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

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?

• 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 Aug 27 '19 at 13:26
• 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 Aug 28 '19 at 7:00
• 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 Aug 28 '19 at 16:53

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