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Elasticity of demand is of three types – price, income and cross.
1. Price Elasticity of Demand:
Price elasticity of demand is defined as the degree of responsiveness of the quantity demanded of a commodity to a certain change in its own price, ceteris paribus.
It is expressed as follows:
Price elasticity of demand is a unit free measures of responsiveness (i.e., how the quantity demanded responds to a change in price), unlike the slope of the demand curve ∆q/∆p, i.e., the absolute change in quantity demanded of a commodity divided by the absolute change in its price.
By using symbols price elasticity of demand is expressed as:
Price elasticity of demand is the ratio of price to quantity multiplied by the reciprocal of the slope of the demand function. The value of e which is called the co-efficient of price elasticity of demand, is, negative since price change and quantity change are in the opposite direction. The convention is to ignore the negative sign and work with absolute values of ep.
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It measures the degree of responsiveness of the quantity demanded to price. It is always a pure number because it is the ratio of two percentage changes. If the value of ep is greater than one, demand is said to be elastic, if it is exactly equal to one, unitary elastic and if it is less than one, inelastic. Demand is said to be elastic if a certain percentage fall (rise) in p leads to more than proportionate fall (rise) in q.
Commodities which have numerical high elasticities are called luxuries, whereas those with small elasticities are called necessities. Price elasticities are pure numbers such as 1/2, 1, 2, 3, etc. independent of the units in which prices and quantities demanded are measured. The’ value of ep is negative if the corresponding demand curve is downward sloping.
Possibilities of Substitution:
The main determinant of price elasticity of demand is the number and closeness of substitutes available. If two commodities are perfect substitutes such as red pencil and black pencil and if the price of red pencil rises by 1%, its sale will fall to zero and the demand for black pencils, will be very elastic.
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If a good has may close substitutes, its demand curve would be very response to its price change. On the other hand, if there are a few close substitutes for a good, its demand will be inelastic.
The Elasticity at a Point of a Linear Demand Curve:
If the demand curve is linear such as p = a – bq, as shown in Fig. 14.7 its slope is constant, – b. Here a is the intercept.
For computing point price elasticity it is convenient to express quantity as a function of price:
We know that ep = (dq/dp)/ (q/p) = (-1/b)/ (q/p)
We now show that price elasticity is different at every point along the same linear demand curve. At the mid-point, where p = a/2 and q = a/2b, we have [because the curve which passes through, point n has double the slope];
For points above the mid-point p > (a/2) and q < (a/2b); so the denominator falls and demand is elastic. For points below the mid-point where p < (a/2) and q > (a/2b), the denominator increases and ep < 1, i.e. demand is inelastic.
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When p = 0 and q = a/b, ep = 0. When p = a and q = 0, ep tends to infinity. Thus we see that price elasticity varies from point to point along a straight line demand curve. The value falls for a movement from left to right.
This point may be intuitively explained. In the elastic zone of the demand curve (where ep > 1) the original price is high and the original quantity is small. Therefore, a small drop in price will lead to a more than proportionate increase in quantity. In contrast in the lower end of the demand curve the original price is low and the original quantity is high. As a result a large drop in price leads to a very small increase in quantity. So demand is inelastic.
The Expenditure Share Weighted Sum of Price Elasticities:
An important deduction from price elasticity of demand is the following:
If two commodities are neither substitutes nor complements, then the expenditure share weighted sum of own price elasticities of demand is always – 1.
Proof:
We know that in a simple two-commodity world the budget constraint is;
m = p1x1 + p2x2 ….. (1)
Differentiating equation (1) partially with respect to ‘p1’ we get;
Elasticity and Total Revenue:
Total revenue (R) is the product of p and q: R = pq
The effect of price change on total revenue depends on how q responds to a change in p. Thus revenue depends on the relative magnitude of changes in p and q or on price elasticity of demand. If the demand for the product of a firm is unitary elastic price change will have no effect on total revenue.
This important result may be proved as follows:
Thus if total revenue is constant, ep has to be equal to 1. In this case the demand curve will be a rectangular hyperbola. See Fig. 14.8, where the demand curve has unitary elasticity throughout. In this case the firm does not gain by changing the price of his product because marginal revenue will be zero whether price rises or falls.
This point may be proved by using the total outlay method of measuring price elasticity of demand. The behaviour of total outlay in response to price change indicates whether demand is price elastic, unitary elastic or inelastic.
This point is discussed in detail below:
If ep < 1 i.e., demand is inelastic a price cut will lead to a fall in total revenue.
The converse is also true. This point may be proved as follows:
If the own-price elasticity of demand is unitary at all points on a demand curve, then;
Thus unitary price elasticity of demand means that total expenditure is constant at all points on a demand curve. If we start at any point (q1, p1) and reduce price (dp1 < 0), the increase in quantity q1 is exactly enough to keep expenditure constant.
But if demand is elastic (p1/q1)(dq1/dp1) > – 1, so that starting from the same price-quantity combination if we reduce price by the same amount as before, the increase in quantity dq1 must be greater than when demand is unit-elastic and, therefore, total expenditure must increase.
Hence if demand is elastic at all points on the demand curve, every reduction in price must increase total expenditure. Similarly, if demand is inelastic (p1/q1)(dq1/dp1) < – 1, a given reduction in price will increase quantity by less than when demand is unit-elastic, so that total expenditure will fall.
Example:
A farmer sells 4 tons of sugarcane at Rs. x per ton. If the demand for sugarcane has unitary (price) elasticity, in what proportion will he have to change his price to be able to sell 6 tons?
So total revenue will fall if output (sales volume) increases. The reason is easy to find out. If demand is not much responsive to price, then a firm has to cut price substantially to increase output (sales). So total revenue falls. Since an increase in q implies a fall in p, how revenue changes with a change in p depends on price elasticity of demand.
Importance of Elasticity for a Profit-Seeking Firm:
If the objective of a firm is to maximise profit (revenue minus cost) it should never operate on the inelastic part of the demand curve. In other words, the firm will never set a price at a level where demand is inelastic.
The reason for this is not far to seek. If demand is inelastic and if a firm raises it price, its total revenue will increase even though the sales volume will fall. But if the sales volume falls then its cost of production must also fall or, at least it cannot increase. So the firm s overall profit will surely rise. This clearly proves that operating at an inelastic part of the demand curve is not consistent with the profit-maximisation objective.
Marginal Revenue Curves:
We know that MR is given by the following formula:
When q = 0, MR=p. This means that the MR and demand (average revenue) curves have the same intercept on the vertical axis. For the first unit of a good sold, MR = p. But if the demand curve is downward sloping, a firm has to reduce p to increase q.
This reduction in p reduces the revenue it receives on all the units the firm was already selling. Thus marginal revenue (the extra revenue the firm receives by selling an extra unit) will be less than p that it receives by selling one extra unit. In other words MR < p except for the first unit.
Inverse Demand Curves and MR:
We may consider the special case of the linear (inverse) demand curve:
The MR curve is shown in Fig. 14.9(a). The MR curve has the same vertical intercept (a) as the demand curve, but has double the slope. MR is negative when q > a/2b When ep– 1, q = a/2b. At any larger value of q, ep < 1, which implies that MR is negative.
The constant elasticity demand curve is another special case of the marginal revenue curve. See Fig. 14.9(b). If ep is constant, then the MR curve will have the form
Since the expression within the bracket is constant the MR curve is some constant fraction of the inverse demand curve. When ep = 1, the MR is constant at zero. When ep > 1, the MR curve lies below the inverse demand curve. When ep < 1, MR is negative.
2. Income Elasticity of Demand:
Income elasticity of demand (henceforth IED) shows how the quantity demanded of a commodity responds to a change in income of buyers, prices remaining constant.
It is expressed as follows:
Since for a normal good an increase income (m) leads to an increase in demand, IED is positive. In case of superior (luxury) goods, IED > 1: a certain % change in m leads to more than proportionate change in q. In case of an inferior good IED is negative because an increase in m leads to a fall in demand.
Expenditure Share Weighted Elasticity of Income:
In general income elasticities tend to move around 1. The reason for this can be found out by examining the budget constraint.
Let us suppose the budget constraints for two different levels of income are the following:
P1x1 + p2x2 = m
P1x1’ + P2x’2 = m’
If we subtract the second equation from the first one and denote the difference as d, we get;
P1dx1 + p2dx2 = dm
Now by multiplying and dividing price i by xi/xi and dividing both sides by m we get;
p1/m dx1/x1 + p2x2/m dx2/x2 = dm/m
Finally, be dividing both sides by dm/m, and using si = pixi/m to denote the expenditure share of good i, we arrive at the following equation:
s1 (dx1/x1)/(dm/m) + s2 (dx2/x2)/(dm/m) = 1
or s1 em1 + s2 em2 = 1
This equation states that the weighted sum of the income elasticities is 1, where the weights are the expenditure shares of different goods purchased. Luxury goods having an income elasticity greater than 1 must be counterbalanced by other (mainly essential) goods that have income elasticities less than 1, so that the sum of the expenditure share weighted income elasticities is equal to 1.
Theorem 1:
In a two-commodity world both goods cannot be inferior simultaneously.
Proof:
A standard theorem of elasticity of demand is that the expenditure share weighted sum of income elasticities of demand is equal to 1.
Therefore, in a 2-commodity world the sum of em can be expressed as:
s1em1 + s2em2 = 1 ……. (1)
where em1 and em2 are the income elasticities of demand for the commodities x1 and x2, respectively. Since and s2 are the proportions of total expenditure for the two goods their sum is also equal to 1.
s1 + s2 = 1…. (2)
Now, substituting (s1 + s2) from equation (2) to the right hand side of equation (1) we get;
s1em1 + s2em2 = (s1 + s2) …. (3)
Rearranging the above equation we get;
s1 (em1 – 1) – s2 (1 – em2) = 0, or s1 (em1 – 1) = s2 (1 – em2)…… (4)
Therefore, from equation (4) we see that in a two-commodity world, both cannot be inferior at the same time. If x1 is normal good with em2 > 0, x2 has to be inferior, with em1 < 0.
Theorem 2:
Income elasticity of an inferior good is always negative.
Example:
An individual spends all his income on two goods x1 and x2. He spends 1/4th of his income on good x1, whose income elasticity is 5. Hence x2 is an inferior good True/ False.
Solution:
The sum of the expenditure share weighted income elasticities of demand in this case is 1/4(5) + 3/4(-1/3). Therefore, x2 is an inferior good.
Income Elasticity in Case of Separable Utility Function:
Income elasticity of demand is also related to the nature of the utility function. In this context we prove the following theorem of the Marshallian (cardinal) approach:
Theorem 3:
If the utility of each good is independent of the quantity of other goods, then all goods must have positive income elasticities.
Corollary:
If the utility function is separable, increasing and strictly concave none of the goods can be inferior. This means that each good has positive income elasticity.
Proof:
Consider an individual with separable utility function u (x1, x2, x3) = f1(x1) + f2(x2) + f3(x3), where fi(xi), i = 1, 2, 3 is increasing and strictly concave. Now we can show that none of the goods can be inferior.
When preferences exhibit additive separability the form of the utility function is;
u = f [u1(x1) + u2(x2) + ….. + un(xn)], f’ [.] > 0
i.e., any positive monotonic transformation of a sum of individual utility functions. This form underlies the cardinal utility based demand theory of Alfred Marshall. But it has an undesirable implication. It rules out the existence of inferior goods.
Given the assumption of increasing and strictly concave utility function, we have:
It m increases, purchase of at least one good must increase, so that all the three goods cannot be inferior at the same time.
Let x1 increases. Then u’1(x1) must fall. This implies that;
To restore the equi-marginal utility condition u’2(x2) and u’3(x3) have to fall. This is possible if and only if both x2 and x3 increase when m increases. Therefore, none of the goods can be inferior. So separable utility functions also rule out the existence of Giffen goods.
An Alternative Interpretation of Price and Income Elasticities:
The own price elasticity of demand for x1(ep) is defined as the proportionate rate of change of x1 divided by the proportionate change of its own price with p2 and m held constant.
A high value for ep implies that quantity is proportionately very responsive to price changes.
The consumer’s expenditure on x1 is p1x1 and;
This means that the consumer’s expenditure on x1 will increase with p1 if ep > – 1, remain unchanged if ep = – 1 and fall if ep < – 1. The cross-price elasticity of demand for the ordinary demand function refers to the proportionate change in x1 to a proportionate change in p2, i.e., the price of x2. So it is expressed as;
Cross-price elasticities may be either positive or negative.
In a two-commodity world the consumer’s budget constraint is expressed as;
m = p1x1 + p2x2 …… (3)
If we take the total differential of this equation and set;
dm = dp2 = 0, we get
p1dx1 + x1dp1 + p2dx2 = 0
Multiplying through by p1x1x2/mx1x2dp1 and rearranging terms, we get;
s1ep + s2ec = – s1 …… (4)
where = p1x1/m and s2 = p2x2/m are the proportions of total expenditures on the two goods. If we know ep, i.e., the own-price elasticity of demand for x1, (4) can be used to evaluate the cross-price elasticity of demand for x2. If ep = -1, ec = 0. If ep < – 1, ec> 0, and if ep > – 1, ec < 0.
If the utility function appears in the multiplicative form such as u = x1x2, then the price and cross elasticities for the ordinary demand function are;
This result is not generally true of ordinary demand curves. This means that all demand functions do not have unit price and zero cross elasticities or even constant elasticities.
As a general rule, elasticities are a function of p1, p2, and m. For an ordinary demand function, income elasticity is defined as the proportionate change in the quantity of a commodity such as x1 demanded in response to a proportionate change in income with prices (p1 and p2 held constant);
where em1 denotes the income elasticity of demand for x1.
Income elasticities can be positive, negative or zero, but are normally assumed to be positive.
Taking the total differential of the budget constraint (3) we get;
P1dx1 + p2dx2 = dm
Now multiplying the first term on the left by x1/x1, the second by x2/x2 and dividing through by dm, we get;
s1em1 + s2em2 = 1 … … (7)
This means that the sum of the income elasticities weighted by total expenditure shares equals unity.
3. Cross Elasticity of Demand:
Cross (price) elasticity of demand is defined as the degree of responsiveness of the quantity demanded of a commodity such as x1 to a certain percentage change in the price of another commodity such as x2.
We use the concept to study cross-price-quantity relationships, i.e., how the quantity demanded of a commodity is affected by a change in the market price of another commodity, its own price and income of the buyer(s) remaining the same.
It is expressed as:
where all terms have their usual meaning.
Cross elasticity is positive if x1 and x2 are substitutes of each other and is negative if they are complements. It is zero in case of unrelated goods, i.e., if x1 and x2 are neither substitutes nor complements.
Theorem 1:
In a two-commodity world if a fall in p1 is followed by a decrease in the purchase of x2, then own price elasticity of demand for x1 is negative.
Theorem 2:
Cross elasticity of demand between two goods is zero if the utility function is of the Cobb-Douglas type.
Proof:
The Cobb-Douglas utility function is expressed as:
u(x1, x2) = xα1xβ2
This can be converted into a linear form by taking logarithms:
u(x, X2) = α log x1 + β log X2
Both the forms (exponential and linear) yield identical demand functions for goods x1 and x2. Here we work with the linear form.
To find the demand function for x1 and x2, given the usual budget constraint, we first form the Lagrangian:
Thus the demand for each good depends only on the price of the good and on income, not on the price of the other good. Thus, the cross-price elasticities of demand are zero.
The Sum of Price, Cross and Income Elasticities of Demand:
We have already noted that in some special cases, i.e., where two commodities are neither substitutes nor complements the own price elasticity of demand is -1, and cross elasticity of demand is zero. We have also proved that the sum of the expenditure share weighted income elasticities of demand is 1.
So we arrive at the following interesting result:
Price elasticity + cross elasticity + income elasticity = -1 + 0 + 1 = 0.
This interesting result may now be proved as follows.
An important property of the demand functions is that they are homogeneous of degree zero in all prices and the level of income. This is because in all prices and money income are increased by a proportion k (> 1) the budget equation of the consumer becomes km = kp1x1 + kp2x2 which is the same as the original budget equation m = p1x1 + p2x2. So the quantity demanded of x1 and x2 remain un-effected. Since x1 = f1 (p1, p2, m) and x2 = f1 (p1, p2, m) we have;
x1k = f1 (kp1, kp2, km)
x2k = f2 (kp1, kp2, km)
This property can also be expressed differently. The sum of the three partial elasticities of the demand functions will be equal to zero.
Let us consider the demand function:
x1 = f1(P1, P2, m).
Since, this is homogeneous of degree zero in prices and income, by applying the Euler’s theorem we get;
Thus for each demand function the sum of three partial elasticities is equal to zero.
Elasticity of Compensated Demand Curve:
It is also possible to define price and cross elasticities of demand for compensated demand functions.
If we take the total differential of the utility function:
u = f(x1, x2) and let du = 0, we get;
MU1 dx1 + MU2 dx2 = 0,
Using the first-order condition p1/p2 = MU1/MU2, multiplying through by p1x1x2/mx1x2dp1, and rearranging terms
α1kp = α2kc = 0 ….. (5)
where the kp and kc are the compensated price and cross elasticities, respectively. Since kp < 0, it follows from (5) that kc > 0.
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