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"# Lucas in the lab\n",
"## Prof. JP Rabanal [EC 398]"
]
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"### The model\n",
"- There is a fixed supply of an infinitely durable asset (trees) and shares which yield a dividend (fruit) in amount $d_t$ per period\n",
"- Dividends are paid in units of the single non-storable consumption good at the beginning of each period. \n",
"- Let $s^i_t$ denote the number of asset shares agent $i$ owns at the beginning of period $t$\n",
"- Let $p_t$ denote the price of an asset share in period $t$\n",
"- Each agent also receives an exogeneous endowment of the consumption good $y^i_t$ at the beginning of each period. \n",
"- Agent $i$ seeks to max utility (we omit $i$ for simplification)\n",
"\n",
"\\begin{equation}\n",
"\\max \\Sigma \\beta u(c_t)\n",
"\\end{equation}\n",
"\n",
"s.t. \n",
"\\begin{equation}\n",
"c_t + p_t s_{t+1} \\leq y_t + d_t s_t + p_t s_t\n",
"\\end{equation}\n"
]
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"### Solution\n",
"Let's find the value function,\n",
"$$ V(s_{t}) = \\max_{c_t, s_{t+1}}\\{u(c_t)+\\beta E[V(s_{t+1})] \\} $$$$ V(s_{t}) = \\max_{s_{t+1}}\\{u(y_t + d_t s_t + p_t (s_t-s_{t+1}))+\\beta E[V(s_{t+1})] \\} $$\n",
"\n",
"FOC \n",
"$$ p_t u'(c_t) = \\beta E_t[V_s(s_{t+1})] $$\n",
"\n",
"Envelop theorem $$ V_s(s_t) = u'(c_t)(d_t + p_t) $$\n",
"\n",
"Envelop condition \n",
"$$ E[V_s(s_{t+1})] = E[u'(c_{t+1})(d_{t+1} + p_{t+1})] $$\n",
"\n",
"\n",
"Stochastic Euler equation\n",
"$$ p_t u'(c_t) = \\beta E[u'(c_{t+1})(d_{t+1} + p_{t+1})] $$\n",
"\n",
"We can rewrite this equation as \n",
"\n",
"$$ u'(c_t) = \\beta E[u'(c_{t+1})\\frac{(d_{t+1} + p_{t+1})}{p_t}] $$\n",
"\n",
"***The LHS is telling us what is our utility of consuming today and the RHS is telling us what is our utility of foregoing our consumption today for tomorrow by holding the asset***\n",
"\n",
"- Let $\\mu_{t+1}$ define as the **stochastic discount factor**, $\\mu_{t+1}=\\beta \\frac{u'(c_{t+1})}{u'(c_t)}$. \n",
"\n",
"- The *asset pricing* equation is \n",
"$$ p_t = \\beta E[\\mu_{t+1}(d_{t+1} + p_{t+1})] $$\n",
"\n",
"- If we assume the aggregate endowments of francs (y) and assets is constant across periods, and the divided $d$ is equal to a constant value $d_t = \\bar{d}$, we can rewrite our equation as \n",
"\n",
"\n",
"$$ p^* = \\frac{\\bar{d}}{\\frac{1}{\\mu_{t+1}}-1} $$\n",
"\n",
"- Since aggregate endowment is constant, strict monotonicity of prefences implies that there can be no growth or decay in consumption for all players in equilibrium. Thus it must be the case that in a steady state CE each play perfecty smoothes her consumption, $c_{t}=c_{t+1}$. Thus, the **fundamental price equation** is\n",
"\n",
"$$ p^* = \\frac{\\beta}{1-\\beta} \\bar{d}$$\n",
"\n",
"\n",
"\n",
"\n"
]
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"source": [
"from IPython.core.display import HTML\n",
"def css_styling():\n",
" import os\n",
" styles = open(os.path.expanduser(\"custom.css\"), \"r\").read()\n",
" return HTML(styles)\n",
"css_styling()"
]
},
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