MICro: cost curves for two plants

[Mauli]

MICRO: A monopoly faces the demand curve P=16-Q where Q is the total demand. and it has two plants and both have identical cost functions C(qi)=15+2qi^2 for qi>0 and zero otherwise..should monopoly produce in both the plants or only one plant..what are the profit maximizing levels of q1,q2.


guess answer
 min
  q1,q2            c(q1) +c(q2)
                   15+2q1^2 + 15+2q2^2
such that
q1+q2=Q

MIn  15+2q1^2 +15 +2(Q-q1)^2
q1    
        which gives us
4q1 -4(Q-q1)=0
q1= Q/2
since the question is symmetrical we get q2= Q/2.

if the monopolist uses only  one plant say plant 1.
then he max profits according to
(16-Q)Q-[15+2[(Q/2)^2].
i am getting Q= 16/3 =5.33


if we use both the plants ,
we max
(16-Q)Q-[ 15+2(Q/2)^2 + 15 +2(Q/2)^2]
  the profit in the first case is > than when both the plants are in operation.
also, if we see that though the marginal costs in both the plants are the same , the total cost faced by the monopolist rises if he uses both the plants due to the presence of TFC.
so he produces in plant 1.
hence the profit max levels are (q1,q2) = (5.33 , 0) or
q1,q2 = 0,5.33 .

please tell me , is it the correct way to do this problem.
PS: i was going through the old threads and this was a question asked during the time of ISI interview. i found it interesting and was curious to know the answer.

[Amit Goyal]

A monopoly faces the demand curve P=16-Q where Q is the total demand. and it has two plants and both have identical cost functions C(qi)=15+2qi^2 for qi>0 and zero otherwise..should monopoly produce in both the plants or only one plant..what are the profit maximizing levels of q1,q2.

Possibility 1 - Use only one plant in the best possible way:
MR = 16-2Q
MC = 4Q

Solving MC = MR for Q, we get
Q = 16/6 = 8/3

Profit  =  (16 - 8/3)(8/3) - 15 - 2(8/3)(8/3)  = 19/3  

Possibility 2 - Using both the plants in the best possible way:
MR = 16-2Q
MC1 = 4r
MC2 = 4s
r + s = Q
Solving MC1 = MC2 = MR for Q, we get
Q = 4, and r = s = 2.

Profit  =  (16 - 4)(4) - 30 - 4^2  =  48 - 30 - 16 = 2

Since 19/3 > 2, monopolist will use only one plant and produce 8/3 units.

[Mauli]

 
it seems i made some calculation error while calculating the cost when only 1 firm operates .
thankyou sir :)

[Mauli]

no! observing my answer again.. i find i made a big mistake that i took q/2 instead of q when only one firm operates.
my bad indeed:(

[Sumit]

Hi Mauli,
I think you doing It correct way...

We can get the answer of where should monopolist will produce???? By looking in which case he will earn higher profit:
Case-1) If he choose to produce in both plants....
Case-2 If he choose to produce in one plant only....

Let see what profit he will earn in first case-1).
min
  q1,q2        c(q1) +c(q2)
             ==>15+2q1^2 + 15+2q2^2

such that     q1+q2=Q

In this case we are getting q1=q2=Q/2

Now,
MR=16-2Q
MC=Diff TC i.e c(q1) +c(q2) after putting q1 n q2 you will get=2Q

He Max Profit where MR=MC
16-2Q=2Q
Q=4..from this q1=q2=2.....Also put Q=4 in Demand function to Get P=12.

NOw, Profit When he choose to produce in both plant=TR-TC
==>PQ-TC
==>12(4)-[30+2q1^2+2q2^2]
==>48-[30+8+8]=2.

Now,Let see what profit he will earn in case-2) when he choose to produce in one plant only.

Max    PQ-c(q1)
q1      (16-Q)Q-[15+2q1^2]

Such That Q=q1.

In this case we get q1=2.666 n By putting this in Demand fn p=13.33.

N profit=(16-q1)q1-15+2q1^2=6.30(approx)..Which is greater than If he choose to produce in both plant..

So, the Answer will be---- He will choose to produce in one plant only bcoz only after doing so he can maximize his profit (n Mauli you reasoning that  the total cost faced by the monopolist rises if he uses both the plants due to the presence of TFC n due to this his profit in case-1 reduced is absolutly correct) also profit maximizing levels of q1,q2=(0,2.66) or (2.66,0)...

PS: You made a little mistake while calculating Q when monopolist choose to produce in one plant only...you take q2=Q/2...Instead this it should be Q=q2...he will now redecide which quantity will max his profit...q2=Q/2 which you take is Profit Max quantity when he is producing in both plant....

Also thanks for sharing I was also looking for this type of questions...n If you found any question just like this where cost of producing in different plant is different plz do share.....

[Sumit]

Wow!!!...Sir posted it while I was writing all this...
anyways thank you sir....
Also Sir, could you post some question like this....It would be a great help...
Thanks in advance...

[Mauli]

sumit i second you:)
sir please post some questions on this topic.

[Sumit]

Sir plz help us in below question too.

consider 2 firms facing dd curve P=50-5q where q =q1+q2. the firm cost fns are C1=20+10q1 and C2=10+12q2
a)suppose that both firms have entered the industry what is the join profit maximizing level of output?how much will each firm produce?how would your ans change if the firms have not yet entered the industry?
b)how much should firm1 be willing to pay to purchase firm 2 if collusion is illegal but a takeover is not?
in a part how will we decide that which firm will produce how much?

PS: Hey Mauli Have you tried this question???? this question is posted by Madhur1987 a few weeks back...Under cournot oligopoly questions....

[Mauli]

hi sumit:)

this is what madhur said to u regarding that post.
 
thanx sumit:)
but in que 1 q1=2.8 and q2=2.4 are cournot eqillibrium qty's(when they are competing). and profit of firm 1 is 19.2 and that of 2nd firm is 18.8. but in que we are asked joint profit maximizing level of output which is coming out Q=2.8 only. now joint will be max when firm 2 produce nothing out of 2.8. but my doubt is is it possible when 2 firms collude 1 will produce nothing. if not how will we decide which firm will produce how much share of 2.8.
and same doubt if one firm takeover other in which plant production takes place??

[Mauli]

i tried to do it step by step and here is what i could conclude
when collusion is asked in the question.
it means
max      p(q1,q2) *(q1+q2) - c1(q1)-c2(q2)
q1,q2
           

[Mauli]

as per the question given:
[50-5(q1+q2)]*(q1+q2)-(20+10q1)-(10+12q2)

MR1=MC1
AND MR2=MC2.
according to which
for firm 1
50-10(q1+q2)=10
q1+q2=4
and for firm 2
50-10(q1+q2)=12
which gives q1+q2=3.8
MC2<MC1

[Mauli]

please check my calculations . coz i am very bad at them and madhur has got q1+q2 as 2.8 and not 3.8 .
i dont know how he got that.
according to me since MC2<MC1
firm 2 will definitely produce more when they both collude.
after this i am lost.
maybe someone can share their inputs too on this.
Sir pleease help.!!!

[Amit Goyal]

Consider 2 firms facing dd curve P=50-5q where q =q1+q2. The firms' cost functions are
C1=20+10q1 and
C2=10+12q2
a)Suppose that both firms have entered the industry what is the joint profit maximizing level of output?
Joint profit maximizing level of output is given by the following condition:
MC of the firm with lower marginal cost = MR
10 = 50 - 10 Q, this gives us
Q = 4
How much will each firm produce?
Firm 1 will produce 4 units and Firm 2 will produce 0 units.
How would your answer change if the firms have not yet entered the industry?
If the firms have not entered the industry fixed cost is not sunk cost. Only one of the two firms will enter. Firm 1 will produce 4 units if it enters and makes a profit of 30(4)-20-10(4) = 60.
Firm 2 will produce 3.8 units if it enters and makes a profit of 32(3.8) - 10 - 12(3.8) = 66
In this case, its more profitable for the firm 2 to enter and produce 3.8 units.

[Amit Goyal]

b) How much should firm 1 be willing to pay to purchase firm 2 if collusion is illegal but a takeover is not?
To find the willingness to pay, find the Cournot equilibrium and hence Cournot profits of firm 1. And compare it with profits when only firm 1 operates. The difference between the two is the willingness to pay.

[Sumit]

Thank you sir...
But could you plz explain more why Only one of the two firms will enter If the firms have not yet entered the industry..????

Also would not Firm 2 will make a profit 31(3.8) - 10 - 12(3.8) = 62.2..Coz P=50-5(3.8)=31

[Amit Goyal]

If you want to maximize joint profits, then both firms entering would mean that they both incur fixed costs and since only one of the firm will operate post entry (as we have seen above) so it does not make any sense for the other firm to enter. So, we compute profits for both and compare them.
Regarding your other query, you are right. There is a typo in my answer. The price is indeed 31 and not 32.

[Sumit]

got it....