Dear all,

I am trying to reproduce Florin Bilbiie’s THANK (Treatable Heterogeneous Agents New Keynesian) model, which you can find at the following link:

I am having trouble log-linearizing the following equation (eq 41 from linked paper, at pg. 55):

FOC_{\omega}: \frac{1}{C_t^S} = \beta E_t \{ \frac{(v_{t+1}+D_{t+1})}{v_t} ) \frac{ 1}{ C_{t+1}^S} \}

where

\omega = firms’ shares detained by household;

v_t = price of firms’ shares at time t;

D_t = profits of firms at time t;

C_t^S = consumption by household of type S at time t.

The problem is that at Steady State (SS): v=0, D=0. I have computed and verified this.

So, I cannot log-linearize normally, because I would get: \hat{x} = log(\frac{x}{\bar{x}})= log(\frac{x}{0})= \inf.

Elsewhere, in the paper (pg. 10), Bilbiie uses, instead of \hat{d}_t=log(\frac{D_t}{\bar{D}}), \hat{d}_t=log(\frac{D_t}{\bar{Y}}). So, he uses the percentage deviation of D_t with respect to aggregate income in SS.

So, my idea would be: instead of using, as an expression of percentage deviation, \hat{d}_t=log(\frac{D_t}{\bar{D}}) e \hat{v}_t=log(\frac{v_t}{\bar{D}}), (I would use) \hat{d}_t=log(\frac{D_t}{\bar{Y}}) and \hat{v}_t=log(\frac{v_t}{\bar{Y}}).

Moreover, even if at SS I would have \frac{\bar{v} + \bar{D}}{\bar{ v}} = 0 / 0 = indet, my idea would be:

I will ignore this fact, because the multiplicative coefficient \frac{\bar{v} + \bar{D}}{\bar{ v}} of every summed element cancels out dividing both sides of the equation by the same coefficient.

I hope I have made myself clear,

I am ready to clarify any doubt,

Thank you,

Paolo