<p>You have the following utility function <span class="math-container">$u(x_1,x_2)= (x_1)^a · (x_2)^{1-a}$</span></p>
<p>On first step you must maximize utility fixed to budget constraint: <span class="math-container">$m≤p_1·x_1 + p_2·x_2$</span> which tell you that the expenditure of the consumer is restricted by his money <span class="math-container">$(m)$</span>. In the long term you could suppose <span class="math-container">$=$</span> instead of <span class="math-container">$≤$</span>, because all money would be expended.</p>
<p>Then, take the Lagrange procedure to optimization:</p>
<p><span class="math-container">$\max(u(x_1,x_2))$</span> fixed to <span class="math-container">$m≤p_1·x_1 + p_2·x_2$</span>:</p>
<p><span class="math-container">$ α = u(x_1,x_2) -λ(m - (p_1·x_1 + p_2·x_2))$</span></p>
<p>You take the first order derivatives of <span class="math-container">$α$</span> on <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span> and <span class="math-container">$λ$</span>, simplify the gradient of <span class="math-container">$α$</span> and get:</p>
<p><span class="math-container">$UMg(x_1)/p_1 = UMg(x_2)/p_2$</span> where <span class="math-container">$UMg(x_i)$</span> is the marginal utility of the good <span class="math-container">$x_i$</span></p>
<p>Then you put <span class="math-container">$x_i$</span> or <span class="math-container">$p_i$</span> from <span class="math-container">$UMg(x_1)/p_1 = UMg(x_2)/p_2$</span> into <span class="math-container">$m≤p_1·x_1 + p_2·x_2$</span> and you get the Marshallian demand.</p>

CC BY-SA 4.0
2021-10-02T23:48:49.493Z
Tell me if this worked for you, if not I will find other approach.
DPedro
CC BY-SA 4.0
2021-09-28T13:53:08.887Z
2021-09-28T19:33:55.617Z
2021-09-28T19:33:55.617Z
DPedro
<p>would like to just check how one can obtain the Marshallian demand for Cobb Douglas function such as u(x1,x2) = x1^a x2^(1-a) ?</p>
<p>Was attempting the question but got somewhat stuck trying to differentiate the FOC of the above function...</p>
<p>Some have recommended implicit differentiation methodology for Cobb Douglas function differentiation but would it be possible to differentiate and FOC for Cobb douglas without implicit differentiation?</p>
<p>Thank you!</p>

CC BY-SA 4.0
2021-09-28T04:19:49.680Z
Please show your work, where did you get stuck?
Giskard
<consumer-theory>
Marshallian demand for Cobb douglas function
Day312
# Marshallian demand for Cobb douglas function

(Asked by: username_0 on 2021-09-28)

would like to just check how one can obtain the Marshallian demand for Cobb Douglas function such as u(x1,x2) = x1^a x2^(1-a) ?

Was attempting the question but got somewhat stuck trying to differentiate the FOC of the above function...

Some have recommended implicit differentiation methodology for Cobb Douglas function differentiation but would it be possible to differentiate and FOC for Cobb douglas without implicit differentiation?

Thank you!

**Tags:** consumer-theory
> **Comments:**
> **username_1**: Please show your work, where did you get stuck?  

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## 1 answers

### Answer by username_2 on 2021-09-28. Score: 0

You have the following utility function $u(x\_1,x\_2)= (x\_1)^a · (x\_2)^{1-a}$

On first step you must maximize utility fixed to budget constraint: $m≤p\_1·x\_1 + p\_2·x\_2$ which tell you that the expenditure of the consumer is restricted by his money $(m)$. In the long term you could suppose $=$ instead of $≤$, because all money would be expended.

Then, take the Lagrange procedure to optimization:

$\max(u(x\_1,x\_2))$ fixed to $m≤p\_1·x\_1 + p\_2·x\_2$:

$ α = u(x\_1,x\_2) -λ(m - (p\_1·x\_1 + p\_2·x\_2))$

You take the first order derivatives of $α$ on $x\_1$, $x\_2$ and $λ$, simplify the gradient of $α$ and get:

$UMg(x\_1)/p\_1 = UMg(x\_2)/p\_2$ where $UMg(x\_i)$ is the marginal utility of the good $x\_i$

Then you put $x\_i$ or $p\_i$ from $UMg(x\_1)/p\_1 = UMg(x\_2)/p\_2$ into $m≤p\_1·x\_1 + p\_2·x\_2$ and you get the Marshallian demand.

> **Comments:**
> **username_2**: Tell me if this worked for you, if not I will find other approach.  

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