Gravity Equation Interpretation I have a question about the gravity equation. In the Feenstra textbook, on page 144, it is stated that 


  In the monopolistic competition model [...] the countries are completely specialized in different product varieties. Trade in these product varieties is referred to as intraindustry trade...


On page 145, it is stated that 


  Then it follows that a good produced in any country is sent to all other countries in proportion to the purchasing country's GDP. 


To formalize this, consider a multicountry framework where $i,j=1...C$
  denotes countries and $k=1,...N$
  denotes products. Let $y_{k}^{i}$
  denote country $i's$
  production of good $k.$
  Since prices are the same across all countries, we normalize them to unity, so $y_{i}^{k}$
 actually measures the value of production. The total GDP in each country is measured by $$Y^{i}=\sum_{k=1}^{N}y_{k}^{i}$$
  and world GDP is $$Y^{W}=\sum_{i=1}^{C}y^{i}.$$

Here is where I am confused. I was under the impression that each country produces a single variety of the differentiated good. In that case, the total GDP in each country should just be $$Y^{i}=y_{k}^{i}$$
  where $k$
  is specific to $i$.
  In other words, I understand the statement as each country producing just one variety of the same good. I believe this is an incorrect interpretation for two reasons: 


If that were the case, the number of goods should actually be $C$,
not $N$.
(It could be that $C=N$
but the author does not specify this)
The GDP
by the product would be $y_{k}^{i}$
but by the expenditure approach would be $Y^{i}=\sum_{k=1}^{N}y_{k}^{i}$. 


Confronted with the likely possibility that my interpretation is fallacious, I don't know what the alternative interpretation is. What else could the statement “countries are completely specialized in different product” varieties mean?
My understanding is that the gravity equation can be derived from different settings.

You can assume that each country

1/ produces only one product. This is knwon as the "Armington assumption" as in Armington (1969) and allows a simple derivation of the gravity equation as in Anderson and Van Wincoop (2003).

2/ is specialized in different products (or varieties of a product, given that each variety of a good counts as a distinct product). This is the case worked out by Feenstra, where products are differentiated by their country of origin. So, each country produces many distinct products.

Then the question is to know if the number of products is fixed, as in Armington (1969), or endogeneous and varies due to free entry, as in the monopolistic competition models.

References 


ANDERSON, J. E., AND E. VAN WINCOOP (2003): “Gravity with Gravitas: A Solution to the Border Puzzle,” The American Economic Review, 93(1), 170–192.
ARMINGTON, P. S. (1969): “A Theory of Demand for Products Distinguished by Place of Production,” International Monetary Fund Staff Papers, 16, 159–178.