Elasticity of substitution between leisure in two periods

Question

This is a basic question, but I am new to macro models. The question is from Romer's text.

Assume a household only has one member and has no initial wealth, and the household lives for two periods. We have

$$U=\ln(c_1)+b\ln(1-l_1)+e^{-\rho}[\ln(c_2)+b\ln(1-l_2)],$$ where $c_i=w_il_i$ and $\rho$ is a discount rate in $(0,1)$.

The text says "Because of the logarithmic functional form, the elasticity of substitution between leisure in the two periods is $1$." I tried to verify that the elasticity is $1$ as the following:

Let U_1 denote partial derivative of U with respect to $l_1$. Similar for $U_2$. Then,

$$E_{21}=\frac{d\ln(l_2/l_1)}{d\ln(U_1/U_2}=(\frac{dl_2}{l_2}-\frac{dl_1}{l_1})/(\frac{dU_1}{U_1}-\frac{dU_2}{U_2}),$$ where $U_1=\frac{1}{U_1}-\frac{b}{1-l_1}$ and $dU_1=(-\frac{1}{l_1^2}-\frac{b}{(1-l_1)^2})dl_1$. How chould $E_{21}=1$? In fact, I got something super messy. Therefore, I doubt if my approach is correct or not. Maybe my formula is not correct? Thanks in advance!

Answer 1

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

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

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.