Question 2.6: LINEAR DEMAND FUNCTION The demand function is given by the e......

(a) The slope and vertical intercept may be deduced as follows:

\begin{matrix} P & = & 100 & + & (−0.5) & Q \\ \vdots && \vdots& & \vdots & \vdots \\ P & = & a &+ & (−b) & Q \\ \vdots && \vdots& & \vdots & \vdots \\ y & &c&+ & m & x \end{matrix}

The vertical intercept is 100. This means that P = 100 when Q = 0.
Alternatively when the price P = 100 there is no demand for the good. The slope
ΔP/ΔQ = −0.5. This indicates that the price drops by 0.5 units for each successive unit increase in quantity demanded.
The horizontal intercept is calculated by substituting P = 0 into the equation of the demand function, that is,

P = 100 − 0.5Q → 0 = 100 = 0.5Q → 0.5Q = 100 → Q = \frac{100} {0.5} = 200

Therefore, the horizontal intercept is at (Q = 200, P = 0). We could describe this situation by saying that the quantity demanded Q = 200 when P = 0, so Q = 200 when the good is free!

(b) The quantity demanded when P = 5 is calculated by substituting P = 5 into the demand function

P = 100 − 0.5Q
5 = 100 − 0.5Q
0.5Q = 100 − 5
0.5Q = 95
Q =\frac{95} {0.5}= 190

(c) Method A: To plot the demand function over the range 0 ≤ Q ≤ 200, choose various quantity values within this range. In Table 2.3 equal intervals of 40 units of quantity are used. Substitute these quantity values into the equation of the demand function to derive corresponding values for P. Plot these points and graph the demand function as in Figure 2.18. Table 2.3 and Figure 2.18 are easily set up using Excel.

Method B: Plot the horizontal and vertical intercepts calculated in part (a). Draw a line through these points.

(d) The demand function Q = f(P) is graphed in Figure 2.19. Its equation is derived as

P = 100 − 0.5Q            in general form as P = a − bQ

0.5Q = 100 −P                                            bQ = a − P

Q = 200 − 2P                                                 Q = \frac{a} {b}  –  \frac{1} {b}P