<p>I do not understand how Mas-Collel, Whinston, and Green derive the Hicksian demand functions in Example 3.E.1 in their textbook. </p>

<p>Allow me to give further background regarding the problem: The authors give us a Cobb-Douglas utility function for <span class="math-container">$L = 2$</span>, where <span class="math-container">$L$</span> refers to the number of commodities. The function is <span class="math-container">$u(x_1, x_2) = kx_1^{α} \cdot x_2^{1-α}$</span>, where <span class="math-container">$k$</span> is some positive constant and alpha is some number between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. They note that the function is strictly increasing on <span class="math-container">$x_i &gt; 0$</span>, where <span class="math-container">$i=1,2$</span>. Thereafter, they apply an increasing transformation of <span class="math-container">$u(x_1,x_2)$</span> - namely, the natural logarithm. They note that since natural logarithm is strictly concave, it is strictly quasi-concave, which tells us that the Walrasian Demand correspondence will in fact be a function. Moreover, since the utility function is increasing, we know that the consumer has monotone preferences, thus her preferences satisfy local non-satiation, thus the budget constraint holds with equality; that is: <span class="math-container">$p_1x_1 + p_2x_2 = w$</span> (wealth). </p>

<p>However, when the authors go on to derive the Hicksian demand functions, I am lost. I tried setting up the first-order conditions for minimization, by setting up the Lagrangian function; that is, take the objective function (the utility function) plus Lambda times (u minus (the utility function)). I set the partial derivative of the Lagrangian with respect to the choice variables and set them greater than or equal to zero. Next, there is the partial derivative of the Lagrangian function with respect to lambda, which I set less than or equal to zero. Lastly, there are the complementary slackness conditions.    </p>

<p>References: Mas-Colell, Andreu, et al. Microeconomic Theory. Oxford University Press, 1995.</p>

<p>This is not a homework problem, I just wish to have elucidation on this example. </p>

CC BY-SA 3.0
2018-01-08T02:02:45.820Z
Welcome to EconSE. Not everyone has a copy of MWG on hand, so you should include all the relevant details from example 3.E.1 in your question.
Theoretical Economist
CC BY-SA 3.0
2018-01-08T15:53:28.443Z
Please don't link to pirated material on this site. If you want to quote from a source then you can do so in accordance with normal academic citation conventions.
Ubiquitous
CC BY-SA 3.0
2018-01-08T20:21:10.463Z
Please write down explicitly what you've tried when trying to solve the minimisation problem, and use [MathJax](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) when doing so.
Theoretical Economist
CC BY-SA 3.0
2018-01-10T21:47:35.210Z
Somebody help - please - anybody
Casablancas
CC BY-SA 4.0
2018-11-12T10:33:57.823Z
Try to use MathJax and give the book or that page reference,since everybody doesn't have the book.
Henam
<microeconomics><consumer-theory>
MWG Example 3.E.1
Casablancas
# MWG Example 3.E.1

(Asked by: username_0 on 2018-01-08)

I do not understand how Mas-Collel, Whinston, and Green derive the Hicksian demand functions in Example 3.E.1 in their textbook.

Allow me to give further background regarding the problem: The authors give us a Cobb-Douglas utility function for $L = 2$, where $L$ refers to the number of commodities. The function is $u(x\_1, x\_2) = kx\_1^{α} \cdot x\_2^{1-α}$, where $k$ is some positive constant and alpha is some number between $0$ and $1$. They note that the function is strictly increasing on $x\_i > 0$, where $i=1,2$. Thereafter, they apply an increasing transformation of $u(x\_1,x\_2)$ - namely, the natural logarithm. They note that since natural logarithm is strictly concave, it is strictly quasi-concave, which tells us that the Walrasian Demand correspondence will in fact be a function. Moreover, since the utility function is increasing, we know that the consumer has monotone preferences, thus her preferences satisfy local non-satiation, thus the budget constraint holds with equality; that is: $p\_1x\_1 + p\_2x\_2 = w$ (wealth).

However, when the authors go on to derive the Hicksian demand functions, I am lost. I tried setting up the first-order conditions for minimization, by setting up the Lagrangian function; that is, take the objective function (the utility function) plus Lambda times (u minus (the utility function)). I set the partial derivative of the Lagrangian with respect to the choice variables and set them greater than or equal to zero. Next, there is the partial derivative of the Lagrangian function with respect to lambda, which I set less than or equal to zero. Lastly, there are the complementary slackness conditions.

References: Mas-Colell, Andreu, et al. Microeconomic Theory. Oxford University Press, 1995.

This is not a homework problem, I just wish to have elucidation on this example.

**Tags:** microeconomics, consumer-theory

> **Comments:**
> **username_1**: Welcome to EconSE. Not everyone has a copy of MWG on hand, so you should include all the relevant details from example 3.E.1 in your question.  
> **username_2**: Please don't link to pirated material on this site. If you want to quote from a source then you can do so in accordance with normal academic citation conventions.  
> **username_1**: Please write down explicitly what you've tried when trying to solve the minimisation problem, and use [MathJax](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) when doing so.  
> **username_0**: Somebody help - please - anybody  
> **username_3**: Try to use MathJax and give the book or that page reference,since everybody doesn't have the book.  

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## 0 answers

