It is early 1980’s, and the company, Happy, has just developed a new machine whi

Question

It is early 1980’s, and the company, Happy, has just developed a new machine which in a matter of seconds can improve an individual’s mood. Happy’s marketing department estimated that the market demand for the machine can be represented by the function y = 150 p, where y is the quantity of machines in thousands and p is the price per machine. The total cost of producing a machine is C(y) = 30y.

(a) After enjoying the status as a monopolist for a while, Happy learns that a competing firm, Smiley, is about to enter the market. Happy hires you as a consultant (who helps Happy maximize profits) and tells you that, after Smiley enters the market, Happy and Smiley will be choosing their outputs simultaneously. (Smiley has the same cost function of C(y) = 30y.)

i. How many machines would you recommend Happy to produce (yH )? And how many machines would you anticipate Smiley to produce (yS)?

ii. What are the profits of Happy and Smiley? 2,0 1,1 4,2 3,4 1,2 2,3 1,3 0,2 3,0 1

(b) Consider another scenario in which Happy tells you that there is a mole from Smiley in the company. The management at Happy do not know who the mole is, and they will have to accept the fact that Smiley will be seeing Happy’s output decision before selecting theirs.

i. How many machines would you recommend Happy to produce in this case (yH)? And how many machines would you anticipate Smiley to produce in this case (yS)?

ii. Demonstrate that now that Happy knows about it, the mole from Smiley is indeed helping Happy but hurting Smiley.

Explanation / Answer

answer a) prosits are maximized when

y = 150 p (demand function)

C(y) = 30y. (cost function)

y=q

q=150-p

p=150-q

TR=p.q

=150-q *q

=150q-q2

MR=change in TR/change in q

=150-2q

Similarly MC= change in TC/change in q

=30y/y

=30

MR=MC

150-2q=30

150-30=2q

120=2q

q=60

thus eq price=

150-p=y

150-p=60

150-60=p

90=p

answer for i and II

p=150-q

here q=y

MC=30 as calculated above

TR=P.Q

=(150-q) qh

=150qh-(qh+qs)

=150qh-q2h-qs*qh

MR= change inTR/change in output by happy

=150-2qh-qs

MRh=MC

150-2qh-qs=30

120=2qh+qs

(120-qs )/2=qh

qh=60-1/2qs

above is the reaction function of happy

similarly reaction function of smiley=

60-1/2qh=qs

solving two eqns by putting the value of one insecond we get

60-1/2(60-1/2qh)= qh

qh=40

thus quantity produced by happy=40

thus smiley must produce 40

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