How to create a transition matrix between two columns?

Question

I have a dataframe in the following format:

    customer_id  prev_tier  tier
0             4         -1     1
1             7         -1     0
2            16         -1     2
3            22          0    -1
4            23          0    -1
5            28          0     1
6            36          0     1
7            38          0     2
8            40          0     2
9            81          1    -1
10           84          1    -1
11           95          1    -1
12           96          1     0
13           97          1     0
14           98          1     2
15           99          1     0
16          108          1     2
17          115          1     2
18          139          2    -1
19          165          2    -1
20          169          2    -1
21          182          2    -1
22          183          2     0
23          184          2     0
24          185          2     0
25          186          2     0
26          199          2     1
27          206          2     1
28          211          2     1
29          225          2     1

I want to create a transition matrix which shows the number of transitions to each tier from each prev_tier, such as this:

     tier    -1    0    1    2
prev_tier
       -1     0    1    1    1
        0     2    0    2    2
        1     3    3    0    3
        2     4    4    4    0

What is an elegant way to achieve this in Pandas?