<p>I think there are two problems with your calculations. First, as <span class="math-container">$l_i$</span> is the time devoted to work, you should care about <span class="math-container">$\frac{d\ln\frac{1-l_2}{1-l_1}}{d\ln (U_1/U_2)},$</span> rahter than the <span class="math-container">$\frac{d\ln\frac{l_2}{l_1}}{d\ln (U_1/U_2)}$</span> you consider. Second, I don't think your <span class="math-container">$U_i$</span> are correct (perhaps it's caused by first). I get</p>

<p><span class="math-container">$$\begin{matrix} U_1=\frac{b}{1-l_1} &amp; U_2=\frac{be^{-\rho}}{1-l_1}\end{matrix},$$</span></p>

<p>that results in the marginal rate of substitution being</p>

<p><span class="math-container">$$MRS=\frac{U_1}{U_2}=\frac{1-l_2}{1-l_1}\frac{1}{e^{-\rho}}.$$</span></p>

<p>If you take the logarithm, you get </p>

<p><span class="math-container">$$\ln MRS = \ln \frac{1-l_2}{1-l_1}+\ln\frac{1}{e^{-\rho}},$$</span></p>

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

<p><span class="math-container">$$ d\ln MRS = d\ln \frac{1-l_2}{1-l_1}+0,$$</span></p>

<p>and you get the desired result.</p>

CC BY-SA 4.0
2019-01-10T12:46:05.580Z
2019-01-10T12:52:19.983Z
2019-01-10T12:52:19.983Z
Patricio
<p>This is a basic question, but I am new to macro models. The question is from Romer's text. </p>

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

<p><span class="math-container">$$U=\ln(c_1)+b\ln(1-l_1)+e^{-\rho}[\ln(c_2)+b\ln(1-l_2)],$$</span>
where <span class="math-container">$c_i=w_il_i$</span> and <span class="math-container">$\rho$</span> is a discount rate in <span class="math-container">$(0,1)$</span>.</p>

<p>The text says "Because of the logarithmic functional form, the elasticity of substitution between leisure in the two periods is <span class="math-container">$1$</span>." I tried to verify that the elasticity is <span class="math-container">$1$</span> as the following:</p>

<p>Let U_1 denote partial derivative of U with respect to <span class="math-container">$l_1$</span>. Similar for <span class="math-container">$U_2$</span>. Then,</p>

<p><span class="math-container">$$E_{21}=\frac{d\ln(l_2/l_1)}{d\ln(U_1/U_2}=(\frac{dl_2}{l_2}-\frac{dl_1}{l_1})/(\frac{dU_1}{U_1}-\frac{dU_2}{U_2}),$$</span>
where <span class="math-container">$U_1=\frac{1}{U_1}-\frac{b}{1-l_1}$</span> and <span class="math-container">$dU_1=(-\frac{1}{l_1^2}-\frac{b}{(1-l_1)^2})dl_1$</span>. How chould <span class="math-container">$E_{21}=1$</span>? 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!</p>

CC BY-SA 4.0
2019-01-10T09:39:16.790Z
Have a look here https://en.wikipedia.org/wiki/Elasticity_of_intertemporal_substitution
caverac
<macroeconomics><microeconomics><labor-economics><elasticity><elasticity-of-substitution>
Elasticity of substitution between leisure in two periods
Vivian Miller
# Elasticity of substitution between leisure in two periods

(Asked by: username_0 on 2019-01-10)

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!

**Tags:** macroeconomics, microeconomics, labor-economics, elasticity, elasticity-of-substitution

> **Comments:**
> **username_1**: Have a look here https://en.wikipedia.org/wiki/Elasticity_of_intertemporal_substitution  

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## 1 answers

### Answer by username_2 on 2019-01-10. Score: 0 | Accepted answer

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.


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