第七讲 动态最优简介与数学基础.pdf

1Ôù Ä •`{0†êÆÄ: •% E ŒÆ²LÆ •% (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ 1 / 24 ù̇SN 1. Ä •`¯K{0 1.1 Ÿo´Ä •`¯Kº 1.2 ¯K©a!«~Ú¦)•{ 2. êÆÄ:£˜¤µ ©£‡©¤•§ 2.1 •ŸoI‡ ©£‡©¤•§º 2.2 •§| )†²-5^‡ 3. êÆÄ:£ ¤µ‘ÅL§ 3.1 ‘ÅL§Ä Vg 3.2 ²-†š²-žmS 3.3 ê •% ‰ÅL§ (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ 2 / 24 1. Ä •`¯K{0 1.1 Ÿo´Ä •`¯Kº Ÿo´Ä •`¯Kº I · •`µ‰½8I¼êÚ å^‡§ÀJ Ï•` ûü5K§½•`) x∗" M ax U (x, z; θ) x∈X s.t. G (x, z; θ) = 0 I Ä •`µ‰½8I¼êÚ å^‡§•)3, ž•7L‡ˆ G §ÀJ ∗ ˜^•`žm´» x (t) " M ax U (x (t) , z (t) ; θ) x(t)∈XT s.t. G (x (t) , z (t) ; θ) = 0 •% (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ 3 / 24 1. Ä •`¯K{0 1.1 Ÿo´Ä •`¯Kº -‡Vg I x0 = A, xT = Z ¡•>.^‡½î ^ ‡£transversality condition¤ h (x, T ) = 0¶ I x (t) •G ½G Cþ£state variable¤§b ˜m£state space¤ XT š˜§= •3#N´»¶ I o • I ^!o|d ¢Š¼ê V : XT → R ¡•dŠ¼ê£value function¤¶ å^‡¥•)£ãG =£ ››• §§½››Cþ£control variable¤ u (t) = g [x (t) , x (t0 ) , z (t) ; θ] , t0 > t •% (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ 4 / 24 1. Ä •`¯K{0 1.2 ¯K©a!«~Ú¦)•{ ¯K©a I lÑžm vs. ëYžm¶ I k•Ï. vs. ÕÏ.¶ I (½5¯K vs. ‘Å5¯K¶ ½ª(: vs. ŒCª(:µ I /Щžm¨Ð©G 0Ú/ª(žm¨ª(G eüöþ²(‰½§=• 0P•kSé (0, A) Ú (T, Z)§ ½ª(:¯K¶•‰½ (0, A)§(T, Z) ؽ•ŒCª (:¯K§dž·‚I‡˜‡î ^‡ h (x, T ) = 0 5“O£¿••`´»XÛ /BL0ª(‚¤ "ŒCª(:qŒ©•n«œ¹µ R†ª(‚ •% (E ŒÆ²LÆ ) Y²ª(‚ êþ²LÆ£1Ôù¤ ª(-‚ 5 / 24 1. Ä •`¯K{0 1.2 ¯K©a!«~Ú¦)•{ ¯K«~ 1. )·±Ï;O RT M ax 0 e−ρt ln (c) dt s.t. dk ≡ k̇ = w + rk − c dt k (0) = k (T ) = 0 2. (½5ÚØ(½5œ¹e M ax U0 = ∞ P •`O• M ax E (U0 ) = E β t ln ct t=0 k0 = 1, lim β t u0 (ct ) kt+1 = 0  β u (ct ) s.t. ct + kt+1 = zt f (kt ) + (1 − δ) kt   k0 = 1, lim E0 β t u0 (ct ) kt+1 = 0 t→∞ t→∞ ln zt = ρ ln zt−1 + εt , z0 = 1, ρ ∈ (0, 1) 3. óŠ|Ï  M ax (E ŒÆ²LÆ ) t t=0 s.t. ct + kt+1 = zf (kt ) + (1 − δ) kt •% ∞ P x , β 1−β Z ∞  ydF (y) 0 êþ²LÆ£1Ôù¤ 6 / 24 1. Ä •`¯K{0 1.2 ¯K©a!«~Ú¦)•{ ¦)•{ ±(½5 ½ª(:¯K•~ £x (0) = A, x (T ) = Z¤ µ ¦)•{ ¯K£ã .‚KF M ax V = M ax V = •`›› Ä t=0 F (xt , xt+1 ) V (x) = î.•§ RT s.t. x0 = g (t, x, u) 5y Ù¥µG PT F (t, x, x0 ) dt t=0 RT M ax V = t=0 F (t, x, u) dt ;C© êÆóä M ax x0 ∈g(t,x,u) F (x) + βV (x0 ) Ç—  ù•§ Cþ x !››Cþ u!dŠ¼ê V !üѼꣽ=£•§¤g§±9 Ï£ ¼ê F" •% (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ 7 / 24 2. êÆÄ:£˜¤ µ ©£‡©¤•§ 2.1 •ŸoI‡ ©£‡©¤•§º •ŸoI‡ ©£‡©¤•§º ±(½5 •`O•¯K•~µ ∞ P M ax∞ U0 = β t ln ct t=0 {kt+1 }t=0 s.t. ct + kt+1 = zf (kt ) + (1 − δ) kt k0 = 1, lim β t u0 (ct ) kt+1 = 0 t→∞ duµ d˜ ∞ X M ax∞ U0 = β t ln [zf (kt ) + (1 − δ) kt − kt+1 ] {kt+1 }t=0 t=0 0 =0Œ ^‡ dkdU t+1 î.•§£Euler equation¤µ   ct+1 = β zf 0 (kt+1 ) + 1 − δ ct •% (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ 8 / 24 2. êÆÄ:£˜¤ µ ©£‡©¤•§ 2.1 •ŸoI‡ ©£‡©¤•§º •ŸoI‡ ©£‡©¤•§º î.•§ÚýŽ 啧éá ( c t+1 ct ˜ ©•§=•)^‡µ = β [zf 0 (kt+1 ) + 1 − δ] ct + kt+1 = f (kt ) + (1 − δ) kt ·‚'% ´µ I ‰½˜‡< ) Щ] •þ k0 §¦AT3z˜Ïž¤õ !;Oõ §±¢y˜ ^•Œz§=•`´»£optimal path¤ {ct , kt+1 }∞ t=0 ´Ÿoº I ²L•Ï Ä üzUĈ I l Ñu§²L•- cG - £steady states¤ lim kt = k∗ , lim ct = c∗ º t→∞ LÞò„ÌN t→∞ =£Ä £transitional dynamics¤L§º nþ§3lÑžmÄ •`¯K¥§)^‡´ ^‡Ò´‡©•§|§Ïd •% (E ŒÆ²LÆ ) ©•§|§éuëYžm¯K§) ©£‡©¤•§´·‚©ÛÄ êþ²LÆ£1Ôù¤ •`¯K Ä óä" 9 / 24 ˜ ˜ yt 2. êÆÄ:£˜¤ µ ©£‡©¤•§ ©•§ )†²-5^‡ Ä L§Œ±L㕘 2.2 •§| )†²-5^‡ 1 ˜
©•§£Ù¥ εt *N 0, σ 2 , y0 = µ + ε0 ¤ µ yt = µ + φyt−1 + εt = µ t X φs + s=0 ∂yt = φt § du ∂ε 0 ù žm t → ∞ ž§yt t X φs εt−s s=0 ²-5¿›X Ï ‘ÅÀÂK•k•§ du |φ| < 1" ÄK φ > 1 ž yt ¬üNuѧφ < −1 ž¬ ~X3 Solow-Swan È\Œ±Lã•š‚5˜ .¥§˜‡I[ ˜ uÑ" ] ©•§µ Kt+1 = (1 − δ) Kt + sYt = (1 − δ) Kt + sAKtα ‰½ëê δ = 0.05, s = 0.4, A = 1, α = 0.3§ mãЫ K0 = 1 Ú K0 = 5 ü«œ¹e] Ä ü?L§" Kt 1 3d-:?Ø‚5 ©•§§'u‡©•§ •% (E ŒÆ²LÆ ) ¦)Œë„÷8,!¢ï êþ²LÆ£1Ôù¤ £2014¤ " 10 / 24 2. êÆÄ:£˜¤ µ õ ˜ ©£‡©¤•§ 2.2 •§| )†²-5^‡ ©•§| )†²-5^‡ ˜ p ©•§ yt = µ + φ1 yt−1 + φ2 yt−2 ... + φp yt−p + εt Œ±^ü {U •õ  ˜ • ©•§| Yt = µ̃ + ΦYt−1 + ε̃t µ         yt µ φ1 φ2 · · · φp yt−1 εt           yt−1   0   1     0 ··· 0         yt−2   0   = . + .  .  +  .  . . . .    ..   .. .. .. .. ..   ..   ..           yt−p+1 0 0 ··· 1 0 yt−p 0 þ㕧|•Œ± Š O • (I − ΦL) Yt = µ̃ + ε̃t §ŠâÇ— î(1999)§ Φ ¤kA Ñ u1 £|λi | < 1, (i = 1, 2...p)¤ž§•§| )•µ Yt = (I − ΦL)−1 (µ̃ + ε̃t ) Ù¥ (I − ΦL)−1 = I + ΦL + (ΦL)2 + ... = +∞ P (ΦL)j" j=0 •% (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ 11 / 24 2. êÆÄ:£˜¤ µ ± AR (2) •~ ©£‡©¤•§ 2.2 •§| )†²-5^‡ -½5^‡ AR (2) L§ yt = µ + φ1 yt−1 + φ2 yt−2 + εt Œ • ˜ •§|µ " # " #" # " # " # yt φ1 φ2 yt−1 µ εt + = + yt−1 0 1 0 yt−2 0 ¦) Φ A •§ |λI − Φ| = A Šµ λ − φ1 −φ2 −1 λ 2 = λ − φ1 λ − φ2 = 0 √ φ ± φ2 1 +4φ2 ⇒ λ1,2 = 1 2 yt ²-5^‡ |λ1,2 | < 1 I‡Óž÷vµ −1 ≤ φ2 ≤ 1!φ2 < 1 − φ1!φ2 < 1 + φ1 § =(φ1 , φ2 )á3m㤫 ùÚn .«• S" •% (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ 12 / 24 3. êÆÄ:£ ¤µ‘ÅL§ 3.1 ‘ÅL§Ä Vg ‘ÅL§ ½Â†êiA I ‘ÅL§£stochastic process¤´‘ÅCþUžm?ü t ∈ T §Y (t, ω) Ñ´˜‡‘ÅCþ§K¡e¡ 8Ü" éz‡ëê ‘ÅCþx•‘ÅL§µ YT = {Y (t, ω) |Y : T × Ω → R} I êiA µ êiA ½Â þŠ µt E (Yt ) • V ar (Yt ) • γt,s gƒ'Xê £ACF¤ ρt,s Š T 1 P Ȳ = T h i E (Yt − µt )2 E [(Yt − µt ) (Ys − µs )] 1 s2Y = T −1 1 T −l−1 √ γt,s √ γt,t γs,s (E ŒÆ²LÆ ) T P Yt − Ȳ t=1 Yt − Ȳ

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2 t=1 TP −l t=1 •% TP −l Yt t=1 v u u (Yt −Ȳ )2 t , s=t+l T P (Ys −Ȳ )2 s=l+1 13 / 24 3. êÆÄ:£ ¤µ‘ÅL§ 3.1 ‘ÅL§Ä Vg ‘ÅL§ ©aµžmëê†G ˜m 1. žmëê t ´ÄëYµS I t ∈ {... − 2, −1, 0, 1, 2...} •Œ I t ∈ (−∞, ∞) •ØŒ 2. G •% £series¤vs. L§£process¤ lÑCþž§¡ {Yt } •žmS ¶ ëYCþž§¡ {Y (t)} •‘ÅL§" ˜m Yt ∈ S ´ÄëY I lÑG ˜mµS = {S1 , S2 , ...}§Xê ‰Åó" I ëYG ˜mµS = R§X‘BL§Úê ‰ÅL§ " (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ 14 / 24 3. êÆÄ:£ ¤µ‘ÅL§ 3.1 ‘ÅL§Ä Vg ‘ÅL§ ©aµVÇA 3. éÜ©Ù´ÄCzµ²-£stationary¤vs. š²-£non-stationary¤ I ∀t1 , t2 ...tN ∈ T Ú s > 0§XJ Yt1 +s , Yt2 +s ...YtN +s Ú Yt1 , Yt2 ...YtN kƒÓ éÜ©Ù§K YT î²-" I ∀t ∈ T §þŠ E (X) = µ Ú • Cov (Yt , Yt+j ) = γj Ø‘žm t Cz§K YT f²-£½°²-! I • ²-¤¶ Ø÷vþãü«½Â ‘ÅL§§´š²- " 4. ^‡VÇ´ÄÕá"~Xê ‰Åó ½Â•µ P (Yt+1 = st+1 |Y1 = s1 , Y2 = s2 , . . . , Yt = st ) = P (Yt+1 = st+1 |Yt = st ) ‘BL§Úê ‰ÅL§•äkT5Ÿ§ •% (E ŒÆ²LÆ ) êþ²LÆ£1Ôù¤ AR (2) ÒØä ê¼5Ÿ" 15 / 24 3. êÆÄ:£ ¤µ‘ÅL§ 3.2 ²-†š²-žmS ²-žmS ©a I AR (p)µp g£8£autoregressive¤ Yt = φ0 + φ1 Yt−1 + ... + φp Yt−p + εt , εt *N 0, σ 2 I M A (q)µq
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