A market contains two consumers, Laurence and Bart. Laurence’s demand for the product is defined by P = 30 – 2qL, where P is the price and qL is his quantity demanded. Similarly, the demand curve for Bart is defined by P = 30 – 0.5qB, where P is the price and qB is his quantity demanded. Which of these defines the market demand curve?
i) P = 30 – 2/5Q
ii) P = 30 – 3Q
iii) P = 30 – 3/2Q
iv) P =30 -5Q
Or is it none of the above?
The market demand curve is the sum of all the individual demand curves in the market. We call the variable market demand for the product Q and the individual demands q, so that the sum of the individual q's is equal to Q. Here there are two individual demands for the consumers Laurence and Bart, which we call qL and qB respectively. We know that qL + qB = Q. We call the variable price of the product P, where P will be the same for all consumers, ie both Laurence and Bart.
Write the individual demand curves as q = f(P)
P = 30 - 2qL becomes after manipulation qL = (30-P)/2 (1)
P = 30 - 0.5qB becomes after manipulation qB = (30-P) x 2 (2)
Using that the market demand Q is the sum of the individual demands qL and qB we have that
Q = qL + qB
Substituting for qL and qB as defined in equations (1) and (2) respectively, we then have that
Q = (30-P)/2 + (30-P) x 2
To rearrange this to an equation of the form P = f(Q) we do addition, subtraction, multiplication and division to both sides of the equation - that is, algebraic manipulation.
Q = 30/2 - P/2 + 30 x 2 - 2P (multiplying out the brackets)
Q = (15 + 60) - (P/2 + 2P) (gathering like terms - constants, P's)
Q = 75 - 5P/2 (simplifying)
5P/2 = 75 - Q (moving addition terms across '=' sign)
P = (2/5) x 75 - (2/5)Q (moving multiplication terms across '=' sign)
P = 150/5 - (2/5)Q (simplifying)
P = 30 - (2/5)Q
We have then that the market demand curve is given by
i) P = 30 - (2/5)Q