%%%%%%%%%%%%%%%%%%%% 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] 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 "SwSLID" "cobbdouglas.m", "netVestCB.m", "TnetVestCB.m", "VestCB" % % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% global A alpha theta s; % s = Savings rate s=str2num(VAsc); A=10; alpha=0.32; theta=0.09; % Depreciation rate % Find k* for which netVest.m is zero kstar=fzero(@netVestCB,[1 100],0.0001); % Savings, Investment. k1=linspace(0,100,100); Inv=VestCB(k1); dgn=theta*k1; subplot(2,2,1); xlabel(['Capital k. k* = ', num2str(kstar)]); ylabel('Invest.'); plot(k1,Inv,'b',k1,dgn,'r'); xlabel(['Capital k. k* = ', num2str(kstar)]); ylabel('Invest.'); title(['Solow. sf(k), \thetak, for s=', num2str(s)]); grid; hold on; Dstar=theta*kstar; plot(kstar,Dstar,'r.','markersize',20); hold on; lineStar=linspace(0,Dstar,30); Qaxis=length(lineStar); lineQ=ones(1,Qaxis).*kstar; plot(lineQ,lineStar,'k-.'); hold off; % Steady state k=linspace(0,kstar,100); NInv=netVestCB(k); Nstar=netVestCB(kstar); subplot(2,2,2); plot(k,NInv); xlabel(['Capital k. k* = ', num2str(kstar)]); ylabel('Net Invest.'); title('"sf(k)-\thetak" with a Cobb-Douglas'); grid; hold on; plot(kstar,Nstar,'r.','markersize',20); hold off; Ini_Stat=1; Tini=0; Tf=100; [t,y]=ode45(@TnetVestCB,[Tini Tf],Ini_Stat); subplot(2,2,3); plot(t,y); xlabel('Time'); ylabel('Capital (k_0=1)'); title('Time Evolution of sf(k(t))-\thetak(t)'); grid; hold on; edy=y(end,:); plot(Tf,edy,'k.','markersize',20); hold off; Ini_StatA =0.5; Ini_StatB =100; Tini=0; Tf=100; [tA,yA]=ode45(@TnetVestCB,[Tini Tf],Ini_StatA); [tB,yB]=ode45(@TnetVestCB,[Tini Tf],Ini_StatB); subplot(2,2,4); 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 initial values k_0=1 k_0=100'); 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 9 March 2010. %"Complex and Chaotic Nonlinear Dynamics. % Advances in Economics and Finance, % Mathematics and Statistics" % T.Vialar, Springer 2009. % Copyright(c).