Row-wise operation in Pandas data frame

You can first reshape by set_index with unstack and then divide by div:

print (df)
  country  year      indicatorName  value
0     USA  1970  Agricultural Land     10
1     USA  1970    Crop production      2
2     USA  2000  Agricultural Land     10
3     USA  2000    Crop production      3
4  Mexico  1970  Agricultural Land     10
5  Mexico  1970    Crop production      5
6  Mexico  2000  Agricultural Land     10
7  Mexico  2000    Crop production      4  

df = (df.set_index(['country','year','indicatorName'])['value']
       .unstack()
       .assign(crop_prod_density=lambda x: x['Crop production'].div(x['Agricultural Land'])))
print (df)
indicatorName  Agricultural Land  Crop production  crop_prod_density
country year                                                        
Mexico  1970                  10                5                0.5
        2000                  10                4                0.4
USA     1970                  10                2                0.2
        2000                  10                3                0.3

Then reshape back by stack:

df1 = df.stack().reset_index(name='value')
print (df1)
   country  year      indicatorName  value
0   Mexico  1970  Agricultural Land   10.0
1   Mexico  1970    Crop production    5.0
2   Mexico  1970  crop_prod_density    0.5
3   Mexico  2000  Agricultural Land   10.0
4   Mexico  2000    Crop production    4.0
5   Mexico  2000  crop_prod_density    0.4
6      USA  1970  Agricultural Land   10.0
7      USA  1970    Crop production    2.0
8      USA  1970  crop_prod_density    0.2
9      USA  2000  Agricultural Land   10.0
10     USA  2000    Crop production    3.0
11     USA  2000  crop_prod_density    0.3

For new column to original append to index new column, but last is necessary change order of columns by reindex:

df2 =(df.set_index(['crop_prod_density'], append=True)
        .stack()
        .reset_index(name='value')
        .reindex(columns=['country','year','indicatorName','value','crop_prod_density']))
print (df2)
  country  year      indicatorName  value  crop_prod_density
0  Mexico  1970  Agricultural Land     10                0.5
1  Mexico  1970    Crop production      5                0.5
2  Mexico  2000  Agricultural Land     10                0.4
3  Mexico  2000    Crop production      4                0.4
4     USA  1970  Agricultural Land     10                0.2
5     USA  1970    Crop production      2                0.2
6     USA  2000  Agricultural Land     10                0.3
7     USA  2000    Crop production      3                0.3

And last remove unnecessary columns and create columns from MultiIndex:

df3 = (df.drop(['Crop production','Agricultural Land'], axis=1)
        .reset_index()
        .rename_axis(None, 1))
print (df3)
  country  year  crop_prod_density
0  Mexico  1970                0.5
1  Mexico  2000                0.4
2     USA  1970                0.2
3     USA  2000                0.3