<p>I am working on a problem 14.2.5 from EMEA by Sydsaeter, Hammond and Strom.</p>
<blockquote>
<p>Consider the consumer demand problem:
<span class="math-container">$$ \max_{x,y} U(x,y) = \alpha \ln(x-a) + \beta \ln(y-b) \text{ s.t. } px+qy=m \tag{*} $$</span>
where <span class="math-container">$\alpha, \beta, a, b, p, q$</span> and <span class="math-container">$m$</span> are positive constants with <span class="math-container">$\alpha + \beta =1 $</span>, and moreover, with <span class="math-container">$m&gt;ap+bq$</span>. Show that if <span class="math-container">$x^*, y^*$</span> solve problem (*),  then expenditure on the two goods is given by the two linear functions:
<span class="math-container">$$ px^*= \alpha m+ pa -\alpha(pa+qb), qy^* = \beta m+ qb - \beta (pa + qb)$$</span></p>
</blockquote>
<p>My attempt:</p>
<p>I set up Lagrangian:</p>
<p><span class="math-container">$$ L= \alpha \ln(x-a) + \beta \ln(y-b) - \lambda (px+qy-m) $$</span></p>
<p>and get FOCs:</p>
<p><span class="math-container">$$ \frac{\alpha}{x-1} = p \lambda$$</span></p>
<p><span class="math-container">$$ \frac{\beta}{y-1} = q \lambda$$</span></p>
<p><span class="math-container">$$ px+qy = m$$</span></p>
<p>I tried to solve the system above and then my plan was to multiply <span class="math-container">$x^*,y^*$</span> with <span class="math-container">$p$</span> and <span class="math-container">$q$</span> but that gives me different consumption function. Book has a hint that I should solve for <span class="math-container">$px$</span> and <span class="math-container">$qy$</span>:</p>
<p><span class="math-container">$$px = pa + \frac{\alpha}{\lambda}$$</span></p>
<p><span class="math-container">$$qy = qb + \frac{\beta}{\lambda}$$</span></p>
<p>Then the book hints I should use budget constraint to eliminate lambda so I plug it there:|</p>
<p><span class="math-container">$$ pa + \frac{\alpha}{\lambda} + qb + \frac{\beta}{\lambda}  = m $$</span></p>
<p><span class="math-container">$$ \lambda = \frac{\alpha + \beta}{m- pa - qb} $$</span></p>
<p>When I plug the solution for lambda back to equation for <span class="math-container">$px$</span> I just get:</p>
<p><span class="math-container">$$ px^* = pa + \frac{\alpha m - \alpha pa - \alpha bq}{\alpha +\beta} $$</span></p>
<p>And this is again not the consumption expenditure function the textbook says I should get. I don't get why I can't end with the same consumption expenditure as the book. Does anyone know how to solve this?</p>

CC BY-SA 4.0
2021-05-04T18:21:13.387Z
As $\alpha + \beta = 1$, I think these are indeed the same solutions.
tdm
CC BY-SA 4.0
2021-05-04T18:21:59.663Z
@tdm I don't get it, can you please explain why?
WilliamT
CC BY-SA 4.0
2021-05-04T18:25:13.637Z
If you put $\alpha + \beta = 1$ in your solution, you get $px = pa + \alpha m - \alpha(pa + bq)$ as in the assignment. There is a typo though, because $1 b$ should be $q b$.
tdm
CC BY-SA 4.0
2021-05-04T19:00:39.550Z
@tdm ooooo I see now, I am so dumb... if you want you can post it as an answer I will accept it. Yes, that was just typo it was qb
WilliamT
CC BY-SA 4.0
2021-05-05T05:06:45.727Z
No problem. By the way, the utility function in you example is called a Stone-Geary utility function [wiki](https://en.wikipedia.org/wiki/Stone%E2%80%93Geary_utility_function). It is very similar to a Cobb-Douglas utilities function, with the addition of subsistence levels $a$ and $b$ that provide the minimal levels of both goods that need to be consumed.
tdm
<mathematical-economics><consumer-theory><self-study><lagrangian>
How to derive consumer expenditures in EMEA 14.2.5
WilliamT
# How to derive consumer expenditures in EMEA 14.2.5

(Asked by: username_0 on 2021-05-04)

I am working on a problem 14.2.5 from EMEA by Sydsaeter, Hammond and Strom.

> Consider the consumer demand problem:
> $$ \max\_{x,y} U(x,y) = \alpha \ln(x-a) + \beta \ln(y-b) \text{ s.t. } px+qy=m \tag{\*} $$
> where $\alpha, \beta, a, b, p, q$ and $m$ are positive constants with $\alpha + \beta =1 $, and moreover, with $m>ap+bq$. Show that if $x^\*, y^\*$ solve problem (\*), then expenditure on the two goods is given by the two linear functions:
> $$ px^\*= \alpha m+ pa -\alpha(pa+qb), qy^\* = \beta m+ qb - \beta (pa + qb)$$

My attempt:

I set up Lagrangian:

$$ L= \alpha \ln(x-a) + \beta \ln(y-b) - \lambda (px+qy-m) $$

and get FOCs:

$$ \frac{\alpha}{x-1} = p \lambda$$

$$ \frac{\beta}{y-1} = q \lambda$$

$$ px+qy = m$$

I tried to solve the system above and then my plan was to multiply $x^\*,y^\*$ with $p$ and $q$ but that gives me different consumption function. Book has a hint that I should solve for $px$ and $qy$:

$$px = pa + \frac{\alpha}{\lambda}$$

$$qy = qb + \frac{\beta}{\lambda}$$

Then the book hints I should use budget constraint to eliminate lambda so I plug it there:|

$$ pa + \frac{\alpha}{\lambda} + qb + \frac{\beta}{\lambda} = m $$

$$ \lambda = \frac{\alpha + \beta}{m- pa - qb} $$

When I plug the solution for lambda back to equation for $px$ I just get:

$$ px^\* = pa + \frac{\alpha m - \alpha pa - \alpha bq}{\alpha +\beta} $$

And this is again not the consumption expenditure function the textbook says I should get. I don't get why I can't end with the same consumption expenditure as the book. Does anyone know how to solve this?

**Tags:** mathematical-economics, consumer-theory, self-study, lagrangian

> **Comments:**
> **username_1**: As $\alpha + \beta = 1$, I think these are indeed the same solutions.  
> **username_0**: @username_1 I don't get it, can you please explain why?  
> **username_1**: If you put $\alpha + \beta = 1$ in your solution, you get $px = pa + \alpha m - \alpha(pa + bq)$ as in the assignment. There is a typo though, because $1 b$ should be $q b$.  
> **username_0**: @username_1 ooooo I see now, I am so dumb... if you want you can post it as an answer I will accept it. Yes, that was just typo it was qb  
> **username_1**: No problem. By the way, the utility function in you example is called a Stone-Geary utility function [wiki](https://en.wikipedia.org/wiki/Stone%E2%80%93Geary_utility_function). It is very similar to a Cobb-Douglas utilities function, with the addition of subsistence levels $a$ and $b$ that provide the minimal levels of both goods that need to be consumed.  

---

## 0 answers

