第二讲 数学基础(I)：函数分析.pdf

1 ù êÆÄ:£I¤µ¼ê©Û •% E ŒÆ²LÆ •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 1 / 26 ù̇SN 1. Ä:¼ê 1.1 ^¼ê£utility function¤ 1.2 ) ¼ê£production function¤ 2. ~^•§ 2.1 ½Â•§ 2.2 1••§ 3. ¼ê©Û¥ Ä úªÚ½n 3.1 )¼ê •35 3.2 )¼ê 5Ÿ 3.3 ¼ê¦) 4. ‘ Š’ •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 2 / 26 1. Ä:¼ê 1.1 ^¼ê£utility function¤ ^¼ê ½Âµ˜«û¬ U = u(x), x ≥ 0 I ž¤ š dU dx 0 Ú5£non-saturated¤§=>S ^£marginal utility¤• µ = u (x) > 0 I >S 0 ^4~£diminishing marginal utility¤µ du = u00 (x) < 0 dx Source: http://economicsconcepts.com/total_utility_and_marginal_utility.htm •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 3 / 26 1. Ä:¼ê 1.1 ^¼ê£utility function¤ ^¼ê ½Âµõ«û¬ U = u(x) = u(x1 , x2 , ...xN ), xi ≥ 0 I à É-‚£indifference curve¤µ UU < UX = UY < UZ I >SO“Ç£Marginal rate of substitution¤µ ^ O\1ü û¬ j û¬ ž¤¤˜ï i ž¤þ" M RSij = − •% ±ØCž§z (E ŒÆ²LÆ ) dxi dxj êþ²LÆ£1 ù¤ 4 / 26 1. Ä:¼ê 1.1 ^¼ê£utility function¤ ^¼ê ~^/ª 1. CobbõDouglasµ u (c) = N Y cβi i i=1 2. StoneõGearyµ u (c) = N Y (ci − ci )βi , γi ≥ 0 i=1 3. Exponential utilityµ ( u (c) = −e−αc , α (α > 0) c, (α = 0) 4. Power utility functionµ ( u (c) = •% (E ŒÆ²LÆ ) c1−γ , 1−γ (γ > 0, γ 6= 1) ln c, êþ²LÆ£1 ù¤ (γ = 1) 5 / 26 1. Ä:¼ê Ø(½5^‡e •Ä <‚éºx 1.1 ^¼ê£utility function¤ ^¼ê Ú5;£risk aversion¤ § CÛÚø.AJÑ ü«ºx5;§Ý ÿÝ•I £Arrow-Pratt measure¤ µ I ýéºx5;£absolute risk-aversion§ARA¤ α=− u00 (c) u0 (c) I ƒéºx5;£relative risk-aversion§RRA¤ γ=− þãü«•IØC cu00 (c) u0 (c) ^¼ê©O¡•~ýéºx5; £CARA¤Ú~ƒéºx5;£CRRA¤ ^¼ê" öSµOŽ•ê ^£Exponential utility¤Ú˜Æ ^£Power utility¤¼ê ºx5;Xê •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 6 / 26 1. Ä:¼ê 1.2 ) ¼ê£production function¤ ) ¼ê ½Â y = f (x1 , x2 ...xN ), xi ≥ 0 I >S ∂y ∂xi Ñ£marginal product¤• µ = fi > 0 I >S Ñ4~£diminishing marginal ∂fi product¤µ ∂x = fii < 0 i I >S Ñdþ eBL²þ product¤ȳi = xyi Ñ£average •p:" Source: https://en.wikipedia.org/ wiki/Production_function •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 7 / 26 1. Ä:¼ê 1.2 ) ¼ê£production function¤ ) ¼ê ~^/ª 1. Leontief production function y = min {x1 , x2 , ...xN } 2. CobbõDouglasµ y= N Y xβi i , (βi > 0, i=1 N X βi = 1) i=1 3. Constant elasticity of substitution£CES¤µ y= "N X #1/ρ ai xρi , (ρ ∈ (−∞, 1], ai > 0, i=1 •% (E ŒÆ²LÆ ) N X ai = 1) i=1 êþ²LÆ£1 ù¤ 8 / 26 2. ~^•§ 2.1 ½Â•§ ½Â•§ I ‡*µ I ž¤öýŽ åµ N X pi ci ≤ W i=1 I ‚ûýŽ åµ N X pi xi ≤ C i=1 •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 9 / 26 2. ~^•§ 2.1 ½Â•§ ½Â•§ I ‡*µ I ž¤öýŽ åµ N X pi ci ≤ W i=1 I ‚ûýŽ åµ N X pi xi ≤ C i=1 I ÷*µ I IS) oŠµ Y = C + I + G + NX I •ÏýŽ åµ −W0 = •% (E ŒÆ²LÆ ) ∞ X N Xt (1 + r)t t=1 êþ²LÆ£1 ù¤ 9 / 26 2. ~^•§ 2.1 ½Â•§ A^µ²LØŽ I ŠâIS) oŠØŽúªµY = C + I + G + N X§PCþ x O•ÇÚÓ' ©O•µ gx = xt − xt−1 , xt−1 sx = xt−1 , (x ∈ {C, I, G, N X}) Yt−1 Kzc Ñ O•ÇŒ©)•eª§,«I¦é ÑO• ±^5lI¦ ݵd˜‡I[ ²LO•Äå" gY = X zÇ• sXgYgX §Œ sx gx = sC gC + sI gI + sG gG + sN X gN X x •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 10 / 26 2. ~^•§ 2.1 ½Â•§ A^µ²LØŽ I ŠâIS) oŠØŽúªµY = C + I + G + N X§PCþ x O•ÇÚÓ' ©O•µ gx = xt − xt−1 , xt−1 sx = xt−1 , (x ∈ {C, I, G, N X}) Yt−1 Kzc Ñ O•ÇŒ©)•eª§,«I¦é ÑO• ±^5lI¦ ݵd˜‡I[ ²LO•Äå" gY = X zÇ• sXgYgX §Œ sx gx = sC gC + sI gI + sG gG + sN X gN X x I ‰½oþ) ¼ê Y = AF (K, L) §aq/k gY = gA + sK gK + sL gL"b½o þ) ¼ê•…Ù- ‚.d/ª Yt = At Ktα L1−α §Kkµ t gY = gA + αgK + (1 − α) gL EâÚ›Ý?Úé²LO• [ •% zÇ• gA /gY §ddŒ±lø‰ ݵd˜‡I ²LO•Ÿþ" (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 10 / 26 2. ~^•§ 2.1 ½Â•§ Jones(2014)µ{I²LO•ØŽ Source: Table 6.2 in Macroeconomics by Jones(2014). •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 11 / 26 2. ~^•§ 2.2 1••§ 1••§µ±®Çû½nØ•~ ïå²d£Purchase Power Parity, PPP¤nØ@•§À1 d I Š3uÙ ïå§Ïd ISû¬½|?uþïG ž§®ÇY² Ò ûuØÓÀ1é/˜@fŒn´û¬0 E= ïåƒ'" eP P∗ I |Dzd£Interest-rate Parity, IP¤nØ@•§^ØÓÀ1Od 7K] Œ±JøƒÓ ýÏÂÃÇž§ ®½|?uþïG § eãIP^‡¤áµ Ef − 1 = i − i∗ E •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 12 / 26 2. ~^•§ 2.2 1••§ PPPÚIP ¢yu ã¡5 µFeenstra and Taylor(2014). •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 13 / 26 2. ~^•§ 2.2 1••§ 1••§µ±ž¤•~ I · µ ž¤ö •% (E ŒÆ²LÆ ) û¬I¦µci = αW pi êþ²LÆ£1 ù¤ 14 / 26 2. ~^•§ 2.2 1••§ 1••§µ±ž¤•~ I · µ ž¤ö I Ä µ I p d û¬I¦µci = αW pi ž¤nصž¤ ûu ÏÂ\Y²" CtK = C0 + CYK · Yt I #p0æZ )·±Ïb`£LCH§Life Cycle Hypothesis¤µˆÏž¤ ûu˜) oÂ\Y²§˜) ož¤ uoÂ\" CtL = CYL · Y L I 6p ù [ÈÂ\b`£PIH§Permanent income hypothesis¤µˆÏ \•)ÅÄ Œ 6ž5Â\Ú •-½ [È5Â\Yt = YtT + YtP §ž ¤• ûu[È5Â\Ü©" CtP = CYP · YtP •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 14 / 26 2. ~^•§ A^µ 2.2 1••§ ‚ ½Æ Wiki: I Engel’s law is an observation in economics stating that as income rises, the proportion of income spent on food falls, even if actual expenditure on food rises. I The law was named after the statistician Ernst Engel (1821õ1896). Source: https://en.wikipedia.org/ wiki/Engel%27s_law •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 15 / 26 2. ~^•§ 2.2 1••§ 1978*2013c¥I¢ŠØ¬[Ì êâ5 µI S/¢ŠØ¬[Ì<þÂ\9 ‚ •% (E ŒÆ²LÆ ) Xꩇ0 "î¶ü êþ²LÆ£1 ù¤ ‚ Xê §p¶ü %" 16 / 26 3. ¼ê©Û¥ Ä úªÚ½n •`) 3.1 )¼ê •35 •35 I Weierstrass½n£½4Š½n§Extreme value theorem¤ µ½Â3š˜; 8 1 X þ ëY¢Š¼ê f : X → R k4Œ£Ú4 ¤Š" I ±ž¤¯K•~§MWG£2001¤½n 3.D.1µ e p > 0§… u ëY§KUMP¯Kk)" [1]µÝþ˜m¥ •% k.48•;8"48 X ¥ z‡4•: x ∈ X" (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 17 / 26 3. ¼ê©Û¥ Ä úªÚ½n •`) 3.2 )¼ê 5Ÿ •˜5 I ±ž¤¯K•~§MWG£2001¤½n 3.D.2µ x∗ •3ž§XJ ‡à8¶XJ •% (E ŒÆ²LÆ ) Ð % ´à £ ^¼ê u []¤ §K x∗ ´˜ дî‚à £ ^¼êî‚[]¤ §K x∗ •˜" êþ²LÆ£1 ù¤ 18 / 26 3. ¼ê©Û¥ Ä úªÚ½n 3.2 )¼ê 5Ÿ ëY5†Œ‡5 I •Œz½n£Theorem of the Maximum¤ µ 8I¼ê u (x; θ) ëY§ å^‡ G (x; θ) = 0 ¤Û¹ å éA g (θ) : Θ → X •;Š!ëYéAž§•ŒŠ:éA x∗ (θ) : Θ → X •3…þŒëY§•üŠéAž•ŒŠ¼ê x∗ (θ) : Θ → R ÚƒA dŠ¼ê u (x∗ (θ) ; θ) = u∗ (θ) : Θ → R Ñ ëY" •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 19 / 26 3. ¼ê©Û¥ Ä úªÚ½n 3.2 )¼ê 5Ÿ ëY5†Œ‡5 I •Œz½n£Theorem of the Maximum¤ µ 8I¼ê u (x; θ) ëY§ å^‡ G (x; θ) = 0 ¤Û¹ å éA g (θ) : Θ → X •;Š!ëYéAž§•ŒŠ:éA x∗ (θ) : Θ → X •3…þŒëY§•üŠéAž•ŒŠ¼ê x∗ (θ) : Θ → R ÚƒA dŠ¼ê u (x∗ (θ) ; θ) = u∗ (θ) : Θ → R Ñ ëY" I ••B©Û§˜„b½ x∗ (θ) Ú u (x; θ) Œ‡" •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 19 / 26 3. ¼ê©Û¥ Ä úªÚ½n 3.3 ¼ê¦) Vúª†•`¯K ¦)^‡ I Vúªµb½¼ê f (x) 3 x∗ «•S k f (x) = f (x∗ ) + f 0 (x∗ ) (x − x∗ ) + ... + •% (E ŒÆ²LÆ ) Œ §Kµ
f k (x∗ ) k (x − x∗ ) + o (x − x∗ )k k! êþ²LÆ£1 ù¤ 20 / 26 3. ¼ê©Û¥ Ä úªÚ½n 3.3 ¼ê¦) Vúª†•`¯K ¦)^‡ Vúªµb½¼ê f (x) 3 x∗ I «•S k f (x) = f (x∗ ) + f 0 (x∗ ) (x − x∗ ) + ... + Œ §Kµ
f k (x∗ ) k (x − x∗ ) + o (x − x∗ )k k! I ±ž¤ö¯K•~§b½8I¼ê u (c) éûüCþ c 0 00 Œ‡§… ∗ u > 0, u < 0 "¦) M ax u (c) ¯K§·‚é u (c) 3 c NC‰ c VÐmµ 2 u (c) = u (c∗ ) + u0 (c∗ ) (c − c∗ ) +   u00 (c∗ ) (c − c∗ ) 2 + o (c − c∗ ) 2 c∗ ••`)§= u (c) ≤ u (c∗ )§…•S:) •% I ˜ ^‡£FOC¤: u0 (c∗ ) = 0 I ^‡£SOC¤µu00 (c∗ ) ≤ 0 (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 7‡^‡•µ 20 / 26 3. ¼ê©Û¥ Ä úªÚ½n 3.3 ¼ê¦) A^µ•`Ý]|Ü I ²L¯KµXJ• úx] ÂÃÇ rf !ºx] Ú• ÂÃÇ µ, σ §±9 ^¼ê 2 / ªÚëê§@oºx] 3‡< o] ¥AÓkõŒ'~º I êƣ㵠1 M ax U = θµ + (1 − θ)rf − Aθ2 σ 2 θ 2 ⇒θ= •% (E ŒÆ²LÆ ) µ − rf Aσ 2 êþ²LÆ£1 ù¤ 21 / 26 3. ¼ê©Û¥ Ä úªÚ½n 3.3 ¼ê¦) ꊦ) Úî-.ÅÖ•{ ¦)•§ ∇f (x) = 0 Úî-.ÅÖ•{£Newton-Raphson method¤µb½¼ê ∇f (x) 3½Â• [a, b] S˜ Œ §…•3 x0 ∈ [a, b] ÷v ∇f (x0 ) 6= 0 … ∇2 f (x0 ) 6= 0" dµ


∇f (x) ≈ ∇f x0 + ∇2 f x0 x − x0 = 0 Œ µ 
−1 
 x = x0 − ∇2 f x0 ∇f x0 XJ ∇f (x) 6= 0 … ∇2 f (x) 6= 0§KO† x0 = x -EþãÚ½ †– ∇f (x) = 0" •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 22 / 26 4. ‘ Š’ 1. ž¤† ’( C[ b½ž¤ö¡ e¡ •`z¯K£c > 0¤µ M ax u (c1 , c2 ) = α ln(c1 − c) + (1 − α) ln c2 c1 ,c2 s.t. (1) G (c1 , c2 ) = w − p1 c1 − p2 c2 = 0 Ñþã¯K )^‡•§" (2) ž¤öATXÛûüº (3) XJ c ´ ¬ (4) \ú •% ù‡ •$ž¤§þãÀJ´ÄÎÜ ‚ ½Æº .UÄ^5ýÿ˜‡I[²LO•L§¥ ’( (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ Czº 23 / 26 4. ‘ Š’ 2. ž¤);Oûü 3˜‡üÏ OLG£overlapping generation¤ .¥§b½˜)Œ±©• c” y ÚcP o üϧ‡<‰Ñž¤) ;Oûü§±•Œz˜) o ^µ M ax U =u (cy ) + βu (co ) s.t. cy + s = wy = 1 co = (1 + r) s = Rs Ù¥µβ ∈ (0, 1) •žmbyǧwy = 1 • 5z c”ž Â\§r • )‰½ |ÇY²"ž£‰±e¯Kµ Ñ . ˜ (1) ^‡¶ (2) b k CRRA /ª 1−γ ^¼ê u (c) = c1−γ §Ù¥ƒéºx5;Xê γ > 1"¦Ñ cy Ú s§©ÛK•;OÇ Ïƒ" •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 24 / 26 4. ‘ Š’ 3. ) ¼ê†Â\© ‰½|ÇÚó]Y² w, r§b½è’ ¬d‚ 5z• P = 1§•`z ¯K•£ρ < 1, 0 < α < 1¤µ M ax Π = Y − rK − wL K,L s.t. Y = A [αK ρ + (1 − α)Lρ ] (1) ѱe<þ/ª 1/ρ µ )^‡•§£x = X L¤ •`z¯K M ax π = y − rk − w k s.t. (2) è’ •% •`<þ] (E ŒÆ²LÆ ) 1/ρ y = A [αk ρ + 1 − α] •þ´õ º êþ²LÆ£1 ù¤ 25 / 26 4. ‘ Š’ 3. ) ¼ê†Â\© £Y¤ rk 0 < ρ < 1 ž§] Â\Ó' sK ≡ rK Y = y ¬É = (3) (4) Šâe¡‡ƒO“ 5 S σ K•º ½Â£Ù¥ M Px ≡ dY dx , (x = K, L) •‡ƒ > Ѥ:
d ln K  L  σ≡− PK d ln M M PL 1 y²é CES ) ¼ê ó§k σ= 1−ρ " (5) )º(J£3¤ ²L¹Â" •% (E ŒÆ²LÆ ) êþ²LÆ£1 ù¤ 26 / 26