In our discussion of the Melitz (2003) model, we assumed that utility U of a consumer was given by

U=[c()(1)/d]/(1) (1)

Note that we have introduced the CES utility function defined over a continuous set of varieties when discussing the Melitz (2003) model, but nothing of our derivations is specific to the Melitz model. We could use the same utility function in a Krugman (1980) model and nothing would change

Question: Show that the price index for a continuous CES utility function is given by

P=[p()1d]1/(1) (2)

Hints: To show that Equation (2) is the price index given that consumers have CES utility, write down the budget constraint of the consumer, i.e., income equals expenditure. Expenditure on variety is p()c(), so to get total expenditure across all varieties we have to integrate over this expression. Taking this into account, the budget constraint can be written as

I=p()c()d. (3)

Demand for variety is given by

c()=(Pp())P (4)

Plug Equation (4) into the budget constraint and solve for P

b.Show that this price index is the ideal price index, i.e., P 1 = U when using the wage as the numeraire. Hint: An ideal price index P transforms individuals' income I into their utility level U, i.e.,

U=I/P

In the Melitz (and Krugman) model, a consumer's income is equal to their wage, i.e. I = w. Hence, in the models we consider in this course, utility is simply another term for "real wage". We can further simplify I = w = 1 as we use the wage as the numeraire. Solve Equation (4) for p() to get the indirect demand function. Plug your result into P¹ using Equation (2) and show that your result is identical to U as given in Equation(1).