Unformatted text preview: Intermediate Microeconomic Analysis
Demand Instructor: Bin Xie
Spring 2015 Deriving Demand Functions In Chapter 2, we take demand functions as ad hoc.
Now we want to know: why it takes this form
Qs = Qs (p1 , p2 , Y )?
Derive it from the consumer theory (Utility maximization)
How a consumer's choice changes when the price changes? Deriving Demand Functions (Cont.)
Once the maximization problem is solved, demand functions
are automatically derived. Example a 1
CobbDouglas Function U = q1 q2 −a , Budget Constraint p1 q1 + p2 q2 = Y Example Perfect Substitute U = q1 + q2 Example Perfect Complements U = min(q1 , q2 ) Deriving Demand Curves Graphically When the own price changes?
The slope of the budget line changes.
It yields a new optimal bundle.
Priceconsumption curve: the line through all optimal
bundles when the price changes. Use all the new optimum bundles, we can trace out points
along the demand curve. Eects of Change in Income When the income changes?
The increase (decrease) of the income causes the parallel shift
out (in) of the budget constraint.
Incomeconsumption curve: the line through all optimal
bundles when the income changes.
Engle curve: with income on the vertical axis, show the
positive relationship between income and quantity demanded. A change in an individual's income, holding tastes and prices
constant, causes a shift of the demand curve. Income Eect and Income Elasticity Income Elasticity of Demand: ε = ∂Q Y
∆Q/Q
=
∆Y /Y
∂Y Q Normal goods, those goods that we buy more of when our
income increases, have a positive income elasticity.
Luxury goods are normal goods with an income elasticity
greater than 1.
Necessity goods are normal goods with an income elasticity
between 0 and 1. Inferior goods, those goods that we buy less of when our
income increases, have a negative income elasticity. Income Eect and Income Elasticity (Cont.) The shape of the incomeconsumption curve for two goods
tells us the sign of their income elasticities.
The shape of the incomeconsumption and Engle curves can
change in ways that indicate goods can be both normal and
inferior, depending on an individual's income level.
When the income elasticity is positive, the income eect is
positive (higher the income, more of the good is demanded). Eects of a Price Change
, an change
in the price of a good has two eects on an individual's
demand:
Substitution eect: the change in quantity demanded when
the good's price changes, holding other prices
and consumer utility constant.
Income eect: the change in quantity demanded when income
changes, holding prices constant.
When the price of a good increases, the total change in
quantity demanded is the sum of the substitution and income
eects.
Holding tastes, other prices, and income constant Substitution Eect A increase in the price of the good makes the consumer prefers
the other good that becomes relatively cheaper.
It is a movement along an indierence curve.
Less of a good is consumed when its price rises, given that
consumer is hypothetically compensated (for some money) to
stay on the original indierence curve. Income Eect A increase in price reduces the consumer's real buying power
although the monetary income does not change.
It reduces the consumer's real income or oppurtunity set in
terms of good the consumer could purchase.
It yields the same eect as the eect of a pure change in
income.
The change of the quantity demanded is based on whether the
good is normal good or Inferior good. Total Eect Total Eect is the sum of Substitution Eect and Income
Eect
Suppose the price increases:
Normal Good: Substitution Eect? (+ or ) Income Eect?
(+ or )
Inferior Good: Substitution Eect? (+ or ) Income Eect?
(+ or )
Gien Good: Income Eect > Substitution Eect Compensated Demand Curve
The demand curve illustrated above is Uncompensated
Demand Curve (Marshallian Demand Curve). It allows
the utility to vary as the price of the good changes.
Utility falls when the price of the good increases.
Both income eect and substitution eects are accounted for. A Compensated Demand Curve (Hicksian Demand
Curve) shows the change of quantity demanded when the
utility is held constant.
Only the pure substitution eect of the price change is
represented.
An individual must be compensated with extra income as the
price rises in order to hold utility constant. Compensated Demand Curve (Cont.)
Compensated Demand Function: q1 = H(p1 , p2 , U)
1. Solve the expenditure minimization E = E (p1 , p2 , U). Dierentiate with respect to p1 , according to Shephard's
Lemma:
∂E
= H(p1 , p2 , U) = q1
∂p1 2. Use the indirect utility function. Derive the demand functions of all goods qi = qi (p1 , p2 , Y ),
i = 1, 2.
Plug the demand functions into the utility function
U = U(q1 , q2 ) to get U = U(p1 , p2 , Y ) (this is the indirect
utility function).
Rewrite the indirect utility function as Y = Y (p1, p2 , U).
Y is essentially the expenditure E and the utility level will be
¯
givenU = U . Example 0
0
A consumer has a CobbDouglas utility function: U = q1 .4 q2 .6 ,
derive the compensated demand function for good q1 . Slutsky Equation
Total eect could be decomposed as substitution eect and
income eect mathematically as well.
Total eect of a price change is the price elasticity of demand
ε on an uncompensated demand function.
Substitution eect of a price change is the price elasticity of
demand ε∗ on a compensated demand function.
The income eect is the income elasticity ξ times the share of
the budget spent on the good θ. Slutsky Equation: ε = ε∗ + (−θξ) Example 0
0
A consumer has a CobbDouglas utility function: U = q1 .4 q2 .6 ,
show ε = ε∗ + (−θξ). Exercises True, False or Uncertain; explain your answer. When income
rises and the price of x falls, the consumer will always buy
more units of x .
ρ
ρ
CES Utility Function U = q1 + q2 , derive the Engel Curve. (ρ
is a constant.)
Utility Function U = q1 + q1 q2 + q2
1. What is the expenditure function?
2. Derive the uncompensated demand curve.
3. Derive the compensated demand curve....
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 Fall '10
 Raven
 Hicksian Demand, income e1Bect

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