diff --git a/django/db/backends/__init__.py b/django/db/backends/__init__.py
index 89131a4fc4..c3468f0c5f 100644
--- a/django/db/backends/__init__.py
+++ b/django/db/backends/__init__.py
@@ -677,6 +677,9 @@ class BaseDatabaseFeatures(object):
     # What kind of error does the backend throw when accessing closed cursor?
     closed_cursor_error_class = ProgrammingError
 
+    # Does 'a' LIKE 'A' match?
+    has_case_insensitive_like = True
+
     def __init__(self, connection):
         self.connection = connection
 
diff --git a/django/db/backends/postgresql_psycopg2/base.py b/django/db/backends/postgresql_psycopg2/base.py
index 91e16a1b76..5b5ca4ae4c 100644
--- a/django/db/backends/postgresql_psycopg2/base.py
+++ b/django/db/backends/postgresql_psycopg2/base.py
@@ -62,6 +62,7 @@ class DatabaseFeatures(BaseDatabaseFeatures):
     supports_combined_alters = True
     nulls_order_largest = True
     closed_cursor_error_class = InterfaceError
+    has_case_insensitive_like = False
 
 
 class DatabaseWrapper(BaseDatabaseWrapper):
@@ -83,6 +84,11 @@ class DatabaseWrapper(BaseDatabaseWrapper):
         'iendswith': 'LIKE UPPER(%s)',
     }
 
+    pattern_ops = {
+        'startswith': "LIKE %s || '%%%%'",
+        'istartswith': "LIKE UPPER(%s) || '%%%%'",
+    }
+
     Database = Database
 
     def __init__(self, *args, **kwargs):
diff --git a/django/db/backends/sqlite3/base.py b/django/db/backends/sqlite3/base.py
index 6c9728889f..e55973ea39 100644
--- a/django/db/backends/sqlite3/base.py
+++ b/django/db/backends/sqlite3/base.py
@@ -334,6 +334,11 @@ class DatabaseWrapper(BaseDatabaseWrapper):
         'iendswith': "LIKE %s ESCAPE '\\'",
     }
 
+    pattern_ops = {
+        'startswith': "LIKE %s || '%%%%'",
+        'istartswith': "LIKE UPPER(%s) || '%%%%'",
+    }
+
     Database = Database
 
     def __init__(self, *args, **kwargs):
diff --git a/django/db/models/lookups.py b/django/db/models/lookups.py
index e736c0332e..00cbcd5173 100644
--- a/django/db/models/lookups.py
+++ b/django/db/models/lookups.py
@@ -158,12 +158,30 @@ class In(BuiltinLookup):
 default_lookups['in'] = In
 
 
-class StartsWith(BuiltinLookup):
+class PatternLookup(BuiltinLookup):
+    def get_rhs_op(self, connection, rhs):
+        # Assume we are in startswith. We need to produce SQL like:
+        #     col LIKE %s, ['thevalue%']
+        # For python values we can (and should) do that directly in Python,
+        # but if the value is for example reference to other column, then
+        # we need to add the % pattern match to the lookup by something like
+        #     col LIKE othercol || '%%'
+        # So, for Python values we don't need any special pattern, but for
+        # SQL reference values we need the correct pattern added.
+        value = self.rhs
+        if (hasattr(value, 'get_compiler') or hasattr(value, 'as_sql')
+                or hasattr(value, '_as_sql')):
+            return connection.pattern_ops[self.lookup_name] % rhs
+        else:
+            return super(PatternLookup, self).get_rhs_op(connection, rhs)
+
+
+class StartsWith(PatternLookup):
     lookup_name = 'startswith'
 default_lookups['startswith'] = StartsWith
 
 
-class IStartsWith(BuiltinLookup):
+class IStartsWith(PatternLookup):
     lookup_name = 'istartswith'
 default_lookups['istartswith'] = IStartsWith
 
diff --git a/tests/expressions/tests.py b/tests/expressions/tests.py
index 0ea4df818c..320271b1dc 100644
--- a/tests/expressions/tests.py
+++ b/tests/expressions/tests.py
@@ -3,7 +3,7 @@ from __future__ import unicode_literals
 from django.core.exceptions import FieldError
 from django.db.models import F
 from django.db import transaction
-from django.test import TestCase
+from django.test import TestCase, skipIfDBFeature
 from django.utils import six
 
 from .models import Company, Employee
@@ -231,6 +231,18 @@ class ExpressionsTests(TestCase):
         queryset = Employee.objects.filter(firstname__iexact=F('lastname'))
         self.assertQuerysetEqual(queryset, ["<Employee: Test test>"])
 
+    @skipIfDBFeature('has_case_insensitive_like')
+    def test_ticket_16731_startswith_lookup(self):
+        Employee.objects.create(firstname="John", lastname="Doe")
+        e2 = Employee.objects.create(firstname="Jack", lastname="Jackson")
+        e3 = Employee.objects.create(firstname="Jack", lastname="jackson")
+        self.assertQuerysetEqual(
+            Employee.objects.filter(lastname__startswith=F('firstname')),
+            [e2], lambda x: x)
+        self.assertQuerysetEqual(
+            Employee.objects.filter(lastname__istartswith=F('firstname')).order_by('pk'),
+            [e2, e3], lambda x: x)
+
     def test_ticket_18375_join_reuse(self):
         # Test that reverse multijoin F() references and the lookup target
         # the same join. Pre #18375 the F() join was generated first, and the