Vertically flatten multiple columns and merge based on multiple columns in Python

Question

I have a dataframe semble this:

   id city  2018  2019
0   1   bj   2.0   3.0
1   2   sh   3.0   3.5
2   3   gz   2.5   4.0

I need to vertically flatten column 2018 and 2019 to year and price, and then merge based on id and city.

The output I want is like this:

   id city  year  price
0   1   bj  2018    2.0
1   1   bj  2019    3.0
2   2   sh  2018    3.0
3   2   sh  2019    3.5
4   3   gz  2018    2.5
5   3   gz  2019    4.0

How can I do that? Thanks.