While researching for my current diploma thesis on “new new” trade theory I realized that fully understanding (or rather: knowing by heart) the Dixit/Stiglitz (1977) model is a necessary precondition to be able to read most of the relevant papers. Unfortunately, the original Dixit/Stiglitz is not written for undergraduate students (intermediate steps are mostly omitted), and the obsolete Chamberlinian terminology does not exactly promote understanding (obsolete by Austrian university standards, that is).

Consequently I’ve compiled this step-by-step walkthrough of the first and most widely used model (the CES case) covered in Dixit/Stiglitz (1977). This should be sufficient for most of the “new” and “new new” trade theory models. You can download it here.

Please let me know if you find any mistakes.

**Update (Dec. 21, 2009):** Minor corrections.

**References:**

*Dixit, Avinash K. and Stiglitz, Joseph E. (1977): “Monopolistic Competition and Optimum Product Diversity”. In: American Economic Review 67(3), 297–308.*

Wow! Just what I was looking for. Many thanks.

This article is quite useful in fully understanding dixit-stiglitz type model.

Many thanks.

Dude, Awesome

I find your article more interesting. Thank you for breaking down the crucial steps of this ‘un-penetrable soup’ of Dixit-Stiglitz model.

thanks! This article really helps me understanding Dixit-Stiglitz.

However, you probably made two errors:

First, in page 6, in developing V(n), the final result is V(n)=n^(1/(rho-1)).(nx)

I tried to redo the equation and I have V(n)=n^((1-rho)/rho).(nx)

Consequently (in the following paragraph), “the term n^(1/(rho-1))>1” must be replaced by “the term n^((1-rho)/rho)>1”.

Second, in the page 3 equation (5):

X0=(1-s(q))I

is probably a unfinished equation. We need to add /q so we have

X0=(1-s(q))I/q

cheers,

Hey,

regarding the first equation, both are right, because notice that

there are no parenthesis, therefore: \[ n^{1/\rho-1} = n^{\frac{1-\rho}{\rho}}\]

Second, the expression for \(x_0\) is correct because we have normalized the price of \(x_0\) to 1. Hence it is clear that the expenditure on \(x_0\), \((1-s(q))I\), must equal the quantity of \(x_0\) consumed.

thanks! Yes, you are right.

I was a bit confused by the equations since it has no parenthesis.

Thanks a lot, this saved me quite a bit of work!

Hi, Your guide to Dixit& Stiglitz is totally awesome. I think I will do something similar for other papers, starting with Grossman and Helpman 1994. I think I found two typos, though. On page three, I think it should be a +qy in the F.O.C. for the lagrange. On page 5, when you compare the ratio of prices to the ratio of quantities, I think you must invert one of the two. I am sorry if I am mistaken. When I have my webpage uploaded, I will comment it here… I just learned how to do such things. Many thanks!

By the way, with internet, there is no reason why articles should not have a longer, but detailed or complete, presentation of math, in an online version. Informally, this is already happening: some people refer to their own homepages for more detailed math results. But I guess it should be allocated to journals, just like they have to provide data verses with datasets.

Pedro, thanks for your comment! You were right on the first one, there was a sign mistake. The ratio was correct, though, because the exponent was \(\frac{1}{\rho-1}\). I switched around the price ratio and changed to exponent to \(\frac{1}{1-\rho}\), I guess this version is more common.

I am happy to help! Yep, I had used the inverted exponents… But I agree that inverting both exponents and the fraction helps. People understand faster when positive exponents are used, and it also helps to have x_1 and p_1 either multiplying each other or in different signs of an equation AND in different positions of a fraction. But of course published papers do not care that much about being didactic…

Again, great job!

I found this pretty useful, too! Thanks much~

I greatly appreciated your generosity to share this useful document. Many thanks.

Thank you very much for this document ! More than useful ! 🙂

The first equation on page 3, in text, has a minus in front of qy, instead of a plus.

How would it be possible to find the optimal number of firms?

Great paper btw. Congrats!

This is the year of 2017. Thank you very much for your sharing the note. It helps me understanding Dixit-Stiglitz aggregators in much of the DSGE literature.