Questions tagged [log-linearization]
The log-linearization tag has no usage guidance.
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Log-linearisation of the Euler equation
I am struggling to log-linearize the following around the steady state:
$
U'\left(e^{z_t} F(k_t, 1) + (1 - \delta)k_t - k_{t+1}\right) - \beta E_{z_t}U'\left(e^{z_{t+1}} F(k_{t+1}, 1) + (1 - \delta)k_{...
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a question about second-order log-linear approximation
I have a very simple question.
I don't understand OA6 equation. Where is the 2, the denominator of $\frac{v_{11}}{2}$, gone?
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Help with the log-linearization of a difficult term
I'm currently trying to retrace a log-linearization done in this paper. I want to log-linearize around the steady-state, as it is commonly done for DSGE models (see here). $\bar{x_t}$ are steady-state ...
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Log-linearizing a second order term around the steady-state
I'm currently trying to retrace a log-linearization done in a paper. I want to log-linearize around the steady-state, as it is commonly done for DSGE models (see here) and I want to disregard all ...
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Log-linearization in RBC model
Can anyone show how by applying log-linearization we can turn this equation:
$\frac{X_t}{C_t} = \left( (1 - \vartheta) + \vartheta \left( \frac{L_t}{C_t} \right)^{1-\nu} \right)^\frac{1}{1-\nu}$
into ...
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how to log-linearize a utility function?
This is a part of Online Appendix to Business Cycle Dynamics under Rational Inattention (Mackowiak and Wiederholt, 2015).
I don't understand why C_j is supposed to be a denominator. Could anyone give ...
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Log-linearization with first-order Taylor approximation
I am having trouble with a log-linearization replication that I stumbled across in the appendix of the Chari, Kehoe & McGrattan paper "Business Cycle Accounting". More specifically the ...
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Log linearization with sums
I have the following equation for a variable $s$:
$$
s = -\beta\frac{\alpha(1-w)^{\zeta-1}-(1-\alpha)w^{\zeta-1}}{\alpha(1-w)^{\zeta}+(1-\alpha)w^{\zeta}}
$$
where $\zeta$, $\alpha$, and $\beta$ are ...
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What is the typical utility function of the standard loglinear demand function?
What is the typical utility function of this demand function?
$$x_1 = \ln(x_2) - \beta \ln(p_1) + \gamma \ln(y).$$
With budget constraint $y = p x_1 + x_2.$
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Log Linearising CES demand
I have been trying to log-linearise the demand function that follows from a standard two-good CES-utility maximamalisation problem. That is:
Maximise
\begin{eqnarray}
U(h,c)= \left(G_1^{\rho}+ G_2^{\...
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log linearize exponential expressions
How do I log-linearize an expression like:
$e^{(\theta*(x_t-\bar{x}))}$
Where $\theta$ is any constant and $\bar{x}$ is the steady-state of $x$
Generally when we log-linearize say $x_t$, we write it ...
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288
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New Keynesian Phillips Curve: log-linearization
I am facing some problems in log-linearizing the following equation:
which is the New Keynesian Phillips Curve (NKPC).
The expected result, once log-linearized, is:
which is described by equation 46 ...
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Percent change vs difference in logs: Which to prefer when forcasting non-stationary series with rare large excursions?
One of the most common ways to convert a non-stationary time series into a stationary one is to take the difference in logs. This is approximately equal to the percentage change for small changes, and ...
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Log-linearization of CES production function
I am trying to recover the Log-linearisation of a CES production function in a paper. Although I am fairly confident with Log-linearisations, I simply do not find the supposed result.
The production ...
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First order condition of log functions in general and interpretation
In the following, a first order condition for a log function is calculated.
I know how the left part was calculated, however, I am a little bit confused where the gamma on the right site comes from ...
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Log deviation from steady state - understanding a journal paper
I hope a question like this is fair game on this website!
I'm doing some research for my thesis, and have come across what seems to be a pretty simple model - two countries, A (representing the USA) ...
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Linearization around the steady state (Asset pricing application)
I am working through a linearization example from Colacito and Croce (2011). In the paper the following expression is derived:
\begin{align}
& (v^{i}_{c,t})^{\theta}=E_{t}[\delta e^{\Delta c^{i}_{...
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Log- Linearization of equations
can somebody help me understand how it is possible to log-linearize this equation?
$$\frac{1}{1+i_{t}}=\beta E^i_{t}\left[\frac{P_{t}}{P_{t+1}} \cdot \frac{U_{c}\left(C_{t+1}^{i}, \xi_{t+1}\right)}{U_{...
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Log linearising EUler equation
I am trying to solve a problem that asks to log linearise following Euler equation of the New Keynesian model:
$$C^{-\sigma}_t=\beta E_tC^{-\sigma}_{t+1}(1+i_t)/(1+\pi_{t+1}).$$
The solution is ...
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Are typical macro DSGE (RBC or New Keynesian DSGE) models linear, non-linear, or log (linearized)?
In Carl Hommes 2015 book on Expectations, it seems he considers DSGE models (being it either RBC DSGE or New Keynesian DSGE) to be linear, or (log)linearized models, on page 3 of the introduction. He ...
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Log-linearizing a non-separable utility function around the steady state
I've started reading Jordi Galí's Monetary Policy, Inflation and the Business Cycle (2nd ed., 2015). In section 2.5.2, Galí considers an example with the following non-separable period utility ...
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Constant elasticity proof for log-linear demand curve
From Perloff 2008e solved 2.2:
Q: Show that the elasticity of demand is a constant e if the demand function is log-linear, ln Q=ln A+e ln p.
A: Differentiating with respect to p, we find that (dQ/dp)/...
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How do you use a Log-linear model when you have negative Xs?
I am trying to us a Log-linear model to derive an elasticity. However, some of my Xs are negative numbers. Being as the model relies on the natural log of the Ys and the Xs, how can this model work ...
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log linearization around steady state (price dynamics)
I'm stuck with a derivation from Gali's book.
In the version 2008, I can't get eq. (7) from chapter 3. All is ok until the following:
$$
(1-\epsilon)\log\pi_{t}=\log(\theta+(1-\theta)\exp[(1-\...
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