# Covariance of Log-Normal Variables

In Obstfeld and Rogoff (2000), formula (12) states the following:

$$W = (\frac{\phi}{\phi-1}) \frac{E\{K(L^\nu)\}}{E\{\frac{L}{P}C^{-\rho}\}}$$

where $\phi$, $\rho$ and $\nu$ are parameters, $E$ is the expectation operator, and $K$, $L$, $P$,$C$ are endogenous variables jointly log-normally distributed.

They state that given the log-normality it is equivalent to write equation (12) as:

$$W = (\frac{\phi}{\phi-1}) \frac{E\{K\}E\{L\}^{\nu-1})}{E\{C\}^{-\rho} E\{ \frac{1}{P} \} } \exp{\psi}$$

where:

$$\psi = \frac{\nu(\nu-1)}{2} \sigma_l^2 - \frac{\rho(\rho+1)}{2} \sigma_c^2 + \nu \sigma_{kl} + \rho \sigma_{cl} - \rho \sigma_{cp} + \sigma_{lp}$$

I tried to derive the expression for psi, I get something slightly different:

$$\psi = \frac{\nu}{2} \sigma_l^2 - \frac{\rho}{2} \sigma_c^2 + \nu \sigma_{kl} + \rho \sigma_{cl} - \rho \sigma_{cp} + \sigma_{lp}$$

I don't understand where those extra terms come from.

Any help?

• Is $\sigma_{kl}$ the co-variance between $K$ and $L$? – Gordon Mar 30 '16 at 16:28
• Yes, indeed. And $\sigma_{lp}$ is the covariance between $L$ and $P$ and so on and so forth. – phdstudent Mar 30 '16 at 16:44