Econ 201: Problem Set 2 Answers
|
|
- Marilyn Nichols
- 5 years ago
- Views:
Transcription
1 Econ 0: Poblem Set Anses Instucto: Alexande Sollaci T.A.: Ryan Hughes Winte 08 Question (a) The fixed cost is F C = 4 and the total vaiable costs ae T CV (y) = 4y. (b) To anse this question, let x = (x,..., x n ) be the vecto of inputs used in this poduction n function. Recall that y = F (x) and that T V C(y) = p i x i, hee p i is the pice of facto i. Then i= T V C(y) = 4 [F (x,..., x n )] = n p i x i. i= F (x,..., x n ) = n p i x i and it follos that F (x) has deceasing etuns to scale. (c) T V C() = = 8, T V C() = = 0, T V C(4) = = 68. See figue fo the plot. (d) AF C(Y ) = F C/y = 4/y. Theefoe, AF C() = 4, AF C() = and AF C(4) =. See figue fo the plot. (e) AV C(y) = T V C(y)/y = 4y. Theefoe, AV C() = 4, AV C() = 8 and AV C(4) = 6. (f) AT C(y) = AT C(y)/y = 4/y +4y. Theefoe AT C() = 8, AT C() = 0 and AT C(4) = 7. (g) As shon above, AT C(y) = 4/y + 4y. We can check that this is a convex function in the elevant ange of y by taking its second deivative: AT C (y) = 4 4y = AT C (y) = y > 0 y > 0. No that e kno that AT C(y) is convex, e kno that taking the FOC ill be sufficient to find a minimum. Then, i= [y] : 4 4y = 0 = y = ±. Since y 0, e can discad the y = solution above, and e ill fund that y = is the value at hich AT C(y) achieves its minimum value.
2 Figue : Total, Vaiable and Fixed Cost. Figue : Aveage Costs. (h) The maginal cost is found by taking the deivative of T C(y)..t. y. Hence, MC(y) = 8y. It is easy to check that maginal cost cuve cosses the aveage total cost cuve at the point hee AT C(y) is at its minimum. See figue fo the plot.
3 To undestand hy this is tue, it is useful to conside an example using test scoes. Imagine that eveyone in a class takes a test and e compute the aveage scoe. If a ne student joins the class and this student s scoe is loe than the class aveage, then the class aveage ill dop; convesely, if the ne student s scoe is highe the class aveage, then the class aveage ill ise. In othe ods: if MC(y) is belo AT C(y), then AT C(y) is deceasing in y; if MC(y) is above AT C(y), then AT C(y) is inceasing in y. It follos that heneve MC(y) cosses AT (y)c (and theefoe becomes lage than AT C), AT C(y) changes fom a deceasing function to an inceasing function, hich means the point hee they coss is the point hee AT C(y) eaches its smallest value. Figue : Aveage and Maginal Cost.
4 Question (a) See figue 4: Figue 4: Plot of the poduction set, using A =, α = /4, β = / and fixing K = Note: if e used gaphs ith the x-, y-, and z-axes, e could epeat the above execise ithout holding K constant. Gaphically, e get figue 5. (b) MP K = f(k, L) K = αakα L β MP L = f(k, L) L = βak α L β The economic intepetation of MP K is the quantity by hich output inceases hen e incease capital, holding labo fixed. Since α < and β > 0, it is clea fom the expession above that MP K is deceasing on K and inceasing on L. Similaly, the intepetation fo MP L is the quantity by hich output inceases hen e incease labo, holding capital fixed. Since β < and α > 0, it is clea that MP L is deceasing on L and inceasing on K. (c) The poduction function has inceasing etuns to scale: f(tk, tl) = A(tk) α (tl) β = t α+β Ak α L β = t α+β f(k, L). Since α + β > and f is homogeneous of degee α + β, it follos that f is homogeneous of degee >, hich means that it has inceasing etuns to scale. f does not violate the La of Diminishing Retuns. As shon in pat (b), both maginal poducts ae deceasing on thei espective facto of poduction (MP K is deceasing on K and MP L is deceasing on L). 4
5 Figue 5: Suface plot of the poduction function F (K, L) = K 4 L. (d) The anses to pat (c) ae not contadictoy. It is a common mistake to think that Diminishing Retuns and Retuns to Scale ae elated, but they ae vey diffeent things. Diminishing Retuns ae about hat happens to output hen e incease one facto of poduction hile keeping all othes fixed. Retuns to Scale ae about hat happens to output hen e incease all factos of poduction hile keeping thei atios constant. (e) T RS = MP L = βakα L β MP K αak α L β = β K α L. The TRS epesents the tadeoff beteen K and L hile holding output constant. Fo example, if L = L and K = K and e ant to incease L by an infinitesimal amount ε, e ill have to decease K by β α ε to be able to keep output constant. K L (f) Yes, the aveage cost must be deceasing on y. Recall that the fim s total cost is given by Then, aveage cost is T C(y) = T C(f(K, L)) = L + K. AT C(y) = T C(y) y = L + K f(k, L) The numeato of the expession above is homogeneous of degee, hile the denominato is homogeneous of degee α + β. It follos that, if e inceased costs by a facto of t, e ould incease y by a facto of t α+β. Hence, AT C(t α+β y) = t(l + K) L + K t α+β = t α β f(k, L) f(k, L) Since t α+β y > y, this poves that AT C(y) is deceasing on y. < AT C(y) t >. 5
6 Question (a) If α + β =, then the fim has CRS; if α + β < the fim DRS. (b) It is easonable to think that a fim ith CRS ould have highe pofits, since it is moe efficient i.e. fo the same esouces, it can poduce moe, povided that both K and L ae geate than. (c) The fim solves Taking FOC s, e get π(, ) = max K,L Kα L β K L. [K] : [L] : αk α L β = βk α L β = Multiplying both sides of [K] by K and both sides of [L] by L, e get that K = αk α L β and L = βk α L β. Since α + β =, it immediately follos that K + L = K α L β, hich implies that pofits ae zeo. (d) The fim s poblem emains identical, so e can use the same FOC s fom pat (c). [K]/[L] gives us α L β K = o K = α β L. Plugging this into [L] yields Afte some algeba, e find β ( ) α α β L L β = ( α ) α L = No plugging this back into [K]/[L], e get It no follos that ( α ) α( β) K α L β = α β ( β ( α ) β K = α β ( β α β ( β ) αβ α β ( α ) αβ α β ( β ) α α β. ) β α β. ) ( α)β α β = ( α ) α α β ( β Again fom pat (c) e had that K = αk α L β and L = βk α L β, so pofits ae ( α ) α π(, ) = ( α β) Note that π(, ) > 0 because α β > 0. α β ( β ) β α β. ) β α β. The esult e found is that the pofit made by the fim ith DRS is highe than the pofit made by the fim ith CRS, contay to hat might be intuitive on pat (b). The takeaay fom this execise is that etuns to scale and efficiency/pofitability ae vey diffeent concepts, even though they may appea to measue elated qualities of the poduction function. 6
7 (e) This claim is a diect application of Eule s Theoem (fom Poblem Set ). Recall that a poduction function ith CRS is homogeneous of degee. Fom Eule s Theoem, if g is homogeneous of degee, then fo any (x, y). No let us look at the fim s poblem. It ants to The FOc s ae g(x, y) = xg x (x, y) + yg y (x, y). (*) max g(x, y) p xx p y y x,y [x] : [y] : g x (x, y ) = p x g y (x, y ) = p y It immediately follos fom those to equations that Theefoe, e can eite the pofit as x g x (x, y ) = x p x and y g x (x, y ) = y p y π(p x, p y ) = max x,y g(x, y) p xx p y y = g(x (p x, p y ), y (p x, p y )) p x x (p x, p y ) p y y (p x, p y ) π(p x, p y ) = g(x, y ) x g x (x, y ) y g y (x, y ), hee the aguments of x and y ee omitted to save space. Using equation ( ), it follos that π(p x, p y ) = 0. Note that this esult can easily be extended to a poduction function ith any numbe of aguments. Question 4 (a) The fim s maximization poblem is The FOC s ae max pk L K L. K,L [L] : p L K = 0 = = p L K () [K] : p K L = 0 = = p K L () Equations () and () imply Plugging back into () K L = = K = L = p L ( L ) = p L 7
8 ) ˆL(,, p) = (A) No plugging this into the expession fo K above, K = ) ) ˆK(,, p) = (B) Finally, [ (p ) ] [ ( ] p ) ˆπ(,, p) = p ) ) [ (p ) ] ) ˆπ(,, p) = p ) ˆπ(,, p) = [ ] It ill also be useful to have ) ˆπ(,, p) =. (C) [ (p ) ] [ ( ] p ŷ(,, p) = ) ) ŷ(,, p) = (D) It is staightfoad fom (A), (B) and (C) that if p, and ae positive, then both facto demands and pofits ae positive as ell. (b) We ant to sho that ˆL(t, t, tp) = ˆL(,, p) and ˆK(t, t, tp) = ˆK(,, p) Fom (A), e have and fom (B), ˆL(t, t, tp) = ( ) tp ( (t)(t) = p ) t t = ˆL(,, p) ˆK(t, t, tp) = ( ) tp ( (t) (t) = p ) t t = ˆK(,, p) The intuition hee is a vaiant of the idea that only elative pices matte. Since multiplying all pices by the same constant does not affect elative pices, demands ae also unchanged. Anothe ay of undestanding this esult is by ealizing that the unit ith hich e measue 8
9 pices should not affect ou decisions. Fo example, if e ee to use cents (instead of dollas) to measue all pices (so t = 00), ould you expect the demands fo inputs of fims to change? No! nothing in the eal economy has changed, so demands shouldn t change. This same idea also caies ove to exchange ates: it does not matte if pices ae in dollas, pounds o pesos; fims should alays make the same decisions. (c) Just take deivatives: ˆL(,, p) ˆK(,, p) ) = < 0 ) = < 0 Both the expessions above ae stictly negative, since (,, p) 0. This esult is analogous to the substitution effect fom consume theoy: if the pice of one facto inceases, then fims ill use less of that facto in poduction. (d) Fom (C), e have ˆπ(t, t, tp) = ( ) tp ( (t)(t) = p ) t t = tˆπ(,, p). The intuition hee is simila to pat (b): suppose e stated measuing pices in cents instead of in dollas. We d bette also measue pofits in cents! (e) We have the pofit function in (C). Taking deivatives, ˆπ(,, p) ˆπ(,, p) = ) ) = = ˆL(,, p), fom (A). = ) ) = = ˆK(,, p), fom (B). ˆπ(,, p) p = p ) = p ( ) ) = = ŷ(,, p), fom (D). (f) Hotelling s Lemma is a diect application of the Envelope Theoem, hich e poved in Poblem Set (except this is even an easie case, since the maximization has no constaints). Question 5 (a) The fim solves The Lagangian is Taking FOC s min K,L K + L s.t y = K L L = K + L + λ[y K L ]. Fo any vecto x, x 0 means that evey enty in the vecto is stictly positive. 9
10 [L] : λl K = 0 = = λ L K () [K] : λk L = 0 = = λ K L (4) [λ] : y K L = 0 = y = K L (5) ()/(4) = K L = K = L (6) (6) into (5) yields y = ( L ) L L (,, y) = y ( ) Plugging this into (6), e get K (,, y) = y ( ) and C (,, y) = y ( ) ( + y ) (b) Plugging L and K into () e get ( = λ y C (,, y) = y (). (7) ( ) ) ( y ( ) ) ( = λy ) ( y ) 6 And fom (7), λ (,, y) = y (8) C (,, y) y = y () = λ (,, y) Thus, the Lagange multiplie equals the maginal cost. Again, this is a diect application of the Envelope Theoem of the fim s poblem in pat (a). 0
11 (c) The fims no chooses y to maximize its pofit, π = py C (,, y): max {py y () } y FOC: SOC: [y] : p y () = 0 [yy] : y () < 0 Note fom the FOC that e have p = y () }{{} C y =λ (9) Fom pats (a) and (b), e had that Plugging this into (9), e get C y = λ = y (). λ (,, y) = p, hich immediately implies that () () and () (4) that is, the cost minimization poblem and the pofit maximization poblem geneate the same fist ode conditions. Question 6 (a) The poblem ith this poduction function is that it is not smooth it has a kink at ak = bl, hich means that the deivative of F at that point does not exist and theefoe the FOC s ae not ell defined. To solve this poblem, e instead use the same intuition fo pefectly complementay goods in consume theoy. Fist, note that the fim ill alays optimally choose K and L such that ak = bl. To see this, suppose that a fim chose instead ˆK and ˆL such that a ˆK > bˆl. The fim s pofit ould be: min{a ˆK, bˆl} ˆK ˆL = bˆl ˆK ˆL Clealy, this fim could educe its demand fo capital and incease it s pofits. In fact, it could do so up to the point hee a ˆK = bˆl; if it educed moe, then min{a ˆK, bˆl} = a ˆK and it ould have the same poblem ith hiing too much labo. Since ak = bl, it also follos that y = min{ak, bl} = ak = bl, hich implies that the conditional facto demands ae K = y/a and L = y/b. (b) Technically speaking, thee is nothing ong ith this poduction function meaning that all of its deivatives ae ell defined. Hoeve, ou usual techniques (taking deivatives and equating them to zeo) ae not helpful. The poblem in this case is that one of the fist ode conditions ill in geneal not be equal to zeo. Instead, e can use economic intuition just like in the pevious case. The main issue hee is that capital and labo ae pefectly substitutable in poducing output. Hence, the fim ill Fo anyone inteested, e can solve this poblem using linea pogamming o the Kuhn-Tucke Conditions instead.
12 alays choose the input that is elatively cheape. The thought pocess goes like this: to poduce unit of output, y, the fim has to hie /a units of capital o /b units of labo. In tun, /a units of capital costs /a dollas and /b units of labo costs /b dollas. It follos that if /a < /b, then it cheape to poduce each unit of output using only capital; convesely, if /b < /a, then it is cheape to poduce output using only labo. If /b = /a, then the mix does not matte. Hence, the conditional facto demands ae y/a if a < b K(, ) = [0, y/a] if a = b 0 if a > b y/b if a > b and L(, ) = [0, y/b] if a = b 0 if a < b
Solution to Problem First, the firm minimizes the cost of the inputs: min wl + rk + sf
Econ 0A Poblem Set 4 Solutions ue in class on Tu 4 Novembe. No late Poblem Sets accepted, so! This Poblem set tests the knoledge that ou accumulated mainl in lectues 5 to 9. Some of the mateial ill onl
More informationReview of the H-O model. Problem 1. Assume that the production functions in the standard H-O model are the following:
Revie of the H-O model Poblem 1 Assume that the poduction functions in the standad H-O model ae the folloing: f 1 L 1 1 ) L 1/ 1 1/ 1 f L ) L 1/3 /3 In addition e assume that the consume pefeences ae given
More informationA Comment on Increasing Returns and Spatial. Unemployment Disparities
The Society fo conomic Studies The nivesity of Kitakyushu Woking Pape Seies No.06-5 (accepted in Mach, 07) A Comment on Inceasing Retuns and Spatial nemployment Dispaities Jumpei Tanaka ** The nivesity
More informationSuggested Solutions to Homework #4 Econ 511b (Part I), Spring 2004
Suggested Solutions to Homewok #4 Econ 5b (Pat I), Sping 2004. Conside a neoclassical gowth model with valued leisue. The (epesentative) consume values steams of consumption and leisue accoding to P t=0
More informationDo Managers Do Good With Other People s Money? Online Appendix
Do Manages Do Good With Othe People s Money? Online Appendix Ing-Haw Cheng Haison Hong Kelly Shue Abstact This is the Online Appendix fo Cheng, Hong and Shue 2013) containing details of the model. Datmouth
More informationHandout: IS/LM Model
Econ 32 - IS/L odel Notes Handout: IS/L odel IS Cuve Deivation Figue 4-4 in the textbook explains one deivation of the IS cuve. This deivation uses the Induced Savings Function fom Chapte 3. Hee, I descibe
More informationNotes on McCall s Model of Job Search. Timothy J. Kehoe March if job offer has been accepted. b if searching
Notes on McCall s Model of Job Seach Timothy J Kehoe Mach Fv ( ) pob( v), [, ] Choice: accept age offe o eceive b and seach again next peiod An unemployed oke solves hee max E t t y t y t if job offe has
More informationac p Answers to questions for The New Introduction to Geographical Economics, 2 nd edition Chapter 3 The core model of geographical economics
Answes to questions fo The New ntoduction to Geogaphical Economics, nd edition Chapte 3 The coe model of geogaphical economics Question 3. Fom intoductoy mico-economics we know that the condition fo pofit
More informationMacro Theory B. The Permanent Income Hypothesis
Maco Theoy B The Pemanent Income Hypothesis Ofe Setty The Eitan Beglas School of Economics - Tel Aviv Univesity May 15, 2015 1 1 Motivation 1.1 An econometic check We want to build an empiical model with
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationTest 2, ECON , Summer 2013
Test, ECON 6090-9, Summe 0 Instuctions: Answe all questions as completely as possible. If you cannot solve the poblem, explaining how you would solve the poblem may ean you some points. Point totals ae
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationF-IF Logistic Growth Model, Abstract Version
F-IF Logistic Gowth Model, Abstact Vesion Alignments to Content Standads: F-IFB4 Task An impotant example of a model often used in biology o ecology to model population gowth is called the logistic gowth
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationMATH 415, WEEK 3: Parameter-Dependence and Bifurcations
MATH 415, WEEK 3: Paamete-Dependence and Bifucations 1 A Note on Paamete Dependence We should pause to make a bief note about the ole played in the study of dynamical systems by the system s paametes.
More informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let
More informationPHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased
PHYS 0B - HW #7 Sping 2004, Solutions by David Pace Any efeenced euations ae fom Giffiths Poblem statements ae paaphased. Poblem 0.3 fom Giffiths A point chage,, moves in a loop of adius a. At time t 0
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationEffect of no-flow boundaries on interference testing. in fractured reservoirs
Effect of no-flo boundaies on intefeence testing in factued esevois T.Aa. Jelmet 1 1 epatement of petoleum engineeing and applied geophysics,, Noegian Univesity of Science and Tecnology, NTNU. Tondheim,
More informationNumerical Integration
MCEN 473/573 Chapte 0 Numeical Integation Fall, 2006 Textbook, 0.4 and 0.5 Isopaametic Fomula Numeical Integation [] e [ ] T k = h B [ D][ B] e B Jdsdt In pactice, the element stiffness is calculated numeically.
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationMEASURING CHINESE RISK AVERSION
MEASURING CHINESE RISK AVERSION --Based on Insuance Data Li Diao (Cental Univesity of Finance and Economics) Hua Chen (Cental Univesity of Finance and Economics) Jingzhen Liu (Cental Univesity of Finance
More informationPsychometric Methods: Theory into Practice Larry R. Price
ERRATA Psychometic Methods: Theoy into Pactice Lay R. Pice Eos wee made in Equations 3.5a and 3.5b, Figue 3., equations and text on pages 76 80, and Table 9.1. Vesions of the elevant pages that include
More informationLecture 28: Convergence of Random Variables and Related Theorems
EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An
More informationC/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22
C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationChapter 5 Linear Equations: Basic Theory and Practice
Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and
More informationAppendix B The Relativistic Transformation of Forces
Appendix B The Relativistic Tansfomation of oces B. The ou-foce We intoduced the idea of foces in Chapte 3 whee we saw that the change in the fou-momentum pe unit time is given by the expession d d w x
More informationAs is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.
Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.
More information2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8
5 CHAPTER Fundamentals When solving equations that involve absolute values, we usually take cases. EXAMPLE An Absolute Value Equation Solve the equation 0 x 5 0 3. SOLUTION By the definition of absolute
More information2 Governing Equations
2 Govening Equations This chapte develops the govening equations of motion fo a homogeneous isotopic elastic solid, using the linea thee-dimensional theoy of elasticity in cylindical coodinates. At fist,
More informationME 3600 Control Systems Frequency Domain Analysis
ME 3600 Contol Systems Fequency Domain Analysis The fequency esponse of a system is defined as the steady-state esponse of the system to a sinusoidal (hamonic) input. Fo linea systems, the esulting steady-state
More informationTHE CONE THEOREM JOEL A. TROPP. Abstract. We prove a fixed point theorem for functions which are positive with respect to a cone in a Banach space.
THE ONE THEOEM JOEL A. TOPP Abstact. We pove a fixed point theoem fo functions which ae positive with espect to a cone in a Banach space. 1. Definitions Definition 1. Let X be a eal Banach space. A subset
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G-type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this investigation
More informationNOTE. Some New Bounds for Cover-Free Families
Jounal of Combinatoial Theoy, Seies A 90, 224234 (2000) doi:10.1006jcta.1999.3036, available online at http:.idealibay.com on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment
More informationCHAPTER 3. Section 1. Modeling Population Growth
CHAPTER 3 Section 1. Modeling Population Gowth 1.1. The equation of the Malthusian model is Pt) = Ce t. Apply the initial condition P) = 1. Then 1 = Ce,oC = 1. Next apply the condition P1) = 3. Then 3
More informationBrad De Long è Maury Obstfeld, Petra Geraats è Galina Hale-Borisova. Econ 202B, Fall 1998
uggested olutions to oblem et 4 Bad De Long Mauy Obstfeld, eta Geaats Galina Hale-Boisova Econ 22B, Fall 1998 1. Moal hazad and asset-pice bubbles. a The epesentative entepeneu boows B on date 1 and invests
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationWhen two numbers are written as the product of their prime factors, they are in factored form.
10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position
More informationAn Application of Fuzzy Linear System of Equations in Economic Sciences
Austalian Jounal of Basic and Applied Sciences, 5(7): 7-14, 2011 ISSN 1991-8178 An Application of Fuzzy Linea System of Equations in Economic Sciences 1 S.H. Nassei, 2 M. Abdi and 3 B. Khabii 1 Depatment
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationCompactly Supported Radial Basis Functions
Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically
More informationUnobserved Correlation in Ascending Auctions: Example And Extensions
Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More information9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.
Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this
More informationDonnishJournals
DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş
More informationLinear Algebra Math 221
Linea Algeba Math Open Book Eam Open Notes Sept Calculatos Pemitted Sho all ok (ecept #). ( pts) Gien the sstem of equations a) ( pts) Epess this sstem as an augmented mati. b) ( pts) Bing this mati to
More information33. 12, or its reciprocal. or its negative.
Page 6 The Point is Measuement In spite of most of what has been said up to this point, we did not undetake this poject with the intent of building bette themometes. The point is to measue the peson. Because
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationAppendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk
Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating
More information( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.
9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can
More informationTo Feel a Force Chapter 7 Static equilibrium - torque and friction
To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on
More information15 Solving the Laplace equation by Fourier method
5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the
More informationElectrostatics (Electric Charges and Field) #2 2010
Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when
More information1 Similarity Analysis
ME43A/538A/538B Axisymmetic Tubulent Jet 9 Novembe 28 Similaity Analysis. Intoduction Conside the sketch of an axisymmetic, tubulent jet in Figue. Assume that measuements of the downsteam aveage axial
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More information18.06 Problem Set 4 Solution
8.6 Poblem Set 4 Solution Total: points Section 3.5. Poblem 2: (Recommended) Find the lagest possible numbe of independent vectos among ) ) ) v = v 4 = v 5 = v 6 = v 2 = v 3 =. Solution (4 points): Since
More informationHomework 7 Solutions
Homewok 7 olutions Phys 4 Octobe 3, 208. Let s talk about a space monkey. As the space monkey is oiginally obiting in a cicula obit and is massive, its tajectoy satisfies m mon 2 G m mon + L 2 2m mon 2
More informationOLYMON. Produced by the Canadian Mathematical Society and the Department of Mathematics of the University of Toronto. Issue 9:2.
OLYMON Poduced by the Canadian Mathematical Society and the Depatment of Mathematics of the Univesity of Toonto Please send you solution to Pofesso EJ Babeau Depatment of Mathematics Univesity of Toonto
More informationThe Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
More informationLab 10: Newton s Second Law in Rotation
Lab 10: Newton s Second Law in Rotation We can descibe the motion of objects that otate (i.e. spin on an axis, like a popelle o a doo) using the same definitions, adapted fo otational motion, that we have
More informationPES 3950/PHYS 6950: Homework Assignment 6
PES 3950/PHYS 6950: Homewok Assignment 6 Handed out: Monday Apil 7 Due in: Wednesday May 6, at the stat of class at 3:05 pm shap Show all woking and easoning to eceive full points. Question 1 [5 points]
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationHopefully Helpful Hints for Gauss s Law
Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux
More informationIntroduction Common Divisors. Discrete Mathematics Andrei Bulatov
Intoduction Common Divisos Discete Mathematics Andei Bulatov Discete Mathematics Common Divisos 3- Pevious Lectue Integes Division, popeties of divisibility The division algoithm Repesentation of numbes
More informationAnalysis of simple branching trees with TI-92
Analysis of simple banching tees with TI-9 Dušan Pagon, Univesity of Maibo, Slovenia Abstact. In the complex plane we stat at the cente of the coodinate system with a vetical segment of the length one
More informationequilibrium in the money market
Sahoko KAJI --- Open Economy Macoeconomics ectue Notes I I A Review of Closed Economy Macoeconomics We begin by eviewing some of the basics of closed economy macoeconomics that ae indispensable in undestanding
More information. Using our polar coordinate conversions, we could write a
504 Chapte 8 Section 8.4.5 Dot Poduct Now that we can add, sutact, and scale vectos, you might e wondeing whethe we can multiply vectos. It tuns out thee ae two diffeent ways to multiply vectos, one which
More informationHOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS?
6th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE HOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS? Cecília Sitkuné Göömbei College of Nyíegyháza Hungay Abstact: The
More informationGauss Law. Physics 231 Lecture 2-1
Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationEfficiency Loss in a Network Resource Allocation Game: The Case of Elastic Supply
Efficiency Loss in a Netwok Resouce Allocation Game: The Case of Elastic Supply axiv:cs/0506054v1 [cs.gt] 14 Jun 2005 Ramesh Johai (johai@stanfod.edu) Shie Manno (shie@mit.edu) John N. Tsitsiklis (jnt@mit.edu)
More informationA generalization of the Bernstein polynomials
A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationof the contestants play as Falco, and 1 6
JHMT 05 Algeba Test Solutions 4 Febuay 05. In a Supe Smash Bothes tounament, of the contestants play as Fox, 3 of the contestants play as Falco, and 6 of the contestants play as Peach. Given that thee
More informationCircular Orbits. and g =
using analyse planetay and satellite motion modelled as unifom cicula motion in a univesal gavitation field, a = v = 4π and g = T GM1 GM and F = 1M SATELLITES IN OBIT A satellite is any object that is
More informationMath 2263 Solutions for Spring 2003 Final Exam
Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate
More informationI. CONSTRUCTION OF THE GREEN S FUNCTION
I. CONSTRUCTION OF THE GREEN S FUNCTION The Helmohltz equation in 4 dimensions is 4 + k G 4 x, x = δ 4 x x. In this equation, G is the Geen s function and 4 efes to the dimensionality. In the vey end,
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More informationINTRODUCTION. 2. Vectors in Physics 1
INTRODUCTION Vectos ae used in physics to extend the study of motion fom one dimension to two dimensions Vectos ae indispensable when a physical quantity has a diection associated with it As an example,
More informationInternational Journal of Mathematical Archive-3(12), 2012, Available online through ISSN
Intenational Jounal of Mathematical Achive-3(), 0, 480-4805 Available online though www.ijma.info ISSN 9 504 STATISTICAL QUALITY CONTROL OF MULTI-ITEM EOQ MOEL WITH VARYING LEAING TIME VIA LAGRANGE METHO
More informationOn the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
More informationEcon 201: Problem Set 3 Answers
Econ 20: Problem Set 3 Ansers Instructor: Alexandre Sollaci T.A.: Ryan Hughes Winter 208 Question a) The firm s fixed cost is F C = a and variable costs are T V Cq) = 2 bq2. b) As seen in class, the optimal
More informationRigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018
Rigid Body Dynamics 2 CSE169: Compute Animation nstucto: Steve Rotenbeg UCSD, Winte 2018 Coss Poduct & Hat Opeato Deivative of a Rotating Vecto Let s say that vecto is otating aound the oigin, maintaining
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationASTR415: Problem Set #6
ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal
More informationMATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates
MATH 417 Homewok 3 Instucto: D. Cabea Due June 30 1. Let a function f(z) = u + iv be diffeentiable at z 0. (a) Use the Chain Rule and the fomulas x = cosθ and y = to show that u x = u cosθ u θ, v x = v
More informationResearch Design - - Topic 17 Multiple Regression & Multiple Correlation: Two Predictors 2009 R.C. Gardner, Ph.D.
Reseach Design - - Topic 7 Multiple Regession & Multiple Coelation: Two Pedictos 009 R.C. Gadne, Ph.D. Geneal Rationale and Basic Aithmetic fo two pedictos Patial and semipatial coelation Regession coefficients
More informationMultiple Experts with Binary Features
Multiple Expets with Binay Featues Ye Jin & Lingen Zhang Decembe 9, 2010 1 Intoduction Ou intuition fo the poect comes fom the pape Supevised Leaning fom Multiple Expets: Whom to tust when eveyone lies
More informationReview Exercise Set 16
Review Execise Set 16 Execise 1: A ectangula plot of famland will be bounded on one side by a ive and on the othe thee sides by a fence. If the fame only has 600 feet of fence, what is the lagest aea that
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationQualifying Examination Electricity and Magnetism Solutions January 12, 2006
1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
More informationExploration of the three-person duel
Exploation of the thee-peson duel Andy Paish 15 August 2006 1 The duel Pictue a duel: two shootes facing one anothe, taking tuns fiing at one anothe, each with a fixed pobability of hitting his opponent.
More informationThe geometric construction of Ewald sphere and Bragg condition:
The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in
More informationPhysics 121 Hour Exam #5 Solution
Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More information