diff --git a/BCG_complete_mkts.ipynb b/BCG_complete_mkts.ipynb index d2a711f..3f1c066 100644 --- a/BCG_complete_mkts.ipynb +++ b/BCG_complete_mkts.ipynb @@ -1406,7 +1406,7 @@ } ], "metadata": { - "date": 1618812137.2846944, + "date": 1619230074.5343747, "filename": "BCG_complete_mkts.rst", "kernelspec": { "display_name": "Python", diff --git a/BCG_incomplete_mkts.ipynb b/BCG_incomplete_mkts.ipynb index e2a7635..e45ab9c 100644 --- a/BCG_incomplete_mkts.ipynb +++ b/BCG_incomplete_mkts.ipynb @@ -648,7 +648,7 @@ "$ C $’s that appear in the pricing functions, then\n", "\n", "- consumers’ Euler equations are satisfied if little $ c $’s are\n", - " equated to Big C’s \n", + " equated to Big $ C $’s \n", "- firms’ first-order necessary conditions for $ k, b $ are\n", " satisfied. \n", "- $ \\check q = q(K,B) $ and\n", @@ -676,7 +676,7 @@ " $ \\epsilon^* \\equiv \\log\\left(\\frac{b}{Ak^\\alpha}\\right) $. \n", "1. (In this step we abuse notation by freezing $ V, k, b $ and in\n", " effect temporarily treating them as Big $ K,B $ values. Thus, in\n", - " this step 6 little k, b are frozen at guessed at value of K, B.)\n", + " this step 6 little $ k, b $ are frozen at guessed at value of $ K, B $.)\n", " Fixing the values of $ V $, $ b $ and $ k $, compute\n", " optimal choices of consumption $ c^i $ with consumers’ FOCs.\n", " Assume that only agent 2 holds debt: $ \\xi^2 = b $ and that both agents\n", @@ -2215,7 +2215,7 @@ } ], "metadata": { - "date": 1618812137.5370972, + "date": 1619230074.8008502, "filename": "BCG_incomplete_mkts.rst", "kernelspec": { "display_name": "Python", diff --git a/about_lectures.ipynb b/about_lectures.ipynb index 7ab874e..a29f31b 100644 --- a/about_lectures.ipynb +++ b/about_lectures.ipynb @@ -28,7 +28,7 @@ } ], "metadata": { - "date": 1618812137.6949542, + "date": 1619230074.956999, "filename": "about_lectures.rst", "kernelspec": { "display_name": "Python", diff --git a/additive_functionals.ipynb b/additive_functionals.ipynb index 1e5152f..7b12f98 100644 --- a/additive_functionals.ipynb +++ b/additive_functionals.ipynb @@ -1498,7 +1498,7 @@ } ], "metadata": { - "date": 1618812137.7941382, + "date": 1619230075.0592115, "filename": "additive_functionals.rst", "kernelspec": { "display_name": "Python", diff --git a/amss.ipynb b/amss.ipynb index a05c1cf..91c8748 100644 --- a/amss.ipynb +++ b/amss.ipynb @@ -972,7 +972,7 @@ "\n", "class RecursiveAllocationAMSS:\n", "\n", - " def __init__(self, model, μgrid, tol_diff=1e-4, tol=1e-4):\n", + " def __init__(self, model, μgrid, tol_diff=1e-7, tol=1e-7):\n", "\n", " self.β, self.π, self.G = model.β, model.π, model.G\n", " self.mc, self.S = MarkovChain(self.π), len(model.π) # Number of states\n", @@ -1489,7 +1489,7 @@ "outputs": [], "source": [ "# Initialize μgrid for value function iteration\n", - "μ_grid = np.linspace(-0.7, 0.01, 200)\n", + "μ_grid = np.linspace(-0.7, 0.01, 300)\n", "\n", "time_example = CRRAutility()\n", "\n", @@ -1781,7 +1781,7 @@ } ], "metadata": { - "date": 1618874888.5793111, + "date": 1619230075.2505703, "filename": "amss.rst", "kernelspec": { "display_name": "Python", diff --git a/amss2.ipynb b/amss2.ipynb index 4baecac..ad8ce7d 100644 --- a/amss2.ipynb +++ b/amss2.ipynb @@ -567,7 +567,7 @@ "\n", "class RecursiveAllocationAMSS:\n", "\n", - " def __init__(self, model, μgrid, tol_diff=1e-4, tol=1e-4):\n", + " def __init__(self, model, μgrid, tol_diff=1e-7, tol=1e-7):\n", "\n", " self.β, self.π, self.G = model.β, model.π, model.G\n", " self.mc, self.S = MarkovChain(self.π), len(model.π) # Number of states\n", @@ -1608,7 +1608,7 @@ } ], "metadata": { - "date": 1618812138.1547377, + "date": 1619230075.4204001, "filename": "amss2.rst", "kernelspec": { "display_name": "Python", diff --git a/amss3.ipynb b/amss3.ipynb index e16a72e..771e1de 100644 --- a/amss3.ipynb +++ b/amss3.ipynb @@ -433,7 +433,7 @@ "\n", "class RecursiveAllocationAMSS:\n", "\n", - " def __init__(self, model, μgrid, tol_diff=1e-4, tol=1e-4):\n", + " def __init__(self, model, μgrid, tol_diff=1e-7, tol=1e-7):\n", "\n", " self.β, self.π, self.G = model.β, model.π, model.G\n", " self.mc, self.S = MarkovChain(self.π), len(model.π) # Number of states\n", @@ -1647,7 +1647,7 @@ } ], "metadata": { - "date": 1618812138.3407247, + "date": 1619230075.6273031, "filename": "amss3.rst", "kernelspec": { "display_name": "Python", diff --git a/arellano.ipynb b/arellano.ipynb index cd2549b..fb609f6 100644 --- a/arellano.ipynb +++ b/arellano.ipynb @@ -1037,7 +1037,7 @@ } ], "metadata": { - "date": 1618812138.67823, + "date": 1619230075.9571571, "filename": "arellano.rst", "kernelspec": { "display_name": "Python", diff --git a/arma.ipynb b/arma.ipynb index a716fab..355f704 100644 --- a/arma.ipynb +++ b/arma.ipynb @@ -1183,7 +1183,7 @@ } ], "metadata": { - "date": 1618812138.9135163, + "date": 1619230076.177326, "filename": "arma.rst", "kernelspec": { "display_name": "Python", diff --git a/black_litterman.ipynb b/black_litterman.ipynb index 6ca885f..1c18f16 100644 --- a/black_litterman.ipynb +++ b/black_litterman.ipynb @@ -1582,7 +1582,7 @@ } ], "metadata": { - "date": 1618812139.1406567, + "date": 1619230076.4078171, "filename": "black_litterman.rst", "kernelspec": { "display_name": "Python", diff --git a/calvo.ipynb b/calvo.ipynb index 7ff1ef1..84402f4 100644 --- a/calvo.ipynb +++ b/calvo.ipynb @@ -1667,7 +1667,7 @@ } ], "metadata": { - "date": 1618812139.3637989, + "date": 1619230076.6404426, "filename": "calvo.rst", "kernelspec": { "display_name": "Python", diff --git a/cattle_cycles.ipynb b/cattle_cycles.ipynb index bc3fda0..4a17f0b 100644 --- a/cattle_cycles.ipynb +++ b/cattle_cycles.ipynb @@ -548,7 +548,7 @@ } ], "metadata": { - "date": 1618812139.499189, + "date": 1619230076.7742686, "filename": "cattle_cycles.rst", "kernelspec": { "display_name": "Python", diff --git a/chang_credible.ipynb b/chang_credible.ipynb index 642550e..299ee4f 100644 --- a/chang_credible.ipynb +++ b/chang_credible.ipynb @@ -1541,7 +1541,7 @@ } ], "metadata": { - "date": 1618812139.8675833, + "date": 1619230077.1273336, "filename": "chang_credible.rst", "kernelspec": { "display_name": "Python", diff --git a/chang_ramsey.ipynb b/chang_ramsey.ipynb index 8820625..9685ee2 100644 --- a/chang_ramsey.ipynb +++ b/chang_ramsey.ipynb @@ -1817,7 +1817,7 @@ } ], "metadata": { - "date": 1618812140.0972116, + "date": 1619230077.354034, "filename": "chang_ramsey.rst", "kernelspec": { "display_name": "Python", diff --git a/classical_filtering.ipynb b/classical_filtering.ipynb index 5e80896..d2a3097 100644 --- a/classical_filtering.ipynb +++ b/classical_filtering.ipynb @@ -1542,7 +1542,7 @@ } ], "metadata": { - "date": 1618812140.3285382, + "date": 1619230077.5866334, "filename": "classical_filtering.rst", "kernelspec": { "display_name": "Python", diff --git a/coase.ipynb b/coase.ipynb index 0c40c54..85eb0b7 100644 --- a/coase.ipynb +++ b/coase.ipynb @@ -854,7 +854,7 @@ } ], "metadata": { - "date": 1618812140.5184815, + "date": 1619230077.7798908, "filename": "coase.rst", "kernelspec": { "display_name": "Python", diff --git a/cons_news.ipynb b/cons_news.ipynb index 211fd0f..aba0608 100644 --- a/cons_news.ipynb +++ b/cons_news.ipynb @@ -1004,21 +1004,19 @@ " simulation of $ \\{\\epsilon_t\\}_{t=0}^T $ where $ T $ is a big\n", " integer $ \\epsilon_t = \\sigma_\\epsilon v_t $, $ v_t $ is a\n", " standard normal scalar, $ y_0 =100 $, and \n", - "\n", - "\n", - "$$\n", - "y_{t+1} - y_t =-\\beta^{-1} \\epsilon_t + \\epsilon_{t+1} .\n", - "$$\n", - "\n", + " \n", + " $$\n", + " y_{t+1} - y_t =-\\beta^{-1} \\epsilon_t + \\epsilon_{t+1} .\n", + " $$\n", + " \n", "1. We take the **same** $ \\{y_t\\} $ realization generated in step 1\n", " and form an innovation process $ \\{a_t\\} $ from the formulas \n", - "\n", - "\n", - "$$\n", - "\\begin{aligned} a_0 & = 0 \\cr\n", - "a_t & = \\sum_{j=0}^{t-1} \\beta^j (y_{t-j} - y_{t-j-1}) + \\beta^t a_0, \\quad t \\geq 1 \\end{aligned}\n", - "$$\n", - "\n", + " \n", + " $$\n", + " \\begin{aligned} a_0 & = 0 \\cr\n", + " a_t & = \\sum_{j=0}^{t-1} \\beta^j (y_{t-j} - y_{t-j-1}) + \\beta^t a_0, \\quad t \\geq 1 \\end{aligned}\n", + " $$\n", + " \n", "1. We throw away the first $ S $ observations and form the sample\n", " $ \\{y_t, \\epsilon_t, a_t\\}_{S+1}^T $ as the realization that\n", " we’ll use in the following steps. \n", @@ -1074,7 +1072,7 @@ } ], "metadata": { - "date": 1618812140.7132404, + "date": 1619230077.970549, "filename": "cons_news.rst", "kernelspec": { "display_name": "Python", diff --git a/discrete_dp.ipynb b/discrete_dp.ipynb index c58351f..3ee11a7 100644 --- a/discrete_dp.ipynb +++ b/discrete_dp.ipynb @@ -1629,7 +1629,7 @@ } ], "metadata": { - "date": 1618812141.1828506, + "date": 1619230078.4446177, "filename": "discrete_dp.rst", "kernelspec": { "display_name": "Python", diff --git a/dyn_stack.ipynb b/dyn_stack.ipynb index 43a3f07..3af6f02 100644 --- a/dyn_stack.ipynb +++ b/dyn_stack.ipynb @@ -1804,7 +1804,7 @@ } ], "metadata": { - "date": 1618812141.5564651, + "date": 1619230078.8227255, "filename": "dyn_stack.rst", "kernelspec": { "display_name": "Python", diff --git a/estspec.ipynb b/estspec.ipynb index 017e337..fe77db3 100644 --- a/estspec.ipynb +++ b/estspec.ipynb @@ -684,7 +684,7 @@ } ], "metadata": { - "date": 1618812141.7593346, + "date": 1619230079.0286222, "filename": "estspec.rst", "kernelspec": { "display_name": "Python", diff --git a/growth_in_dles.ipynb b/growth_in_dles.ipynb index cc56ef5..f2c9f17 100644 --- a/growth_in_dles.ipynb +++ b/growth_in_dles.ipynb @@ -880,7 +880,7 @@ } ], "metadata": { - "date": 1618812141.8836367, + "date": 1619230079.1558185, "filename": "growth_in_dles.rst", "kernelspec": { "display_name": "Python", diff --git a/hs_invertibility_example.ipynb b/hs_invertibility_example.ipynb index 610cf57..4e9e9de 100644 --- a/hs_invertibility_example.ipynb +++ b/hs_invertibility_example.ipynb @@ -489,7 +489,7 @@ } ], "metadata": { - "date": 1618812142.0289655, + "date": 1619230079.302092, "filename": "hs_invertibility_example.rst", "kernelspec": { "display_name": "Python", diff --git a/hs_recursive_models.ipynb b/hs_recursive_models.ipynb index d30f403..ad52915 100644 --- a/hs_recursive_models.ipynb +++ b/hs_recursive_models.ipynb @@ -356,7 +356,8 @@ "\\begin{aligned}\n", " x_t &= Ax_{t-1} + Cw_t \\\\\n", "&= A^2 x_{t-2} + ACw_{t-1} + Cw_t \\\\\n", - "&= \\Bigl[\\sum_{\\tau=0}^{t-1} A^\\tau Cw_{t-\\tau} \\Bigr] + A^t x_0 \\end{aligned}\n", + "&= \\Bigl[\\sum_{\\tau=0}^{t-1} A^\\tau Cw_{t-\\tau} \\Bigr] + A^t x_0\n", + "\\end{aligned}\n", "$$\n", "\n", "\n", @@ -394,7 +395,8 @@ "$$\n", "\\begin{aligned}\n", " v_1 &= CC^\\prime \\\\\n", - " v_j &= CC^\\prime + A v_{j-1} A^\\prime, \\quad j \\geq 2 \\end{aligned}\n", + " v_j &= CC^\\prime + A v_{j-1} A^\\prime, \\quad j \\geq 2\n", + "\\end{aligned}\n", "$$" ] }, @@ -458,8 +460,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " z_{t+1} &=A_{22} z_t + C_2 w_{t+1}\n", - "\\\\ b_t &= U_b z_t \\\\ d_t &= U_d z_t .\\end{aligned}\n", + "z_{t+1} &=A_{22} z_t + C_2 w_{t+1}\n", + "\\\\ b_t &= U_b z_t \\\\ d_t &= U_d z_t .\n", + "\\end{aligned}\n", "$$\n", "\n", "We can now briefly summarize other components of our economies, in particular\n", @@ -480,8 +483,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " \\Phi_c c_t + \\Phi_g g_t + \\Phi_i i_t &=\\Gamma k_{t-1} + d_t \\\\\n", - "k_t &=\\Delta_k k_{t-1} + \\Theta_k i_t \\\\ g_t \\cdot g_t &=\\ell_t^2 \\end{aligned}\n", + "\\Phi_c c_t + \\Phi_g g_t + \\Phi_i i_t &=\\Gamma k_{t-1} + d_t \\\\\n", + "k_t &=\\Delta_k k_{t-1} + \\Theta_k i_t \\\\ g_t \\cdot g_t &=\\ell_t^2\n", + "\\end{aligned}\n", "$$\n", "\n", "Here $ \\Phi_c, \\Phi_g, \\Phi_i, \\Gamma, \\Delta_k, \\Theta_k $ are all matrices conformable to the vectors they multiply and\n", @@ -503,8 +507,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " s_t &= \\Lambda h_{t-1} + \\Pi c_t \\\\\n", - " h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t \\end{aligned}\n", + "s_t &= \\Lambda h_{t-1} + \\Pi c_t \\\\\n", + "h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t\n", + "\\end{aligned}\n", "$$\n", "\n", "where $ \\Lambda, \\Pi, \\Delta_h, \\Theta_h $ are matrices that pin down the household technology.\n", @@ -609,7 +614,8 @@ "c_t &=\\gamma k_{t-1} + d_{1t} ,\\ \\ \\gamma > 0 \\\\\n", "\\phi_1 i_t &= g_t + d_{2t}, \\ \\ \\phi_1 > 0 \\\\\n", "\\ell^2_t &= g^2_t \\\\\n", - "k_t &= \\delta_k k_{t-1} + i_t ,\\ 0< \\delta_k < 1 \\end{aligned}\n", + "k_t &= \\delta_k k_{t-1} + i_t ,\\ 0< \\delta_k < 1\n", + "\\end{aligned}\n", "$$\n", "\n", "Set\n", @@ -870,7 +876,8 @@ "$$\n", "\\begin{aligned}\n", "z_{t+1} &=A_{22} z_t + C_2 w_{t+1}\n", - "\\\\ b_t &= U_b z_t \\\\ d_t &= U_d z_t \\end{aligned}\n", + "\\\\ b_t &= U_b z_t \\\\ d_t &= U_d z_t\n", + "\\end{aligned}\n", "$$\n", "\n", "**Production Technology**\n", @@ -878,7 +885,8 @@ "$$\n", "\\begin{aligned}\n", "\\Phi_c c_t &+ \\Phi_g g_t + \\Phi_i i_t = \\Gamma k_{t-1} + d_t \\\\\n", - "k_t &=\\Delta_k k_{t-1} + \\Theta_k i_t \\\\ g_t \\cdot g_t &=\\ell_t^2 \\end{aligned}\n", + "k_t &=\\Delta_k k_{t-1} + \\Theta_k i_t \\\\ g_t \\cdot g_t &=\\ell_t^2\n", + "\\end{aligned}\n", "$$\n", "\n", "**Household Technology**\n", @@ -886,7 +894,8 @@ "$$\n", "\\begin{aligned}\n", "s_t &=\n", - "\\Lambda h_{t-1} + \\Pi c_t \\\\ h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t \\end{aligned}\n", + "\\Lambda h_{t-1} + \\Pi c_t \\\\ h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t\n", + "\\end{aligned}\n", "$$\n", "\n", "**Preferences**\n", @@ -918,13 +927,14 @@ "\n", "$$\n", "\\begin{aligned}\n", - " \\Phi_c c_t &+ \\Phi_g \\, g_t + \\Phi_i i_t = \\Gamma k_{t-1} + d_t,\n", + "\\Phi_c c_t &+ \\Phi_g \\, g_t + \\Phi_i i_t = \\Gamma k_{t-1} + d_t,\n", "\\\\\n", "k_t &= \\Delta_k k_{t-1} + \\Theta_k i_t , \\\\\n", "h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t , \\\\\n", "s_t &=\\Lambda h_{t-1} + \\Pi c_t , \\\\\n", " z _{t+1} &= A_{22} z_t + C_2 w_{t+1} , \\ b_t = U_b z_t, \\ \\hbox{ and } \\\n", - "d_t = U_d z_t \\end{aligned}\n", + "d_t = U_d z_t\n", + "\\end{aligned}\n", "$$\n", "\n", "and initial conditions for $ h_{-1}, k_{-1} $, and $ z_0 $.\n", @@ -963,14 +973,15 @@ "\n", "$$\n", "\\begin{aligned}\n", - " {\\mathcal L} &= - E \\sum_{t=0}^\\infty \\beta^t \\biggl[\n", + "{\\mathcal L} &= - E \\sum_{t=0}^\\infty \\beta^t \\biggl[\n", "\\Bigl( {1 \\over 2} \\Bigr) [ (s_t - b_t) \\cdot (s_t - b_t) + g_t\n", "\\cdot g_t] \\\\ &+ {\\cal M}_t^{d \\prime} \\cdot ( \\Phi _cc_t +\n", "\\Phi_gg_t + \\Phi_ii_t - \\Gamma k_{t-1} - d_t ) \\\\ &+{\\cal M}_t^{k\n", "\\prime} \\cdot (k_t - \\Delta_k k_{t-1} - \\Theta_k i_t ) \\\\ &+ {\\cal\n", "M}_t^{h \\prime} \\cdot (h_t - \\Delta_h h_{t-1} - \\Theta_h c_t) \\\\ &+\n", "{\\cal M}_t^{s \\prime} \\cdot (s_t - \\Lambda h_{t-1} - \\Pi c_t )\n", - "\\biggr] \\Bigl| J_0 \\end{aligned}\n", + "\\biggr] \\Bigl| J_0\n", + "\\end{aligned}\n", "$$\n", "\n", "The planner maximizes $ {\\mathcal L} $ with respect to the quantities $ \\{c_t, i_t, g_t\\}_{t=0}^\\infty $\n", @@ -982,15 +993,16 @@ "\n", "$$\n", "\\begin{aligned}\n", - " - \\Phi_c^\\prime {\\cal M}_t^d &+\\Theta_h^\\prime {\\cal\n", + "-\\Phi_c^\\prime {\\cal M}_t^d &+\\Theta_h^\\prime {\\cal\n", "M}_t^h + \\Pi^\\prime {\\cal M}_t^s = 0 , \\\\\n", "&- g_t - \\Phi_g^\\prime {\\cal M}_t^d = 0 , \\\\\n", - "- {\\cal M}_t^h &+ \\beta E ( \\Delta_h^\\prime {\\cal M}^h_{t+1} +\n", + "-{\\cal M}_t^h &+ \\beta E ( \\Delta_h^\\prime {\\cal M}^h_{t+1} +\n", "\\Lambda^\\prime {\\cal M}_{t+1}^s ) \\mid J_t = 0 , \\\\\n", "&- \\Phi_i^\\prime {\\cal M}_t^d + \\Theta_k^\\prime {\\cal M}_t^k = 0 , \\\\\n", - "- {\\cal M}_t^k &+ \\beta E ( \\Delta_k^\\prime {\\cal M}^k_{t+1} + \\Gamma^\\prime\n", + "-{\\cal M}_t^k &+ \\beta E ( \\Delta_k^\\prime {\\cal M}^k_{t+1} + \\Gamma^\\prime\n", "{\\cal M}_{t+1}^d) \\mid J_t = 0 , \\\\\n", - "&- s_t + b_t - {\\cal M}_t^s = 0 \\end{aligned}\n", + "&- s_t + b_t - {\\cal M}_t^s = 0\n", + "\\end{aligned}\n", "$$\n", "\n", "for $ t=0,1, \\ldots $.\n", @@ -1001,10 +1013,11 @@ "\n", "$$\n", "\\begin{aligned}\n", - " \\lim_{t \\to \\infty}& \\beta^t E [ {\\cal M}_t^{k \\prime} k_t ]\n", + "\\lim_{t \\to \\infty}& \\beta^t E [ {\\cal M}_t^{k \\prime} k_t ]\n", "\\mid J_0 = 0 \\\\\n", - " \\lim_{t \\to \\infty}& \\beta^t E [ {\\cal M}_t^{h \\prime} h_t ]\n", - "\\mid J_0 = 0\\end{aligned}\n", + "\\lim_{t \\to \\infty}& \\beta^t E [ {\\cal M}_t^{h \\prime} h_t ]\n", + "\\mid J_0 = 0\n", + "\\end{aligned}\n", "$$\n", "\n", "The system formed by the FONCs and the transition equations can be handed over to Python.\n", @@ -1015,13 +1028,13 @@ "\n", "$$\n", "\\begin{aligned}\n", - " - \\Phi_c^\\prime {\\cal M}_t^d &+\\Theta_h^\\prime {\\cal\n", + "-\\Phi_c^\\prime {\\cal M}_t^d &+\\Theta_h^\\prime {\\cal\n", "M}_t^h + \\Pi^\\prime {\\cal M}_t^s = 0 , \\\\\n", "&- g_t - \\Phi_g^\\prime {\\cal M}_t^d = 0 , \\\\\n", "- {\\cal M}_t^h &+ \\beta E ( \\Delta_h^\\prime {\\cal M}^h_{t+1} +\n", "\\Lambda^\\prime {\\cal M}_{t+1}^s ) \\mid J_t = 0 , \\\\\n", "&- \\Phi_i^\\prime {\\cal M}_t^d + \\Theta_k^\\prime {\\cal M}_t^k = 0 , \\\\\n", - "- {\\cal M}_t^k &+ \\beta E ( \\Delta_k^\\prime {\\cal M}^k_{t+1} + \\Gamma^\\prime\n", + "-{\\cal M}_t^k &+ \\beta E ( \\Delta_k^\\prime {\\cal M}^k_{t+1} + \\Gamma^\\prime\n", "{\\cal M}_{t+1}^d) \\mid J_t = 0 , \\\\\n", "&- s_t + b_t - {\\cal M}_t^s = 0 \\\\\n", "\\Phi_c c_t &+ \\Phi_g \\, g_t + \\Phi_i i_t = \\Gamma k_{t-1} + d_t,\n", @@ -1029,8 +1042,9 @@ "k_t &= \\Delta_k k_{t-1} + \\Theta_k i_t , \\\\\n", "h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t , \\\\\n", "s_t &=\\Lambda h_{t-1} + \\Pi c_t , \\\\\n", - " z _{t+1} &= A_{22} z_t + C_2 w_{t+1} , \\ b_t = U_b z_t, \\ \\hbox{ and } \\\n", - "d_t = U_d z_t \\end{aligned}\n", + "z _{t+1} &= A_{22} z_t + C_2 w_{t+1} , \\ b_t = U_b z_t, \\ \\hbox{ and } \\\n", + "d_t = U_d z_t\n", + "\\end{aligned}\n", "$$\n", "\n", "The Lagrange multipliers or **shadow prices** satisfy\n", @@ -1092,12 +1106,13 @@ "\n", "$$\n", "\\begin{aligned}\n", - " \\Phi _cc_0 &+ \\Phi_g g_0 + \\Phi_ii_0 = \\Gamma k_{-1} + d_0 ,\\\\\n", + "\\Phi _cc_0 &+ \\Phi_g g_0 + \\Phi_ii_0 = \\Gamma k_{-1} + d_0 ,\\\\\n", "k_0 &= \\Delta_k k_{-1} + \\Theta_k i_0 , \\\\\n", "h_0 &= \\Delta_h h_{-1} + \\Theta_h c_0 , \\\\\n", "s_0 &= \\Lambda h_{-1} + \\Pi c_0 , \\\\\n", - " z_1 &= A_{22} z_0 + C_2 w_1,\\ b_0 = U_b z_0 \\ \\hbox{ and }\\ d_0 =\n", - "U_d z_0 \\end{aligned}\n", + "z_1 &= A_{22} z_0 + C_2 w_1,\\ b_0 = U_b z_0 \\ \\hbox{ and }\\ d_0 =\n", + "U_d z_0\n", + "\\end{aligned}\n", "$$\n", "\n", "Because this is a linear-quadratic dynamic programming problem, it turns out that the value function has the form\n", @@ -1185,9 +1200,10 @@ "\n", "$$\n", "\\begin{aligned}\n", - " P_{j+1} &= R + \\beta A^\\prime P_{j} A - (\\beta\n", + "P_{j+1} &= R + \\beta A^\\prime P_{j} A - (\\beta\n", "A^\\prime P_{j} B + W) (Q + \\beta B^\\prime P_{j} B)^{-1} (\\beta B^\\prime P_{j}\n", - "A + W')\\\\ \\rho_{j+1} &=\\beta \\rho_{j} + \\beta \\ {\\rm trace} \\ P_{j} C C^\\prime\\end{aligned}\n", + "A + W')\\\\ \\rho_{j+1} &=\\beta \\rho_{j} + \\beta \\ {\\rm trace} \\ P_{j} C C^\\prime\n", + "\\end{aligned}\n", "$$\n", "\n", "We can now state the planning problem as a dynamic programming problem\n", @@ -1205,19 +1221,20 @@ "\n", "$$\n", "x_t = \\begin{bmatrix} h_{t-1} \\\\ k_{t-1} \\\\ z_t \\end{bmatrix} , \\qquad\n", - " u_t = i_t\n", + "u_t = i_t\n", "$$\n", "\n", "where\n", "\n", "$$\n", "\\begin{aligned}\n", - " A &=\\begin{bmatrix} \\Delta_h & \\Theta_h U_c [ \\Phi_c \\ \\\n", + "A &=\\begin{bmatrix} \\Delta_h & \\Theta_h U_c [ \\Phi_c \\ \\\n", "\\Phi_g]^{-1} \\Gamma & \\Theta_h U_c [ \\Phi_c \\ \\ \\Phi_g]^{-1} U_d \\\\ 0\n", "& \\Delta_k & 0 \\\\ 0 & 0 & A_{22} \\\\ \\end{bmatrix} \\\\\n", "B &= \\begin{bmatrix} - \\Theta_h U_c [ \\Phi_c \\ \\ \\Phi_g]^{-1} \\Phi_i\n", "\\\\ \\Theta_k \\\\ 0 \\end{bmatrix} \\ ,\\ C = \\begin{bmatrix} 0 \\\\ 0 \\\\\n", - "C_2 \\end{bmatrix} \\end{aligned}\n", + "C_2 \\end{bmatrix}\n", + "\\end{aligned}\n", "$$\n", "\n", "$$\n", @@ -1252,7 +1269,8 @@ "{\\mathcal M}_t^k &= M_k x_t\\ \\hbox{ and }\\ {\\cal M}_t^h = M_h\n", "x_t \\ \\hbox{ where } \\\\\n", "M_k &= 2 \\beta [ 0 \\ I \\ 0 ] P A^o \\\\\n", - "M_h &= 2 \\beta [ I \\ 0 \\ 0 ] P A^o \\end{aligned}\n", + "M_h &= 2 \\beta [ I \\ 0 \\ 0 ] P A^o\n", + "\\end{aligned}\n", "$$\n", "\n", "$$\n", @@ -1521,9 +1539,10 @@ "\n", "$$\n", "\\begin{aligned}\n", - " L^2_t &= [\\{y_s\\}^\\infty_{s=t} : \\ y_s \\ \\hbox{ is a random variable\n", + "L^2_t &= [\\{y_s\\}^\\infty_{s=t} : \\ y_s \\ \\hbox{ is a random variable\n", "in }\\ J_s\\ \\hbox{ for }\\ s \\geq t \\\\\n", - "&\\hbox {and } E\\, \\sum^\\infty_{s=t}\\, \\beta^{s-t}\\ y^2_s \\mid J_t < + \\infty] .\\end{aligned}\n", + "&\\hbox {and } E\\, \\sum^\\infty_{s=t}\\, \\beta^{s-t}\\ y^2_s \\mid J_t < + \\infty] .\n", + "\\end{aligned}\n", "$$\n", "\n", "$$\n", @@ -1574,8 +1593,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " x_{t+1} &= A^o x_t + Cw_{t+1} \\\\\n", - "y_t & = Gx_t + v_t\\end{aligned}\n", + "x_{t+1} &= A^o x_t + Cw_{t+1} \\\\\n", + "y_t & = Gx_t + v_t\n", + "\\end{aligned}\n", "$$\n", "\n", "where $ v_t $ is a martingale difference sequence of measurement\n", @@ -1591,7 +1611,8 @@ "$$\n", "\\begin{aligned}\n", "\\hat x_{t+1} &=A^o \\hat x_t + K_t a_t \\\\\n", - "y_t &= G \\hat x_t + a_t,\\end{aligned}\n", + "y_t &= G \\hat x_t + a_t,\n", + "\\end{aligned}\n", "$$\n", "\n", "where $ a_t = y_t - E[y_t | y^{t-1}], E a_t a_t^\\prime \\equiv \\Omega_t = G \\Sigma_t G^\\prime + R $.\n", @@ -1621,8 +1642,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " \\Sigma_{t+1} &= A^o \\Sigma_t A^{o \\prime} + CC^\\prime \\\\\n", - "&- A^o \\Sigma_t G^\\prime (G \\Sigma_t G^\\prime + R)^{-1} G \\Sigma_t A^{o \\prime}\\end{aligned}\n", + "\\Sigma_{t+1} &= A^o \\Sigma_t A^{o \\prime} + CC^\\prime \\\\\n", + "&- A^o \\Sigma_t G^\\prime (G \\Sigma_t G^\\prime + R)^{-1} G \\Sigma_t A^{o \\prime}\n", + "\\end{aligned}\n", "$$\n", "\n", "**Innovations Representation as Whitener**\n", @@ -1631,8 +1653,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " a_t &=y_t - G \\hat x_t \\\\\n", - "\\hat x_{t+1} &= A^o \\hat x_t + K_t a_t \\end{aligned}\n", + "a_t &=y_t - G \\hat x_t \\\\\n", + "\\hat x_{t+1} &= A^o \\hat x_t + K_t a_t\n", + "\\end{aligned}\n", "$$\n", "\n", "can be used recursively to construct a record of innovations\n", @@ -1643,15 +1666,17 @@ "\n", "$$\n", "\\begin{aligned}\n", - " \\Sigma &= A^o \\Sigma A^{o \\prime} + CC^\\prime \\\\\n", + "\\Sigma &= A^o \\Sigma A^{o \\prime} + CC^\\prime \\\\\n", "&- A^o \\Sigma G^\\prime (G \\Sigma G^\\prime + R)^{-1} G \\Sigma A^{o \\prime} \\\\\n", - " K &= A^o \\Sigma_t G^\\prime (G \\Sigma G^\\prime + R)^{-1}\\end{aligned}\n", + "K &= A^o \\Sigma_t G^\\prime (G \\Sigma G^\\prime + R)^{-1}\n", + "\\end{aligned}\n", "$$\n", "\n", "$$\n", "\\begin{aligned}\n", - " \\hat x_{t+1} &= A^o \\hat x_t + K a_t \\\\\n", - "y_t &= G \\hat x_t + a_t\\end{aligned}\n", + "\\hat x_{t+1} &= A^o \\hat x_t + K a_t \\\\\n", + "y_t &= G \\hat x_t + a_t\n", + "\\end{aligned}\n", "$$\n", "\n", "where $ E a_t a_t^\\prime \\equiv \\Omega = G \\Sigma G^\\prime + R $." @@ -1668,10 +1693,11 @@ "\n", "$$\n", "\\begin{aligned}\n", - " f(y_T, y_{T-1}, \\ldots, y_0 )&=\n", - " f_T(y_T \\vert y_{T-1}, \\ldots, y_0) f_{T-1}(y_{T-1} \\vert\n", - " y_{T-2}, \\ldots, y_0) \\cdots f_1(y_1 \\vert y_0) f_0(y_0 ) \\\\\n", - " &= g_T(a_T) g_{T-1} (a_{T-1}) \\ldots g_1(a_1) f_0(y_0).\\end{aligned}\n", + "f(y_T, y_{T-1}, \\ldots, y_0 )&=\n", + " f_T(y_T \\vert y_{T-1}, \\ldots, y_0) f_{T-1}(y_{T-1} \\vert\n", + " y_{T-2}, \\ldots, y_0) \\cdots f_1(y_1 \\vert y_0) f_0(y_0 ) \\\\\n", + "&= g_T(a_T) g_{T-1} (a_{T-1}) \\ldots g_1(a_1) f_0(y_0).\n", + "\\end{aligned}\n", "$$\n", "\n", "Gaussian Log-Likelihood:\n", @@ -1721,7 +1747,8 @@ "\\begin{aligned}\n", " & G (zI - A^o)^{-1} C C^\\prime (z^{-1} I - A^{o\\prime})^{-1} G^\\prime + R = \\\\\n", "& [G(zI-A^o)^{-1}K +I] [G \\Sigma G^\\prime + R] [K'(z^{-1} I -A^{o\\prime})^{-1}\n", - "G^\\prime + I] .\\end{aligned}\n", + "G^\\prime + I] .\n", + "\\end{aligned}\n", "$$\n", "\n", "**Key Insight:** The zeros of the polynomial\n", @@ -1792,9 +1819,10 @@ "\n", "$$\n", "\\begin{aligned}\n", - " h_t &=\\Delta_h h_{t-1} + \\Theta_h c_t \\\\\n", + "h_t &=\\Delta_h h_{t-1} + \\Theta_h c_t \\\\\n", " s_t &= \\Lambda h_{t-1} + \\Pi c_t \\\\\n", - " b_t &=U_b z_t\\end{aligned}\n", + " b_t &=U_b z_t\n", + "\\end{aligned}\n", "$$\n", "\n", "**Definition:** A household service technology\n", @@ -1815,9 +1843,10 @@ "\n", "$$\n", "\\begin{aligned}\n", - " c_t &= - \\Pi^{-1} \\Lambda h_{t-1} + \\Pi^{-1} s_t\\\\\n", + "c_t &= - \\Pi^{-1} \\Lambda h_{t-1} + \\Pi^{-1} s_t\\\\\n", "h_t &= (\\Delta_h - \\Theta_h\\Pi^{-1} \\Lambda) h_{t-1} + \\Theta_h \\Pi^{-1}\n", - "s_t \\end{aligned}\n", + "s_t\n", + "\\end{aligned}\n", "$$\n", "\n", "The restriction on the eigenvalues of the matrix\n", @@ -1844,7 +1873,8 @@ "$$\n", "\\begin{aligned}\n", "s_{i,t}&= \\Lambda h_{i,t-1} \\\\\n", - "h_{i,t}&= \\Delta _h h_{i,t-1}\\end{aligned}\n", + "h_{i,t}&= \\Delta _h h_{i,t-1}\n", + "\\end{aligned}\n", "$$\n", "\n", "where $ h_{i,-1} = h_{-1} $.\n", @@ -1863,11 +1893,12 @@ "\n", "$$\n", "\\begin{aligned}\n", - " c_t &= -\\Pi^{-1} \\Lambda h_{t-1} + \\Pi ^{-1} b_t\n", - " - \\Pi^{-1} \\mu_0^w E_t \\{ \\Pi^{\\prime\\, -1} - \\Pi^{\\prime\\, -1}\\Theta_h ' \\\\\n", + "c_t &= -\\Pi^{-1} \\Lambda h_{t-1} + \\Pi ^{-1} b_t\n", + "-\\Pi^{-1} \\mu_0^w E_t \\{ \\Pi^{\\prime\\, -1} - \\Pi^{\\prime\\, -1}\\Theta_h ' \\\\\n", "& \\qquad [I - (\\Delta_h ' - \\Lambda ' \\Pi^{\\prime \\, -1} \\Theta_h ')\\beta L^{-1}]\n", - " ^{-1} \\Lambda ' \\Pi^{\\prime -1} \\beta L^{-1} \\} p_t^0 \\\\\n", - " h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t \\end{aligned}\n", + "^{-1} \\Lambda ' \\Pi^{\\prime -1} \\beta L^{-1} \\} p_t^0 \\\\\n", + "h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t\n", + "\\end{aligned}\n", "$$\n", "\n", "This system expresses consumption demands at date $ t $ as\n", @@ -1922,8 +1953,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " s_{i,t}&= \\Lambda h_{i,t-1} \\\\\n", - "h_{i,t}&= \\Delta _h h_{i,t-1},\\end{aligned}\n", + "s_{i,t}&= \\Lambda h_{i,t-1} \\\\\n", + "h_{i,t}&= \\Delta _h h_{i,t-1},\n", + "\\end{aligned}\n", "$$\n", "\n", "where now $ h_{i,t-1} = h_{t-1} $. Define time $ t $ wealth\n", @@ -1935,17 +1967,18 @@ "\n", "$$\n", "\\mu^w_t = {E_t \\sum^\\infty_{j=0} \\beta^j \\rho^t_{t+j}\\cdot\n", - " (b_{t+j} -s_{i,t+j}) - W_t \\over E_t \\sum^\\infty_{t=0}\n", + "(b_{t+j} -s_{i,t+j}) - W_t \\over E_t \\sum^\\infty_{t=0}\n", "\\beta^j \\rho^t_{t+j} \\cdot \\rho^t_{t+j}}\n", "$$\n", "\n", "$$\n", "\\begin{aligned}\n", - " c_t &= -\\Pi^{-1} \\Lambda h_{t-1} + \\Pi ^{-1} b_t\n", - " - \\Pi^{-1} \\mu_t^w E_t \\{ \\Pi^{\\prime\\, -1} - \\Pi^{\\prime\\, -1}\\Theta_h ' \\\\\n", - " & \\qquad [I - (\\Delta_h ' - \\Lambda ' \\Pi^{\\prime \\, -1} \\Theta_h ')\\beta L^{-1}]\n", - " ^{-1} \\Lambda ' \\Pi^{\\prime -1} \\beta L^{-1} \\} p_t^t \\\\\n", - " h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t \\end{aligned}\n", + "c_t &= -\\Pi^{-1} \\Lambda h_{t-1} + \\Pi ^{-1} b_t\n", + "-\\Pi^{-1} \\mu_t^w E_t \\{ \\Pi^{\\prime\\, -1} - \\Pi^{\\prime\\, -1}\\Theta_h ' \\\\\n", + "& \\qquad [I - (\\Delta_h ' - \\Lambda ' \\Pi^{\\prime \\, -1} \\Theta_h ')\\beta L^{-1}]\n", + "^{-1} \\Lambda ' \\Pi^{\\prime -1} \\beta L^{-1} \\} p_t^t \\\\\n", + "h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t\n", + "\\end{aligned}\n", "$$" ] }, @@ -2030,7 +2063,8 @@ " \\mu_0^w E_t \\{ \\Pi^{\\prime\\, -1} - \\Pi^{\\prime\\, -1}\\Theta_h' \\\\\n", " & \\qquad[I - (\\Delta_h' - \\Lambda' \\Pi^{\\prime\\, -1} \\Theta_h')\\beta\n", " L^{-1}]^{-1} \\Lambda' \\Pi^{\\prime -1} \\beta L^{-1} \\} p_t \\\\\n", - " h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t \\end{aligned}\n", + " h_t &= \\Delta_h h_{t-1} + \\Theta_h c_t\n", + "\\end{aligned}\n", "$$\n", "\n", "From material described earlier in this lecture, we know how to reverse engineer preferences that generate this demand system\n", @@ -2065,7 +2099,7 @@ " % x_{t+1}& = A^o x_t + C w_{t+1} \\\\\n", " % d_t& = S_d x_t \\\\\n", " % p_t& = M_c x_t\n", - " \\end{aligned}\n", + "\\end{aligned}\n", "$$" ] }, @@ -2087,7 +2121,8 @@ "\\begin{aligned}\n", " c_t &= \\gamma k_{t-1} \\\\\n", " k_t &= \\delta_k k_{t-1} + i_t \\\\\n", - " g_t &= f_1 i_t + f_2 d_t \\end{aligned}\n", + " g_t &= f_1 i_t + f_2 d_t\n", + "\\end{aligned}\n", "$$\n", "\n", "where $ d_t $ is a cost shifter, $ \\gamma> 0 $, and\n", @@ -2114,7 +2149,8 @@ "\\begin{aligned}\n", " R_t &= b_t + \\alpha h_t \\\\\n", " p_t &= E_t \\sum_{\\tau =0}^\\infty (\\beta \\delta_h)^\\tau\n", - " R_{t+\\tau} \\end{aligned}\n", + " R_{t+\\tau}\n", + "\\end{aligned}\n", "$$\n", "\n", "where $ h_t $ is the stock of housing at time $ t $\n", @@ -2196,7 +2232,8 @@ " -(\\gamma_0 \\tilde h_t) (g \\tilde x_{t-1}) - (\\gamma_1 \\tilde h_t)\n", " (g \\tilde x_{t-2}) - m_t c_t \\\\\n", " &- \\Psi(\\tilde x_t, \\tilde x_{t-1},\n", - " \\tilde x_{t-2}, c_t) \\}\\end{aligned}\n", + " \\tilde x_{t-2}, c_t) \\}\n", + "\\end{aligned}\n", "$$\n", "\n", "where\n", @@ -2290,7 +2327,7 @@ "0 & 0 & 0 & \\cdots & 0 \\end{bmatrix} \\begin{bmatrix}h_{1t-1}\\\\ h_{2t-1}\\\\ \\vdots\\\\ h_{k,t-1} \\\\\n", " h_{k+1,t-1}\\end{bmatrix} + \\begin{bmatrix}0\\\\ 0\\\\ \\vdots\\\\ 0\\\\ 1\\\\\\end{bmatrix} c_t \\\\\n", "%b_t &= \\epsilon_{1t}\n", - " \\end{aligned}\n", + "\\end{aligned}\n", "$$\n", "\n", "This specification sets Rosen’s $ N_t = h_{1t-1}, n_t = c_t,\n", @@ -2329,7 +2366,8 @@ "$$\n", "\\begin{aligned}\n", " N_{st+k} &=\\delta_N N_{st+k-1} + n_{st} \\\\\n", - " N_{ut} &=\\delta_N N_{ut-1} + n_{ut} ; \\end{aligned}\n", + " N_{ut} &=\\delta_N N_{ut-1} + n_{ut} ;\n", + "\\end{aligned}\n", "$$\n", "\n", "where $ N_{st}, N_{ut} $ are stocks of the two types of labor, and\n", @@ -2345,7 +2383,8 @@ " v_{st} &= E_t \\beta^k \\sum_{j=0}^\\infty (\\beta \\delta_N)^j\n", " w_{st+k+j} \\\\\n", " v_{ut} &=E_t \\sum_{j=0}^\\infty (\\beta \\delta_N)^j\n", - " w_{ut+j}\\end{aligned}\n", + " w_{ut+j}\n", + "\\end{aligned}\n", "$$\n", "\n", "where $ w_{ut}, w_{st} $ are wage rates for the two occupations;\n", @@ -2382,8 +2421,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " \\phi_c \\cdot c_t+i_t&=\\gamma k_{t-1}+e_t \\\\\n", - " k_t&= k_{t-1} + i_t \\end{aligned}\n", + "\\phi_c \\cdot c_t+i_t&=\\gamma k_{t-1}+e_t \\\\\n", + " k_t&= k_{t-1} + i_t\n", + "\\end{aligned}\n", "$$\n", "\n", "$$\n", @@ -2418,8 +2458,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " {\\mathcal M}_t^k &= E ({\\mathcal M}_{t+1}^k | J_t) \\\\\n", - "{\\mathcal M}_t^e &= E ({\\mathcal M}_{t+1}^e | J_t) \\end{aligned}\n", + "{\\mathcal M}_t^k &= E ({\\mathcal M}_{t+1}^k | J_t) \\\\\n", + "{\\mathcal M}_t^e &= E ({\\mathcal M}_{t+1}^e | J_t)\n", + "\\end{aligned}\n", "$$\n", "\n", "and\n", @@ -2532,9 +2573,10 @@ "\n", "$$\n", "\\begin{aligned}\n", - " c_{jt} &= - \\Pi^{-1} \\Lambda h_{j,t-1} + \\Pi^{-1}s_{jt} \\\\\n", + "c_{jt} &= - \\Pi^{-1} \\Lambda h_{j,t-1} + \\Pi^{-1}s_{jt} \\\\\n", "h_{jt} &= (\\Delta_h - \\Theta_h \\Pi^{-1}\\Lambda) h_{j,t-1} + \\Pi^{-1}\n", - " \\Theta_h s_{jt} \\end{aligned}\n", + " \\Theta_h s_{jt}\n", + "\\end{aligned}\n", "$$\n", "\n", "Here $ h_{j,-1} $ given." @@ -2559,8 +2601,9 @@ "\n", "$$\n", "\\begin{aligned}\n", - " s_{it} &= \\Lambda_i h_{i t-1} + \\Pi_i\\, c_{it} \\\\\n", - "h_{it} &=\\Delta_{h_i}\\, h_{i t-1} + \\Theta_{h_i} c_{it}\\ ,\\ i=1,2 .\\end{aligned}\n", + "s_{it} &= \\Lambda_i h_{i t-1} + \\Pi_i\\, c_{it} \\\\\n", + "h_{it} &=\\Delta_{h_i}\\, h_{i t-1} + \\Theta_{h_i} c_{it}\\ ,\\ i=1,2 .\n", + "\\end{aligned}\n", "$$\n", "\n", "$$\n", @@ -2595,11 +2638,12 @@ "\n", "$$\n", "\\begin{aligned}\n", - " & - {1\\over 2}\\, \\lambda E_0 \\sum^\\infty_{t=0}\\, \\beta^t [ (s_{1t}\n", - "- b_{1t})\\cdot (s_{1t} - b_{1t}) + \\ell^2_{1t}]\\\\\n", - " &\n", - "- {1\\over 2}\\, (1-\\lambda) E_0 \\sum^\\infty_{t=0}\\, \\beta^t [ (s_{2t} -\n", - "b_{2t}) \\cdot (s_{2t} - b_{2t}) + \\ell^2_{2t}] \\end{aligned}\n", + "& - {1\\over 2}\\, \\lambda E_0 \\sum^\\infty_{t=0}\\, \\beta^t [ (s_{1t}\n", + "-b_{1t})\\cdot (s_{1t} - b_{1t}) + \\ell^2_{1t}]\\\\\n", + "&\n", + "-{1\\over 2}\\, (1-\\lambda) E_0 \\sum^\\infty_{t=0}\\, \\beta^t [ (s_{2t} -\n", + "b_{2t}) \\cdot (s_{2t} - b_{2t}) + \\ell^2_{2t}]\n", + "\\end{aligned}\n", "$$\n", "\n", "**Mongrel Aggregation: Static**\n", @@ -2646,12 +2690,13 @@ "\n", "$$\n", "\\begin{aligned}\n", - " p_t &= (\\mu_{01} \\Pi_1^{-1} \\Pi_1^{-1 \\prime} + \\mu_{02}\n", + "p_t &= (\\mu_{01} \\Pi_1^{-1} \\Pi_1^{-1 \\prime} + \\mu_{02}\n", " \\Pi_2^{-1} \\Pi_2^{-1 \\prime})^{-1}\n", " (\\Pi_1^{-1} b_{1t} + \\Pi_2^{-1} b_{2t}) \\\\\n", - " &- (\\mu_{01} \\Pi_1^{-1} \\Pi_1^{-1 \\prime} +\n", + "&- (\\mu_{01} \\Pi_1^{-1} \\Pi_1^{-1 \\prime} +\n", " \\mu_{02} \\Pi_2^{-1} \\Pi_2^{-1 \\prime}\n", - " )^{-1} c_t\\end{aligned}\n", + " )^{-1} c_t\n", + "\\end{aligned}\n", "$$\n", "\n", "**Punchline:** choose $ \\Pi $ associated with the aggregate ordering to\n", @@ -2687,9 +2732,10 @@ "\n", "$$\n", "\\begin{aligned}\n", - " h_{jt} &= \\Delta_{hj}\\, h_{jt-1} + \\Theta_{hj}\\, c_{jt}, j=1,2\\\\\n", + "h_{jt} &= \\Delta_{hj}\\, h_{jt-1} + \\Theta_{hj}\\, c_{jt}, j=1,2\\\\\n", "s_{jt} &=\\Delta_j h_{jt-1} + \\Pi_j c_{jt}\\ , j=1,2\\\\\n", - "c_{1t} + c_{2t} &=c_t\\end{aligned}\n", + "c_{1t} + c_{2t} &=c_t\n", + "\\end{aligned}\n", "$$\n", "\n", "subject to $ (h_{1, -1},\\, h_{2, -1}) $ given and\n", @@ -2721,7 +2767,7 @@ } ], "metadata": { - "date": 1618812142.2629051, + "date": 1619230079.5436962, "filename": "hs_recursive_models.rst", "kernelspec": { "display_name": "Python", diff --git a/index.ipynb b/index.ipynb index e7fb729..cac3aad 100644 --- a/index.ipynb +++ b/index.ipynb @@ -60,7 +60,7 @@ } ], "metadata": { - "date": 1618874888.6762455, + "date": 1619230079.6939697, "filename": "index.rst", "kernelspec": { "display_name": "Python", diff --git a/index_asset_pricing.ipynb b/index_asset_pricing.ipynb index 2712d5c..f665d42 100644 --- a/index_asset_pricing.ipynb +++ b/index_asset_pricing.ipynb @@ -50,7 +50,7 @@ } ], "metadata": { - "date": 1618812142.6370249, + "date": 1619230079.9295764, "filename": "index_asset_pricing.rst", "kernelspec": { "display_name": "Python", diff --git a/index_classic_linear_models.ipynb b/index_classic_linear_models.ipynb index af005cb..0ae97ff 100644 --- a/index_classic_linear_models.ipynb +++ b/index_classic_linear_models.ipynb @@ -36,7 +36,7 @@ } ], "metadata": { - "date": 1618812142.6581662, + "date": 1619230079.9506888, "filename": "index_classic_linear_models.rst", "kernelspec": { "display_name": "Python", diff --git a/index_dynamic_programming_squared.ipynb b/index_dynamic_programming_squared.ipynb index 1e4bc7e..a606393 100644 --- a/index_dynamic_programming_squared.ipynb +++ b/index_dynamic_programming_squared.ipynb @@ -95,7 +95,7 @@ } ], "metadata": { - "date": 1618874888.707577, + "date": 1619230079.9915779, "filename": "index_dynamic_programming_squared.rst", "kernelspec": { "display_name": "Python", diff --git a/index_hs_recursive_models.ipynb b/index_hs_recursive_models.ipynb index cea2bcf..db7106e 100644 --- a/index_hs_recursive_models.ipynb +++ b/index_hs_recursive_models.ipynb @@ -66,7 +66,7 @@ } ], "metadata": { - "date": 1618812142.7338166, + "date": 1619230080.0270367, "filename": "index_hs_recursive_models.rst", "kernelspec": { "display_name": "Python", diff --git a/index_lq_control.ipynb b/index_lq_control.ipynb index 4db09aa..b54642b 100644 --- a/index_lq_control.ipynb +++ b/index_lq_control.ipynb @@ -88,7 +88,7 @@ } ], "metadata": { - "date": 1618812142.771562, + "date": 1619230080.0675685, "filename": "index_lq_control.rst", "kernelspec": { "display_name": "Python", diff --git a/index_multi_agent_models.ipynb b/index_multi_agent_models.ipynb index eb97995..000e307 100644 --- a/index_multi_agent_models.ipynb +++ b/index_multi_agent_models.ipynb @@ -60,7 +60,7 @@ } ], "metadata": { - "date": 1618812142.8028126, + "date": 1619230080.1008086, "filename": "index_multi_agent_models.rst", "kernelspec": { "display_name": "Python", diff --git a/index_time_series_models.ipynb b/index_time_series_models.ipynb index 7c47fc8..9f286d1 100644 --- a/index_time_series_models.ipynb +++ b/index_time_series_models.ipynb @@ -72,7 +72,7 @@ } ], "metadata": { - "date": 1618812142.8379095, + "date": 1619230080.1389678, "filename": "index_time_series_models.rst", "kernelspec": { "display_name": "Python", diff --git a/index_toc.ipynb b/index_toc.ipynb index ff5ea85..febbe0f 100644 --- a/index_toc.ipynb +++ b/index_toc.ipynb @@ -76,7 +76,7 @@ } ], "metadata": { - "date": 1618812142.920322, + "date": 1619230080.2282295, "filename": "index_toc.rst", "kernelspec": { "display_name": "Python", diff --git a/index_tools_and_techniques.ipynb b/index_tools_and_techniques.ipynb index 6119600..0bf4fe5 100644 --- a/index_tools_and_techniques.ipynb +++ b/index_tools_and_techniques.ipynb @@ -63,7 +63,7 @@ } ], "metadata": { - "date": 1618812142.9540548, + "date": 1619230080.2633407, "filename": "index_tools_and_techniques.rst", "kernelspec": { "display_name": "Python", diff --git a/irfs_in_hall_model.ipynb b/irfs_in_hall_model.ipynb index 2ee5f5a..f56b296 100644 --- a/irfs_in_hall_model.ipynb +++ b/irfs_in_hall_model.ipynb @@ -445,7 +445,7 @@ } ], "metadata": { - "date": 1618812143.006659, + "date": 1619230080.3171942, "filename": "irfs_in_hall_model.rst", "kernelspec": { "display_name": "Python", diff --git a/knowing_forecasts_of_others.ipynb b/knowing_forecasts_of_others.ipynb index eb909ce..ff9316d 100644 --- a/knowing_forecasts_of_others.ipynb +++ b/knowing_forecasts_of_others.ipynb @@ -1927,7 +1927,7 @@ } ], "metadata": { - "date": 1618812143.2310474, + "date": 1619230080.5478446, "filename": "knowing_forecasts_of_others.rst", "kernelspec": { "display_name": "Python", diff --git a/lqramsey.ipynb b/lqramsey.ipynb index da30927..bba59a7 100644 --- a/lqramsey.ipynb +++ b/lqramsey.ipynb @@ -1241,7 +1241,7 @@ } ], "metadata": { - "date": 1618812143.4996943, + "date": 1619230080.8263125, "filename": "lqramsey.rst", "kernelspec": { "display_name": "Python", diff --git a/lu_tricks.ipynb b/lu_tricks.ipynb index d582c10..682f246 100644 --- a/lu_tricks.ipynb +++ b/lu_tricks.ipynb @@ -1525,7 +1525,7 @@ } ], "metadata": { - "date": 1618812143.870546, + "date": 1619230081.2107198, "filename": "lu_tricks.rst", "kernelspec": { "display_name": "Python", diff --git a/lucas_asset_pricing_dles.ipynb b/lucas_asset_pricing_dles.ipynb index 803f135..c5b15c4 100644 --- a/lucas_asset_pricing_dles.ipynb +++ b/lucas_asset_pricing_dles.ipynb @@ -456,7 +456,7 @@ } ], "metadata": { - "date": 1618812143.9664783, + "date": 1619230081.32767, "filename": "lucas_asset_pricing_dles.rst", "kernelspec": { "display_name": "Python", diff --git a/lucas_model.ipynb b/lucas_model.ipynb index 63f6320..2dd8eab 100644 --- a/lucas_model.ipynb +++ b/lucas_model.ipynb @@ -732,7 +732,7 @@ } ], "metadata": { - "date": 1618812144.1012716, + "date": 1619230081.4723663, "filename": "lucas_model.rst", "kernelspec": { "display_name": "Python", diff --git a/markov_jump_lq.ipynb b/markov_jump_lq.ipynb index f47fa74..26c2229 100644 --- a/markov_jump_lq.ipynb +++ b/markov_jump_lq.ipynb @@ -1304,7 +1304,7 @@ } ], "metadata": { - "date": 1618812144.285447, + "date": 1619230081.660601, "filename": "markov_jump_lq.rst", "kernelspec": { "display_name": "Python", diff --git a/matsuyama.ipynb b/matsuyama.ipynb index 7c29400..da34a58 100644 --- a/matsuyama.ipynb +++ b/matsuyama.ipynb @@ -956,7 +956,7 @@ } ], "metadata": { - "date": 1618812144.4850667, + "date": 1619230081.861952, "filename": "matsuyama.rst", "kernelspec": { "display_name": "Python", diff --git a/muth_kalman.ipynb b/muth_kalman.ipynb index 431ba04..ddfdb9c 100644 --- a/muth_kalman.ipynb +++ b/muth_kalman.ipynb @@ -508,7 +508,7 @@ } ], "metadata": { - "date": 1618812144.6067169, + "date": 1619230081.9816062, "filename": "muth_kalman.rst", "kernelspec": { "display_name": "Python", diff --git a/opt_tax_recur.ipynb b/opt_tax_recur.ipynb index ee67511..032a398 100644 --- a/opt_tax_recur.ipynb +++ b/opt_tax_recur.ipynb @@ -2210,7 +2210,7 @@ } ], "metadata": { - "date": 1618812144.8438509, + "date": 1619230082.2241015, "filename": "opt_tax_recur.rst", "kernelspec": { "display_name": "Python", diff --git a/orth_proj.ipynb b/orth_proj.ipynb index 690708a..0480072 100644 --- a/orth_proj.ipynb +++ b/orth_proj.ipynb @@ -1083,7 +1083,7 @@ } ], "metadata": { - "date": 1618812145.279637, + "date": 1619230082.657284, "filename": "orth_proj.rst", "kernelspec": { "display_name": "Python", diff --git a/permanent_income_dles.ipynb b/permanent_income_dles.ipynb index 33524a7..113bfd2 100644 --- a/permanent_income_dles.ipynb +++ b/permanent_income_dles.ipynb @@ -369,7 +369,7 @@ } ], "metadata": { - "date": 1618812145.4191415, + "date": 1619230082.7995365, "filename": "permanent_income_dles.rst", "kernelspec": { "display_name": "Python", diff --git a/rob_markov_perf.ipynb b/rob_markov_perf.ipynb index 401e052..d4c835e 100644 --- a/rob_markov_perf.ipynb +++ b/rob_markov_perf.ipynb @@ -1233,7 +1233,7 @@ } ], "metadata": { - "date": 1618812145.5584552, + "date": 1619230082.9321942, "filename": "rob_markov_perf.rst", "kernelspec": { "display_name": "Python", diff --git a/robustness.ipynb b/robustness.ipynb index e811131..48fd1e7 100644 --- a/robustness.ipynb +++ b/robustness.ipynb @@ -1364,7 +1364,7 @@ } ], "metadata": { - "date": 1618812145.7768016, + "date": 1619230083.1516995, "filename": "robustness.rst", "kernelspec": { "display_name": "Python", diff --git a/rosen_schooling_model.ipynb b/rosen_schooling_model.ipynb index e373b32..3c96a03 100644 --- a/rosen_schooling_model.ipynb +++ b/rosen_schooling_model.ipynb @@ -459,7 +459,7 @@ } ], "metadata": { - "date": 1618812145.90924, + "date": 1619230083.2857695, "filename": "rosen_schooling_model.rst", "kernelspec": { "display_name": "Python", diff --git a/smoothing.ipynb b/smoothing.ipynb index b70fbe9..90be282 100644 --- a/smoothing.ipynb +++ b/smoothing.ipynb @@ -1108,7 +1108,7 @@ } ], "metadata": { - "date": 1618812146.0672584, + "date": 1619230083.4401689, "filename": "smoothing.rst", "kernelspec": { "display_name": "Python", diff --git a/smoothing_tax.ipynb b/smoothing_tax.ipynb index 92233cb..50a5f6e 100644 --- a/smoothing_tax.ipynb +++ b/smoothing_tax.ipynb @@ -1194,7 +1194,7 @@ } ], "metadata": { - "date": 1618812146.238838, + "date": 1619230083.6120484, "filename": "smoothing_tax.rst", "kernelspec": { "display_name": "Python", diff --git a/stationary_densities.ipynb b/stationary_densities.ipynb index 73b8798..4b46cdd 100644 --- a/stationary_densities.ipynb +++ b/stationary_densities.ipynb @@ -1264,7 +1264,7 @@ } ], "metadata": { - "date": 1618812146.6919732, + "date": 1619230084.0615132, "filename": "stationary_densities.rst", "kernelspec": { "display_name": "Python", diff --git a/tax_smoothing_1.ipynb b/tax_smoothing_1.ipynb index 6672dec..51f8811 100644 --- a/tax_smoothing_1.ipynb +++ b/tax_smoothing_1.ipynb @@ -666,7 +666,7 @@ } ], "metadata": { - "date": 1618812146.8789053, + "date": 1619230084.2506893, "filename": "tax_smoothing_1.rst", "kernelspec": { "display_name": "Python", diff --git a/tax_smoothing_2.ipynb b/tax_smoothing_2.ipynb index 7770cf8..2e57ec2 100644 --- a/tax_smoothing_2.ipynb +++ b/tax_smoothing_2.ipynb @@ -967,7 +967,7 @@ } ], "metadata": { - "date": 1618812147.012526, + "date": 1619230084.38447, "filename": "tax_smoothing_2.rst", "kernelspec": { "display_name": "Python", diff --git a/tax_smoothing_3.ipynb b/tax_smoothing_3.ipynb index 066c92d..f117a38 100644 --- a/tax_smoothing_3.ipynb +++ b/tax_smoothing_3.ipynb @@ -415,7 +415,7 @@ } ], "metadata": { - "date": 1618812147.1151218, + "date": 1619230084.4846237, "filename": "tax_smoothing_3.rst", "kernelspec": { "display_name": "Python", diff --git a/troubleshooting.ipynb b/troubleshooting.ipynb index 2647b26..e4ab639 100644 --- a/troubleshooting.ipynb +++ b/troubleshooting.ipynb @@ -97,7 +97,7 @@ } ], "metadata": { - "date": 1618812147.1735072, + "date": 1619230084.5401595, "filename": "troubleshooting.rst", "kernelspec": { "display_name": "Python", diff --git a/von_neumann_model.ipynb b/von_neumann_model.ipynb index 9e4f889..0474c7a 100644 --- a/von_neumann_model.ipynb +++ b/von_neumann_model.ipynb @@ -1246,7 +1246,7 @@ } ], "metadata": { - "date": 1618812147.3013396, + "date": 1619230084.6687143, "filename": "von_neumann_model.rst", "kernelspec": { "display_name": "Python", diff --git a/zreferences.ipynb b/zreferences.ipynb index 8c1430f..603ca4e 100644 --- a/zreferences.ipynb +++ b/zreferences.ipynb @@ -260,7 +260,7 @@ } ], "metadata": { - "date": 1618812147.473953, + "date": 1619230084.821457, "filename": "zreferences.rst", "kernelspec": { "display_name": "Python",