Log-linear modeling

In economics, a model economy can be approximated with a log-linear system of equations. The log-linearized system is often easier to solve and easier to estimate with statistics.

An equilibrium of an economy can usually be defined as a vector that satisfies a set of equations. These equations are often hard to solve. Or working backwards, the parameters of the equations are hard to estimate with statistics.

Linear equations are easy to solve with Gaussian elimination, and easy to estimate statistically with ordinary least squares. Many economics models have expressions with multivariate polynomials, such as the Cobb-Douglas production function that split into linear terms when considering the logarithms of the variables. Log-linearization transforms a system of equations by taking logs of all variables, and approximates the result with linear equations. The resulting system is often a good approximation due to the polynomial structure, and is easy to study, since it is linear.

Log-linearizing a function
Log-linearizing a function is analogous to linearizing a function with a Taylor expansion. First, log-linearization requires a function $$f : \mathbb{R}^n \rightarrow \mathbb{R},$$ and a point $$x^*$$ in the domain to log-linearize around. For illustration, this article begins with the univariate case.

Define
 * $$g(x) = f(e^x)$$.

The Taylor expansion of $$g$$ around $$\log {x^*}$$ is
 * $$g(x) \approx g(\log x^*) + g'(\log x^*)(x - \log x^*).$$

Then, setting $$\hat{x} = \log x - \log x^*,$$ we have
 * $$f(x) = g(\log x) \approx f(x^*) + x^* f'(x^*)\hat{x}.$$

This final expression is the log-linear approximation of $$f$$. It is linear (or more precisely, affine) in $$\hat{x}$$.

Examples of Log-linearized Functions

 * $$x^n \approx x^{*n} (1 + n \hat{x})$$
 * $$x^m y^n \approx x^{*m} y^{*n} (1 + m \hat{x} + n \hat{y})$$
 * $$e^{x+y} \approx e^{x^* + y^*} (1 + x^* \hat{x} + y^* \hat{y})$$