midterm 1 review - university of california, san diegoeconweb.ucsd.edu/~vleahmar/pdfs/econ 100a -...

Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips

Midterm 1 ReviewECON 100A - Fall 2013

Vincent Leah-Martin1

October 20, 2013

[email protected] Leah-Martin Midterm 1 Review

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

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, ...)

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.

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.

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

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

First Order Conditions

Solving this problem yields that for any goods i and j :

MRSi ,j =MUi

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.

MRSi ,j =MUi

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 )

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 )

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 )

Elasticity

Definition

Elasticity a measure how a percent change in an independentvariable affects a dependent variable in percentage terms.

Formula

εy ,x =∂y

Elasticity

Definition

Elasticity a measure how a percent change in an independentvariable affects a dependent variable in percentage terms.

Formula

εy ,x =∂y

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 .

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 .

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

∂u(x1,x2)∂x1

∂u(x1,x2)∂x2

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

∂u(x1,x2)∂x1

∂u(x1,x2)∂x2

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.

Demand Functions

Mathematically

In Economics

Demand Functions

Mathematically

In Economics

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.

Engel Curve

Definition

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.

Engel Curve

Definition

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.

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 ), ...)

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 ), ...)

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).

Economic Bads

Definition

Intuition

Implication

Economic Bads

Definition

Intuition

Implication

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?

Monotonicity

Homogeneity (Scale Properties)

Homogeneous of Degree 0

A function is homoegeneous of degree 0 if for every λ ∈ R:

f (λx) = f (x)

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)

f (λx) = f (x)

f (λx) = λf (x)

f (λx) = λk f (x)

f (λx) = f (x)

f (λx) = λf (x)

f (λx) = λk f (x)

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).

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).

Cobb-Douglas Utility

u(x1, x2) = Axα1 xβ2

Must have x1 > 0 andx2 > 0

Satisfies diminishing MRS

One good could beinferior (not both)

Leontief Utility

u(x1, x2) = min{x1

}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

Slope of optimization line (line goingthrough kinks) is A

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”

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

Indirect Utility Functions

V (p1, p2, I )

Must be homogeneous of degree 0∂V∂pi≤ 0

∂V∂I≥ 0

midterm 1 review - university of california, san diegoeconweb.ucsd.edu/~vleahmar/pdfs/econ 100a -...

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