<p><strong>Hint:</strong> I won't give a full answer since this is a very specific question, but I can give you a long outline here. A Lagrangian is set-up with an objective function (in this case utility) plus or minus (depending on Kuhn Tucker constraints, which we won't worry about) a constraint function (in this case, income) which is set to zero and then multiplied by the Lagrange multiplier, which we will denote $\lambda$.</p>

<p>So we set up the Lagrangian function. (Plug in the utility function yourself.)</p>

<p>$$\mathcal{L} = u(x_1, x_2, x_3) - \lambda (p_1x_1 + p_2x_2 + p_3x_3 - m)$$</p>

<p>Solve for the partial derivatives $\frac{\partial \mathcal{L}}{\partial x_1}$, $\frac{\partial \mathcal{L}}{\partial x_2}$, and $\frac{\partial \mathcal{L}}{\partial x_3}$ and set them equal to zero (You can also solve for the partial of $\lambda$, but it just gives you back the budget constraint, which I skip to). For example, you get for your first derivative:</p>

<p>$$\frac{\partial \mathcal{L}}{\partial x_1} = \alpha (x_1 + z_1)^{\alpha - 1} - \lambda p_1 = 0$$</p>

<p>Once you solve the three partials, the easiest thing for you to do is solve for $\lambda$ with respect to the other variables for all three partials, and then figure out how to substitute them into the budget constraint. You should solve for each $x$ in terms of the prices, maybe some $z$'s, the greek letters $(\alpha, \beta, \gamma)$ and income $m$. (I tried working some of it out; enjoy your algebra.)</p>

CC BY-SA 3.0
2016-01-06T03:23:07.967Z
2016-01-08T02:33:05.790Z
2016-01-08T02:33:05.790Z
Kitsune Cavalry
<p>The marshallian demand functions are solutions to the utility maximization problem. That is, the solution to
$$
\max u(\textbf{x})\\
s.t. \textbf{p}\cdot \textbf{x} = m
$$
Where boldface is used to represent vectors.</p>

<p>In your problem, you would use your utility function, and $\textbf{p}\cdot \textbf{x} = p_1x_1 + p_2x_2 + p_3x_3$</p>

<p>The other answer provides a link that I believe shows how to derive the formulas. I just wanted to explicitly state that marshallian demands are solutions to the utility maximization problem. If you know how to solve a maximization problem (using FOC and/or lagrangians), you can derive the marshallian demands.</p>

CC BY-SA 3.0
2016-01-05T23:01:25.343Z
2016-01-05T23:01:25.343Z
majmun
<p>The Marshallian demand functions are basically partial derivatives of the Cobb-Douglas utility function. You should consider that you want to maximize spending first, then derive the functions to get the optimal prices, demand and a equilibrium with both - not sure if I used the correct words.</p>

<p>Nevertheless, here's an example with 2 products, <a href="https://economics.stackexchange.com/a/5030/6832">here</a>, from SE.</p>

CC BY-SA 3.0
2016-01-06T09:07:48.477Z
Marshallian demand functions are not partial derivatives of the utility function. (Not even in the Cobb-Douglas case.) Additional steps are needed.
Giskard
CC BY-SA 3.0
2016-01-06T17:05:18.573Z
Poor misuse of words. But the example I gave there is useful to understand the steps. There's more exemplifications in that question and they're related because the problem is fairly similar.
gsilvapt
CC BY-SA 3.0
2016-01-05T21:41:52.230Z
2016-01-05T21:41:52.230Z
2017-04-13T12:51:38.253Z
gsilvapt
<p>Consider a consumer with a Cobb-Douglas utility function over three goods of the form $u=(x_1+z_1)^\alpha \cdot (x_2+z_2)^\beta \cdot (x_3+z_3)^\gamma$ with $\alpha, \ \beta, \ \gamma, \ z_1, \ z_2, \ z_3&gt;0$, and $\alpha+\beta+\gamma=1$. The consumer has income $m$, and pays prices $p_1, p_2, p_3$ for the three goods.</p>

<p>Find the Marshallian demands for goods 1, 2 and 3 for this consumer, using the
Lagrangian method.</p>

CC BY-SA 3.0
2016-01-05T19:12:56.290Z
Please edit your question according to this policy http://meta.economics.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions
emeryville
CC BY-SA 3.0
2016-01-05T19:22:19.123Z
I cannot understand why the result is x1=(alpha+y)/p1-z1, where y=p1z1+p2z2+p3z3
Michele Galbusera
<microeconomics>
Marshallian demand help
Michele Galbusera
# Marshallian demand help

(Asked by: username_0 on 2016-01-05)

Consider a consumer with a Cobb-Douglas utility function over three goods of the form $u=(x\_1+z\_1)^\alpha \cdot (x\_2+z\_2)^\beta \cdot (x\_3+z\_3)^\gamma$ with $\alpha, \ \beta, \ \gamma, \ z\_1, \ z\_2, \ z\_3>0$, and $\alpha+\beta+\gamma=1$. The consumer has income $m$, and pays prices $p\_1, p\_2, p\_3$ for the three goods.

Find the Marshallian demands for goods 1, 2 and 3 for this consumer, using the
Lagrangian method.

**Tags:** microeconomics
> **Comments:**
> **username_1**: Please edit your question according to this policy http://meta.economics.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions  
> **username_0**: I cannot understand why the result is x1=(alpha+y)/p1-z1, where y=p1z1+p2z2+p3z3  

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## 3 answers

### Answer by username_2 on 2016-01-06. Score: 2

**Hint:** I won't give a full answer since this is a very specific question, but I can give you a long outline here. A Lagrangian is set-up with an objective function (in this case utility) plus or minus (depending on Kuhn Tucker constraints, which we won't worry about) a constraint function (in this case, income) which is set to zero and then multiplied by the Lagrange multiplier, which we will denote $\lambda$.

So we set up the Lagrangian function. (Plug in the utility function yourself.)

$$\mathcal{L} = u(x\_1, x\_2, x\_3) - \lambda (p\_1x\_1 + p\_2x\_2 + p\_3x\_3 - m)$$

Solve for the partial derivatives $\frac{\partial \mathcal{L}}{\partial x\_1}$, $\frac{\partial \mathcal{L}}{\partial x\_2}$, and $\frac{\partial \mathcal{L}}{\partial x\_3}$ and set them equal to zero (You can also solve for the partial of $\lambda$, but it just gives you back the budget constraint, which I skip to). For example, you get for your first derivative:

$$\frac{\partial \mathcal{L}}{\partial x\_1} = \alpha (x\_1 + z\_1)^{\alpha - 1} - \lambda p\_1 = 0$$

Once you solve the three partials, the easiest thing for you to do is solve for $\lambda$ with respect to the other variables for all three partials, and then figure out how to substitute them into the budget constraint. You should solve for each $x$ in terms of the prices, maybe some $z$'s, the greek letters $(\alpha, \beta, \gamma)$ and income $m$. (I tried working some of it out; enjoy your algebra.)


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### Answer by username_3 on 2016-01-05. Score: 0

The marshallian demand functions are solutions to the utility maximization problem. That is, the solution to
$$
\max u(\textbf{x})\\
s.t. \textbf{p}\cdot \textbf{x} = m
$$
Where boldface is used to represent vectors.

In your problem, you would use your utility function, and $\textbf{p}\cdot \textbf{x} = p\_1x\_1 + p\_2x\_2 + p\_3x\_3$

The other answer provides a link that I believe shows how to derive the formulas. I just wanted to explicitly state that marshallian demands are solutions to the utility maximization problem. If you know how to solve a maximization problem (using FOC and/or lagrangians), you can derive the marshallian demands.


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### Answer by username_4 on 2016-01-05. Score: -2

The Marshallian demand functions are basically partial derivatives of the Cobb-Douglas utility function. You should consider that you want to maximize spending first, then derive the functions to get the optimal prices, demand and a equilibrium with both - not sure if I used the correct words.

Nevertheless, here's an example with 2 products, [here](https://economics.stackexchange.com/a/5030/6832), from SE.

> **Comments:**
> **username_5**: Marshallian demand functions are not partial derivatives of the utility function. (Not even in the Cobb-Douglas case.) Additional steps are needed.  
> **username_4**: Poor misuse of words. But the example I gave there is useful to understand the steps. There's more exemplifications in that question and they're related because the problem is fairly similar.  

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