Economic growth and economic development 584

Introduction to Modern Economic Growth
Denote the prices of these two firms by p1 and p2 . Show that the price
difference that would make the consumer indifferent between purchasing
from the two firms satisfies
p1 − p2 = (2x − z1 − z2 ) t
with
t (z1 − x) + p1 ≤ R.
(2) Suppose that p1 and p2 satisfy the above inequality. Then show that all
x0 ∈ [z2 , x) strictly prefer to buy from firm 2 and all x0 ∈ (x, z1 ] strictly
prefer to buy from firm 1.

(3) Now assume that there are three firms along the circle at locations z1 >
z2 > z3 . Show that firm 2’s profits are given by
à
ả
p1 p2 z1 z2 p3 p2 z2 − z3
π2 (p1 , p2 , p3 | z1 , z2 , z3 ) = (p2 − ψ)
+
+
+
2t
2
2t
2
and calculate its profit maximizing price.
(4) Now look at the location choice of firm 2. Suppose that p1 = p3 . Show that
it would like to locate half way between z1 and z3 .
(5) Prove that in a symmetric equilibrium with N firms, the distance between
any two firms will be 1/N.
(6) Show that the symmetric equilibrium price with N equity-distant firms is
t

p=ψ+ .
N
(7) Explain why the markup here is a decreasing function of the number of
firms, whereas it was independent of the number of firms in the DixitStiglitz model.

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