LK, represents the total amount of labor and capital available in the economy, P, P denote the prices

Size: px
Start display at page:

Download "LK, represents the total amount of labor and capital available in the economy, P, P denote the prices"

Transcription

1 Fall 1 Econ 455 Answers - Proble Set 3 Due Noveber 8, 1 Harvey Lapan 1. Consder a spled verson o the Heckscher-Ohln odel wth the ollowng technology: To produce ood: 1 unt o labor and 3 unts o captal are requred or each unt o F. To produce anuactures (M): 3 unts o labor and 1 unt o captal are requred or each unt o M. LK, represents the total aount o labor and captal avalable n the econoy, P, P denote the prces o output and W, R denote the prces o labor and captal respectvely. a) Fnd producton costs and output prce n ters o actor prces. The hnt gves the answer or anuactures; jt apply the sae reasonng to ood. 3 3 ; Pr ( ) 3 TC M Q W R MC M W R ce M MC P W R For ood, each unt o output requres one worker and 3 unts o captal; hence: 3 3 ; Pr ( ) 3 TC ood Q W R MC ood W R ce F MC P W R. Fnd actor prces n ters o output prce. Show how an ncrease n P aects W,R. Fro above: P 3 W R; P W 3R. Ths s lke two lnear equatons n two unknowns; one can nvert ths relatonshp and solve or R: P 3W R R P 3W ; substtute ths nto the relatonshp or ood: 3P P P W 3R W 3P 3W 3P 8W or W 8 3P P 3P P Use the soluton or W to solve or R: R P 3W P P P 3P P W ; R 8 8 How does an ncrease n P aect actor prces?. Suarzng: W 1 R P 3P ; 1 ; P 3 P R 3P P R P c 1 6 Th an ncrease n the prce o ood lowers the return to labor and rases the real return to captal (snce ood s captal-ntensve and anuactures are labor-ntensve). Moreover, the return on captal ncreases by ore (n % ters) than does the prce o ood so the real return on captal ncreases n ters o ether ood or anuactures. 1

2 b) Fnd the producton possblty ronter (pp). Let Q, Q denote the outputs o good M and F, respectvely. The resource constrants are: Labor: (1) 3Q Q L snce the technology ples: L 3Q and L Q Captal: () Q3Q K snce the technology ples: K Q and K 3Q The ollowng gure shows the producton possblty ronter or ths econoy; the ponts on, or below, the lne labeled labor constrant nsure that labor eployed s no larger than avalable labor (wth ull eployent on that lne), whle the lne labeled captal constrant has the sae nterpretaton. For ths spled econoy, the only output level where both nputs are ully eployed s where the two lnes ntersect, at pont V, where output s: 3L K 3K L Q ; Q 8 8 Food Output L Labor constrant Z K/3 New captal constrant V Captal constrant L/3 K Manuactures The easble producton set s the regon bounded by: {,(K/3),V,(L/3)}, and the producton possblty ronter s the lne segents descrbed by: {(K/3),V,(L/3)}. ()Show how an ncrease n the supply o captal shts the pp. An ncrease n captal shts the captal constrant outward, as shown by the dotted lne n the gure. The pont Z represents the new output level where both actors are ully eployed (n ths sple verson, there s a unque producton pont that represents ull eployent o both nputs).

3 Note that an ncrease n K leads to an ncrease n output o the captal ntensve good (F) and a decrease n output o the labor ntensve good as descrbed n class and n the text. c) Assung the Japan s captal abundant and the US s labor abundant (but they have dentcal tastes and technology), copare autarky prces, then nd the pattern o trade and dscs ts consequences. As shown above, gven prces, an ncrease n the supply o captal (n Japan) ncreases output o the captal ntensve good (F) and decreases output o the labor ntensve good (M). At gven prces, ths wll create an excess supply o good F and an excess deand or good M. Hence, as the supply o captal ncreases wthn an econoy, the equlbru prce o the captal-ntensve good (ood) decreases. Slarly, an ncrease n the supply o labor (US) caes the equlbru autarky relatve prce o the labor ntensve good (anuactures) to all. Hence, the autarky relatve prce o good M wll be hgher n Japan than n the US. Ths, ro part (a) ples that the wage rate wll be hgher n Japan and the return on captal wll be hgher n the US (.e., Japan US P P n autarky W W and R R P P Japan US Japan US Th, wth trade, the US wll export M (the labor-ntensve good) and port F (the captal-ntensve good). As a result o trade, P P rses n the US and alls n Japan. But, ro (a), ths ples that the wage rate rses n the US and alls n Japan, whle the return on captal (R) alls n the US and rses n Japan. Fnally, ree trade equalzes coodty prces and both goods are produced n both countres, t t equalze actor prces (see equatons deternng actor prces n (a)), provded technology s the sae n the two countres. Ths s the actor prce equalzaton theore. d) Mody the above odel by assung Japan s productvty n both sectors double, whle US technology reans unchanged. In Japan: Food requres: 1/ unt o labor and 3/ unts o captal or each unt o ood produced. Manuactures requres: 3/ unts o labor and 1/ unt o captal or each unt o M produced. ) Show how doublng o productvty n Japan aects autarky output prces and actor prces. In Rcardan ters, whle Japan has an absolute advantage (technologcally) n both goods, there s no coparatve advantage due to technology. To see ths speccally, we can re-derve the producton possblty ronter or Japan, wth the new technology: Labor: (1a)3 Q1 Q L as the technology ples: L 3Q and L 1Q Captal: (a)1 Q 3 Q K as the technology ples: K 1Q and K 3 3L K 3K L Ths yelds the ull eployent pont o: Q ; Q 4 4 Th, at ull eployent, output o both goods double and th the relatve supply s unchanged. Hence, deand or both goods also doubles (becae ncoe doubles) so that relatve deand s Q 3

4 unchanged, the doublng o productvty n both sectors wll not aect autarky relatve goods prces and hence wll not aect the pattern o trade between the US and Japan. Turnng to nput prces, ng the logc o part (a) o the answer, or Japan we have: 3 1 ; 1 3 P MC W R P MC W R ; solvng or nput prces (n Japan) n ters o output prces we have: 3P P 3P Japan Japan ; P W R, whereas or the US (ro part a): 4 4 3P P W ; 8 3P P R 8 Th, we see that gven output prces the doublng o productvty n both sectors n Japan leads to a doublng o the real return to both actors. ) Wll ree trade equalze actors prces and reove the pressure or actor graton? Snce the doublng o productvty n both sectors n Japan leaves relatve autarky prces unchanged, t ollows that the pattern o trade s stll deterned by actor endowents so Japan wll export ood and the US anuactures. Trade stll lowers the real return to labor (rases the real return to captal) n Japan snce t lowers the prce o the labor-ntensve good, whereas the opposte happens n the US. However, trade wll not lead to actor prce equalzaton becae technologes are derent. As the above exaple shows, trade equalzes goods prces, then the real return to both actors wll be twce as hgh n Japan as n the US. Th, there wll stll be pressure or actor oveents.. (Factor oveents) There s a sngle good (e.g., ood), produced ng land and labor. The aount o land n a country s xed; labor ay be oble across countres. US and Mexcan technology and resources are: US: Mexco: Q 64 T L ; T 15; L Q 5 T L ; T 64; L 64 a) For each country, nd and sketch the labor deand curve. Also, calculate the equlbru wage, return on land and per capta ncoe n each country (all easured n ters o output) Q 64 T 64 T deand 64P US : MPLabor P W L T L 3 L 3 L 3W PQ WL 18 L Q US land rent: R P P T 3 T T 4

5 Snce there s only one good, you can set P 1, snce GNP, wages and rental rates are all easured n ters o ths good. Evaluatng at T 15, L 64 : W ; 3 Slarly or Mexco: R Q 5 T 5 T deand 5 Mexco : W L T L 3 L 3 L 3W Mexcan land rent: Q 5 L R T 3 T Evaluatng at T 64, L 64 : W ; R Below you see the labor deand curve or the US, and how graton shts the doestc labor supply. The labor deand curve or Mexco looks slar; though egraton ro Mexco (to the US) would reduce the labor supply n Mexco. (w/p) A L s L d L s +I (w/p) e (w/p) E G L Wage Deternaton The returns to land can be easured, n ths dagra, as the consuer surpl snce we can thnk o landowners as rentng labor, and hence a lower labor prce benets landowners. In the dagra land 5

6 rents, at the ntal wage, would be gven by {A,E,(w/P) e }; wages all, as shown, land rents ncrease by the area {(w/p) e,e,g,(w/p) } b) Assue the US allows soe Mexcan workers to enter. Let V stand or the axu nuber o guest workers the US allows (the nuber o vsas), and let I represent the actual nuber who choose to coe, where I V. The US labor orce becoes L I and the Mexcan labor orce L I.. Show how ()US wages, ()the return on US land, ()Mexcan wages, (v)the return on Mexcan land; and (v)world output are deterned as a uncton o the nuber o Vsas (V). Assung workers wll want to coe to the US (and only ) US wages are hgher than Mexcan wages, then the nuber who would coe there were no restrctons s deterned by choosng the nuber o grants (I) to equalze wages T 5 T Fro earler, wages are deterned by: W ; W 3 L I 3 L I Where, L I, L I are the nuber o workers. Settng wages n L L s the populaton, and the two countres equal to each other, and ng the values or land, we have T 5 T 64 T T W W 3 L I 3 L I 5 L I L I L I L I 9I 8L L 448; 15 L I L I * 448 I Th, or V, US wages wll be hgher and all the vsas wll be ed. Hence T T W W ; ; 53 3 L V V 9L V W T T W ; 53 3 L V V 9L V US wages all and Mexcan wages rse as the nuber o vsas ssued ncrease, untl wages are equalzed, beyond whch ssung ore vsas wll not atter. For the return on land, opposte results hold: 6

7 R L V L V R ; ; 3 T V T L V L V R R ; 3 T V 9 T Fnally, total output ncreases as long as US wages are hgher; th: L V L V L V L V T Q Q Q T L V T L V Q V T L V L V 8L V L V 9V 8L L Whch s the sae condton we saw above to nsure US wages are hgher than Mexcan wages... As above, the nuber o vsas s set very hgh, then not all wll be ed and wages wll be equalzed. Calculate how U.S. ncoe, Mexcan ncoe, and world output change as V ncreases. Fro the proble set: Y Q W V 64 T L V W V; T 15; L 64; I V Y Q W V 5 T L V W V; T 64; L W V V Y T W W V 3 L V V V 3 5 W V W W V W W Y T W W V 3 L V V V W as V {o course, once wages are equalzed, ssung ore vsas wll have no eect} Th, snce the US pays the Mexcan workers ther argnal value product n the US (as the wage), havng ore workers coe n lowers the wage they receve and the US net ncoe rses as V ncreases. However, or Mexco, snce ther workers n the US receve the US wage, as ore Mexcans enter the US labor arket, that hurts Mexcan workers already there (reducng ther wage). Mexco s lke a onopoly seller o labor and would gan by placng ts own restrctons on egraton, provded 7

8 Y Mexcan workers n the US receve the US wage. (that s, or W W ). V We calculated n the prevo part that world output ncreases wth labor oveents as long as the argnal product o labor n the US s hgher than n Mexco. v. Suppose the U.S. governent auctons o the work vsas. P s the prce pad at aucton or each vsa, so Mexcans who work n the U.S. receve net ncoe o W P, whle Mexcans who stay n Mexco receve net ncoe o W. Show how the ollowng vary wth V: (1)P; ()U.S. net ncoe and (3)Mexcan net ncoe change as V ncreases. Ths s essentally the reverse o the prevo proble; Mexco gans by ncreasng V up to the pont where argnal products are equalzed, whle the US has an ncentve to restrct graton snce the prce o vsas s reduced as V ncreases ; Y Q W P V T L V W P V Y Q W P V 5 T L V W P V; Where P W W. Hence: Y 64 T L V W V; Y W W W W V W W V V V as Y Q W P V 5 T L V W V; Y V W W W V V Snce Mexcan workers n the US essentally receve the Mexcan wage, Mexco has no reason to lt out-graton but the US has an ncentve to lt graton. World output s axzed when there s ree labor oblty but that does not ean ndvdual countres don t have an econoc ncentve to restrct oblty. v. Does unrestrcted labor oblty axze US ncoe? Essentally dscsed above. Also, the grants becoe peranent resdents o the US so there are no wage payents that leave the country then graton rases US ncoe but wll lower per capta ncoe (paradoxcally, both orgnal resdents o the US and the grants can both be better o but stll per captal ncoe can all becae the enterng grants had a uch lower per capta ncoe n ther hoe country). 8

9 c) Suppose each worker n the U.S. receves or ree soe benets (e.g., edcal care or schoolng). Assue the guest worker pays no taxes. I a guest worker progra allows workers to reely choose where to work (and Mexco has no such benets or workers), wll ree worker oveent between the two countres axze total output o the two countres? I the Mexcan worker values the (edcal) benets at $B per year, then they wll copare: W B to W They wll grate to the US untl the total wage, ncludng benets, s equalzed across countres. Th, n equlbru they wll settle or a lower eployer wage they receve ree edcal benets ro the governent. Ths, n turn, eans n equlbru that the argnal product o labor wll be lower n the US, assung the wage pad by rs equals the (value o the) argnal product o labor. Hence, too uch labor oveent wll occur and world output s not axzed. 3. Consder a sall country (Ncaragua) wth the ollowng deand and supply curves or sugar: Supply = 6P s ; Deand = 4P s Ncaragua can export sugar at a gven world prce o: Ps 4. a) Show how: doestc prce, consupton and producton change as the export tar t ncreases. Calculate how consuer surpl, producer surpl, and governent tar revenue, and overall welare, change as t ncreases. Gven the world prce, the net revenue ro exportng or a r s 4 t ; ths wll be the doestc prce trade occurs. Snce the autarky prce s, or t the tar s prohbtve: there are no exports, and the doestc prce equals the autarky prce o. For t, we have: d d d P 4 t ; D 4P 4 4 t; S 6P 4 6 t; X S D 1t s s s Consupton ncreases, producton decreases and exports all as t ncreases. Fro the gure below, one can see that, due to the tar, producer surpl decreases by area {4,B,B,(4-t)}, consuer surpl ncreases by area: {4,A,A,(4-t)}. Tar revenue s area {A,K,J,B }. Th: PS 1 t 4 4 6t 4t 3 t ; CS 1 t 4 4 4t 4t t 1 1 TR tx t t t t s tar revenue. Producer surpl decreases wth the tar, consuer surpl ncreases (only t s relevant), whereas tar revenue ncreases wth the tar or t 1, and then decreases thereater. Overall: Welare TR PS CS 5t so that the tar lowers overall welare. Ths loss s the su o the areas o the two deadweght loss trangles {A,A,K} and {B,B,J} 9

10 P 4 A K J B 4-t A B S A E 4 4+4t 1 4-6t Fgure 1 4 D Q ) Even though the tar hurts Ncaragua copared to ree trade, t s stll better o wth the export tar then under autarky. The gan, copared to autarky, s area {A,B,E} pl the tar revenue {A,K,J,B }. Soe trade s better than no trade. ) I t, the tar s prohbtve and no trade occurs. b) Copare the doestc equlbru when t=1 to the case where there s no tar, but there s an export quota o 1 unts. Fro part (a), wth t=1,exports X 1t 1. Th, a quota o 1 and a tar o 1 have dentcal eects on doestc prce, consupton, producton and exports; and hence they have dentcal eects on consuer and producer surpl. The only possble derence s the tar revenue (whch s 1, under the tar); under the quota, exporters ake 1 on each unt exported (snce they can buy at the doestc prce, 3, and resell on the world arket at a prce o 4), and hence they wll earn excess prots o 1, unless the quota lcenses are auctoned o, n whch case the two polces are dentcal. I, or soe reason, the quotas were gven to oregn porters, then the revenue ro the tar s lost to the country, and so the quota, n that case, would be neror to the tar or the exportng country. c) Suppose the governent subsdzes exports at a rate o s per unt o port. Show how ths export subsdy aects the ollowng (see Fgure below): ()doestc prce doestc prce or both consuers and producers ncreases to {4+s} snce an exporter receves 4 ro the oregn buyer and s ro the governent. Hence, the doestc prce s less than {4+s} exporters wll try to export all doestc output, drvng up prce to {4+s}. ()Consuer surpl decreases - due to a hgher prce - by area next to deand curve between two s prces area {4,A,A,(4+s)} n the gure; hence: CS 4 4 4s 4s s 1

11 ()Producer surpl ncreases due to the hgher prce, by the area next to supply between two prces s area {4,B,B,(4+s)}. Hence: PS 4 4 6s 4s 3s (v)governent expendtures: Cost o subsdy s s*exports = area {A B ML}; exports are S D 1s,hence: Cost to governent = s 1s s 1s (v)ipact overall welare: CS PS Governent Expendtures = 5s, s Ths s area o two deadweght loss trangles, {A,A,L} and {B,M,B}. P 4+s A B S 4 L A B M E 4-4s s D Q. No export quota can orce exports to ncrease over the ree trade level so there s no quanttatve polcy that has the sae pact as an export subsdy.. Unlke the case o a tar, the export subsdy by encouragng too uch trade can ake Ncaragua worse o than under autarky the subsdy s hgh enough. d) I the goal s to ncrease doestc sugar producton, an export subsdy accoplshes ths by rasng doestc prce; but the export subsdy also caes doestc consupton to decrease. Wth a producton subsdy o 1, there s no change n consuer surpl copared to ree trade, whereas the change n producer surpl s the sae as wth a export subsdy o 1. Th, the welare consequences o a producton subsdy o 1: CS ; PS s 1 4s 3s 7 Subsdy Cost = 3 where output s 3 when subsdy = 1 11

12 Welare Loss = -3. For the export subsdy, wth s=1, the welare loss s 5. Th, the producton subsdy whch results n the sae level o doestc producton as the export subsdy has a lower welare cost becae we do not have the loss assocated wth the reducton n consupton. 4. (15 ponts extra credt) (More sophstcated verson o H-O odel). There are two goods (M and F) and two nputs (K and L). The producton unctons are: where Q K L ; Q K L K,L are the nputs (captal, labor) ed n sector M and K,L are the nputs ed n sector F. Let W denote the wage rate (prce o L) and R the rental rate (cost o ng K, captal). Fnally, let P, P denote the output prces o goods M and F, respectvely. (a) There are two ways to derve the cost uncton one by substtuton, and one by ng non-lnear prograng (whch nvolves ng the Lagrangan uncton). By substtuton: 1 1 Q K L. Solvng or labor yelds: L Q K Q K. Let Total costs are: 1 TC WL RK RK W Q K (1) Equaton (1) expresses total costs as a uncton o actor prces, output and captal nputs. Cost nzaton eans choose the captal nput that nzes ths expresson. Th: d TC dk R W Q K 1 1 () snce: d K dk K 1 A sucent condton or an nteror nu s that the rst dervatve o the uncton be zero and that the second dervatve be postve; t s readly seen that the second dervatve s postve. Solvng () or K yelds: 1 W (3) * W 1 K Q Q R R where * K denotes the soluton. Substtutng back or L yelds: * R L Q W 1 (4) Note that the choce o nputs depends on relatve actor prces, not absolute actor prces. Also, note that when 1, the nput e s proportonal to output. Fnally, substtutng back nto the cost 1

13 uncton (1) yelds nu costs: 1 * * * C Q W R WL RK W R Q,, where: 1 (you need to substtute and then sply the expresson; t s a bt tedo, but you should get the result above): Ths result can be checked becae the dervatve o the cost uncton wth respect to nput prce should gve you back the optal nput e. Hence: 1 * dc 1 W R W Q 1 Q dr R whch s * K. Slarly, derentatng wth respect to W gves you * L : 1 * dc 1 R 1 R W Q Q dw W Use o the Lagrangean uncton gves the sae results, o course. Brely, the Lagrangean s: WL RK Q L K where s the Lagrangean ultpler. Partally derentatng yelds, or an nteror soluton: 1 R K L K 1 W K L L Q L K (1a) (a) (3a) Takng the rato o (1a) to (a) yelds: L R L R K W K W (4a) Hence, the labor ntensty depends on actor prces and n ters o the orgnal producton uncton s ncreasng n the paraeter on L and decreasng n the paraeter on K. Usng (4a) to solve or L n ters o K, and then substtutng ths nto (3a) yelds the optu captal nput, whch wll be the sae as above. Then, ng ths soluton or captal, the soluton or labor s ound ro (4a), and the cost curve by pluggng back nto the objectve uncton. You do not need to solve or, but you do you get the ollowng ro (1a)

14 1 * R K R W L 1 Q 1 1 (5a) Lookng back at the cost uncton derved above and coparng to (5a), we see that (5a) represents the argnal cost uncton. Ths s no concdence; the Lagrangean ultpler n ths proble wll always yeld the argnal cost uncton. Fnally, or the specc unctons gven: 1 * * * C Q W R WL RK W R Q,, Q K L,, 1 so: TC Q, W, R W R Q * Q K T,, 1 so: TC Q, W, R W R Q 4 4 * where: Clearly, good M s labor ntensve and F s captal ntensve as we have ro (4a) above: L R K K ; 3 K W L 3 L where: W R (b) Gven output prces, show how an ncrease n the avalable supply o labor changes output. Fro the cost curves above we have: MC R W P and MC W R P (1b) We can e these two equatons to solve or actor prces n ters o output prce. Takng the rato o argnal costs and settng ths equal to the prce rato (relatve prces) yelds: MC W R W P MC W R R P where: ; P W (b) P R Pluggng ths back nto (1b) and solvng gves the level o actor prces: 14

15 P R P 3 1 W P (3b) P 3 1 Fro (3) and (4) n part (a) you have the optal aount o nputs n each sector: 1 34 * W * * 14 ; 3 K Q K Q K Q (4b) R * R * 3 * 1 L Q L Q, L Q (5b) W 3 P You can express these nput deands n ters o output prce by substtutng or n ters o. P Dong so and wrtng the resource constrants yelds: L L L 3 Q 3 Q L (6b) K K K 3 Q 3 Q K (7b) Gven prces, equatons (6b) and (7b) are jt lke proble #1 (.e., the labor and captal e per unt output are xed) and can be solved or output levels. Dong so yelds: ; 3 83 Q L K Q K L Th, gven prces, an ncrease n L wll ncrease the output o good M, the labor ntensve good, and decrease the output o good F. Also, note that output prces do not change, nput prces do not change snce, ro equaton (3b) above, actor prces can be deterned n ters o only output prces. Th, gven prces, an ncrease n L caes the supply o good M to ncrease and that o good F to decrease. But total ncoe ncreases as L ncreases, so deand or both goods ncreases. Hence, to restore equlbru, the prce o good F t ncrease (relatve prce o M decreases). Th, the autarky (relatve) prce o good F (the captal ntensve good) s an ncreasng uncton o the country s relatve labor abundance. Consequently, gven the sae deands and technology, the labor abundant country wll export the labor ntensve good (M) and port the captal ntensve good (F). Further, snce the wage rate decreases, and the rental rate ncreases, as the prce o good F ncreases (.e., the relatve prce o good M decreases), ths eans that the autarky wage rate wll be lower, and the autarky rental rate on captal hgher, n the labor abundant country. Hence, derences n actor supples lead to derences n autarky output prces, whch lead to the derences n nput prces one would expect.. As dscsed above, gven output prces, nput prces are deterned and hence changes actor supples wll not change actor prces. 15

16 (c) To show how actor prces change wth output prces, look back at equaton (3b). P R P 3 1 P W. Th P R 3 P R P R P 1 3 1; P P P ; 3 R P P P 3 W P W P W P W , 3 P P P P W P P These results ply that an ncrease n P, the captal ntensve good, lowers the real return to labor and rases the real return to captal n ters o ether good (a 1% ncrease n P, ncreases the rental rate R by 1.5%, and hence RP ncreases as P ncreases). Slarly, an ncrease n the prce o the labor ntensve good (M) lowers the real return to captal and rases the real return to labor n ters o ether good. The results show that, as P ncreases, W ncreases proportonately ore than P (.e., ncreases wth P ). Ths s the Stolper-Sauelson result. W P (d) Use your answer to parts (b) and (c) to predct the pattern o trade between a labor-abundant country (lke Chna) and a captal-abundant country (lke the U.S.). How wll trade aect the dstrbuton o ncoe n each country? Wll everybody gan ro trade? Explan. Snce Chna s labor-abundant, the autarky relatve prce o M wll be lower n Chna than n the U.S., the real wage wll be lower n Chna and the real return on captal wll be hgher. I trade, s allowed, the US wll export F and Chna wll export M. The relatve prce o M rses n Chna (alls n the US), so the real wage rses n Chna and the return on captal alls. O course, n the US the real wage alls and the real return on captal rses. Not everybody gans ro trade, though the ganers could copensate the losers. (e) Assue the U.S. s captal-abundant. Whch group n the U.S. s lkely to avor port tars and whch group s lkely to oppose trade restrctons? Explan. I the H-O odel correctly explans trade, the workers n the U.S. wll avor port tars and captalsts wll oppose the. I ports are labor-ntensve, ports rase the doestc prce o the labor ntensve good, ncreasng the real wage and reducng the real return on captal. 16

= 2W + 2R R = ; substitute this in the relationship for manufactures:

= 2W + 2R R = ; substitute this in the relationship for manufactures: Fall 0 Eon 455 Answers - Proble Set 3 Due Noveber 4, 0 Harvey Lapan. Consder a splfed verson of the Heksher-Ohln odel wth the followng tehnology: To produe loth (C): unts of labor and unts of aptal are

More information

Preference and Demand Examples

Preference and Demand Examples Dvson of the Huantes and Socal Scences Preference and Deand Exaples KC Border October, 2002 Revsed Noveber 206 These notes show how to use the Lagrange Karush Kuhn Tucker ultpler theores to solve the proble

More information

Problem Set #2 Solutions

Problem Set #2 Solutions 4.0 Sprng 003 Page Proble Set # Solutons Proble : a) A onopolst solves the followng proble: ( Q ) Q C ( )= 00Q Q 0Q ax P Q wth frst-order condton (FOC) b) Gven the results fro part a, Q 90 Q = 0 Q P =

More information

A FURTHER GENERALIZATION OF THE SOLOW GROWTH MODEL: THE ROLE OF THE PUBLIC SECTOR

A FURTHER GENERALIZATION OF THE SOLOW GROWTH MODEL: THE ROLE OF THE PUBLIC SECTOR FURTHER GENERLIZTION OF THE SOLOW GROWTH MODEL: THE ROLE OF THE PUBLIC SECTOR Oscar Bajo-Rubo (Unversdad Públca de Navarra) bstract We develop n ths paper an augented verson of the Solow (956) growth odel,

More information

COMP th April, 2007 Clement Pang

COMP th April, 2007 Clement Pang COMP 540 12 th Aprl, 2007 Cleent Pang Boostng Cobnng weak classers Fts an Addtve Model Is essentally Forward Stagewse Addtve Modelng wth Exponental Loss Loss Functons Classcaton: Msclasscaton, Exponental,

More information

Endogenous timing in a mixed oligopoly consisting of a single public firm and foreign competitors. Abstract

Endogenous timing in a mixed oligopoly consisting of a single public firm and foreign competitors. Abstract Endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors Yuanzhu Lu Chna Economcs and Management Academy, Central Unversty o Fnance and Economcs Abstract We nvestgate endogenous

More information

What is LP? LP is an optimization technique that allocates limited resources among competing activities in the best possible manner.

What is LP? LP is an optimization technique that allocates limited resources among competing activities in the best possible manner. (C) 998 Gerald B Sheblé, all rghts reserved Lnear Prograng Introducton Contents I. What s LP? II. LP Theor III. The Splex Method IV. Refneents to the Splex Method What s LP? LP s an optzaton technque that

More information

Fermi-Dirac statistics

Fermi-Dirac statistics UCC/Physcs/MK/EM/October 8, 205 Fer-Drac statstcs Fer-Drac dstrbuton Matter partcles that are eleentary ostly have a type of angular oentu called spn. hese partcles are known to have a agnetc oent whch

More information

1 Review From Last Time

1 Review From Last Time COS 5: Foundatons of Machne Learnng Rob Schapre Lecture #8 Scrbe: Monrul I Sharf Aprl 0, 2003 Revew Fro Last Te Last te, we were talkng about how to odel dstrbutons, and we had ths setup: Gven - exaples

More information

EconS 501 Micro Theory I 1 Recitation #12 Imperfect Competition

EconS 501 Micro Theory I 1 Recitation #12 Imperfect Competition EconS 50 Mcro Theory I Rectaton # Iperfect Copetton Exercse 5.3 (NS). [ON YOUR OWN] Ths exercse analyzes Cournot copetton when frs have dfferent argnal costs. Ths departure fro dentcal frs allows the student

More information

System in Weibull Distribution

System in Weibull Distribution Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co

More information

AN ANALYSIS OF A FRACTAL KINETICS CURVE OF SAVAGEAU

AN ANALYSIS OF A FRACTAL KINETICS CURVE OF SAVAGEAU AN ANALYI OF A FRACTAL KINETIC CURE OF AAGEAU by John Maloney and Jack Hedel Departent of Matheatcs Unversty of Nebraska at Oaha Oaha, Nebraska 688 Eal addresses: aloney@unoaha.edu, jhedel@unoaha.edu Runnng

More information

PHYS 1443 Section 002 Lecture #20

PHYS 1443 Section 002 Lecture #20 PHYS 1443 Secton 002 Lecture #20 Dr. Jae Condtons for Equlbru & Mechancal Equlbru How to Solve Equlbru Probles? A ew Exaples of Mechancal Equlbru Elastc Propertes of Solds Densty and Specfc Gravty lud

More information

On Pfaff s solution of the Pfaff problem

On Pfaff s solution of the Pfaff problem Zur Pfaff scen Lösung des Pfaff scen Probles Mat. Ann. 7 (880) 53-530. On Pfaff s soluton of te Pfaff proble By A. MAYER n Lepzg Translated by D. H. Delpenc Te way tat Pfaff adopted for te ntegraton of

More information

CHAPTER 6 CONSTRAINED OPTIMIZATION 1: K-T CONDITIONS

CHAPTER 6 CONSTRAINED OPTIMIZATION 1: K-T CONDITIONS Chapter 6: Constraned Optzaton CHAPER 6 CONSRAINED OPIMIZAION : K- CONDIIONS Introducton We now begn our dscusson of gradent-based constraned optzaton. Recall that n Chapter 3 we looked at gradent-based

More information

Chapter 13. Gas Mixtures. Study Guide in PowerPoint. Thermodynamics: An Engineering Approach, 5th edition by Yunus A. Çengel and Michael A.

Chapter 13. Gas Mixtures. Study Guide in PowerPoint. Thermodynamics: An Engineering Approach, 5th edition by Yunus A. Çengel and Michael A. Chapter 3 Gas Mxtures Study Gude n PowerPont to accopany Therodynacs: An Engneerng Approach, 5th edton by Yunus A. Çengel and Mchael A. Boles The dscussons n ths chapter are restrcted to nonreactve deal-gas

More information

Page 1. SPH4U: Lecture 7. New Topic: Friction. Today s Agenda. Surface Friction... Surface Friction...

Page 1. SPH4U: Lecture 7. New Topic: Friction. Today s Agenda. Surface Friction... Surface Friction... SPH4U: Lecture 7 Today s Agenda rcton What s t? Systeatc catagores of forces How do we characterze t? Model of frcton Statc & Knetc frcton (knetc = dynac n soe languages) Soe probles nvolvng frcton ew

More information

Xiangwen Li. March 8th and March 13th, 2001

Xiangwen Li. March 8th and March 13th, 2001 CS49I Approxaton Algorths The Vertex-Cover Proble Lecture Notes Xangwen L March 8th and March 3th, 00 Absolute Approxaton Gven an optzaton proble P, an algorth A s an approxaton algorth for P f, for an

More information

,..., k N. , k 2. ,..., k i. The derivative with respect to temperature T is calculated by using the chain rule: & ( (5) dj j dt = "J j. k i.

,..., k N. , k 2. ,..., k i. The derivative with respect to temperature T is calculated by using the chain rule: & ( (5) dj j dt = J j. k i. Suppleentary Materal Dervaton of Eq. 1a. Assue j s a functon of the rate constants for the N coponent reactons: j j (k 1,,..., k,..., k N ( The dervatve wth respect to teperature T s calculated by usng

More information

Physics 123. Exam #1. October 11, 2006

Physics 123. Exam #1. October 11, 2006 hyscs Exa # October, 006 roble /0 roble /0 roble /0 roble 4 /0 roble 5 /0 roble 6 /0 roble 7 /0 roble 8 /0 roble 9 /0 roble 0 /0 Total /00 Free-Response robles: lease show all work n order to receve partal

More information

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. ) with a symmetric Pcovariance matrix of the y( x ) measurements V

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. ) with a symmetric Pcovariance matrix of the y( x ) measurements V Fall Analyss o Experental Measureents B Esensten/rev S Errede General Least Squares wth General Constrants: Suppose we have easureents y( x ( y( x, y( x,, y( x wth a syetrc covarance atrx o the y( x easureents

More information

Supporting Information for: Two Monetary Models with Alternating Markets

Supporting Information for: Two Monetary Models with Alternating Markets Supportng Informaton for: Two Monetary Models wth Alternatng Markets Gabrele Camera Chapman Unversty & Unversty of Basel YL Chen St. Lous Fed November 2015 1 Optmal choces n the CIA model On date t, gven

More information

LECTURE :FACTOR ANALYSIS

LECTURE :FACTOR ANALYSIS LCUR :FACOR ANALYSIS Rta Osadchy Based on Lecture Notes by A. Ng Motvaton Dstrbuton coes fro MoG Have suffcent aount of data: >>n denson Use M to ft Mture of Gaussans nu. of tranng ponts If

More information

total If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions.

total If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions. Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Last te we used ewton s second law to deelop the pulse-oentu theore. In words, the theore states that the change n lnear oentu

More information

CHAPTER 7 CONSTRAINED OPTIMIZATION 1: THE KARUSH-KUHN-TUCKER CONDITIONS

CHAPTER 7 CONSTRAINED OPTIMIZATION 1: THE KARUSH-KUHN-TUCKER CONDITIONS CHAPER 7 CONSRAINED OPIMIZAION : HE KARUSH-KUHN-UCKER CONDIIONS 7. Introducton We now begn our dscusson of gradent-based constraned optzaton. Recall that n Chapter 3 we looked at gradent-based unconstraned

More information

University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2013

University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2013 Lecture 8/8/3 Unversty o Washngton Departent o Chestry Chestry 45/456 Suer Quarter 3 A. The Gbbs-Duhe Equaton Fro Lecture 7 and ro the dscusson n sectons A and B o ths lecture, t s clear that the actvty

More information

Elastic Collisions. Definition: two point masses on which no external forces act collide without losing any energy.

Elastic Collisions. Definition: two point masses on which no external forces act collide without losing any energy. Elastc Collsons Defnton: to pont asses on hch no external forces act collde thout losng any energy v Prerequstes: θ θ collsons n one denson conservaton of oentu and energy occurs frequently n everyday

More information

Computational and Statistical Learning theory Assignment 4

Computational and Statistical Learning theory Assignment 4 Coputatonal and Statstcal Learnng theory Assgnent 4 Due: March 2nd Eal solutons to : karthk at ttc dot edu Notatons/Defntons Recall the defnton of saple based Radeacher coplexty : [ ] R S F) := E ɛ {±}

More information

XII.3 The EM (Expectation-Maximization) Algorithm

XII.3 The EM (Expectation-Maximization) Algorithm XII.3 The EM (Expectaton-Maxzaton) Algorth Toshnor Munaata 3/7/06 The EM algorth s a technque to deal wth varous types of ncoplete data or hdden varables. It can be appled to a wde range of learnng probles

More information

Quantum Particle Motion in Physical Space

Quantum Particle Motion in Physical Space Adv. Studes Theor. Phys., Vol. 8, 014, no. 1, 7-34 HIKARI Ltd, www.-hkar.co http://dx.do.org/10.1988/astp.014.311136 Quantu Partcle Moton n Physcal Space A. Yu. Saarn Dept. of Physcs, Saara State Techncal

More information

A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegian Business School 2011

A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegian Business School 2011 A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegan Busness School 2011 Functons featurng constant elastcty of substtuton CES are wdely used n appled economcs and fnance. In ths note, I do two thngs. Frst,

More information

Supporting Materials for: Two Monetary Models with Alternating Markets

Supporting Materials for: Two Monetary Models with Alternating Markets Supportng Materals for: Two Monetary Models wth Alternatng Markets Gabrele Camera Chapman Unversty Unversty of Basel YL Chen Federal Reserve Bank of St. Lous 1 Optmal choces n the CIA model On date t,

More information

Solutions for Homework #9

Solutions for Homework #9 Solutons for Hoewor #9 PROBEM. (P. 3 on page 379 n the note) Consder a sprng ounted rgd bar of total ass and length, to whch an addtonal ass s luped at the rghtost end. he syste has no dapng. Fnd the natural

More information

1.3 Hence, calculate a formula for the force required to break the bond (i.e. the maximum value of F)

1.3 Hence, calculate a formula for the force required to break the bond (i.e. the maximum value of F) EN40: Dynacs and Vbratons Hoework 4: Work, Energy and Lnear Moentu Due Frday March 6 th School of Engneerng Brown Unversty 1. The Rydberg potental s a sple odel of atoc nteractons. It specfes the potental

More information

On the number of regions in an m-dimensional space cut by n hyperplanes

On the number of regions in an m-dimensional space cut by n hyperplanes 6 On the nuber of regons n an -densonal space cut by n hyperplanes Chungwu Ho and Seth Zeran Abstract In ths note we provde a unfor approach for the nuber of bounded regons cut by n hyperplanes n general

More information

Online Appendix: Reciprocity with Many Goods

Online Appendix: Reciprocity with Many Goods T D T A : O A Kyle Bagwell Stanford Unversty and NBER Robert W. Stager Dartmouth College and NBER March 2016 Abstract Ths onlne Appendx extends to a many-good settng the man features of recprocty emphaszed

More information

Temperature. Chapter Heat Engine

Temperature. Chapter Heat Engine Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the

More information

OPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming

OPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or

More information

COS 511: Theoretical Machine Learning

COS 511: Theoretical Machine Learning COS 5: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #0 Scrbe: José Sões Ferrera March 06, 203 In the last lecture the concept of Radeacher coplexty was ntroduced, wth the goal of showng that

More information

Chapter 12 Lyes KADEM [Thermodynamics II] 2007

Chapter 12 Lyes KADEM [Thermodynamics II] 2007 Chapter 2 Lyes KDEM [Therodynacs II] 2007 Gas Mxtures In ths chapter we wll develop ethods for deternng therodynac propertes of a xture n order to apply the frst law to systes nvolvng xtures. Ths wll be

More information

Welfare Properties of General Equilibrium. What can be said about optimality properties of resource allocation implied by general equilibrium?

Welfare Properties of General Equilibrium. What can be said about optimality properties of resource allocation implied by general equilibrium? APPLIED WELFARE ECONOMICS AND POLICY ANALYSIS Welfare Propertes of General Equlbrum What can be sad about optmalty propertes of resource allocaton mpled by general equlbrum? Any crteron used to compare

More information

Economics 2450A: Public Economics Section 10: Education Policies and Simpler Theory of Capital Taxation

Economics 2450A: Public Economics Section 10: Education Policies and Simpler Theory of Capital Taxation Economcs 2450A: Publc Economcs Secton 10: Educaton Polces and Smpler Theory of Captal Taxaton Matteo Parads November 14, 2016 In ths secton we study educaton polces n a smplfed verson of framework analyzed

More information

f(x,y) = (4(x 2 4)x,2y) = 0 H(x,y) =

f(x,y) = (4(x 2 4)x,2y) = 0 H(x,y) = Problem Set 3: Unconstraned mzaton n R N. () Fnd all crtcal ponts of f(x,y) (x 4) +y and show whch are ma and whch are mnma. () Fnd all crtcal ponts of f(x,y) (y x ) x and show whch are ma and whch are

More information

ASYMMETRIC TRAFFIC ASSIGNMENT WITH FLOW RESPONSIVE SIGNAL CONTROL IN AN URBAN NETWORK

ASYMMETRIC TRAFFIC ASSIGNMENT WITH FLOW RESPONSIVE SIGNAL CONTROL IN AN URBAN NETWORK AYMMETRIC TRAFFIC AIGNMENT WITH FLOW REPONIVE IGNAL CONTROL IN AN URBAN NETWORK Ken'etsu UCHIDA *, e'ch KAGAYA **, Tohru HAGIWARA *** Dept. of Engneerng - Hoado Unversty * E-al: uchda@eng.houda.ac.p **

More information

Limited Dependent Variables

Limited Dependent Variables Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages

More information

Let p z be the price of z and p 1 and p 2 be the prices of the goods making up y. In general there is no problem in grouping goods.

Let p z be the price of z and p 1 and p 2 be the prices of the goods making up y. In general there is no problem in grouping goods. Economcs 90 Prce Theory ON THE QUESTION OF SEPARABILITY What we would lke to be able to do s estmate demand curves by segmentng consumers purchases nto groups. In one applcaton, we aggregate purchases

More information

Problem Set 3. 1 Offshoring as a Rybzcynski Effect. Economics 245 Fall 2011 International Trade

Problem Set 3. 1 Offshoring as a Rybzcynski Effect. Economics 245 Fall 2011 International Trade Due: Thu, December 1, 2011 Instructor: Marc-Andreas Muendler E-mal: muendler@ucsd.edu Economcs 245 Fall 2011 Internatonal Trade Problem Set 3 November 15, 2011 1 Offshorng as a Rybzcynsk Effect There are

More information

Notes on Kehoe Perri, Econometrica 2002

Notes on Kehoe Perri, Econometrica 2002 Notes on Kehoe Perr, Econometrca 2002 Jonathan Heathcote November 2nd 2005 There s nothng n these notes that s not n Kehoe Perr NBER Workng Paper 7820 or Kehoe and Perr Econometrca 2002. However, I have

More information

,, MRTS is the marginal rate of technical substitution

,, MRTS is the marginal rate of technical substitution Mscellaneous Notes on roducton Economcs ompled by eter F Orazem September 9, 00 I Implcatons of conve soquants Two nput case, along an soquant 0 along an soquant Slope of the soquant,, MRTS s the margnal

More information

Revision: December 13, E Main Suite D Pullman, WA (509) Voice and Fax

Revision: December 13, E Main Suite D Pullman, WA (509) Voice and Fax .9.1: AC power analyss Reson: Deceber 13, 010 15 E Man Sute D Pullan, WA 99163 (509 334 6306 Voce and Fax Oerew n chapter.9.0, we ntroduced soe basc quanttes relate to delery of power usng snusodal sgnals.

More information

General Tips on How to Do Well in Physics Exams. 1. Establish a good habit in keeping track of your steps. For example, when you use the equation

General Tips on How to Do Well in Physics Exams. 1. Establish a good habit in keeping track of your steps. For example, when you use the equation General Tps on How to Do Well n Physcs Exams 1. Establsh a good habt n keepng track o your steps. For example when you use the equaton 1 1 1 + = d d to solve or d o you should rst rewrte t as 1 1 1 = d

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

CHAPTER 10 ROTATIONAL MOTION

CHAPTER 10 ROTATIONAL MOTION CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The

More information

Least Squares Fitting of Data

Least Squares Fitting of Data Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2014. All Rghts Reserved. Created: July 15, 1999 Last Modfed: February 9, 2008 Contents 1 Lnear Fttng

More information

Dummy variables in multiple variable regression model

Dummy variables in multiple variable regression model WESS Econometrcs (Handout ) Dummy varables n multple varable regresson model. Addtve dummy varables In the prevous handout we consdered the followng regresson model: y x 2x2 k xk,, 2,, n and we nterpreted

More information

Polynomial Barrier Method for Solving Linear Programming Problems

Polynomial Barrier Method for Solving Linear Programming Problems Internatonal Journal o Engneerng & echnology IJE-IJENS Vol: No: 45 Polynoal Barrer Method or Solvng Lnear Prograng Probles Parwad Moengn, Meber, IAENG Abstract In ths wor, we study a class o polynoal ordereven

More information

Multipoint Analysis for Sibling Pairs. Biostatistics 666 Lecture 18

Multipoint Analysis for Sibling Pairs. Biostatistics 666 Lecture 18 Multpont Analyss for Sblng ars Bostatstcs 666 Lecture 8 revously Lnkage analyss wth pars of ndvduals Non-paraetrc BS Methods Maxu Lkelhood BD Based Method ossble Trangle Constrant AS Methods Covered So

More information

Physics 2A Chapter 3 HW Solutions

Physics 2A Chapter 3 HW Solutions Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C

More information

How Strong Are Weak Patents? Joseph Farrell and Carl Shapiro. Supplementary Material Licensing Probabilistic Patents to Cournot Oligopolists *

How Strong Are Weak Patents? Joseph Farrell and Carl Shapiro. Supplementary Material Licensing Probabilistic Patents to Cournot Oligopolists * How Strong Are Weak Patents? Joseph Farrell and Carl Shapro Supplementary Materal Lcensng Probablstc Patents to Cournot Olgopolsts * September 007 We study here the specal case n whch downstream competton

More information

MAE140 - Linear Circuits - Fall 13 Midterm, October 31

MAE140 - Linear Circuits - Fall 13 Midterm, October 31 Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator

More information

Chapter One Mixture of Ideal Gases

Chapter One Mixture of Ideal Gases herodynacs II AA Chapter One Mxture of Ideal Gases. Coposton of a Gas Mxture: Mass and Mole Fractons o deterne the propertes of a xture, we need to now the coposton of the xture as well as the propertes

More information

Our focus will be on linear systems. A system is linear if it obeys the principle of superposition and homogenity, i.e.

Our focus will be on linear systems. A system is linear if it obeys the principle of superposition and homogenity, i.e. SSTEM MODELLIN In order to solve a control syste proble, the descrptons of the syste and ts coponents ust be put nto a for sutable for analyss and evaluaton. The followng ethods can be used to odel physcal

More information

Production Function Estimation

Production Function Estimation Producton Functon Estmaton Producton functon L: labor nput K: captal nput m: other nput Q = f (L, K, m ) Example, Cobb-Douglas Producton functon Q = AL α K β exp(ɛ ) ln(q ) = ln(a) + αln(l ) + βln(k )

More information

Description of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t

Description of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set

More information

Excess Error, Approximation Error, and Estimation Error

Excess Error, Approximation Error, and Estimation Error E0 370 Statstcal Learnng Theory Lecture 10 Sep 15, 011 Excess Error, Approxaton Error, and Estaton Error Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton So far, we have consdered the fnte saple

More information

CS286r Assign One. Answer Key

CS286r Assign One. Answer Key CS286r Assgn One Answer Key 1 Game theory 1.1 1.1.1 Let off-equlbrum strateges also be that people contnue to play n Nash equlbrum. Devatng from any Nash equlbrum s a weakly domnated strategy. That s,

More information

Online Appendix to The Allocation of Talent and U.S. Economic Growth

Online Appendix to The Allocation of Talent and U.S. Economic Growth Onlne Appendx to The Allocaton of Talent and U.S. Economc Growth Not for publcaton) Chang-Ta Hseh, Erk Hurst, Charles I. Jones, Peter J. Klenow February 22, 23 A Dervatons and Proofs The propostons n the

More information

The Parity of the Number of Irreducible Factors for Some Pentanomials

The Parity of the Number of Irreducible Factors for Some Pentanomials The Party of the Nuber of Irreducble Factors for Soe Pentanoals Wolfra Koepf 1, Ryul K 1 Departent of Matheatcs Unversty of Kassel, Kassel, F. R. Gerany Faculty of Matheatcs and Mechancs K Il Sung Unversty,

More information

Linear Momentum. Center of Mass.

Linear Momentum. Center of Mass. Lecture 16 Chapter 9 Physcs I 11.06.2013 Lnear oentu. Center of ass. Course webste: http://faculty.ul.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.ul.edu/danylov2013/physcs1fall.htl

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

More information

Complex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: Shih-Yuan Chen

Complex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: Shih-Yuan Chen omplex Varables hapter 8 Integraton n the omplex Plane March, Lecturer: Shh-Yuan hen Except where otherwse noted, content s lcensed under a BY-N-SA. TW Lcense. ontents ontour ntegrals auchy-goursat theorem

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

International Trade and Public Finance with Public Goods. Wenming Wang. Graduate School of Economics

International Trade and Public Finance with Public Goods. Wenming Wang. Graduate School of Economics Internatonal Trade and Publc Fnance wth Publc Goods Wenmng Wang Graduate School of Economcs Nagoya Unversty, Japan Internatonal Trade and Publc Fnance wth Publc Goods By Wenmng Wang A Dssertaton Submtted

More information

Mathematics Intersection of Lines

Mathematics Intersection of Lines a place of mnd F A C U L T Y O F E D U C A T I O N Department of Currculum and Pedagog Mathematcs Intersecton of Lnes Scence and Mathematcs Educaton Research Group Supported b UBC Teachng and Learnng Enhancement

More information

Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extension Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

More information

Applied Mathematics Letters

Applied Mathematics Letters Appled Matheatcs Letters 2 (2) 46 5 Contents lsts avalable at ScenceDrect Appled Matheatcs Letters journal hoepage: wwwelseverco/locate/al Calculaton of coeffcents of a cardnal B-splne Gradr V Mlovanovć

More information

One-sided finite-difference approximations suitable for use with Richardson extrapolation

One-sided finite-difference approximations suitable for use with Richardson extrapolation Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,

More information

Final Exam Solutions, 1998

Final Exam Solutions, 1998 58.439 Fnal Exa Solutons, 1998 roble 1 art a: Equlbru eans that the therodynac potental of a consttuent s the sae everywhere n a syste. An exaple s the Nernst potental. If the potental across a ebrane

More information

CIE4801 Transportation and spatial modelling Trip distribution

CIE4801 Transportation and spatial modelling Trip distribution CIE4801 ransportaton and spatal modellng rp dstrbuton Rob van Nes, ransport & Plannng 17/4/13 Delft Unversty of echnology Challenge the future Content What s t about hree methods Wth specal attenton for

More information

Least Squares Fitting of Data

Least Squares Fitting of Data Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2015. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

Department of Economics, Niigata Sangyo University, Niigata, Japan

Department of Economics, Niigata Sangyo University, Niigata, Japan Appled Matheatcs, 0, 5, 777-78 Publshed Onlne March 0 n ScRes. http://www.scrp.org/journal/a http://d.do.org/0.6/a.0.507 On Relatons between the General Recurrence Forula o the Etenson o Murase-Newton

More information

ACTM State Calculus Competition Saturday April 30, 2011

ACTM State Calculus Competition Saturday April 30, 2011 ACTM State Calculus Competton Saturday Aprl 30, 2011 ACTM State Calculus Competton Sprng 2011 Page 1 Instructons: For questons 1 through 25, mark the best answer choce on the answer sheet provde Afterward

More information

Economics 101. Lecture 4 - Equilibrium and Efficiency

Economics 101. Lecture 4 - Equilibrium and Efficiency Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of

More information

x i1 =1 for all i (the constant ).

x i1 =1 for all i (the constant ). Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by

More information

Chapter 3 Differentiation and Integration

Chapter 3 Differentiation and Integration MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton

More information

In the figure below, the point d indicates the location of the consumer that is under competition. Transportation costs are given by td.

In the figure below, the point d indicates the location of the consumer that is under competition. Transportation costs are given by td. UC Berkeley Economcs 11 Sprng 006 Prof. Joseph Farrell / GSI: Jenny Shanefelter Problem Set # - Suggested Solutons. 1.. In ths problem, we are extendng the usual Hotellng lne so that now t runs from [-a,

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

Mathematical Economics MEMF e ME. Filomena Garcia. Topic 2 Calculus

Mathematical Economics MEMF e ME. Filomena Garcia. Topic 2 Calculus Mathematcal Economcs MEMF e ME Flomena Garca Topc 2 Calculus Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca . Unvarate Calculus Calculus Functons : X Y y ( gves or each element X one element

More information

Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum

Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum Recall that there was ore to oton than just spee A ore coplete escrpton of oton s the concept of lnear oentu: p v (8.) Beng a prouct of a scalar () an a vector (v), oentu s a vector: p v p y v y p z v

More information

FEEDBACK AMPLIFIERS. v i or v s v 0

FEEDBACK AMPLIFIERS. v i or v s v 0 FEEDBCK MPLIFIERS Feedback n mplers FEEDBCK IS THE PROCESS OF FEEDING FRCTION OF OUTPUT ENERGY (VOLTGE OR CURRENT) BCK TO THE INPUT CIRCUIT. THE CIRCUIT EMPLOYED FOR THIS PURPOSE IS CLLED FEEDBCK NETWORK.

More information

Physics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall

Physics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall Physcs 231 Topc 8: Rotatonal Moton Alex Brown October 21-26 2015 MSU Physcs 231 Fall 2015 1 MSU Physcs 231 Fall 2015 2 MSU Physcs 231 Fall 2015 3 Key Concepts: Rotatonal Moton Rotatonal Kneatcs Equatons

More information

Force = F Piston area = A

Force = F Piston area = A CHAPTER III Ths chapter s an mportant transton between the propertes o pure substances and the most mportant chapter whch s: the rst law o thermodynamcs In ths chapter, we wll ntroduce the notons o heat,

More information

Irreversible Work of Separation and Heat-Driven Separation

Irreversible Work of Separation and Heat-Driven Separation J. Phys. Che. B 004, 08, 6035-604 6035 Irreversble Wor of Separaton and Heat-Drven Separaton Anatoly M. Tsrln and Vladr Kazaov*, Progra Syste Insttute, Russan Acadey of Scence, set. Botc, PerejaslaVl-Zalesy,

More information

Online Appendix. t=1 (p t w)q t. Then the first order condition shows that

Online Appendix. t=1 (p t w)q t. Then the first order condition shows that Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 1 Onlne Appendx Appendx E: Proofs Proof of Proposton 1 Frst we derve the equlbrum when the manufacturer does not vertcally ntegrate

More information

Thermodynamics and Gases

Thermodynamics and Gases hermodynamcs and Gases Last tme Knetc heory o Gases or smple (monatomc) gases Atomc nature o matter Demonstrate deal gas law Atomc knetc energy nternal energy Mean ree path and velocty dstrbutons From

More information

Which Separator? Spring 1

Which Separator? Spring 1 Whch Separator? 6.034 - Sprng 1 Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng 3 Margn of a pont " # y (w $ + b) proportonal

More information

MAE140 - Linear Circuits - Fall 10 Midterm, October 28

MAE140 - Linear Circuits - Fall 10 Midterm, October 28 M140 - Lnear rcuts - Fall 10 Mdterm, October 28 nstructons () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator

More information

PROBLEM SET 7 GENERAL EQUILIBRIUM

PROBLEM SET 7 GENERAL EQUILIBRIUM PROBLEM SET 7 GENERAL EQUILIBRIUM Queston a Defnton: An Arrow-Debreu Compettve Equlbrum s a vector of prces {p t } and allocatons {c t, c 2 t } whch satsfes ( Gven {p t }, c t maxmzes βt ln c t subject

More information

Optimal Marketing Strategies for a Customer Data Intermediary. Technical Appendix

Optimal Marketing Strategies for a Customer Data Intermediary. Technical Appendix Optal Marketng Strateges for a Custoer Data Interedary Techncal Appendx oseph Pancras Unversty of Connectcut School of Busness Marketng Departent 00 Hllsde Road, Unt 04 Storrs, CT 0669-04 oseph.pancras@busness.uconn.edu

More information