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" 4-., " ".,.,, (, ),.

, -.

:

H. Varian. Microeconomic Analysis.

D. Krebs. A Course in Microeconomic Theory.

" " " 1,2 ();,. : 1) ; 2) ; 3) ; 4).

:

. : ( ).-.,, 1985.

. :.-.,, 1985.

..,.. : ;.-.," 1992.

1 1 3 1.1...................... 8 2 () 10 2.1............................... 11 2.2 -,................. 17 2.3,.................................. 20 3 22 3.1................................ 22 3.2.... 23 3.3................ 28 4 30 4.1...................... 30 4.2 ()..................... 33 4.3 ( )......... 35 4.4.......... 38 4.5 -.............................. 40 4.6 .......... 43 5, 44 5.1.................................. 44 5.2.......................... 46 5.3......................... 47 5.4....................................... 48 6 52 6.1........................... 52 6.2.......................... 55 7 58 7.1................................... 58 7.2.......... 60 7.3 ( )................... 61 7.4...................... 64 8 66 8.1................... 67 8.2..................... 1 (.. ) ()., -.

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

, "".

() (1) :

( ), (, ) ().,,, G := I, (Xi)I, (ui)I, I := {1,..., m} i, X := (Xi)I := Xi , i u := (ui)I = (ui)iI ( ui : Xi I,, (xj)jI).

R ( ), Bi(xj=i) Xi xj=i. G := I, (Xi)I, (Bi)I, (ui)I, x X.

(, ). :.,,. :

x-i := (xj)jI\{i} i,.

., "", "" ( ). :

:.

,,.

1:

, j I () 1. (Xi)I I " DE MM "" () 2. (ui)I\{j} SE " P NE "Perfect Nash Equilibrium" 3. (xi)I\{j} NE 4. NEm " " 5., - - StE " " 6. "" 1.0.1 xi Xi i yi Xi, x-i X-i ui(xi, x-i) ui(yi, x-i), x-i X-i ui(xi, x-i) > ui(yi, x-i), -i := I \ {i}, X-i := (Xj)j=i.

xi, yi ui(xi, x-i) = ui(yi, x-i) x-i,,,.

Xi :

1.0.2 xi Xi i ( ) :

yi Xi, x-i X-i ui(xi, x-i) ui(yi, x-i).

i I Di.

( ) i Di.

1.0.3 I := I DE Di.

iI, ..,, NE, MME, SE, StE.

: I " DE " ".

1.0.4 ( ) NE := {x X| ui(xi, x-i) = max ui(yi, x-i) ) (i I)}.

yiXi,, PNE ( "Perfect Equilibrium")., x P NE, ( ).

, - (, ).,,.

(NE), ().

1.0.5 G Xi = {1,..., ni} (i i I) i 3 i = (k)n = i k= ni ni i (k(xk))n Xim := {i I | R+ k=1 k = 1} i i k=1 i "" xk; Gm := i I, (Xim)I, (Ui)I, Ui() := ui(x)(1(x1)2(x2)...n(xn)).

xXI NEm (i)I,,, ;

1 arg max u1(x)(1(x1) 2(x2) ... n(xn)).

xXI (),,.

1.0.6 MM := {x X| ui(xi, x-i) = sup ( inf ui(yi, z-i) ) (i I)}.4 (2) yiXi z-iX-i : ( ). ;

.

1.0.7 Sad := MME NE. (),.

1.0.8 (,, ),.., ui(x) = 0, x X. iI 0 Sad, 0 := sup inf u1(x1, x2) = inf sup u1(x1, x2).

x2X2 x1Xx1X1 x2X.

,. " ",,.

1.0.9 G1, G2,..., Gt,..., : Xt+1 := Dt (t = 1, 2,...) ( ). G1 :

W SE := Dt = Dt-1 (t 1). x SE x W SE, Gt.

(Stackelberg) () -, (), NE,.

1.0.10 1-, xNR-1(x1) := {x-1 X-1| ui(x) = maxx Xi ui(xi, x-i), (i = 1)}, :

i () N x StEP1, arg maxx X1 minx NR-1(x1) (, ), - 1 - arg minx NR-1(x1) (, ).

- - -() x StEO arg maxx X1 maxx NR-1(x1) (, ), - 1 - arg maxx NR-1(x1) (, ) - - -,, () (""). StEO ( NR), StEP1 ; "", StEO StEP.

.

1.0.11 () (-, " -") (), P := {x| x X : u(x) > u(x)}, (: I ); - W P := {x| x X : u(x) u(x)}.

, (,.40). StE ( StEOi, StEPi ).

> =, .

- ,.

1.0.12 C (, ) T I, () : T x X xi Xi (i T ), T x,..

uT (xT, x-T ) uT (x) x-T T.. 1.0.1 -, () -:

Cs P, C W P.

,.

(. ).

1 (, 1951) (i I) Xi, ui(.) xi, NE =,.

1.1 Xi, NEm =,.

F (3).

NEm..1.1 (),, NEm. 1- ( ) 0 0, , ni 0, k = 1, ak 0 (k = 1,..., n2), ak I Rn k= (ak) := (u1(xj, xk)). 1j 1 ().,. Gm,.

2 2 NEm, ( ) NRi(x-i).

1.0.2 1) Xi, W SE =. 2) Xi SE Sad, SE =,.

Cs, ( uT (xT, x-T ) > uT (x) uT (xT, x-T ) uT (x) ).

:,,, SE ( ).

1.0.3 Xi ui, 1) DMME =, P = ; 2) I MME, NE, NE NEm; 3) I =, DE, DE MME I = D = Sad = SE NE, DE (3) :,.

1.1 I =.

DE 1.1 " " (R.Luce, H.Raiffa,1957, ).

.

.,, 1, 10.

, 7.,, 3.

(.2),, .

2:

-7 - -7 - -10 --1 -, .,,, 7,, 10. "".,,.. ( ),,,, .

.. - :, - (u1 = -3, u2 = -3)..

, (-3, -3), -.

("") : ( ),..

W SE. ;

.

,,,.

( ) Fi(.) - i x-i :

Fi(x-i) := NRi(x-i) = arg max ui(xi, x-i) xiXi,.

NE :

x..., xi Fi(x-i) xi = Fi(x-i) i I. (3) Fi(.),.

NE, StE.

1.2 ( ) . (w) 0 w 3, (l..) 0 l 1. :

(w, l) = wl - 2l2, 2l2 . :

(w, l) = 2 l - wl.

.

1) (MME). : ;

MME.,, .,.,, ., = 2 l - 3l.

, l :

1/ l - 3 = 0.

, l = 1/9 0 w 3.

w = 3,.

2) (NE). (w = 3). f(l) = 3.

l:

g(w) = 1/w2.

{w = 3, l = 1/w2, } (w, l) = (3, 1/9),,.

3) (StE) ( ).,, = wg(w) - 2g(w)2 = 1/w - 2/w4.

, 2, g(2) = 1/4.

4) - (P)., -.

, - : l = 1/ 16. 0 w 3., -.

5) (C). :

(.. -), ( )... (l, w) : l = l, = w- 2 (w, 0) = 0 = 2 l - w (3, 1/9) = 1/3.

l l2 l 2 (), :

1) :.

2) : ( ).

3) "Costless trade":,, " ",.

4) :,,, ( ).

,,,.

, -. 1)4) -, " " 1) ; 2) ; 3) ; 4),.

2.1..

() 9 G := I, XI, uI, I, wI, J, YJ, (4) I, XI, uI,,.

I := {1,..., m} , J := {1,..., n} (), K := {1,..., l} (). : xk i- i k k- (k K), yj j- k- (k ), pk k-. wi ( ) k i.

. i I ( ) ui(.),,, xi = {xk}kK. i xi., xi Xi I ("", Rl l : Xi = I R+)., :

pxi = pkxk i(.). i(.) ().

kK i, i(p, wi) = pwi., wi.,, ( ),.

, i(.) = i.,.. (.. ) Xi(p, i). " ".

:

Xi(p, i) := {xi Xi| ui(xi) = max ui(xi)}, (5) xiBi(p) Bi(.) :

Bi(.) := {xi Xi| pxi i(.)}. (6) ( ).

( ) " " ". : A.Smith-1776, D.Rickardo-1817, L.Walras-1874,1883, K.Arrow & G.Debreu-1953.

,, (, ) xi Xi.

..

(, x - (x) max (7) r(x) 0 r = 1,..., r (8),, x, r(r = 1,..., r) , L(, x) := (x) + rr(x) x, r r, (), r- : r(x) > 0, r ( ).

( ) (.), k(.), x ( ), x.

i I.

1 (). Xi, ui(.) (.. ui(tx + (1 - t)y) tui(x) + (1 - t)ui(y) t [0, 1], x, y), - ( ).

; ( ),,, u(x, y) = xy v(x, y) = ln(u(x, y)) = ln(x)+ln(y),, -.,, " ". " ". - u(.),, xi xi {xi Xi| ui(xi) ui(xi)} ( : [ui(tx + (1 - t)y) min{ui(x), ui(y)} x, y, t [0, 1] ]).

,., ( ).

2 (). - xi ( ) (xi int(Xi)), grad ui(xi) = 0.

xi Xi ( -, ), ui(xi) max ; pxi i, xi Xi (9) xi ) k(x), "" (.. x - " "), x.., (1,..., m) 0,.

L = ui(xi) + i(i - pxi), i . dL/dxk(xi) = 0 k, ( uk i i k) uk(xi) = ipk (k K). (10) i ( ) i, xi p (.. ):

pk/ps = uk(xi)/us(xi) (k, s K). (11) i i, uk = 0, pk = 0, i xi.

uk/us ( ) i i k s., - ( ).

,... grad ui = 0 x X,., (dL(xi, i)/di = 0) :

pxi = i (12) (11), (12), x () - - ()., (11), (12)., 2.1.1 (), () (9) xi Xi, 0) (11), (12), x ; : (11), (12).

, -.

k. yj = {yj }kK j J Yj I Rl.

() fj(yj): Yj := {yj I fj(yj) 0}. Rl| ( h) -h -h h k yj gj(yj ), yj := (yj )k =h ( ), h. h -h Y, fj(yj) = -yj + gj(yj ) 0.

"" k = pyj = pkyj., kK,,. (, ) Yj(.):

Yj(p) := {yj Yj| pyi = max pyj}. (13) yjYj..

3 (). Yj, j fj(yj) 0, j.

4 (). yj grad fj(yj) = 0.

(13) L = pyj + jfj(yj), j . p = k k yj dL(yj)/dyj = 0, pk = f (yj)j (k K) ( j k f k yj). j, j = 0, j, yj:

2 1 2 1 pk = 0 pk /pk = fk (yj)/fk (yj) (k1, k2 K). (14) j j 1 fk /fk j j k1 k2., - ( "").

h fj(yj) = gj(yj) - yj h (, ) (k) k -1/j ( k), k -1/j = ph/pk.

fj(yj) = 0., (),.

, (h), c(.). -h h yj gj(yj ). k yj -pkyj h k =h k ( yj < 0 k = h, ,, -h ) yj = gj(yj ). p-h h h. -h k cj(yj p-h) := arg miny( -pkyj | yj = gj(yj )).

h, h k =h ph yj, h h k j = phyj - cj(yj, p-h). h h ph = dcj(yj, p-h)/dyj,..

j h.

() (), (general equilibrium models).

.

1.. l.,, w I R+.

Y := {y} := {w }.

i(.) = di 0. : xi y = w ( ).

iI. l, w. i I di ( ).,, (, ).. (, ), t pk(t) k {1,..., l},,,.

"" (. tatonement).

p ( ),.,,.

2... l wi I R+,. i(p) = pwi. xi wi iI iI,. ( I), wi (, ),, ( ), tatonnement..

( II) ( Xi ), i I (,,, ) wi;, ( ). tatonnement. ( ) ( ).

3. -.,. wi, ij.

i(p, y) = pwi + ijpyj + di, jJ ij [0, 1] i j J, di "" ;

di = 0 (i I) -. :

xi wi + yj.

iI iI jJ, II.

, i(.),,...

, (, i(.) , ).

, -,,.

2.1.1 ( ) (p, x, y), :

1) (x, y) p,..

x X (p), y Y(p). (15) 2) :

xi wi + yj, (16) iI iI jJ 3) ( " "):

p( xi - wi - yj) = 0. (17) iI iI jJ W E(d, w, ).

(x, y) (16), (p, x, y), W E=(d, w, ).

W E(d, w, ),, "" W E(d), "" W E(w), - W E(w, ).

2.1.2 () (Partial Equilibrium) k p (x, y),, (15) (16) k ( ), P Ek(p).

W E NE,, I, X, u, J, Y, d, w, G ( ),. W E NE.

W E, W E C,,.,, (General equilibrium), "" "" (Partial equilibrium).

, "" -,..,.

2 "..

,. ( ) tatonnement (.,.5,.149),.

( pk(t)/ t) = k( (xi(t) - wi) - yj(t) ). (18) i j .

,,,, (,..

). ;.

2.2 -, "",.., xi Xi i I, yj Yj j J (16).

2.2.1 - (x, ), (x, y), (ui(xi))iI = (ui(xi))iI.

-, -,..

,.

2.2.1, (x, ), -,, i0 {1,..., m}) ui (xi ) max (19) 0 xX,yY ui(xi) i = ui(xi) (i I \ {i0}), (20) fj(yj) 0 (j J) (21) k k xk wi + yj. (22) i i i j.

, W E P,, ( l Xi = I R+).

.

,, 1) - ( " "), 2) : - (), (,,.); -.

5 (1): i I ui Xi, xi Xi V (xi) xi V (xi) Xi : u(xi) > u(xi)).

,, xi - ui l xi I ( R+ ""), Xi l Xi = Xi + I R+.

2 (1 W E(d,, )). (p, x, y) , (1), (x, y) -...:

( )&[(p, x, y) W E(d,, )] (x, y) P. (23) 1- 2-, -,.

- 1 W E(d). : (x, ),,, u(x) u(x). t I =, x : ut(x) > ut(x).

1),,, (6),.. pxt > dt., xt : xt Bt(p, d) (15), xt, xt, x. pxt > dt.

pxi di (i I)., (1) xi Xi, pxi di,, ui(xi) > ui(xi),,, xi.

pxi > di. (24) i i 2), (6) : pxi di. i (24) pxi > di pxi. (25) i i i 3) x, xi w. p i (17) ( ), (25):

pxi pw = pxi. (26) i i.

[[[]]]], -, 2) i(.) = (pwi + ijpyj) (pwi + ijpj), (, i i j i j )., pyj pyj,.

jJ jJ, -.

6 (). i I Xi, ui,16.. ui(tx + (1 - t)y) tui(x) + (1 - t)ui(y) t [0, 1] x, y.

Yj j.

7 (). (x, ) (.. xi int(Xi), i I), grad ui(xi) = 0, i.

Yj := {yj| fj(yj) 0}, grad(f(x)) = 0.

() 2. -,. -, grad ui 0 xi int(Xi). -,, () : [ K+, x.

l 1), Xi I & xk > 0 k K+, 2) R+ i k K+.] (" ").

2 -.

3 (2 W E(d, w, ).) - (x, ) P w, (), (), l (d, w, ) 0 p I R+, ij = 1 (j J), wi = wi, (p, x, ) , iI iI iI ( )&( )&[(p, x, y) P] (p, d, w, ) : (x, y) W E(d,, ).

n, I : [di = id, wi = iw, ij = i (j)] (i I).

R+, -.

- 2 W E(d). 1) 2.2.1, x -, m ( s = 1,..., m) (19) :

us(xs) max ; (27) xX ui(xi) = ui(xi) (i I \ {s}), (28) xk yk = wk (k K). (29) i i 2) (27) -, "" (.. x ).,, " ".

():,,.

(. ).

3) (27) -, i 0, i = 1,.., m (28) k 0, k = 1,.., l (22) :

iuk(xi) - k = 0 (i I, k K). (30) i uk(xi) := ui/xk, s := 1., s = 1, i i grad us = 0 = 0, i > 0 (i I).

4) : p := I Rl, - : di := pxi.,.

, x p, d,.. (9). (12). i = 1/i ( i > 0, (30)),, p = i xi (10). () x.

(), (), xi Xi(p, di) (i I).

"" - x, - (27)., pk = k - k, (), (17).

.

[[[]]]], -, (w, ), i(w, p, ) (pxi), x.

i = (pxi/ pxj).

jI, (30) - ( ) k, t :

k/t = uk(x)/uk(x) (i I), k/t = fk()/fk(x) (j J).. (31) i i j f 12,. :

,.

2.3,, ui wi.,, (15), (5) ( ) Xi. Xi, Xi(p). 0 p,,, p1 = 1, l - 1 p2,..., pl.

k i : ui/xk > 0, i,, (16) . l l p1,..., pl, : (Xi(p) - wi) = 0.

I, (17), i (6), ( )., l -, l - 1 p2,..., pl.

x = X (p).

( ),, 2 ( -). n (l - 1) (p1,..., pl), (x1,..., xl)I. n i i, (),.

, -.

.

,,.,.

2.1 (" ").,. -:

u1 = ln x1 + 3 ln x2 u2 = 3 ln x1 + ln x2. 1 1 2 w1 = w2 = (2, 2).

. 1:

xxP C x w p xx1 2, 1, 2, : u1(x)/u2(x) = x2/(3x1) u1(x)/u2(x) = 3x2/ 1 1 1 1 2 2 x1. -, x2x1 = 9x1x2.

2 1 2 1 x1 + x1 = 4 x2 + x2 = 4.

1 2 1 -: x2 = 9x1/(1 + 2x1).

1 1., x1 = (1, 3), x2 = (3, 1), p = (1, 1).

-,,., : u1 = ln x1 + 3 ln x2 u1(2, 2) = 4 ln 2, u2 = 1 3 ln x1 + ln x2 u2(2, 2) = 4 ln 2 x2 = 9x1/(1 + 2x1).

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

3. 3.1.1,,,, ui / Xi : ui = ui(x, y), Xi = Xi(x-i, y) ( ).

, (), Yj : Yj = Yj(y-j, x); fj, fj = fj(yj, y-j, x) 0.

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

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

() 17.

3.1 (" ")18. m - i {1,..., m} yi 0., (yi/y )- " " f(y ), f(.) y := yi., f(.) > 0, i f(.) < 0, ( ).

, : "".

. David Hume,1790 ().

p, 1 (, ), i y-i i(yi, y-i) = (p yi f(yi + y-i))/(yi + y-i) - yi.

, = pf(y ) - y.

, m > 1, -, (" " ).

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