# Log linearisation - struggling!

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

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}

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

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.

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

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

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

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.