Question

Consider an industry composed of four firms which produce a homogeneous good with inverse demand P(Y) 100-Y where Y - Notice that this set-up implies the firms compete in quantities. The firms have similar c(y) 20y production costs defined as a) Compute firm profits from the Nash equilibrium. b) Show that a merger between firms one and two is not profitable. c) Suppose that a merger between firms one and two will generate cost synergies s where s is a percent bounded between zero and one. Hence, the new cost function for the merged firm is c(v) 20 s)y. What value of s is required to make the merger profitable? Hint: the firms not involved in the merger are symmetric, which should simplify things.

Answer 1

1) First find the firms best response functions .firm 1's profit is,

y₁(100-y₁-y₂)-20y₁

Taking the derivative of this profit with respect to y₁ (holding y₂ constant) and setting the derivative equal to zero we obtain

100-2y1-Y_2-20=0

Y_{1 =}(80-Y₂)/2

WE NOW NEED TO FIND A PAIR (Y1,Y2) OF OUTOUT WITH THE PROPERTY THAT

Y1 =B1(y2) and y2 =b₂(y₁)

THAT IS ,

Y₁ = (80-Y₂)/2 AND Y₂ = (80-Y₁)/2

SUBSTITUTING ONE EQUATION IN THE OTHER WE OBTAIN Y₁ = (80- (80-Y₁)/2)/2 , SO THAT

Y₁=20

SUBSTITUTING ONE EQUATION IN THE OTHER WE OBTAIN Y ₂ = (80-(80-Y₂)/2)/2, SO THAT

Y₂ =20

WE CONCLUDE THAT THERE IS A UNIQUE NASH EQUILIBRIUM ,IN WHICH THE OUTPUT OF EACH FIRM IS 20.

EACH FIRM PROFIT IS =(20)(100-20-20)-(20)(20) =800