1 Deriving the ideal price index for a CES utility function In our discussion of the Melitz (2003) model, we assumed that utility U of a consumer was given by a/(0-1) U = c ( w ) 10 - 1)/0 dw (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. (a) Show that the price index for a continuous CES utility function is given by P = / P ( w ) " - aw " ( -0 ) . (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. Ex- penditure on variety w is p(w)c(w), so to get total expenditure across all varieties wehave to integrate over this expression. Taking this into account, the budget constraint can be written as I = [ p(w)c(w)dw. (3) Demand for variety w is given by c ( w ) = p(w) W P P (4) Plug Equation (4) into the budget constraint and solve for P.(h) Show that this price index is the ideal price index, i.e., I3"1 = U when using the wage as the numrajre. Hint: An ideal price index P transforms individuals' income I into their utility level U, 1.e., U = - {9) In the Melitz (and ngman} model, a consumer's income is equal to their wage, i.e. I = 21:. 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(m} to get the indirect demand function. Plug your result into P'l using Equation (2) and show that your result is identical to U as given in Equation (1)