Finding equilibrium price and quantity

[Asd1995]

 Roses, once in full bloom, have to be picked up and sold on the same day. On any day the market demand function for roses is given by            P  =  α - Q         (Q is number of roses ; P is  price of a rose). It is also given that the cost of growing roses, having been incurred by any owner of a rose garden long ago, is not a choice variable for him now.

( a ) Suppose, there is only one seller in the market and he finds 1000 roses in full bloom on a day. How many roses should he sell on that day and at what price?

( b ) Suppose there are 10 sellers in the market, and  each finds in his garden 100 roses in full bloom ready for sale on a day. What will be the equilibrium price and the number of roses sold on that day? (To answer this part assume α ≥ 1100).

( c )  Now suppose, the market is served by a large number of price taking sellers. However, the total availability on a day remains unchanged at 1000 roses. Find the competitive price and the total number of roses sold on that day.

[Algaeconomics]

(a) Profit fn = (a-Q)*Q

Maximising we get a-2Q=0 or Q= a/2

Now P= a-Q = a-a/2= a/2

Therefore, if a>2000 then Q= 1000 and P= a-1000
                  a< 2000 then Q= a/2 and P= a/2

(b) Profit fn for firm 1= (a-Q)*q1
                              = (a- (q1+q2+...+q10))q1
           Maximizing we get
                                      (a-(q2+q3+..+q4))/2 = q1

Now by symmetry all the best response functions look the same so

 solving for q==q1=q2=...=q10
we get (a-9q)/2=q
or a=11q
or q= a/11
and p=(a-10*q)
= a-10/11a
= a/11

and since a>= 1100
q>= 1100/11 = 100
so they can and will sell q=100 and p= 100

(c) p= MC = 0
therefore a-Q=0
or Q=a