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Mat 540 Quiz

Satisfactory Essays

representative household maximise expected lifetime utility \begin{equation} \label{eq:household_utility} E_t \sum_{t=0}^{\infty} \beta^t U(C_t, N_t) \end{equation} where $\beta^t$ is an exponential discount factor and $C_t$ is a consumption index given by \begin{equation} \label{eq:household_C_t} C_t = ( \int^1_0 C_t(i)^\frac{\varepsilon-1}{\varepsilon} di)^{\frac{\varepsilon}{\varepsilon-1}}) \end{equation} $C_t(i)$ represents the quantity of good $i$ consumed by the household in period $t$. Assume there exist a continuum of goods represented by the interval [0,1]. Household maximises (\ref{eq:household_utility}) with respect to the budget constraint given by \begin{equation} \label{eq:household_budgetconstrained} \int^1_0 P_t(i) C_t(i) di …show more content…

Letting aggregate output be defined as $Y_t = ( \int^1_0 Y_t(i)^\frac{\varepsilon-1}{\varepsilon} di)^{\frac{\varepsilon}{\varepsilon-1}})$, it follows that \begin{equation} Y_t = C_t \end{equation} for all $t$. Then, combining the goods market market clearing condition with the consumer's Euler in equation (\ref{eq:eulerequationc_t}) yields, \begin{equation} \label{eq:household_y_t} \widehat{Y}_t = E_t \{\widehat{Y}_{t+1}\} - \frac{1}{\sigma} (r_t - E_t\{\pi_{t+1}\} - r^n_t)\end{equation} natural interest rate is described as \begin{align*}r^n_t = \sigma \frac{1+\varrho}{\sigma + \varrho} (z_{t+1} - z) \end{align*} \subsubsection*{Dynamic IS} Rewriting (\ref{eq:household_y_t}) in terms of output-gap yields the first key equation in describing the model, referred to as Dynamic IS equation (DIS). \begin{equation} \label{eq:household_DIS} \widehat{Y}_{gt} = {\widehat{y}_{gt+1}} - \frac{1} {\sigma} (r_t - \pi_{t+1} - r^n_t) + d_t \end{equation} where I add an exogenous demand shock $d_t$ which follows an AR(1) process: \begin{align*} d = \rho_d d_{t-1} + \epsilon_d

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