Timing convention for capital adjustment costs

I have a couple of questions to ask. Apologies if they appear “lame”.

The cost of adjusting investment is usually equal to phi*(i/k- delta)^2k. When you program it with dynare and using k = (1-delta)k(-1) + i, do you program the cost in your program as phi(i/k(-1) -delta)^2k?

Also, if you want to do log-linearization, would you program your costs like this: phi*(exp(i)/exp(k(-1)) - delta)^2*k, considering your i and k are logs.

You need to be consistent.


you mean that k uses the stock at the beginning of period timing convention.

When using Dynare’s stock at the end of period timing, this becomes

phi*(i/k(-1) - delta)^2*k(-1).
That is, you cannot selectively alter the timing of only some occurrences of k without altering the meaning.

Regarding log-linearization, again you need to be consistent. You propose doing a variable substitution where exp() is used to redefine variables as their logs. In this case

phi*(exp(i)/exp(k(-1)) - delta)^2*exp(k(-1)).
See Pfeifer(2013): “A Guide to Specifying Observation Equations for the Estimation of DSGE Models” sites.google.com/site/pfeiferecon/Pfeifer_2013_Observation_Equations.pdf for more on this.

Thank you so much Johannes for your helpful suggestions.