# Log Linearising CES demand

I have been trying to log-linearise the demand function that follows from a standard two-good CES-utility maximamalisation problem. That is:

Maximise $$\begin{eqnarray} U(h,c)= \left(G_1^{\rho}+ G_2^{\rho} \right)^{1/\rho} \end{eqnarray}$$ Subjected to: $$\begin{eqnarray} Y = G_1 + pG_2 \end{eqnarray}$$
Where I normalised the price of the first good to 1. From this maximization it follows that the demand of good 2 can be written as: $$\begin{eqnarray}\label{VAR} G_2 = \frac{ p^{\frac{1}{\rho-1}}Y}{ 1 + p^ {\frac{\rho}{\rho-1}} } \end{eqnarray}$$ When I try to log-linearise this equation I am not sure how to work with the denominator. Thus far, I have this:

$$\begin{eqnarray} \ln G_2 = \frac{1}{\rho-1} \ln p+ \ln y - \ln { (1 + p^ {\frac{\rho}{\rho-1}}) } \end{eqnarray}$$

$$\begin{eqnarray} \ln G_2^* +\frac{1}{G_2^*}(G_2-G_2^*) = \frac{1}{\rho-1} \ln p^* +\frac{\frac{1}{\rho-1}}{p^*}(p-p^*) + \ln y^* +\frac{1}{y^*}(y-y^*) - ??? \end{eqnarray}$$ Yet, I fail to log-linearise the last term with the summation and exponent. I tried to find similar log-linearizations online, but I was only able to find papers that log-linearise the production function of a CES-function, which was not helpful for my problem.

## 1 Answer

Here's my guess.

Let use the notation $$\tilde x_t \approx \ln(x_t) - \ln(x) \approx \dfrac{x_t - x}{x}.$$ If we take logs on both sides we get: $$\ln(G_t) = \frac{1}{1 -\rho} \ln(p_t) + \ln(y_t) - \ln(1 + p_t^{\frac{\rho}{\rho-1}})$$ Subtracting the steady state gives: $$\tilde G_t = \frac{1}{1 - \rho} \tilde p_t + \tilde y_t - \left[\ln(1 + p^{\frac{\rho}{\rho - 1}}) - \ln(1 + p^{\frac{\rho}{\rho - 1}})\right]$$

Taking a Taylor expansion of the last term gives: \begin{align*} \ln(1 + p_t^{\frac{\rho}{\rho - 1}}) - \ln(1 + p^{\frac{\rho}{\rho - 1}}) &\approx \frac{1}{1 + p^{\frac{\rho}{\rho - 1}}}\frac{\rho}{\rho - 1}p^{\frac{\rho}{\rho - 1}}\frac{(p_t- p)}{p},\\ &\approx \frac{p^{\frac{\rho}{\rho- 1}}}{1 + p^{\frac{\rho}{\rho - 1}}} \frac{\rho}{\rho - 1} \tilde p_t \end{align*} So we get: $$\tilde G_t \approx \left(\frac{1}{1 - \rho}- \frac{p^{\frac{\rho}{\rho - 1}}}{1 + p^{\frac{\rho}{\rho - 1}}} \frac{\rho}{\rho - 1} \right) \tilde p_t + \tilde y_t,\\ = \left(\frac{1}{1 - \rho}- \frac{G}{Y} \frac{\rho}{\rho - 1} \right) \tilde p_t + \tilde y_t,\$$

• Thank you, that was really helpful!
– Rosa
Oct 1, 2021 at 7:20