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M p(yM) + p (yM)yM = c (yj ). (75) j,,.

,,.,, M (yj = yj )..

, : dyj > 0, M M M (75), dj = (p(yM) + p (yM)yj - c (yj )))dyj = -p (yM)(yM - yj )dyj > 0.

j.

p(.) y. yj,, y-j, p(.). j- y-j :

Yj(y-j) := arg max j(y) := arg max p(y )yj - cj(yj). (76) yj yj 5.4.1 -34 y, yj Yj(y-j) j = 1,..., n.

p(y )yj - cj(yj) maxy 0, :

j N N p(yN) + p (yN)yj = c (yj ) = c0. (77) j.,.

, - ( ) (.3 ). (1(y1, y2) = const 2(y1, y2) = const) (y1 = Y1(y2) y2 = Y2(y1)),,. - (yN).,,,, - ( ).

5.4 : p(y) = a-by, cj(yj) = dyj (j = 1,.., n), j = (a - by )yj - dyj.

n a-d yN = (, n+1 b,.

2(a-d) a-d yN = ). yM =, 3b 2b (, ). a-d yC =, b ( ).

.

, (),, (76), :

1 = y1p(y1 + Y2(y1))y1 - c1(y1). (78) 5.5 5.4 a-d-by1 a-d b Y2(y1)) =, 1 = y1 - (y1)2.

2b 2 3(a-d) a-d a-d y1 =., y2 =, y =, 2b 4b 4b,,,.

( 1) p, (j = 2,..., n),, Yj(p) := arg maxy pyj - cj(yj).

j. n, : y1 = (D(p) - Yj(p)).

j=, n n 1 = [(D(p) - Yj(p))p - c1(D(p) - Yj(p))] max. (79) p j=2 j= pj., ( ) p-j. :

1),,, : yj = 0.

2), -,,, yj = D(pmin)/(k + 1),, yj = y = D(pj).

,,,., -,.,, -.

,.,., "", pj.

,.,.

yj = yj(pj, p-j), pj p-j., dyi pj (jj < 0), (ij = > dpj yi i = j)35.,.

,, pj(1 - 1/|jj|) = c. (80) j,..

.. ( 5.2), 1 ij pj(1 - ) - pi = c. (81) j |jj| |jj| i =j 5.6 p2 p y1(p1, p2) =, y2(p1, p2) =, p1+1 p2+ cj(yj) = yj (,, > 0, ). 11 = 22 = -( + 1) (80), (+1), p1 = p2 =.

y1 = y2 = ((+1) )1+-.

,. 3.

,,.,,,,.

,, 12 = 21 =, (81) (+1-), pj =,, y1 = y2 = ((+1-) )1+-.

,, "", (), ().

pj = pj(yj, y-j)36,..

5.7 1 p1(y1, y2) = p2(y1, y2) =,, .

y1 y2 y2 y p1 = p2 = y1 = y2 = (1- )1+-. 1-,,,,.

,,, ( ).

6, ..,,,,, . "" "".

,,.

6.1 (),.

: (. unit tax) (. ad valorem).,, : (pH) (pL).. : pH - pL = t, t .., pH = pL(1 + ),, pL = pH(1 - ) .. 4.

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

.

, t t (.. 4, ). pL = pH(1 - ), pH = pL(1 + ), =.

1,,.,,.,..

,,.

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

,,,.,,,,.

p p S S S pH pH p t p p pL pL D D D D y y y y y y a) ). 4: ), ).

, .,,,.

,.

,,,,.

.

,,,,.

.

,,,,.

.

,,,,.

.

:,,, 37:

k tk (|1 + ) | k D S =. (82) pLk 1 + |k | D k , tk , pLk , |k | D , k , S,.

, ( ),..,, .

, ad valorem (), :,,.

.

6.1, pH = a - by, (d), pL = pH - t.

,,, a-t-d a+t+d a-t+d a-d a+d y =, pH = pL =. y = p =.

2b 2 2 2b, .

37 tk 1, +. pk |k | k D S.

(a-d) (p-d)(yC -y) =, 2 8b a-d yC = . :

b (a+t-d)2 (a-d)(pH - d)(yC - y) = > 2 8b 8b.

, (ad valorem),, (unit tax).,. :

ad valorem unit tax, ( " ").

" ".

.,, .

,,,,,.

6.2.

() T,., d. ( ):

X (p, t,, T ) = arg maxx0 xB(p,t,,T )u(x). (83),. :

(p + t)x d, (84) (1 + k)pkxk d, (85) kK px d - T. (86),, -. -.,...

,,,.

,., T.

-. (tk/pk = const) (k = const).,,,., k = (85) d (86) T = = px, x 1+.

,,.,, u ( ).

p, p (pk = pk +tk pk = pk(1+ k) ).,, p1/p2 > p1/p2. (87),.. x px = d (88) uk(x) pk 1 = k1, k2 K. (89) uk(x) pk 2,, : p(x+dx) = d-(p-p)x. dx1 > 0 dx2 = -p dx1. :

pu1(x) p1 p du = u1(x)dx1 + u2(x)dx2 = u2(x)dx1( - ) = u2(x)dx1(p1/p2 - ) > 0. (90) u2(x) p2 p.

,, (), (.. 5 ). (B B) (x x). (B), "" (B),.,,,. -.

().

,, :

(income effect), (substitution effect), (, ).

: ( ), ( ).,.,.

x2 xx x I I I I I B B B xx x1 d/p1 d/p1 x a) ). 5: : ) : ).

,.

,.,.., ( ).

u (, ) ,., u,,..,.

, (.5 ).

,,.,,,. "".

,, (, ),, ( ).

,...

.

7, :, :

,..,,,.,.

7.1,.

Q.

., k K q Q.

i- xi = {xkq}.,, i.

Ui(.). i., (, ),.

,,,.

7.1.1 (), U(.) (),,.

, U,, :

U(x) = qu(xq), (91) qQ q [0, 1], q = 1 q Q, q u(xq) : I I , Rl R,.,,.

-, ().

(. XI.6 ), :, ( "" ), -, u..

-.

7.1.2 i U, u() (),,, ().

7.1.3 u(.), (0, 1) u(x + (1 - )x ) u(x ) + (1 - )u(x ).

u(.) 38, (0, 1) u(x + (1 - )x ) min{u(x ), u(x )}.

, u U, ( ). u U,.

, ., q (xq). ( ).

7.1.4 () :

E(x) = qxq. (92) qQ 7.1.5 x, : xq = E(x).

7.1.6 39 x x, :

U(x) = qu(xq) = U(x) = u(E(x)). (93) qQ 7.1.7 x x, E(x) - x x:

U(x) = u(E(x) - x). (94),,,..

.

..

. certainty equivalent.

7.1. 6 ( )., (A B) (A B). x = (xA, xB), xA xB ,, A B.

, (E(x), U(x)), (xA, u(xA)) (xB, u(xB)) B A. E(x) , U(x) .,, U(x) u(E(x)).

x, u(x) = U(x). x.

7.2,.,. -.

kq pxi = pkqxkq pkqwi = pwi. (95) i qQ kK qQ kK :

Ui(xi) max. (96) xiBi(p),,..

7.2 ( ). (), w1, 1 (), 2 w2 < w1.

[0, 1], y, y, y.

,, x1 = w1 - y,, x2 = w2 - y + y. (95), y:

(1 - )x1 + x2 (1 - )w1 + w2.

, " 1" " 2" p1/p2 = (1 - )/.

, - U = (1 - )u(x1) + u(x2), u(.) (.. ), . U/x1 1 - =. (97) U/xu u(x) r u(xB) u(E(x)) U(x) aau(xA) ax aax xA x E(x) xB ) ). 6: ). ).

1 ( - 1)u(x1) = ( - 1)u(x2)., u(.) ,.

= ( ), x1 = x2, w1 - w2.

( > ),, x1 > x2,., < ,.

7.3 ( ) (k = 0, 1,..., l). ( ) q Q q rk q., w () k = 0 ( ), r0., w = (w0,..., wl) : wk = w.

k zk zk 0, zk w. ( r0, k zk 0 ). q q xq = zkrk.

k q U(x) = qu(xq) = qu( zkrk) q q k. w ( ), k k = zk/w. :

q U(x) max xq = krk, k 0, k 1 (98) k k u(.), (, x = r0). U(.) U = c0 + c1r - c22, (99) ci > 0 -, r = E(x) , 2 = var(x) ( ). : r = krk, rk k- k q (rk = qrk), 2 = k k k k2, k (q k1 k2 1 2 q ) k- (k = q(rk -rk)2), k k2 q q q (k k2 = q(rk - rk)(rk - rk)).

k1 k2 q 1 (),, r. p p, rp, ( = 0).,. (.) p = (0,..., ), + 1.

1) k : rp = krk.

k 2) p = (0, 1), (k = 0) (), (, ), p = 11., (0, r0) (1, r1).

,, (0, r0). / ., (0, r0),., - , .

3) : k k2 = 1 (k1, k2 = 0).

a = kk (, ), k, (k, rk), k = 0,..., l.

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

(), , k = (rk - r0)/k (k = 0).

3), ( k k2 = 2 k1 = k2), = kk.

k,.

,,. ( ). (.) 4),.,, 12. = 2 2 2 2 11 - 1212 + 22 = |11-22|. = 0, 1 =, 2 =.

1+2 1+5), () -,,. P (),. P, ( !) "". 7.3.1 ("Mutual Fund Theorem"),,, ( ). (,, ).

, = (r - r0)/.

,., "",, k,.,,,.

,,,, "".

( ).

7.3.2 - u(.), 0 (, - ).,..

(rk > r0),.. k > 0.

q -. : L = qu( krk)+(1- k)+ q k k kk, 0 k 1, 0 k =0 k k q k 0. dL/d0 = 0 r0 q qu (xq) =, dL/dk = 0 qrku (xq) = q - k.

,. q.. k = 0 rk xq, q q rk u (xq). qrk q qu (xq) = rk q qu (xq) = - k q r0 q qu (xq). qu (xq) > 0,, rk > r0.

[[[]]]] 7.4 42, ( ).,, 43.,,.

,., m l. Q = {1,..., q} ., "" "".

,,,,.

( ""), k K q Q. pkq,, i xkq.,,.

i i q q lq wi I, wi I Rl, R+., (95). Ui(xi)..

, (16) q Q :

.

,, (k, q)..

,. XI "".

. : Knight F. Risk, Uncertainty and Profit, 1921.

7.4.1 ;, -.

7.3 ,, w1 = (1, 3), w2 = (3, 1) : 1 1$ R (), S (), ,.,,, 1$ 3$, (3,1). 1 2, Ui(xi) = 0.5ln(xR) + 0.5ln(xS).

i i ". (""),.

q (q Q) i q i.,.

: U1(x1) = 0.25ln(x1) + 0.75ln(x2), U2(x2) = 0.75ln(x1) + 0.25ln(x2). 1 2 ; . : wi = (2, 2).

.,. 7.4.1, -.

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

..,,,.

, - q1 q2., -...

,.

7.4 (1 2) (x y).

" " (B) " " (G) (G = B = 1/2). u1 = x1 + y1 u2 = ln x2 + ln y2.,,. x, y. 6, 2.

,. ( " "),.. u1 = x1 + y1 pxx1 + pyypxwx, u2 = ln x2 + ln y2 pxx2 + pyy2 pywy, (wx, wy) = (2, 2) (wx, wy) = (6, 6).

(x1, y1, x2, y2) = (1, 1, 1, 1) (x1, y1, x2, y2) = (3, 3, 3, 3).

-,. ( ). B G B G U1 = 1/2u1(xB, y1 ) + 1/2u1(xG, y1 ) = 1/2(xB + y1 + xG + y1 ) 1 1 1 B G B G U2 = 1/2u2(xB, y2 ) + 1/2u2(xG, y2 ) = 1/2(ln (xB) + ln (y2 ) + ln (xG) + ln (y2 )).

2 2 2 - (x, ) B G : xB = 2 = xG = 2., 2.

,,.,. B G pxBxB+pyBy1 +pxGxG+pyGy1 pxBwxB+pxGwxG 1 B G pxBxB + pyBy2 + pxGxG + pyGy2 pyBwyB + pyGwyG.

2,.., - xB = y2 = xG = y2.

B G.

2 (0, 0), (2, 2). (4, 4), (2, 2).

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

8.,. 44.

,, " ",.

8.1 (, ),..,,,....

8.1 ( ),, : ("") (""45),., "" "", l, g > l.

l l, g > g,,.

, 1 - .,, p, p p = (1 - )g + l., g,,.

p < g,,,,. = .,.. :,,,,,, -, () !,,.,, c,.,. " " (),. :

" " ( ).

.,.

.,.

,., "".

8.2 ( ) : "" (H) "" (L). limon " ".

aH aL < aH.,,., xi 0, "" cHxH > 0, "" cLxL, cL > cH.

x,, (wL),, (wH). "" x, "" 0,. -:

,, cHxH > 0 .

.,. (moral hazard)..,,,.,,.

.,.,,,.

8.2 (principal) (agent)46., y = y(x, ) x X, X,.

. R(.) ,,. y, x, s, s ,.., x /. y, x, : R = R(y)..

u(x, R), x, R., "" u0, (, ).

"principal-agent problem" 8.2.1 R R(y, x,, s) R, E() = E(y(x, ) - R(y(x, ), x,, s)) (100) x :

1), :

E(u(x, R(y(x, ), x,, s))) u0, (101) 2) x R(.):

E(u(x, R(y(x, ), x,, s))) E(u(x, R(y(x, ), x,, s))) (x X). (102).

1) " " (take it or leave it) , r x.

2), : R = a + bx.

3), R = y-b, b ,, b.,,.

4) R = y, < 1 .

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

( ).,.,,.,, u(x, R) = R - (x). E() = E(y(x, )) - (x) - u0 maxxX.

R, E(u) = R - (x) = u0.,..

.

,, (. shirking)...,,.,.,,, (principal) ( )..

:

() :vspace2mm ()., ( ).-.:, 1985.

().,.-.:, 1985.

().,.,.-.:,, 1992.

()..,..,.-., 1994.

().., :.-.:, 1992.

: vspace2mm.1,2,3;.6.-.4;.15;..15.6;.4..9.2;.17.1,2;.17;..9.3,4;.17.3;..3.9, 4.3,8, 7.6,7;.10,11,12;.8,9;.10,.4.1;.3.2, 5.3;.4.6, 5.6, 5., 10..11;.5;.13;.5.

.12;.16;.+ :

>.2;.3;.5;.>.2.7;.4.2;.5.2;.4.>.3;.6,7,8;.6,7;.6,7,>.2, 4.>.4.4,.9;.5.3;

.5.6, 5., 8.5, 12., 4,, -, 6,, -, 17, 18,,,,,,,,,,,,,,, (moral hazard), (principal-agent problem),,,,, (unit tax),, (ad valorem),,, 15, 35,,, -,,,,,,,,,,,,,, (shirking),,,,,,,,,,,,,,,, 13,,, -,,,,,,,,, -,,, 16,, 34 -,, 4, 9,,, 45,,,, (deadweight loss),, -,,,,,,,,,,,,,,,,,,,,,, 59,,,,,, ,,,, (Mutual Fund Theorem),,,,,, -,,,

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