AEM 4240
Problem Set #3 Answer Key
1. Rad and Bug (Bertrand competition with differentiated products)
a) What is RAD's best response price as a function of the price that BUG charges?
Writing out RAD’s profit function in terms of P
R
and P
B
,
П
R
= P
R
*[1,000  10 P
R
 (10,000/P
B
)]  10[1,000  10 P
R
 (10,000/P
B
)]
=
1,000P
R

10 P
R
2
10,000 P
R
/
P
B
– (10,000100
P
R

100,000/P
B
)
and taking the derivative with respect to P
R
, setting that equal to zero, gives RAD’s best
response function:
∂
П
R
/
∂
P
R
=
1,000
 20
P
R
– (
10,000
/
P
B
100) = 0
20
P
R
= 1,100 
10,000
/
P
B
P
R
= 55 – 500/P
B
. Likewise, BUG’s best response function is P
B
= 55 – 500/P
R
.
b) Show both firms charging P
R
= P
B
= 43.51 is a Nash equilibrium.
Plug 43.51 as the prices in both reaction functions. You will see that the two functions are
equal to each other at that value and thus
P
R
= P
B
= 43.51
is a Nash equilibrium. You do
not need to solve to get 43.51. But, read on if you want to know how to derive the value
of 43.51.
Substituting BUG's best response function into RAD's yields
We know by symmetry of the two best response functions that P
B
= P
R,
so we can simply
use the same variable, P
B
, to represent both:
P
B
= 55 – 500/P
B
Multiply through by P
B
:
P
B
2
= 55 P
B
– 500
And rearrange to get into standard quadratic form:
P
B
2
 55 P
B
+ 500 = 0
Then use the quadratic formula:
P
B
= b +/
√
(b
2
 4ac), where in this case a = 1, b = 55, and c = 500, to get
[55 +
√
(3025  2000)]/2
≈
43.51
[55 
√
(3025  2000)]/2
≈
11.49
We can substitute these two possible solutions into the profit function and see that 43.51
gives a higher value.
P
B
= 43.51, P
R
= 43.51.
c) If BUG found a way to produce its product (exactly the same product) with a MC=0,
would it change the demand that either firm faced? Would it change either firm's reaction
function? Would it change the Nash equilibrium?
It would not change the demand functions that either firm faced. It would change BUG's
best response function, but it would not change RAD's best response function. BUG's
best response to any given price of RAD would now be lower than before: