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Demand Theory
and
Analysis
Introduction
The concept of demand is one of the most useful notions from applied microeconomic
theory.
The concept of demand is the desire for a good backed up by the purchasing power to
make the desire effective, and the willingness to part with the purchasing power.
The demand function describes the relationship that exists (during some period of time)
between the number of units of a good or service that consumers are willing to buy and a
given set of conditions that influence the willingness to purchase. The period of time can
be a year, a month, or any other convenient measure.
The conditions influencing the willingness of consumers to buy include the price of the
good in question, consumer income levels, prices of substitute products, advertising
expenditures, and future price expectations.
Demand analysis serves two major managerial objectives: (1) it provides the insights
necessary for the effective manipulation of demand, and (2) it aids in forecasting sales
and revenues.
The Demand Schedule and the Demand Curve
Illustrate
Income and Substitution Effects
Economists have identified two basic reasons for the increase in quantity demanded as a
result of a price reduction.
First, when the price of a commodity declines, the effect of this decline is that the real
income of the consumer has increased. With the increased real income, the consumer can
buy more than before and thus there is an increase in quantity demanded. This has been
called the income effect.
The second reason which has been identified to explain the inverse relationship between
price and quantity is that a decline in the price of one good, A, makes it more price
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attractive in relation to other goods. Consequently, the rational consumer should switch
some portion of his or her total expenditure from the relatively higher-priced items to the
relatively lower-priced item. This has been called the substitution effect.
Law of Diminishing Marginal Utility
One of the basic assumptions of economic theory is that consumers are rational. As
rational individuals they seek to maximize the satisfaction gained from their consumption
or expenditure decisions. This satisfaction may be defined as utility.
In order to maximize utility it can be shown that consumers should allocate their
consumption in such a manner that they receive the same marginal, or additional,
satisfaction for the last dollar spent on each commodity they consume.
Specifically, marginal utility may be defined as the rate of change in total utility per unit
change in the consumption of a given commodity, while the quantity of other
commodities is held constant.
In order to maximize utility, the ratio of marginal utility MU to Price P for all
commodities purchased must be equal:
If the condition above does not hold, it would be possible to reallocate expenditure
dollars in a manner that would result in an increase in total satisfaction.
If, for example, , it would be possible for the consumer to increase
his/her utility by transferring consumption from product B to product A and vice versa.
The process will continue until the equality condition is reached.
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The Demand Function and Shits in Demand Curve
The demand curve is a graphic representation of a tabular demand schedule or algebraic
demand function. It specifies a relationship between prices and the quantity of a
commodity that will be demanded at those prices at some point in time, holding constant
the influence of other factors. A number of these factors may effect a change in the shape
as well as the position of the demand curve as time passes. In other words, we may say
that quantity demanded is a function of a number of factors in addition to price. These
factors may include the firm’s advertising budget, the size of the sales force, price of
related goods, changes in the taste and choice of consumers, promotional expenditures by
the competitors, and many others.
Changes in price of the commodity will result only in movement along the demand
curve, whereas changes in any of the other independent variables in the demand function
are likely to result in a shift of that curve.
Elasticity of Demand
From a decision making perspective the firm needs to know the effect of changes in any
of the independent variables in the demand function on the quantity demanded. Some of
these variables are under the control of management, such as price, advertising, product
quality and customer service. For these variables, management must know the effects of
changes on quantity in order to assess the desirability of instituting a change. Other
variables, including income, prices of competitors’ products, and expectations of
consumers regarding future prices, are outside the direct control of the firm.
Nevertheless, effective forecasting of demand requires that the firm be able to measure
the impact of changes in these variables on the quantity demanded.
The most commonly used measure of the responsiveness of quantity demanded to change
in any of the variables that influence the demand function is elasticity. In general,
elasticity may be thought of as a ratio of the percentage change in one quantity (or
variable) to the percentage change in another, ceteris paribus. In other words, how
responsive is some dependent variable to changes in a particular independent variable?
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Price Elasticity of Demand
Price elasticity of demand may be defined as the ratio of a percentage change in quantity
demanded to a percentage change in price:
ED = %∆Q/%∆P
Where, ∆Q = Change in quantity demanded
∆P = Change in price.
Because of the normal inverse relationship between price and quantity demanded, the
sign of the price elasticity coefficient will usually be negative. Occasionally, demand
elasticities are referred to as absolute values.
Price elasticity may be point price elasticity or arc price elasticity.
Point Price Elasticity
The elasticity of demand is measured on a particular point along the demand curve. It is
given as –
Ed =
Where, = the partial derivative of quantity with respect to price
Qd = the quantity demanded at price P
P = the price at some specific point on the demand curve.
Example: Assume the following demand function -
Qd = 150 – 10P
What is the point elasticity of demand when P = 5 and Qd = 100.
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Arc Price Elasticity
The arc price elasticity of demand is calculated between two prices. It indicates the effect
of a change in price, from P1 to P2 for example, on the quantity demanded. It can be
measured in the following manner:
Ed =
Where, Ed = Price elasticity of demand
Q1 = Quantity sold prior to a price change
Q2 = Quantity sold after a price change
P1 = Original price
P2 = Price after a price change.
Example: Given the following demand schedule, calculate price elasticity and interpret
the result:
Price, (P) : 20 19 18 17 16 12 11
Quantity, Qd: 12 14 16 18 20 28 30
Example: Consider the X Corporation, which had monthly sales of 100 units (at Tk.10
per unit) prior to a price cut by its major competitor. After this competitor’s price
reduction, the company’s sales declined to 80 units a month. From the past experience the
company has estimated the price elasticity of demand to be about -0.2 in this price
quantity range. If the company wishes to restore its sales to 100 units a month, what price
must be charged?
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Interpreting the Elasticity Measure
Once the price elasticity of demand has been calculated, it is necessary to interpret the
meaning of the number obtained. The elasticity coefficient may take on absolute values
over the range from 0 to ∞. Values in the indicated ranges are shown below:
Range Description
Ed = 0 Perfectly inelastic
0< Ed <1 Inelastic
Ed = 1 Unit elastic
1 < Ed < ∞ Elastic
Ed = ∞ Perfectly elastic
When demand is unit elastic, a percentage change in price P is matched by an equal
percentage change in quantity demanded Qd. When demand is elastic, a percentage
change in P is exceeded by the percentage change in Qd. For inelastic demand, a
percentage change in P results in a smaller percentage change in Qd. The two extremes
are perfectly elastic and perfectly inelastic.
One of the more important relationships that may be derived from the demand elasticity
concept is the effect a change in price will have on the total revenue that is generated.
Since total revenue TR is equal to price times the number of units sold, Qd, we may
ascertain from our knowledge of demand elasticity the effect on total revenue when price
changes.
When demand elasticity is less than 1 or inelastic, an increase (decrease) in price will
result in an increase (decrease) in total consumer expenditures (P . Qd). The reason this
occurs is that an inelastic demand indicates that, for example, a percentage increase in
price results in a smaller percentage decrease in quantity sold, the net effect being an
increase in the total expenditure, which is the total revenue from producer’s point of
view.
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On the other hand, when demand is elastic – that is, Ed >1 – a percentage increase
(decrease) in price is more than offset by a larger percentage decrease (increase) in
quantity sold.
When demand is unit elastic, a percentage change in price is exactly offset by the same
percentage change in quantity demanded, the net result being a constant total consumer
expenditure.
When the price elasticity of demand is equal to 1 (or is unit elastic), the total revenue
function is maximized.
Example: Given the following demand function, show that price elasticity of demand is
1 at the output level where total revenue is maximum.
Factors Affecting the Price Elasticity of Demand
Luxury goods vs. Necessities
Luxury items tend to be more price elastic than necessity items. For example, the demand
for wheat, a necessity with few good substitutes, is very price inelastic. In contrast, the
demand for luxury items such as automobile tends to be much more price elastic.
Availability of Substitutes
The greater the number of substitute goods, the more price elastic the demand for a
product. This is to because a customer can easily shift to the substitute good if the price
of the product in question increases. The availability of substitutes related not only to
different products, but also to the availability of the same product from the different
producers.
Durable Goods
The demand for durable goods tends to be more price elastic than the demand for non-
durables. This is true because of the ready availability of a relatively inexpensive
substitute in many cases i.e., repairing a worn-out durable good, such as a TV, car, or
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refrigerator, rather than buying a new one. Consumers of durable goods are often in a
position to wait for a more favorable price, a sale, or a special deal when buying these
items.
Price Level
The demand for high-priced goods tends to be more price elastic than the demand for
inexpensive items. This is true because expensive items account for a greater portion of a
person’s income and potential expenditures than do low-priced items. Consequently, one
would expect the demand for automobiles to be more price elastic than the demand for
children’s toys.
Time Frame of Analysis
Over time, the demand for many products tends to become more elastic. This happens
because of the increase in the number of effective substitutes which become available.
Income Elasticity of Demand
The income elasticity of demand is a measure of the responsiveness of a change in
quantity demanded of some commodity to a change in the consumer’s disposable income.
It may be expressed as
Ey = , ceteris paribus
Where, ∆Y = change in disposable personal income
∆ QD = change in quantity demanded of some product.
Arc Income Elasticity
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The equation for calculating the arc income elasticity is
Ey =
Where, Q2 = quantity sold after an income change
Q1 = quantity sold before an income change
Y2 = new level of income
Y1 = original level of income
Example: Assume that an increase in disposable personal income from $100 million to
$110 million is associated with an increase in sales from 50,000 units to 60,000 units.
What is the income elasticity over this range? Interpret the result.
Point Income Elasticity
The arc income elasticity measures the responsiveness of quantity demanded to changes
in income levels over a range. In contrast, the point income elasticity provides a measure
of this responsiveness at a specific point on the demand function.
The point income elasticity is defined as
Ey =
Where, Y = disposable personal income
QD = quantity demanded of some commodity
= the partial derivative of quantity with respect to disposable
personal income.
Interpretation of Income Elasticity Coefficients
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Those goods having a calculated elasticity that is negative are called ‘inferior
goods’.
Income elasticity is typically defined as being low when it is between 0 and 1, and
high if it is greater than 1.
Those commodities that are normally considered luxury items generally have a
high-income elasticity, whereas goods that are necessities have low income
elasticities.
Cross Elasticity of Demand
Cross elasticity of demand is a measure of responsiveness of changes in the quantity
demanded of Product A to price changes for Product B (PB). The equation for cross
elasticity is given by
Ex = , ceteris paribus.
Where, ∆QDA = change in quantity demanded of Product A
∆PB = change in price of Product B
Arc Cross Elasticity
The arc cross elasticity is computed as
Ex =
Where, QA2 = quantity demanded of A after a price change in B
QA1 = original quantity demanded of A
PB2 = new price for Product B
PB1 = original price for Product B
Example: Suppose the price of Coffee PB increases from Tk.100 to Tk.150 per pound. As
a result, the quantity demanded of Tea QA increases from 500 pounds to 600 pounds a
month at a local grocery store. Compute and interpret arc cross elasticity of demand.
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Point Cross Elasticity
In similar fashion, the point cross elasticity between Products A and B may be computed
as Ex =
Where, PB = Price of Product B
QA = Quantity demanded of Product A when the price of Product
B equals PB
= The partial derivative of QA with respect to PB.
Interpretation of Cross Elasticity
If the cross elasticity measured between items A and B is positive, the two
products are referred to as substitute for each other. The higher the cross
elasticity, the closer the substitute relationship.
A negative cross elasticity, on the other hand, indicates that the two products are
complementary.
Other Elasticity Measures
Advertising Elasticity
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Advertising elasticity measures the responsiveness of sales to changes in advertising
expenditures. It is measured by the ratio of the percentage change in sales to a percentage
change in advertising expenditures.
Elasticity of Price Expectations
In an inflationary environment, the elasticity of price expectations may provide helpful
insights. It is defined as the percentage change in future prices expected as a result of
current percentage price changes. When the coefficient exceeds unity, it indicates that
buyers expect future prices to rise (or fall) by a greater percentage amount than current
prices.
Price Elasticity of Supply
The price elasticity of supply measures the responsiveness of quantity supplied by
producers to changes in prices.
Mathematical Problems
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Problem – 1:
The Photomac Range Corporation manufactures a line of ovens costing $ 500 each. Its
sales have averaged about 3,000 units per month during the past year. In August,
Photomac’s closest competitor, Spring City Works, cut their price for a closely
competitive model from $600 to $ 450. Photomac noticed that its sales volume declined
to 2,500 units per month after Spring City announced its price cut.
Instructions:
a. What is the arc cross elasticity of demand between Photomac’s oven and the
competitive Spring city model?
b. Would you say that two firms are very close competitors? What other factors
could have influenced the observed relationship?
c. If Photomac knows that the arc price elasticity of demand for its ovens is -3.0,
what price would Photomac have to charge in order to sell the same number of
units it did before the Spring City price cut?
Problem – 2:
Olympus Camera Co. manufactures an automatic camera that currently sells for
Tk.10,000. Sales volume is about 100,000 cameras per month. A close competitor, Sony
Ltd., has cut the price of a similar camera it makes from Tk.11,000 to Tk.9,000. The
Olympus company’s economist has roughly estimated the cross elasticity of demand
between the two firms’ products at about 0.4, given current income and price level. What
impact, if any, will the action by Sony have on total revenue generated by Olympus, it
leaves its current price unchanged? What price would Olympus have to charge to earn the
same revenue it earned before the competitor’s price cut, if the arc price elasticity is -3.0?
Problem – 3:
The demand for mobile is given by
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QD = 250,000 – 25P.
(a) How many mobile would be demanded at a price of Tk.2,000? Tk.4,000?
Tk.6,000?
(b) What is the arc price elasticity of demand between Tk.2,000 and Tk.4,000?
Between Tk.4,000 and Tk.6,000?
(c) What is the point price elasticity of demand at Tk.2,000,Tk.4,000, and Tk.6,000?
(d) If 25,000 mobiles were sold last year, what would you expect the average price to
have been?
Problem – 4:
A number of empirical studies of automobile demand have been made yielding the
following estimates of income and price elasticities:
Study Income elasticity Price elasticity
A +3.0 -1.2
B +2.5 -1.4
C +2.5 -1.5
D +3.9 -1.2
Assume also that income and price effects on automobile sales are independent and
additive.
Assume also that the auto companies intend to increase the average price of an
automobile by about 4 percent in the next year, and that next year’s disposable personal
income is expected to be 2 percent higher than this year’s. If this year’s automobile sales
were 11 million units, how many would you expect to be sold under each pair of price
and income elasticity estimates?
Problem – 5:
The generalized linear demand for good X is estimated to be
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Q = 250,000 – 500 P – 1.5 M – 240 PR
Where, P is the price of good X, M is average income of consumers who buy good X, and
PR is the price of related good R. The values of P, M and PR are expected to be Tk.200,
Tk.60,000 and Tk.100 respectively. Use these values at this point on demand to make the
following computations:
(a) Compute the quantity of good X demanded for the given values of P, M and PR.
(b) Calculate the price elasticity of demand, ED. At this point on the demand for X, is
demand elastic, inelastic or unitary elastic? How would increasing the price of X
affect total revenue? Explain.
(c) Calculate income elasticity of demand EM. Is good x normal or inferior? Explain
how a 4% increase in income would affect demand for X, ceteris paribus.
(d) Calculate the Cross-price elasticity EXR. Are the goods X and R substitutes or
complements? Explain how a 5% decrease in the price of related good R would
affect demand for X, ceteris paribus.
Problem – 6:
The demand for the product of ‘X’ Corporation is given by –
Qx = 12,000 – 5,000 Px + 5 I + 500 Pc
Where, Qx = Quantity demanded
Px = Price per unit
I = Income per capita
Pc = The price of good from competitors.
The initial values of Px, I and Pc are Tk.5, Tk.10,000 and Tk.6 respectively.
(a) Determine what effect a price increase would have on total revenues.
(b) Evaluate how sale of the products would change during a period of rising
incomes.
(c) Assess the probable impact if competitors raise their prices.
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