Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule."

Transcription

1 Text questions, Chpter 5, problems 1-5: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed by recognizing tht mximum world output of C is just the verticl sum of mximum ntionl outputs. From tht point, if there is ny demnd for F, it will be provided by the country with comprtive dvntge in F production. In the text, this is the Foreign country. Thus, the slope of the world PPF will be identicl to tht of the Foreign country s PPF. Foreign will continue to provide F (s well s C, Home is specilized to C production), until it is specilized in the production of F, t point 1. From tht point, dditionl demnd for F must come from the Home country, which will be producing both goods until it too is specilized in production of F. b. ht is the menu of the worst combintions of outputs? Show tht it is mde up of two liner segments. The dshed line in the bove digrm is the menu of ntionlly efficient but globlly inefficient points. [i.e. the truly worst point would be the origin zero output from either country.] This is generted by perverse speciliztion. Tht is, from the point of globlly mximum output of C, if demnd for F is first served by the Home country, until it is speicilized, t point 2, nd then by the Foreign country, we construct the dshed frontier. 2. In the discussion of Section 5.6 suppose tht both Home nd Foreign countries produce commodity 3. Recll tht the sitution in section 5.6 involves lrge country producing 3 goods (1,2,3),

2 nd second country producing good 3, nd possibly other goods.. ht cn be sid bout the distribution of income between countries if Home demnd switches smll mount from commodity 2 to some commodity produced brod? Answering these questions involves using the zero-profit conditions i.e. P j # Lj w nd P j * # LJ *w* nd recognizing tht if good 3 is produced in common P 3 = P 3 *. Using these two fcts, nd fixed Lj, we get key impliction: P 3 = P 3 * Y Lj w = LJ *w*, so if price of good 3 is unchnged between equilibri, wges must be unchnged. Thus, there re two cses to consider. First, the incresed demnd for Foreign lbor (in the form of incresed demnd for Foreign goods) rises w*/w. But this must result in L3 w < L3 *w*, nd shift of ll good 3 production to the Home country. Alterntively, the Home country is lrge nd Foreign is specilized to production of good 3, before nd fter the chnged demnd. Becuse reltive prices re determined by production conditions in the lrge (i.e. Home) country, reltive prices re unchnged, both countries continue to produce good 3, so w*/w is unchnged. b. If Home technology for producing commodity 1 improves? Fix the price of good 3 t $1. Since there is no chnge in technology in the production of good 3, there will be no chnge in wges in either country. The price of good 1 will then fll by n mount equl to the mount of the productivity increse in production of good 1. This will led to gins for both countries, with the gins being proportionl to the mount of good 1 consumed. 3. Suppose costs of production depend only on lbor costs, nd tht to produce unit of ech commodity in ech country tkes the number of lbor-hours shown: LA LB Home Foreign In which commodity does the Home country possess the gretest comprtive dvntge? For mking comprisons with n > 2 goods, it is useful to recll tht unit cost for good j is c j = Lj w. Furthermore, with trde the lw of one price implies tht there cn only be one price for good j nd p j = c j in ll countries with positive output. Thus, if Lj w < Lj *w*, then only the Home country will produce good j. Rerrnging, we hve condition which sttes tht if:

3 Lj * Lj * w < Home produces j. w Similrly, if Lj w > Lj *w*, then only the Foreign country will produce good j, nd we hve: Lj * Lj * w > Foreign produces j. w Tking these together mens tht we cn crete chin, ordering industries by degree of Home dvntge. Using the dt in the bove tble, we hve: 10 LB 10 LA = < = < = * * *. LB LA 3 which implies tht the Home country hs the gretest comprtive dvntge in the production of commodity C. Note tht these rtios re bsolute dvntge comprisons nd re not informtive bout production nd trde without wge dt. But we cn use this chin to identify the extreme members. b. If the Foreign wge rte is $1 per lbor hour nd free-trde equilibrium is reched, wht is the most tht the Home wge rte cn be? hy? Recll tht which countries produce which goods is determined by the reltive wge in the two countries w*/w. ith Foreign wge of $1, the Home country must be ble to produce t lest one good, so the w*/w cnnot be less thn 10/7. ith Foreign wge of $1, the Home wge rte cn be t most 7/10. c. If the Foreign wge rte is $1 per lbor-hour, wht would possible Home wge rte be so tht the Home country cn produce only one commodity? hich commodity would tht be? This question sks to to find Home wge, given Foreign wge of $1, such tht Home is just priced out of producing good B. This will occur if w*/w > 2. Thus, the Home country cn produce only one commodity, C, for ny wge greter thn ½.

4 4. In Ricrdin world, with lbor the only fctor of production being pid, the following tble gives the constnt lbor costs per unit of producing different commodities for countries " nd $. het L Crs Tnkers LT Rectors LR Trctors LX " $ In which commodities does country " hve n bsolute dvntge? hy? Country " hs n bsolute dvntge in the production of rectors nd trctors. Recll tht bsolute dvntge involves direct comprison of lbor productivities. Since LR < LR * nd LX < LX * we hve our nswer. Note tht we could compre lbor productivities directly by looking t MPPLs: " LR > " LR * nd " LX > " LX *. b. In which commodities does country $ hve comprtive dvntge? hy? As in the previous question, we need to construct the chin: α LX β LX α α α LR LT < β < β < β < LR LT < < < < But, without informtion on reltive wges, we cnnot sy where the chin will be broken. $ will certinly hve comprtive dvntge in whet, but beyond tht we cnnot sy. c. hich country would export tnkers? Explin. Country " will export trctors, if it exports nything. Tnkers, in the middle of the chin cnnot be predicted without informtion on w $ /w ". 5. Consider the world to consist of two countries (Home nd Foreign) mde up of individuls with identicl nd homothetic tste ptterns. Portry these by set of smoothly bowed-in indifference curves. Suppose production in the Home country requires two lbor hours per unit of food nd only one lbor-hour per unit of clothing, wheres the Foreign country s figures for food nd clothing re just the opposite. The Foreign country s lbor force consists of 1 million lbor hours.. If the Home country is smll reltive to the Foreign country, one of the countries will produce both goods. hich country? ht will be food s reltive price? α L β L

5 It is useful to strt by explicitly showing the opportunity cost rtios: Home Home 2 1 = > = = 1 2 For For. The smll country lwys specilizes. Since the smll country is price tker, the world price is fixed by the lrge country s utrky price rtio. Thus, if Foreign is lrge, the world reltive price of Food is ½ nd the Home country will specilize in clothing. The Foreign country will produce both goods. b. If the Home country is lrge reltive to the Foreign country, one of the countries will produce both goods. hich country? ht will be food s reltive price? The sitution here is just reversed. The world price is now set by the, now lrge, Home country s utrky reltive price of food, which is 2. The, now smll, Foreign country will specilize in its comprtive dvntge industry, in this cse food. And the Home country will produce both goods. c. Construct world supply nd demnd schedules for food, with p = P F /P C on the verticl xis nd world food output on the horizontl. * * p w S The supply curve hs two horizontl components nd one verticl. For D # þ F, e.g. 1, tht demnd will be stisfied by the country with comprtive dvntge in food, i.e. the Foreign country. At the Foreign speciliztion point, þ F, there re rnge of prices consistent with ech country being specilized in its comprtive dvntge good e.g. world demnd curve 2. If demnd is sufficiently gret tht output of food from the Home country is lso required, i.e. D $ þ F (e.g. t demnd curve 3), the world price will be set by technicl conditions in the Home country. This determines the other horizontl component. þ* F Q + Q* F F

6 orkbook questions, 1, 2, 4, nd 8 1. Absolute versus Comprtive Advntge: Suppose tht in the US four mn-hours re required to produce ech unit of clothing nd ech unit of food. In Cnd, one mn-hour is required for ech unit of clothing nd two mn-hours re required for unit of food.. hich country hs n bsolute dvntge in ech good? To strt with, it is useful to collect these ssumptions into simple tble: US 4 4 Cnd 1 2 Note tht bsolute dvntge comprisions involve bilterl comprisons of productivity by industry. Thus, Cnd hs n bsolute dvntge in the production of both goods, i.e.: 1 < 4, nd 2 < 4. b. hich country hs comprtive dvntge in ech good? Recll tht comprtive dvntge comprisons involve compring reltive productivity. Thus, we need to compre: US US 4 1 = = 1 > = = 4 2 Cn Cn. This comprison shows tht Cnd hs lower opportunity cost/utrky price of clothing. Thus, Cnd hs comprtive dvntge in clothing. c. Assuming tht ech country hs 40 mn-hours of lbor vilble for production, drw the production possibilities frontiers for ech country. (Put food on the verticl xis). ht do the slopes of these frontiers indicte?

7 As we derived in clss, the slopes of these curves show the opportunity costs of clothing in terms of food. Tht is, the number of units of food tht must be given up to secure one more unit of clothing. d. Drw the world production possibilities frontier. ht does it s slope indicte? As drwn, the world production possibilities frontier shows output being produced by the most efficient country. Just s in the individul countries, the slope shows the opportunity cost of clothing in terms of food, in this cse to the world economy s whole. Since equilibrium occurs on the prt of the world PPF whose slope is determined by Cndin production conditions, this mens tht the world terms-of-trde will be equl to the

8 utrky MRT in Cnd. This implies, by the wy, tht ll gins from trde will ccrue to the US. e. If consumers in both countries hve Leontief preferences, consuming clothing nd food in fixed proportion of one-to-one, wht is the trde pttern? If we drw in the income-consumption line, which is fixed for Leontief preferences, we see tht Cnd will produce both goods, while the US will be specilized to food. Thus, the US will export food nd import clothing, while Cnd does the reverse. f. If the lbor force in the US increses by fctor of 20, will nything hppen to the trde pttern? This increse in the endowment of lbor will result in new world equilibrium, in which the US is producing both goods nd Cnd is specilized to clothing, with the world price determined by the opportunity cost rtio in the US. However, the trde pttern remins the sme: US will export food nd import clothing, while Cnd does the reverse. This is determined by comprtive dvntge, fct determined by technology, which ws unchnged by the increse in US popultion. 2. Commodity prices nd fctorl terms of trde: Suppose the lbor coefficients in the food nd clothing industries re s follows: PC P F * * 4 1 PC 10 = = = nd * = = = 8 2 P 5 = 4 * = 10 = 8 * = 5 Suppose lso tht the Home country hs 16 units of lbor nd the Foreign country hs 30.. Drw the production possibilities frontier for ech country nd clculte ech country s utrky prices. Since utrky reltive prices will be equl to the opportunity cost rtio (i.e. the MRT) we cn clculte the prices s: F 2 *. 1 Thus, utrky price of clothing in the Home country is ½ unit of food, while the utrky price of clothing in the Foreign country is 2 units of food. The PPFs for the two countries re shown in the following grph.

9 b. Drw the world production possibilities frontier. ht must the world reltive price of food in terms of clothing be in order for both countries to specilize in food? In clothing? Over wht rnge of prices will countries specilize in the good in which ech hs comprtive dvntge? PF > 2 P C will result in both countries specilizing in food, nd countries to specilize in clothing. If 1 2 PF < < P C 2 PF < 1 P 2 C for both ech country will specilize to its

10 comprtive dvntge good. c. Suppose tht the world price rtio is equl to 1. Clculte the reltive wges, w/w*. If prices re equl, both countries re specilized to their comprtive dvntge goods nd P C = w = *w* = P F *, so 4w = 5w* nd w/w* = 5/4. d. Now suppose tht there is n increse in world demnd for clothing so tht the reltive price of clothing rises to 3/2. ht hppens to the reltive wge between the Home nd Foreign country? Ptterns of production nd speciliztion re unltered, i.e. both countries re specilized in their comprtive dvntge good. However, now commodity prices re not equl, but re relted s 3/2. Thus the reltive wge, w/w* = 15/8. 4. Compring models: Define n increse in demnd for clothing s shift in the indifference curves towrd the clothing xis. This is illustrted below s shift from curve A to curve B.. Consider first n endowment economy with n lloction fo food nd clothing shown by point E nd the indifference curve A. Given our definition of demnd shock, suppose there is n increse in demnd for clothing. ht hppens to the reltive (utrky price of clothing)? ht hppens to the supply of clothing.

11 From the bove digrm it is esy to see tht t tstes given by B the reltive price of food (determined by the MRS t E) is greter thn when the tstes re given by A. Becuse this is n endowment economy, there is no chnge in supply. b. Now suppose the economy hs bowed out production possibilities frontier. ht hppens to the reltive price of clothing when demnd for clothing increses? ht hppens to the supply of clothing? Now the finl price is determined by the tngency between n indifference curve nd the PPF t the point where MRS = MRT. At this new equilibrium point the reltive price of food hs incresed, but so hs the output of food. The output of clothing hs, of course,

12 fllen. c. Finlly consider Ricrdin model of production. Now wht hppens to the reltive price of clothing when the demnd for clothing increses? ht hppens to the supply of clothing? In this cse, becuse we re t n interior equilibrium both before nd fter the chnge in tstes (i.e. both goods re produced in both equilibri), the reltive price is unchnged. Recll tht in the Ricrdin model, reltive price t interior equilibri is completely determined by technologicl conditions. Adjustment occurs only on the output mrgin, with output of food rising nd output of food flling. d. Since we know tht utrky prices determine the pttern of trde (i.e. vi the lw of comprtive dvntge), in wht sense re the endowment model nd the Ricrdin model extreme cses? (Hint: Think bout the role of supply nd demnd in determining utrky prices.) In the endowment model utrky prices re determined completely by the MRS, while in the Ricrdin model they re determined completely by the MRT. Tht is, demnd conditions determine utrky reltive price in the first cse nd supply conditions determine utrky reltive price in the ltter cse. In the stndrd neoclssicl cse, the interction of these two forces determine the utrky reltive price. 8. Interntionl wge comprisons: The following tble of wge rtes in different countries ws published in the ll Street Journl. Hourly Py Levels (For production workers in US Dollrs)

13 United Sttes $12.96 $13.19 $13.44 $13.62 Jpn $6.47 $9.47 $11.44 $13.80 est Germny $9.56 $13.35 $16.87 $20.19 Itly $7.40 $10.01 $12.33 $14.77 Frnce $7.52 $10.27 $12.42 $14.03 Britin $6.19 $7.50 $8.96 $11.06 In view of wht you hve lerned from the Ricrdin model of production, how would you respond to the following comments?. The high cost of hiring production workers in the US hs undermined its competitive position in world mrkets. No. US workers re pid more becuse they re more productive. b. The high wges in the US compred to those found brod re due to lbor unions forcing their levels bove wht would be implied by lbor s productivity in competitive Ricrdin world. Possible, but not likely. If wges were held bove competitive levels, US goods would be unble to compete interntionlly, cusing the eventul shutdown of industries supporting such wges. Also, note tht this might explin the reltively high wges in 1985, but if this explntion is correct, the dt show wekening of unions in the US reltive to the other countries, where much stronger wge growth is seen. c. The increse in Jpn s wges implies tht Jpn is losing its competitive edge in world mrkets. No. It implies tht the productivity of Jpnese workers is incresing. d. The low wges in Britin re n indiction tht British workers re being underpid. No. Their productivity is lower. e. Allowing importtion of goods from workers with lower wges is unfir to Americn workers. No. The lower pid worker is lso the less productive one.

Homework #6: Answers. a. If both goods are produced, what must be their prices?

Homework #6: Answers. a. If both goods are produced, what must be their prices? Text questions, hpter 7, problems 1-2. Homework #6: Answers 1. Suppose there is only one technique tht cn be used in clothing production. To produce one unit of clothing requires four lbor-hours nd one

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Written Homework 6 Solutions

Written Homework 6 Solutions Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Chapter 5: Elasticity. measures how strongly people respond to changes in prices and changes in income.

Chapter 5: Elasticity. measures how strongly people respond to changes in prices and changes in income. Chpter 5: Elsticity Elsticity responsiveness mesures how strongly people respond to chnges in prices nd chnges in income. Exmples of questions tht elsticity helps nswer Wht hppens to ttendnce t your museum

More information

Curve Sketching. 96 Chapter 5 Curve Sketching

Curve Sketching. 96 Chapter 5 Curve Sketching 96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

Tests for One Poisson Mean

Tests for One Poisson Mean Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution

More information

Week 7 - Perfect Competition and Monopoly

Week 7 - Perfect Competition and Monopoly Week 7 - Perfect Competition nd Monopoly Our im here is to compre the industry-wide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between

More information

Labor Productivity and Comparative Advantage: The Ricardian Model of International Trade

Labor Productivity and Comparative Advantage: The Ricardian Model of International Trade Lbor Productivity nd omrtive Advntge: The Ricrdin Model of Interntionl Trde Model of trde with simle (unrelistic) ssumtions. Among them: erfect cometition; one reresenttive consumer; no trnsction costs,

More information

Solutions to Section 1

Solutions to Section 1 Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this

More information

Answer, Key Homework 8 David McIntyre 1

Answer, Key Homework 8 David McIntyre 1 Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your

More information

Perfect competition model (PCM)

Perfect competition model (PCM) 18/9/21 Consumers: Benefits, WT, nd Demnd roducers: Costs nd Supply Aggregting individul curves erfect competition model (CM) Key ehviourl ssumption Economic gents, whether they e consumers or producers,

More information

Lecture 1: Introduction to Economics

Lecture 1: Introduction to Economics E111 Introduction to Economics ontct detils: E111 Introduction to Economics Ginluigi Vernsc Room: 5B.217 Office hours: Wednesdy 2pm to 4pm Emil: gvern@essex.c.uk Lecture 1: Introduction to Economics I

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Quadratic Equations. Math 99 N1 Chapter 8

Quadratic Equations. Math 99 N1 Chapter 8 Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree

More information

The Quadratic Formula and the Discriminant

The Quadratic Formula and the Discriminant 9-9 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt

More information

1 Numerical Solution to Quadratic Equations

1 Numerical Solution to Quadratic Equations cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

More information

AP QUIZ #2 GRAPHING MOTION 1) POSITION TIME GRAPHS DISPLACEMENT Each graph below shows the position of an object as a function of time.

AP QUIZ #2 GRAPHING MOTION 1) POSITION TIME GRAPHS DISPLACEMENT Each graph below shows the position of an object as a function of time. AP QUIZ # GRAPHING MOTION ) POSITION TIME GRAPHS DISPLAEMENT Ech grph below shows the position of n object s function of time. A, B,, D, Rnk these grphs on the gretest mgnitude displcement during the time

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

1. 1 m/s m/s m/s. 5. None of these m/s m/s m/s m/s correct m/s

1. 1 m/s m/s m/s. 5. None of these m/s m/s m/s m/s correct m/s Crete ssignment, 99552, Homework 5, Sep 15 t 10:11 m 1 This print-out should he 30 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. The due time

More information

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function. Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while

More information

120/30=4 (c) What is the opportunity cost of 1 bookcase in terms of guitars?

120/30=4 (c) What is the opportunity cost of 1 bookcase in terms of guitars? Exercise 1 uose you cn roduce one guitr ith 120 hours of lbor or bookcse ith 30 hours of lbor. You re on your F ith stock of 10 guitrs nd 30 bookcses. () Ho mny units of lbor do you hve? From F, e cn get

More information

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl

More information

Net Change and Displacement

Net Change and Displacement mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions

not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the x-coordintes of the points where the grph of y = p(x) intersects the x-xis.

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Version 001 Summer Review #03 tubman (IBII20142015) 1

Version 001 Summer Review #03 tubman (IBII20142015) 1 Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Double Integrals over General Regions

Double Integrals over General Regions Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

Lecture 5. Inner Product

Lecture 5. Inner Product Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Mechanics Cycle 1 Chapter 5. Chapter 5

Mechanics Cycle 1 Chapter 5. Chapter 5 Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies

More information

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Chapter 9: Quadratic Equations

Chapter 9: Quadratic Equations Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

More information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values) www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

More information

Chapter 6 Solving equations

Chapter 6 Solving equations Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of

More information

r 2 F ds W = r 1 qe ds = q

r 2 F ds W = r 1 qe ds = q Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study

More information

Plotting and Graphing

Plotting and Graphing Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured

More information

4.0 5-Minute Review: Rational Functions

4.0 5-Minute Review: Rational Functions mth 130 dy 4: working with limits 1 40 5-Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

More information

Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and

Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

Theory of Forces. Forces and Motion

Theory of Forces. Forces and Motion his eek extbook -- Red Chpter 4, 5 Competent roblem Solver - Chpter 4 re-lb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics - Everything

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

Uniform convergence and its consequences

Uniform convergence and its consequences Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,

More information

3 The Utility Maximization Problem

3 The Utility Maximization Problem 3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best

More information

Rotating DC Motors Part II

Rotating DC Motors Part II Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE

UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10-438 UNVERSTY OF NOTTNGHAM Discussion

More information

Introduction to Mathematical Reasoning, Saylor 111

Introduction to Mathematical Reasoning, Saylor 111 Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

1. 0 m/s m/s m/s m/s

1. 0 m/s m/s m/s m/s Version PREVIEW Kine Grphs PRACTICE burke (1111) 1 This print-out should he 30 questions. Multiple-choice questions m continue on the next column or pge find ll choices before nswering. Distnce Time Grph

More information

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π. . Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for

More information

Let us recall some facts you have learnt in previous grades under the topic Area.

Let us recall some facts you have learnt in previous grades under the topic Area. 6 Are By studying this lesson you will be ble to find the res of sectors of circles, solve problems relted to the res of compound plne figures contining sectors of circles. Ares of plne figures Let us

More information

Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers

Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers Redistributing the Gins from Trde through Non-liner Lump-sum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lump-sum trnsfer rules to redistribute the gins from

More information

AAPT UNITED STATES PHYSICS TEAM AIP 2010

AAPT UNITED STATES PHYSICS TEAM AIP 2010 2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS - 75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD

More information

14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes

14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes he Men Vlue nd the Root-Men-Squre Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men

More information

Sirindhorn International Institute of Technology Thammasat University at Rangsit

Sirindhorn International Institute of Technology Thammasat University at Rangsit Sirindhorn Interntionl Institute of Technology Thmmst University t Rngsit School of Informtion, Computer nd Communiction Technology COURSE : ECS 204 Bsic Electricl Engineering L INSTRUCTOR : Asst. Prof.

More information

Small Businesses Decisions to Offer Health Insurance to Employees

Small Businesses Decisions to Offer Health Insurance to Employees Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employer-sponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults

More information

Week 11 - Inductance

Week 11 - Inductance Week - Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n

More information

Chapter G - Problems

Chapter G - Problems Chpter G - Problems Blinn College - Physics 2426 - Terry Honn Problem G.1 A plne flies horizonlly t speed of 280 mês in position where the erth's mgnetic field hs mgnitude 6.0µ10-5 T nd is directed t n

More information

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS 4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem

More information

Lecture 3 Basic Probability and Statistics

Lecture 3 Basic Probability and Statistics Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The

More information

The Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx

The Chain Rule. rf dx. t t lim  (x) dt  (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the single-vrible chin rule extends to n inner

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

Volumes of solids of revolution

Volumes of solids of revolution Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the x-xis. There is strightforwrd technique which enbles this to be done, using

More information

Version 001 CIRCUITS holland (1290) 1

Version 001 CIRCUITS holland (1290) 1 Version CRCUTS hollnd (9) This print-out should hve questions Multiple-choice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Small Business Networking

Small Business Networking Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer

More information

Unit 29: Inference for Two-Way Tables

Unit 29: Inference for Two-Way Tables Unit 29: Inference for Two-Wy Tbles Prerequisites Unit 13, Two-Wy Tbles is prerequisite for this unit. In ddition, students need some bckground in significnce tests, which ws introduced in Unit 25. Additionl

More information

An Off-Center Coaxial Cable

An Off-Center Coaxial Cable 1 Problem An Off-Center Coxil Cble Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Nov. 21, 1999 A coxil trnsmission line hs inner conductor of rdius nd outer conductor

More information

Arc Length. P i 1 P i (1) L = lim. i=1

Arc Length. P i 1 P i (1) L = lim. i=1 Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x

More information