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4 1 1.1 a 1 a x a x x f(x =a x a x a >1 0 <a< r 1 n (1 + r n 1 n (1 + r n n 1 1 (1+r 1 n 1+r discount factor m, n (1 a 0 =1 ( a 1 = a (3 a m a n = a m+n (4 a m /a n = a m n (5 (a m n = a mn y = a x M M = a x x a M log a M 1.. (1 3 =8 log 8=3 ( 3 1 = 1 3 log = 1 a 1 f(x = log a x x a M,N > 0 p (1 log a 1=0 ( log a a =1 (3 log a MN = log a M + log a N (4 log a M N = log a M log a N (5 log a M p = p log a M 4

5 1.3. z = x α y β log a z = α log a x + β log y X = log a x, Y = log a y, Z = log a z 1 a n = Z = αx + βy ( 1+ 1 n n n e ( e = lim 1+ 1 n. n n e e = e log x ln x 1. x, y x y 1 y x function y x y = f(x y = f(x x y f(x x x y y = f(x x f(x domain x y range 1.4. (1 a x ( log a x (3 x (4 1 x 0 5

6 y = f(x y y = f(x x x = g(y x y y = g(x f(x inverse function f 1 (x 1.5. (1 f(x = 1 x +3 f 1 (x =x 6 ( f(x =a x a>0,a 1 f 1 (x = log a x (3 f(x = x (x 0 f 1 (x =x (x 0 (4 f(x =x a x = a x ± a (5 demand function law of demand inverse demand function f(x,g(x y = f(x, z = g(y g(y y = f(x z = g(f(x f g composite function (g f(x 1.6. (1 f(x =x 1, g(x =x +1 (g f(x =x 1, (f g(x =4x +4x (g f(x (f g(x ( indirect utility function 1.3 f(x x a a f(x α lim f(x =α x a α x a f(x f(x α x f(x x,f(x x f(x x,f(x 6

7 lim x x = lim 1 1 =0 lim x x x 0 x f(x a lim f(x =f(a x a f(x x = a f(x f(x 1.8. f(x x =0 f(x = { 0 x<0 1 x ( (. f(x [a, b] f(a f(b f(a f(b k f(c =k c a c b 1.9 (. D(p S(p f(p =D(p S(p p f(p (i p f(p > 0 (ii p f(p < 0 f(p f(p =0 p f(p =0 D(p =S(p p 7

8 1.4 U U f(x U a f(a + h f(a lim h 0 h f a a f f (a f U f f U x f (x f f (x f f(x = x x =0 x (. (1 {f(x g(x} = f (x g(x+f(x g (x ( {f 1 (x} = 1 f (x (3 (g f (x =g (f(x f (x ( 1 (4 = g (x g(x {g(x} ( f(x (5 = f (xg(x f(xg (x g(x {g(x} 1.4 (. (1 f(x =e x f (x =e x ( f(x = log x f (x = 1 x (3 f(x =a x f (x =a x log a (1 f(x = x ( f(x =(x 4x(3x 1 (3 f(x = 1 x +3 (4 f(x =(x (5 f(x = 3x x +5 (6 f(x =x log x x

9 (1 (log f(x = f (x f(x ( (log a x = 1 x log a 1.4. e e =lim h 0 (1 + h 1 h ( lim 1+ r n = e r n n 9

10 .1.1 (. f(x a, b (a <b f (c = f(b f(a b a a b c b = x f(x =f(a+f (c (x a a x f(x x 1 1 f (c f(x f(x 1 f (c x. f(x f (x f (x f f (x.1. (1 f(x =ax + bx + c f (x =ax + b, f (x =a ( f(x =e x f (x =f (x =e x (3 f(x = log x f (x = 1 x, f (x = 1 x C r r> f r f (r (x r f r f C r.3.. f(x C a x (x a x a c f(x =f(a+f (a(x a+ 1 f (c(x a 10

11 f(x x k k k =1, f(x f (c x.. f(x =e x a =0 f(x =1+x + ec x.4 1 f(x f(x f(x f(x f(x f(x 1 f(x x f (x =0 f (x > 0 f(x x x x f(x f(x f (x < 0 x x f(x f(x f(x f(x.3 (1. (1 f(x x f (x =0 ( f(x x f (x =0 1 f (x =0 1 first-order condition 1 f(x f (x =0 f (x =0 f(x y = f(x f(x.1 (. * 1 *1 f(x 11

12 (1 f(x concave function a, b 0 t 1 t tf(a+(1 tf(b f (ta +(1 tb ( f(x convex function a, b 0 t 1 t tf(a+(1 tf(b f (ta +(1 tb.3. x, log x x, e x 1 ax x 3 +3x.4 (. f(x C (1 f(x x f (x 0 ( f(x x f (x 0 f(x x 1 f (x =0. x f(x =f(x +f (x (x x + 1 f (c(x x = f(x + 1 f (c(x x f(x c f (c 0 1 f (c(x x 0 x f(x f(x f(x.5 (. (1 f(x f(x x f (x =0 ( f(x f(x x f (x =0 1

13 .5 f(x f (x =0 (1.4 f(x ( f (x =0.4 (. y x y = x p w f(x f(x =p x wx f (x = p x w, f (x = p x 4x p >0 x>0 f (x < 0 f(x.5 f(x x f (x = p x w =0 x = p 4w p, w factor demand function y = p 4w = p, w supply function p w (1 f(x =x +x +1 ( f(x = x 3 +3x (3 f(x = x (4 f(x = e 3x (5 f(x = log(x + 1 (6 f(x = log x x.3. f (x =0f(x.4. f(x =ax + bx + c a 0 a<0 13

14 .5..4 (1 f(x = log x ( f(x =e x (3 f(x =x 4.6. f(x =e x ex.7. y = log(x + 1 y x p w p >w>0 p w.8. p = 0 x p x c =5y c y f(x =px 5x = (0 xx 5x (1 f(x ( f(x x 14

15 3 3.1 x 1,x (x 1,x R R x, y x, y x + y x + y =(x 1 + y 1,x + y α α x αx =(αx 1, αx x x x = x 1 + x x x, y x y x, y x y x y x y = x 1 y 1 + x y 3.1. p 1,p x 1,x p 1 x 1 + p x p =(p 1,p x =(x 1,x p x 3.1. x, y θ x y = x y cos θ 3.1 x, y =0x y =0 cos θ =0 cos θ =0x y 15

16 R a 1 x 1 + a x =0 a 1,a a =(a 1,a a x =0 a x =(x 1,x R l a 1 x 1 + a x + b =0 l a 1 x 1 + a x =0 a l a l normal vector 3.. p 1,p, m p 1 x 1 + p x = m (x 1,x budget line p =(p 1,p 3. n n (x 1,,x n n 0 0 n x, y α x ± y α x x ± y =(x 1 ± y 1,,x n ± y n, αx =(αx 1,, αx n. n n R n n x x x = x x n n x, y x y = x 1 y x n y n R n x y = x y cos θ a 1 x a n x n = b hyperplane a =(a 1,,a n normal vector 16

17 p 1,p,p 3, m p 1 x 1 + p x + p 3 x 3 = m (x 1,x,x 3 n p 1,,p n, m p 1 x p n x n = m (x 1,,x n n p = (p 1,,p n (i 0 (ii x ± y (iii αx 3 vector space 3.4 (. [0, 1] C[0, 1] C[0, 1] 0 0(x 0(x C[0, 1] f,g α f + g αf (f + g(x =f(x+g(x, (αf(x =αf(x f + g, αf [0, 1] f + g αf C[0, 1] 1 m n x (1, x (,, x (m 1 linear combination m α 1,, α m α 1 x (1 + + α m x (m α α m =1 α i 0 i =1,...,m convex combination 1 m n x (1, x (,, x (m 1 linearly independent m α 1,, α m α 1 x (1 + + α m x (m = 0 α 1 = = α m =0 1 1 linearly dependent x (1, x (,, x (m 1 α 1 x (1 + + α m x (m = 0 17

18 α i 0 x (i = α 1 α i x (1 α i 1 α i x (i 1 α i+1 α i x (i+1 α m α i x (m x (i m 1 x (1,...,x (i 1, x (i+1,...,x (m R e 1 =(1, 0, e =(0, 1 1 x =(x 1,x 0 e 1, e, x 1 x 1 e 1 + x e x = 0 dimension 1 R n n 3.. m n x (1, x (,, x (m R n 1 m n R n 1 n n 3.6. R n n e 1 =(1, 0,...,0, e =(0, 1, 0,...,0,...,e n =(0,...,0, n x =(x 1,...,x n e 1,...,e n 1 x = x 1 e x n e n. e 1,...,e n C[0, 1] f (n (x =x n n =1,,... {f (1 (x,f ( (x,...} 1 R n {x (1,...,x (n } R n basis (B1 {x (1,...,x (n } 1 (B n {x (1,...,x (n } {e 1,...,e n } R n standard basis R n R {e 1, e } {(1, 1, ( 1, 1} 1 n R n 18

19 3.3 R n R m R n R m mapping f : R n R m f : R n R m n f 1,,f m m x R n f(x =f(x 1,,x n = f 1 (x 1,,x n. f m (x 1,,x n f i m i (coordinate function f (vector-valued function 3.7 (. n aggregate execess demand functions R n ++ Rn p p z(p z(p = z 1 (p 1,,p n. z n (p 1,,p n R n ++ n. R n ++ = {x R n x i > 0 i =1,...,n}. 0 f : R n R m linear n x, y α f(x + y =f(x+f(y, f(αx =αf(x m = n 1 linear transformation f f(0 = (x 1,x 1 (1 x 1 (x 1, x ( ( x 1, x (3 θ 19

20 R n n e 1 =(1, 0,...,0, e =(0, 1, 0,...,0,...,e n =(0,...,0, xy- 1 f(x, y =ax + by (a, b =

21 4 4.1 matrix m n m n m n a 11 a 1n A =..... a m1 a mn a ij A i j m = n A n m n n m m n n n 1 x = x 1. x n. m n A n m A transposed matrix A a 11 a m1 A =..... a 1n a mn 4.1. A = ( ( A 14 = 5 36 m n A, B A + B A, B m n A B a 11 + b 11 a 1n + b 1n A + B =..... a m1 + b m1 a mn + b mn m n A n l B A B m l n k=1 a n 1kb k1 k=1 a 1kb kl A B =..... n k=1 a n mkb k1 k=1 a mkb kl 1

22 A B i j A i B j n 1 n n 1 4. (. A = ( ( 1 34, B = ( ( ( A B = = B A = 3 4 ( ( = ( A B B A C = ( ( 10 00, D = ( ( ( C D = = C, D ( ( C D = ( 1. ax + by = p cx + dy = q (a, b, c, d, p, q ( 19 = ( 3 34 = = ( ,. ( a b c d ( x y ( p = q 4. A A A 1 = A 1 A = E A 1 A E E = ( ( 5 1 ( 5 1 ( 1 5 ( 1 5 = ( 1 5 ( 5 1 = (

23 ( a b A = c d det A det A = ad bc 4.1. A det A (1 A α det A α ( A (3 A det A A = ( ab cd det A 0 A 1 A 1 = 1 ( d b = det A c a ( d ad bc b ad bc c a ad bc ad bc A = ( A 1 = ( 1 5 { ax + by = p cx + dy = q (a, b, c, d, p, q A = ( ab cd ( x A y ( p = q A A 1 A 1 ( ( A 1 x A = A 1 p y q ( x = y ( dp bq ad bc aq cp ad bc A 1 ( p q B1, B ( ( p b a p B 1 =, B q d =, c q 3

24 ( x y = ( det B1 det A det B det A A 1 n A A A 1 = A 1 A = E n n A 1 A E n n 1 0 E n = m n A 1 A rank A 4.3. n A (i A (ii rank A = n (iii Ax = 0 x = 0 ( 4.3 A = ab cd Ax = 0 (4.1 A (4.1 A 1 x = 0 (4.1 x 1 = x =0 (i (iii (4.1 x 1 ( a c + x ( b d = x 1,x x 1 = x =0 (a, c (b, d 1 1 rank A = (iii (ii rank A = A n A i j (n 1 (n 1 ( 0 0 A ij A det A i. det A =( 1 i+1 a i1 det A i1 +( 1 i+ a i det A i + +( 1 i+1 a in det A in 4

25 j det A =( 1 1+j a 1j det A 1j +( 1 +j a j det A j + +( 1 n+j a nj det A nj A det A A det A ij ( a11 a 1 a A = a 1 a a 3 1 a 31 a 3 a 33 det A = a 11 det A 11 a 1 det A 1 + a 11 det A 13 = a 11 (a a 33 a 3 a 3 a 1 (a 1 a 33 a 3 a 31 +a 13 (a 1 a 3 a a 31 det A = a 1 det A 1 + a det A a 3 det A 3 = a 1 (a 1 a 33 a 3 a 31 +a (a 11 a 33 a 13 a 31 a 3 (a 11 a 3 a 13 a n n A det A 0 A 1 det A 11 det A det A 1 det A 1 det A det A ( 1 n+1 det A n1 det A ( 1 1+n det A 1n det A ( 1 n det A nn det A A 1 i j ( 1 j+i det A ji det A A =

26 ( ( ( A 11 =, A 1 1 =, A 1 13 =, 1 1 ( ( ( A 1 =, A 1 =, A 1 3 =, 1 1 ( ( ( A 31 =, A 1 3 =, A =, 1 det A 11 =deta =deta 33 =3, det A 1 =deta 1 =deta 3 =deta 3 =1, det A 13 =deta 31 = 1 1 det A = det A 11 1 det A 1 +1 det A 13 = ( 1 = A 1 = det A 11 det det A 1 det A 31 det A det det A 1 det A det A det det A 3 det A det A 13 det A det A 3 det A 33 det A det A = y 1 = ax 1 + bx y = cx 1 + dx (x 1,x (y 1,y 4.5 ( y1 y = ( a b c d ( x1 x 6

27 m n n m 4.5. f : R n R m f(x =Ax m n A a 11 a 1n f(x =Ax =..... a m1 a mn x 1.. x n = a 11 x a 1n x n. a m1 x a mn x n 4.5 f(x =Ax A {e 1,...,e n } a 1j. a mj = f(e j, j =1,...,n a 11 a 1n A =..... a m1 a mn A j f(e j 4.3 n x e 1,...,e n 1 f f(x =f(x 1 e x n e n =x 1 f(e x n f(e n a 11 a 1n = x 1. a m1 + + x n. a mn = a 11 a 1n =..... a m1 a mn x 1. x n = Ax a 11 x a 1n x n. a m1 x a mn x n A 7

28 ( 0 1 (1 1 0 (4 ( (7 (x 1 x ( 1 0 ( ( ( x1 x ( (5 ( ( ( ( ( ( ( ( 0 1 (6 ( ( ( (1 ( (3 ( { ax + by = p cx + dy = q (a, b, c, d, p, q A = ( ab cd 4.4. ( (1 ( (3 ( ( ( y1 y ( a b = c d ( x1 x 4.6. x =(x 1,x θ y =(y 1,y f θ f : R R 4.5 R n L subspace 8

29 (i 0 L. (ii x, y L x ± y L. (iii x L α αx L. L 1 L dimension diml L R n diml n 4.8. (1 R x 1 = x L L {(1, 1} L dim L =1 ( R 3 x 3 =0 L L {e 1, e } L dim L = f R n R m f m n A f m f image Imf Imf = {y R m y = f(x x R n } R R f(x =(x, x R 45 Imf = {(x 1,x R x 1 = x }. Imf R {(1, 1} f 0 n f kernel kerf kerf = {x R n f(x =0}. kerf Ax = R R f(x =a 1 x 1 + a x a 0 R a 1 a kerf = {(x 1,x R a 1 x 1 + a x =0}. kerf R {(a, a 1 } 4.9, 4.10 Imf, kerf y = f(x, y = f(x y + y = f(x+f(x =f(x + x 9

30 y + y Imf α αy = αf(x =f(αx αy Imf Imf f(x =f(x =0 f(x + x =f(x+f(x =0 x + x ker f α f(αx =αf(x =0 αx ker f ker f 4.6. f R n R m (i Imf R m (ii kerf R n f : R n R m Imf ker f R m, R n 4.7 (. f R n R m dim kerf +dimimf = n. (4. f A dim Imf = ranka f(x =x 1 a 11. a m1 + + x n a 1n. a mn Imf A Imf A 1 (4. rank A = n dim kerf. (4.3 f R n R n R n y y = f(x x f inverse transformation f 1 f f 1 x, y f 1 (f(x = x, f ( f 1 (y = y (4.4 30

31 f 1 f 1 1 y = f(x, y = f(x f y + y = f(x+f(x =f(x + x f 1 (y+f 1 (y =x + x = f 1 (y + y α, f αy = αf(x =f(αx αf 1 (y =αx = f 1 (αy 1 f(x = Ax f f 1 (y =By B (4.4 x, y BAx = x, ABy = y BA = AB = E n B = A 1 A A 1 A 1 n A A f(x =Ax f 1 (y =A 1 y (a y R n y = f(x x R n (b x x f(x f(x f (a surjection (b injection (a (b bijection (a Imf = R n (b ker f = {0} f f(0 =0 ker f 0 ker f = {0} x x x x 0 f(x x 0 f(x f(x dim Imf = rank A A 1 rank A = n dim ker f =0 (4.3 rank A = n dim ker f =0 (i A 1 (ii rank A = n (iii Ax = 0 x

32 5 5.1 A α 0 x Ax = αx (5.1 α A eigen value x α A eigen vector (5.1 x, y x + y,kx k 5.1. A = ( 4 13 α =, 5 α = ( 1 1 ( ( 1 1 = ( 4 1+ ( ( 1 = ( ( 1 = 1. α =5 ( 1 ( ( 1 = ( = ( 10 5 ( = ( 1 1 ( 1 1 A α, β x, y x y 1 0 k x = ky αx = Ax = Aky = kβy = βx α β x y A α, β ( p q, ( rs P ( p r P =. q s det P 0 P 3

33 1 f A f A (5.1 f A α, β x, y 5.1 x, y 1 z a, b z = ax + by Az = A(ax + by =aax + bay = aαx + bβy 5.. A = ( 0 03 x =(x1,x ( ( ( 0 x1 x1 Ax = = 0 3 x 3x Ax A 3 e 1 = ( 1 0, e = ( 0 1 x = x 1 e 1 + x e ( ( x1 1 Ax = =x 3x 1 0 Ax =x 1 e 1 +3x e +3x ( 0 1 A = ( ab cd (5.1 ( ( ( α a b x1 0 = c α d 0 x α A (α ax 1 bx =0 cx 1 +(α dx =0 x 1 = x =0 ( α a b det c α d =0 ( α a b det c α d = α (a + dα + ad bc characteristic polynomial α A α α (a + dα + ad bc =0 (5. 33

34 (5. A characteristic equation characteristic root A A 5.3. A = ( ( 4 13 det α 4 1 α 3 = α 7α + 10 α =, 5 α 7α + 10 = 0 A n (5.1 (A αe n x = 0 E n n 1 0 A f A (α =det(a αe n α n det(a αe n =0 A n n n A b = c α (a + dα + ad b =0 (a + d 4(ad b =(a d +4b 0 A 5.. n 5.4. A = ( 1 1 ( α =1, 3 α 4α +3=0 1 ( 3 34

35 A = ( ab cd α, β x = ( pq (, y = rs Ax = αx, Ay = βy ( ( ( a b p r p r = c d q s q s ( α 0 0 β. P = ( pr qs P P 1 AP = ( α 0 0 β 5.5. A = ( ( 4 13, (, 1 ( P = det P =3 0P P 1 = 1 3 ( P 1 AP = 1 3 ( ( ( = ( A P 5.6. A = ( 1 1 ( ( P = P 1 = ( 1, 3 P 1 AP = ( , ( P P 1 P P A P 1 = P 5.3. n A n α 1,...,α n n P P 1 AP = α α n P A A P 1 = P P 35.

36 5.3 A = ( ab cd α P A = P ( α 0 0 α P 1 = αp ( P 1 = α ( = ( α 0 0 α A A = ( α 0 0 α 5. quadratic form 1 ax a 0 ax 1 + bx 1 x + cx A = ( a b/ b/ c ax1 + bx 1 x + cx x Ax ( x a 1 Ax =(x 1 x 1 b c b ( x1 x = ax 1 + bx 1 x + cx A = ( ab bc x Ax n a ij x i x j i j n n 1 ax a >0 0 x ax a<0 0 x ax a 5.1. x Ax (1 0 x x Ax > 0 A positive definite ( 0 x x Ax < 0 A negative definite (3 x x Ax 0 A positive semidefinite (4 x x Ax 0 A negative semidefinite (5 (1-(4 indefinite 36

37 5.7. (1 f(x 1,x =x 1 + x ( f(x 1,x = x 1 x = (x 1 + x (3 f(x 1,x =(x 1 x x 1 = x (x 1,x f(x 1,x =0 (4 f(x 1,x = (x 1 x x Ax = ax 1 +bx 1 x + cx a =0 x = ( x 1 0 x Ax =0 a 0 ax 1 +bx 1 x + cx ( = a x 1 + b a x ac b + x a x Ax a >0,ac b > 0 x Ax a <0,ac b > 0 a >0,ac b 0 a <0,ac b 0 ac b A 5.4. A = ( ab bc x Ax (= ax 1 +bx 1 x + cx (1 A a >0 det A = ac b > 0 ( A a <0 det A = ac b > 0 (3 A a 0, c 0, det A = ac b 0 (4 A a 0, c 0, det A = ac b (1 x 1 + x = x ( x 1 > 0, =1> 0 ( x 1 x = x ( x 1 < 0, ( 1 ( 1 0 0=1> 0 (3 (x 1 x = x 1 x 1 x + x = x ( x 1 > 0, 1 1 ( 1 ( 1 = 0 37

38 (4 (x 1 x = x 1 +x 1 x x = x ( x 1 < 0, ( 1 ( = 0 x Ax A P P AP A 5.3 Ã x Ax x Ãx 5.5. n A n (1 x Ax A ( x Ax A (3 x Ax A (4 x Ax A ( 4 1 (1 1 ( ( ( a b 0 c a c 5.3. α A x y α ax+by α a, b 5.4. (1 x 1 +3x ( x 1 x (3 x 1 +4x 1 x +4x (4 (x 1 + x 5.5. A = ( 1 1 x Ax (1 A ( A Ã 38

39 (3 A Ã 39

40 6 6.1 f(x 1,x a =(a 1,a f f(a 1 + h 1,a f(a 1,a (a = lim h 1 0 h 1 f f(a 1,a + h f(a 1,a (a = lim x h 0 h a f n f(x =f(x 1,...,x n a =(a 1,...,a n f f(a + he i f(a (a = lim x i h 0 h f(x a x i f x i (a a x i f(x x f x i (x f x i f(x x i 6.1. f(x 1,x =x 1 3 x f (x 1,x =3x 1 x, f x (x 1,x =x 1 3 x f, f x x 1,x f x i (x x j ( f x i x j (x f f x i x j (x i = j f x i (x 40

41 6.. f(x 1,x =x 1 3 x f (x =6x 1x, f x (x = f x (x =6x 1 x, f x (x =x f x (x = f x (x f(x x i x j x j x i 6.1 (. f(x f x i x j (x = f x j x i (x C r f(x r f(x C r C smooth C 6. x =(x 1,...,x n (gradient vector f(x f(x = 6.3. f(x =x 1 3 x ( f (x,, f (x. x n f(x = ( 3x 1 x, x 1 3 x D f(x x (Hessian matrix D f(x = f f (x... x 1 x n (x..... f x n (x f x (x n. 41

42 6.4. f(x =x 1 3 x D f(x = ( f (x x 1 f x (x f x (x f (x x = ( 6x1 x 6x 1 x 6x 1 x 3 x 1 1 f(x x = a y f(a =f (a(x a y f(a f(x df (a x a x dx df (a =f (adx f(x a 1 1 f (a f(x 1,x a =(a 1,a f 1 df (a =v 1 dx 1 + v dx v 1,v f a a =(a 1,a f(a + h f(a vh lim =0 h 0 h v =(v 1,v f a v x v(x =(v 1 (x,v (x 6.. f(x =f(x 1,x f v 1 (x = f (x, v (x = f x (x f df (x = f (xdx 1 + f (xdx 4

43 x = a df (a a f(x 1 f(a+ f (a(x 1 a 1 + f (a(x a dx 1,dx f(x df (a =f(a + dx f(a dx 1,dx 1 f (adx 1 + f (adx x f(x x = a f(a f(x a f(x df (a = f(adx. f(a dx 3. dx f(a f(x f(x a f(a 6.3 (. f(x =f(x 1,x g 1 (t,g (t F (t =f(g 1 (t,g (t x =(g 1 (t,g (t F (t = f (xg 1(t+ f (xg (t 6.5. f(x 1,x =x 1 3 x,g 1 (t = t, g (t = 1 t t >0 f (x =3x 1 x, f (x =x 3 1 x, g x 1(t = 1 t, g (t = 1 t F (t =f(g 1 (t,g (t F (t = 3( t ( 1 1 t t + ( t 3 1 ( 1 t t t = t 43

44 6.3 1 f(x 1,x x =(x 1,x h =(h 1,h t 1 g(t g(t =f(x + th =f(x 1 + th 1,x + th. g(t t =0 g(1 = g(0 + g (0 + 1 g (θ (6.1 θ g (0, g (θ g (0 = f (xh 1 + f x (xh g (θ = f x (x + θhh 1 + f (x + θhh 1 h 1 x + f (x + θhh h 1 + f (x + θhh x g (θ h 1,h f g (θ =(h 1 h ( f = h D f(x + θhh (x + θh x 1 f x (x + θh x f x (x + θh ( h1 f (x + θh x h (6.1 f(x + h =f(x+ f(xh + 1 h D f(x + θhh n 6.4 (. f(x C h 0 1 θ f(x + h =f(x+ f(x h + 1 h D f(x + θhh f(xh 1 1 hd f(x + θhh 1 n h 44

45 (1, 1 (1 f(x =x 1 x ( f(x = x 1 x (3 f(x =x 1 + x (4 f(x = (x 1 + x 6.. (0, 0 h 1,h θ (1 f(x =x 1 +x 1 x + x ( f(x =e x 1+x (3 f(x = 1 3 log(x log(x f(x =x 1 + x, g 1 (t =t, g (t = log t t>0 F (t =f(g 1 (t,g (t 6.4. u(x 1,x x 1 (p 1,p,m, x (p 1,p,m p 1,p m indirect utility function v(p 1,p,m 6.3 v(p 1,p,m=u (x 1 (p 1,p,m,x (p 1,p,m. v m (p 1,p,m= u (x 1,x m (p 1,p,m+ u (x 1,x x x m (p 1,p,m 45

46 n n f(x 1,,x n x 1,,x n x =(x 1,...,x n n 1 f(x n x f(x x f(x 0 x i f x i (x > 0 x i x f i f(x x i (x < 0 x i x i f(x f(x (1. f(x C 1 f(x (7.1 1 first-order conditions f x i (x =0, i =1,,n (7.1 (7.1 f(x =0.3 (7.1 (7.1 x 7.1. f(x 1,x = x 1 x (0, 0 0 f(x 1,x = ( x 1, x f(0, 0 = (0, 0 f(x 1,x =x 1 x f(x 1,x =(x 1, x f(0, 0 = (0, 0 f(0, 0 = 0 a f(a, 0 = a > 0,f(0,a= a < f(a =0 D f(a 46

47 f(x a f(a + h =f(a+ 1 hd f(a + θhh f(x C h h D f(a + θh hd f(a + θhh < 0 f(a 7. (. f(x C (1 f(a =0 D f(a f(a ( f(b =0 D f(b f(b f(x 1,x =x 3 1 x +6x 1 x f(x 1,x =(3x 1 +6x, x +6x 1 f(x 1,x =(0, 0 (x 1,x =(0, 0, ( 6, 18 D f(x 1,x = ( 6x1 6 6 D f( 6, 18 f(x 1,x ( 6, D f(0, 0 f(0, 0 7. y = f(x 1,x y y = f(x 1,x y n 7.1 (. n f(x =f(x 1,,x n (i f(x concave function n a, b 0 t 1 t tf(a+(1 tf(b f (ta +(1 tb (ii f(x convex function n a, b 0 t 1 t tf(a+(1 tf(b f (ta +(1 tb 47

48 f(x C 1 (1 f(x a, b f(b f(a+ f(a(b a ( f(x a, b f(b f(a+ f(a(b a f(x 1,x C 1 y = f(x 1,x S S (a 1,a y = f(a 1,a + f (a 1,a (x 1 a 1 + f x (a 1,a (x a S y S S y (a 1,a, (b 1,b f(b 1,b f(a 1,a + f (a 1,a (b 1 a 1 + f x (a 1,a (b a f(b f(a+ f(a(b a S a, b f(b f(a+ f(a(b a f(x C (1 f(x x D f(x ( f(x x D f(x 48

49 f(x f(a + h =f(a+ f(ah + 1 hd f(a + θhh 7.3 f(x a, h f(a + h f(a+ f(ah a, h hd f(a + θhh 0 x D f(x 7.3. f(x 1,x = x 1 x D f(x = ( 0 0 f(x 1,x 7.3 concave programming convex programming 6. f(x f(x =0 7.3 x f(x f(x + f(x (x x f(x =0 f(x f(x f(x 7.5. f(x C 1 (1 f(x f(x x f(x =0 ( f(x f(x x f(x =0 7.5 f (x =0,, f (x =0 x n 49

50 7.4. f(x 1,x = (x 1 1 (x (x 1,x =(1, D f(x = ( 0 0 f(x1,x 7.5 f(x 1,x (x 1,x f (x 1,x = x 1 =0, f x (x 1,x = x 4=0 (x 1,x =(1, 7.4 y = f(x 1,x y x 1,x p w 1,w π π(x 1,x =py (w 1 x 1 + w x =pf(x 1,x w 1 x 1 w x π (x 1,x =0, π x (x 1,x =0 f (x 1,x = w 1 p, f (x x 1,x = w p (7. (7. D π(x 1,x =pd f(x 1,x f(x 1,x π(x 1,x (x 1,x p, w 1,w factor demand function x 1 (p, w 1,w,x (p, w 1,w supply function y(p, w 1,w y(p, w 1,w =f(x 1 (p, w 1,w,x (p, w 1,w. p, w 1,w y 7.5. f(x 1,x = x 1 + x π(x 1,x π(x 1,x =p ( x 1 + x (w 1 x 1 + w x. 50

51 1 π (x 1,x = π x (x 1,x = p x 1 w 1 =0 p x w =0 x 1 = p 4w 1,x = p 4w (7.3 ( D f(x 1,x = x 1 4x x 4x f(x 1,x (7.3 (7.3 p y = 4w + p 1 4w = w 1 + w p w 1 w (1 f(x = x 1 x ( f(x =x 1 + x (3 f(x = e x 1 + log x (4 f(x =x 1 x 7.. (1 f(x,g(x f(x+g(x ( 1 f(x =a 1 x a n x n (a 1,...,a n 7.3. f(x 1,x =x 1 +x 1 x +3x x 1 +x +3 (1 f(x 1,x ( f(x 1,x 51

52 7.4. f(x 1,x = x 1 + x (x 1,x p w 1,w x 1,x,p,w 1,w > 0 (1 f(x 1,x ( π(x 1,x =pf(x 1,x (w 1 x 1 + w x (3 π(x 1,x p, w 1,w (4 w 1 = w = t n f(x 1,...,x n k positively homogeneous of degree k x 1,...,x n t> f(tx 1,...,tx n =t k f(x 1,...,x n (7.4 (1 1 k f(x =ax k a 0 ( 1 f(x 1,x =ax 1 + bx 1 f(x 1,x =ax 1 + bx 1 x + cx (3 f(x 1,x =x 1 α x β α + β f(tx 1,tx =(tx 1 α (tx β = t α+β x 1 α x β = t α+β f(x 1,x (4 f(x 1.x =min{ax 1,bx } 1 f(tx 1.x =min{atx 1,btx } = t min{ax 1,bx } = tf(x 1,x (5 CES f(x 1,x =(ax ρ 1 + bxρ 1 ρ 1 (7.4 t t =1 7.6 (. f(x 1,...,x n k f(x 1,,x n x f(x 1,,x n x n x n = kf(x 1,...,x n 5

53 economies of scale t t>1 t decreasing returns to scale t constant returns to scale t increasing returns to scale f(x 1,,x n k k<1 k =1 k> n = f(x 1,x (1 f(x 1,x =x α 1 x β α + β < 1 α + β =1 α + β > 1 ( f(x 1.x =min{ax 1,bx } (3 CES f(x 1,x =(ax ρ 1 + bxρ 1 ρ Y K L Y = F (K, L F (K, L 1 K, L 1/L Y 1/L Y L = 1 ( K F (K, L =F L L, 1 y = Y/L, k = K/L f(k =F (k, 1 y = F (k, 1 y = f(k k f(k

54 Y = F (L =al a>0 p w π(l π(l =pf (L wl = pal L =(pa wl (i pa w>0 a> w p L π(l (ii a< w p L =0 (iii a = w p L a = w p f(x 1,,x n 1 f x i (x 1,,x n = w i p, i =1,,n. 7.6 pf(x 1,,x n (w 1 x w n x n [ f(x = p x f(x ] x n (w 1 x w n x n x n =(w 1 x w n x n (w 1 x w n x n =0 F (K, L Y = F (K, L K F (K, L K + L (7.5 L 1 F (K, L K = r, F (K, L L r w (7.5 = w Y = rk + wl (7.6 54

55 Y rk wl 1 (7.6 55

56 f(x 1,,x n g(x 1,,x n g(x 1,,x n =0 (x 1,,x n f(x 1,,x n MPE MPE max f(x sub.to g(x = 0 (MPE x =(x 1,,x n n 1 x 1,x p 1,p p 1,p > 0 u(x 1,x m m>0 p 1 x 1 + p x = m (x 1,x u(x 1,x max u(x 1,x sub.to p 1 x 1 + p x = m (UMP UMP x = p 1 p x 1 + m p. (8.1 1 U(x 1 ( U(x 1 =u x 1, p 1 x 1 + m. p p u = x 1 x ( U(x 1 =x 1 p 1 x 1 + m = p 1 x 1 + m x 1 p p p p U(x 1 (x 1,x U(x 1 x 1 = x 1.3 U (x 1=0 56

57 U (x λ = u (x 1,x 1 p 1 U (x 1= u (x 1,x u x (x 1,x p 1 p =0 u (x x 1,x 1 = u (x 1 p 1 x 1,x 1 p = u x (x 1,x 1 p u (x 1,x =λ p 1, u x (x 1,x =λ p (x 1,x λ u (x x 1,x =λ p 1 1 (8. u (x x 1,x =λ p (8.3 p 1 x 1 + p x = m (8.4 (8. (8.3 (8.-(8.4 u(x =λ p (8.5 px = m (8.6 (8.5 1 MPE (x 1,x MPE (8.1 g(x 1,x =0 (x 1,x x = g x (x 1,x 0 (x 1,x g(x 1,x =0 (x 1,x x = h(x 1 1 F (x 1 F (x 1 =f (x 1,h(x 1. 57

58 (x 1,x MPE F (x 1 F (x 1 1 F (x 1= f (x 1,x + f x (x 1,x h (x 1=0 h (x 1= λ f (x 1,x g (x 1,x = g (x 1,x g x (x 1,x f x (x 1,x g x (x 1,x f (x x 1,x =λ g (x i x 1,x, i =1, i n 8.1 (1. x MPE g(x 0 (R f(x =λ g(x (8.7 g(x =0 (8.8 λ 8.1 R g(x =0 (8.7 f g MPE Step 1: MPE n +1 L L(x, λ =f(x λg(x L Lagrangean λ Lagrange multiplier 58

59 x l ( f (x 1,x, f x (x 1,x ( g (x 1,x, g x (x 1,x x f(x 1,x =f(x 1,x o x 1 g(x 1,x =0 x 1 Step : L L (x, λ =0,, L (x, λ =0, x n L (x, λ =0. λ f (x =λ g (x. f x n (x =λ g x n (x g(x =0 (8.9 Step 3: (8.9 (x, λ =(x 1,,x n, λ 8.1 g(x 0 x MPE 8.1. UMP L(x 1,x, λ L(x 1,x, λ =u(x 1,x λ(p 1 x 1 + p x m L(x 1,x, λ L 7.1 L (x 1,x, λ =0, L x (x 1,x, λ =0, (8., (8.3, (8.4 L λ (x 1,x, λ =0 8.1 MPE max f(x 1,x sub.to g(x 1,x =t. 59

60 t t = t x =(x 1,x x t x 1 = x 1 (t, x = x (t f g 1 F G F (t =f (x 1 (t,x (t, G(t =g (x 1 (t,x (t t. F (t t G(t 0 F (t G(t F (t = f (x dx 1 dt (t + f x (x dx dt (t (8.10 G (t = g (x dx 1 dt (t + g x (x dx dt (t 1. ( f (x =λ g (x f, x (x =λ g x (x (8.10 [ g F (t =λ (x dx 1 dt (t + g (x dx ] x dt (t G(t G (t =0(8.11 g (x dx 1 dt (t + g x (x dx 1 dt (t =1 F (t [ g F (t =λ (x dx 1 dt (t + g (x dx ] x dt (t = λ t F (t 8. (. t max f(x,t sub.to. g(x,t = 0 x(t v(t =f(x(t,t v (t = f (x,t λ g t t (x,t 60

61 8.3 (8. (8.3 1 (8., (8.3, (8.4 u (x 1,x u x (x 1,x = p 1 p UMP p 1,p m demand functions x 1 (p 1,p,m,x (p 1,p,m UMP indirect utility function v(p 1,p,m v(p 1,p,m=u (x 1 (p 1,p,m,x (p 1,p,m. 8. v m (p 1,p,m=λ 8.. u(x 1,x =x 1 x max x 1 x sub.to, p 1 x 1 + p x = m x 1,x 0 p 1,p,m L(x 1,x, λ =x 1 x λ(p 1 x 1 + p x m 61

62 1 3 (8.1 (8.13 L =x 1 x λp 1 =0 (8.1 L = x 1 λp =0 x (8.13 L λ = p 1x 1 p x + m =0 (8.14 x x 1 = p 1 p (x 1,x (8.1-(8.14 x 1 = m 3p 1 (8.15 x = m 3p (8.16 λ = 4m 9p 1 p (8.17 (8.15 ( (8.15 (8.16 v(p 1,p,m ( m m v(p 1,p,m= = 3p 1 3p v m (p 1,p,m= 4m 9p 1 p = λ 4m3 7p 1 p f(x 1,x w 1,w y CMP (x 1,x 7.1 min w 1 x 1 + w x sub.to. f(x 1,x =y. (CMP w 1 = λ f (x x 1,x 1 (8.18 w = λ f (x x 1,x (8.19 f(x 1,x =y (8.0 6

63 λ (8.18 (8.19 f (x 1,x f x (x 1,x = w 1 w 1 CMP w 1,w y conditional factor demand functions x 1 (w 1,w,y,x (w 1,w,y CMP cost function c(w 1,w,y CMP w 1 x 1 + w x c(w 1,w,m=w 1 x 1 (w 1,w,y+w x (w 1,w,y. y c y (w 1,w,y=λ 8.3. f(x 1,x = x 1 + x min w 1 x 1 + w x sub.to. x1 + x = y x 1,x 0, w 1,w,y L 1 3 L(x 1,x, λ = (w 1 x 1 + w x λ(y x 1 x. L = w 1 + λ 1 =0 x 1 (8.1 L = w + λ 1 x =0 x (8. L λ = y + x 1 + x =0. (8.3 63

64 (8.1 (8. x x1 = w 1 w (8.1-(8.3 ( w y x 1 = (8.4 w 1 + w ( w1 y x = (8.5 w 1 + w λ = w 1w y (8.6 w 1 + w (8.4 ( c(w 1,w,y (8.4 (8.5 w 1 x 1 + w x c(w 1,w,y=w 1 ( w y w 1 + w = w 1w w 1 + w y. + w ( w1 y w 1 + w c y (w 1,w,y= w 1w w 1 + w y = λ 8.4 m g 1 (x =0,,g m (x =0 x f(x m n m<n m>n 64

65 Step 1: L Step : L(x, λ =f(x m λ j g j (x x =(x 1,,x n n λ =(λ 1,, λ m m j=1 1 L (x, λ =0,, L (x, λ =0,, L L (x, λ =0,, (x, λ =0 x n λ 1 λ m f (x = m j=1 λ j g j (x. f x n (x = m j=1 λ j g j x n (x g 1 (x =0. g m (x =0 (8.7 Step 3: (8.7 (x, λ g 1 (x,, g m (x 1 x x 1,x p 1,p u = x 1 x =(x 1 x 1 m max x 1 x sub.to. p 1 x 1 + p x = m x 1,x,p 1,p,m (1 8.1 ( 1 x 1,x p 1,p,m (3 u = x α 1 x β α, β > 0 x 1,x p 1,p,m 8.. y = x 1 x y x 1,x w 1,w w 1,w,y min w 1 x 1 + w x sub.to. y = x 1 x 65

66 (1 7.1 ( 1 x 1,x w 1,w,y (3 ( w 1 x 1 + w x c(w 1,w,y (4 (3 c(w 1,w,y y (1 x = p 1 p x 1 + m p U(x 1 ( U(x 1 x 1 x (1 y = x 1 + x x x = ( (1 w 1 x 1 + w x C(x 1 C(x 1 (3 C(x 1 x 1 x x 1 + x =1 (1 u =x 1 + x 8.1 x 1,x ( u = x 1 + x 8. x 1,x 8.6 u (x 1,x p 1 x 1 + p x min p 1 x 1 + p x sub.to u(x 1,x =u (EMP EMP (x 1,x 7.1 u p 1 = λ (x 1,x (8.8 u p = λ (x 1,x x (8.9 u(x 1,x =u (8.30 λ (8.8 (8.9 u (x 1,x u x (x 1,x = p 1 p 66

67 EMP (x 1,x p 1,p u compensated demand functions x H 1 (p 1,p,u,x H (p 1,p,u EMP p 1,p u expenditure function e(p 1,p,u EMP p 1 x 1 + p x e(p 1,p,u=p 1 x H 1 (p 1,p,u+p x H (p 1,p,u u(x 1,x = x 1 x min p 1 x 1 + p x sub.to. x1 x = u L L(x 1,x, λ = (p 1 x 1 + p x λ(u x 1 x. 1 3 (8.31 (8.33 x L = p 1 + λ =0 (8.31 x 1 x1 L = p + λ x =0 x (8.3 L λ = u + x 1 x =0. (8.33 p x 1 = u (8.34 p 1 p1 x = u (8.35 p (8.34 (8.35 p 1 x 1 + p x ( p p1 e(p 1,p,u=p 1 (u + p u =u p 1 p. p 1 p 67

68 v (p,m= u (p,m (p,m+ u (p,m x (p,m p i p i x p i 1 (8., (8.3 [ ] v x (p,m=λ p 1 (p,m+p (p,m. p i p i p i p 1 x 1 (p,m+p x (p,m=m p i x i (p,m+p 1 p i (p,m+p x p i (p,m=0 v p i (p,m= λx i (p,m v m (p,m=λ 8.3 (. x i (p,m= v p i (p,m v m (p,m 8.5. u(x 1,x = x 1 x x 1 (p 1,p,m= m p 1, x (p 1,p,m= m p, 8.1 v(p 1,p,m= m p 1 p v m (p 1,p,m= 1 p 1 p, v p i (p 1,p,m= m 4p i p1 p (i =1, v p i (p 1,p,m v m (p = m = x i (p 1,p,m,i=1, 1,p,m p i 68

69 e p i (p,u=x H i (p,u+p 1 x H 1 p i (p,u+p x H p i (p,u 1 (8.8, (8.9 [ e u (p,u=x H i (p,u+λ (p,u xh 1 (p,u+ u ] (p,u xh (p,u p i p i x p i u ( x H 1 (p,u,x H (p,u = u p i u (p,u xh 1 p i (p,u+ u (p,u xh p i (p,u=0 * 8.4 (. e p i (p,u=x H i (p,u u(x 1,x = x 1 x x H p 1 (p 1,p,u=u, x H p1 (p 1,p,u=u p 1 p e(p 1,p,u=u p 1 p 8.4 e p (p 1,p,u=u = x H 1 (p,u, p 1 p 1 e p1 (p 1,p,u=u = x H (p,u p p UMP EMP p m UMP x 1 x u = u(x EMP x EMP 1 x EMP * 69

70 p u EMP x 1 x m = px UMP x UMP 1 x UMP 8.5 (. (1 x i (p,e(p,u = x H i (p,u, i =1,. x H i (p,v(p,m = x i (p,m, i =1,. ( u = v(p,m, m = e(p,u u = v(p,m m = e(p,u 8.7. u(x 1,x = x 1 x x i (p 1,p,m= u p 1 p p i pj = u = x H i (p 1,p,u, i =1, p i x H m i (p 1,p,u= pj u = m = x i (p 1,p,m, i =1, p 1 p p i p i u = v(p 1,p,m= m p 1 p m m =u p 1 p = e(p 1,p,u m = e(p 1,p,u=u p 1 p u u = m p 1 p = v(p 1,p,m 70

71 money metric utility p x m(p, x =e(p,u(x p m(p, x u(x u(x µ(p; q,m =e(p,v(q,m (p 0,m 0 (p 1,m 1 v(p 1,m 1 v(p 0,m 0 equivalent variation EV EV = µ(p 0 ; p 1,m 1 µ(p 0 ; p 0,m 0 =µ(p 0 ; p 1,m 1 m 0. compensation variation CV CV = µ(p 1 ; p 1,m 1 µ(p 1 ; p 0,m 0 =m 1 µ(p 1 ; p 0,m 0. H. Varian Microeconomic Analysis 71

72 9 9.1 x i 0 i =1,,n p 1 x 1 + p x m g 1 (x 1,,x n 0,,g l (x 1,,x n 0 (x 1,,x n f(x 1,,x n MPI MPI (MPI max f(x sub.tog j (x 0 (j =1,,...,l. 9. MPI f(x g j (x concave programming g j (x 0 j =1., l f(x, g 1 (x,, g l (x MPI MPI 9.1. f(x 1,x =x 1 + x, g 1 (x 1,x =x 1 1, g (x 1,x =x 1 n = l = MPI max x 1 + x sub.to. x 1 1,x 1. 7

73 4 x 1,x, λ 1, λ 1 λ 1 =0 (9.1 1 λ =0 (9. λ 1 (x 1 1 = 0 (9.3 λ (x 1 = 0 (9.4 (x 1,x, λ 1, λ =(1, 1, 1, 1 (x 1,x =(1, 1 MPI MPI g 1 (x 1,x =g (x 1,x = 0, λ 1, λ > 0 (9.1 (9. (x, λ f g 1 g (x 1,x λ 1 (x 1,x λ (x 1,x =0 f g 1 g (x 1,x λ 1 (x 1,x λ (x 1,x =0 x x x f(x =λ 1 g 1 (x+λ g (x f(x =λ 1 g 1 (x +λ g (x x MPI f(a > f(x a h = a x f(x 6.4 f(x h > 0 f(x h = [ λ 1 g 1 (x +λ g (x ] h = λ 1 g 1 (x h + λ g (x h > 0 λ 1, λ > 0 g 1 (x h > 0 g 1 (x h > 0 g 1 (x, g (x 7.4 g j (x h > 0 g j (a =g j (x + h >g j (x =0 a x MPI n m 73

74 9.1 (. f(x g j (x (j =1,,...,l x m f(x = λ j g j (x (9.5 j=1 λ j g j (x =0 (j =1,, l (9.6 g j (x 0 (j =1,, l (9.7 λ j 0 (j =1,, l (9.8 λ =(λ 1,...,λ l x MPI Karush-Kuhn-Tucker conditions KKT (9.5 1 first-order conditions L(x, λ L(x, λ =f(x m λ j g j (x. (9.5 L(x, λ x i =0 j=1 L x i (x, λ =0 (i =1,,n. (9.6 complementary slackness conditions g j (x < 0 λ j =0 (9.5 1 (9.7 x (9.8 MPI KKT 9.. max f(x 1,x =x 1 sub.to. g 1 (x 1,x =x 1 +(x 1 1 0, g (x 1,x =x 1 +(x (0, 1 1 (0, 1 1 (0, 0 (0, 0 f(0, 0 = (1, 0, g 1 (0, 0 = (0, 1, g (0, 0 = (0, 1 1 f(0, 0 = λ 1 g 1 (0, 0 + λ g (0, 0 λ 1, λ 74

75 9. KKT (0, 0 MPI KKT KKT KKT 9. (. x MPI (i g j (x (j =1,,, l (ii g j ( x < 0(j =1,,, l x KKT λ KKT constraint qualification 9. (i, (ii Slater condition 9. (ii KKT f(x g 1 (x,...,g m (x g 1 ( x < 0,...,g m ( x < 0 x MPI KKT MPI Step 1: f(x g 1 (x,...,g m (x Step : g 1 ( x < 0,...,g m ( x < 0 x Step 3: 1 (9.5, (9.6 n + l Step 4: Step 3 (9.7 ( (x, λ x MPI 9. MPI 9.3 u(x 1,x m p 1,p max u(x 1,x sub.to. p 1 x 1 + p x m, x 1 0, x 0 75

76 U1: u(x 1,x C U: p 1 > 0,p > 0,m>0 1 U ˆx 1 = m 4p 1,ˆx = m 4p ˆx 1 > 0, ˆx > 0 p 1ˆx 1 + p ˆx = m <m U1 U 9.1, 9. KKT u (x 1,x =λ 1 p 1 λ u (x 1,x =λ 1 p λ 3 x λ 1 (p 1 x 1 + p x w =0, λ x 1 =0, λ 3 x =0 p 1 x 1 + p x m, x 1 0, x (x 1,x, λ 1, λ, λ 3 KKT (x 1,x 9.3 (. u = x α 1 x 1 α (0 < α < 1 m p 1,p p 1 > 0,p > 0,m>0 max x α 1 x 1 α sub.to. p 1 x 1 + p x m, x 1 0, x 0 KKT x 1 = αm p 1, αx α 1 1 x 1 α = λ 0 p 1 λ 1, (1 αx a 1x α = λ 0 p λ, λ 0 (p 1 x 1 + p x m =0, λ 1 x 1 =0, λ x =0, p 1 x 1 + p x m, x 1 0, x 0, λ 0 0, λ 1 0, λ 0. x = (1 αm ( α a ( 1 α 1 α, λ 0 =, λ1 = λ =0 p p 1 p ( D α(α 1x α u = 1 x 1 α α(1 αx α 1 1 x α α(1 αx α 1 1 x α α(α 1x α 1 x α 1 76

77 α(α 1x α 1 x 1 α 0, α(α 1x α 1 x α 1 0, det D u =0 D u u 6.1, 6.5 p 1 > 0,p > 0,m > 0 p 1 x 1 + p x < m, x 1 > 0, x > 0 (x 1,x x 1 = αm p 1, x = (1 αm p 9.4 (. u = x 1 + x 0 <p 1 <p,m>0 max x 1 + x sub.to. p 1 x 1 + p x m, x 1 0, x 0 KKT 1=λ 0 p 1 λ 1, 1=λ 0 p λ, λ 0 (p 1 x 1 + p x m =0, λ 1 x 1 =0, λ x =0, p 1 x 1 + p x m, x 1 0, x 0, λ 0 0, λ 1 0, λ 0. x 1 = m p 1, x =0, λ 0 = 1 p 1, λ 1 =0, λ = p p 1 p 1. 1 p 1 > 0,p > 0,m > 0 p 1 x 1 + p x <m, x 1 > 0, x > 0 (x 1,x x 1 = m p 1, x =0 f(x 1,x w 1,w ȳ min w 1 x 1 + w x sub.to. f(x 1,x ȳ, x 1 0, x 0 77

78 C1: f(x 1,x C C: f(ˆx 1, ˆx > ȳ (ˆx 1, ˆx w 1 x 1 + w x 1 (w 1 x 1 + w x C1 C 9.1, 9. C f(x 1,x ȳ ȳ = f( x 1, x ( x 1, x KKT w 1 = λ 1 f (x 1,x +λ w = λ 1 f x (x 1,x +λ 3 λ 1 (ȳ f(x 1,x = 0, λ x 1 =0, λ 3 x =0 f(x 1,x y, x 1 0, x 0 λ 1 0, λ 0, λ , 9. (x 1,x, λ 1, λ, λ 3 KKT (x 1,x 9.5 (. y = x 1 x w 1,w w 1 > 0,w > 0 ȳ>0 KKT min w 1 x 1 + w x sub.to. x1 x ȳ, x 1 0, x 0 w 1 = λ 0 x 4x 1 + λ 1 w = λ 0 x1 4x + λ λ 0 (ȳ x 1 x =0, λ 1 x 1 =0, λ x =0 x1 x ȳ, x 1 0, x 0 λ 0 0, λ 1 0, λ 0 w w1 x 1 =ȳ, x =ȳ, λ 0 = w 1 w, λ 1 = λ =0 w 1 w f D f f x1 x > ȳ, x 1 > 0, x > 0 (x 1,x w w1 x 1 =ȳ, x =ȳ w 1 w 78

79 (i p 1 >p > 0 (ii p 1 = p > ȳ p 1,p x 1,x m u = x x +1 u (1 p 1 = p =1,m=4 KKT ( p 1 =1,m =3 0 p 9.4. y = x 1 + x y x 1,x p w 1,w (1 p w 1 = 1 4,w = 1 6 max p ( x 1 + ( 1 x 4 x x y p ( y w 1 = 1 4,w = 1 6 y min 1 4 x x sub.to x 1 + x y (3 ( C(y C(y (4 p y p max p y C(y 79

80 R n C (convex set C a, b C αa +(1 αb C 9.6. (1 R R R + = {(x 1,x R x 1 0 x 0} R \ R + = {(x 1,x R x 1 < 0 x < 0} ( 1 D D D D D D (3 u(x 1,x =x 1 x x 1,x 0 u(x 1,x u(1, 1 (x 1,x (1, f(x (quasi-concave function f(a f(b = f(a f(αa +(1 αb f(x (quasi-convex function f(x u(x 1,x 9.3. f(x a f(x f(a x 9.6(3 u(x 1,x =x 1 x x 1,x f(x C 1 f(x f(a f(b = f(a (b a 0 80

81 9.5. f(x C 1 a f(a 0 f(x f(a <f(b = f(a(b a > 0 (9.9 a, b (9.9 pseudo concave function 9.6. f(x C f(x 0 f(x x f(xh =0= hd f(xh 0 f(x g j (x

82 ax + by + c =0 (x, y b 0 y = a b x c b 1 y = a b x c b x (x, y y b =0a 0 x = b a y c a f(x, y = x + y 1=0 1 x = y = y = x = (x, y y 10.1 y >0 y = 1 x x >0 y x x = 1 y x y f(x, y =0 implicit function theorem 10.1 (. f(x, y (x,y (10.1 f(x, y =0 (10.1 8

83 (1 f y (x,y 0 x U x 1 g(x U x x f(x, g(x = 0 g(x x g (x = f x (x,y f y (x,y ( f x (x,y 0 y U y 1 h(y U y y f(h(y,y=0 h(y y h (y = f y (x,y f x (x,y 10.. (1 f(x, y =ax + by + c f y (x, y =b 0 x f(x, g(x = ax + b ( f(x, y =x + y 1 ( ( g (x =, x 1 x g(x = a b x c b ( a b x c + c =0 b g (x = a f(x,y b = x f(x,y y, f(x, y =0 g(x = 1 x y = g(x ( g = 1 = ( f x ( f y,, 83

84 u(x 1,x (x 1,x u(x 1,x =a a (x 1,x u(x 1,x =a u(x 1,x a =0 u x (x 1,x g u(x 1,g(x 1 = a *3 (x 1,x x = g(x 1 (x 1,x (x 1,x marginal rate of substitution 10.1 g (x 1= u (x 1,x u x (x 1,x = u(x 1,x =x 1 x u (x 1,x =x 1 x, u x (x 1,x =x 1 (x 1,x =(1, 1 u (1, 1 u x (1, 1 = 1 = (x 1,x =(a, b u (a, b u x (a, b = ab a = b a *3 x 1 84

85 10. (. f i (x, y =f i (x 1,...,x n,y 1,...,y m (i =1,...,n n + m f 1 (x 1,...,x n,y 1,...,y m =0. f n (x 1,...,x n,y 1,...,y m =0 (10. (x, y (10. *4 det f 1 f 1 (x, y x n (x, y..... f n (x, y f n x n (x, y 0 y U n m g 1 (y,,g n (y (1 U y =(y 1,,y m f 1 (g 1 (y,...,g n (y, y =0. f n (g 1 (y,...,g n (y, y =0 ( g 1,,g n y =(y 1,,y m g 1 y j (y. g n y j (y = f 1 f 1 (x, y x n (x, y..... f n (x, y f n x n (x, y 1 f 1 y j (x, y. f n y j (x, y (j =1,,m 10. exogeneous variables endogeneous variable comparative statics *4 (Jacobian matrix (Jacobian 85

86 1 f(p, t p t t p f(p,t =0 (p,t f(p, t =0 t 9.1 f p (p,t 0 p = g(t g (t = f t (p,t f p (p,t f 1 (p 1,p,a= p 1 + p + a f (p 1,p,a=3p 1 p a p 1,p a a =1 p 1 = p =1 * 5 Df p = ( f1 f 1 p 1 p f f p 1 p Df p = ( 1 3, det Df p =1 0 ( Df 1 1 p = 3 a p 1,p 10. p 1 = g 1 (a,p = g (a ( g 1 (1 g (1 ( f1 1 = Df p a f a, ( 1 = 3 ( 1 1 = ( 1 1 g 1(1 = 1 > 0, g (1 = 1 > 0. *5 (p 1,p,a=(1, 1, 1 86

87 f(x, y =x 3xy + y 3 f(x, y =7 (10.3 (1 (x, y =(4, 3 (10.3 ( ( g (4,h ( u(x 1,x =x α 1 x 1 α (, 1 x =(x 1,x x 1,x a x 1 + ax 1 x + x 1=0 (10.4 x 1 + x a +3=0 (10.5 (1 a = (x 1,x =(0, 1 (10.4,(10.5 ( a (0,

88 f(x y t+1 = f(y t, t =0, 1,, ( first-order difference equation y 0 (11.1 y 0,y 1,,y t, y = f(y y stationary point y t = y (11.1 stationary solution y t+1 = ay t + b (a, b (11. b =0 homogeneous b 0 nonhomogeneous a 1 y = b 1 a (11. y 0 y 1 = ay 0 + b y = ay 1 + b = a y 0 + b + ab y 3 = ay + b = a 3 y 0 + b + ab + a b. y t = a t y 0 + b + ab + a b + + a t 1 b b a y 0 (11. y t = { a t y 0 + (1 at b 1 a y 0 + bt a 1 a =1 (

89 y t+1 y t+1 = y t y t+1 = ay t + b b 1 a y y 1 b E O y 0 y 1 y b 1 a y t 0 <a<1,b > 0 (11.3 y 0 45 y t+1 = y t y 1,y, y 0,y 1,y, y = b 1 a (11.3, a < 1 t y t b 1 a 11.. a < 1 (11. y = b 1 a (1 y t+1 = 1 y t +1 y = 1 y +1 y = y t = 1 t (y 0 + t y 0 y t ( y t+1 =y t 1 y =y 1 y =1 y 0 (i y 0 =1, y t =1 y t = t (y (ii y 0 1, t y t ± (3 y t+1 = y t + y = y + y =1 y 0 (i y 0 =1, y t =1 (ii y 0 1 y t =( 1 t (y y t = { y 0 t y 0 + t. 89

90 y t { y t+1 = a 11 y t + a 1 z t + b 1 z t+1 = a 1 y t + a z t + b t =0, 1,,. a 11,a 1,a 1,a,b 1,b x t = ( ( ( yt a11 a, A = 1 b1, b = z t a 1 a b x t+1 = Ax t + b, t =0, 1,,. (11.4 (11.4 x = Ax + b x = ( y z stationary point xt = x t =0, 1, (7.8 (11.4 b = 0 b 0 b = 0 (11.4 x 0 = ( y 0 z 0 (7.8 x 1 = Ax 0 x = Ax 1 = A x 0 x 3 = Ax = A 3 x 0. (11.4 x t = A t x 0 (11.5 A α, β 7.3 ( α 0 0 β = P 1 AP A = P ( α 0 0 β P 1 90

91 P A = AA ( ( α 0 = P P 1 α 0 P 0 β 0 β ( α 0 = P 0 β P 1 A 3 = A ( A α 0 = P 0 β = P ( α β 3 P 1 A t = P ( P 1 α 0 P 0 β ( α t 0 0 β t P 1 P 1 P 1 (11.5 (11.4 α < 1, β < 1 t α t, β t 0 x t (11.4 b = 0 A α, β x 0 (7.8 ( α t 0 x t = P 0 β t P 1 x 0 (t =1,, α < 1, β < 1 x t t 0 b 0 x ( x t = x t x yt y = z t z x t+1 = A x t, t =0, 1,, (11.6 x t (t =0, 1, (7.8 (7.10 x t+1 = A x t x t+1 x = A(x t x x t+1 x = Ax t Ax x t+1 = Ax t + b (11.4 ( (

92 11.4. (11.4 x A α, β x 0 (11.4 ( α t 0 x t = P 0 β t P 1 (x 0 x + x (t =1,, α < 1, β < 1 x t t x A 11.3 f(x 1,x y t+ = f(y t+1,y t, t =0, 1,, (11.7 second-order difference equation y 0 y 1 (11.7 y t+ = ay t+1 + by t + c, t =0, 1, (11.8 a, b, c c =0 c 0 y = ay + by + c y stationary point y t = y t =0, 1, (11.8 a + b 1 (11.8 c =0 { y t+ = ay t+1 + by t y t+1 = y t+1 y t = ( yt+1 y t ( a b, A = 1 0 y t+1 = Ay t (11.9 A α, β 11.3 y 0 = ( y 1 y 0 (11.9 y t = P ( α t 0 0 β t P 1 y 0 (

93 ( p r P = q s P 1 = 1 ps rq ( s r q p (11.10 y t+1 = p(ry 0 sy 1 ps rq α t + r(py 0 qy 1 β t ps rq c =0 (11.8 M,N (11.11 c 0 y t = Mα t 1 + Nβ t 1 (11.11 ỹ t = y t y ỹ t+ = aỹ t+1 + bỹ t (11.1 y t t =0, 1, (11.8 (11.1 ỹ t+ = aỹ t+1 + bỹ t y t+ y = a(y t+1 y +b(y t y y t+ = ay t+1 + by t + c (11.8 (11.1 (11.11 (11.1 (11.8 ỹ t = Mα t 1 + Nβ t 1 y t = Mα t 1 + Nβ t 1 + y (11.13 A 93

94 y t+1 = 1 3 y t +4 (1 ( y 0 (3 t { y t+1 = 5 4 y t 3 4 z t z t+1 = 3 4 y t z t t =0, 1,,. (1 ( ( (3 y 0 = z 0 t y t+ = 5 6 y t y t +1 (1 ( ( 5/6 1/6 1 0 (3 y 0,y 1 94