Creating a count inversion array of an unsorted array of unique integers
Thanks for an interesting, well-posed question.  Before we get to issues of performance, let me make some other suggestions for your code.


The way you have broken down the code into three functions is reasonable and logical.  Nice!
The functions you wrote don't have docstrings, so it is hard to know how to use them.  I couldn't get your count_inversion() function to run initially, for example, because I didn't know what N was supposed to be.  Adding docstrings would make this clear.  
For the specific case of the N parameter in count_inversion, why do you need it?  When I used the function I did it like this:

arr = [2, 3, 1, 4]
arr2 = [2, 1, 4, 3]
arr3 = [20]
arr4 = [1, 2, 3, 4, 5, 6]
arr5 = [87, 78, 16, 94]

arrs_to_test = [arr, arr2, arr3, arr4, arr5]

[count_inversion(test, len(test)) for test in arrs_to_test]


That to me suggests that you don't need N as a parameter, instead just do something like:

def count_inversion(arr):
    # docstring goes here
    N = len(arr)
    # &lt;&lt;rest of code&gt;&gt;

I usually dislike when people are sticklers for PEP8 variable-naming conventions in mathematically oriented code, but I think your naming could use some work.  For example I had to read dct = {i:0 for i in arr} multiple times to understand that i was not an index but the data.  So dct = {el:0 for el in arr} would have been more natural to me.  Plus dct isn't the best name either.  If I understand the code correctly, perhaps result would be better?
Possible bug: Related to the above, do you really want to create a dictionary keyed on the elements of arr?  Doing so means that the behavior when integers are repeated in the input is probably not what you want:

&gt;&gt;&gt; repeated = [3, 2, 1, 0, 3, 4, 5]
&gt;&gt;&gt; count_inversion(repeated, len(repeated))
 '3 2 1 0 3 0 0'


Is the fifth element in this array really supposed to be "3"?
Right now the code functions because of the mutability of dct.  I'm not 100% sure it could be done cleanly, but if possible I'd suggest re-writing merge_sort and merge to return dct instead of returing None.  That way the initialization of dct could happen in those functions too, which feels more natural to me.
You are using your (mysteriously named) dct variable in a way that looks like a Counter, so you might consider using this built-in datatype of Python.
Now, to issues of performance.  In Python, line-profiler is an easy to use package for assessing the performance of your code.  I use this package in a Jupyter notebook like this:

from random import randint
%load_ext line_profiler

big_test = [randint(0, 10000) for _ in range(1000)]
%lprun -f count_inversion -f merge_sort -f merge count_inversion(big_test, len(big_test))


The output of this operation is at the end of my answer.  It shows that you are right, and the slowest part of the operation is indeed the nested for loops in merge().  
Dictionaries, unlike lists, can't be mutated using slice notation.  The only reason you need the for x in xrange(index, left_hi+1): nested loop is because you can't change whole swaths of the dictionary at once.  With lists, you can.  Thus, if you agree that the possible bug I described above is in fact a bug, you can switch to storing the output values in a list instead of a dictionary, and get rid of the nested loop.  The speedup is very small for short input arrays but grows with array size.  On my machine it led to a ~2-10&times; improvement at input arrays of 10,000 elements.  I'll put the output of my line profiling below too.
Using numpy for applications like this makes sense because python doesn't have a mutable fixed-type data structure.  Interesting, using numpy only to represent the dct variable in your original code, while leaving in place all other for loops, speeds up the execution another 2&times; or so for lists of 10,000 elements.

for index in res:
        sublist_length = left_hi+1 - index
        out[index:left_hi+1] += np.ones(sublist_length, dtype = int)





Original code timing

from random import randint
big_test = [randint(0, 100) for _ in range(10000)]
%lprun -f merge count_inversion(copy(big_test), len(big_test))

Results in:

Timer unit: 1e-06 s

Total time: 26.9413 s
File: &lt;ipython-input-1-77a541281305&gt;
Function: merge at line 4

Line #      Hits         Time  Per Hit   % Time  Line Contents
==============================================================
     4                                           def merge(arr, left_lo, left_hi, right_lo, right_hi, dct):
     5      9999         4407      0.4      0.0      startL = left_lo
     6      9999         4315      0.4      0.0      startR = right_lo
     7      9999         5812      0.6      0.0      N = left_hi-left_lo + 1 + right_hi - right_lo + 1
     8      9999         8330      0.8      0.0      aux = [0] * N
     9      9999         4401      0.4      0.0      res = []
    10    143615        72509      0.5      0.3      for i in range(N):
    11                                           
    12    133616        60250      0.5      0.2          if startL &gt; left_hi: 
    13      5778         2956      0.5      0.0              aux[i] = arr[startR]
    14      5778         2700      0.5      0.0              startR += 1
    15    127838        57380      0.4      0.2          elif startR &gt; right_hi:
    16      7503         3822      0.5      0.0              aux[i] = arr[startL]
    17      7503         3507      0.5      0.0              startL += 1
    18    120335        64095      0.5      0.2          elif arr[startL] &lt;= arr[startR]:
    19     61505        31935      0.5      0.1              aux[i] = arr[startL]
    20     61505        28853      0.5      0.1              startL += 1
    21                                                       # print aux
    22                                                   else:
    23     58830        30780      0.5      0.1              aux[i] = arr[startR]
    24     58830        34221      0.6      0.1              res.append(startL)
    25     58830        28077      0.5      0.1              startR += 1
    26                                                       # print aux
    27                                           
    28     68829        35529      0.5      0.1      for index in res:
    29  24750922     11838525      0.5     43.9          for x in range(index, left_hi+1):
    30  24692092     14464132      0.6     53.7              dct[arr[x]] += 1
    31                                           
    32    143615        76619      0.5      0.3      for i in range(left_lo, right_hi+1):
    33    133616        73794      0.6      0.3          arr[i] = aux[i - left_lo]
    34      9999         4362      0.4      0.0      return


Improved Code Timing (list slicing)

Timer unit: 1e-06 s

Total time: 3.11468 s
File: &lt;ipython-input-2-224c772db490&gt;
Function: new_merge at line 4

Line #      Hits         Time  Per Hit   % Time  Line Contents
==============================================================
     4                                           def new_merge(arr, left_lo, left_hi, right_lo, right_hi, out):
     5                                               # docstring goes here
     6      9999         4566      0.5      0.1      startL = left_lo
     7      9999         4491      0.4      0.1      startR = right_lo
     8      9999         6601      0.7      0.2      N = left_hi-left_lo + 1 + right_hi - right_lo + 1
     9      9999         7904      0.8      0.3      aux = [0] * N
    10      9999         4571      0.5      0.1      res = []
    11    143615        69607      0.5      2.2      for i in xrange(N):
    12                                           
    13    133616        62294      0.5      2.0          if startL &gt; left_hi: 
    14      5778         3044      0.5      0.1              aux[i] = arr[startR]
    15      5778         2804      0.5      0.1              startR += 1
    16    127838        59349      0.5      1.9          elif startR &gt; right_hi:
    17      7503         3945      0.5      0.1              aux[i] = arr[startL]
    18      7503         3673      0.5      0.1              startL += 1
    19    120335        67720      0.6      2.2          elif arr[startL] &lt;= arr[startR]:
    20     61505        33280      0.5      1.1              aux[i] = arr[startL]
    21     61505        30419      0.5      1.0              startL += 1
    22                                                       # print aux
    23                                                   else:
    24     58830        31818      0.5      1.0              aux[i] = arr[startR]
    25     58830        34578      0.6      1.1              res.append(startL)
    26     58830        28988      0.5      0.9              startR += 1
    27                                                       # print aux
    28                                           
    29     68829        37996      0.6      1.2      for index in res:
    30     58830        31690      0.5      1.0          sublist_length = left_hi+1 - index
    31     58830       164834      2.8      5.3          ones = [1]*sublist_length
    32     58830      2264227     38.5     72.7          out[index:left_hi+1] = map(add, out[index:left_hi+1], ones)
    33                                           
    34    143615        73228      0.5      2.4      for i in xrange(left_lo, right_hi+1):
    35    133616        78612      0.6      2.5          arr[i] = aux[i - left_lo]
    36      9999         4440      0.4      0.1      return


Improved Code Timing (numpy)

Timer unit: 1e-06 s

Total time: 0.979072 s
File: &lt;ipython-input-33-e2db83e49c93&gt;
Function: d_merge at line 4

Line #      Hits         Time  Per Hit   % Time  Line Contents
==============================================================
     4                                           def d_merge(arr, left_lo, left_hi, right_lo, right_hi, out):
     5                                               # docstring goes here
     6      9999         4439      0.4      0.5      startL = left_lo
     7      9999         4360      0.4      0.4      startR = right_lo
     8      9999         5728      0.6      0.6      N = left_hi-left_lo + 1 + right_hi - right_lo + 1
     9      9999         8283      0.8      0.8      aux = [0] * N
    10      9999         4511      0.5      0.5      res = []
    11    143615        69717      0.5      7.1      for i in xrange(N):
    12                                           
    13    133616        62436      0.5      6.4          if startL &gt; left_hi: 
    14      5778         2955      0.5      0.3              aux[i] = arr[startR]
    15      5778         2779      0.5      0.3              startR += 1
    16    127838        59843      0.5      6.1          elif startR &gt; right_hi:
    17      7503         3916      0.5      0.4              aux[i] = arr[startL]
    18      7503         3645      0.5      0.4              startL += 1
    19    120335        66024      0.5      6.7          elif arr[startL] &lt;= arr[startR]:
    20     61505        32834      0.5      3.4              aux[i] = arr[startL]
    21     61505        29695      0.5      3.0              startL += 1
    22                                                       # print aux
    23                                                   else:
    24     58830        31913      0.5      3.3              aux[i] = arr[startR]
    25     58830        34546      0.6      3.5              res.append(startL)
    26     58830        28552      0.5      2.9              startR += 1
    27                                                       # print aux
    28                                           
    29     68829        36887      0.5      3.8      for index in res:
    30     58830        31918      0.5      3.3              sublist_length = left_hi+1 - index
    31     58830       303818      5.2     31.0              out[index:left_hi+1] += np.ones(sublist_length, dtype = int)
    32                                           
    33    143615        72165      0.5      7.4      for i in xrange(left_lo, right_hi+1):
    34    133616        73672      0.6      7.5          arr[i] = aux[i - left_lo]
    35      9999         4436      0.4      0.5      return


Note:

All the code I used (both mine and the original) is in a Jupyter notebook hosted at GitHub.