Question

suppose in a market there are three individual consumer with three different demand function such as A,B and C whose demand function are given below Qa=40-2P Qb= 25.5-0.75P and
Qc = 36.5- 1.25P. Interpret the individual derivative for Qb and deduce the market function.

Answer 1

$$Solution$$

a. ) To interprate the individual derivative for $Q_b$

$Q_b=25.5-0.75P \implies \frac{\delta Q}{\delta P}=-0.75$

$Q_b$ has a negative slope implying the demand decreases as price increases

b. ) By connecting the three points with a line, we can approximate the actual demand curve.

Let’s consider these three different demand functions:

$$Q_a=40-2P\\ Q_b=25.5-0.75P\\ Q_c=36.5-1.25P$$

We want total demand, which is the sum of all the quantities at every price or $Q_{TOTAL} =Q_a + Q_b + Q_c$ . To get this, we can simply add up the left and right and sides of the equations above. If we did so we would get:

$$Q_{TOTAL}=102-4P$$

The total demand function might not be correct. For instance;

Consider a price which reveals a problem with our total demand function. Notice that at a price of $25:

$$Q_a=0,\ Q_b=6.75,\ Q_c=5.25$$

So the total demand is 12. Note that quantity demanded cannot have a negative number.

If we put p = $25 into the aggregate function we get:

$$Q_{TOTAL}=102-4(25)=2$$

In this case, we have not accounted for the fact that $Q_a$  has stopped at zero. So we have not quite accurately described the total demand.

Note that the each individual demand function has a different y-intercept. So we have to account for the case where a demand goes to zero.

$Q_a=40-2P \implies 2P=40-Q_a \implies P=20 \ at\ Q_a=0\\ Q_b=25.5-0.75P \implies 0.75P=25.5-Q_b \implies P=34 \ at\ Q_b=0\\ Q_c=36.5-1.25P \implies 1.25P=36.5-Q_c \implies P=29.2 \ at\ Q_c=0$

For customer A this is at p=$20, for customer B this is at p=$34, and for customer C this is at p=$29.2.

It’s important to note that for prices above $20 there are only two consumers who still demand, consumers B and C. And for prices above $29.2, only consumer B demands. We can express the demand curve by the function:

$$Q_{TOTAL}=\\102-4P, for\ P\leq 20,\\ 62-2P,for\ 20 <P \leq 29.2, \\25.5-0.75P, for\ 29.2<P$$