You can sell 130 pet chias per week if they are marked at $1 each, but only 100

Question

You can sell 130 pet chias per week if they are marked at $1 each, but only 100 each week if they are marked at $2/chia. Your chia supplier is prepared to sell you 40 chias each week if they are marked at $1/chia, and 110 each week if they are marked at $2 per chia.
(a) Write down the associated linear demand and supply functions.

demand function     q(p) =
supply function     q(p) =

(b) At what price should the chias be marked so that there is neither a surplus nor a shortage of chias? HINT [See Example 4.]
$  

Explanation / Answer

Solution:

(a)
The quantity functions q(p) you are told are linear so they must be of the form
q(p) = mp+c, for some values m and c which we have to determine.

Demand function:
We are told q(1) = 130, and q(2) = 100
So substituting this into q(p)=mp+c tells us
130 = m+c
100 = 2m+c
We have a pair of simultaneous equations which we can solve to get m = -30 and c = 160
q(p) = -30p+160

Supply function:
Similarly we know q(1) = 40, and q(2) = 110
so
40 = m+c
110 = 2m+c
Therefore m = 70, c = -30
q(p) = 70p - 30

(b)
We want supply to equal demand
i.e. 70p - 30 = -30p+160
100p = 190
p = 1.9
$1.90

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