We limit the investigation to curves of form y = m/(ax-h),
with
m > 0,
a > 0,
h >= 0, and
x > h/a
Suppose two distinct curves, y = m/(ax-h)
and y = M/(Ax-H), do intersect. Then there exists
a point (x1,y1) such that
y1 = m/(ax1-h) = M/(Ax1-H).
Case 1: h = 0 and H = 0.
y1 = m/ax1 = M/Ax1 <=> m/a
= M/A <=> M/m = A/a.
The two curves are identical. Contradiction.
Case 2: h = 0, H > 0.
Then m/ax1 = M/(Ax1-H)
<=> x1 = mH/(mA-Ma).
But then x1 > 0 <=> mA > Ma
<=> A/a > M/m.
Case 3: h > 0, H = 0.
Then m/(ax1-h) = M/Ax1
<=> x1 = Mh/(Ma-mA)
But then x1 > 0 <=> Ma > mA
<=> A/a < M/m.
Case 4: h > 0, H > 0.
Then m/(ax1-h) = M/(Ax1-H)
<=> mAx1-mH = Max1-Mh
<=> (mA-Ma)x1 = (mH-Mh)
Case 4-1: (mA-Ma) = 0.
Then M/m = A/a, but also
x1 > 0 => (mH-Mh) = 0, so
H/h = M/m = A/a,
implying the two curves are identical. Contradiction.
Case 4-2: (mA-Ma) > 0
Then immediately we have M/m < A/a.
Also, x1 = (mH-Mh)/(mA-Ma) and x1
> 0
<=> (mH-Mh) > 0.
But we also require x1 > h/a, so
(mH-Mh)/(mA-Ma) > h/a
<=> amH-aMh > hmA-hMa
<=> m(aH-hA) > M(ah-ha)
= 0
<=> aH > hA
<=> H/h > A/a.
Similarly,
(mH-Mh)/(mA-Ma) > H/A
<=> AmH-AMh > HmA-HMa
<=> m(AH-HA) > M(Ah-Ha)
<=> (Ah-Ha) < 0
<=> H/h > A/a, confirming the same
conclusion.
Combining results, M/m < A/a < H/h.
Case 4-3:
(mA-Ma) < 0
Then immediately we have M/m > A/a.
Also, x1 = (mH-Mh)/(mA-Ma) and x1
> 0
<=> (mH-Mh) < 0.
But we also require x1 > h/a, so
(mH-Mh)/(mA-Ma) > h/a
<=> amH-aMh < hmA-hMa
<=> m(aH-hA) < M(ah-ha)
= 0
<=> (aH-hA) < 0
<=> A/a > H/h.
Similarly,
(mH-Mh)/(mA-Ma) > H/A
<=> AmH-AMh < HmA-HMa
<=> m(AH-HA) < M(Ah-Ha)
<=> (Ah-Ha) > 0
<=> A/a > H/h, confirming the same
conclusion.
Combining results, M/m > A/a > H/h.
Collecting all of this information, the condition for two distinct indifference
curves of form y = m/(ax-h) and y = M/(Ax-H)
to cross each other is that:
h = 0, H > 0, and A/a > M/m,
or
h > 0, H = 0, and A/a < M/m,
or
h > 0, H > 0, and M/m < A/a
< H/h,
or
h > 0, H > 0, and M/m > A/a
> H/h.
Note that these are strict inequalities.
Expressing the same information the opposite way around, the conditions
for two such hyperbolas to serve as indifference curves in the same preference
map are:
H = h = 0,
or
M/m = A/a,
or
H/h = A/a (h > 0)
or
M/m < A/a and H/h < A/a
(h > 0),
or
M/m > A/a and {H/h > A/a
or h = 0}.
Summarizing, if h > 0, one of the ratios M/m and H/h must equal A/a, or both must be on the same side of it.
For the special case A = a = 1 (curves y = m/(x-h)
and y = M/(x-H)), the conditions are especially
simple:
H = h,
or
M = m,
or
M < m and H < h,
or
M > m and {H > h or h = 0}.
Here are two pairs of otherwise acceptable curves which do cross (pairwise, they fail to satisfy the conditions just shown):
y = 4/x and y = 2/(x-4):
a = A = 1, m = 4, M = 2, h = 0, H
= 4, so M < m but H > h.
Crossing is at (x1,y1) = (8, 1/2).
y = 4/(x-8) and y = 12/(x-4):
a = A = 1, m = 4, M = 12, h = 8, H
= 4, so M > m but H < h.
Crossing is at (x1, y1) = (10, 2).