Elasticity of substitution between leisure in two periods
I think there are two problems with your calculations. First, as $l_i$ is the time devoted to work, you should care about $\frac{d\ln\frac{1-l_2}{1-l_1}}{d\ln (U_1/U_2)},$ rahter than the $\frac{d\ln\frac{l_2}{l_1}}{d\ln (U_1/U_2)}$ you consider. Second, I don't think your $U_i$ are correct (perhaps it's caused by first). I get

$$\begin{matrix} U_1=\frac{b}{1-l_1} &amp; U_2=\frac{be^{-\rho}}{1-l_1}\end{matrix},$$

that results in the marginal rate of substitution being

$$MRS=\frac{U_1}{U_2}=\frac{1-l_2}{1-l_1}\frac{1}{e^{-\rho}}.$$

If you take the logarithm, you get 

$$\ln MRS = \ln \frac{1-l_2}{1-l_1}+\ln\frac{1}{e^{-\rho}},$$

where it is useful to observe that the last term of the RHS is a constant. Now, the total differential is

$$ d\ln MRS = d\ln \frac{1-l_2}{1-l_1}+0,$$

and you get the desired result.