<ol>
<li><p>As we have discussed in the comments, the utility function used in the book is Cobb-Douglas: <span class="math-container">$U = CM^{\mu}A^{1 - \mu}$</span>. The well known fact mentioned in equation (3.3) on page 57 - the one that you highlighted - is that the Cobb-Douglas utility function is homothetic - the share of total income spent on any of the goods is <strong><em>constant</em></strong> over all possible incomes. More specifically here a share of <span class="math-container">$\mu$</span> of the expenditure is always spent on manufacturing goods and a share of <span class="math-container">$1-\mu$</span> is spent on the agriculture good. <span class="math-container">$\sigma$</span> and <span class="math-container">$p_i$</span> only determine how the <span class="math-container">$\mu y$</span> units of income that the representative consumer spends on manufacturing output is allocated among the distinct manufacturing goods. </p></li>
<li><p>Although the output goods are different most such models assume that both the jobs and the employees are identical and there is a perfectly competitive domestic labor market. Hence the identical wage rate.</p></li>
</ol>

<p>Note: <span class="math-container">$y$</span> is personal income and <span class="math-container">$L$</span> is the number of employees, thus the total income of the economy is <span class="math-container">$yL$</span>. (E.g. the last sentence on page 56). </p>

<p>Thus the total expenditure <span class="math-container">$E = yL$</span> - the total income, as the model is static. The expenditure on manufactured goods is <span class="math-container">$\mu E = \mu yL = PM \Leftrightarrow M = \frac{\mu yL}{P}$</span>.</p>

CC BY-SA 4.0
2019-08-30T19:06:43.230Z
2019-08-30T21:46:52.233Z
2019-08-30T21:46:52.233Z
Grada Gukovic
<p>A specialized question for those, who excel in monopolistic competition and modern trade theories. I am interested in a derivation of an aggregate demand function for a model of monopolistic competition presented in the <a href="https://press.princeton.edu/titles/8765.html" rel="nofollow noreferrer">(text-)book by Combes, Mayer and Thisse (2008)</a>.</p>

<p><strong>Setup of consumer problem</strong></p>

<p><em>Utility of an individual consumer</em>: <span class="math-container">$$U_i = CM^{\mu}$$</span>
Where <span class="math-container">$M$</span> stands for a composite manufacturing good defined a a sum of "manufacturing" goods consumed by a representative agent: <span class="math-container">$$M \equiv \left(\Sigma_i^n q_i^{\frac{\sigma - 1}{\sigma}}\right)^\frac{\sigma}{\sigma-1} $$</span>
<em>Budget constraint</em>: 
<span class="math-container">$$PM \leq y$$</span>
Where <span class="math-container">$P$</span> is a composite price index for a composite manufacturing good defined as follows: <span class="math-container">$$P \equiv  \left(\Sigma_i^n p_i^{-(\sigma - 1)}\right)^{-\frac{1}{\sigma-1}} $$</span>
Here comes the punchline (p. 57). Without any additional derivations, authors write the following:</p>

<blockquote>
  <p>In this case, it is well-known that the aggregate demand function take the form: <span class="math-container">$$ M = \frac{\mu L y}{P} $$</span> My comment: <span class="math-container">$L$</span> stands for a total labor supply in a country. </p>
</blockquote>

<p>Two questions:</p>

<ol>
<li><p>This well-known result is totally unknown to me. Could someone kindly show its derivation (or maybe hint to a textbook) in detail?</p></li>
<li><p>Throughout the chapter, authors treat <span class="math-container">$y$</span> as an exogenous value albeit the model they present is in a general equilibrium setting. This implies that <span class="math-container">$y$</span> should be a sum of wages in the economy that feature in variable costs of firms: <span class="math-container">$C(q_i) = f + wq_i$</span>. I suppose that they assume a perfect competition on a labour market among the identical workers and therefore the wage rates could be treated as fixed. Is my intuition true?</p></li>
</ol>

CC BY-SA 4.0
2019-08-27T13:26:41.983Z
Artem, Ive skimmed over the part of the chapter that you're talking about. I think  your attempt to remove agricultural production ***A*** to simplify the question is rather unlucky. The utility function in the book is $CM^{\mu}A^{1-\mu}$ thus to remove ***A*** from the utility function you assume that it is a ***positive*** constant. On the other hand in the budget constraint you assume that ***A*** is zero to remove it from there. I dont think that you can do any of these but whatever  the true value of ***A*** is it must be the same in both equations.
Grada Gukovic
CC BY-SA 4.0
2019-08-28T07:00:04.780Z
You are right, I have simplified the question but I don't think it introduces any dramatic changes. Assume any positive integer for $A$ and assume the initial budget constraint is $PM + p_A A \leq I$, where $I$ is an exogenous income. Rearrange such that $PM$ is on the left-hand side alone: $PM \leq I - p_A A$. Now just define the Netto income after deducting agricultural goods as follows: $y \equiv I - p_A A$. Substitute $y$ back in the equation and you obtain the budget constraint the way I have defined it: $PM \leq y$.
Artem Kochnev
CC BY-SA 4.0
2019-08-28T16:53:48.733Z
Taken together $PM\leq y$ and $M = \frac{\mu Ly}{P}$ imply $\mu L=1$ under the usual assumption of monotonic preferences. Im not sure that this is true. Further $\mu'$s job in the Cobb-Douglas utility function in the original model is to determine the marginal rate of substitution between manufacturing and agricultural goods in the optimal choice bundle. By removing **A** you remove this entire consideration from the simplyfied model. And  $M= \frac{\mu Ly}{P}$ is obvioulsy a relust of the maximization in the initial model that takes the substitution between **A** and **M** into account.
Grada Gukovic
<microeconomics><mathematical-economics><international-trade><international-economics><monopolistic-competition>
Derivation of aggregate demand function for Monopolistic Competition (based on Combes et. al, 2008)
Artem Kochnev
# Derivation of aggregate demand function for Monopolistic Competition (based on Combes et. al, 2008)

(Asked by: username_0 on 2019-08-23)

A specialized question for those, who excel in monopolistic competition and modern trade theories. I am interested in a derivation of an aggregate demand function for a model of monopolistic competition presented in the [(text-)book by Combes, Mayer and Thisse (2008)](https://press.princeton.edu/titles/8765.html).

**Setup of consumer problem**

*Utility of an individual consumer*: $$U\_i = CM^{\mu}$$
Where $M$ stands for a composite manufacturing good defined a a sum of "manufacturing" goods consumed by a representative agent: $$M \equiv \left(\Sigma\_i^n q\_i^{\frac{\sigma - 1}{\sigma}}\right)^\frac{\sigma}{\sigma-1} $$
*Budget constraint*:
$$PM \leq y$$
Where $P$ is a composite price index for a composite manufacturing good defined as follows: $$P \equiv \left(\Sigma\_i^n p\_i^{-(\sigma - 1)}\right)^{-\frac{1}{\sigma-1}} $$
Here comes the punchline (p. 57). Without any additional derivations, authors write the following:

> In this case, it is well-known that the aggregate demand function take the form: $$ M = \frac{\mu L y}{P} $$ My comment: $L$ stands for a total labor supply in a country.

Two questions:

1. This well-known result is totally unknown to me. Could someone kindly show its derivation (or maybe hint to a textbook) in detail?
2. Throughout the chapter, authors treat $y$ as an exogenous value albeit the model they present is in a general equilibrium setting. This implies that $y$ should be a sum of wages in the economy that feature in variable costs of firms: $C(q\_i) = f + wq\_i$. I suppose that they assume a perfect competition on a labour market among the identical workers and therefore the wage rates could be treated as fixed. Is my intuition true?

**Tags:** microeconomics, mathematical-economics, international-trade, international-economics, monopolistic-competition
> **Comments:**
> **username_1**: Artem, Ive skimmed over the part of the chapter that you're talking about. I think  your attempt to remove agricultural production ***A*** to simplify the question is rather unlucky. The utility function in the book is $CM^{\mu}A^{1-\mu}$ thus to remove ***A*** from the utility function you assume that it is a ***positive*** constant. On the other hand in the budget constraint you assume that ***A*** is zero to remove it from there. I dont think that you can do any of these but whatever  the true value of ***A*** is it must be the same in both equations.  
> **username_0**: You are right, I have simplified the question but I don't think it introduces any dramatic changes. Assume any positive integer for $A$ and assume the initial budget constraint is $PM + p_A A \leq I$, where $I$ is an exogenous income. Rearrange such that $PM$ is on the left-hand side alone: $PM \leq I - p_A A$. Now just define the Netto income after deducting agricultural goods as follows: $y \equiv I - p_A A$. Substitute $y$ back in the equation and you obtain the budget constraint the way I have defined it: $PM \leq y$.  
> **username_1**: Taken together $PM\leq y$ and $M = \frac{\mu Ly}{P}$ imply $\mu L=1$ under the usual assumption of monotonic preferences. Im not sure that this is true. Further $\mu'$s job in the Cobb-Douglas utility function in the original model is to determine the marginal rate of substitution between manufacturing and agricultural goods in the optimal choice bundle. By removing **A** you remove this entire consideration from the simplyfied model. And  $M= \frac{\mu Ly}{P}$ is obvioulsy a relust of the maximization in the initial model that takes the substitution between **A** and **M** into account.  

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## 1 answers

### Answer by username_1 on 2019-08-30. Score: 3 | Accepted answer

1. As we have discussed in the comments, the utility function used in the book is Cobb-Douglas: $U = CM^{\mu}A^{1 - \mu}$. The well known fact mentioned in equation (3.3) on page 57 - the one that you highlighted - is that the Cobb-Douglas utility function is homothetic - the share of total income spent on any of the goods is ***constant*** over all possible incomes. More specifically here a share of $\mu$ of the expenditure is always spent on manufacturing goods and a share of $1-\mu$ is spent on the agriculture good. $\sigma$ and $p\_i$ only determine how the $\mu y$ units of income that the representative consumer spends on manufacturing output is allocated among the distinct manufacturing goods.
2. Although the output goods are different most such models assume that both the jobs and the employees are identical and there is a perfectly competitive domestic labor market. Hence the identical wage rate.

Note: $y$ is personal income and $L$ is the number of employees, thus the total income of the economy is $yL$. (E.g. the last sentence on page 56).

Thus the total expenditure $E = yL$ - the total income, as the model is static. The expenditure on manufactured goods is $\mu E = \mu yL = PM \Leftrightarrow M = \frac{\mu yL}{P}$.


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