In Gali (2015)'s textbook Monetary Policy, Inflation, and the Business Cycle, variables in levels are denoted with capital letters, logged variables with lowercase letters.
However, when a log-linearisation is performed, say of the household optimality condition, why is the result in logs and not in percentage deviations, (lower case letters with a hat, for instance)?
Is there an intermediate step/reason for why we can go from percentage deviations to logs? Thanks!
We can easily switch between logged equations and log-deviations.
To see this, note that a given equation always holds, and therefore it also holds in steady state. Hence, you can just subtract the steady state values of both sides of an equation to get log deviations, if you wish.
The reason you want logs, in this case, can be seen in (9). You want $i_t - E_t\{\pi_{t+1}\}$, the real interest rate, to deviate around $\rho$. This makes more sense economically than deviations around their individual steady states. Mathematically they are equivalent.
$$\frac{x_{t+1}}{x_t} \equiv \frac{x_{t}+Δx_{t+1}}{x_t} = 1+\frac{Δx_{t+1}}{x_t}$$
Further,
$$\ln (1+a) \approx a,$$
an acceptable approximation for $|a| <0.1$ and certainly for $|a| <0.05$.
