Studying the Ramsey Model

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<p> 1Ramsey Model</p> <p>OLG </p> <p>1</p> <p>1 t1 max J = I(x, u, t)dt + F (x1 , t1 ), (1.0.1){u(t)} t0</p> <p>x = f (x, u, t), x(t0 ) = x(t1 ) = {u(t)} x0 , x1 , U.</p> <p>(1.0.2) (1.0.3) (1.0.4) (1.0.5)</p> <p>x state u control x u t x y co-state H </p> <p>H(x, u, y, t) = I(x, u, t) + yf (x, u, t).</p> <p>(1.0.6)</p> <p>H u</p> <p>= 0,</p> <p>(1.0.7) (1.0.8) (1.0.9) (1.0.10) (1.0.11) (1.0.12)</p> <p>H , y H , y = x x(t0 ) = x0 , x = x(t1 ) = y(t1 ) = x1 F . x1</p> <p>1 </p> <p>Intriligator 1971 </p> <p>1</p> <p>2</p> <p>Social Planner : </p> <p> Blanchard and Fisher (1989) 2 Social Planner </p> <p>2.1</p> <p>Kt t Nt t Yt </p> <p>Yt = F (Kt , Nt ) Yt Ct dKt /dt dKt . (2.1.1) dt Yt = Ct + n n Constant Returns to Scale:CRS f a x af (x) = f (ax) </p> <p>Yt F (Kt , Nt ) Kt N t Kt = = F( , ) = F ( , 1). Nt Nt Nt Nt Ntdt t Nt = K dKt</p> <p>(2.1.2)</p> <p> yt = Yt /Nt kt = Kt /Nt yt = f (kt ) = F (kt , 1)dkt dt +nkt</p> <p> Ct /Nt = ct </p> <p>2</p> <p>yt = f (kt ) = ct +</p> <p>dk + nkt . dt</p> <p>(2.1.3)</p> <p>dk = f (kt ) ct nkt . (2.1.4) dt f kt kt f (kt ) 0kt 0 f (kt ) </p> <p>2k</p> <p>t</p> <p>=</p> <p>Kt Nt</p> <p> kt Nt = Kt t </p> <p>dkt Nt dt</p> <p>+ k dNt = dt</p> <p>dKt /dt dKt N dt t</p> <p>=</p> <p>dkt dt</p> <p>+ nkt</p> <p>2</p> <p>2.2</p> <p> k0 (&gt; 0) T </p> <p>lim kT eT 0</p> <p>(2.2.1)</p> <p> T kT +1 </p> <p>2.3</p> <p>Social Planner </p> <p>Social Planner kt ct kt state ct control &gt; 0 U 3 </p> <p> U=0</p> <p>u(ct )et dt.</p> <p>(2.3.1)</p> <p>U {ct } </p> <p> U =0</p> <p>u(ct )et dt</p> <p>(2.3.2) (2.3.3) (2.3.4)</p> <p> kt = f (kt ) ct nkt k0 &gt; 0t t</p> <p>lim kt e</p> <p>0</p> <p>(2.3.5)</p> <p>state kt co-state qt </p> <p>H(kt , ct , qt ) = u(ct )et qt (nkt + ct f (kt )). t = et qt 4 </p> <p>(2.3.6)</p> <p>H(kt , ct , t (qt )) = et {u(ct ) t (nkt + ct f (kt ))}.ct , kt , qt H qt </p> <p>(2.3.7)</p> <p>H k(t) = = f (kt ) nkt ct . qtct </p> <p>(2.3.8)</p> <p>H = et (u (ct ) t ) = 0. ct</p> <p>(2.3.9)</p> <p>3 et u(c ) U &lt; t 4 q t t </p> <p>3</p> <p>u (ct ) = t .kt </p> <p>(2.3.10)</p> <p>q= t </p> <p>H = qt (n f (kt )) = et t (n f (kt )). kt d t (e qt ) = et qt + et qt . dt</p> <p>(2.3.11)</p> <p>t = </p> <p>(2.3.12)</p> <p>t </p> <p>= et qt + et et t (n f (kt )) = t + t (n f (kt )) = t ( + n f (kt )).</p> <p>(2.3.13) (2.3.14) (2.3.15)</p> <p> kt , ct , t 1 c implicit </p> <p> kt</p> <p>=</p> <p>f (kt ) nkt ct t t ( + n f (kt )).</p> <p>(2.3.16) (2.3.17) (2.3.18)</p> <p>u (ct ) = t =</p> <p>2.4</p> <p> lim kt et = 0 F (xT , T ) kT eT </p> <p>F = T = T et kT complementary slackness condition </p> <p>T kT eT = 05 T = u (cT )eT </p> <p>kT u (cT )eT = 0 T T </p> <p>lim kT u (cT )eT = 0</p> <p>6 </p> <p>5 6 </p> <p>U </p> <p>4</p> <p>2.5</p> <p> 2 t </p> <p>t = u (ct )ct . 3 </p> <p>(2.5.1)</p> <p>u (ct )ct = u (ct )( + n f (kt )), kt ct ct kt = u (ct ) ( + n f (kt )), u (ct )</p> <p>(2.5.2)</p> <p>(2.5.3) (2.5.4)</p> <p>= f (kt ) nkt ct ,</p> <p> (ct , kt ) </p> <p>33.1</p> <p> Nt wt kt rt t = F (Kt , Nt ) rt Kt wt Nt </p> <p>t Kt t Nt</p> <p>= FKt rt = 0 = FNt wt = 0.</p> <p>(3.1.1) (3.1.2)</p> <p>FK</p> <p>= = = =</p> <p>F (K + , N ) F (K, N ) = lim 0 0 lim0</p> <p>1 N (F (K</p> <p>+ , N ) F (K, N )) 1 N F ( K , 1) N</p> <p>(3.1.3) (3.1.4) (3.1.5) (3.1.6)</p> <p>lim</p> <p>F ( K+ , N ) F ( K , N ) N N N N N</p> <p>= lim</p> <p>F(K + N</p> <p>0</p> <p> N , 1) N</p> <p>0</p> <p>lim</p> <p>F (k +</p> <p> N , 1) N</p> <p> F (k, 1)</p> <p>= lim</p> <p>f (k +</p> <p>0</p> <p> N) N</p> <p> f (k)</p> <p>f (k + ) f (k) = f (k). 0 lim</p> <p> = /N 0 0 F 7 </p> <p>F (K, N ) = KFK + N FN . N F (K, N )/N = f (k)</p> <p>(3.1.7)</p> <p>K FK + FN = kf (k) + FN . N7g</p> <p>(3.1.8)g x</p> <p> k g(x) = kx </p> <p>g . x</p> <p> k = 1 g(x) = x </p> <p>5</p> <p>FN = f (k) kf (k).</p> <p>(3.1.9)</p> <p>f (kt ) = rt f (kt ) kt f (kt )</p> <p>(3.1.10) (3.1.11)</p> <p>= wt</p> <p>3.2</p> <p> K0 </p> <p>r w C = A dA/dt </p> <p>dAt = wt Nt + rt At . dt Ct + ct +</p> <p>(3.2.1)</p> <p>dat + nat = wt + rt at , dt</p> <p>(3.2.2)</p> <p>3.3</p> <p>NPG </p> <p> at </p> <p>ct +</p> <p>dat = wt + (rt n)at , dt</p> <p>(3.3.1)</p> <p>rt n 8 aT 9 RT0</p> <p>aT eT </p> <p>(rs n)ds 0</p> <p>(3.3.2)</p> <p>lim aT e</p> <p>RT0</p> <p>(rs n)ds 0</p> <p>(3.3.3)</p> <p> No Ponzi Game 10 Rt</p> <p>8 </p> <p>rt a0 e(rn)t a0 e RT 9 a = a e 0 (rs n+)ds , &gt; 0 t 0 a0 eRT0</p> <p>0 (rn)ds</p> <p>(rs n + )ds e</p> <p>RT0</p> <p>(rs n)ds = a e 0</p> <p>RT0</p> <p>ds</p> <p>= a0 eT ,</p> <p>T .</p> <p>10 </p> <p>6</p> <p>NPG NPG at (rt n)at = wt ct . eRt0</p> <p>(3.3.4)(rs n)ds</p> <p>0</p> <p>at e (at eRt0</p> <p>Rt0</p> <p>(rs n)ds</p> <p> (rt n)at eRt0</p> <p>Rt0</p> <p>(rs n)ds</p> <p>= (wt ct )eRt0</p> <p>Rt</p> <p>(rs n)ds</p> <p>.</p> <p>(3.3.5)</p> <p>(rs n)ds</p> <p>) = at e </p> <p>(rs n)ds</p> <p> (rt n)at eT 0</p> <p>(rs n)ds</p> <p> 0 T</p> <p>[ ]T Rt 0 (rs n)ds at e =0</p> <p>(wt ct )e</p> <p>Rt0</p> <p>(rs n)ds</p> <p>dt,</p> <p>(3.3.6)</p> <p> t = 0 a0 T RT Rt aT e 0 (rs n)ds a0 = (wt ct )e 0 (rs n)ds dt0</p> <p>(3.3.7)</p> <p> NPG T RT RT 0 (rs n)ds ct e dt = wt e 0 (rs n)ds dt + a0 , (3.3.8)0 0</p> <p>3.4</p> <p> U =0 </p> <p>u(ct )et dt</p> <p>(3.4.1) (3.4.2) (3.4.3)</p> <p>at = wt ct + (rt n)at a0 0t Rt</p> <p>lim at e</p> <p>(rs n)ds 0 0</p> <p>(3.4.4)</p> <p>at co-state qt </p> <p>H(ct , at , qt ) = u(ct )et + qt (wt ct + (rt n)at ).</p> <p>(3.4.5)</p> <p>H ct at qt </p> <p>= u (ct )et qt = 0, H = wt ct + (rt n)at , qt H = = qt (rt n). at =</p> <p>(3.4.6) (3.4.7) (3.4.8)</p> <p>ct qt qt = u (ct )ct et u (ct )et = (u (ct )ct u (ct ))et . 3 </p> <p>(3.4.9)</p> <p>u (ct )ct = u (ct )( rt + n). 7</p> <p>(3.4.10)</p> <p>3.5</p> <p> t </p> <p>rt kt</p> <p>= f (kt ), = at ,</p> <p>(3.5.1) (3.5.2)</p> <p> 1 </p> <p>wt = f (kt ) kt f (kt ),</p> <p>(3.5.3)</p> <p> kt</p> <p>= f (kt ) kt f (kt ) ct + (f (kt ) n)kt , = f (kt ) ct nkt , u (ct ) ( f (kt ) + n). = u (ct )</p> <p>(3.5.4) (3.5.5) (3.5.6)</p> <p>ct </p> <p> Social Planner </p> <p>c</p> <p>E c=0</p> <p>E k=0O 1: </p> <p>k</p> <p>8</p> <p>44.1</p> <p>Degree of Relatvie Risk Aversion: RRA(ct ) </p> <p>(ct ) = ctct </p> <p>u (ct ) . u (ct )</p> <p>(4.1.1)</p> <p>ct = </p> <p>ct (f (kt ) n). (ct )</p> <p>(4.1.2)</p> <p>ct = 0 ct = 0 f (kt ) = + n 11 ct kt + n kt = 0 ct = f (kt ) nkt 12 </p> <p>4.1.1</p> <p>CRRA </p> <p>u(c) =</p> <p>c1 , 1</p> <p>(0 &lt; &lt; 1)</p> <p>(4.1.3)</p> <p>u (c) = c , u (c) = c1 </p> <p>(c) = c</p> <p>c1 = . c</p> <p>Constant Relative Risk Averse CRRA </p> <p>f (k) = k </p> <p>(0 &lt; &lt; 1)</p> <p>f (k) = k 1 </p> <p>ct kt</p> <p>=</p> <p>ct (k 1 n), </p> <p>(4.1.4) (4.1.5)</p> <p> = kt ct nkt .</p> <p>4.1.2</p> <p>CARA </p> <p>1 u(c) = et , 11 k</p> <p>( &gt; 0)</p> <p>0 &gt; 0, u (c) (c 0) ct = 0 12 </p> <p>9</p> <p> (Constant Absolute Risk Aversion) CARA u (c)/u (c) CARA </p> <p>CARA ct ct kt = 1 (k 1 n), (4.1.6) (4.1.7)</p> <p> = kt ct nkt .</p> <p>4.2</p> <p> n n &lt; 0 (rt n) n 13 </p> <p>c</p> <p>E c=0</p> <p>E k=0</p> <p>E k=0O 2: </p> <p>k</p> <p>13 </p> <p>10</p> <p>55.1</p> <p> t t gt gt </p> <p>ct + at + nat = wt + rt at t . </p> <p>(5.1.1)</p> <p>at = wt + (rt n)at t ct , at co-state qt </p> <p>H(ct , at , qt ) = u(ct )et + qt (wt + (rt n)at t ct )</p> <p>H ct qt at </p> <p>= u (ct )et qt = 0, H = qt (rt n), at = wt + (rt n)at t ct . = </p> <p>(5.1.2) (5.1.3) (5.1.4)</p> <p> kt = f (kt ) nkt t ct , kt = 0 ct = f (kt ) nkt t ,</p> <p>(5.1.5)</p> <p>(5.1.6)</p> <p>t = k </p> <p>NPG Rt 0 (rs n)ds ct e dt =0 0 </p> <p>wt e</p> <p>RT0</p> <p>(rs n)ds</p> <p> dt 0 RT0</p> <p>t e</p> <p>RT0</p> <p>(rs n)ds</p> <p>dt + a0 .RT0</p> <p>(5.1.7) dt </p> <p> t = gt </p> <p>0</p> <p>t e</p> <p>(rs n)ds</p> <p>dt =</p> <p>0</p> <p>gt e</p> <p>(rs n)ds</p> <p> Rt RT RT 0 (rs n)ds 0 (rs n)ds ct e dt = wt e dt gt e 0 (rs n)ds dt + a0 .0 0 0</p> <p>(5.1.8)</p> <p>11</p> <p>5.2</p> <p> bt bt t bt rt </p> <p> bt + nbt = gt t + rt bt , kt </p> <p>(5.2.1)</p> <p>NPG RT RT t e 0 (rs n)ds dt = gt e 0 (rs n)ds dt + b0 ,0 0</p> <p>(5.2.2)</p> <p> vt = at + bt ct + vt + nvt = wt + rt vt t RT 0 (rs n)ds dt = ct e0 0 RT0</p> <p>wt e</p> <p>(rs n)ds</p> <p> dt 0</p> <p>t e</p> <p>RT0</p> <p>(rs n)ds</p> <p>dt + v0 ,</p> <p>(5.2.3)</p> <p> RT RT ct e 0 (rs n)ds dt = wt e 0 (rs n)ds dt 0 0 0</p> <p>gt e</p> <p>RT0</p> <p>(rs n)ds</p> <p>dt + a0</p> <p>(5.2.4)</p> <p>5.3</p> <p> rt t zt </p> <p>ct +</p> <p>dat + nat = wt + (1 t )rt at + zt . dt</p> <p>(5.3.1)</p> <p>H(ct , at , qt ) = u(ct )et + qt (wt + ((1 t )rt n)at + zt ct ),</p> <p>(5.3.2)</p> <p>H ct qt at </p> <p>= u (ct )et qt = 0, = H = qt (n (1 t )rt ), at = wt + ((1 t )rt n)at + zt ct . 12</p> <p>(5.3.3) (5.3.4) (5.3.5)</p> <p>u (ct )ct = + n (1 t ) u (ct ) ct = 0 </p> <p>(5.3.6)</p> <p>rt =</p> <p>n+ , 1 t</p> <p>(5.3.7)</p> <p> rt = f (kt ) t rt 1 1 f k k </p> <p>k</p> <p>1</p> <p>kn+f (k)</p> <p>n+ 1- 3: </p> <p> kt (r = f (k), w = f (k) kf (k)) k = f (k) ( f (k) + n)z c, k = 0 c = f (k) + z nk f (k)k,</p> <p>(5.3.8)</p> <p>(5.3.9)</p> <p> z f (k)k </p> <p>13</p> <p>c</p> <p> c=0</p> <p>z O k=0 4: </p> <p>k</p> <p>14</p> <p>6</p> <p> Blanchard, Olivier and Stanley Fischer, A Lecture Note on Macroeconomics, MIT Press, 1989. Intriligator, Michael D., Mathematical Optimization and Economic Theory, Prentice-Hall, 1971. Barro, Robert J and Xavier Sala-i-Martin, Economic Growth, MIT Press, 1995.</p> <p>15</p>