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"# Consumption and production economies\n",
"## Prof. Rabanal [EC398]\n",
"\n"
]
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"We are going to study a textbook neoclassical growth model in the lab (see Li and Noussair, AER, 2000). \n",
"\n",
"\n",
"### Objective function\n",
"\n",
"\\begin{align}\n",
"\\max \\sum^{\\infty}_{t=0} (1+\\rho)^{-t} u(c_t)\n",
"\\end{align}\n",
"\n",
"### Constraints\n",
"\\begin{align}\n",
"c_t + k_{t+1}\\leq f(k_t) + (1-\\delta) k_{t} \n",
"\\end{align}\n",
"\\begin{align}\n",
"k_{t+1} \\geq (1-\\delta) k_t\n",
"\\end{align}\n",
"\n",
"where $\\rho$ is the discount rate, $u(.)$ is a concave utility function, $k_t$ is the stock of capital, $f$ is the production function, and $\\delta$ is depreciation. \n",
"\n",
"### Solution\n",
"\n",
"Let's find the value function, \n",
"\n",
"\\begin{equation}\n",
"V(k_{t}) = \\max_{c_t, k_{t+1}}\\{u(c_t)+(1+\\rho)^{-t} E[V(k_{t+1})] \\} \n",
"\\end{equation}\n",
"\n",
"\\begin{equation}\n",
"V(k_{t}) = \\max_{k_{t+1}}\\{u(f(k_t) + (1-\\delta) k_{t}-k_{t+1})+(1+\\rho)^{-1} E[V(k_{t+1})] \\} \n",
"\\end{equation}\n",
"\n",
"FOC\n",
"\\begin{equation}\n",
"u'(c_t) = (1+\\rho)^{-1} E_t[V_k(k_{t+1})]\n",
"\\end{equation}\n",
"\n",
"Envelop theorem\n",
"\\begin{equation}\n",
"V_k(k_t) = u'(c_t)(1-\\delta+f')\n",
"\\end{equation}\n",
"\n",
"Envelop condition\n",
"\\begin{equation}\n",
"E_t [V_k(k_{t+1})] = E_t[u'(c_{t+1})(1-\\delta+f') ] \n",
"\\end{equation}\n",
"\n",
"Stochastic Euler equation\n",
"\n",
"\\begin{equation}\n",
"u'(c_t) = (1+\\rho)^{-1} E_t[u'(c_{t+1})(1-\\delta+f') ] \n",
"\\end{equation}\n",
"\n",
"Steady-state $c_{t+1}=c_t=\\bar{c}$ and $k_{t+1}=k_t=\\bar{k}$ \n",
"\\begin{equation}\n",
"f' = \\rho + \\delta\n",
"\\end{equation}\n",
"\\begin{align}\n",
"\\bar{c}=f(\\bar{k})-\\delta \\bar{k}\n",
"\\end{align}\n",
"\n",
"\n"
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