1 (utility) 1.1 x u(x) x i x j u(x i ) u(x j ) u (x) 0, u (x) 0 u (x) x u(x) (Marginal Utility) 1.2 Cobb-Daglas 2 x 1, x 2 u(x 1, x 2 ) max x 1,x 2 u(

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1 1 (utilit) 1.1 x u(x) x i x j u(x i ) u(x j ) u (x) 0, u (x) 0 u (x) x u(x) (Marginal Utilit) 1.2 Cobb-Daglas 2 x 1, x 2 u(x 1, x 2 ) x 1,x 2 u(x 1, x 2 ) s.t. P 1 x 1 + P 2 x 2 (1) (P i :, : ) u(x 1, x 2 ) = ū (x 1, x 2 ) ( : Indifference Curve) x 1 x 2 dx 2 dx 1 (Marginal Rate of Substitution :MRS) MRS du(= 0) = u x 1 dx 1 + u x 2 dx 2 dx 2 dx 1 = u/ x 1 u/ x 2 ( ) P 1 x 1 + P 2 x 2 = 1 2 Cobb-Daglas u(x 1, x 2 ) = x α 1 x 1 α 2 (0 < α 1) P 1 = u/ x 1 = αxα 1 1 x 1 α 2 P 2 u/ x 2 (1 α)x α 1 x α 1 2 = α 1 α x 2 x 1

2 x 1, x 2 (x 1, x 2) αp 2 x 2 = (1 α)p 1 x 1 P 1 x 1 = α(p 2 x 2 + P 1 x 1 ) = α x 1 = α, x (1 α) 2 = P 1 P (1) α ln x 1 + (1 α) ln x 2 + λ[ P 1 x 1 P 2 x 2 ](= L) x 1,x 2 *1 λ (FOC) L x 1 = α x 1 λp 1 = 0, L = 1 α λp 2 = 0, x 2 x 2 L λ = P 1x 1 P 2 x 2 = 0 x 1 = α P 1, x 2 = (1 α) P 2, λ = 1 x(p, ) Warlas 1.4 min x 1,x 2 P 1 x 1 + P 2 x 2 s.t. u(x 1, x 2 ) = ū min x 1,x 2 P 1 x 1 + P 2 x 2 + µ[ū u(x 1, x 2 )] h(p, ū) Hicks 1.5 Warlas x 1, x 2 u V (P, ) Cobb-Daglas V (P 1, P 2, ) = α ln α (1 α) + (1 α) ln P 1 P 2 P 1, P 2, V P 1 = α P 1 < 0, V = 1 α < 0, P 2 P 2 V = 1 > 0 Hicks E(P 1, P 2, ū) Warlas Hicks V (P 1, P 2, ) = ū = E(P 1, P 2, ū) V E *1 (x 1, x 2 ) 2

3 E(P, V (P,)) = V (P, E(P,u)) = u x(p, ) = h(p, V (P,)) h(p, u) = x(p, E(P,u)) ( V ) ( E u ) ( Warlas V Hicks ) ( u Hicks E Warlas ) P 1 h(p 1, P 2, u) = x 1 (P 1, ) h = x 1 + x 1 x 1 = h x 1 P 1 P 1 P 1 P 1 P 1 x 1 (Slutzk Equation) h P 1 x 1 x x 1 P 1 x 1 x 2 ( ) x 1, x 2 ( ) (A B) (B C) Π Π = p wx (p :, :, w :, x : ) 2.1 f(x) f (x) > 0, f (x) < 0 = w p x + Π p (2) x (2) x f(x) (2) (2) f(x) Π x x f(x) = x a (o<a<1) w p = f (x) = ax a 1 x = ( w ap ) 1 a 1 Π = x a = ( w ap ) a a 1, Π = p wx = w 1 a a ( w ap ) 1 a 1 (p) 3 3 f(x)

4 3 ( ) 1 * p c() (c(): ) { FOC p c ((p)) = 0 1 c ((p)) (p) = 0 SOC() c 0 (p) > 0 c () (Marginal Cost:MC) c (Variable Cost:VC) (Fixed Cost:FC) ( c V () F ) p c V () F F p cv () (p) cv () (Average Variable Cost:AVC) p MC AVC 2 c () = cv () c () cv () 2 ( = c V ) () = 0 *3 MC AVC AVC 4 c() = MC AVC AVC p = c () p(= c ()) <AVC = 0 (4) *2 ( ) *3 4

5 3.2 X(p) = i x i (p) = Y (p) = i i (p) n X(p) = n (p(n)) X p p n = (p(n)) + n p p n p n = (p(n)) X (p(n)) n (p(n)) < 0 *4 p <AVC p=avc=c () 0 u() + z p z 1 m,z u() + z s.t. p + z = m u() + m p FOC u () p = 0 p = u () 3.3 p (Consumer Surplus:CS) (Producer Surplus:PS) CS+PS W CS = 0 PS = p u ()d p = u() p 0 c ()d = p c() = Π W = CS + PS = u() c() Π = CS PS E ( ) (Representive Agent) u() + z c() + z w,z u() + z s.t. c(x) + z = w u() + w c() FOC u () c () = 0 u () = c () = p W () = u () c () = 0 *4 x (p) < 0, (p) > 0 5

6 4 4.1 t (p t p E )X t (p t :, p E :, X t : ) (6) (Dead Weight) (p t p E )X t (Import Demand) (Export Suppl) p D m S ex 2 A B A B p p 7 A ( ) B * 5 5 *5 (A PS ) 6

7 5.1 (Monopolist) p() p() c() FOC d d p c () = p() + p () c () = 0 d d p (Marginal Revenue:MR) ε d/ dp/p *6 FOC ( c () = p() 1 + dp ) ( = p() ) d p ε p W = u ()d p M + p M c ()d = u( ) c( ) 0 0 W = u () c () = p() c () = p () > 0 W 8 MR = MC ( ) (Time Inconsistans) (strateg) (profit) *7 Nash u i (s 1, s 2,, s i, ) u i(s 1, s 2,, s i, ) (s 1, s 2,, s i, ) Nash Nash Nash *6 ε *7 ( ) 7

8 (Cournot Competition) p(y ) = p( ) c 1 ( 1 ), c 2 ( 2 ) i p(y ) i c i ( i ) (i=1,2) { FOC p(y ) + p (Y ) i c i ( i) = 0 SOC 2p (Y ) + p (Y ) i c i ( i) Π 1 ( 1 ( 2 ), 2 ) FOC Π = 0 d Π = 2 Π 1 d d Π d 2 = 0 d 1 = 2 Π/ d 2 2 Π/ *8 1 = 1( 2 ) 1 = 2 i ( j) 2 = 2( 1 ) 1 = 2 1 = 2 *9 Cournot-Nash n p(y ) = a by, c i ( i ) = c i FOC ( p(y ) + p (Y ) i c = 0 p(y ) 1 + dp dy (Market Share) s i i Y ( p(y ) 1 + dp ) Y dy p s i ( = p(y ) 1 + s i ε ) i = c p(y ) ) = c MC c s i = 1 n p(y ) ( nε) = c n c(=mc) (Bertrand Competition) ( ) ( ) ( c i ( i ) = c i ) 0 c 1 = a 1 b 1 p 1 + cp 2, 2 = a 2 b 2 p 2 + cp 1 (a 1, b 1, c > 0) * 10 c i ( i ) = 0 1 p 1 (a 1 b 1 p 1 + cp 1 )p 1 FOC a 1 2b 1 p 1 + cp 2 = 0 p 1 = a 1 + cp 2 2b 1 () *8 Π () < 0, 2 Π = p 1 (Y ) + p (Y ) *9 Nash *10 c 8

9 ( p 2 2a1 b 2 + a 2 c 4b 1 b 2 c 2, 2a ) 2b 1 + a 1 c 4b 1 b 2 c 2 0(= c) (Stackelberg Competition) 1st stage 1 1 2nd stage 2 2 stage game 1 2 stage (Backward Induction) 2 p(y ) = a by, c i ( i ) = c i 1 2 p(y ) 2 c 2 ( 2 ) FOC a b 1 2b 2 c = 0 2 = a c b 1 2b 1 ( ( a b 1 + a c b 1 2b )) 1 c 1 FOC 1 2 (a c 2b 1) = 0 1 = a c 2b, 2 = a c 4b (a c)2 (a c)2 Π 1 =, Π 2 = stage game 8b 16b 1st stage s advantage = a c (a c)2, Π = 3b 9b (Collusion) Π Π 1 + Π 2 Π = p( ) ( ) c 1 ( 1 ) c 2 ( 2 ) FOC Π = 0 Π = 0 1, p( ) + p ( ) ( ) c 1( 1 ) = 0 p( ) + p ( ) ( ) c 2( 2 ) = 0 c 1 = c 2 1, 2 1 Π 1 = p( ) 1 c 1 ( 1 ) Π 1 1 = p( ) + p ( ) 1 c 1( 1 ) = p ( ) 2 > ( ) p = a b( ), Π i = p i c i, Π = Π 1 + Π 2 ( 1 = 2 = a c 3b, Π 1 = Π 2 = (a c)2 9b ) Π FOC Π = Π = a c 2b( ) = 0 1, MC c ( 1 = 2) a c 4b 1 = 0 1(= 2) = a c 4b, Π (a c)2 1(= Π 2 ) = 8b 9

10 1 1 p 1 c 1 s.t. 2 = a c 4b FOC 3(a c) 4 2b 1 = 0 1 = 3(a c) 9(a c)2, Π 1 = 8b 64b (Trigger Strateg) n n n 1 Π c Π a r r δ δ(< 1) (Discount Factor) Π c Π a : Π c = : Π a = t=0 t (a c)2 δ 8b 9(a c)2 64b + t=1 = 1 (a c) 2 1 δ 8b t (a c)2 δ 9b = 9(a c)2 64b Π c Π a > 1 δ > δ (a c) 2 = (a c)2 (81 17δ) 1 δ 9b 576b(1 δ) * 11 6 (General Equibrium) 6.1 ( 1,2) (x, ) i x, x i, i X, Y X = x 1 + x 2, Y = *11 δ 10

11 X Y 1 2 Edgewarth Box 6.2 Edgewarth Box 9 9 Edgewarth Box A 2 1 (Pareto Efficient) * 12 Edgewarth Box (Contract Curve) x( ) ( x) x 10 (u 1 = u 2 = x α 1 α ) 6.3 Warlas Warlas X p n n x i = w i (x i :,w i : ) x x px > px i=1 i=1 (X, p) Warlas (X, p) Warlas X Warlas *12 11

12 : 1 I. 1. A) B) C) D)A B 2. A) B) C) D)A B E)B C 3. A) B) C) D) E)B C 4. Left Right Top 3,5 5,10 Bottom 1,30 2,100 A){Top,Left} B){Top,Right} C){Bottom,Left} D){Bottom,Right} II , 2 () c 1 ( 1 ) = 2 1, c 2 ( 2 ) = 2 p(y ) = a by Y = I. q 3 A,B,C A B C q p(q) = 100 Q, Q = q A + q B + q C MC A = 20, MC B = 7, MC C = 7 12

13 1. A B C 2. (1) A B C A B C II. P (Q) = 100 Q Q C(Q) = 40Q ( ) (1) I. 1. A() 2. D(A B ) 3. E(B C ) 4. 3 > 1 5 > 2 Top 10 > 5, 100 > 30 Right {Top,Right} II (a b( )) FOC Π 1 1 = a 2b 1 b 2 2 = 0 2 (a b( )) 2 2 FOC 2 Π 2 2 = a b 1 2b 2 1 = 0 1 = a 3 3b, 2 = a 3b 3. p(y ) = a by = a ( ) p(y ) = a (2) I. 1.A: p(q) q A MC A q A B: p(q) q B MC B q B C: p(q) q C MC C q C q A q B q B 2. B C MC B FOC 100 q A 2q B q B 7 = 0 q B(= q C) = 31 q A 3 Q = 62 + q A 3 A FOC 3.p(Q) = 100 (q A + q B + q c ) = q A 3 20 = 0 q A = 27, q B = q C = A = 243 B C

14 B C II. 1. Q 2.FOC 100 2Q 40 = 0 Q = 30 3.(100 30) = = = ( ) ( ) : 14

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P 6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P