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
1 l() = 1 (f() + g() f() g() ) 2 2 1 45 = 2 e 1 log 1: 2
3 1 = = ( ) 2: = = = = 0 { if 0 = if < 0 2 = 2, 0 = 0, 2 = 2 3
= 2 0 3: 1 2 1 0.5 2 = 2 0 4: f 1 1 1 1 1 4
= 0 5: 4 = 6 0 = 0 0 6: 1 2 3 5
= 3 0 7: 3 = 0 5 2 2 z = f(, ) z = 2 + 2 z = 0 0 = 2 + 2 (0, 0) z = 1 1 = 2 + 2 1 z = 2 2 = 2 + 2 2 z (, ) ( ) 3 1 2 (,, z) = ( 1 1 2, 2, 1) 1 6
z 2 1 1 1 ( 1 2, 1 2 ) 8: 3 1 2 6 ( ) a b f b a f()d sum s a b f() 2 7
z 1 ( 1 2, 1 2 ) 9: s 2 1 5 5 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
z 1 ( 1 2, 1 2 ) 10: = 2 = 2 13 2 2 ( ) 2 ( n ) = n n 1 n 9
f a b 11: f() = 2 5 i=1 i2 4 1 5 2 12: 2 ( ) 2 13: 10
8 1 f : X Y X 8.1 2 ( 1, 1 ) ( 2, 2 ) l l ( 1, 1 ) ( 2, 2 ) 0 14: = = l l = 2 1 2 1 = 2 1 2 1 ( 1 ) + 1 2 3 11
(1) m = m( 1 ) + 1 ( 1, 1 ) (2) m = m + b b (3) ( 1, 1 ) = 2 1 ( 1 ) + 1 2 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
1 = 1 2 0 = 0 1 = 1 2 = 1 3 1 = 1 = 2 + 2 = 2 + 2 16 = 2 + 2 = 2 + 2 = 1 = 1 2 0 16: = 1 2 2 = 0 = 4 8.3 0 = 0 + f( 0 ) f() f ( 0 ) f( 0 ) 13
f() f( 0 ) + f ( 0 ) f ( 0 ) f( 0 ) 0 17: f ( 0 ) 0 f() f( 0 ) + f ( 0 ) 2 (1.03) 2 2 2 f() = 2 0 = 1 = 0.03 f () = 2 f() f( 0 ) + f ( 0 ) = 2 0 + 2 0 = 1 2 + 2 1 0.03 = 1.06 (1.03) 2 = 1.0609 14
3 2.01 2 13 (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 0 0 18 15
= 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 0 2 18 0 0 3 f I f () > 0 ( I) = f I f () < 0 ( I) = f I f () = 0 ( I) = f I 10 19 = 0 1 (local maimum) = 0 f 0 16
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 ) = 0 18 2 17
4 (1) = 3 3 (2) = 3 + 12 2 11 ( ) 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
3 Q C(Q) C(Q) = Q 2 + 10Q + 9 3 MC = C (Q) = (Q 2 + 10Q + 9) = 2Q + 10 F C = 9 V C = Q 2 + 10Q AC = C(Q) Q = Q + 10 + 9 Q AV C = V C Q = Q + 10 MC = dc dq = 2Q + 10 21 ( ) MC AC AV C 0 Q 21: 5 MC(3) = AC(3) 19
4 Q C(Q) C(Q) = Q 3 4Q 2 + 7Q + 64 4 MC = C (Q) = 3Q 2 8Q + 7 F C = 64 V C = Q 3 4Q 2 + 7Q AC = C(Q) Q = Q2 4Q + 7 + 64 Q AV C = V C Q = Q2 4Q + 7 20 6 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
1 = 5 p C(q) = q 2 /2 5 ( ) q 2 = 1 2 2 2q = q p q = p 7 Q C(Q) C(Q) = Q 2 + 10Q + 9 12 f,g f/g (f/g)() = f() g() g() 0 h h() = f() g() (2) 21
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 2 8 2 3 + 5 2 + 1 9 2 2 22
10 ( ) 11 a + b 13 21 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
( ) 1 = g g g 2 g() 0 6 1/ 6 (1/) = () 2 = 1/ 2 1/ = 1 1/ 2 = 2 12 1 3 + 3 2 13 15 2 1 0 1 2 2 = 1 2 1 = 1 0 = 1 2 n = 1 n (2 3 ) 2 = 2 3 2 3 = (2 2 2 ) (2 2 2 ) = 2 6 (2 3 ) 2 = 2 3 2 = 2 6 2 ( a ) b = a b 24
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) 14 1 6 15 ( : f()/g() f() 1/g() ) 16 1. 2. 3. 4. 5. 6. 25
n f() = n f () = n n 1 17 (1) 1/ (2) 1 2 (3) 2 16 (production function) 8 155Kg 155 2 = 0 8 9 22: 1 (marginal product of labor) MPL (marginal product) 1 26
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 2 = 1 2 L (diminishing returns) 8 9 10 5 170 165 155 = 5 = 10 0 8 9 10 23: L Y = F (L) = L p w π(l) = pf (L) wl (3) 27
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 1 1000 100 1000/100 = 10 1 10 2 = 24 F (L) Y Y = w p L + π p Y = L Y E Y = w p L + π p L L 24: 28
π L Y π = py wl Y Y = w p L + π p (4) w/p L (4) ulπ E 29