An even more interesting way to look at the puzzle of determining point elasticities of demand for a linear demand function uses a diagram. Suppose there are three points along a linear demand function for which we would like to calculate elasticities. In Figure 3.3, these three points are called A, B and C.
Further, suppose that point B exactly bisects the linear demand function. This means that the linear demand function is cut exactly in half. Note that the demand function intersects the quantity axis at Q = 20, and the price axis at P = $40. Drop a vertical line at point B to the quantity (horizontal) axis, and call that point MB. Now draw a horizontal line from point B to the Price(vertical axis, and call that point RB. Since point B exactly bisects the linear demand function and the linear demand function intersects the quantity axis at Q = 20, the point RB should be at
$0.00
$8.00
$16.00
$24.00
$32.00
$40.00
0 4 8 12 16 20
Price
Quantity Figure 3.3 Point Elasticities for a LInear Demand Function
Elasticity varies from point to point
MC
Demand A
B RA
RB
RC
C
MA MB T
exactly Q = 20/2 = 10. On the price (vertical) axis, point RB should be exactly $40/2 = $20 since the demand function intersects the price axis at $40.
Assuming that a vertical line drawn from point A intersects the quantity axis at 5 units, (that is, MA = 5), Then the point RA should be 40 ‐ 2 × 5 = 30 (remember that the slope of the demand function, dP/dQ, is ‐2).
Assuming that a vertical line drawn from point C intersects the quantity axis at 5 units, (that is, MC = 15), Then the point Rc should be 40 ‐ 2 × 15 = 10 (remembering once again that the slope of the demand function, dP/dQ, is ‐2).
But by just observing where vertical lines drawn from points A, B and C intersect the horizontal (quantity), we can get an accurate estimate of the elasticity of demand at each of the three points can be obtained. In particular, since the vertical line drawn from point B exactly bisects the quantity axis, I know that the elasticity of demand at point B will be exactly unitary, or
‐1. Further, point A will be elastic or more negative than ‐1. Finally, point C will be inelastic and still negative, but less negative than ‐ 1.
Let us call the origin of Figure 3.2 (P = $0, Q = 0) point O. Notice that the point where the demand function intersects the quantity axis is labeled as point T. Then we can always use the following method to determine the demand elasticity at any point along the demand function.
Pick any point you would like on the demand function, and drop a vertical line to the quantity (horizontal) axis. Label that point as M. Also, draw a horizontal line to the price (vertical) axis, and call that point R. The distance from the origin (point O) to point M is OM. The distance from the origin (point O) to the point where the demand function intersects the horizontal axis is OT. The distance from the origin to the point where the line drawn from the point you selected to the vertical (price) axis is OR.
The formula for the exact point elasticity of demand for any point selected is given by the formula Ed = ‐ (MT/OM). To find the elasticity, we need to compare two measurements along the horizontal (quantity) axis. How large is the distance from where the line dropped from the point selected (point M) to the point where the demand function intersects the horizontal axis (point T) relative to the distance of the quantity axis from the origin of the graph (point O) to the point where the vertical line from your point cuts the quantity axis (point M).
Look again at Figure 3.3. Note that at point A, MAT > OMA. Hence point A is an elastic point, more negative than ‐1. We can calculate the exact point elasticity at A because we know that MT = 20 ‐ 5 = 15 and OM = 5 ‐ 0 = 5. So the elasticity of demand at point A is ‐ 15/5 = ‐3, an elastic point.
Now note that at point C, McT < OMc. Hence point C is an inelastic point, negative but less negative than ‐1. We can calculate the exact point elasticity at C because we know that MT = 20 ‐ 15 = 5 and OM = 15 ‐ 0 = 15. So the elasticity of demand at point A is ‐ 5/15 = ‐1/3, an inelastic point.
Finally note that at point B, MBT = OMB. Hence, point B is exactly of unit or unitary elasticity, equal to ‐1. We can calculate the exact point elasticity at C because we know that MT = 20 ‐ 10 = 10 and OM = 10 ‐ 0 = 10. So the elasticity of demand at point A is ‐ 10/10 = ‐1, unit elasticity.
Why does this method work so well? Think of an elasticity of demand as being comprised of two parts that are multiplied together. Part number 1 is the inverse slope of the function, dQd/dP, where the slope of the demand function is 1/(inverse slope).
This means also that dQd/dP = 1/(dP/dQd) when dP/dQd is the slope of the demand function.
For any selected point on the linear demand function, price is the distance OR and quantity is the distance MT. Hence, the slope of the demand function at the point we selected is a triangle OR/MT which is dP/dQd. But the inverse slope is OM/OR, which is equivalent to dQd/dP = MT/OR, the first component of our elasticity formula.
The second component of the elasticity formula is P/Qd, the ratio of price to quantity at the particular point we selected. At the particular point we selected P = OR and Q = OM, so P/Qd = OR/OM.
Ed = ‐ dQd/dP × P/Qd. Ed = ‐ MT/OR × OR/OM.
The two OR values cancel and we are left with the simple formula for the point elasticity of demand of Ed = ‐ MT/OM.
What about nonlinear demand functions? Let’s suppose the demand function descends in a nonlinear fashion, such as the one depicted in Figure 3.4.
Pick an arbitrary point on the nonlinear demand function, and call that point A. Now draw a line tangent to point A and let that line intersect both the price (vertical) and the quantity (horizontal) axis. Label the point of intersection on the quantity axis as point T. Now drop a vertical line from point A to the quantity axis, and label that point M. Draw a horizontal line from point A to the price axis and label that point R. Assume that the origin 0, 0 for the graph is point O.
Now apply the same formula as before, that is, Ed = ‐ MT/OM. Note that MT is shorter than OM so Ed at point A is inelastic.
0
5 10 15 20 25 30 35 40
0 5 10 15 20
Price
Quantity Figure 3.2 Point Elasticity on a Nonlinear Demand Function
Demand A
Line drawn tangent to point A R
M T
The formula Ed = ‐ MT/OM can be used to find elasticities at specific points along nonlinear demand functions as easily as point elasticities can be found for linear demand functions.