Midterm 2 Review - econweb.ucsd.edueconweb.ucsd.edu/~vleahmar/pdfs/ECON 100A - F13 MT2 Review...
Transcript of Midterm 2 Review - econweb.ucsd.edueconweb.ucsd.edu/~vleahmar/pdfs/ECON 100A - F13 MT2 Review...
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Midterm 2 ReviewECON 100A - Fall 2013
Vincent Leah-Martin1
UCSD
November 12, 2013
[email protected] Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
First Order Conditions
Solving this problem yields that for any goods i and j :
MRSi ,j =MUi
MUj=
pipj
or
MU1
p1=
MU2
p2= ...for all goods
If these conditions are not satisfied then the consumer can dobetter by buying more of whatever good gives him moremarginal utility per dollar.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
First Order Conditions
Solving this problem yields that for any goods i and j :
MRSi ,j =MUi
MUj=
pipj
or
MU1
p1=
MU2
p2= ...for all goods
If these conditions are not satisfied then the consumer can dobetter by buying more of whatever good gives him moremarginal utility per dollar.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
First Order Conditions
Solving this problem yields that for any goods i and j :
MRSi ,j =MUi
MUj=
pipj
or
MU1
p1=
MU2
p2= ...for all goods
If these conditions are not satisfied then the consumer can dobetter by buying more of whatever good gives him moremarginal utility per dollar.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Corner Solutions
Suppose instead we have that for some goods i and j :
MUi
pi>
MUj
pj
Then we want to buy more of good i . If this holds no matterhow much good i we buy (for instance with perfectsubstitutes) then we will spend all income on good i .
x∗i =I
pi
x∗j = 0
Remember: the demand function will depend on the relativeprices. For some prices we buy all of good j !
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Corner Solutions
Suppose instead we have that for some goods i and j :
MUi
pi>
MUj
pj
Then we want to buy more of good i . If this holds no matterhow much good i we buy (for instance with perfectsubstitutes) then we will spend all income on good i .
x∗i =I
pi
x∗j = 0
Remember: the demand function will depend on the relativeprices. For some prices we buy all of good j !
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Corner Solutions
Suppose instead we have that for some goods i and j :
MUi
pi>
MUj
pj
Then we want to buy more of good i . If this holds no matterhow much good i we buy (for instance with perfectsubstitutes) then we will spend all income on good i .
x∗i =I
pi
x∗j = 0
Remember: the demand function will depend on the relativeprices. For some prices we buy all of good j !
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Demand Functions
With some algebra we can use the first order conditions andthe budget constraint to solve for the optimal value of eachgood as a function of prices and income:
x∗(p1, p2, ...I )
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Demand Function Properties
x∗i (p1, p2, I )
Must be homogeneous of degree 0 (pure inflation has noeffect on demand)∂x∗i∂pi≤ 0 in most cases
Income Properties∂x∗i∂I ≥ 0⇔ Normal or superior good∂x∗i∂I < 0⇔ Inferior good
Income Elasticity Propertiesεxi ,I > 1⇔ Superior goodεxi ,I ∈ [0, 1]⇔ Normal goodεxi ,I < 0⇔ Inferior good
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Elasticity
Definition
Elasticity a measure how a percent change in an independentvariable affects a dependent variable in percentage terms.
Formula
εy ,x =∂y
∂x
x
y
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Elasticity
Definition
Elasticity a measure how a percent change in an independentvariable affects a dependent variable in percentage terms.
Formula
εy ,x =∂y
∂x
x
y
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Monotonicity
Monotonically Increasing
A function is monotonically increasing in x if for every x ′ > x ,f (x ′) ≥ f (x).
Monotonically Decreasing
A function is monotonically decreasing in x if for every x ′ > x ,f (x ′) ≤ f (x).
How does this relate to the derivative?
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Monotonicity
Monotonically Increasing
A function is monotonically increasing in x if for every x ′ > x ,f (x ′) ≥ f (x).
Monotonically Decreasing
A function is monotonically decreasing in x if for every x ′ > x ,f (x ′) ≤ f (x).
How does this relate to the derivative?
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Monotonicity
Monotonically Increasing
A function is monotonically increasing in x if for every x ′ > x ,f (x ′) ≥ f (x).
Monotonically Decreasing
A function is monotonically decreasing in x if for every x ′ > x ,f (x ′) ≤ f (x).
How does this relate to the derivative?
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Homogeneity (Scale Properties)
Homogeneous of Degree 0
A function is homogeneous of degree 0 if for every λ ∈ R:
f (λx) = f (x)
This is no returns to scale.
Homogeneous of Degree 1
A function is homogeneous of degree 1 if for every λ ∈ R:
f (λx) = λf (x)
This is constant returns to scale.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Homogeneity (Scale Properties)
Homogeneous of Degree 0
A function is homogeneous of degree 0 if for every λ ∈ R:
f (λx) = f (x)
This is no returns to scale.
Homogeneous of Degree 1
A function is homogeneous of degree 1 if for every λ ∈ R:
f (λx) = λf (x)
This is constant returns to scale.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
General Homogeneity (Scale Properties)
Homogeneous of Degree k
A function is homogeneous of degree k if for every λ ∈ R:
f (λx) = λk f (x)
This is increasing returns to scale for k > 1. This is decreasingreturns to scale for k ∈ (0, 1).
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Regular Demand
Function
x∗i (p, I )
Properties
Can be increasing or decreasing in pj . Generallydecreasing in pi .
Can be increasing or decreasing in I .
Homogeneous of degree 0
Interpretation
Outputs quantity of a good i demanded at prices p andincome I which maximizes utility.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Regular Demand
Function
x∗i (p, I )
Properties
Can be increasing or decreasing in pj . Generallydecreasing in pi .
Can be increasing or decreasing in I .
Homogeneous of degree 0
Interpretation
Outputs quantity of a good i demanded at prices p andincome I which maximizes utility.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Regular Demand
Function
x∗i (p, I )
Properties
Can be increasing or decreasing in pj . Generallydecreasing in pi .
Can be increasing or decreasing in I .
Homogeneous of degree 0
Interpretation
Outputs quantity of a good i demanded at prices p andincome I which maximizes utility.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Compensated Demand
Function
h∗i (p, u)
Properties
Increasing in pj .
Decreasing in pi .
Homogeneous of degree 0 in p
Interpretation
Outputs quantity of a good i demanded at prices p such thatthe consumer obtains utility u.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Compensated Demand
Function
h∗i (p, u)
Properties
Increasing in pj .
Decreasing in pi .
Homogeneous of degree 0 in p
Interpretation
Outputs quantity of a good i demanded at prices p such thatthe consumer obtains utility u.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Compensated Demand
Function
h∗i (p, u)
Properties
Increasing in pj .
Decreasing in pi .
Homogeneous of degree 0 in p
Interpretation
Outputs quantity of a good i demanded at prices p such thatthe consumer obtains utility u.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Indirect Utility
Function
V (p, I ) ≡ u(x∗1 (p, I ), (x∗2 (p, I ), ...(x∗n (p, I ))
Properties
Nonincreasing in p.
Nondecreasing in I .
Homogeneous of degree 0.
Interpretation
Outputs the maximum amount of utility the consumer obtainsat prices p and income I .
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Indirect Utility
Function
V (p, I ) ≡ u(x∗1 (p, I ), (x∗2 (p, I ), ...(x∗n (p, I ))
Properties
Nonincreasing in p.
Nondecreasing in I .
Homogeneous of degree 0.
Interpretation
Outputs the maximum amount of utility the consumer obtainsat prices p and income I .
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Indirect Utility
Function
V (p, I ) ≡ u(x∗1 (p, I ), (x∗2 (p, I ), ...(x∗n (p, I ))
Properties
Nonincreasing in p.
Nondecreasing in I .
Homogeneous of degree 0.
Interpretation
Outputs the maximum amount of utility the consumer obtainsat prices p and income I .
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Expenditure Function
Function
e(p, u)
Properties
Nondecreasing in p.
Homogeneous of degree 1 in p.
hi(p, u) = ∂e(p,u)∂pi
Interpretation
Outputs the minimum amount of income needed at prices p toobtain utility u. This is the solution to the expenditureminimization problem.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Expenditure Function
Function
e(p, u)
Properties
Nondecreasing in p.
Homogeneous of degree 1 in p.
hi(p, u) = ∂e(p,u)∂pi
Interpretation
Outputs the minimum amount of income needed at prices p toobtain utility u. This is the solution to the expenditureminimization problem.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Expenditure Function
Function
e(p, u)
Properties
Nondecreasing in p.
Homogeneous of degree 1 in p.
hi(p, u) = ∂e(p,u)∂pi
Interpretation
Outputs the minimum amount of income needed at prices p toobtain utility u. This is the solution to the expenditureminimization problem.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
What Engel Curves Are...
We are graphing regular demand as income changes - that is,we are fixing p and graphing how x∗i (p, I ) changes as Ichanges.
I on the horizontal axis.x∗i on the vertical axis.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
What Engel Curves Tell Us...
The slope of the Engel curve is simply:
∂x∗i (p, I )
∂I
That is, the Engel curve is upward sloping for income rangesover which xi is a normal good. and downward sloping forincome ranges over which xi is an inferior good.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Comparing Engel Curves of Different Goods
A useful way to think of the slope of the Engel curve:Suppose you get $1. The slope of the Engel curve tells youhow much more (or less) of xi you buy with that $1.
⇒ If I know prices, this tells me how much of that $1 I spendon xi .
pi∂x∗i (p, I )
∂I⇒ This then tells me how much of that $1 I spend on theother goods.⇒ If there are only two goods, I can divide that amount byhow much the other good costs to obtain:
∂xj∂I
which is the slope of the other goods’ Engel curve.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Comparing Engel Curves of Different Goods
A useful way to think of the slope of the Engel curve:Suppose you get $1. The slope of the Engel curve tells youhow much more (or less) of xi you buy with that $1.⇒ If I know prices, this tells me how much of that $1 I spendon xi .
pi∂x∗i (p, I )
∂I
⇒ This then tells me how much of that $1 I spend on theother goods.⇒ If there are only two goods, I can divide that amount byhow much the other good costs to obtain:
∂xj∂I
which is the slope of the other goods’ Engel curve.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Comparing Engel Curves of Different Goods
A useful way to think of the slope of the Engel curve:Suppose you get $1. The slope of the Engel curve tells youhow much more (or less) of xi you buy with that $1.⇒ If I know prices, this tells me how much of that $1 I spendon xi .
pi∂x∗i (p, I )
∂I⇒ This then tells me how much of that $1 I spend on theother goods.
⇒ If there are only two goods, I can divide that amount byhow much the other good costs to obtain:
∂xj∂I
which is the slope of the other goods’ Engel curve.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Comparing Engel Curves of Different Goods
A useful way to think of the slope of the Engel curve:Suppose you get $1. The slope of the Engel curve tells youhow much more (or less) of xi you buy with that $1.⇒ If I know prices, this tells me how much of that $1 I spendon xi .
pi∂x∗i (p, I )
∂I⇒ This then tells me how much of that $1 I spend on theother goods.⇒ If there are only two goods, I can divide that amount byhow much the other good costs to obtain:
∂xj∂I
which is the slope of the other goods’ Engel curve.Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Own-Price, Cross-Price, and Income Elasticity Sum
Formula
εxi ,pi +∑j 6=i
εxi ,pj + εxi ,I = 0
Interpretation
Adding together the own-price elasticity, cross-priceelasticities, and income elastiticies for a particular goodresults in 0.
This is a result of demand functions being HD0 andfollows from the Euler Theorem.
This is useful because if we know some properties of xiwe can potentially infer other properties which are notgiven from knowing this formula.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Own-Price, Cross-Price, and Income Elasticity Sum
Formula
εxi ,pi +∑j 6=i
εxi ,pj + εxi ,I = 0
Interpretation
Adding together the own-price elasticity, cross-priceelasticities, and income elastiticies for a particular goodresults in 0.
This is a result of demand functions being HD0 andfollows from the Euler Theorem.
This is useful because if we know some properties of xiwe can potentially infer other properties which are notgiven from knowing this formula.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Regular Demand → Compensated Demand
Formula
x∗i (p, I ) = hi(p,V (p, I ))
Interpretation
How much a consumer demands to maximize utility at prices pand income I is the same as how much a consumer demandsat prices p such that he gets utility V (p, I ) which is themaximum utility he can get at prices p and with income I .
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Regular Demand → Compensated Demand
Formula
x∗i (p, I ) = hi(p,V (p, I ))
Interpretation
How much a consumer demands to maximize utility at prices pand income I is the same as how much a consumer demandsat prices p such that he gets utility V (p, I ) which is themaximum utility he can get at prices p and with income I .
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Compensated Demand → Regular Demand
Formula
hi(p, u) = x∗i (p, e(p, u))
Interpretation
How much a consumer demands at prices p such that he getsutility u is the same as how much the consumer demands atprices p to maximize utility when given enough money to getat most utility u at prices p.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Compensated Demand → Regular Demand
Formula
hi(p, u) = x∗i (p, e(p, u))
Interpretation
How much a consumer demands at prices p such that he getsutility u is the same as how much the consumer demands atprices p to maximize utility when given enough money to getat most utility u at prices p.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Expenditure Function of Indirect Utility
Formula
e(p,V (p, I )) = I
Interpretation
The minimum amount of income needed at prices p to get themost utility you can get at prices p when given income I is I .
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Expenditure Function of Indirect Utility
Formula
e(p,V (p, I )) = I
Interpretation
The minimum amount of income needed at prices p to get themost utility you can get at prices p when given income I is I .
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Utility of Expenditure Function
Formula
V (p, e(p, u)) = u
Interpretation
The maximum amount of utility that can be obtained at pricesp when given the least amount of money needed to obtainutility u at prices p is u.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Utility of Expenditure Function
Formula
V (p, e(p, u)) = u
Interpretation
The maximum amount of utility that can be obtained at pricesp when given the least amount of money needed to obtainutility u at prices p is u.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Study Recommendations
Recommended Practice Problems: Comparative Statics ofDemand 1, 5, 6, 7, 10-14, 17, 28, 30, 33
Most of the above problems relate to Engel Curves
Know exactly what each function is and how to derive it.There aren’t a lot of problems related to this.
Review MT1 material, make sure you have a solidunderstanding of this.
Exam will not cover income effect, substitution effect, orSlutsky equation.
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Additional Problems
Demand functions:
p. 39 : 10, 14p. 41 : 35p. 42 : 40, 43, 48p. 51 : 78, 81
Engel Curves:
p. 45 : 11, 14p. 46 : 28, 30p. 49 : 54, 56
Identities:
p. 45 : 12
Elasticities:
p. 39 : 16p. 42 : 45p. 43 : 53p. 44 : 10p. 47 : 39, 42p. 49 : 61p. 52 : 92
Vincent Leah-Martin Midterm 2 Review
Pre-Midterm 1 Important Functions Engel Curves Identities Exam Tips
Good luck!
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once you know what the question actually is, you’ll know whatthe answer means.”
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Vincent Leah-Martin Midterm 2 Review