ECN 2115: Lecture 2.

CHANGES IN PRICE AND DEMAND

1. Changes in Goods Own Price.

You saw that the consumer maximizes utility subject to the budget constraint. From there we
can determine the consumer’s optimum bundle. Here we look at what happens to the consumer’s
optimal bundle when own price changes, other factors held constant. That is that we hold the other
price and income constant. Our initial equilibrium will be as depicted in figure 2.3.1.

Figure 2.4.1. Consumer equilibrium

The consumer maximizes the utility function, U (X, Y) subject to the budget constraint M 0 = Px X + Py Y.
This is for a given income, M0 ; and prices Px and Py.

Thus the consumer will buy a bundle consisting of X 0 and Y0. Now what happens when we change
the price of X , but hold the price of X and income constant.

Please recollect that when we change the price of X , hold the level of income and the price of Y
constant, we have a tilt in the budget line. This is depicted in figure 2.4.2.

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Figure 2.4.2: Change in the price of X

The implication of this is that the consumer will move to a higher indifference curve, from A to B. In this
case, the price of X decreases, the budget line tilts from LM to L N. The consumer bundle moves from A
to B. This is called the price effect. The price effect can be defined as the change in the consumer
bundle due to the change in price. We shall see later that the consumer’s response to the change in
price depends on whether the good is a normal good or inferior good.

Now if we continue changing the price of X, holding other factors constant, we can trace the behaviour
of the consumer. We can derive the price consumption curve or price expansion path. This is depicted in
Figure 2.4.3.

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Figure 2.4.3: Price Consumption Curve

In this case, we continue decreasing the price of X. The budget line tilts from LM to LN and LP. The
consumer will move from A to B and to C optimal bundles as the price of X changes. Remember, the
consumer’s tastes, income and the price of Y are constant. We can trace a path from A to C. This path
is referred to as the price consumption curve. We can therefore define the price consumption curve
(PCC) as the locus of points representing various combinations of two goods purchased by the consumer
at different prices of X, all other things remaining the same.

Derivation of Consumer Demand Curve

The consumer demand curve for a good, say, good X, can be derived by using the indifference curve
technique. We have derived the price consumption curve. The curve gives us the necessary data that we
can use to derive the demand curve. To draw the demand curve for good X, we need data on different
prices of good X and the corresponding quantities of good X consumed. This information is provided by
the price consumption curve. We show this in Figure 2.4.4 and Figure 2.4.5.

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Figure 2.4.4: Price Consumption Curve

We first derive the PCC. This will give us the various quantities of X we consume as we change the price
of X. We can derive this from Figure 2.4.4. So if we have three prices of X: P 0, P1, and P2 , this will give
us the three quantities of good X consumed, X 0 , X1 , and X2 . We can depict this in Figure 2.4.5.

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Figure 2.4.5: Demand Curve

This gives us the individual demand curve. This gives us a negative or inverse relationship between price
per unit and quantities of goods consumed.

Income and Substitution Effects of Price Change

We have noticed above that , a change in the price of good X, other factors constant, causes a change in
the demand for good X. We called this change in demand the price effect. The total price effect consists
of two direct effects of a price change on consumer choice. These are the income effect and the
substitution effect.

Income effect is caused by the change in consumer’s real income or purchasing power caused by the
change in price. A rise in price reduces the consumer’s real income and a fall in price increases the
consumer’s real income or purchasing power. But a change in real income causes a change in the
consumer bundle. This is the income effect of a price change.

Substitution effect arises due to a change in the relative price. When the price of good X changes and the
price of good Y is constant, there is a change in relative price. So when the price of one good decreases,
it becomes relatively cheaper than the other good. Conversely, when the price of one good increases, it
becomes relatively costlier than the other good. Consumers have an inherent tendency to substitute
cheaper goods for relatively costlier ones. This is the substitution effect.

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Thus, the total price effect is composed of the income effect and substitution effect. We use
indifference curve analysis to decompose these effects. We try to show this in Figure 2.4.6 below.

Figure 2.4.6: Income and Substitution Effect

Suppose the initial consumer budget is given by budget line AB. Consumer equilibrium is at E 2. The
consumers purchases OX3 of good X. When the price of good X increases, the budget line shifts to AD.
The consumer moves to E1 and purchases X 1 on a lower indifference curve.

The consumer’s consumption of good X decreases from X 3 to X1 . This is the total price effect.

We need to decompose this into the income and substitution effects. To do this we compensate the
consumer of the loss of income due to the price change. We do this by drawing a budget line parallel to
the AD budget line and drag it so that it is tangent to the original indifference curve. This will be budget
line HC , tangent to the original indifference curve at bundle E 2 . This is consumer equilibrium after
compensation.

The movement from E1 to E3 shows an increase in the consumption of good X by O X 2 - O X1 . This rise
in the consumption of good X is a result of the compensation of income. This is the income effect.

Now if we can find the substitution effect. We know that;

Total Effect = Substitution Effect + Income Effect.

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We can then subtract the income effect from the total effect to find the substitution effect. We shall
find that to be OX3 - O X2 in the figure.

2. Changes in the Price of another Good

As we saw above, a change in the price of X also affects the quantity of Y demanded. We can derive the
equation;

∂Y/ ∂Px = ∂Y / ∂Px - X ( ∂ Y / ∂ M)

∂Y / ∂Px is the effect of the change in the price of X on the demand for Y. The first term is the
substitution effect and the second term is the income effect. The income effect here represents the
change in income as a result of changing the price of good X.

This clearly highlights the importance of utility maximization. Whenever, a price changes this will affect
all goods demanded in the bundle, not only the good whose price has changed. These changes can
always be broken down into substitution and income components. One has to analyse the strength and
direction of these components to determine the total change in the quantity demanded.

If we have two goods, we can say the two goods are substitutes if one good, may as a result of changed
conditions, replace the other good in use. Some examples are: tea and coffee, burgers and hotdogs and
drugs and alcohol. Complements on the other hand are goods which “go together” such as: coffee and
milk, fish and chips or brandy and cigars.

So if ∂ Y / ∂ Px > 0 the good Y is a substitute to good X, and

If ∂Y /∂ Px < 0 , good Y is a complement to good X

3. Demand Functions.

We have derived the demand curve. The individual demand curve expresses the relationship of
the per unit and the quantities demanded of a good. If we have two gods X and Y. The demand curve for
good X will give us the relationship between the price per unit of good X and the quantities demanded
of good X. This relationship gives us the law of demand which states that, holding other factors constant,
the price per unit is negatively related to the quantity demanded. This is depicted in the figure below:

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Figure 2.4. 7 : Demand Curve.

We can also express the demand curve as a demand function. Thus we have a utility maximization
problem subject to a budget constraint. Any two goods, X and Y ; income M, and P x and Py . We can
express the demand curve as a demand function. Thus we have a utility maximization problem subject
to a budget constraint. Any two goods, X and Y; income M and prices, Px and Py.

We wish to Maximize U = U (X, Y) subject to the constraint M = X Px + Y Py

We have a direct utility function defined over two goods, X and Y. Our utility maximization problem will
yield demand functions for good X and Y or for any number of goods in the consumer bundle. Thus we
have:

X = f (M, P x, P y) for good X

Y = f (M, Px, Py) for good Y

These demand functions are derived from the first order conditions.

Let us take a specific example.

We have three goods, X 1 , X2 , X 3 ; Income, M and prices; P1 , P2 , P3.

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And utility function U = U( X1 , X2 , X3 ) = 5 log X 1 + 3 log X 2 + 2log X3

We form a Langrangian and maximize the function

L = 5 log X1 + 3 log X2 + 2 log X3 + λ [ M – P1 X1 – P2 X2 – P3 X 3]

∂L/ ∂X 1 = 5/ X 1 - λP1 = 0 (1)

∂L/∂ X2 = 3/X2 – λP2 = 0 (2)

∂L/∂X 3 = 2/ X3 - λP3 = 0 (3)

∂L/∂λ = M – P1 X1 – P2 X2 – P3 X 3 (4)

We solve for X1 , X2 , X 3 and λ . We have;

P 1 X1 = 5/ λ ; P2 X 2 = 2/ λ ; P3 X 3 = 5/ λ (5 )

We substitute these into the budget constraint, we get;

M = 5/λ + 3/ λ + 2/ λ (6 )

And λ = 10/M

We use the information in (5) to get the demand functions. We get;

X1 = M / 2P1

X 2 = 3M/ 10P2

X 3 = M/ 5 P3

These are individual demand functions for the consumer.

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4. Market Demand

We have derived the individual demand curve. In a market, there are many consumers. From
the utility maximization problem, we are able to drive the demand curve for the individual consumer. So
the different consumers in the market will have different individual demand curves.

To get the market demand curve, we need to sum up all individual demand curves for a single good. If
we assume two individuals. Individual I and individual II. The market demand curve for the two
individuals is a horizontal sum of each individual’s demand curve. This is shown in Figure 2.4.8. and
Figure 2.4.9 . below.

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Figure 2.4.8: Individual demand Curves

From the two individual demand curves in figure 2.4.8, we can derive the market demand curve. This
shown below.

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Figure 2.4.9: Market Demand Curves

5. The Concept of Elasticity

We are interested in how quantity demanded responds to changes in a good’s price. A common way of
summarizing this response is by using the concept of elasticity.

The price elasticity of demand for a good is defined to be the percentage change in the quantity of a
good purchased divided by the concomitant percentage change in the good’s price.

Thus we have Q and P. The elasticity of Q with respect to P is defined as

E QP = percentage change in Q / percentage change in P.

= ( ∆ Q/Q ) / ( ∆ P/ P )

= ∆Q / ∆P . P / Q

= ∂ Q/∂ P . P/Q

Since in the usual case ∂Q/ ∂P < 0 , elasticity will usually be negative.

We classify elasticity in three categories: less than, equal to and greater than minus 1. We depict this in
the table below:

Value of E at a point Curve at this point is said to be:

E < -1 Elastic
E = -1 Unit elastic
E > -1 Inelastic

Table 2.4.1: Some Price Elasticity Estimates from US Economy

Good or Service Price Elasticity

Electricity -1.2
Beer -1.9
Movies -0.87
Air Travel (Foreign) -0.77
Shoes -0.70

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Elasticity and Total Expenditure

One of the most important relationship in all of economics is the relationship between price elasticity
and total expenditure. “If the price of a good changes, how will the total expenditure on the good be
affected?”

The general rule is this:

A price reduction will increase total revenue if and only if the absolute value of the price elasticity of
demand is greater than 1.

An increase in price of the good will increase total revenue if and only if the absolute value of price
elasticity is less than 1.

Cross Price Elasticity.

The quantity of a good purchase in the market depends on its price consumer income and prices of
related goods. Cross price elasticity of demand is the percentage change in the quantity demanded of
one good caused by a 1 percent change in the price of another good. Thus for any two goods, X and Y,
the cross price elasticity of demand is

Z = ∆ Q x/ Qx) /∆ Py / P y

We can then define the elasticity for complements and substitutes. If elasticity is positive, it is
substitutes. If elasticity is negative. They are complements.

Table 2.4.2: Cross Price Elasticities for Selected Pairs of Goods in the USA

Good or Service Good or Service with Price Cross Price Elasticity

Change
Butter Margarine + 0.81
Margarine Butter +0.67
Natural Gas Fuel Oil +0.44
Electricity Natural Gas +0.20
Entertainment Food - 0.72
Cereals Fresh Fish - 0.87

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6. Consumer Surplus

When exchange takes place voluntarily, economists generally assume that it makes all participants
better off. Otherwise they would not have engaged in the exchange. It is therefore useful to have a
Kwacha measure of the extent to which people benefit from a transaction. Such a measure is called
Consumer Surplus.

The concept is used a lot in evaluating potential government programmes. For example, if the State
wants to build a road, it is straightforward to measure the costs of building a new road. But one has to
make an estimate of the extent to which consumers will benefit from it.

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