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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