<p>Lucas (1972) Neutrality of Money section 3 has following set up.
<span class="math-container">$U(c,n)$</span> is Utility function of consumption and labor (or producing labor output). <span class="math-container">$U_c (c,n) &gt;0$</span> and <span class="math-container">$U_n (c,n)&lt;0$</span>. Further, the noninferiority conditions hold for leisure, <span class="math-container">$U_{cc} +U_{cn}&lt;0$</span> and <span class="math-container">$U_{nn} +U_{nc}&lt;0$</span> . We have following Kuhn-Tucker conditions for the objective function.</p>
<ol>
<li><span class="math-container">$U_c(c,n)-p\mu&lt;= 0$</span>, with equality if <span class="math-container">$c$</span>&gt;0,</li>
<li><span class="math-container">$U_n(c,n)+p\mu&lt;= 0$</span>, with equality if <span class="math-container">$n$</span>&gt;0, and</li>
<li><span class="math-container">$p(n-c)-\lambda &gt;=0$</span>, with equality if <span class="math-container">$n$</span>&gt;0</li>
</ol>
<p>These three conditions can solve for <span class="math-container">$c,n,$</span> and <span class="math-container">$p\mu$</span> as a function of <span class="math-container">$\lambda/p.$</span> The function <span class="math-container">$h(\lambda/p)$</span> is the solution for <span class="math-container">$p\mu$</span>, which is positive, <strong>increasing</strong>, and continuously differentiable.
I have  trouble understating that <span class="math-container">$h(\lambda/p)$</span> can be an increasing function. We can infer from conditions 1 and 2 that the function is positive, and continuous. However, since marginal utility function is a decreasing function, I can not put my head around the idea why <span class="math-container">$h(\lambda/p)$</span> be an increasing function. Moreover, I tried substituting Cob-Douglas utility function <span class="math-container">$U(c,n) = c^\alpha n^{\alpha -1}$</span>, where <span class="math-container">$\alpha\in(0,1)$</span>, which will satisfy concavity, continuity, and noninferiority conditions. By substituring the Cob-Douglas type utility function, the derived <span class="math-container">$h(\lambda/p)$</span> is positive, <strong>decreasing</strong>, and continuous function.</p>

CC BY-SA 4.0
2024-08-01T13:41:48.487Z
Are you sure you substituted the correct Cobb-Douglas? It can be a typo, but in your Cobb Douglas the exponent of $n$ should be $1-\alpha$, not $\alpha-1$
BakerStreet
CC BY-SA 4.0
2024-08-02T14:22:27.963Z
I agree that usual Cobb-Douglas function is of the form $\alpha, 1-\alpha$. However, I wanted to set up the $U_n(C,n)<0$, and $U_C(C,n) >0$. For that reason, I set up the function $U(C,n) = C^{\alpha} n^{\alpha-1}$.
ps12345
<money><expected-utility><proof>
Lucas 1972 Neutrality of Money Proof of derivation
ps12345
# Lucas 1972 Neutrality of Money Proof of derivation

(Asked by: username_0 on 2024-07-30)

Lucas (1972) Neutrality of Money section 3 has following set up.
$U(c,n)$ is Utility function of consumption and labor (or producing labor output). $U\_c (c,n) >0$ and $U\_n (c,n)<0$. Further, the noninferiority conditions hold for leisure, $U\_{cc} +U\_{cn}<0$ and $U\_{nn} +U\_{nc}<0$ . We have following Kuhn-Tucker conditions for the objective function.

1. $U\_c(c,n)-p\mu<= 0$, with equality if $c$>0,
2. $U\_n(c,n)+p\mu<= 0$, with equality if $n$>0, and
3. $p(n-c)-\lambda >=0$, with equality if $n$>0

These three conditions can solve for $c,n,$ and $p\mu$ as a function of $\lambda/p.$ The function $h(\lambda/p)$ is the solution for $p\mu$, which is positive, **increasing**, and continuously differentiable.
I have trouble understating that $h(\lambda/p)$ can be an increasing function. We can infer from conditions 1 and 2 that the function is positive, and continuous. However, since marginal utility function is a decreasing function, I can not put my head around the idea why $h(\lambda/p)$ be an increasing function. Moreover, I tried substituting Cob-Douglas utility function $U(c,n) = c^\alpha n^{\alpha -1}$, where $\alpha\in(0,1)$, which will satisfy concavity, continuity, and noninferiority conditions. By substituring the Cob-Douglas type utility function, the derived $h(\lambda/p)$ is positive, **decreasing**, and continuous function.

**Tags:** money, expected-utility, proof
> **Comments:**
> **username_1**: Are you sure you substituted the correct Cobb-Douglas? It can be a typo, but in your Cobb Douglas the exponent of $n$ should be $1-\alpha$, not $\alpha-1$  
> **username_0**: I agree that usual Cobb-Douglas function is of the form $\alpha, 1-\alpha$. However, I wanted to set up the $U_n(C,n)<0$, and $U_C(C,n) >0$. For that reason, I set up the function $U(C,n) = C^{\alpha} n^{\alpha-1}$.  

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## 0 answers

