### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Microecon Theory I ECON 7005

The U

GPA 3.81

### View Full Document

## 24

## 0

## Popular in Course

## Popular in Economcs

This 13 page Class Notes was uploaded by Chelsie Cronin on Monday October 26, 2015. The Class Notes belongs to ECON 7005 at University of Utah taught by Gabriel Lozada in Fall. Since its upload, it has received 24 views. For similar materials see /class/230028/econ-7005-university-of-utah in Economcs at University of Utah.

## Reviews for Microecon Theory I

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/26/15

Mathematical Prerequisites for Econ 7005 Microeconomic Theory I and Econ 7007 MacroeconomicTheory I at the University of Utah by Gabriel A Lozada The contents of Sections 176 below are required for basic consumer and producer theory which is usually taught at the very beginning of Econ 7005 The contents of Section 7 are not required until the topic of uncertainty is reached Sections 174 are a shortened version of a more complete treatment described in lecture notes I wrote for Econ 7001 those notes are available upon request The terms and notation needed to understand these notes include for Section 1 letters of the Greek alphabet commonly used in mathematics ma trix matrix determinant symmetric matrix suf cient condition gt nec essary condition lt necessary and suf cient condition ltgt iff if and only if converse contrapositive Cquot function f Rquot gt R177 gradient vector Vfx Hessian matrix Vg x positive de nite positive semide nite negative de nite negative semide nite convex function strictly con vex function concave function strictly concave function quasiconcavity quasiconvexity convex set open set and closed set In Section 2 local minimum local maximum In Section 3 binding and nonbinding inequality constraints strict local minimum strict local maximum In Section 4 global minimum global maximum unique global minimum unique global maxi mum In Section 5 endogenous variables exogenous variables dependent variables independent variables structural form of a system of equations differential of a function of multiple variables matrix inverse Cramer s Rule In Section 6 no additional new terms or notation In Section 7 probability of an event x 677 Sim f day fair random process intervals of the real line 11 11 1717 and 0117 1 Convexity Quadratic Forms and Minors Let A denote a matrix It does not have to be square A minor of A of order r is obtained by deleting all but r rows and r columns of A then taking the determinant of the resulting r X r matrix Now let A denote a square matrix A principal minor of A of order r77 is obtained by deleting all but r rows and the corresponding r columns of A then taking the determinant of the resulting r X r matrix For example if you keep the rst third and fourth rows then you have to keep the rst third and fourth columns A principal minor of A of order r is denoted by AT of A Again let A denote a square matrix A leading principal minor of A of order r77 is obtained by deleting all but the rst r rows and the rst r columns of A then taking the determinant of the resulting r X r matrix A leading principal minor of A of order r is denoted by Dr of A A square matrix of dimension n X n has only 1 leading principal minor of order r for r 1 n 1 2 3 4 Exam 16 Su ose A i 5 6 7 8 This matrix is not s mmetric P 39 pp 9 10 11 12 39 y 39 13 14 15 16 Usually one is interested in the minors only of symmetric matrices but there is nothing wrong With nding the minors of this non symmetric matrix 0 The leading principal minor of order 1 of A is D1 There are four principal minors of order 1 of A they are the Al s 1 D1 6 11 and 16 There are sixteen minors of A of order 1 O The leading principal minor of order 2 of A is D2 z There are six principal minors of order 2 of A they are the Ag s D2 from rows and columns 1 and 2 131 from rows and columns 1 and 3 113 146 from rows and columns 1 and 4 160 11 from rows and columns 2 and 3 11 12 and 4 and 15 16 There are thirty six minors of A of order 2 6 8 1416 from rows and columns 2 from rows and columns 3 and 4 1 2 3 o The leading principal minor of order 3 of A is D3 5 6 7 9 10 11 There are four principal minors of order 3 of A they are the Ag s 1 2 3 1 2 4 5 6 7 D3 from rows and columns 1 2 and 3 5 6 s from 9 10 11 13 14 16 1 3 4 rows and columns 1 2 and 4 9 11 12 from rows and columns 1 3 1 5 16 6 7 8 and 4 and 10 11 12 from rows and columns 2 3 and 4 There are sixteen minors of A of order 3 O The leading principal minor of order 4 of A is D4 There is only one principal minor of order 4 of A it is A4 and it is equal to There is only one minor of order 4 of A it is End of Example Let f be a 02 function mapping Rquot into R1 Denote the Hessian matrix of by V2fx this matrix has dimension n X 71 Let Dr of V2fx77 denote the rth order leading principal minor of the Hessian of Let AT of V2fx77 denote all the rth order principal minors of the Hessian of Proposition 1 One has DrofV2fXgt0forr1nandforallX S 1 ltgt V2fx is positive de nite for all X E S 2 is strictly convex on S 3 Also All the AT of V2fx 2 0 forr 1 n andforaHXE S 4 ltgt V2fx is positive semide nite for all X E S V m V ltgt is convex on S 6 V If V2fx is replaced by an arbitrary symmetric matrix it is still true that 1 ltgt 2 and 4 ltgt 5 The typical procedure is to check 1 rst If 1 doesn t apply because one of the signs was strictly negative then the contrapositive of 4 iff 677 tells you that the function is not convex This is because each D 6 Ala Proposition 1 Similarly D of V2fx alternate in sign beginning with lt O for 7 1 n and V x E S 1 ltgt V2fx is negative de nite for all x E S 2 gt is strictly concave on S 3 Also All the A of V2fx alternate in sign beginning with g 0 for 7 1 n and V x E S 4 ltgt V2fx is negative semide nite for all x E S 5 ltgt is concave on S 6 The following proposition is a test for pseudoconvexity and pseudo concavity77 but for all practical purposes you should assume that pseudo convexity is the same as quasiconvexity and pseudoconcavity is the same as quasiconcavity so I Will not even bother to de ne pseudoconvexity and pseudoconcavity Proposition 2 Test of Pseudoconvexity Let f be a 02 function de ned in an open convex set S in Rquot De ne the bordered Hessian determinants 54X r 1 n by 0 fiX 00 W mix six i 1700 fHX MK MK A suf cient condition for f to be pseudoconveX is that 34X lt 0 for r 2 nandallX S Proposition 2 Similarly a suf cient condition for f to be pseudoconcave is that 5x alternate in sign beginning with gt O for 7 2 n and all x E S 2 FirstOrder Conditions Proposition 3 Suppose that f In hj and 91 gi are C1 func tions of 71 variables Suppose that Xquot E Rquot is a local minimum of on the constraint set de ned by thej equalities and k inequalities h1X 01 hjX Cj 7 9100 2517 m7 974quot 1 8 Form the Lagrangian j k X7A7 H NO ZN Ci hiXl EM bi 001 9 i1 i1 Then under certain conditions I omit here there exist multipliers 3 and Hquot such that 1 83XApai 0 for all i 1 j This is equivalent to mixquot Ci for all i 1 2 83xAp8wi 0 for all 239 1 n 3112 0 gix 7 hi 2 0 and M gix 7 hi 0 for all i 1 k These three conditions are often called the Kuhn Tucker conditions The last condition is sometimes called the complementary slackness condition77 Proposition 3 For a maximum change 8 to 91X b17 my 9kx bk 8 Then the Lagrangian is formed in the same Way Condition 1 is unchanged Condition 2 is unchanged Condition 3 becomes it 2 O gix 7 hi g 0 and it gix 7 hi O for alli 1 19 3 SecondOrder Conditions Local Let f be the Lagrangian of the optimization problem In Section 2 I named the Lagrange multipliers X7 if they were associated With one of the j equality constraints and M if they were associated With one of the k inequality constraints In this section a ignore all the nonbinding inequal ity constraints at quot7 X M and b rename the Lagrange multipliers of the binding inequality constraints M11 M12 A7 Where m is the number of equality constraints plus the number of bind ing inequality constraints It is allowed to have m 0 if m 0 then there are no Lagrange multipliers Denote the in binding Lagrange multipliers collectively by A Let there be 71 variables with respect to which the optimization is occurring denote these variables collectively by X Let V23 be the following particular Hessian of the Lagrangian rst dif ferentiate f with respect to all the Lagrange multipliers then differentiate it with respect to the original variables X fAlh 3M W fAlAm l 3AM 3A1 w 3A1 EAQAI EAQAQ H EAQAW l fk h 3A2 quot 3amp1quot fAmAl 3M 3mm l fAmmi 3 3AMquot v92 L L L L L L L L 931M 931M i fmlAm l 921921 921922 quot 92192 932M 932M quot 39 femMn l 922921 922922 92292quot fmnAl finb H fmnAm l fawn 9qu fmnmn If you do this right V23 should have an m X in zero matrix in its upper left hand corner 3 f 0 f 2 7 AA Am 7 Am where fxm is an m X 71 matrix and where a T superscript denotes the transpose One has the following result Proposition 4 A suf cient condition for the point x identi ed in Proposition 3 to be a strict local minimum is that 71m has the same Sign as all of the following when they are evaluated at quot7 X ng1 of V23 D2m2 ongf Dmn ongf If m 0 this is equivalent to the condition that V23 which in such a case equals V2fx be positive de nite which occurs iff is strictly convex Proposition 4 Similarly one will have a strict local maximum if when they are evaluated at x X the following alternate in sign beginning with the sign of 71m1 D2m1 of V23 D2m2 of V23 Dm of V21 There is a second order necessary condition for a minimum also Proposition 5 Forz lt m de ne 3 of V23 to be the subset of Ai of V23 formed by only considering those Ai of V23 Which retain parts of the rst m rows and rst m columns of V23 If m 0 there is no difference between the A s and the 87s Then a necessary condition for the point X X identi ed in Proposi tion 3 to be a local minimum is that 71m or zero have the same sign as all of the following When they are evaluated at X X 82m of V23 32 of V22 am of V23 The typical procedure is to check Proposition 4 rst If Proposition 4 doesn t apply because one of the signs was strictly the same as 71m1 then Proposition 5 tells you that X is not a local minimum point This is because each D E Proposition 5 The version of Proposition 5 for a local maximum requires that the following if they are evaluated at xX alternate in sign beginning with the sign of 71 1 or zero77 then having the sign of 71 2 or zero77 and so forth 2m1 of V22 at of V22 gm of Viz 4 SecondOrder Conditions Global Let X X be a point identi ed in Proposition 3 Let j be the number of equality constraints and k be the number of inequality constraints 1 Ifj k 0 an unconstrained problem and fx is convex then x is a global minimum point off in S The converse also holds Furthermore ifj k 0 and is strictly convex then X is the unique global minimum point off in S The converse also holds 1 If k 0 only equality constraints and fx is convex in X then X is a global constrained minimum point off Furthermore if k 0 and fx is strictly convex in x then x is the unique global constrained minimum point of Aside Similarly using the problem de ned in Proposition 3 and 8 1 lfj k 0 an unconstrained problem and is concave then x is a global maximum point of f in S The converse also holds Furthermore ifj k O and is strictly concave then x is the unique global maximum point of f in S The converse also holds 1 If k 0 only equality constraints and 3x is concave in x then x is a global constrained maximum point of Furthermore if k O and 3x is strictly concave in x then x is the unique global constrained maximum point of 5 Comparative Statics Let X E Rquot denote the endogenous or dependent variables in a model and let y E Rm denote the exogenous or independent variables in that model Suppose the model is described by a general system of structural equations of the form f1X7 y f2X7 10 fn X7 339 0 Taking the differential of both sides of each equation results in 3f 1 3f 1 3f 1 3f1 7d 7d 7d 7d 0 8x1 031 8xquot 56quot ayl 241 gym ym 11 3fquot 3fquot 3fquot 3fquot 7d d 7d d 0 8x1 031 8xquot 56quot ayl 241 gym ym Moving the last m terms in each equation to the right and rewriting in matrix form results in 8961 i i 9x7 dxl 9241 824m 3 3 3 dy1 f dym 12 8x1 9x7 9271 824m Let J denote the matrix on the left hand side of 12 This matrix is a Jacobian matrix If J is invertible then we can solve for 1031 d502 dayquot as a function of dyl dyg dym as follows u in dfcl 5241 924m J 1 E dy1J71 E dym 13 dayquot 3241 5y Alternatively and more commonly 12 is solved using Cramer s Rule es pecially in the many problems in which most of the dy s are zero Consider the common problem of determining the sign of dwidyj from 12 The easiest way to do this if 10 are the rst order conditions of an optimization problem which in microeconomics is usually the case is usually to solve 12 using Cramer s Rule Then dwidyj would have the form numerator W In such cases the second order conditions of the optimization problem usu 14 ally determine the sign of lJl then all that remains in order to determine the sign of dwidyj is to nd the sign of the numerator of 14 6 The Value Function and the Envelope Theorem Consider the problem of maximizing a function f over endogenous vari ables X given exogenous variables C and constraints 711X7 C 0 hg X7 C 0 hjX C 0 The optimized value function for this problem is de ned as fc mmax fx7 c such that 15 h1xc0 h2xc0 hjxc0 Equivalently if Xquot is the solution to the maximization problem in 15 then f0 fXC7 C 16 Let f be the Lagrangian function 9 for the maximization problem in 15 The Envelope Theorem states that 8 i 837 80139 7 80139 17 Where 3 is 3 evaluated at quot7 C Consider the special case of 15 in Which fxc does not depend on C and in Which the constraints take the form h1ltXgt Cl 7 Cj Then 17 implies that 8 837 80139 7 80139 7 AZ This is often used to give an interpretation of M 7 Probability Theory All the probability theory that is required for this course is an understanding of hoW to compute the expected value of a discrete or continuous random variable A super cial understanding Will su ice but some students might be interested in a more careful treatment Which I give below However I still Will not be giving a fully satisfactory treatment because that requires mea sure theory Borel sets and other advanced mathematics such a treatment is given for example in Chapter 1 of Malliaris and Brock s 1982 textbook Stochastic Methods in Economics and Finance Let the set of possible outcomes of an uncertain event be called the sample space and be denoted by Q We Will rst suppose that the number of elements in Q is nite or countably in nite With each element 0 E Q associate a real number Xw For example if Q is a deck of playing cards and each 0 is one card then Xw might be 1 When w is the 2 of Hearts 10 When w is the Jack of Hearts 14 When w is the 2 of Spades and so forth The function X Q gt R is called a discrete random variable Let Prw denote the probability that 0 occurs Let the function R gt 07 1 be de ned by Prw Xw The function f is called the probability distribution of the discrete random variable X One has 2 we 1 16R 10 The expected value of the random variable also called the mean of the random variable or the average of the random variable is de ned to be EX Z xfx 16R For example consider the outcome of a roll of a die The set of outcomes in no particular order is Q 371574672 Let the rst outcome be 0J1 3 the second outcome be 0J2 1 and so forth so the sixth outcome is 0J6 2 De ne the random variable Xw in the following way Xw1 32 331 Xwg 12 332 Xw6 22 336 If in addition the die is fair so all the outcomes occur with probability 16 then the expected value of X is 6 Z 50139 f 50139 i1 32 12 52 42 62 22 1 1 76912516364916156 For another example again consider the outcome of a roll of a die This time write the set of outcomes as Q 172374576 Let the rst outcome be 0J1 1 the second outcome be 0J2 2 and so forth so the sixth outcome is 0J6 6 De ne the random variable Yw in the following way Yw1 1 y1Ywg 2 yg Yw6 6 36 If in addition the die is fair so all the outcomes occur with probability 16 then the expected value of Y is 6 1 1 1 1 7 7 E yi yl 62 6 6 6 7 12345621635 This completes our treatment of the case when the number of elements in Q is nite or countably in nite Now suppose instead that the number of elements in Q is uncountably in nite Furthermore suppose that to each element 0 E 9 we can associate a real number Xw For example if 0 is the color of paint in a paint can which we nd together with many other paint cans in an abandoned building then 11 Q is the set of all possible colors in the abandoned cans and if red is one s favorite color then Xw might be the grams of red pigment contained in the rst abandoned paint can The function X Q gt R is called a continuous random variable Let the function R gt 07 1 be de ned by Prw Xw cc The function F is called the cumulative probability density function or CDF of the continuous random variable X In the example CDFx is the probability that the paint can Will have less than or equal to 0 grams of red pigment One has 1 The function dFlt a we 7 d is called the probability density function or PDF of the continous random variable X One has 1fxdx1 The probability that the value of X is between a and b Where a 7 b is day The probability that the value of X is exactly equal to a is not given by do 0 because then X could never take on any value Instead frequency With Which the value of X is exactly equal to any particular value 1 goes to zero in the limit as the number of draws from the distribution goes to in nity The expected value of the random variable also called the mean of the random variable or the average of the random variable is de ned to be EX x m doc For example ifQ 07 12 for the outcome of the spin ofa fair arroW centered on the face of a clock if 0 is de ned be the number that the arroW points to on the clock face and ifXw is de ned to equal 0 so X3 3 then the CDF of the arroW is 025 at a 3 075 at a 9 and in general is equal to 5012 The PDF in this example is 1 112 12 and the expected value is 12 x 0 1 E 1337

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over $500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.