Log linearisation - struggling!

General DSGE Modeling
1

Hi,

I’m having trouble log-linearising this equation -
1/C_t = \alpha (M_t^{-\sigma}) + \beta (1/(C_{t+1}V_{t+1}))

I conducted the first order Taylor approximation and then tried to expand the fractions, but I’m still not able to really get a satisfactory answer, especially because I can’t figure out how to remove the sigma as an exponent.

the most I’m able to get to is
-\sigma \alpha M^{-\sigma} \hat m_t + (\beta/C_{t+1}V_{t+1}) \hat c_t

2

I get

\begin{gathered} \frac{1}{{{C_{t + 1}}}} = \alpha M_t^{ - \sigma } + \beta \frac{1}{{{C_{t + 1}}{V_{t + 1}}}} \hfill \\ - \frac{1}{{{C^2}}}\left( {{C_{t + 1}} - C} \right) = \alpha \left( { - \sigma } \right){M^{ - \sigma - 1}}\left( {{M_t} - M} \right) + \beta \left( { - 1} \right)\frac{1}{{{C^2}V}}\left( {{C_{t + 1}} - C} \right) \hfill \\ + \beta \left( { - 1} \right)\frac{1}{{C{V^2}}}\left( {{V_{t + 1}} - V} \right) \hfill \\ - \frac{1}{{{C^2}}}C\frac{{\left( {{C_{t + 1}} - C} \right)}}{C} = \alpha \left( { - \sigma } \right){M^{ - \sigma - 1}}M\frac{{\left( {{M_t} - M} \right)}}{M} + \beta \left( { - 1} \right)\frac{1}{{{C^2}V}}C\frac{{\left( {{C_{t + 1}} - C} \right)}}{C} \hfill \\ + \beta \left( { - 1} \right)\frac{1}{{C{V^2}}}V\frac{{\left( {{V_{t + 1}} - V} \right)}}{V} \hfill \\ - \frac{1}{C}{c_{t + 1}} = \alpha \left( { - \sigma } \right){M^{ - \sigma }}{m_t} - \beta \frac{1}{{CV}}{c_{t + 1}} - \beta \frac{1}{{CV}}{v_{t + 1}} \hfill \\ \end{gathered}
3

Yes, I tried some alternative steps I found in other resources, and no matter what, I can’t get the -SIGMA exponent to disappear. That can’t be used, can it?
I did see some papers which listed ‘linearised’ equations yet these equations had exponents as well. Im not sure if I should just proceed as is

4

You are confusing something here. Only the objects with a time index are variables. The rest are parameters. Thus, the equation is actually linear despite the exponents.

5

Oh I see, that’s interesting. Is that why we assign steady state values as we assign parameter values?

6

Yes, because these are constants for the purpose of the model.

7

wait so how would we enter these as constants in the code? I’ve not been taught about constants being anything other than the greek parameters

8

The steady states are just functions of the “greek parameters” that can be computed. For example, the steady state interest rate is 1/\beta-1.