 # Loglinearizing Euler equation for firms' shares (Steady State values are zero) - Heterogenous Agents

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

It depends what you want. Typically, variables with steady state 0 are linearized, not log-linearized.

Thank you for the suggestion: I will try to linearize dividends D and price of shares v and I will reply to your answer once I have done that.

What I want is that the log-linearized version of the Euler equation above is coherent with the log-linearization of the equation for dividends in the model. This is what I am struggling with.

I am struggling with this because, in the paper, Bilbiie log-linearizes only the equation for dividends, but not the Euler equation for firms’ shares detained.

Equation for dividends, psi is the parameter for cost of inflation and pit is net inflation at time t:

D_t = Y_t - W_t*N_t - (\frac{\psi}{2})\pi_t^2Y_t

Log-linearized equation for dividends, where d_t = ln(\frac{Dt}{\bar{Y}}):

d_t = -w_t

This is done considering that the production function is Y_t = N_t, so, log-linearizing, y_t = n_t.

Do you have any ideas as to how I should make log-linearization coherent in the two cases?

It is also possible that Bilbiie does this simplification (d_t=ln(\frac{Dt}{\bar{Y}})), instead of doing a proper log-linearization, only to show a particular result, while in the model it would be proper to linearize, as you say. What do you think about it? Is it possible?

In the attempt to reconcile the Euler equation for firms’ shares with the equation for dividends, I have made the following linearization of the Euler equation for firms’ shares:

\hat{v}_t = \beta E_t \{ \hat{v}_{t+1} + \hat{d}_{t+1} \}

where \hat{v}_t=\frac{v_t}{\bar{Y}} and \hat{d}_t=\frac{D_t}{\bar{Y}}.

I have derived it so:

\frac{1}{C_t^S} = \beta E_t\left\{ \frac{v_{t+1}+D_{t+1}}{v_t} \frac{1}{C_{t+1}^S} \right\} % Euler equation

\frac{1}{C_t^S \bar{Y}} = \beta E_t\left\{ \frac{v_{t+1}+D_{t+1}}{v_t} \frac{1}{C_{t+1}^S \bar{Y}} \right\} % Euler eq divided by Yss

Moving on, the log-linearized expression for C are eliminated because they are multiplied by zeros: vss= 0 and vss + Dss = 0; so I linearize the variables v and D:

\frac{1}{\bar{C}}\frac{v_t}{ \bar{Y}} = \beta E_t\left\{ \frac{1}{\bar{C}} \frac{v_{t+1}}{\bar{Y}} + \frac{1}{\bar{C}} \frac{D_{t+1}}{\bar{Y}}\right\}

\frac{v_t}{ \bar{Y}} = \beta E_t\left\{ \frac{v_{t+1}}{\bar{Y}} + \frac{D_{t+1}}{\bar{Y}}\right\}

\hat{v}_t = \beta E_t \{ \hat{v}_{t+1} + \hat{d}_{t+1} \}

where \bar{C}=\bar{C}^S, i.e. the steady state consumption of savers (S) is equal to \bar{C}.

What do you think about my derivation?

Also, I have the following supposition about the log-linearization performed by Bilbiie for the equation of dividends:

D_t = Y_t - W_tN_t - \frac{\psi}{2}\pi_t^2Y_t

He has actually linearized D, so (divide everything by Yss and subtract 1 from both sides):

\frac{D_t}{\bar{Y}} -1 = \frac{Y_t}{\bar{Y}} - \frac{W_tN_t}{\bar{Y}} - \frac{\psi}{2\bar{Y}}\pi_t^2Y_t -1

After linearizing on the lhs and log-linearizing on the rhs, (-1) cancels out on the rhs, so you have:

\frac{D_t-\bar{Y}}{\bar{Y}} = -\hat{w}_t

In the paper (pg. 10 of the linked paper), Bilbiie states \hat{d}_t = ln (\frac{D_t}{\bar{Y}}), because one can approximate ln (\frac{D_t}{\bar{Y}}) = ln (1 + \frac{Dt}{\bar{Y}} - 1) = ln( 1 + \frac{D_t-\bar{Y}}{\bar{Y}}) \approx \frac{D_t-\bar{Y}}{\bar{Y}}, if \frac{D_t-\bar{Y}}{\bar{Y}} is small enough.

However, it is also correct to do as I do, i.e. simply linearize D_t and do not subtract one on both sides. So, also the following equation for the linearization of dividends is correct:

\frac{D_t}{\bar{Y}} = -\hat{w}_t

Could you please use the \LaTeX-capabilities of the forum to make the posts readable. Simply put Dollar signs to start the code, e.g. $\LaTeX$