Beispiel #1
0
                    output (GDP)
    C_path        = [T+S-2,] vector, equilibrium time path of aggregate
                    consumption
    EulErr_path   = [S, T+S-1] matrix, equilibrium time path of the
                    Euler errors for all the savings decisions
    tpi_time      = scalar, number of seconds to compute TPI solution
    ResDiff       = [T-1,] vector, errors in the resource constraint
                    from period 1 to T-1. We don't use T because we are
                    missing one individual's consumption in that period
    --------------------------------------------------------------------
    '''
    if TPI_solve == True:
        print 'BEGIN EQUILIBRIUM TIME PATH COMPUTATION'
        Gamma1 = 0.9 * b_ss
        # Make sure init. period distribution is feasible in terms of K
        K1, K_constr_tpi1 = ssf.get_K(Gamma1)
        rparams = (A, alpha, delta)
        r1 = ssf.get_r(rparams, K1, L)
        BQ1 = (1 + r1) * Gamma1[-1]
        if K1 <= 0 and Gamma1[-1] > 0:
            print 'Initial savings distribution is not feasible because K1<=0. Some element(s) of Gamma1 must increase.'
        elif K1 <= 0 and Gamma1[-1] <= 0:
            print 'Initial savings distribution is not feasible because K1<=0 and b_{S+1,1}<=0. Some element(s) of Gamma1 must increase.'
        elif K1 > 0 and Gamma1[-1] <= 0:
            print 'Initial savings distribution is not feasible because b_{S+1,1}<=0. b_{S+1,1} must increase.'
        else:
            # Choose initial guess of path of aggregate capital stock
            # and total bequests. Use parabola specification
            # aa*x^2 + bb*x + cc

            # Initial aggregate capital path
Beispiel #2
0
                    output (GDP)
    C_path        = [T+S-2,] vector, equilibrium time path of aggregate
                    consumption
    EulErr_path   = [S, T+S-1] matrix, equilibrium time path of the
                    Euler errors for all the savings decisions
    tpi_time      = scalar, number of seconds to compute TPI solution
    ResDiff       = [T-1,] vector, errors in the resource constraint
                    from period 1 to T-1. We don't use T because we are
                    missing one individual's consumption in that period
    --------------------------------------------------------------------
    '''
    if TPI_solve == True:
        print 'BEGIN EQUILIBRIUM TIME PATH COMPUTATION'
        Gamma1 = 0.9 * b_ss
        # Make sure init. period distribution is feasible in terms of K
        K1, K_constr_tpi1 = ssf.get_K(Gamma1)
        rparams = (A, alpha, delta)
        r1 = ssf.get_r(rparams, K1, L)
        BQ1 = (1 + r1) * Gamma1[-1]
        if K1 <= 0 and Gamma1[-1] > 0:
            print 'Initial savings distribution is not feasible because K1<=0. Some element(s) of Gamma1 must increase.'
        elif K1 <= 0 and Gamma1[-1] <= 0:
            print 'Initial savings distribution is not feasible because K1<=0 and b_{S+1,1}<=0. Some element(s) of Gamma1 must increase.'
        elif K1 > 0 and Gamma1[-1] <= 0:
            print 'Initial savings distribution is not feasible because b_{S+1,1}<=0. b_{S+1,1} must increase.'
        else:
            # Choose initial guess of path of aggregate capital stock
            # and total bequests. Use parabola specification
            # aa*x^2 + bb*x + cc

            # Initial aggregate capital path
Beispiel #3
0
def TPI(params, Kpath_init, BQpath_init, Gamma1, nvec, Lpath, b_ss,
  graphs):
    '''
    Generates steady-state time path for all endogenous objects from
    initial state (K1, BQ1, Gamma1) to the steady state.

    Inputs:
        params      = length 17 tuple, (S, T, beta, sigma, chi_b, L, A,
                      alpha, delta, K1, K_ss, BQ_ss, C_ss maxiter_TPI,
                      mindist_TPI, xi, TPI_tol)
        S           = integer in [3,80], number of periods an individual
                      lives
        T           = integer > S, number of time periods until steady
                      state
        beta        = scalar in [0,1), discount factor for each model
                      period
        sigma       = scalar > 0, coefficient of relative risk aversion
        chi_b       = scalar > 0, scale parameter on utility of bequests
        L           = scalar > 0, exogenous aggregate labor
        A           = scalar > 0, total factor productivity parameter in
                      firms' production function
        alpha       = scalar in (0,1), capital share of income
        delta       = scalar in [0,1], model-period depreciation rate of
                      capital
        K1          = scalar > 0, initial period aggregate capital stock
        K_ss        = scalar > 0, steady-state aggregate capital stock
        BQ_ss       = scalar > 0, steady-state total bequests
        maxiter_TPI = integer >= 1, Maximum number of iterations for TPI
        mindist_TPI = scalar > 0, Convergence criterion for TPI
        xi          = scalar in (0,1], TPI path updating parameter
        TPI_tol     = scalar > 0, tolerance level for fsolve's in TPI
        Kpath_init  = [T+S-1,] vector, initial guess for the time path
                      of the aggregate capital stock
        BQpath_init = [T+S-1,] vector, initial guess for the time path
                      of total bequests
        Gamma1      = [S,] vector, initial period savings distribution
        nvec        = [S,] vector, exogenous labor supply n_{s,t}
        Lpath       = [T+S-1,] vector, exogenous time path for aggregate
                      labor
        b_ss        = [S,] vector, steady-state savings distribution
        graphs      = boolean, =True if want graphs of TPI objects

    Functions called:
        c8ssf.get_r
        c8ssf.get_w
        get_cbepath
        c4ssf.get_K

    Objects in function:
        start_time   = scalar, current processor time in seconds (float)
        iter_TPI     = integer >= 0, current iteration of TPI
        dist_TPI     = scalar >= 0, distance measure for fixed point
        Kpath_new    = [T+S-2,] vector, new path of the aggregate
                       capital stock implied by household and firm
                       optimization
        BQpath_new   = [T+S-1,] vector, new path of total bequests
                       implied by household and firm optimization
        r_params     = length 3 tuple, parameters passed in to get_r
        w_params     = length 2 tuple, parameters passed in to get_w
        cbe_params   = length 6 tuple. parameters passed in to
                       get_cbepath
        rpath        = [T+S-2,] vector, equilibrium time path of the
                       interest rate
        wpath        = [T+S-2,] vector, equilibrium time path of the
                       real wage
        cpath        = [S, T+S-2] matrix, equilibrium time path values
                       of individual consumption c_{s,t}
        bpath        = [S, T+S-1] matrix, equilibrium time path values
                       of individual savings b_{s+1,t+1}
        EulErrPath   = [S, T+S-1] matrix, equilibrium time path values
                       of Euler errors corresponding to individual
                       savings b_{s+1,t+1} (first column is zeros)
        Kpath_constr = [T+S-2,] boolean vector, =True if K_t<=0
        Kpath        = [T+S-2,] vector, equilibrium time path of the
                       aggregate capital stock
        Y_params     = length 2 tuple, parameters to be passed to get_Y
        Ypath        = [T+S-2,] vector, equilibrium time path of
                       aggregate output (GDP)
        Cpath        = [T+S-2,] vector, equilibrium time path of
                       aggregate consumption
        elapsed_time = scalar, time to compute TPI solution (seconds)

    Returns: bpath, cpath, BQpath, wpath, rpath, Kpath, Ypath, Cpath,
             EulErrpath, elapsed_time
    '''
    start_time = time.clock()
    (S, T, beta, sigma, chi_b, L, A, alpha, delta, K1, K_ss, BQ_ss,
        C_ss, maxiter_TPI, mindist_TPI, xi, TPI_tol) = params
    iter_TPI = int(0)
    dist_TPI = 10.
    Kpath_new = Kpath_init
    BQpath_new = BQpath_init
    r_params = (A, alpha, delta)
    w_params = (A, alpha)
    cbe_params = (S, T, beta, sigma, chi_b, TPI_tol)

    while (iter_TPI < maxiter_TPI) and (dist_TPI >= mindist_TPI):
        iter_TPI += 1
        Kpath_init = xi * Kpath_new + (1 - xi) * Kpath_init
        BQpath_init = xi * BQpath_new + (1 - xi) * BQpath_init
        rpath = ssf.get_r(r_params, Kpath_init, Lpath)
        wpath = ssf.get_w(w_params, Kpath_init, Lpath)
        cpath, bpath, EulErrPath = get_cbepath(cbe_params, nvec, rpath,
                                   wpath, BQpath_init, Gamma1, b_ss)
        Kpath_new = np.zeros(T+S-2)
        Kpath_new[:T], Kpath_constr = ssf.get_K(bpath[:, :T])
        Kpath_new[T:] = K_ss
        Kpath_constr = np.append(Kpath_constr, np.zeros(S-1, dtype=bool))
        Kpath_new[Kpath_constr] = 1
        BQpath_new = np.zeros(T+S-2)
        BQpath_new[:T] = ssf.get_BQ(rpath[:T], bpath[S-1, :T])
        BQpath_new[T:] = BQ_ss

        # Check the distance of Kpath_new and BQpath_new
        Kdist_TPI = (Kpath_new[1:T] - Kpath_init[1:T]) / Kpath_init[1:T]
        BQdist_TPI = (BQpath_new[1:T] - BQpath_init[1:T]) / BQpath_init[1:T]
        dist_TPI = np.absolute(np.append(Kdist_TPI, BQdist_TPI)).max()
        print ('iter: ', iter_TPI, ', dist: ', dist_TPI, ', max Eul err: ',
            np.absolute(EulErrPath).max())

    if iter_TPI == maxiter_TPI and dist_TPI > mindist_TPI:
        print 'TPI reached maxiter and did not converge.'
    elif iter_TPI == maxiter_TPI and dist_TPI <= mindist_TPI:
        print 'TPI converged in the last iteration. Should probably increase maxiter_TPI.'
    Kpath = Kpath_new
    BQpath = BQpath_new
    Y_params = (A, alpha)
    Ypath = ssf.get_Y(Y_params, Kpath, Lpath)
    Cpath = np.zeros(T+S-2)
    Cpath[:T-1] = ssf.get_C(cpath[:, :T-1])
    Cpath[T-1:] = C_ss
    elapsed_time = time.clock() - start_time

    if graphs == True:
        # Plot time path of aggregate capital stock
        tvec = np.linspace(1, T+S-2, T+S-2)
        minorLocator   = MultipleLocator(1)
        fig, ax = plt.subplots()
        plt.plot(tvec, Kpath)
        # for the minor ticks, use no labels; default NullFormatter
        ax.xaxis.set_minor_locator(minorLocator)
        plt.grid(b=True, which='major', color='0.65',linestyle='-')
        plt.title('Time path for aggregate capital stock')
        plt.xlabel(r'Period $t$')
        plt.ylabel(r'Aggregate capital $K_{t}$')
        # plt.savefig('Kt_Chap8')
        plt.show()

        # Plot time path of total bequests
        tvec = np.linspace(1, T+S-2, T+S-2)
        minorLocator   = MultipleLocator(1)
        fig, ax = plt.subplots()
        plt.plot(tvec, BQpath)
        # for the minor ticks, use no labels; default NullFormatter
        ax.xaxis.set_minor_locator(minorLocator)
        plt.grid(b=True, which='major', color='0.65',linestyle='-')
        plt.title('Time path for total bequests')
        plt.xlabel(r'Period $t$')
        plt.ylabel(r'Total bequests $BQ_{t}$')
        # plt.savefig('BQt_Chap8')
        plt.show()

        # Plot time path of aggregate output (GDP)
        tvec = np.linspace(1, T+S-2, T+S-2)
        minorLocator   = MultipleLocator(1)
        fig, ax = plt.subplots()
        plt.plot(tvec, Ypath)
        # for the minor ticks, use no labels; default NullFormatter
        ax.xaxis.set_minor_locator(minorLocator)
        plt.grid(b=True, which='major', color='0.65',linestyle='-')
        plt.title('Time path for aggregate output (GDP)')
        plt.xlabel(r'Period $t$')
        plt.ylabel(r'Aggregate output $Y_{t}$')
        # plt.savefig('Yt_Chap8')
        plt.show()

        # Plot time path of aggregate consumption
        tvec = np.linspace(1, T+S-2, T+S-2)
        minorLocator   = MultipleLocator(1)
        fig, ax = plt.subplots()
        plt.plot(tvec, Cpath)
        # for the minor ticks, use no labels; default NullFormatter
        ax.xaxis.set_minor_locator(minorLocator)
        plt.grid(b=True, which='major', color='0.65',linestyle='-')
        plt.title('Time path for aggregate consumption')
        plt.xlabel(r'Period $t$')
        plt.ylabel(r'Aggregate consumption $C_{t}$')
        # plt.savefig('Ct_Chap8')
        plt.show()

        # Plot time path of real wage
        tvec = np.linspace(1, T+S-2, T+S-2)
        minorLocator   = MultipleLocator(1)
        fig, ax = plt.subplots()
        plt.plot(tvec, wpath)
        # for the minor ticks, use no labels; default NullFormatter
        ax.xaxis.set_minor_locator(minorLocator)
        plt.grid(b=True, which='major', color='0.65',linestyle='-')
        plt.title('Time path for real wage')
        plt.xlabel(r'Period $t$')
        plt.ylabel(r'Real wage $w_{t}$')
        # plt.savefig('wt_Chap8')
        plt.show()

        # Plot time path of real interest rate
        tvec = np.linspace(1, T+S-2, T+S-2)
        minorLocator   = MultipleLocator(1)
        fig, ax = plt.subplots()
        plt.plot(tvec, rpath)
        # for the minor ticks, use no labels; default NullFormatter
        ax.xaxis.set_minor_locator(minorLocator)
        plt.grid(b=True, which='major', color='0.65',linestyle='-')
        plt.title('Time path for real interest rate')
        plt.xlabel(r'Period $t$')
        plt.ylabel(r'Real interest rate $r_{t}$')
        # plt.savefig('rt_Chap8')
        plt.show()

        # Plot time path of individual savings distribution
        tgrid = np.linspace(1, T, T)
        sgrid = np.linspace(2, S+1, S)
        tmat, smat = np.meshgrid(tgrid, sgrid)
        cmap_bp = matplotlib.cm.get_cmap('summer')
        fig = plt.figure()
        ax = fig.gca(projection='3d')
        ax.set_xlabel(r'period-$t$')
        ax.set_ylabel(r'age-$s$')
        ax.set_zlabel(r'individual savings $b_{s,t}$')
        strideval = max(int(1), int(round(S/10)))
        ax.plot_surface(tmat, smat, bpath[:, :T], rstride=strideval,
            cstride=strideval, cmap=cmap_bp)
        # plt.savefig('bpath_Chap8')
        plt.show()

        # Plot time path of individual consumption distribution
        tgrid = np.linspace(1, T-1, T-1)
        sgrid = np.linspace(1, S, S)
        tmat, smat = np.meshgrid(tgrid, sgrid)
        cmap_cp = matplotlib.cm.get_cmap('summer')
        fig = plt.figure()
        ax = fig.gca(projection='3d')
        ax.set_xlabel(r'period-$t$')
        ax.set_ylabel(r'age-$s$')
        ax.set_zlabel(r'individual consumption $c_{s,t}$')
        strideval = max(int(1), int(round(S/10)))
        ax.plot_surface(tmat, smat, cpath[:, :T-1], rstride=strideval,
            cstride=strideval, cmap=cmap_cp)
        # plt.savefig('cpath_Chap8')
        plt.show()

    return (bpath, cpath, BQpath, wpath, rpath, Kpath, Ypath, Cpath,
        EulErrPath, elapsed_time)