<p>I have a utility function:
max <span class="math-container">$U = \frac{c_o^{1-\eta}}{1-\eta} + \frac{\beta}{2} c_{1a} + \frac{\beta}{2} c_{1b} $</span>
subject to the budget constraint <span class="math-container">$p_0c_0 + p_{1a}c_{1a} + p_{1b}c_{1b} = p_0e_0 + p_{1a}e_{1a} + p_{1b}e_{1b} = p_0 + p_{1a}(1+\epsilon) + p_{1b}(1-\epsilon)   $</span> and the non-negativity constraints for <span class="math-container">$c_{1a},c_{1b}$</span></p>
<p>How do I solve for the optimal solution <span class="math-container">$(c_0,c_{1a},c_{1b})$</span>?</p>
<p>I try to set up a Lagrangian for this and divide it by cases:
<span class="math-container">$L = \frac{c_o^{1-\eta}}{1-\eta} + \frac{\beta}{2} c_{1a} + \frac{\beta}{2} c_{1b} + \theta_1(p_0c_0 + p_{1a}c_{1a} + p_{1b}c_{1b} - p_0 - p_{1a}(1+\epsilon) - p_{1b}(1-\epsilon)) + \theta_2c_{1a} + \theta_3c_{1b}$</span></p>
<p>The FOCs are:
<span class="math-container">$$ c_o^{-\eta} -\theta_1p_0 = 0 $$</span>
<span class="math-container">$$ \frac{\beta}{2} - \theta_1 p_{1a} + \theta_2 = 0 $$</span>
<span class="math-container">$$ \frac{\beta}{2} - \theta_1 p_{1b} + \theta_3 = 0 $$</span>
<span class="math-container">$$ p_0c_0 + p_{1a}c_{1a} + p_{1b}c_{1b} = p_0 + p_{1a}(1+\epsilon) + p_{1b}(1-\epsilon)  $$</span></p>
<ol>
<li>For corner solution cases, i.e, <span class="math-container">$c_{1a} = 0, c_{1b} &gt; 0, \theta_3 = 0$</span>, I can find a solution, but for interior solution case, i.e, <span class="math-container">$c_{1a} &gt; 0, c_{1a} &gt; 0, \theta_2 = 0, \theta_3 = 0$</span>, I cannot proceed because only the budget constraint has <span class="math-container">$c_{1a},c_{1b}$</span> so I cannot find both.</li>
<li>Even if I can find all three cases (2 corner solutions, 1 interior solution), how do I know which solution is optimal?</li>
</ol>

CC BY-SA 4.0
2021-10-25T09:44:29.237Z
Is this the first utility optimization problem you encounter? Have you by any chance ever had a problem where the goal/utility function was one of perfect complements? If yes, did you solve it by Lagrangian?
Giskard
CC BY-SA 4.0
2021-10-25T11:48:55.277Z
For 1: $c_{1a}$ and $c_{1b}$ are perfect substitutes. There are infinite solutions if their prices are equal. For 2: for each price, there's only one case. If $p_{1a}=p_{2a}$ you have interior solution. If $p_{1a}>p_{2a}$ you have $c_{1a}=0$, if $p_{1a}<p_{2a}$ you have $c_{2a}=0$.
Daniel R
CC BY-SA 4.0
2021-10-25T11:56:46.167Z
Also, for the interior solution, the consumer is indifferent on any combination of $c_{1a}$ and $c_{1b}$, including the combinations where one of them is zero.
Daniel R
<utility>
quasilinear utility maximization with non-negativity constraints
user38396
# quasilinear utility maximization with non-negativity constraints

(Asked by: username_0 on 2021-10-24)

I have a utility function:
max $U = \frac{c\_o^{1-\eta}}{1-\eta} + \frac{\beta}{2} c\_{1a} + \frac{\beta}{2} c\_{1b} $
subject to the budget constraint $p\_0c\_0 + p\_{1a}c\_{1a} + p\_{1b}c\_{1b} = p\_0e\_0 + p\_{1a}e\_{1a} + p\_{1b}e\_{1b} = p\_0 + p\_{1a}(1+\epsilon) + p\_{1b}(1-\epsilon) $ and the non-negativity constraints for $c\_{1a},c\_{1b}$

How do I solve for the optimal solution $(c\_0,c\_{1a},c\_{1b})$?

I try to set up a Lagrangian for this and divide it by cases:
$L = \frac{c\_o^{1-\eta}}{1-\eta} + \frac{\beta}{2} c\_{1a} + \frac{\beta}{2} c\_{1b} + \theta\_1(p\_0c\_0 + p\_{1a}c\_{1a} + p\_{1b}c\_{1b} - p\_0 - p\_{1a}(1+\epsilon) - p\_{1b}(1-\epsilon)) + \theta\_2c\_{1a} + \theta\_3c\_{1b}$

The FOCs are:
$$ c\_o^{-\eta} -\theta\_1p\_0 = 0 $$
$$ \frac{\beta}{2} - \theta\_1 p\_{1a} + \theta\_2 = 0 $$
$$ \frac{\beta}{2} - \theta\_1 p\_{1b} + \theta\_3 = 0 $$
$$ p\_0c\_0 + p\_{1a}c\_{1a} + p\_{1b}c\_{1b} = p\_0 + p\_{1a}(1+\epsilon) + p\_{1b}(1-\epsilon) $$

1. For corner solution cases, i.e, $c\_{1a} = 0, c\_{1b} > 0, \theta\_3 = 0$, I can find a solution, but for interior solution case, i.e, $c\_{1a} > 0, c\_{1a} > 0, \theta\_2 = 0, \theta\_3 = 0$, I cannot proceed because only the budget constraint has $c\_{1a},c\_{1b}$ so I cannot find both.
2. Even if I can find all three cases (2 corner solutions, 1 interior solution), how do I know which solution is optimal?

**Tags:** utility
> **Comments:**
> **username_1**: Is this the first utility optimization problem you encounter? Have you by any chance ever had a problem where the goal/utility function was one of perfect complements? If yes, did you solve it by Lagrangian?  
> **username_2**: For 1: $c_{1a}$ and $c_{1b}$ are perfect substitutes. There are infinite solutions if their prices are equal. For 2: for each price, there's only one case. If $p_{1a}=p_{2a}$ you have interior solution. If $p_{1a}>p_{2a}$ you have $c_{1a}=0$, if $p_{1a}<p_{2a}$ you have $c_{2a}=0$.  
> **username_2**: Also, for the interior solution, the consumer is indifferent on any combination of $c_{1a}$ and $c_{1b}$, including the combinations where one of them is zero.  

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## 0 answers

