., (x, ) i1 1 i1 i,, dui = 0, i2.

[[[]]]], -,, " ti = 0.

-,., ui(y1, x2) := min{a1y1; a2vi(x2,..., xl)}, (66) i i i i i a1 0, a2 0 ,, i i, vi = vi(x2,..., xl) i i i. ( 1982), (52) (66), -.

, : - . 4- "",, 13. 5-,.

, n 1$, 4n$, (t1,..., tn) = 0.,, :, : xk = 0, k = 2,..., l, i I; i.

, -. "" " ".

,, :,,.

4.1 ( ) 3, k = 2 (, ), ui = i ln(y1) + x2 (i = 1, 2, 3),28 i i. i 1: y1 = g(y2) = -y2, 2 (x2 - wi ) = y2., i i (, ),, 1.

ti, y1 = ti.

i ui = i ln(t1 + t2 + t3) + wi - ti ti 0. i/y1 1, (1 - i/y1)ti = 0.,,,.

i/y1 = 1.,,..

y1 = t3 = 3, t1 = t2 = 0., - y1 = 6,.

4.4, ,.,,,.

.,,,,,.

i- ip1y1. i 0 , i = 1. i zi Zi ( Zi, Zi = I, zi := x1, ) R+, i - G(z1,..., zm) ( ), y1 = G(z).

, :

V1) : G = zi/m, i V2) : G = mini zi, V3) : G = maxi zi, V4) : G = med(z1,..., zm), med(.) z1,..., zm, m ., (".),.

(V2)(V4) (.. ),, - G(z)/zi = 0, zi int(Zi), zj ! () zj, "" "" ;.

,, z-i zi ( ). ( (56) ) ui(x1, x2,..., xl) max, (67) i (zi,xi) xi 0, zi Zi (68) l p1ix1 + pkxk i i i k=x1 = G(z1,..., zm).

i 4.4.1 G(.) (z, x, y.p),, (i) (56), (68), (ii), x1 = y1 (i I), (iii) " " ( ), (iv) i : y1 = G(z).

4.4.1 (z, x, y.p) i ( U G(zi, -i)/zi = 0) (4.2.1), (x, y, p, q) qi = ip1,, -.

,.

[[[]]]],, x1 = y1 (i I) .

i, -,,,. V1, V2 V ( ). "", (), (. (4.1) ).

(4.4.1), " " ( ) ( ). "" ( ), ( ).,,.

(, ),,.

,. :

2.

1) p(t) ( t) i(t).

,,.

2) xk (k = 2, l), zi i. x1 = G(z) : x1 = zi/m. i yj.

3) pk(t), k k : pk(t + 1) := pk(t) + (xk - wi ) - yj (k = 1,..., l, > 0) ( i i j ) () :

m m i(t + 1) = [i(t)i(t)/ j(t)j(t)], i := mzi/ zj 0. (69) j=1 j= 1)..

,,,.

,,.,.

4.1 (), V1...

y1 = (z1+z2+z3)/3. ui = i ln((z1+z2+z3)/3)+wi -i(z1+ z2 + z3)/3 zi 0. i/y1 i, (i - i/y1)zi = 0.,,,,.,, : i = 1/3., y1 = 9,. :

zi = 0.

u1/u2 = i/y1. i i ( -).

,, = (1/6, 1/3, 1/2).

V1 zi 0,, V3.

, ( ) med., : med = i/m, i.

4.5, ( ) () -. : 30 (Gibbard, Saterthwait).

Arrow, K.J. (1951) Social Choice and Individual Values..

.

,, ui y1., (y1, t1, t2, t3) : ( ).

,,,, (),, 1 Y (, Y, I -.

R+).

4.5.1 l, ui = i(y1, x2,..., xl-1) + xl, xl i i i i xl = xl 31.

i i, :

4.5.1 l, - ui.,, l = i ( ) i, 1 ( ).

(52) ( ) ui g. (49) ui. xl i i : L/xl = i - l = 0., i ;, i i = 1 (i I)., - :

l k k L(x, y, , ) := ui(.) + (g(.) - y1) + k( yj + wi - xk), (70) i i k=2 j i i,.

l 1(x, ) = -1/l() (71) i i.

.

-.

1) ci(y1) c(y1),,.

c(y1)i = c(y1),, (, ) i i.

2) vi(y1) = i(y1) - ci(y1).

3), :

y1 := G(v) := arg max vi(y1), (72) yi, i- ;

y(i) := G(i)(v) := arg max vj(y1), i I.

yj=i 4), :

i = (vj(y(i)) - vj(y1)), i I, j=i,, (y(i) = y1).

5) vi(y1) - i = i(y1) ci(y1) - i., ().

4.5.2 vi, (I) - (y1 = (1), (i = 0, i I),,,. (II) : ci(y1) = ip1y1 (i = i(1)/ j(1)), jI y1 > 0,.

. I 4.5.1.

II, (y1, w - t) 0 ( i I vi(y1) y1). vi. I I y1,.

[[[]]]] 4.5.3 vi .

., i = 1 1 = v1 1 y1.

,, y, :

1 u1() = v1() - (vj(y(i)) - vj(1)) v1(y) - (vj(y(i)) - vj(y1)).

j=1 j=i vj(y(i)) vj(1) vj(y1), (72).

[[[]]]] jI jI, y1 ,.,,, - -,.,,. ( ),.

( ). t = 2, 3,..., ; t + t., i t.

4.5.4 -, i, t, t. ( ) 4.1 (), - ln(y1) + 2 ln(y1) + 3 ln(y1) - y1 max.

, y1 = 6.

-.,, : i = 1/3(i)., vi = i ln(y1) - y1/3. - : y1 = arg maxy (v1 + v2 + v3) = arg maxy (6 ln(y1) - y1) = 6.

, i- (i = 1, 2, 3):

y1 = arg maxy (5 ln(y1) - 2y1/3) = 7.5, y1 = arg maxy (4 ln(y1) - 2y1/3) = 6, y1 = arg maxy (3 ln(y1) - 2y1/3) = 4.5.

: 1 = (5 ln(7.5) - 7.52/3) - (5 ln(6) - 62/3) 0.12.

2 = 0 3 0.14.

4.6 ,,, ( ).,,,. " ", ,.

,.

p1 i -. (. discrimination ) 32. , ( ), -.,, ( ).

(. lump-sum costs " ") (, ),,.

,., ,. .

( ).,,.

,,, :.,,.,.

5,.,. ":

89. 6.

5.,.,, : p = p(y) ( ).

C.

c(y). :

= p(y)y - c(y) maxy.

, p(y) c(y).

:

p (y)y + p = c (y) p(1 - 1/||) = c (y), (73) dy p = .

dp y,.,,,,,,,.

p(y),,, ( ).

,, p(y) :

p(y) = D-1(y), D(p) Xi(.),..3 ).

5.1.1 c (y)-p(y),,,.

. yM, yC. :

c (yM) - p(yM) = p (yM)yM < c (yC) - p(yC) = 0.

yM < yC pM < pC.

5.1.2 (c (y) > 0), : || > 1.

(73).

5.1 p(y) = a - by, c(y) = dy (a, b, d ). a-d : pC = d, yC =.

b = y(a - by) - dy = (a - d)y - by2. a-d a+d yM = pM = 2b,...

,,..,,.. ,..

.

,.. (),, (),,, -.

,,..,, -.

,,,..

(. deadweight loss).

,,,. -,.,,.

5.,,,, ( ).

,,,,,.

.

:

1) (,, );

2) ;

3) (, ).

5.2 2 : pa(ya) pb(yb). c(ya + yb). = pa(ya)ya + pb(yb)yb - c(ya + yb) ya yb,. : pa(1 - 1/|a|) = c pb(1 - 1/|b|) = c.

:

pa 1 + |a|-= (74) pb 1 + |b|-, ( ),,.

5.3, " ". :

.

1...

2..,,.,,..

3..

,,,..

5.3 :. :.

, :.,,.

,.

,,.

,.

:

(, ) - (0, 0) - - { - (2, 1) { - - (1, 9).

,., (2, 1).

(0, 2),.

,,,.

p y1 = Y1(y2) yS (MS) pM pC yN y2 = Y2(y1) D (AR) MR yy yM yC. 3: ) ) 5.,,.,.., (, ).

:

(I),. (II) :,,.,,, (A), (B).

, (. ).

, n. j j = pjyj - cj(yj).

( ),. ( ), 33.

,.

,,.

= y p(y ) - cj(yj), y := y1 +... + yn j.,. yM:

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