If both AR and MR curves are straight lines, it can be shown that MR curve is half-way from the AR curve on the Y-axis. Consider Fig. 2.21 in which we have straight line AR and MR curves. A point ‘P’ is taken on the Y-axis and a line PS is drawn parallel to the X-axis.
The line PS cuts the MR curve at point ‘R’. To show that MR cuts the distance between Y-axis and the AR curve in the middle implies that in Fig. 2.21, PR = RS.
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Now, from point ‘S’ draws a line SQ perpendicular to the X-axis. We can find the TR received by the seller from the sale of OS unit of the commodity in two ways.
TR = Price x Quantity sold
= OP x QQ
= Area OP SQ
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Also, TR = Total of MR of all the OS units sold = ∑MR
= Area OB AQ
By definition, TR should be the same by the two methods. Equating (2.3) and (2.4), we get
Area OPSQ = Area OBAQ
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From the figure, we can see that
Area (OPSQ) = Area (OPRAQ + ASR)
Area (OBAQ) = Area (OPRAQ + BPR)
From equations (2.5), (2.6) and (2.7), we notice in terms of area that
OPRAQ + ASR = OPRAQ + BPR
Thus,
Area (ASR) = Area (BPR)
This implies that the areas of the triangles ASR and BPR are equal.
Further, In ∆ASR and ∆BPR
BPR = ASR (right-angles)
BRP = SRA (vertically opposite angles)
PBR = SAR (alternate angles)
Because all the three angles of the two triangles are equal to each other, the two triangles ASR and BPR are similar.
It is proved that both the triangles are equal in area and similar too. Thus, they must be congruent. If two triangles are congruent, their corresponding sides are equal. It means that
PR = RS
Hence, if AR and MR curves are straight lines, then, MR curve bisects the distance between the Y-axis and the AR curve. This relationship between the AR and MR curves can be used in drawing the two curves. But, this relationship between AR and MR curves is not true, if the two curves are either convex or concave to the origin.
As shown in Fig. 2.22 (a), if, AR and MR curves are convex to the origin, MR cuts the distance between the Y-axis and AR at less than half-way as measured from the Y-axis (i.e., to the left of the middle point). And, if, the AR and MR curves are concave to the origin, MR curve cuts the distance between the Y-axis and AR at more than half- way as measured from the Y-axis (see Fig. 2.22 (b)) in either of the two cases, MR curve will not lie half way from the Y-axis. The proof of this is beyond the scope of this book.
