# Capital adjustment cost in D’Auria (2015)

I am trying to replicate the paper by D’Auria (2015) (see attached file)
10.1017@S1365100513000874.pdf (363.8 KB)
I’ve already seen one thread regarding this paper, but it doesn’t asnwer my question.
I am a bit confused by the law of motion for capital and the definition of a function describing capital adjustment cost. Specifically, equations (17) and (29) formulate the LOM of capital as:

k_(t+1) = Phi (i_t/k_t)*k_t + (1-delta)*k_t

I have two questions:

1. Is the law of motion specified correctly? It is different compared to a similar model with capital adjustment costs by Sims (https://sites.nd.edu/esims/files/2023/05/rbc_extensions_sp17.pdf), page 14
2. What could be the particular functional form for Phi in this case?
1. If you use that notation, you need to set
predetermined_variables k;

1. You should typically be able to use a quadratic function \frac{\phi}{2} \left(\frac{i_t}{k_t}\right)^2

My confusion was more related to the fact that I expected the equation to look like something along the lines of:
k_(t+1)=i_t - Phi(…)*k_t + (1-d)*k_t
Should I include i_t separately, or with the appropriate functional form of Phi it’s not necessary?
What also confuses me is that the quadratic function that you suggest doesn’t satisfy the properties listed in the paper. After eq. (17) it is said that it should have a positive first derivative and a non-positive second derivative.

My mistake. Have a look at https://github.com/JohannesPfeifer/DSGE_mod/blob/master/Jermann_1998/Jermann1998_Algebra.pdf

I used the functional form that you suggested. In the original paper it is assumed that \Phi(\delta) = \delta and \Phi'(\delta) = 1. Hence, in this case the function becomes:

\Phi(i_t/k_t) = \frac{\delta^a}{1-a} (\frac{i_t}{k_t})^{1-a} + \delta(1 - \frac{1}{1-a}) \\ \Phi'(i_t/k_t) = \delta^a (\frac{i_t}{k_t})^{-a} > 0 \\ \Phi''(i_t/k_t) = -a \delta^a (\frac{i_t}{k_t})^{-a-1} < 0

Then I log-linearized the FOC for investment (eq. 19 in the paper, q = [\Phi'(i_t/k_t)]^{-1}) and got:

\hat{q_t} = \alpha \hat{i_t} - \alpha \hat{k_t}

Finally, I decided to put all equations together in Dynare. Here’s what I got:
model.mod (3.1 KB)
I noticed that in the original paper \hat{w} - \hat{p} doesn’t really make sense, because \hat{w} is already real wage’s deviation from the steady state. Same for returns to capital. So I modified eqs. (32), (33) and (39) to get rid of \hat{p}. Now I have 18 equations and 18 variables. Is this the correct way to proceed?

Another (final, hopefully) question is whether I uderstand correctly that in the wage Phillips curve:

\pi^w_t = \beta E (\pi^w_{t+1}) + k_w (\hat{mrs}_t - \hat{w}_t)

k_w is equal to (1 - \beta \phi) (1-\phi)/\phi, where \phi is the wage rigidity parameter?

Yes, that sound sensible. However, if you linearize yourself, why do you even need a functional form for the adjustment costs? You only need the derivatives evaluated at the steady state, and those are free parameters.

True, I understand that. I just wanted to get to the bottom of it by working out the full nonlinear solution for the model myself.

Thanks a lot for your assistance!