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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?

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    $\begingroup$ 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. $\endgroup$ – Grada Gukovic Aug 27 at 13:26
  • $\begingroup$ 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: $PM \leq y$. $\endgroup$ – Artem Kochnev Aug 28 at 7:00
  • $\begingroup$ 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. $\endgroup$ – Grada Gukovic Aug 28 at 16:53
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  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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