Non linearities : exponential

Stochastic simulations
1

Hello Professor,

I have a question regarding a specific function in Dynare for a DSGE model.

The model contains the following equations:

Oil extraction:

O_r = A_m * N_m^(alpha_m) * S_m^(1-alpha_m);

Reserve accumulation:

S_m = S_m(-1) - O_r(-1) + d_m(-1);

Discovery function:

d_m = A * exp(eta * S_m);

I run different type of simulations (IRFs, perfect foresight,etc…).In my model, S_m has a steady-state value expressed in model units rather than being directly calibrated/normalized, while eta is negative and very small after estimation from time series data .

Therefore, I am wondering whether the term exp(eta * S_m) creates any particular problem for the first-order approximation performed by Dynare.

I know that the first-order Taylor approximation of the discovery function around the steady state (S_m_bar) is:

d_m ≈ d_m_bar + eta * d_m_bar * (S_m - S_m_bar);

And the relative deviation is :
(d_m - d_m_bar) / d_m_bar = A* eta * (S_m - S_m_bar).

So :

1. Does Dynare handle the exponential term exp(eta*S_m) correctly when computing the first-order approximation?

2. What about the interpretation of the IRFs(deviation from the steady state) since we have (S_m - S_m_bar) and not (S_m - S_m_bar)/S_m_bar in deviation when we compute d_m_hat??

I would really appreciate it if you could take a close look at this point, Professor.

Thank you so much.

2
  1. I am not sure what you mean with “handle correctly”. The exponential function will be approximated by a straight line. That is the correct first order approximation and usually what you want.
  2. The interpretation here is a semi-elasticity instead of an elasticity. That is nothing unusual.
3

Thank you for your quick feedback… Really appreciate it..