## 4 Important Methods for Calculating Elasticity of Demand for a Good

The credit for measuring the elasticity of demand by this method goes to Flux. That is why, it is sometimes called Flux’s percentage method. According to this method, elasticity of demand is the ratio of the percentage (or proportionate) change in quantity demanded to a percentage (or proportionate) change in its price.

It should be remembered that (price) elasticity of demand is negative. The reason is that price and quantity demanded of a commodity move in opposite directions. So change in one of these has to be negative.

Hence, whenever we say that the price elasticity is 2.5, strictly speaking it is -2.5. For the sake of convenience only, elasticity of demand is denoted by a positive number. Minus sign before the value of elasticity only shows that price and quantity are inversely related.

It is important to note that the elasticity coefficient is ordered according to absolute value. Hence, an elasticity of ‘-2’ is greater than an elasticity of ‘-1’ though, algebraically, the opposite would be true.

If instead, price rises from Rs. 48 to Rs. 50, as a result of which demand falls from 110 to 100. Then, the value of elasticity will be different due to the changes in value of original price and quantity.

Here,

According to the formula of elasticity of demand, the demand is said to be elastic (e > 1), if the percentage change in demand is greater than the percentage change in price. Similarly, if the percent- age change in demand is less than the percentage change in price, demand will be inelastic (ep < 1).

Further, when price and demand change in the same proportion, demand is said be unitary elastic and so on. The higher the numerical value of the elasticity, the larger is the effect of price change on quantity demanded. Following table gives the description of elasticity coefficients in various situations. The demand curves corresponding to different elasticities are graphically shown in Fig. 2.1.

By looking at the Table 2.1 and Fig. 2.1, we can say that the less steep is the demand curve, more elastic it tends to be and vice-versa.

#### 2. Total Outlay Method:

Total outlay method was suggested by Marshall. In this method, we study elasticity of demand in relation to change in total outlay (expenditure) as a result of change in price and the consequent change in demand for a product. By knowing the expenditure incurred on a commodity before and after the change in the price, we can measure elasticity of demand of the product.

However, this method fails to give the precise value of the elasticity of demand. Hence, the outlay method is restrictive and only a rough measure of elasticity. By this method, we can only know whether the elasticity is equal to one (unit elastic), greater than one (elastic demand) or less than one (inelastic demand).

(a) Elasticity is One:

If the rise or fall in price of a commodity leaves the total outlay unaffected, elasticity is said to be equal to one. In this case, change in price (rise or fall) is neutralized by the change in demand (fall or rise) such that net effect on consumer outlay is zero, i.e., change in price and demand are proportionate.

(b) Elasticity is more than one:

If the fall in price leads to increase in total outlay, while rise in price reduces this total, elasticity is more than one. In this situation, change in demand is proportionately more than the change in price. Here, total outlay spent on the commodity varies inversely with the change in price.

(c) Elasticity is Less than One:

If the fall in price reduces total outlay, while rise in price increases this total, elasticity is less than one. In this case, change in demand is proportionately less than the change in price. Here, total outlay spent on the commodity varies directly with the change in price.

The three cases can be illustrated with the help of the schedule given here in Table 2.2

The relationship shown in Table 2.3 can be graphically shown (Fig. 2.2). Demand is unitary elastic over the price range OP, to OP2 as total outlay does not change with change in price. Demand is inelastic over the range O to OP1, as total outlay rises with prices. Demand is elastic over the price range OP2 to OP3 as total outlay falls with rise in price.

To understand the relation between elasticity and the total expenditure incurred on a commodity we consider three demand curves – one which is elastic (Fig. 2.3), other which is inelastic (Fig. 2.4) and another having unit elastic demand (Fig. 2.5).

In the demand curve shown in Fig 2.3, with the fall in the price from OP1 to OP2, the demand rises from OQ1 to OQ2. Total outlay, the product of the price and the quantity increases substantially from OP, R, Q, to OP2 R2 Q2.

The total outlay, after the fail in the price has increased. It can be explained like this. Amount of total outlay shown by the rectangle OP2 RQ1, continues to be spent in the new situation. That is, this is included in both the outlays before and after the fall in the price. The reduction in the outlay due to the fall in the price is shown by the rectangle P2P1R1 R.

Increase in the quantity demanded is Q1,Q2– Thus, Q1,RR2Q2 is the increase in the total outlay because of the increase in the quantity demanded. Clearly quantity effect Q1, R R2Q2 is greater than price effect P2 P1,R1, R. Hence, Total outlay has increased due to the fall in the price from OP1 to OP2.

Now, see Fig. 2.4. In this figure, price has come down from OP1 to OP2. Consequently, quantity demanded has increased from OQ1 to OQ2 . OP1R1 Q1 is the total outlay before the fall in the price and OQ2R2P2 is the total outlay after the fall in the price. P2 P1R1 R is the reduction in the total outlay resulting from the fall in the price. Q1RR2Q2 is the increase in total outlay resulting from the increase in quantity. Here, reduction in the expenditure is greater than the increase in the expenditure. The total outlay after the fall in the price has decreased.

When the elasticity of demand for a product is equal to one (Fig. 2.5), the total expenditure on the product will remain the same. At OP1 price, total expenditure (or total revenue) is OP1 R1Q2. At OP2 price, total expenditure is OP2R2Q2. It will be seen that the total expenditure is the same. That is, area OP2R2Q2 = area OP2R2Q2 and the shaded area P1R1RP2 (loss of outlay) = shaded area R2RQ1,Q2 (gain of outlay).

We have discussed the relationship between the elasticity of demand for a product and total expenditure on that product, for a fall in price only. The similar relationship will hold for an increase in the price. With an increase in the price of a product, total expenditure on that product will decrease, if elasticity of demand is greater than one. But, it will increase if elasticity of demand is less than one. Total outlay will remain the same, if elasticity of demand is equal to unity.

Now, it should be clear as to why so much importance is given to elasticity of demand equal to unity. It is a dividing line. On the one side of it are demands with elasticity greater than one, e.g., 2,4, or 5.

These are called elastic demands. The proportionate increase in demand is greater than the proportionate fall in the price. It is very responsive to price reductions. Total outlay on the product increases with a fall in the price of the product. Similarly, total outlay on the product decreases with an increase in the price.

On the other side of the unity elasticity of demand are the demands with elasticity less than one, e.g., 1/2, 1/3 or 1/5. These are called inelastic demands. The proportionate increase

in quantity demanded is less than proportionate fall in the price.

It is not very responsive to price reductions. Total outlay on the product decreases with a fall in the price of the product. Similarly, total outlay on the product increases with rise in the price of the product.

On the dividing line itself, elasticity of demand is equal to one. The proportionate increase in demand is equal to proportionate fall in the price of the product. Total outlay on the product remains the same with a fall in the price of the product.

Similarly, for a rise in the price of the product, total outlay will also remain the same. This is because for a rise in the price of the product, proportionate rise in the price of the product will be equal to the proportionate fall in the quantity demanded. Decrease in the total outlay due to decrease in the quantity demanded in just offset by the increase in the total outlay due to increase in the price of the product.

#### 3. Point Method:

This method originally suggested by Marshall, is used to measure elasticity at a point on the demand curve. This method has now become very popular. In Fig. 2.6, DD1, is the straight line demand curve.

Elasticity at a point ‘R’ on the straight line demand curve DD1, is

In this way, we have measured the elasticity at a point ‘R’ in the Fig. 2.6. When the elasticity is measured at a point on the demand curve, it is called the point elasticity of demand. Elasticity of demand on a straight line demand curve is different at different points. This is clear from the Fig. 2.7.

In Fig. 2.7, DD1 is a straight line demand curve. Point ‘A’ is at the middle of the demand curve DD1. Hence, lower and upper segments, AD1 and DA are exactly equal. Thus, elasticity

at point A is one (as AD1/DA = 1). Now, take a point ‘B’ above the middle point ‘A’ of the demand curve DD1. In the figure, BD1 > BD. Hence, elasticity at point ‘B’, which is equal to BD1 / DB, is greater than one. For every point that lies above the middle point ‘A’, the elasticity of demand will be greater than one. AD is the zone of the demand curve, where elasticity is greater than one. As we move further towards ‘D’, the elasticity of demand increases.

This is because as we move towards point ‘D’, lower segment will become smaller and smaller, while the upper segment will be increasing. At point ‘D’, elasticity is equal to infinity, because at this point, there is no upper segment of the demand curve DD1. Elasticity is, thus, equal to DD1/0, which is equal to infinity.

If we move downward from the middle point ‘A’ of the demand curve DD1, the elasticity of demand decreases. The reason is that as we move downward, the lower segment of the demand curve decreases and the upper segment of the demand curve increases. At every point between ‘A’ and D1, the lower segment of the curve is smaller than the upper segment.

Thus, elasticity of demand which is defined as the lower segment divided by upper segment of the demand curve is less than one at all the points between ‘A’ and D1. At point D1, elasticity of demand is equal to zero, because at this point D1, there is no lower segment of demand curve.

Now, we may conclude that at the midpoint of the straight line demand curve, elasticity of demand is equal to unity. At points higher than the middle point, elasticity of demand is greater than unity.

At points lower than the middle point, elasticity is less than unity. At the highest point of the demand curve, where it meets the vertical axis, elasticity is equal to infinity A smallest reduction in price here raises the quantity demanded from zero to some positive amount, i.e., increase in quantity demanded is infinite in percentage terms.

Further, at the lowest point where the demand curve meets the horizontal axis, elasticity of demand is equal to zero. Here, a rise in price from zero to a positive number (i.e., infinite percentage rise in price) is associated with a finite change in the quantity demanded.

In a nutshell, elasticity of demand is high towards the left hand end of the curve and low towards the right hand end of the curve. Elasticity of demand falls as one move down from left to right on the curve. This is explained below in another alternative way.

For a negatively sloped linear demand curve, the slope ?P/?Q is constant. Thus, the reciprocal of slope or ?Q/ ?P is also constant. So the elasticity of demand can be known through the second part in the elasticity formula ?Q/ ?P. P/Q. At the price axis, P/Q is undefined, as Q = 0. Hence, the elasticity of demand is infinity.

As we more downward on the demand curve, price falls and quantity rises reducing P/Q ratio. At the quantity axis, P = 0, making P/Q = 0. Thus, elasticity of demand is zero.

Example: Compete the value of elasticity of demand for

The elasticity of demand at different points (or prices) on a demand curve is different. It is true not only for a straight line demand curve, but also for a non-linear demand curve. To measure elasticity at a point on a curve, a tangent at that point is to be drawn and then elasticity is calculated at that point in the usual way.

In Fig. 2.8, the demand curve is not a straight line. To find the elasticity at point ‘R’ on the demand curve, a tangent to this point is drawn. AB is the tangent to the curve at point ‘R’. Elasticity of demand at point ‘R’ is RB/AR.

Since RB is greater than AR, elasticity at point ‘R’ is greater than unity. Further, elasticity at point R’ is R’ B’/A’ R’. As R’ B’ is smaller than A’R’, elasticity at point ‘R’ is less than unity. Hence, elasticity at points ‘R’ and R’ are different. Similarly, elasticity’s at different points on the demand curve can be found, which will be different.

The point elasticity method is appropriate for very small movements in the price. Further, it is easily amenable to mathematical treatment, since very small movements can be interpreted in terms o derivatives. We can apply differential calculus for finding the values of elasticity of demand at different points of a demand curve. In terms of calculus, the formula for the elasticity can be rewritten as:

Thus, price elasticity of the demand becomes a ratio of marginal demand (dq/dp) to average demand. For a linear demand function, say, q = 20-0.4p, the point elasticity for p = Rs. 5 can be worked out as under. Here,

Q=20-0.4(5) =18

dq/dp=-0.4

=>e=dq/dp/q/p=-0.4/3.6=-1/9

#### 4. ARC Method:

The formula used for the calculation of point elasticity of demand is relevant only if the change in the price is very minute. Here, the demand function is continuous and only marginal changes are calculable. In the calculation of point elasticity of demand, we are concerned with elasticity over a very small (strictly infinitely small) range of the curve.

The formula gives answer with reasonable accuracy only if the changes in price and quantity are not large. But, generally, the change in the price is not too small, so that we have to measure the elasticity over a substantial range of a demand curve.

We have seen that elasticity at different points of a demand curve is different. If there is a single numerical elasticity of demand over the whole length of the curve, there is no problem, but this rarely happens (we shall discuss some curves having uniform elasticity in the later part of this chapter). If the demand curve is like the one shown in Fig 2.9, where the data is discrete, only incremental changes are measurable.

Here, we need a formula to calculate the elasticity of demand over a range of the curve instead of at a point. The formula used for the calculation of the point elasticity of demand is likely to give a wrong answer, as the price elasticity of demand will be changing along the range of the curve with which we are concerned.

To measure elasticity of demand over a range of the curve (called arc elasticity), another formula is used. This formula measures elasticity of demand over a range, or arc of a demand curve like the range R0R1, in Fig. 2.9. In the formula for calculating point-elasticity of demand, we use original quantity and original price.

But, in the calculation of arc elasticity of demand, average of the original and new prices and the average of the original and new quantities are used as denominators. Thus, the formula for calculating arc elasticity of demand (e) is:

Let’s us take an example. If at a price of Rs. 10, 20 units of a commodity are demanded. If the price of the commodity falls to Rs 6 and consequently demand for the commodity increase to 25 units, then

Arc elasticity is an average elasticity, since it considers prices and quantities both before as well as after the change. Unlike the percentage method, here, the answer will remain unaffected, if the original and new prices as well quantities are inter-changed.

Further, the measure of the arc elasticity gives an approximation of the true elasticity of the arc. The more convex to the origin the demand curve is, the poorer the linear approximation attained by this formula. The value of elasticity may be different, when different methods are used for the measurement of the elasticity.

However, for small proportionate changes in price and quantity, percentage or point method may be used. On the other hand, for relatively large changes in price and quantity, total outlay or arc method is recommended.