7. 1 max max min f g h h(x) = max{f(x), g(x)} f g h l(x) l(x) = min{f(x), g(x)} f g 1 f g h(x) = max{f(x), g(x)} l(x) = min{f(x), g(x)} h(x) = 1 (f(x)

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1 7. 1 ma ma min f g h h() = ma{f(), g()} f g h l() l() = min{f(), g()} f g 1 f g h() = ma{f(), g()} l() = min{f(), g()} h() = 1 (f() + g() + f() g() ) 2 1

2 1 l() = 1 (f() + g() f() g() ) = 2 e 1 log 1: 2

3 3 1 = = ( ) 2: = = = = 0 { if 0 = if < 0 2 = 2, 0 = 0, 2 = 2 3

4 = 2 0 3: = 2 0 4: f

5 = 0 5: 4 = 6 0 = 0 0 6:

6 = 3 0 7: 3 = z = f(, ) z = z = 0 0 = (0, 0) z = 1 1 = z = 2 2 = z (, ) ( ) (,, z) = ( 1 1 2, 2, 1) 1 6

7 z ( 1 2, 1 2 ) 8: ( ) a b f b a f()d sum s a b f() 2 7

8 z 1 ( 1 2, 1 2 ) 9: s i=1 i 2 i a 1 b 5 f() i 2 1 i 7 1. ( n ) = n n 1 2. (kf) = kf (k ) 3. (f + g) () = f () + g () 8

9 z 1 ( 1 2, 1 2 ) 10: = 2 = ( ) 2 ( n ) = n n 1 n 9

10 f a b 11: f() = 2 5 i=1 i : 2 ( ) 2 13: 10

11 8 1 f : X Y X ( 1, 1 ) ( 2, 2 ) l l ( 1, 1 ) ( 2, 2 ) 0 14: = = l l = = ( 1 )

12 (1) m = m( 1 ) + 1 ( 1, 1 ) (2) m = m + b b (3) ( 1, 1 ) = 2 1 ( 1 ) ( 2, 2 ) 8.2 f 0 f f 0 f f( 0 ) 0 15: f 0 f ( 0 ) 0 f ( 0 ) 0 f( 0 ) ( 0, f( 0 )) (1) f 0 = f ( 0 )( 0 ) + f( 0 ) f 0 = f ( 0 )( 0 ) + f( 0 ) 12

13 1 = = 0 1 = 1 2 = = 1 = = = = = 1 = : = = 0 = = 0 + f( 0 ) f() f ( 0 ) f( 0 ) 13

14 f() f( 0 ) + f ( 0 ) f ( 0 ) f( 0 ) 0 17: f ( 0 ) 0 f() f( 0 ) + f ( 0 ) 2 (1.03) f() = 2 0 = 1 = 0.03 f () = 2 f() f( 0 ) + f ( 0 ) = = = 1.06 (1.03) 2 =

15 (2.01) 2 0 f( 0 ) + f ( 0 )( 0 ) f() = 0 f() f( 0 ) + f ( 0 ) f ( 0 ) = f() f( 0 ) 0 9 f 0 0 (a, b) a < < 0 = f() f( 0 ), 0 < < b = f( 0 ) f() 16 2 f 0 f ( 0 ) > 0 = f 0 f ( 0 ) < 0 = f

16 = 3 = 3 = 2 f (0) = 0 f f (0) = 0 f f (0) = 0 f 18: f (0) = 0 = 0 f = 0 f = 0 f = 0 f f I f () > 0 ( I) = f I f () < 0 ( I) = f I f () = 0 ( I) = f I = 0 1 (local maimum) = 0 f 0 16

17 19: (a, b) f( 0 ) f() ( (a, b)) = 0 f 0 (a, b) f( 0 ) f() ( (a, b)) = 0 > < f 0 f 4 f 0 f ( 0 ) = 0 f ( 0 ) =

18 4 (1) = 3 3 (2) = ( ) C V C 64 0 F C Q 20: (total cost) TC or C Q Q C(Q) (marginal cost) MC 1 (fied cost) FC (variable cost) VC (AC) (AVC) 20 18

19 3 Q C(Q) C(Q) = Q Q MC = C (Q) = (Q Q + 9) = 2Q + 10 F C = 9 V C = Q Q AC = C(Q) Q = Q Q AV C = V C Q = Q + 10 MC = dc dq = 2Q ( ) MC AC AV C 0 Q 21: 5 MC(3) = AC(3) 19

20 4 Q C(Q) C(Q) = Q 3 4Q 2 + 7Q MC = C (Q) = 3Q 2 8Q + 7 F C = 64 V C = Q 3 4Q 2 + 7Q AC = C(Q) Q = Q2 4Q Q AV C = V C Q = Q2 4Q π(q) = pq C(Q) (1) p Q π(q) 16 0 (1) 0 dπ(q) dq = (pq C(Q)) = p(q) C (Q) = p 1 MC(Q) = p MC(Q) dπ(q) dq = 0 p MC(Q) = 0 p = MC(Q) 20

21 1 = 5 p C(q) = q 2 /2 5 ( ) q 2 = q = q p q = p 7 Q C(Q) C(Q) = Q Q f,g f/g (f/g)() = f() g() g() 0 h h() = f() g() (2) 21

22 h (2) f() = g()h() f () = g ()h() + g()h () h () h () = f () g ()h() g() h() (2) ( ) f () g () f() h g() () = g() = f ()g() g ()f() [g()] 2 ( ) f () = f ()g() g ()f() g (g()) 2 g() 0 g = 1 g = 0 f 1 f ( ) f = f g g f g g

23 10 ( ) 11 a + b dac dq = d dq ( ) C(Q) = C (Q) Q C(Q) (Q) Q = C (Q)Q C(Q) Q 2 Q 2 C (Q)Q C(Q) = 0 C (Q) = C(Q) Q MC(Q) = AC(Q) 14 f() = 1 g() ( ) 1 () = 1 g g() g() ( ) 1 () = g () g (g()) 2 23

24 ( ) 1 = g g g 2 g() 0 6 1/ 6 (1/) = () 2 = 1/ 2 1/ = 1 1/ 2 = = = 1 0 = 1 2 n = 1 n (2 3 ) 2 = = (2 2 2 ) (2 2 2 ) = 2 6 (2 3 ) 2 = = ( a ) b = a b 24

25 d d n = d ( ) 1 d n = n n 1 d d n nn 1 = ( n ) = = n n 1 (2n) = n n 1 2n 2 2n n k d d k = k k 1 n ( n ) = n n 1 (n Z) ( : f()/g() f() 1/g() )

26 n f() = n f () = n n 1 17 (1) 1/ (2) 1 2 (3) 2 16 (production function) 8 155Kg = : 1 (marginal product of labor) MPL (marginal product) 1 26

27 7 L Y Y = F (L) = L (L R + ) 7 MP L = F (L) = d L dl = 1 2 L1/2 1 = L = 1 2 L (diminishing returns) = 5 = : L Y = F (L) = L p w π(l) = pf (L) wl (3) 27

28 L (3) L dπ(l) dl = (pf (L)) (wl) = p(f (L)) w(l) = pf (L) w (3) 0 pf (L) w = 0 F (L) = w p /100 = = 24 F (L) Y Y = w p L + π p Y = L Y E Y = w p L + π p L L 24: 28

29 π L Y π = py wl Y Y = w p L + π p (4) w/p L (4) ulπ E 29

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0 = m 2p 1 p = 1/2 p y = 1 m = 1 2 d ( + 1)2 d ( + 1) 2 = d d ( + 1)2 = = 2( + 1) 2 g() 2 f() f() = [g()] 2 = g()g() f f () = [g()g()] 8. 2 1 2 1 2 ma,y u(, y) s.t. p + p y y = m u y y p p y y m u(, y) = y p + p y y = m y ( ) 1 y = (m p ) p y = m p y p p y 2 0 m/p U U() = m p y p p y 2 2 du() d = m p y 2p p y 1 0 = m 2p 1 p = 1/2 p y

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