内容讲稿2研一上高宏modeldynamics applications

1、Advanced MacroeconomicsRamsey-Cass-Koopmans M: MDynamics andApplicationsOctober 14, 2015 Dynamics of cc(t) = r (t)g = f 0 k(t)gc(t).ckkc =( . 0)c0( . 0)cDynamics of kk (t) = w (t) + r (t)k(t) c(t) (n + g )k(t) =f k(t) c(t) (n + g )k(t)c.(k 0)kPhase Diagram.cc = 0E.k = 0kkSaddle Path.cc = 0AEB.k = 0C

2、FDkk(0)kSaddle PathIs a saddle path unique?Yes. To prove this formally, we need to linearize the dynamic system and find the eigenvalues of the coefficient matrix.(c(t) f 0 k(t)gc(t)=k (t) = f k(t) c(t) (n + g )k(t)Saddle Path(c(t) f 0 k(t)gc(t)=k (t)f k(t) c(t) (n + g )k(t)(00f k c(t)ck(t) k k (t)

3、f 0 k (n + g )k(t) k c(t) cSaddle PathThe purpose is to convert it to X Define Z = X G 1b: Z = G Z= G X bG M = M : Diagonal matrix of eigenvaluesM : columns are eigenvectorsIILet Z =M Y : Y =Y tY11Y1Y1(t)Y1(0)e1 tY2Y2Y (t)Y (0)e2222=.nYnY (t) tYY (0)ennnnSaddle Path#c f 00 k01G =f k (n + g )0Denote

4、the eigenvalues of G as 1 and 212 n = |G | and 1 + 2 + . . . + n = trace(G )I00f k We have: = c 012Hence, we have a unique saddle path.WelfareIs the RCK decentralized equilibrium Pareto-efficient?Pareto-efficient: it is impossible to make anyone better off without making someone worse offIThe answer

5、 is Yes according to the first welfare theoremIf markets are competitive and complete and there are no externalities, then the decentralized equilibrium is Pareto-efficientIIn the Solow M, this question is not well-dened. Why?WelfareSocial Planners Problemmax BRet c(t)1 dtt=01s.t.k (t) = f k(t) c(t)

6、 (n + g )k(t)k(t) 0k(0) is givenSame solution as in the RCK decentralized equilibrium?WelfareSocial Planners ProblemThe present-value HamiltonianIt c(t)1H= e+1 (t)f k(t) c(t) (n + g )k(t)WelfareSocial Planners ProblemHPontryaginsum Principle:= 0 1cetc(t) = (t)Euler Equation:2H = F k(t) hf 0 k(t) (n

7、+ g )i = k (t) = f k(t) c(t) (n + g )k(t)FTranversality Condition: tlim (t)k(t) = 03The RCK decentralized equilibrium is Pareto-efficientNote: f 0 k(t) = r (t) and f k(t) = w (t) + r (t)k(t)IBalanced Growth PathBGP (same as the Solow m)At the steady state, k(t), c(t) and y (t) are constant.I I IIy (

8、t)c(tSaving rate, is constant)y (t)Physical capital can NOTfor a signicant part of growth.Effectiveness of labor (A) remains the only source of persistent growth in output per worker.Difference from the Solow m: k 0k: modified golden-rule capital stockIAn unexpected fall in the discount rateSuppose

9、a RCK economy is on its balanced growth path. Experiment one: an unexpected fall in the discount rate, .ckOLD kNEWkc =E0.k = 0E AAn unexpected fall in the discount rateThe rate of adjustmentccc k k +c c kckkkk k +c c kcAn unexpected fall in the discount rate f 00 kc(t) ck(t) k k (t) f 0 k (n + g )k(

10、t) k c(t) c k(t) k c(t) cDefine c = c c and k = k kc c kk f 00 k k ccc kAn unexpected fall in the discount rateDefine c=cc = f 00 kc 1kf k c00 1 = = 24f 00 kc/ 0.52If the economy is to converge to (k,c), then must be negative (denoted as 1).An unexpected fall in the discount rateAlternative way to c

11、ompute #c f 00 kc f 00 k0101G =f k (n + g )0 are the eigenvalues of G00f k 0 c 24f 00 kc/ 0.5= 0 =21 An unexpected fall in the discount ratek(t) = k + e1t k(0) kc(t) = c + e1t c(0) c.c0kkc =BEA0A.k =BAn unexpected fall in the discount ratek= 0: positive slope; positive vertical interceptk= 0 c(t) =

12、k(t) + (c k) f 0 k = + gc = f k (n + g )kc kc ( n (1 )g )kf k ( + g )k=f k f 0 kk0An unexpected increase in government purchasesSuppose a RCK economy is on its balanced growth path.Experiment two: An unexpected increase in government purchases,G (t)Assumptions:G (t) does not affect the utility from

13、private consumption.G (t) are devoted to public consumption rather than public investment. G (t) is financed by lump-sum taxes and the government budget is balanced.I IIAn unexpected increase in government purchasesAdd government to the mc(t) = f 0 k(t)gc(t)k (t) = f k(t) c(t) G (t) (n + g )k(t)An u

14、nexpected increase in government purchasesPermanent increase in government purchases.ckkc =E0.k = 0E An unexpected increase in government purchasesTemporary increase in government purchases.ckk(a)r (t) + gt0t1(b)Timec =E0.k = 0Wars and Real Interest RatesTemporarily high government purchases cause real interest rates to rise, whereas permanently h