Linearising macroeconomic models

Linearisation of macroeconomic models is common in theoretical and empirical work, so it is worth considering when it is applicable.

A function f of some economic quantity x can be written as

f(x+e) = f(x) + e*f’(x) + O(e^2)

where the dash denotes differentiation and the O term is of order e^2. Linearisation assumes that e is small, so that the O term is negligible, or that there is no O term at all, so we can write

f(x+e) = f(x) + e*f’(x)

The truncation masks misspecified models for small variations of the data, since almost any model has a truncated form like this one. The linearization will not hold for larger variation and misspecified models. The linearization may not be much use unless we have a stable steady state, when small variations in x result in x returning to its initial value. If the model spec is correct, non-locally the linearization will not hold unless it happens to coincide with the full expansion and the O term is identically zero.

Either the quantities of interest are in a stable steady state which is often is probably not true in developing countries and frequently not true in developed countries either, so should be demonstrated but frequently isn’t. Or if the linearization coincides with the full expansion of f and so the O term can be neglected, then evidence of goodness of fit should be given across the full range of inputs, but again frequently isn’t.