# Income and Substitution Effects

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Income and Substitution Effects
Chapter 5 Income and Substitution Effects

Demand Functions The optimal levels of x1,x2,…,xn can be expressed as functions of all prices and income These can be expressed as n demand functions of the form: x1* = d1(p1,p2,…,pn,I) x2* = d2(p1,p2,…,pn,I) xn* = dn(p1,p2,…,pn,I) If there are only two goods (x and y), we can simplify the notation x* = x(px,py,I) y* = y(px,py,I) Prices and income are exogenous

xi* = xi(p1,p2,…,pn,I) = xi(tp1,tp2,…,tpn,tI)
Homogeneity If all prices and income were doubled, the optimal quantities demanded will not change the budget constraint is unchanged xi* = xi(p1,p2,…,pn,I) = xi(tp1,tp2,…,tpn,tI) Individual demand functions are homogeneous of degree zero in all prices and income

Changes in Income An increase in income will cause the budget constraint to shift out in a parallel fashion Since px/py does not change, the MRS will stay constant as the individual moves to higher levels of satisfaction

Effects of a Rise in Income
Income-consumption curve shows how consumption of both goods changes when income changes, while prices are held constant. Engel curve - the relationship between the quantity demanded of a single good and income, holding prices constant.

Effect of a Budget Increase on an Individual’s Demand Curve
Y L 1 2.8 e Budget Line, L 1 I 1 26.7 X I PX Y = - X PY PY E1 Price of X 12 Initial Values PX = price of X = \$12 PY = price of Y = \$35 I = Income = \$419. D 1 26.7 X , Budget I Income goes up! Y 1 = \$419 E * 1 26.7 X

Effect of a Budget Increase on an Individual’s Demand Curve
y L 2 Y L 1 4.8 e 2 2.8 e Budget Line, L 1 I 2 I 1 26.7 38.2 X I PX Y= - X PY PY E E Price of X 12 1 2 Initial Values PX = price of X = \$12 PY = price of Y = \$35 I = Income = \$419. D 2 D 1 26.7 38.2 X , Budget I Income goes up! \$628 Y = \$628 2 E * 2 Y = \$419 1 E * 1 26.7 38.2 X

Effect of a Budget Increase on an Individual’s Demand Curve
2 Y L 1 Income-consumption curve e 3 7.1 4.8 e 2 2.8 e I3 Budget Line, L 1 I 2 I 1 26.7 38.2 49.1 X I PX Y = - X PY PY E E E Price of X 12 1 2 3 Initial Values PX = price of X = \$12 PY = price of Y = \$35 I = Income = \$837. D3 D 2 D 1 26.7 38.2 49.1 X , Budget I Engel curve for X Y = \$837 * 3 E 3 Income goes up again! Y = \$628 2 E * 2 Y = \$419 1 E * 1 26.7 38.2 49.1 X

Increase in Income If both x and y increase as income rises, x and y are normal goods As income rises, the individual chooses to consume more x and y Quantity of y C U3 B U2 A U1 Quantity of x

Normal and Inferior Goods
• If the income consumption curve shows that the consumer purchases more of good x as her income rises, good x is a normal good. • Equivalently, if the slope of the Engel curve is positive, the good is a normal good. • If the income consumption curve shows that the consumer purchases less of good x as her income rises, good x is an inferior good. • Equivalently, if the slope of the Engel curve is negative, the good is an inferior good.

Increase in Income If x decreases as income rises, x is an inferior good As income rises, the individual chooses to consume less x and more y Quantity of y C U3 B U2 A U1 Quantity of x

Normal and Inferior Goods
Example: Backward Bending ICC and Engel Curve – a good can be normal over some ranges and inferior over others

Income Consumption Curve: Normal and Inferior Goods

• • • Price Consumption Curves
Is the set of optimal baskets for every possible price of good x, holding all other prices and income constant. Y (units) PY = \$4 I = \$40 10 Price Consumption Curve PX = 1 PX = 2 PX = 4 X (units) XA=2 XB=10 XC=16 20

• • • Individual Demand Curve PX
Individual Demand Curve For Commodity X PX = 4 PX = 2 Quantity increasing PX = 1 X XA XB XC

Individual Demand Curve: Properties
1) The level of utility that can be attained changes as we move along the curve. 2) At every point on the demand curve, the consumer is maximizing utility by satisfying the condition that MRS of X for Y = the ratio of the prices of X and Y.

Changes in a Good’s Price
A change in the price of a good alters the slope of the budget constraint It changes the MRS at the consumer’s utility-maximizing choices When the price changes, two effects come into play substitution effect income effect

The Substitution Effect
As the price of x falls, all else constant, good x becomes cheaper relative to good y and vice versa. This change in relative prices alone causes the consumer to adjust his/ her consumption basket. This effect is called the substitution effect. The substitution effect always is negative.

Income Effect As the price of x falls, all else constant, purchasing power rises and vice versa. This is called the income effect of a change in price. The income effect may be positive (normal good) or negative (inferior good).

Changes in a Good’s Price: Price of X falls
U1 A Suppose the consumer is maximizing utility at point A. Quantity of y U2 B If the price of good x falls, the consumer will maximize utility at point B. Total increase in x Quantity of x

Changes in a Good’s Price: Price of X falls
Quantity of y To isolate the substitution effect, we hold “real” income constant but allow the relative price of good x to change The substitution effect is the movement from point A to point C A C The individual substitutes x for y because it is now relatively cheaper Substitution effect U1 Quantity of x

Changes in a Good’s Price : Price of X falls
The income effect occurs because “real” income changes when the price of good x changes Quantity of y B The income effect is the movement from point C to point B If x is a normal good, the individual will buy more because “real” income increased Income effect A C U2 U1 Quantity of x Total Effect or Price Effect(AB) = Substitution Effect (AC) Income Effect (CB)

Changes in a Good’s Price: Price of X rises
Quantity of y An increase in the price of good x means that the budget constraint gets steeper C The substitution effect is the movement from point A to point C Substitution effect A Income effect The income effect is the movement from point C to point B B U1 U2 Quantity of x Total Effect or Price Effect (AB) = Substitution Effect (AC) + Income Effect (CB)

Price Changes – Normal Goods
If a good is normal, substitution and income effects reinforce one another when pX : substitution effect  quantity demanded of X income effect  quantity demanded of X  when pX : substitution effect  quantity demanded of X  income effect  quantity demanded of X 

Price Changes – Inferior Goods
If a good is inferior, substitution and income effects move in opposite directions when pX : substitution effect  quantity demanded of X  income effect  quantity demanded of X  when pX : substitution effect  quantity demanded of X  income effect  quantity demanded of X 

INCOME AND SUBSTITUTION EFFECTS: INFERIOR GOOD
Consumer is initially at point A on RS. With a ↓ PFood, the consumer moves to point B. a substitution effect, F1E (associated with a move from A to D), an income effect, EF2 (associated with a move from D to B). In this case, food is an inferior good because the income effect is negative.

A Special Case: The Giffen Good
Giffen good Good whose demand curve slopes upward because the (negative) income effect is larger than the substitution effect. Food: an inferior good Consumer is initially at A After the ↓P of food, moves to B and consumes less food. The income effect F2F1 > the substitution effect EF2, Income effect dominates over the substitution effect The ↓Pfood leads to a lower quantity of food demanded. UPWARD-SLOPING DEMAND CURVE: THE GIFFEN GOOD

For inferior goods, no definite prediction can be made for changes in price the substitution effect and income effect move in opposite directions if the income effect outweighs the substitution effect, we have a case of Giffen’s paradox Giffen Paradox: If the income effect of a price change is strong enough, there could be a positive relationship between price and quantity demanded an decrease in price leads to a increase in real income since the good is inferior, a increase in income causes quantity demanded to fall

The Individual’s Demand Curve
An individual’s demand for x depends on preferences, all prices, and income: x* = x(px,py,I) It may be convenient to graph the individual’s demand for x assuming that income and the price of y i.e (py) are held constant

The Individual’s Demand Curve
Quantity of y As the price of x falls... px x’ px’ U1 x1 I = px’ + py x’’ px’’ U2 x2 I = px’’ + py x’’’ px’’’ x3 U3 I = px’’’ + py x …quantity of x demanded rises. Quantity of x Quantity of x

Shifts in the Demand Curve
Three factors are held constant when a demand curve is derived income prices of other goods (py) the individual’s preferences If any of these factors change, the demand curve will shift to a new position Difference between a change in quantity demanded and change in demand

Compensated Demand Curve
The demand curves shown so far have all been uncompensated, or Marshallian, demand curves. Consumer utility is allowed to vary with the price of the good. Alternatively, we have a compensated, or Hicksian, demand curve. It shows how quantity demanded changes when price increases, holding utility constant.

Compensated Demand Curves
The actual level of utility varies along the demand curve As the price of x falls, the individual moves to higher indifference curves it is assumed that nominal income is held constant as the demand curve is derived this means that “real” income rises as the price of x falls

Compensated Demand Curves
An alternative approach holds real income (or utility) constant while examining reactions to changes in px the effects of the price change are “compensated” so as to force the individual to remain on the same indifference curve reactions to price changes include only substitution effects

Compensated Demand Curves
A compensated (Hicksian) demand curve shows the relationship between the price of a good and the quantity purchased assuming that other prices and utility are held constant The compensated demand curve is a two-dimensional representation of the compensated demand function x* = xc(px,py,U)

Compensated Demand Curves
Holding utility constant, as price falls... Quantity of y px x’ px’ Xc …quantity demanded rises. U2 x’’ px’’ x’’’ px’’’ Quantity of x Quantity of x

Compensated & Uncompensated Demand
x’’ px’’ At px’’, the curves intersect because the individual’s income is just sufficient to attain a certain utility level px x xc Quantity of x

Compensated & Uncompensated Demand
x* x’ px’ At prices above px’’, income compensation is positive because the individual needs more income to remain on a certain utility level px px’’ x xc Quantity of x

Compensated & Uncompensated Demand
px x*** x’’’ px’’’ At prices below px’’, income compensation is negative to prevent an increase in utility from a lower price px’’ X xc Quantity of x

Compensated & Uncompensated Demand
For a normal good, the compensated demand curve is less responsive to price changes than is the uncompensated demand curve the uncompensated demand curve reflects both income and substitution effects the compensated demand curve reflects only substitution effects

The Response to a Change in Price
We will use an indirect approach using the expenditure function minimum expenditure = E(px,py,U) Then, by definition xc (px,py,U) = x [px,py,E(px,py,U)]

The Response to a Change in Price
xc (px,py,U) = x[px,py,E(px,py,U)] We can differentiate the function and get

The Response to a Change in Price
The first term is the slope of the compensated demand curve the mathematical representation of the substitution effect

The Response to a Change in Price
The second term measures the way in which changes in px affect the demand for x through changes in purchasing power the mathematical representation of the income effect

The Slutsky Equation The substitution effect can be written as
The income effect can be written as

The Slutsky Equation The utility-maximization hypothesis shows that the substitution and income effects arising from a price change can be represented by

The Slutsky Equation The first term is the substitution effect
always negative The second term is the income effect if x is a normal good, entire income effect is negative if x is an inferior good, entire income effect is positive

Marshallian Demand Elasticities
Most of the commonly used demand elasticities are derived from the Marshallian demand function x(px,py,I) Price elasticity of demand (ex,px) Income elasticity of demand (ex,I) Cross-price elasticity of demand (ex,py)

Price Elasticity of Demand
The own price elasticity of demand is always negative the only exception is Giffen’s paradox The size of the elasticity is important if ex,px < -1, demand is elastic if ex,px > -1, demand is inelastic if ex,px = -1, demand is unit elastic

Compensated Price Elasticities
It is also useful to define elasticities based on the compensated demand function If the compensated demand function is xc = xc(px,py,U) we can calculate compensated own price elasticity of demand (exc,px) compensated cross-price elasticity of demand (exc,py)

Price Elasticities The Slutsky equation shows that the compensated and uncompensated price elasticities will be similar if the share of income devoted to x is small the income elasticity of x is small

Consumer Welfare Consumer Surplus
Suppose we want to examine the change in an individual’s welfare when price changes Consumer Welfare If the price rises, the individual would have to increase expenditure to remain at the initial level of utility expenditure at px0 = E0 = E(px0,py,U0) expenditure at px1 = E1 = E(px1,py,U0) In order to compensate for the price rise, this person would require a compensating variation (CV) of CV = E(px1,py,U0) - E(px0,py,U0)

Consumer Welfare Suppose the consumer is maximizing
Quantity of y Suppose the consumer is maximizing utility at point A. U1 B If the price of good x rises, the consumer will maximize utility at point B. The consumer’s utility falls from U2 to U1 A U2 Quantity of x

Consumer Welfare Quantity of y
The consumer could be compensated so that he can afford to remain on U2 C CV is the amount that the individual would need to be compensated CV A B U2 U1 Quantity of x

Consumer Welfare When the price rises from px0 to px1,
the consumer suffers a loss in welfare welfare loss px1 x1 px0 x0 xc(px…U0) Quantity of x

Consumer Welfare A price change generally involves both income and substitution effects should we use the compensated demand curve for the original target utility (U0) or the new level of utility after the price change (U1)?

Consumer Welfare Is the consumer’s loss in welfare best described by area px1BApx0 [using xc(...,U1)] or by area px1CDpx0 [using xc(...,U0)]? px Is U1 or U0 the appropriate utility target? C B px1 A px0 D xc(...,U1) xc(...,U0) x1 x0 Quantity of x

Consumer Welfare We can use the Marshallian demand curve as a compromise px The area px1CApx0 falls between the sizes of the welfare losses defined by xc(...,U1) and xc(...,U0) C x(px,…) B px1 A px0 D xc(...,U1) xc(...,U0) x1 x0 Quantity of x

Consumer Surplus We will define consumer surplus as the area below the Marshallian demand curve and above the prevailing market price shows what an individual would pay for the right to make voluntary transactions at this price changes in consumer surplus measure the welfare effects of price changes

Compensating Variation versus Equivalent Variation
p1 rises. Q: What is the least extra income that, at the new prices, just restores the consumer’s original utility level? A: The Compensating Variation.

Compensating Variation
p1=p1’ p2 is fixed. x2 u1 x1

Compensating Variation
p1=p1’ p1=p1” (Price rise) p2 is fixed. x2 u1 u2 x1

Compensating Variation
p2 is fixed. p1=p1’ p1=p1” (Price rise) x2 u1 CV = I2 - I1. u2 x1

Equivalent Variation p1 rises.
Q: What is the least extra income that, at the original prices, just restores the consumer’s final utility level? A: The Equivalent Variation.

Equivalent Variation p1=p1’ p2 is fixed. x2 u1 x1

Equivalent Variation p1=p1’ p1=p1” (Price rise) p2 is fixed. x2 u1 u2

Equivalent Variation EV = I1 - I2. p2 is fixed.
p1=p1’ p1=p1” (Price rise) x2 u1 EV = I1 - I2. u2 x1

CV and EV for a fall in the price of X

The effect on (behaviour and) welfare of a change in a price.
Let us now look at these two variations with quasi-linear preferences....

Compensating vs. Equivalent Variation Quasi –linear Preferences

From Individual Demand to Market Demand
The market demand function is the horizontal sum of the individual (or segment) demands. In other words, market demand is obtained by adding the quantities demanded by the individuals (or segments) at each price and plotting this total quantity for all possible prices.

Market Demand P P P 10 P = 10 - Q P = 4 – 0.2Q 4 Q Q Q Segment 1

Market Demand Two points should be noted:
1.The market demand curve will shift to the right as more consumers enter the market. 2. Factors that influence the demands of many consumers will also affect market demand. The aggregation of individual demands into market becomes important in practice when market demands are built up from the demands of different demographic groups or from consumers located in different areas.

Network Externalities
Up to this point we have assumed that people’s demands for a good are independent of one another. This assumption has led to obtain the market demand curve simply by summing individual’s demand. For some goods, one person’s demand also depends on the demands of other people If one consumer's demand for a good changes with the number of other consumers who purchased the good, there are network externalities Network externality: When each individual’s demand depends on the purchases of other individuals.

Network Externalities
Network externalities can be positive or negative A positive network externality exists if the quantity of a good demanded by a consumer increases in response to an increase in purchases by other consumers Negative network externalities are just the opposite

Positive Network Externalities
Bandwagon effect: A positive network externality that refers to the increase in each consumer’s demand for a good as more consumers buy the good E.g: Toys for kids

Positive Network Externality: Bandwagon Effect
Price (\$ per unit) D20 20 40 D40 60 D60 80 D80 100 D100 When consumers believe more people have purchased the product, the demand curve shifts further to the the right. Quantity (thousands per month)

Positive Network Externality: Bandwagon Effect
Price (\$ per unit) D20 20 40 D40 60 D60 80 D80 100 D100 The market demand curve is found by joining the points on the individual demand curves. It is relatively more elastic. Demand Quantity (thousands per month)

Positive Network Externality: Bandwagon Effect
Price (\$ per unit) D20 20 40 D40 60 D60 80 D80 100 D100 But as more people buy the good, it becomes stylish to own it and the quantity demanded increases further. Suppose the price falls from \$30 to \$20. If there were no bandwagon effect, quantity demanded would only increase to 48,000 \$30 Demand 48 \$20 Bandwagon Effect Pure Price Effect Quantity (thousands per month)

Network Externalities
Snob effect: A negative network externality that refers to the decrease in each consumer’s demand as more consumers buy the good The snob effect refers to the desire to own exclusive or unique goods The quantity demanded of a “snob” good is higher the fewer the people who own it

Network Externality: Snob Effect
Price (\$ per unit) Demand Originally demand is D2, when consumers think 2,000 people have bought a good. \$30,000 4 6 8 D4 D6 D8 However, if consumers think 4,000 people have bought the good, demand shifts from D2 to D6 and its snob value has been reduced. \$15,000 D2 Pure Price Effect Quantity (thousands per month) 2 14

Network Externality: Snob Effect
Quantity (thousands per month) Price (\$ per unit) 2 Demand D2 \$30,000 \$15,000 14 4 6 8 D4 D6 D8 Pure Price Effect The demand is less elastic and as a snob good its value is greatly reduced if more people own it. Net Effect Snob Effect

Revealed Preference Hypothesis
The theory of revealed preference was proposed by Paul Samuelson in the year 1938. Considered major breakthrough in the theory of demand . It establishes law of demand without using IC and their respective assumptions. Assumptions: 1) Rationality 2) Consistency 3) Transitivity 4) Revealed Preference axiom

Revealed Preference Axiom
Consider two bundles of goods: A and B If the individual can afford to purchase either bundle but chooses A, we say that A had been revealed preferred to B The chosen basket A maximizes (axiomatically) his utility and rest of the baskets on the budget line are revealed inferior.

Revealed Preference Hypothesis: Derivation of Demand Curve
Initially consumer at B with budget line RS. A ↓ PF shifts budget line from RS to RT. The new market basket chosen must lie on line segment BT' of budget line R′T' (which intersects RS to the right of B), and the quantity of food consumed must be greater than at B.

The Weak and the Strong Axioms of the Revealed Preference Theory
While listing the assumptions of the RPT, the ordering principles (consistency and transitivity) of consumer preferences are mentioned We combine the ordering principles with the RP axioms to state and establish 1) Weak Axiom of Revealed Preference (WARP) 2) Strong Axiom of Revealed Preference (SARP)

Weak Axiom of Revealed Preference (WARP)
If bundle (x1, y1) is directly revealed preferred to bundle (x2, y2), the two bundles being different from each other, it cannot happen that bundle (x2, y2) would be directly revealed preferred to bundle (x1, Y1). That is: If (x1,y1) is revealed preferred when (x2,y2) was affordable then (x1,y1) is preferred always (or at all prices).

Weak Axiom of Revealed Preference (WARP)
If (x1, y1) is directly revealed preferred to (x2,y2), and the two bundles are not the same, then it cannot happen that (x2,y2) is directly revealed preferred to (x1, y1). This is the weak axiom of revealed preference (WARP) If this occurs then WARP is violated

Strong Axiom of Revealed Preference
If commodity bundle 0 is revealed preferred to bundle 1, and if bundle 1 is revealed preferred to bundle 2, and if bundle 2 is revealed preferred to bundle 3,…, and if bundle K-1 is revealed preferred to bundle K, then bundle K cannot be revealed preferred to bundle 0 If (x1,y1) is revealed preferred to (x3,y3) either directly or indirectly) then (x3,y3) cannot be directly or indirectly revealed preferred to (x1,y1) SARP is a necessary and sufficient condition for observed behaviour to be consistent with the underlying model of consumer choice