Derivation of aggregate demand function for Monopolistic Competition (based on Combes et. al, 2008)

Question

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).

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?

Answer 1

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 $\mu E = \mu yL = PM \Leftrightarrow M = \frac{\mu yL}{P}$.