<p>After substituting for <span class="math-container">$c_{1t}$</span> and <span class="math-container">$c_{2t}$</span>, the problem can be rewritten as :
<span class="math-container">\begin{eqnarray*} \max_{q_t^e} \ &amp; \frac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma} \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma} + \frac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma} \\ \text{s.t. } &amp; q_t^e \in [0, 1]\end{eqnarray*}</span> </p>

<p>which is same as
<span class="math-container">\begin{eqnarray*} \max_{q_t^e} \ &amp; \alpha \left((1-q_t^e)  + z_tq_t^e\right)^{-\gamma}  \\ \text{s.t. } &amp; q_t^e \in [0, 1]\end{eqnarray*}</span>
where <span class="math-container">$\alpha  = \dfrac{(1- \pi) (n\omega_t)^{-\gamma} }{-\gamma}  + \dfrac{\pi (AR\theta\omega_t)^{-\gamma } }{-\gamma z_t^{ -\gamma }} $</span></p>

<p>Observe that this is equivalent to solving :
<span class="math-container">\begin{eqnarray*} \max_{q_t^e} \ &amp;  (1-q_t^e)  + z_tq_t^e  \\ \text{s.t. } &amp; q_t^e \in [0, 1]\end{eqnarray*}</span></p>

<p>This is an objective that is linear in choice variable <span class="math-container">$q_t^e$</span>, which is increasing in <span class="math-container">$q_t^e$</span> when <span class="math-container">$z_t &gt; 1$</span>, decreasing in <span class="math-container">$q_t^e$</span> when <span class="math-container">$z_t &lt; 1$</span>, and is a constant for <span class="math-container">$z_t = 1$</span>. Consequently, the solution is
<span class="math-container">\begin{eqnarray*} q_t^e \in \begin{cases} \{1\} &amp; \text{if } z_t &gt; 1 \\ \{0\} &amp; \text{if } z_t &lt; 1 \\ [0, 1] &amp; \text{if } z_t = 1 \end{cases} \end{eqnarray*}</span></p>

CC BY-SA 4.0
2020-01-08T14:29:52.940Z
Hi @Amit, your solutions are very clever and you solve very well and clearly. I perfectly understand and learn how to solve some type of questions when you solve. Thanks a lot to you. Now, I have a question on job search theory. I guess you can solve it very well. Please Help me to do it. Please teach me how to solve such type of questions. I will be happy if you will help me. Thank you so much. :)  https://economics.stackexchange.com/questions/33549/job-search-theory-in-discrete-time
studentp
CC BY-SA 4.0
2019-11-13T20:51:45.007Z
2019-11-13T20:51:45.007Z
Amit
<p>I have to maximize following expected utility function using Kuhn tucker conditions -</p>

<p><img src="https://i.sstatic.net/3fWH4.jpg" alt="enter image description here"></p>

<p>Since expected utility function are increasing <span class="math-container">$C_{1,t}$</span> and <span class="math-container">$C_{2,t}$</span> so constraints (i) and (ii) will hold with equality. Thus, I substituted these constraints into the objective function. </p>

<p>Note: Here, only <span class="math-container">$q_t^e$</span> is the only variable and expected utility function is maximized subject to <span class="math-container">$q_t^e$</span></p>

<p>For simplification, I denoted M= (1-<span class="math-container">$q_t^e$</span>)<span class="math-container">$w_t$</span> + <span class="math-container">$q_t^e$</span><em><span class="math-container">$w_t$</span></em><span class="math-container">$z_t$</span></p>

<p>This made <span class="math-container">$C_{1,t}$</span> = nM and <span class="math-container">$C_{2,t}$</span>= (ARθ/<span class="math-container">$z_t$</span>)M</p>

<p>I also broke down constraint iii into two constraints <span class="math-container">$q_t^e$</span>≤1 and -<span class="math-container">$q_t^e$</span>≤0</p>

<p>I set up the legrange:</p>

<p>L = (1-π)[<span class="math-container">${(nM)}^{-γ}$</span>]/(-γ) + π[<span class="math-container">${((ARθ/z_t)M)}^{-γ}$</span>]/(-γ) + <span class="math-container">$λ_1$</span>(1-<span class="math-container">$q_t^e$</span>) + <span class="math-container">$λ_2$</span>(0+<span class="math-container">$q_t^e$</span>)</p>

<p>Differentiating with respect to <span class="math-container">$q_t^e$</span></p>

<p>(1-π)[<span class="math-container">${(nM)}^{-γ-1}$</span>][n(<span class="math-container">$z_t$</span>-1)] + π[<span class="math-container">${((ARθ/z_t)M)}^{-γ-1}$</span>][(ARθ/<span class="math-container">$z_t$</span>)(<span class="math-container">$z_t$</span>-1)] = <span class="math-container">$λ_1$</span> - <span class="math-container">$λ_2$</span></p>

<p><span class="math-container">$λ_1$</span>(1-<span class="math-container">$q_t^e$</span>) =0;   <span class="math-container">$λ_1$</span>≥0</p>

<p><span class="math-container">$λ_2$</span>(0+<span class="math-container">$q_t^e$</span>) = 0;  <span class="math-container">$λ_2$</span>≥0</p>

<p>Case 1: <span class="math-container">$λ_1$</span>=0 and <span class="math-container">$λ_2$</span></p>

<p>I got (1-π)[<span class="math-container">${(nM)}^{-γ-1}$</span>][n(<span class="math-container">$z_t$</span>-1)] + π[<span class="math-container">${((ARθ/z_t)M)}^{-γ-1}$</span>][(ARθ/<span class="math-container">$z_t$</span>)(<span class="math-container">$z_t$</span>-1)] =0</p>

<p>Which boils down to M=0</p>

<p>which gave me - - > <span class="math-container">$q_t^e$</span> = 1/(1-<span class="math-container">$z_t$</span>)</p>

<p>But solution provided is of the following form - </p>

<p><img src="https://i.sstatic.net/ViI2G.jpg" alt="enter image description here"></p>

<p>My answer doesn't match the solution provided. Can someone please look at my solution and tell me what did I do wrong? </p>

<optimization><inequality>
Kuhn Tucker Maximization
Elina Gilbert
# Kuhn Tucker Maximization

(Asked by: username_0 on 2019-11-12)

I have to maximize following expected utility function using Kuhn tucker conditions -

![enter image description here](https://i.sstatic.net/3fWH4.jpg)

Since expected utility function are increasing $C\_{1,t}$ and $C\_{2,t}$ so constraints (i) and (ii) will hold with equality. Thus, I substituted these constraints into the objective function.

Note: Here, only $q\_t^e$ is the only variable and expected utility function is maximized subject to $q\_t^e$

For simplification, I denoted M= (1-$q\_t^e$)$w\_t$ + $q\_t^e$*$w\_t$*$z\_t$

This made $C\_{1,t}$ = nM and $C\_{2,t}$= (ARθ/$z\_t$)M

I also broke down constraint iii into two constraints $q\_t^e$≤1 and -$q\_t^e$≤0

I set up the legrange:

L = (1-π)[${(nM)}^{-γ}$]/(-γ) + π[${((ARθ/z\_t)M)}^{-γ}$]/(-γ) + $λ\_1$(1-$q\_t^e$) + $λ\_2$(0+$q\_t^e$)

Differentiating with respect to $q\_t^e$

(1-π)[${(nM)}^{-γ-1}$][n($z\_t$-1)] + π[${((ARθ/z\_t)M)}^{-γ-1}$][(ARθ/$z\_t$)($z\_t$-1)] = $λ\_1$ - $λ\_2$

$λ\_1$(1-$q\_t^e$) =0; $λ\_1$≥0

$λ\_2$(0+$q\_t^e$) = 0; $λ\_2$≥0

Case 1: $λ\_1$=0 and $λ\_2$

I got (1-π)[${(nM)}^{-γ-1}$][n($z\_t$-1)] + π[${((ARθ/z\_t)M)}^{-γ-1}$][(ARθ/$z\_t$)($z\_t$-1)] =0

Which boils down to M=0

which gave me - - > $q\_t^e$ = 1/(1-$z\_t$)

But solution provided is of the following form -

![enter image description here](https://i.sstatic.net/ViI2G.jpg)

My answer doesn't match the solution provided. Can someone please look at my solution and tell me what did I do wrong?

**Tags:** optimization, inequality


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## 1 answers

### Answer by username_1 on 2019-11-13. Score: 3

After substituting for $c\_{1t}$ and $c\_{2t}$, the problem can be rewritten as :
\begin{eqnarray\*} \max\_{q\_t^e} \ & \frac{(1- \pi) (n\omega\_t)^{-\gamma} }{-\gamma} \left((1-q\_t^e) + z\_tq\_t^e\right)^{-\gamma} + \frac{\pi (AR\theta\omega\_t)^{-\gamma } }{-\gamma z\_t^{ -\gamma }} \left((1-q\_t^e) + z\_tq\_t^e\right)^{-\gamma} \\ \text{s.t. } & q\_t^e \in [0, 1]\end{eqnarray\*}

which is same as
\begin{eqnarray\*} \max\_{q\_t^e} \ & \alpha \left((1-q\_t^e) + z\_tq\_t^e\right)^{-\gamma} \\ \text{s.t. } & q\_t^e \in [0, 1]\end{eqnarray\*}
where $\alpha = \dfrac{(1- \pi) (n\omega\_t)^{-\gamma} }{-\gamma} + \dfrac{\pi (AR\theta\omega\_t)^{-\gamma } }{-\gamma z\_t^{ -\gamma }} $

Observe that this is equivalent to solving :
\begin{eqnarray\*} \max\_{q\_t^e} \ & (1-q\_t^e) + z\_tq\_t^e \\ \text{s.t. } & q\_t^e \in [0, 1]\end{eqnarray\*}

This is an objective that is linear in choice variable $q\_t^e$, which is increasing in $q\_t^e$ when $z\_t > 1$, decreasing in $q\_t^e$ when $z\_t < 1$, and is a constant for $z\_t = 1$. Consequently, the solution is
\begin{eqnarray\*} q\_t^e \in \begin{cases} \{1\} & \text{if } z\_t > 1 \\ \{0\} & \text{if } z\_t < 1 \\ [0, 1] & \text{if } z\_t = 1 \end{cases} \end{eqnarray\*}

> **Comments:**
> **username_2**: Hi @username_1, your solutions are very clever and you solve very well and clearly. I perfectly understand and learn how to solve some type of questions when you solve. Thanks a lot to you. Now, I have a question on job search theory. I guess you can solve it very well. Please Help me to do it. Please teach me how to solve such type of questions. I will be happy if you will help me. Thank you so much. :)  https://economics.stackexchange.com/questions/33549/job-search-theory-in-discrete-time  

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