4

Economists (most of them) build their models assuming most of the time stochastic dynamic equilibrium. So Economics does not contrast "dynamic" with "equilibrium" - it synthesizes them. It is stochastic in the sense that random shocks are acknowledged. It is dynamic in the sense that it may revolve around a deterministic or stochastic trend. And it is an ...


4

Rearranging the steady state equation $$ \overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}- \alpha\overline{p}^{\alpha}-\frac{a+1}{\sigma} $$ we get $$ (1 + \alpha)\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}- \frac{a+1}{\sigma}. $$ As $\alpha \in [0,1]$, the left hand side of the equation is increasing in $\overline{p}$ and the right hand side ...


3

You are missing an integration constant $$ \log\left(\frac{p + qF(t)}{1 - F(t)}\right) = (p + q)t + \color{red}{\tilde{C}} $$ This constant you can name it whatever you want, I'm going to name it as $$ \color{red}{\tilde{C}} = \color{blue}{C}(p + q) + \ln q $$ where $C$ is just another constant. So I basically changed one constant for another one (...


3

We have the recurrence relation $$x_{k+1} = \frac{x_{k+8}}{x_{k+1}}$$ If the denominator is nonzero, this recurrence relation can be rewritten as follows $$x_{k+7} = x_k^2$$ Assuming positivity and taking the logarithm of both sides, we obtain a linear recurrence relation $$\ln (x_{k+7}) = 2 \ln (x_k)$$ Shifting, $$\ln (x_{k+1}) = 2 \ln (x_{k-6})$$ ...


2

The OP correctly identified a mistake here. Since the author claims monotonicity for a general function, let's disprove it for the simple linear case. Consider $$x_{t+1} = g(x_t) = -0.5x_t$$ This satisfies all the requirements and conditions stated by the author, and also that the derivative is smaller than unity in absolute values $\forall x$. But it ...


2

Take your second equation, move it forward one period, and rearrange. You get: $$ B_t = \frac{p_{t+1} s_{t+1} + \Delta M_{t+1}}{R_{t+1}} + \frac{B_{t+1}}{R_{t+1}} $$ Then, define the nominal primary deficit as $D_{t+1} = -(p_{t+1} s_{t+1} + \Delta M_{t+1})$. The above transforms into: $$ B_t = -\frac{D_{t+1}}{R_{t+1}} + \frac{B_{t+1}}{R_{t+1}} $$ which ...


2

The following approach seems to work in this case: look for the stead state formula of $T_t$. You can do this by taking (2.8) and the formula for $T_t$, and combine them. Then, get rid of all $t$ index: $$ T = \frac{1-\delta}{p}\left(B - \frac{B}{1+i}\right) + \frac{T}{1+i}$$ Then, rearrange until you get $T$, the steady state formula for $T_j$. You ...


2

You have the government's flow budget constraint (re-written in real terms): $b_{t} + m_{t} + \tau_{t} = g + \frac{m_{t-1}}{\pi_{t}} + R_{t-1}\frac{b_{t-1}}{\pi_{t}}$ (1) Now all you need to do is substitute (2) and the policy rules (also, I don't think Leeper's utility function had a $\delta$ but that's not important) and linearise. I.e linearise: $b_{t} ...


1

The differential equation is of the form $$y' + f(x)y = q(x)$$ The correct answer in our case is $$p = -s$$ so that you know what you are targeting. Namely, it does not depend on $z$. You can verify that it satisfies the differential equation. Then the author just plays around like $$p= - s \implies p -p = s-s \implies p - \frac p z z = \frac s z z - ...


1

There is nothing more to it than the equation $$Y_t - (c_y+v)Y_{t-1} = C^a + I^a$$ This is a linear non-homogeneous first-order difference equation, and it is non-homogeneous because there is a non-zero constant, $C^a + I^a \neq 0$. This is not some "habit in economics", but rather standard mathematical terminology.


1

If you are planning on using Dynare, you do not need to "solve" the model using Sim's method. Dynare takes care of the solution algorithm for you. If you want to get to IRFs quickly, I suggest writing up the linearized version of your model in a .mod file, then from Matlab, simply run dynare model.mod. Here are some example .mod files for you to work ...


1

I think I have managed to solve it. However, not the way I was initially hoping. I simplified the stacked matrices using the given conditions and some assumptions. Here is my solution: Eq. (3) I write as $\pi_{t+1} = \alpha \beta \pi_{t} + \beta \theta_{t} + \eta_{t+1}$ Forwarding equation (4) one period and arranging it in terms on $b_{t+1} = -\varphi_{1} ...


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