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

November 12, 2013

[email protected] Leah-Martin Midterm 2 Review

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.

Vincent Leah-Martin Midterm 2 Review

MRSi ,j =MUi

Corner Solutions

Suppose instead we have that for some goods i and j :

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

x∗j = 0

Remember: the demand function will depend on the relativeprices. For some prices we buy all of good j !

Corner Solutions

x∗i =I

x∗j = 0

Corner Solutions

x∗i =I

x∗j = 0

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 )

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

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

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 homogeneous of degree 0 if for every λ ∈ R:

f (λx) = f (x)

This is no returns to scale.

f (λx) = λf (x)

This is constant returns to scale.

Homogeneity (Scale Properties)

f (λx) = f (x)

This is no returns to scale.

f (λx) = λf (x)

This is constant returns to scale.

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).

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.

Regular Demand

Function

x∗i (p, I )

Properties

Interpretation

Regular Demand

Function

x∗i (p, I )

Properties

Interpretation

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.

Compensated Demand

Function

h∗i (p, u)

Properties

Increasing in pj .

Decreasing in pi .

Interpretation

Compensated Demand

Function

h∗i (p, u)

Properties

Increasing in pj .

Decreasing in pi .

Interpretation

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 .

Indirect Utility

Function

V (p, I ) ≡ u(x∗1 (p, I ), (x∗2 (p, I ), ...(x∗n (p, I ))

Properties

Nonincreasing in p.

Interpretation

Indirect Utility

Function

V (p, I ) ≡ u(x∗1 (p, I ), (x∗2 (p, I ), ...(x∗n (p, I ))

Properties

Nonincreasing in p.

Interpretation

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.

Function

e(p, u)

Properties

Nondecreasing in p.

Interpretation

Function

e(p, u)

Properties

Nondecreasing in p.

Interpretation

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.

What Engel Curves Tell Us...

The slope of the Engel curve is simply:

∂x∗i (p, 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.

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.

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 )

⇒ 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

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

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

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.

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.

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 .

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 .

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.

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.

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 .

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 .

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.

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.

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.

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

Good luck!

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once you know what the question actually is, you’ll know whatthe answer means.”

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Midterm 2 Review - econweb.ucsd.edueconweb.ucsd.edu/~vleahmar/pdfs/ECON 100A - F13 MT2 Review...

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