Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Midterm 1 ReviewECON 100A - Fall 2013
Vincent Leah-Martin1
UCSD
October 20, 2013
[email protected] Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Preferences
We started with a bundle of commodities:
(x1, x2, x3, ...) ≡ (apples, bannanas, beer, ...)
We then suppose a consumer can rank every bundle:
(2, 2, 2, ...) � (1, 1, 1, ...) � (1, 1, 0, ...)
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Preferences
We started with a bundle of commodities:
(x1, x2, x3, ...) ≡ (apples, bannanas, beer, ...)
We then suppose a consumer can rank every bundle:
(2, 2, 2, ...) � (1, 1, 1, ...) � (1, 1, 0, ...)
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Utility Representation
Now we assign a number to each bundle such that for anybundles A and B :
A � B ⇔ u(A) > u(B)
This yields a utility function u(x1, x2, ...). The function isordinal because the it outputs doesn’t matter as long as theorder of ranking bundles is the same.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Utility Representation
Now we assign a number to each bundle such that for anybundles A and B :
A � B ⇔ u(A) > u(B)
This yields a utility function u(x1, x2, ...). The function isordinal because the it outputs doesn’t matter as long as theorder of ranking bundles is the same.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
The Consumer Problem
Given this utility function, we want to model how a consumerbehaves. We say that a consumer wants to maximize theirutility - which just means they want whatever bundle theymost prefer:
maxx1,x2,...u(x1, x2, ...)
However, they are restricted by the price of each commodityand their income:
p1x1 + p2x2 + ... = I
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
The Consumer Problem
Given this utility function, we want to model how a consumerbehaves. We say that a consumer wants to maximize theirutility - which just means they want whatever bundle theymost prefer:
maxx1,x2,...u(x1, x2, ...)
However, they are restricted by the price of each commodityand their income:
p1x1 + p2x2 + ... = I
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions 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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions 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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions 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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions 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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Indirect Utility
The last thing we did was find a function for the most utilitythe consumer can get given prices and income:
u(x∗1 (p1, p2, ...I ), x∗2 (p1, p2, ...I ), ...) ≡ V (p1, p2, ...I )
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Indirect Utility
The last thing we did was find a function for the most utilitythe consumer can get given prices and income:
u(x∗1 (p1, p2, ...I ), x∗2 (p1, p2, ...I ), ...) ≡ V (p1, p2, ...I )
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions 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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions 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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Indifference Curves
Definition
Indifference curves are a plot of all the bundles which theconsumer is indifferent between.
Mathematically
They are a collection of points (x1, x2) for which u(x1, x2) = cfor any given constant c .
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Indifference Curves
Definition
Indifference curves are a plot of all the bundles which theconsumer is indifferent between.
Mathematically
They are a collection of points (x1, x2) for which u(x1, x2) = cfor any given constant c .
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Marginal Rate of Substitution
What it is...
The MRS is the slope of the level curve of a function at agiven point (x1, x2).
Formula
MRS1,2 =MU1
MU2=
∂u(x1,x2)∂x1
∂u(x1,x2)∂x2
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Marginal Rate of Substitution
What it is...
The MRS is the slope of the level curve of a function at agiven point (x1, x2).
Formula
MRS1,2 =MU1
MU2=
∂u(x1,x2)∂x1
∂u(x1,x2)∂x2
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Demand Functions
Mathematically
The demand function is the optimal value of a choice variableas a function of parameters.
In Economics
x∗i (p1, p2, ..., I ) is the amount of good i I consume when facedwith these prices and income.
Note: Hold all pj for j 6= i and I constant, then you can plotthe demand function with xi on the x-axis and pi on they-axis. This is the normal demand curve from ECON 1.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Demand Functions
Mathematically
The demand function is the optimal value of a choice variableas a function of parameters.
In Economics
x∗i (p1, p2, ..., I ) is the amount of good i I consume when facedwith these prices and income.
Note: Hold all pj for j 6= i and I constant, then you can plotthe demand function with xi on the x-axis and pi on they-axis. This is the normal demand curve from ECON 1.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Demand Functions
Mathematically
The demand function is the optimal value of a choice variableas a function of parameters.
In Economics
x∗i (p1, p2, ..., I ) is the amount of good i I consume when facedwith these prices and income.
Note: Hold all pj for j 6= i and I constant, then you can plotthe demand function with xi on the x-axis and pi on they-axis. This is the normal demand curve from ECON 1.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Engel Curve
Definition
The Engle curve for good xi isthe plot of income and xi withxi on the x-axis and I on they-axis.
This tells us if you buy more orless of a good when incomeincreases.Note: You can also plot anEngel curve with the axesswitched. On the exam if it isnot explicitly stated, ask whichvariable is on which axis.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Engel Curve
Definition
The Engle curve for good xi isthe plot of income and xi withxi on the x-axis and I on they-axis.
This tells us if you buy more orless of a good when incomeincreases.
Note: You can also plot anEngel curve with the axesswitched. On the exam if it isnot explicitly stated, ask whichvariable is on which axis.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Engel Curve
Definition
The Engle curve for good xi isthe plot of income and xi withxi on the x-axis and I on they-axis.
This tells us if you buy more orless of a good when incomeincreases.Note: You can also plot anEngel curve with the axesswitched. On the exam if it isnot explicitly stated, ask whichvariable is on which axis.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Indirect Utility
The question in words...
What is the most utility a consumer can get give prices andincome?
The answer in math...
V (p1, p2, ..., I ) ≡ u(x∗1 (p1, p2, ...I ), x∗2 (p1, p2, ...I ), ...)
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Indirect Utility
The question in words...
What is the most utility a consumer can get give prices andincome?
The answer in math...
V (p1, p2, ..., I ) ≡ u(x∗1 (p1, p2, ...I ), x∗2 (p1, p2, ...I ), ...)
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Economic Bads
Definition
We say a commodity xi is an economic bad if MUi < 0.
Intuition
The more the consumer consumes of xi the less happy he is.
Implication
If a good is an economic bad, the consumer will try toconsume as little of it as possible (typically 0).
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Economic Bads
Definition
We say a commodity xi is an economic bad if MUi < 0.
Intuition
The more the consumer consumes of xi the less happy he is.
Implication
If a good is an economic bad, the consumer will try toconsume as little of it as possible (typically 0).
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Economic Bads
Definition
We say a commodity xi is an economic bad if MUi < 0.
Intuition
The more the consumer consumes of xi the less happy he is.
Implication
If a good is an economic bad, the consumer will try toconsume as little of it as possible (typically 0).
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions 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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions 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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions 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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Homogeneity (Scale Properties)
Homogeneous of Degree 0
A function is homoegeneous of degree 0 if for every λ ∈ R:
f (λx) = f (x)
Homogeneous of Degree 1
A function is homoegeneous of degree 1 if for every λ ∈ R:
f (λx) = λf (x)
Homogeneous of Degree k
A function is homoegeneous of degree k if for every λ ∈ R:
f (λx) = λk f (x)
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Homogeneity (Scale Properties)
Homogeneous of Degree 0
A function is homoegeneous of degree 0 if for every λ ∈ R:
f (λx) = f (x)
Homogeneous of Degree 1
A function is homoegeneous of degree 1 if for every λ ∈ R:
f (λx) = λf (x)
Homogeneous of Degree k
A function is homoegeneous of degree k if for every λ ∈ R:
f (λx) = λk f (x)
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Homogeneity (Scale Properties)
Homogeneous of Degree 0
A function is homoegeneous of degree 0 if for every λ ∈ R:
f (λx) = f (x)
Homogeneous of Degree 1
A function is homoegeneous of degree 1 if for every λ ∈ R:
f (λx) = λf (x)
Homogeneous of Degree k
A function is homoegeneous of degree k if for every λ ∈ R:
f (λx) = λk f (x)
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Strictly Diminishing MRS
Intuitive Definition
The slope of the indifferencecurve gets flatter as you movedown the indifference curve(increasing x1 and decreasingx2)
Implication
Take any two points on theindifference curve and theconsumer will prefer any linearcombination of those twopoints (see picture).
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Strictly Diminishing MRS
Intuitive Definition
The slope of the indifferencecurve gets flatter as you movedown the indifference curve(increasing x1 and decreasingx2)
Implication
Take any two points on theindifference curve and theconsumer will prefer any linearcombination of those twopoints (see picture).
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Cobb-Douglas Utility
u(x1, x2) = Axα1 xβ2
Must have x1 > 0 andx2 > 0
Satisfies diminishing MRS
One good could beinferior (not both)
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Leontief Utility
u(x1, x2) = min{x1
A,x2
B
}Must have x1 > 0 and x2 > 0
Positive marginal utility for xi only ifconsuming too much of xj
Also known as “perfect complements”
Always optimize at kinks x1
A= x2
B
Slope of optimization line (line goingthrough kinks) is A
B
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Linear Utility
u(x1, x2) = αx1 + βx2
Unless certain conditions hold,will consume none of one good(corner solution)
Constant marginal utility forboth goods
Also known as “perfectsubstitutes”
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Demand Functions
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 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Indirect Utility Functions
V (p1, p2, I )
Must be homogeneous of degree 0∂V∂pi≤ 0
∂V∂I≥ 0
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Recommended Problems
At this point you should have done a substantial number ofproblems. Here are a handful I think might be useful to makesure you understand:
Elasticity (pg 26-29): 2, 3, 5, 6, 8, 9, 14, 21, 32
Level Curves (pg 29): 1, 3, 4
Constrained Optimization (pg 29-32): 1, 3, 6, 8, 10, 15
Comparative Statics of Solutions Functions (pg 32-34): 6
Consumer Preferences (pg 35-38): 1, 2, 4, 7, 15, 28, 32
Utility Maximization and Demand Functions (pg 38-43):1, 3, 5, 13, 14, 16, 22, 24, 35
Comparative Statics of Demand (pg 44-53): Don’t worryabout this section.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Last Minute Check
Do you have a solid understanding of how optimizationwith two goods works graphically?
Do you know what happens graphically when I change aparameter in the consumer’s problem?
Can you verbally explain all of the definitions?
Can you verbally explain the first order conditions?
Can you explain the economic intuition for everymathematical concept?
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Advice
Final Office Hours
Today, ECON 300, 2pm-4pm - I will go over any questions youhave from the packet or on the concepts.
Monday Night
Once you feel like there is nothing more you can do, stopworrying about the exam and do something relaxing.
During the Exam
Relax, look at the problem. First identify what the problem istalking about then apply what you know about that topic.Nothing will require a very long answer so if you’re writing alot or doing very messy math you might be doing somethingwrong.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Advice
Final Office Hours
Today, ECON 300, 2pm-4pm - I will go over any questions youhave from the packet or on the concepts.
Monday Night
Once you feel like there is nothing more you can do, stopworrying about the exam and do something relaxing.
During the Exam
Relax, look at the problem. First identify what the problem istalking about then apply what you know about that topic.Nothing will require a very long answer so if you’re writing alot or doing very messy math you might be doing somethingwrong.
Vincent Leah-Martin Midterm 1 Review
Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips
Advice
Final Office Hours
Today, ECON 300, 2pm-4pm - I will go over any questions youhave from the packet or on the concepts.
Monday Night
Once you feel like there is nothing more you can do, stopworrying about the exam and do something relaxing.
During the Exam
Relax, look at the problem. First identify what the problem istalking about then apply what you know about that topic.Nothing will require a very long answer so if you’re writing alot or doing very messy math you might be doing somethingwrong.
Vincent Leah-Martin Midterm 1 Review