recitation #1 sai ma...practice and practice questions from textbook recommanded sai ma (nyu)...

Intermediate MicroeconomicsRecitation #1

Sai Ma

New York University

September 10, 2013

Sai Ma (NYU) Recitation #1 September 10, 2013 1 / 21

Introduction

Sai Ma

Second Year Econ PhDEmail: [email protected]¢ ce hour

Depatment of Economics

19 West 4th, 5-8th FloorO¢ ce 819


Course Tips

Check message from blackboard frequently. (Not emailed to everyonesomtimes)

Feel free to email me regarding short question. Long question iswelcomed but appointment is preferred.

Make sure to fully understand the problem sets

Help understand the materialsHelp prepare the midterm and nal

Practice and practice

Questions from textbook recommanded


Question 1

ProblemJoe has the following reference over the co¤ee shops: for any two co¤eeshops A,B, he weakly prefers A to B if a small cup of co¤ee is at leastcheap at A as at B, and also A is at least as close (to his house) as B.Thus if pi , di are the price, distance for co¤ee shop i ,

A � B () pA � pB and dA � dB

Show that his preference relation � may not be complete


Question 1

Lets quickly refresh our memory about the basics of Preference. We usethe following symbols to denote an individuals preferences between anytwo alternatives:

% means weak peference� means strick peference� means indiference

Preferences are rational if % satisies two properties:complete: for any two choices A,B, either A % B or B % A or both

when doing a questionnaire choosing A,B , the answer "I dont know"is not allowed

transitive: if A % B and B % C , then A % C


Question 1

SolutionFor % to be complete, it must be that for any pair A,B, either A % B orB % A or both.This does not hold if there are two co¤ee shops A,B suchthat pA < pB and dA > dB .

Why?

The rst inequality implies that B % A does not hold, since B is moreexpensive; the second implies that A % B does not hold, since A isfurther away.


Question 2

ProblemExplain graphically why indi¤erence curves cannot cross


Question 2

Lets formally dene the indi¤erence curve

DenitionLet % be a preference relation on a set X .The indi¤erence curve IC (x) is aset of all y 2 X for which y � x

In the set notation,

IC (x) = fy 2 X jy � xg


Question 2

SolutionGraph on the Board. Suppose by contradiction that the indi¤erence curvecan cross. Then by IC2, A � B. But according to IC1, A � C and C � B.By transitivity, this implies A � B, contradiction.


Question 3

Problem (3a)

Argue that the utility unction v(x1, x2) = 2 ln x1 + ln x2 represents the

same preference as u(x1, x2) = x231 x

132 .

Problem (3b)Write the equation of an indi¤erence curve which represents thesepreferences and goes through the point (1,1)

Problem (3c)If his consumer has twice as many units of good 1 as good 2, then whatshis marginal rate of substitution (MRS) between the goods?


Question 3

Lets briey talk about utility function.

DenitionWe say that the function U : X ! R represents the preference % if for allx and y 2 X , x % y if and only i¤ U(x) > U(y).If the function Urepresents the preference relation %, we refer to it has a utility function orwe say that % has a utility representation.


Question 3

TheoremIf U represents %, then fo any strictly increasing function f : R ! R,the function V (x) = f (U(x)) represents % as well

Proof?


Question 3

Proof.

x % y() U(x) > U(y) (since U represents % )() f (U(x)) > f (U(y)) (since f is strictly increasing)() V (x) > U(y)


Question 3

Back to our problem

Problem (3a)

Argue that the utility unction v(x1, x2) = 2 ln x1 + ln x2 represents the

same preference as u(x1, x2) = x231 x

132 .

SolutionHere v = ln(u3).This is a strictly increasing function of u (why?)andhence by the theorem, v represents the same preference.


Question 3


SolutionWe want all points (x1, x2) satisfying (x1, x2) � (1, 1) : since u representsthe preferences, this requires

u(x1, x2) = u(1, 1)

, x231 x

132 = 1

Note you can also use v , in that case the equation is2 ln x1 + ln x2 = 0 (verify)

Graph?

plot x2 = 1x 21


Question 3


SolutionWe want all points (x1, x2) satisfying (x1, x2) � (1, 1) : since u representsthe preferences, this requires

u(x1, x2) = u(1, 1)

, x231 x

132 = 1

Note you can also use v , in that case the equation is2 ln x1 + ln x2 = 0 (verify)

Graph?plot x2 = 1x 21


Question 3

For 3(c), lets review marginal rate of substitution (MRS)

tells you the rate at which you would give up y to get a bit more x

Formula in this case:

MRS(x ,y ) =MUxMUy

Negative slope of the indi¤erence curveDiminishing MRS implies preferences are convex


Question 3

Problem (3c)

If his consumer has twice as many units of good 1 as good 2, then whatshis marginal rate of substitution (MRS) between the goods?

Solution

MU1 =∂u∂x1

=23(x2x1)13

MU2 =∂u∂x2

=13(x1x2)23

So, MRS(x1,x2) =MU1MU2

=2x2x1

) MRS(x1,x2) = 1 when x1 = 2x2

Mush less tedious if using v(x1, x2) instead of u.Sai Ma (NYU) Recitation #1 September 10, 2013 17 / 21

Question 4

ProblemJohn always eats his ballpark hotdog in a special way: he uses a footlonghotdog with precisely half a bun, 1 ounce of mustard, and 2 ounces ofpickle relish. His utility depends only on these four items, and any extraamount of a single item (without the others) is worthless to him. Whatform does his utility function have?

Solution?

u(h, b,m, r) = min(h, 2b,m, 0.5r)

(1, 0.5, 1, 2) � (1, 0.5, 800, 2)


Convexity

Denitions

Preferences % are convex if x % y implies αx + (1� α)y % y ,8α 2 (0, 1).

Denitions

Preferences % are strictly convex if a % y , b % y and a 6= bimpliesαa+ (1� α)b � y , 8α 2 (0, 1).


Convexity

In the footnote in chapter 3 of textbook, there is a fancy way tocheck if the preference is convex. Denote f be the utility function,then the represented preference is strictly convex if

f 22 f11 � 2f1f2f12 + f 21 f22 < 0

So if the cross derivative is zero (i.e f12 = 0), then the preference isstrictly convex if f11 < 0 and f22 < 0

Examples

(Cobb-Douglas) u(x , y) = ln x + ln y(Perfect Substitution) u(x , y) = x + y

(CES) u(x , y) = xδ+y δ

δ

You can always use diminishing MRS criterion to check if thepreferences are convex


CES Utility Function

u(x , y) = xδ+y δ

δ

If δ = 1, u(x , y) = x + y , (perfect substitution)

If δ = 0, u(x , y) = ln x + ln y , (Cobb-Douglas). Derivation usingLHôpitals rule

If δ = �∞, u(x , y) = min(x , y) perfect complementDerivation beyond the scope of this course


IntroductionCourse Tips

ExamplesQuestion 1Question 2Question 3Question 4Convexity

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