<p>A shortcut to solve <span class="math-container">$$\min_{x,y} x^{\alpha} y^{\beta}$$</span> s.t. <span class="math-container">$$xp_x + yp_x \leq m$$</span> and <span class="math-container">$$0 \leq x,y$$</span> is to set <span class="math-container">$x = 0$</span> and/or <span class="math-container">$y = 0$</span>, as this results in <span class="math-container">$U(x,y) = 0$</span>, and the function does not map to negative values.</p>

CC BY-SA 4.0
2019-10-25T20:18:34.140Z
2019-10-25T20:18:34.140Z
Giskard
<p>Say a Cobb Douglas like:</p>

<p><span class="math-container">$$\max_{X,Y\: s.t. X \cdot P_x+ Y \cdot P_y=I} U=X^\alpha Y^\beta$$</span></p>

<p>When it comes to maximization I would do the following way (for the fastest result):</p>

<p>x: <span class="math-container">$\alpha/(\alpha + \beta) = r; r*income = r_2$</span>; <span class="math-container">$r_2$</span>/price good = x$</p>

<p>Is there a similar (or really fast) method to solve for minimization problems?</p>

CC BY-SA 4.0
2019-10-25T18:33:54.260Z
What are you minimizing?
Herr K.
CC BY-SA 4.0
2019-10-25T20:13:49.800Z
Some restriction or cost, say for exaplme PriceX + PriceY >= budget restriction. A standard Cobb Douglas minimization.
zhe_charmander
CC BY-SA 4.0
2019-10-25T21:02:16.413Z
I coudn't follow your math so I tried to clean it up with latex. Please look and confirm that the updated version is correct.
BKay
CC BY-SA 4.0
2019-10-26T23:16:44.117Z
I believe you meant maximize utility subject to the constraint. Or did you imply minimize cost subject to utility?
Brennan
<microeconomics><mathematical-economics>
Short cuts to solve Cobb Douglas Utility function (minimization)
zhe_charmander
# Short cuts to solve Cobb Douglas Utility function (minimization)

(Asked by: username_0 on 2019-10-25)

Say a Cobb Douglas like:

$$\max\_{X,Y\: s.t. X \cdot P\_x+ Y \cdot P\_y=I} U=X^\alpha Y^\beta$$

When it comes to maximization I would do the following way (for the fastest result):

x: $\alpha/(\alpha + \beta) = r; r\*income = r\_2$; $r\_2$/price good = x$

Is there a similar (or really fast) method to solve for minimization problems?

**Tags:** microeconomics, mathematical-economics

> **Comments:**
> **username_1**: What are you minimizing?  
> **username_0**: Some restriction or cost, say for exaplme PriceX + PriceY >= budget restriction. A standard Cobb Douglas minimization.  
> **username_2**: I coudn't follow your math so I tried to clean it up with latex. Please look and confirm that the updated version is correct.  
> **username_3**: I believe you meant maximize utility subject to the constraint. Or did you imply minimize cost subject to utility?  

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## 1 answers

### Answer by username_4 on 2019-10-25. Score: 1

A shortcut to solve $$\min\_{x,y} x^{\alpha} y^{\beta}$$ s.t. $$xp\_x + yp\_x \leq m$$ and $$0 \leq x,y$$ is to set $x = 0$ and/or $y = 0$, as this results in $U(x,y) = 0$, and the function does not map to negative values.


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