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z#@6_SE6{|QB42Z8fH^e5yYL54bWxLuZ8*oB&R75QM=y7~``4<t_UzBP_?}A0n+d0Z zfI<9=g?9mi6z{URc@g)HiFt1OKC2?u>~Hb6lridA?#7EeIoWOTsG~u|xlu-5qKpfK SolRL;8OgeDpPs2c>;D1eEHShI diff --git a/deterministic.tex b/deterministic.tex index 91ff3eb..326f910 100644 --- a/deterministic.tex +++ b/deterministic.tex @@ -682,7 +682,7 @@ $$d_T = (B_T-A_TD_{T-1})^{-1}(f(y_{T+1},y_T,y_{T-1},u_T)+A_Td_{T-1})$$ \begin{equation*} \label{eq:social-planner-problem:1} \max_{\{c_{t+j},\ell_{t+j},k_{t+j}\}_{j=0}^{\infty}} - \mathbb{E}_t \sum_{j=0}^{\infty}\beta^ju(c_{t+j},\ell_{t+j}) + \sum_{j=0}^{\infty}\beta^ju(c_{t+j},\ell_{t+j}) \end{equation*} s.t. \begin{gather*} @@ -715,7 +715,7 @@ $$d_T = (B_T-A_TD_{T-1})^{-1}(f(y_{T+1},y_T,y_{T-1},u_T)+A_Td_{T-1})$$ \begin{itemize} \item Euler equation: \begin{equation*} - u_c(c_t,\ell_t) = \beta\, \mathbb E_t\left[ + u_c(c_t,\ell_t) = \beta\, \left[ u_c(c_{t+1},\ell_{t+1})\Bigl(A_{t+1}f_k(k_t,\ell_{t+1}) + 1 -\delta\Bigr) \right] \end{equation*} @@ -1045,7 +1045,7 @@ end; \begin{frame}[fragile] \frametitle{First order conditions} \begin{gather*} - u_c(c_t,\ell_t) - \mu_t = \beta\, \mathbb E_t\left[ + u_c(c_t,\ell_t) - \mu_t = \beta\, \left[ u_c(c_{t+1},\ell_{t+1})\left(A_{t+1}f_k(k_t,\ell_{t+1}) + 1 -\delta\right) - \mu_{t+1}(1-\delta)\right] \\ \frac{u_{\ell}(c_t,\ell_t)}{u_c(c_t,\ell_t)} + A_tf_l(k_{t-1},\ell_t) = @@ -1178,10 +1178,10 @@ occbin_graph y i i_not pie; How to simulate an unexpected shock at a period $t > 1$? \begin{itemize} \item Do a perfect foresight simulation from periods $0$ to $T$ \emph{without the - shock} + unexpected shock in $t$} (but with other expected shocks) \item Do another perfect foresight simulation from periods $t$ to $T$ \begin{itemize} - \item applying the shock in $t$, + \item applying the unexpected shock in $t$ (and keeping expected shocks), \item and using the results of the first simulation as initial condition \end{itemize} \item Combine the two simulations: @@ -1443,7 +1443,7 @@ plot(oo_.endo_simul(ic, 1:21)); 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8ee18c6..b991929 100644 --- a/exercise-zlb.tex +++ b/exercise-zlb.tex @@ -29,7 +29,7 @@ stochastic version. Display the interest rate with \texttt{rplot}. What do you observe? \item Now, transform the previous example by adding a \emph{negative} - (\texttt{i.e.} inflationary) one-standard-deviation productivity shock in + (\textit{i.e.} inflationary) one-standard-deviation productivity shock in period 5, in two different ways. In the first version, the shock will be anticipated; in the second one, the shock will be unexpected. Verify that in the first version the ZLB is -- GitLab