1. Income and Substitution Effects
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2. The (big) Plan Marshallian demand functions (recap) Income effect
Substitution effect
Demand and demand curves
Hicksian (compensated) demand and demand curves
Slutsky decomposition: income- and price effects
Marshallian- and compensated price elasticities
Engel aggregation; Cournot aggregation
[Consumer surplus and compensated variation (CV)]
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3. Marshallian Demand Functions
UMP: The optimal levels of x1,x2,…,xn
Can be expressed as n functions of all prices and income
x1* = x1(p1,p2,…,pn,I)
x2* = x2(p1,p2,…,pn,I) …
xn* = xn(p1,p2,…,pn,I)
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4. Marshallian Demand Functions
Indirect utility function V
V(p1,p2,…,pn,I)=U(x1(p1,p2,…,pn,I),…, xn(p1,p2,…,pn,I))
Maximized utility for any given (p1,p2,…,pn,I)
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5. Marshallian Demand Functions
For only two goods (x and y):
x* = x(px,py,I)
y* = y(px,py,I)
Prices and income are exogenous
The individual has no control over these parameters
Query. In which situations does individual impact on prices?
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6. Cobb-Douglas utility function:
5.1 Homogeneity
Cobb-Douglas utility function:
utility = U(x,y) = x0.3y0.7
The demand functions are: x*=0.3I/px and
y*=0.7I/py
Query. Homogeneity in prices and income
CES utility function:
utility = U(x,y) = x0.5 + y0.5
The demand functions are:
Query. Homogeneity in prices and income
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7. Income Effect: Changes in Income
An increase in income
Will shift the budget line out in a parallel fashion
px/py does not change
The optimal MRS must stay constant as the individual moves to higher levels of satisfaction
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8. Changes in Income Normal good Inferior good Over some range of income
A good xi for which ∂xi/∂I ≥0 over that range of income
Inferior good
A good xi for which ∂xi/∂I < 0 over that range of income
Query. Relate consumption of junk food and high quality food to income!
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9. Effect of an Increase in Income on the Quantities of x and y Chosen
5.1
Effect of an Increase in Income on the Quantities of x and y Chosen I3
Quantity of x
Quantity of y U3 I2 U2 x3 y3 U1 x2 y2 I1 x1 y1
As income increases from I1 to I2 to I3, the optimal (utility-maximizing) choices of x and y are shown by the successively higher points of tangency. Observe that the budget constraint shifts in a parallel way because its slope (given by -px/py) does not change.
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10. An Indifference Curve Map Exhibiting Inferiority
5.2
An Indifference Curve Map Exhibiting Inferiority
Quantity of z
Quantity of y U3 I3 z3 y3 I2 U2 z2 y2 I1 U1 z1 y1
In this diagram, good z is inferior because the quantity purchased decreases as income increases. Here, y is a normal good (as it must be if there are only two goods available), and purchases of y increase as total expenditures increase.
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11. Changes in a Good’s Price
A change in the price of a good
Alters the slope of the budget line (constraint)
Changes the optimal MRS at the consumer’s utility-maximizing choices
Two effects come into play
Substitution effect
Income effect in addition
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12. Changes in a Good’s Price
Substitution effect of a price change
Consumption patterns would be allocated so as to equate the MRS to the new price ratio
Query. What about perfect substitutes?
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13. Changes in a Good’s Price
Income effect of a price change
Arises because a price change necessarily changes an individual’s real income
The individual cannot stay on the initial indifference curve and must move to a new one
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14. 5.3
Demonstration of the Income and Substitution Effects of a Decrease in the Price of x
When the price of x decreases from p1x to p2x , the utility-maximizing choice shifts from x*,y* to x**,y**. This movement can be broken down into two analytically different effects: first, the substitution effect, involving a movement along the initial indifference curve to point B, where the MRS is equal to the new price ratio; and second, the income effect, entailing a movement to a higher level of utility because real income has increased. In the diagram, both the substitution and income effects cause more x to be bought when its price decreases. Notice that point I/py is the same as before the price change; this is because py has not changed. Therefore, point I/py appears on both the old and new budget constraints.
Quantity of x
Quantity of y U2
I/py
I=p2xx+pyy
I=p1xx+pyy U1
y**
x** y* x* B xB
Substitution effect
Income effect
Total increase in x
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15. 5.4
Demonstration of the Income and Substitution Effects of an Increase in the Price of x
When the price of x increases, the budget constraint shifts inward. The movement from the initial utility-maximizing point (x*,y*) to the new point (x**,y**) can be analyzed as two separate effects. The substitution effect would be depicted as a movement to point B on the initial indifference curve (U2). The price increase, however, would create a loss of purchasing power and a consequent movement to a lower indifference curve. This is the income effect. In the diagram, both the income and substitution effects cause the quantity of x to decrease as a result of the increase in its price. Again, the point I/py is not affected by the change in the price of x.
Quantity of x
Quantity of y
I/py
I=p1xx+pyy
I=p2xx+pyy U1 U2 B xB
y**
x** y* x*
Income effect
Substitution effect
Total decrease in x
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16. Changes in a Good’s Price
If a good is normal, substitution and income effects reinforce one another
When p :
Substitution effect: quantity demanded 
Income effect: quantity demanded 
When p :
Substitution effect: quantity demanded 
Income effect: quantity demanded 
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17. Changes in a Good’s Price
If a good is inferior, substitution and income effects move in opposite directions
When p :
Substitution effect: quantity demanded 
Income effect: quantity demanded 
When p :
Substitution effect: quantity demanded 
Income effect: quantity demanded 
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18. Changes in a Good’s Price
Giffen’s paradox
If the income effect of a price change is strong enough, there could be a positive relationship between price and quantity demanded
An increase in price leads to a drop in real income
Since the good is inferior, a drop in income causes quantity demanded to rise
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19. Changes in a Good’s Price
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
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20. The Individual’s Demand Curve
An individual‘s Marshallian demand for x
Depends on preferences, all prices, and income: x* = x(px,py,I)
It is convenient to graph it assuming that income and the price of y (py) are held constant: demand curve
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21. Construction of an Individual’s Marshallian Demand Curve
5.5
Construction of an Individual’s Marshallian Demand Curve
Quantity of y
Quantity of x
Quantity of x px
I = px’’’x + pyy
I = px’x + pyy
I / py
I = px’’x + pyy U3 U2 U1
px’ x x’
px’’
x’’ x’ x”
x”’
Px‘‘‘
x’’’
In (a), the individual’s utility-maximizing choices of x and y are shown for three different prices of x(p’x, p”x, and p’”x). In (b), this relationship between px and x is used to construct the demand curve for x. The demand curve is drawn on the assumption that py, I, and preferences remain constant as px varies.
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22. The Individual’s Demand Curve
An individual demand curve
Shows the relationship between the price of a good and the quantity of that good purchased by an individual
Assuming that all other determinants of demand are held constant
Visually graphs Marshallian demand, holding constant “other determinants”
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23. The Individual’s Demand Curve
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 demand :. quantity demanded
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24. The Individual’s Demand Curve
A change in quantity demanded
Movement along a given demand curve
Caused by a change in the price of the good
A change in demand
Shift in the demand curve
Caused by changes in income, prices of other goods, or preferences
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25. 5.2 Demand Functions and Demand Curves
Cobb-Douglas
The demand functions: x=0.3I/px and y=0.7I/py
For I=$100: x=30/px and y=70/py
Or: pxx=30 and pyy=70
demand curves: simple hyperbolas
An increase in income would shift both demand curves outward
The demand curve for x is not shifted by changes in py and vice versa – Query: How is this possible?
Income effect: neg as higher py lowers real income; subst positive, as higher py lowers y and raises x; both effects exactly cancel.
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26. 5.2 Demand Functions and Demand Curves
CES
For good x:
For I=$100 and py=1: x=100/(px2+px)
General hyperbolic relationship between price and quantity consumed
Increases in I or py would shift the demand curve for good x outward because:
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27. Uncompensated /Compensated Demand Curves
Approach 1: Price change for given income level
Marshallian = uncompensated demand curves
As price increases, income fixed, utility declines
Reactions to price changes include both income and substitution effects
Approach 2: Price change for given utility level
Hicksian = compensated demand curves
As price increases, utility fixed, income increases
Reactions to price changes include only substitution effects
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28. Marshallian Demand Curves and Functions
Actual level of utility
Varies along the demand curve
As the price of x falls, the individual moves to higher indifference curves
nominal income is held constant as the demand curve is derived
real income rises as the price of x falls
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29. Hicksian Demand Curves and Functions
Price change for given utility level
Hold utility constant while examining reactions to changes in px
effects of the price change are “compensated” by income changes so as to force the individual to remain on the same indifference curve
Reactions to price changes include only substitution effects
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30. Compensated Demand Curves and Functions
Compensated (Hicksian) demand curve
Shows the relationship between the price of a good and the quantity purchased
while other prices and utility are held constant (income adjusted)
Is a two-dimensional representation of the compensated demand function (= solution of the EMP)
x* = xc(px,py,U)
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31. Construction of a Compensated Demand Curve
5.6
Construction of a Compensated Demand Curve
Quantity of y
Quantity of x
Quantity of x px U2
Slope = p’x/py x’
Slope = p”x/py xc
px’ x’
Slope = p”’x/py
px’’
x’’ x”
px’’’
x’’’
x”’
The curve xc shows how the quantity of x demanded changes when px changes, holding py and utility constant. That is, the individual’s income is ‘‘compensated’’ to keep utility constant. Hence xc reflects only substitution effects of changing prices.
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32. Hicksian Demand Functions
Expenditure minimization problem (recap)
Min expenditure: pxx+pyy s.t. U(x,y)≥Ū
Lagrangian: ℒ =pxx+pyy+λ[U(x,y)-Ū]
Solution:
Hicksian demand functions: x(px,py, Ū)
expenditure function: E(px,py, Ū) = px x(px,py, Ū) + py y(px,py, Ū)
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33. Expenditure function and Hicksian Demand Functions
Shephard’s lemma
Compensated demand function for a good
Derived from the expenditure function
Envelope theorem:
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34. Compensated / Uncompensated Demand Curves
Relationship between compensated and uncompensated demand curves
Normal good
Compensated demand curve is less responsive to price changes than is the uncompensated demand curve
Uncompensated demand curve reflects both income and substitution effects
Compensated demand curve reflects only substitution effects
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35. Comparison of Compensated and Uncompensated Demand Curves
5.7
Comparison of Compensated and Uncompensated Demand Curves
Quantity of x px
xc(px,py,U)
x(px,py,I)
px’ x’ x*
px’’
x’’
px’’’
x**
x’’’
The compensated (xc) and uncompensated (x) demand curves intersect at p”x because x” is demanded under each concept. For prices above p”x, the individual’s purchasing power must be increased with the compensated demand curve; thus, more x is demanded than with the uncompensated curve. For prices below p”x, purchasing power must be reduced for the compensated curve; therefore, less x is demanded than with the uncompensated curve. The standard demand curve is more price-responsive because it incorporates both substitution and income effects, whereas the curve xc reflects only substitution effects.
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36. 5.3 Compensated Demand Functions
The utility is: utility = U(x,y) = x0.5y0.5
The Marshallian demand functions:
x(px,py,I) = 0.5I/px and y(px,py,I) = 0.5I/py
Expenditure function: E(px,py, Ū)=2px0.5py0.5Ū
Use Shephard’s lemma to calculate the compensated demand functions as
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37. A Mathematical Development of Response to Price Changes
Consider two goods, x and y
Important relationship between uncompensated and compensated demand
Let U be an arbitrary utility level; I be minimum expenditure required to obtain that U: I = E(px,py,U)
xc (px,py,U) = x [px,py,I] = x [px,py,E(px,py,U)]
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38. A Mathematical Development of Response to Price Changes
Differentiate xc (px,py,U) = x [px,py,E(px,py,U)] with respect to px
The substitution effect
The first term: the slope of the compensated demand curve
The income effect
The second term: measures the way in which changes in px affect the demand for x through changes in purchasing power
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39. A Mathematical Development of Response to Price Changes
The Slutsky equation
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40. 5.4 A Slutsky Decomposition
Marshallian demand function for good x
x(px,py,I) = 0.5I/px
Compensated demand function for this good
xc(px,py, U)=px-0.5py0.5U
Total effect of a price change on Marshallian demand
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41. 5.4 A Slutsky Decomposition
Query. Verify that total effect = SE + IE
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42. Elasticities Changes in prices or income ☞ change in demand
absolute changes depend on units of measurement
elasticities: %-measure, independent of units
Elasticities
not restricted to demand changes due to price/income
changes between any two variables
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43. Marshallian Demand Elasticities
Marshallian demand function: x(px,py,I)
1. Price elasticity of demand, ex, px
Measures the proportionate change in quantity demanded
In response to a proportionate change in a good’s own price
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44. Marshallian Demand Elasticities
2. Income elasticity of demand, ex,I
Measures the proportionate change in quantity demanded
In response to a proportionate change in income
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45. Marshallian Demand Elasticities
3. Cross-price elasticity of demand, ex,py
Measures the proportionate change in quantity of x demanded
In response to a proportionate change in the price of some other good (y)
Query. In which case is the cross-price elasticity equal to zero?
If IE = SE (for price increase of other good) – Cobb Douglas case!
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46. Price elasticity of demand
The own price elasticity of demand is always negative
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
Query. Why is size important?
If you increase price to raise revenue – revenue only increases when elasticity is low.
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47. Price Elasticity of Demand
Total spending on x = pxx
If ex,px > -1, demand is inelastic
Price and total spending move in the same direction
If ex,px < -1, demand is elastic
Price and total spending move in opposite directions
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48. Compensated Price Elasticities
Compensated demand function, xc(px,py,U)
Compensated own-price elasticity of demand, exc,px
Measures the proportionate compensated change in quantity demanded
In response to a proportionate change in a good’s own price
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49. Compensated Price Elasticities
2. Compensated cross-price elasticity of demand, exc,py
Measures the proportionate compensated change in quantity of x demanded
In response to a proportionate change in the price of another good, y
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50. Compensated Price Elasticities
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51. Compensated Price Elasticities
Slutsky equation:
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
Sure, if income effect is small then the elasticities are similar
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52. Relationships among Demand Elasticities
Homogeneity
Demand functions are homogeneous of degree zero in all prices and income
Euler’s theorem for homogenous functions shows that
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53. Relationships among Demand Elasticities
Engel aggregation
Engel’s law: income elasticity of demand for food items is <1
Income elasticity of demand for all nonfood items must be >1
Differentiate the budget constraint with respect to income
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54. Relationships among Demand Elasticities
Cournot aggregation
The size of the cross-price effect of a change in px on the quantity of y consumed is restricted because of the budget constraint
Differentiate the budget constraint with respect to px
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55. Relationships among Demand Elasticities
Cournot aggregation
First, multiply with px/I
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56. 5.5 Demand Elasticities: The Importance of Substitution Effects
Cobb-Douglas: U(x,y)=xαyβ, α+β=1
Demand functions: x(px,py,I)=αI/px, and y(px,py,I)=βI/py=(1- α)I/py
Elasticities:
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57. Cobb-Douglas: 5.5 Demand Elasticities Because sx = α and sy = β
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58. 5.5 Demand Elasticities: The Importance of Substitution Effects
Cobb-Douglas:
Using the Slutsky equation in elasticity form to derive the compensated price elasticity:
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59. Application: Welfare Change (CV)
Measure the change in welfare
That an individual experiences if the price of good x increases from p0x to p1x
To reach U0
Expenditure at p0x: E(p0x,py,U0)
Expenditure at p1x: E(p1x,py,U0)
To compensate for the price increase – compensating variation (CV) of:
CV = E(px1,py,U0) - E(px0,py,U0)
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60. 5.8 (a) Indifference curve map
Showing Compensating Variation
Spending on
other goods ($)
Quantity of x
E(p1x,py,U0) CV U0
E(p0x,py,U0) U1
E(p0x,py,U0) y1 x1 y2 x2 y0 x0
If the price of x increases from p0x to p1x, this person needs extra expenditures of CV to remain on the U0 indifference curve. Integration shows that CV can also be represented by the shaded area below the compensated demand curve in panel (b).
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61. 5.8 (b) Compensated demand curve
Showing Compensating Variation
Price
Quantity of x x1 x0
xc(px,…,U0)
p2x
p1x B
p0x A
If the price of x increases from p0x to p1x, this person needs extra expenditures of CV to remain on the U0 indifference curve. Integration shows that CV can also be represented by the shaded area below the compensated demand curve in panel (b).
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62. Relationships among Demand Concepts
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