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**INCOME AND SUBSTITUTION EFFECTS**

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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)

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Demand Functions 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 the individual has no control over these parameters

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

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

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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 Note that the indifference curves do not have to be “oddly” shaped. The assumption of a diminishing MRS is obeyed. B U2 A U1 Quantity of x

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**Normal and Inferior Goods**

A good xi for which xi/I 0 over some range of income is a normal good in that range A good xi for which xi/I < 0 over some range of income is an inferior good in that range

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**Changes in a Good’s Price**

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

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**Changes in a Good’s Price**

Even if the individual remained on the same indifference curve when the price changes, his optimal choice will change because the MRS must equal the new price ratio the substitution effect The price change alters the individual’s “real” income and therefore he must move to a new indifference curve the income effect

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**Changes in a Good’s Price**

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

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**Changes in a Good’s Price**

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

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**Changes in a Good’s Price**

Quantity of y The income effect occurs because the individual’s “real” income changes when the price of good x changes 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

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**Changes in a Good’s Price**

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

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**Price Changes for Normal Goods**

If a good is normal, substitution and income effects reinforce one another when price falls, both effects lead to a rise in quantity demanded when price rises, both effects lead to a drop in quantity demanded

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**Price Changes for Inferior Goods**

If a good is inferior, substitution and income effects move in opposite directions The combined effect is indeterminate when price rises, the substitution effect leads to a drop in quantity demanded, but the income effect is opposite when price falls, the substitution effect leads to a rise in quantity demanded, but the income effect is opposite

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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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**The Individual’s Demand Curve**

An individual’s demand for x depends on preferences, all prices, and income: x* = x(px,py,I) 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

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**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

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**Shifts in the Demand Curve**

A movement along a given demand curve is caused by a change in the price of the good a change in quantity demanded A shift in the demand curve is caused by changes in income, prices of other goods, or preferences a change in demand

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**Demand The Cobb-Douglas utility function is**

U(x,y) = xy (+=1) The demand functions for x and y are

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**Demand Functions and Curves**

If the individual’s income is $100, these functions become Any change in income will shift these demand curves

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**Uncompensated 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 know as an uncompensated or Marshallian Demand

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**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 constrain the individual to remain on the same indifference curve reactions to price changes include only substitution effects Also know as Hicksian Demand

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**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)

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**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

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**Compensated & Uncompensated Demand**

px x’’ px’’ At px’’, the curves intersect because the individual’s income is just sufficient to attain utility level U2 x xc Quantity of x

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**A Mathematical Examination of a Change in Price**

xc (px,py,U) = x[px,py,E(px,py,U)] We can differentiate the compensated demand function and get

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

The income effect can be written as Note that E/px = x a $1 increase in px raises necessary expenditures by x dollars

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

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**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)

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**Elasticities Income elasticity of demand (ex,I)**

Cross-price elasticity of demand (ex,py)

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**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

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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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**Homogeneity Dividing by x, we get**

Any proportional change in all prices and income will leave the quantity of x demanded unchanged

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Engel Aggregation We can see this by differentiating the budget constraint with respect to income (treating prices as constant)

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Engel Aggregation Engel’s law suggests that the income elasticity of demand for food items is less than one this implies that the income elasticity of demand for all nonfood items must be greater than one

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