%%%%%%%%%%%%%%%%%%%% Solow Model (1956) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % Steady State. % % % % [1] Relations of Solow Model (1956): % % dk/dt = s*f(k(t))-theta*k(t) % % netVest = s*f(k)-theta*k % % k(0) = k_0 > 0 % % % % [2] C.E.S FUNCTION: % % (C.E.S = Constant Elasticity of Substitution) % % Y = [a*K^(-gamma)+(1-a)*L^(-gamma)]^(-µ) % % with 0-1 and different from zero. % % % % The per capita formulation of the CES function % % with constant returns to scale: % % y = [a*k^(-gamma)+(1-a)]^(-1/gamma) % % % % [3] COBB-DOUGLAS FUNCTION: % % Y = A*K^(alpha)*L^(beta) % % with alpha>0, beta<1. % % If beta = 1 - alpha, constant returns to scale. Then the (per capita) % % cobb-Douglas function is given by: % % y = A*k^(alpha) % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % % see also "ces.m", "cobbdouglas.m", "netVest.m", "TnetVest2.m", "Vest.m" % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% global A alpha gamma theta s; A=10; alpha=0.32; theta=0.09; % Depreciation rate s=0.12; % Savings rate % In the case gamma=0, we obtain a Cobb-Douglas function. % If gamma is different from 0, we obtain a CES function, and in such a % case we suggest for example gamma=0.4. disp(' ') disp('We suggest gamma=0 to obtain a Cobb-Douglas function') disp('... or gamma=0.4 for a CES function') disp(' ') gamma=input('gamma = '); disp(' ') % Find k* for which "netVest.m" is zero kstar=fzero(@netVest,[10 80],0.0001) % Savings, Investment. k1=linspace(0,80,80); Inv=Vest(k1); dgn=theta*k1; plot(k1,Inv,'b',k1,dgn,'r'); hold on; Dstar=theta*kstar; plot(kstar,Dstar,'r.','markersize',20); xlabel(['Capital: k, where k* =', num2str(kstar)]); ylabel('Invest.'); grid; hold on; lineStar=linspace(0,Dstar,30); Qaxis=length(lineStar); lineQ=ones(1,Qaxis).*kstar; plot(lineQ,lineStar,'k-.'); hold off; if gamma == 0; title('Curves of "sf(k)" and "\thetak" where f(k) is a Cobb-Douglas function'); else title('Curves of "sf(k)" and "\thetak", where f(k) is a C.E.S function'); end; disp(' ') disp(' ') disp('Press Enter To Continue ... ') disp('.. to Stop the Simulation, Type "Ctrl-c" in the Command Window ..') disp(' ') disp(' ') pause % Steady state % By using kstar=fzero(@netVest,[10 80],0.0001) k=linspace(0,kstar,80); NInv=netVest(k); Nstar=netVest(kstar); plot(k,NInv); xlabel(['Capital (k) with k* = ', num2str(kstar)]); grid; hold on; plot(kstar,Nstar,'r.','markersize',20); hold off; if gamma == 0; title('Net Investment: "sf(k)-\thetak", where f(k) is a Cobb-Douglas function'); else title('Net Investment: "sf(k)-\thetak", where f(k) is a C.E.S function'); end; disp(' ') disp('Press Enter To Continue ... ') disp('.. to Stop the Simulation, Type "Ctrl-c" in the Command Window ..') disp(' ') disp(' ') pause Ini_Stat=1; Tini=0; Tf=100; [t,y]=ode45(@TnetVest2,[Tini Tf],Ini_Stat); plot(t,y); xlabel('Time'); ylabel('Capital'); grid; hold on; edy=y(end,:); plot(Tf,edy,'k.','markersize',20); hold off; if gamma == 0; title('Time Evolution of net Invest. sf(k[t])-\thetak[t], when f(k) is a Cobb-Douglas function'); else title('Time Evolution of net Invest. sf(k[t])-\thetak[t], when f(k) is a C.E.S function'); end; disp(' ') disp('Press Enter To Continue ... ') disp('.. to Stop the Simulation, Type "Ctrl-c" in the Command Window ..') disp(' ') disp(' ') pause Ini_StatA =5; Ini_StatB =60; Tini=0; Tf=100; [tA,yA]=ode45(@TnetVest2,[Tini Tf],Ini_StatA); [tB,yB]=ode45(@TnetVest2,[Tini Tf],Ini_StatB); plot(tA,yA); hold on; plot(0,Ini_StatA,'b.','markersize',20); hold on; plot(0,Ini_StatB,'b.','markersize',20); hold on; plot(tB,yB); xlabel('Time'); ylabel('Capital'); title('Time Evolution for 2 different initial values'); grid; hold on; edyA=yA(end,:); edyB=yB(end,:); plot(Tf,edyA,'k.','markersize',20); hold on; plot(Tf,edyB,'k.','markersize',20); hold on; lhA=length(tA); lineA=ones(1,lhA).*kstar; plot(tA,lineA,'k-.'); hold on; lhB=length(tB); lineB=ones(1,lhB).*kstar; plot(tB,lineB,'k-.'); hold off % Last revised on 10 March 2010. %"Complex and Chaotic Nonlinear Dynamics. % Advances in Economics and Finance, % Mathematics and Statistics" % T.Vialar, Springer 2009. % Copyright(c).