I had a question about log-linearization in dynare. How would dynare log linearize an expression like exp(theta*(x(t) - x_bar)) where theta is any constant and x_bar is the steady state value of x(t).
At first order, you should be getting
{e^{\theta \left( {{x_t} - x} \right)}} \approx {e^0 } + 1 \times \theta \times \left( {{x_t} - x} \right)
Note that Dynare only linearizes, it does not loglinearize (unless you use the loglinear-keyword)
Yeah but my original problem is log linearization of an exponential expression. And this is what I have done manually
Generally when we log-linearize say X(t), we write it as X(t) = X_bar * exp(log(X(t)-log(X_bar)). = 1 + x(t) where x(t) = log(X(t)-log(X_bar). But in my current problem, the expression itself is exponential. So is the following right?
exp(theta*(X(t)-X_bar)) = exp(exp(theta*(log(X(t)-log(X_bar))) * exp(exp(-theta*(log(X_bar)-log(X_bar))))
which boils down to exp(exp(thetax(t))) * exp(1) = exp(1 + exp(thetax(t))) = 1 + 1 + exp(theta*x(t))
I am trying to make sure if what I am getting manually is correct.
Why would you even want to log-linearize this expression? I don’t see the point. If you want percentage deviations, you could simply do
{e^0 } + 1 \times \theta \times \left( {{x_t} - x} \right)=1+x\times \frac{x_t - x}{x}
One of my FOCs contain this expression. To get a clear view about the relationship between the variables, I need to log-linearize this.