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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 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 free trde under incomplete informtion, where the government does not know the chrcteristics of individuls. By considering trnsfer rules tht mke the mount of trnsfer dependent on trnsctions of individuls under utrky, I show tht there is no incentivecomptible nd budget-blncing trnsfer rule to redistribute the gins from free trde successfully, if the discount fctor is sufficiently lrge. Also, the pper exmines whether the mount of trnsfer should be dependent on trnsctions under utrky or trnsctions under free trde, nd shows tht the former is better when the discount fctor is smll or the government hs enough budget for redistribution. *Address: Okmoto, Higshind-ku Kobe , Jpn. E-mil: 1

2 1 Introduction It is well known tht free trde cretes winners nd losers. However, with n pproprite lump-sum trnsfer, Preto improvement cn be chieved s country moves from utrky to free trde, s shown by Kemp (1962), Kemp nd Wn (1972), nd Grndmont nd McFdden (1972), mong others. 1 Essentilly, the ide of Preto-improving lump-sum trnsfers is to let wht ech individul consumed under utrky ffordble under free trde, too. 2 Lter, Dixit nd Normn (1986) pointed out prcticl difficulties of implementing such lump-sum trnsfers. In implementing lump-sum trnsfer to mke everyone better off fter free trde, the government hs to figure out not only who re txed nd who re subsidized, but lso how much is txed or subsidized for ech individul. To do so, the government needs to collect informtion bout the chrcteristics of individuls. Moreover, when sked by the government, ech individul my not revel his chrcteristics tht the government is going to use to clculte the mount of trnsfer. As Dixit nd Normn (1986) write, individuls hve n incentive to mnipulte their behvior so s to misled the plnner bout these chrcteristics nd secure lrger net trnsfer. How do they mnipulte their behvior? Does tht mnipultion mtter? 3 Wong (1997) studied these questions in generl setting, nd rgued tht mnipultion does not mtter if the utrkic price does not chnge by such mnipultion. Recently, however, exmining the sme question in simpler setting of two-goods exchnge economy with qusi-liner utility, Ichino (212) showed tht due to consumers mnipultion the utrkic price chnges to the directions unfvorble to those who lose by free trde. As result, lump-sum trnsfer rule tht llows everyone to consume under free trde wht they used to consume under utrky becomes ineffective in redistributing the gins from free trde. Moreover, in Ichino (212) it is shown tht ny trnsfer rules tht re linerly 1 These studies work on models of perfect competition. Under imperfect competition such s oligopoly, whether there re gins from free trde is less unmbiguous. See, for exmple, Dong nd Yun (21) nd Kemp (21). 2 Using such lump-sum trnsfers hs been one of the stndrd techniques to mke welfre comprison. For textbook tretment, see Wong (1995; pp ). 3 Che nd Fcchini (27) studied this kind of mnipultion in context of mrket liberliztion, while we study the mnipultion in the context of trde liberliztion. 2

3 dependent on utrkic consumption cnnot chieve Preto gins from trde. In this pper, we consider more generl clss of lump-sum trnsfer rules, including trnsfer rules non-linerly dependent on utrkic consumption, nd explore if there exist ny trnsfer rules tht cn successfully redistribute the gins from free trde to everyone. For this purpose, we pply contrct-theory technique used by Feenstr nd Lewis (1991) (herefter, we refer it s FL (1991)). As explined bove, since the government does not hve complete informtion on individuls chrcteristics, it hs to sk the individuls to revel such informtion in order to clculte the mount of trnsfer. Then, under such circumstnce, individuls hve n incentive to mnipulte their behvior, or misreport their chrcteristics if doing so increses the mount of trnsfer to be received. To prevent mnipultion or misreporting, the government hs to design trnsfer rule tht induces truth telling. We therefore formulte welfre-mximiztion problem in which trnsfer rules should stisfy the incentive comptible constrint s well s the government budget constrint nd the everyone-is-mde-better-off constrint. By solving the welfre mximiztion problem, we firstpointoutthtiftrnsferrules re to relte the mount of trnsfer to the mount of consumption under utrky, then the utrkic consumption should be distorted to prevent mnipultion or misreporting of the individuls. Then, we show tht there is no incentive-comptible nd budget-blncing trnsfer rules to chieve Preto gins from free trde if the discount fctor is sufficiently lrge. Intuitively, this is becuse the mount of subsidies given to losers of free trde hs to be lrge enough to compenste him not only for his loss from free trde but lso for the loss from distorted consumption under utrky. At the sme time, the mount of txes collected from the giners of free trde is limited becuse of distorted consumption. Also, we compre the outcome of our model with the outcome of FL (1991). As mentioned bove, in this pper, we consider trnsfer rules dependent on the consumption under utrky. On the other hnd, in FL (1991), trnsfer rules re dependent on the consumption under free trde. By compring the resulting welfre of our model nd FL(1991) s, we show tht the former gives higher welfre thn the ltter when the discount fctor is smll or the government hs enough budget for redistribution. 3

4 2 The Model We consider n exchnge economy with two goods, good x nd good y, where good y is the numerire. In this economy, there is continuum of individuls with the popultion normlized to one. The individuls re ssumed identicl, except the endowment of good x. Withω denoting the endowment of good x, the individuls in the economy re indexed by ω. For simplicity, ssume tht ω is uniformly distributed over the rnge of [, 1]. The utility function of n individul hving ω units of the endowment of good x is given by u(x + ω)+y +ȳ where x nd y denote excess demnd of good x nd y respectively, nd ȳ is the endowment of good y (which is the sme for everyone). To hve explicit solution, we ssume tht the sub-utility function u( ) is qudrtic: u (x + ω) =k + (x + ω) b 2 (x + ω)2. where k, >, b>. In this model, we hve two periods: in period, the economy is under utrky; in period 1, the economy is under free trde. The utility mximiztion problem of n individul is given by mx U (x, p, ω) =u(x + ω) px +ȳ x since the budget constrint px + y is lwys binding. The first-order condition is u (x + ω) =p, from which the excess demnd function x (p, ω) =( p) /b ω is derived. Then, the utrkic equilibrium price p is determined from the mrket-clering condition R 1 x (p, ω) dω =.SinceR 1 ωdω =1/2, solving this condition gives p =(2 b) /2. Let x (ω) x (p, ω) denote the excess demnd (or, we cll it trnsction ) of individul ω in the utrkic equilibrium, nd define the indirect utility of individul ω in the utrkic equilibrium U (ω) U (x (ω),p, ω). In period 1, when the economy llows free trde, the price of good x is equl to the 4

5 world price p (we ssume tht the economy is smll). Then, trnsction of individul ω in the free-trde equilibrium is given by x (ω) ( p ) /b ω, nd the indirect utility in the free-trde equilibrium by U (ω) U (x (ω),p, ω). We ssume tht p <p :theworld price of good x is smller thn the utrky price. 4 By this ssumption, x (ω) >x (ω). In ggregte, there re gins from free trde 5 : = = U (ω) dω U (ω) dω [U (x,p ) U (x,p )] dω + [U (x,p ) U (x,p )] dω +(p p ) [U (x,p ) U (x,p )] dω x dω > since the first term is positive (given p, the utility-mximizing trnsction is x )ndthe second term is zero becuse R 1 x dω =. However, the gins from trde do not fll eqully to everyone. Buyers of good x under utrky gin from free trde: U (ω) U (ω) =[U (x,p ) U (x,p )] + [U (x,p ) U (x,p )] =[U (x,p ) U (x,p )] + (p p ) x > since x >. On the other hnd, the sellers of good x under free trde lose from free trde: U (ω) U (ω) =[U (x,p ) U (x,p )] + [U (x,p ) U (x,p )] =(p p ) x +[U (x,p ) U (x,p )] <. provided tht x <. 4 For the prmeters of the utility function nd the world price, we impose the following restriction: (i) 2 >b. Thisistomketheutrkicpricepositive. (ii)p < (2 b) /2. This is from the ssumption of p <p. (iii) ( p ) /b < 1. This is to mke some individuls still sellers of good x under free trde. 5 Here, to simplify the nottion, we use expressions such s U (x,p ) to men U (x (ω),p, ω), by suppressing ω. Herefter, if there is likely no confusion, we write U (x, p) to denote U (x (ω),p,ω). 5

6 3 Lump-sum Trnsfers under incomplete informtion In order to redistribute the gins from trde, the government cn give the following lumpsum trnsfer z (ω) = (p p ) x (ω) to individul ω in period 1 (note tht z (ω) > is subsidy nd z (ω) < is tx). By this trnsfer, the budget constrint of individul ω in period 1 becomes p x + y (p p ) x (ω). It is esy to show tht this trnsfer mkes wht ech individul consumed under utrky ffordble under free trde s well: p x + y (p p ) x (ω) p x + y p x (ω) p x (ω) Therefore, everyone gets better off by free trde. Tht is, U (ω) +z (ω) >U (ω) for ll ω. In ddition, with this lump-sum trnsfer, the government budget is blnced since R 1 z (ω) dω = (p p ) R 1 x (ω) dω =. The discussion bove tht the free trde policy with the lump-sum trnsfer z (ω) mkes everyone better off, however, depends on n implicit ssumption tht the government cn figure out wht z (ω) s re. For the government to clculte z (ω), ithsto know ech individul s endowment. But this is not likely the cse. Without knowing the endowment of ech individul, the government is not ble to clculte x (ω), nd thus cnnot figure out z (ω). Then, the only wy for the government to find the mount of lump-sum trnsfer for ech individul is to observe his trnsction in period (under utrky). However, if ech individul knows tht the government is going to observe his trnsction in period to determine the mount of trnsfer for him, then he my wnt to chnge, or mnipulte his period- trnsction in order to get lrger trnsfer in period 1. So, here is the question: Is it possible to chieve Preto gins from free trde through lump-sum trnsfers even when the government does not hve complete informtion on the chrcteristics of the individuls nd thus there cn be mnipultion of trnsction? To nswer the question formlly, let us now clrify the setting. As mentioned erlier, in the following nlysis we suppose tht the government does not know endowment of ech individul; the government just observes trnsction of ech individul in period (under utrky). In period, ll individuls know tht free trde will be llowed in period 6

7 1. They lso know tht lump-sum trnsfer z (x) will be given in period 1. Here, note tht the trnsfer depends on x, the observed trnsction of n individul in period. In this setting, the utility mximiztion problem of n individul should be n intertemporl one, nmely, mx x,x 1 U (x, p, ω)+δz (x)+δu x 1,p, ω (1) where x nd x 1 re respectively the period- nd the period-1 trnsction of good x, p is the previling price in period, nd δ is discount fctor δ 1 (we ssume everyone hs the sme discount fctor). By inspection of the mximiztion problem (1), it is obvious tht x 1 = x : the trnsction under free trde is the sme s the one we sw in Section 2. But the trnsction under utrky is now different from x becuse of the dependence of the lump-sum trnsfer on x. Thus, the utrkic equilibrium price, too, cn be different from p. We sy tht lump-sum trnsfer rule z (x) cn chieve Preto gins from free trde if the following inequlity holds for ll ω: mx x [U (x, p, ω)+δz (x)] + δu (ω) (1 + δ) U (ω). Ichino (212) exmined the cse of z (x) = (p p ) x, the lump-sum trnsfer rule to mke sure tht ll individuls cn consume under free trde wht they used to consume under utrky, nd found tht due to trnsction mnipultion Preto gins from trde cnnot be chieved by this trnsfer rule. Moreover, in Ichino (212) it is shown tht ny trnsfer rules tht re liner in x cnnot chieve Preto gins from trde. In this pper, we consider more generl clss of lump-sum trnsfers, including trnsfer rules non-linerly dependent on x, nd explore if there re ny trnsfer rules tht cn successfully distribute the gins from free trde to everyone. For this purpose, we pply contrct-theory technique used by FL (1991), where the endowment of good x is tken s type of n individul not known to the government, nd the government is going to solve welfre-mximiztion problem to derive menu {x (ω),z(ω)}, i.e.,pirof trnsction nd trnsfer. 7

8 4 Welfre-Mximizing trnsfers 4.1 Welfre mximiztion problem Given menu {x (ω),z(ω)} for ω 1, if n individul whose endowment is ω picks prticulr pir {x (ω ),z(ω )}, his intertemporl utility is U x ω,p,ω + δz ω + δu (ω). By the reveltion principle, without loss of generlity we cn confine the set of pirs {x (ω),z(ω)} to be considered to those stisfying the following incentive comptibility (IC) constrints: U (x (ω),p,ω)+δz (ω) U x ω,p,ω + δz ω for ll ω, ω. (2) In ddition to the IC constrints, trnsction-trnsfer menu {x (ω),z(ω)} hs to be designed so s to mke everyone better off by free trde. Tht is, the inequlity U (x (ω),p,ω)+δz (ω) U (ω)+δ (U (ω) U (ω)) for ll ω (3) hs to be stisfied. This is nlogous to the individul rtionlity (IR) constrint in stndrd contrct-theory model. So, in the pper we refer this s the IR constrint. Third, trnsfers should stisfy the government s budget (GB) constrint: z (ω) dω B (4) where B is some mount of budget the government cn use for trnsfer policy. Finlly, the mrket-clering condition x (ω) dω = (5) determines the period- equilibrium price p. 8

9 The welfre mximiztion problem of the government is set s follows: mx x(ω),z(ω) [U (x (ω),p,ω)+δz (ω)] dω + δb s.t. (2), (3), (4), nd (5), δz (ω) dω (6) where the objective function is the sum of the individul s utility nd the government surplus. Problem (6) is nturl extension of Ichino (212). However, here we do not solve problem (6). Insted, we solve somewht different problem defined below. mx x(ω),z(ω) [U (x (ω),p, ω)+δz (ω)] dω + δb [δz (ω) (p p ) x (ω)] dω (7) s.t. U (x (ω),p, ω)+δz (ω) U x ω,p, ω + δz ω for ll ω, ω, U (x (ω),p, ω)+δz (ω) U (ω)+δ (U (ω) U (ω)) for ll ω. δz (ω) dω (p p ) x (ω) dω δb. The welfre mximiztion problem (7) is different from problem (6) in how the period- price is determined. In problem (6), the period- price p is determined by the utrkic mrket-clering condition R 1 x (ω) dω =. Therefore, the utrkic equilibrium price my be different from p, depending on the shpe of x (ω). On the other hnd, in problem (7), the period- price is supported by the government t p, the utrkic equilibrium price under the complete informtion cse. To support the price t p, the government impose n import triff (if x (ω) > ) ornexportsubsidy(ifx (ω) < ) withthesizeof(p p ). Thtiswhywehveterm R 1 (p p ) x (ω) dω in the government s budget constrint. Simply put, in problem (6) utrky is mintined by prohibition of interntionl trde, while utrky is mintined by prohibitive triff (or export subsidy) in problem (7). Below, we present n outline of solving problem (7). Problem (6) is solved in similr fshion since it hs the sme structure s problem (7). Lter, we explin why we focus on problem (7) insted of problem (6). To mke expressions simpler, by busing nottion little bit, we define U (ω) 9

10 U (x (ω),p, ω)+δz (ω). As explined by FL (1991), the IC constrints of problem (7) is equivlent to (i) U (ω) =u (x (ω)+ω) nd (ii) x (ω) is nonincresing. By using condition (i) for the IC constrint, we cn construct the following constrined Hmiltonin by treting x (ω) nd z (ω) s control vribles nd U (ω) s stte vrible: H = [U (ω)+δb δz (ω)+(p p ) x (ω)] (8) +γ (ω) u (x (ω)+ω) +μ (ω)(u (ω) U (ω) δ (U (ω) U (ω))) +ρ (ω)(u (x (ω),p, ω)+δz (ω) U (ω)) +λ (δb δz (ω)+(p p ) x (ω)), where the second line is condition (i) of the IC constrint, nd gives the eqution of motion for U (ω) (we confirm tht condition (ii) of the IC constrint is stisfied fter hving derived solution). The third line is the IR constrint, the fourth line defines U (ω), ndthefifth line is the GB constrint. The mximum principle requires tht H x = (p p )+γ (ω) u + ρ (ω) u (x (ω)+ω) p + λ (p p )=, (9) H z = δ + ρ (ω) δ λδ =, (9b) H U = 1 μ (ω)+ρ (ω) =γ (ω), (9c) nd the trnsverslity conditions re γ () [U () U () δ (U () U ())] =, nd γ (1) [U (1) U (1) δ (U (1) U (1))] =. We let x (ω),z (ω) ª denote the solution, nd U (ω) utility level chieved by the solution. Here, we present n outline of deriving the solution. For the detil, see the ppendix. From (9b), ρ (ω) =1+λ is derived. Substituting this into (9c) gives γ (ω) =γ () 1

11 R ω μ (t) dt + λω. Then, pplying these expressions to (9), we cn derive eqution determining the trnsction schedule x (ω) u x (ω)+ω p = γ () R ω μ (t) dt + λω u, (1) 1+λ For the time being, ssume tht the IR constrint is binding only t ω =1(Lter, we explin it is ctully the cse when the government s budget B is lrge enough). Then, γ () = by the trnsverslity condition nd R ω μ (t) dt =since the IR constrint is not binding for ll ω < 1. Thus, (1) is simplified to u x (ω)+ω p = u mω where m = λ/ (1 + λ). Solvingforx, the following expression x (ω) =x n (ω) p b (m +1)ω (11) is derived. Here, we hve defined x n (ω) s solution when the IR constrint is not binding for ll ω < 1 (the superscript n for not binding ). Since the multiplier λ, x n (ω) is decresing in ω, stisfying condition (ii) of the IC constrint. Once x n (ω) is derived, from the IC constrint, nd from the IR constrint binding t ω =1,theperiod- utility is determined s follows: U (ω) =U n (ω) U (1) + δ (U (1) U (1)) ω u (x n (ω)+t) dt. We cn interpret U (ω) s trget level of utility to be chieved, with trnsfer z (ω). On the other hnd, without trnsfer, the period- utility of individul ω is U (x n (ω),p, ω). The trnsfer schedule is thus determined so s to fill the gp between the trget level nd the without-trnsfer level: δz (ω) =δz n (ω) U n (ω) U (x n (ω),p, ω). (12) 11

12 Then, we cn clculte the GB constrint δz n (ω) dω (p p ) x n (ω) dω = δb, nd from this we cn determine m. This completes the solution. Now, we explin why we solved problem (7) insted of problem (6). Lemm 1 shows tht wht is chieved by solving problem (6) is chievble by problem (7). Lemm 1 Consider solution {ˆx (ω), ẑ (ω)} of problem (6) nd the welfre level chieved by the solution. There exists some z (ω) by which the sme welfre level nd the sme ˆx (ω) is implemented such tht {ˆx (ω),z(ω)} stisfies the constrints of problem (7). Proof. Consider solution {ˆx (ω), ẑ (ω)} of problem (6). In problem (6), the utility level of individul ω is equl to U (1) + δ (U (1) U (1)) R 1 ω u (ˆx (t)+t) dt. To chieve this trget level, the trnsfer schedule ẑ (ω) is clculted s follows: δẑ (ω) =U (1) + δ (U (1) U (1)) ω u (ˆx (t)+t) dt U (ˆx (ω),p,ω). With the sme ˆx (ω), the sme trget level of utility cn be chieved in problem (7) by the following trnsfer schedule δz (ω). δz (ω) =U (1) + δ (U (1) U (1)) ω u (ˆx (t)+t) dt U (ˆx (ω),p, ω). Since R 1 ˆx (ω) dω =by the mrket clering condition in problem (6), the ggregte trnsfers re the sme for ẑ (ω) nd z (ω), ndthetriff revenue in problem (7) is zero. Therefore, R 1 δẑ (ω) dω = R 1 δz (ω) dω R 1 (p p )ˆx (ω) dω. 4.2 Chrcteriztion of the solution Now we chrcterize the solution x (ω),z (ω) ª s follows, noticing tht the solution depends on the size of the government s budget B. When B is lrge nd the GB constrint is not binding, the solution is trivil. Since 12

13 m = λ/ (1 + λ) =if the GB constrint is not binding, from (11) we see tht x (ω) = ( p ) /b ω = x (ω). Thtis,thegovernmentcnchievethemostefficient outcome by letting everyone consume good x of the mount under the free trde. However, s the individuls consume x (ω)+ω when fcing p, their period- utility without trnsfer is U (x (ω),p, ω), which is less thn U (ω). Since U (ω) <U (ω)+δ (U (ω) U (ω)) for lrge ω, the government hs to give positive trnsfer to those individuls to stisfy the IR constrint (see Figure 1). In prticulr, individuls with ω =1need to be compensted t lest by U (1) + δ (U (1) U (1)) U (x (1),p, 1) in order for the IR constrint to be stisfied. Suppose tht the government gives the individuls with ω =1exctly this mount of trnsfer so tht the IR constrint is just binding t ω =1. Then, the IC constrint requires tht the trget level of utility U (ω) hs slope of u (x (ω)+ω) =p sinceeveryoneconsumesx (ω) +ω. Nmely, U (ω) hs to be prllel upwrd shift of U (ω) by the mount of (1 + δ)(u (1) U (1)). By this construction, for ll ω the trget level of utility is bove the utility without trnsfer. Therefore, the government hs to give positive trnsfers to everyone to chieve the trget level of utility. The sum of the trnsfers is δz (ω) dω = U (ω) dω +(1+δ)(U (1) U (1)) U (x (ω),p, ω) dω. On the other hnd, becuse x (ω) >x (ω), in ggregte there is imports of good x so tht the government erns triff revenue R 1 (p p ) x (ω). Thus, the overll government expenditure needed to implement the trnsction-trnsfer schedule is = δz (ω) dω (p p ) x (ω) U (ω) dω +(1+δ)(U (1) U (1)) = (1+δ)(U (1) U (1)). U (x,p ) dω (p p ) x (ω) So, when B =(1+δ)(U (1) U (1)), the government hs just enough budget to implement x (ω) with the IR constrint binding only t w =1. In other words, when the 13

14 government hs lrge enough budget such tht B (1 + δ)(u (1) U (1)), themost efficient trnsction x (ω) is implemented, with the GB constrint not binding. As B flls below (1 + δ)(u (1) U (1)), letting everyone consume x (ω)+ω is no longer comptible with the IR constrint. To stisfy the IR constrint with the smller mount of the budget vilble to the government, now the trnsfer hs to be mde smller for the individuls with low ω (since they re the ones who gin from free trde, there is room to shve the trnsfer for them). Then, with smller trnsfer for low ω, if x (ω) =x (ω) were offered, the individuls with low ω would hve n incentive to pretend tht his ω is higher thn wht ctully is in order to get lrge trnsfer. To prevent it, trnsction schedule x (ω) hs to be distorted in such wy tht consumption x (ω) +ω is no longer the sme for everyone but smller for high ω. Thisisseenby eqution (11): consumption x (ω) +ω is equl to x n (ω) +ω, which is distorted from x (ω)+ω by mω. Put differently, we cn explin the following trde-offs of distorting consumption. Tke prticulr individul with < ω 1. As consumption is distorted to be smller thn the free trde level, his period- utility without trnsfer is decresed. This mkes trnsfer needed to be given to this prticulr individul lrger. However, such distortion steepens the slope of the trget utility u (x n (ω)+ω), which lowers the trget utility levelofthell individuls below ω. This mkes trnsfers needed to be given to those individuls smller. In ggregte, the ltter effect outweighs the former effect, nd thus the government expenditure for trnsfers is mde smller by such distortion. In prticulr, for the individuls with lrge ω, the ltter effect is dominting. This is why distortion is lrge for lrge ω, s illustrted in Figure 4. The trget utility level is depicted in Figure 2. It is now steeper thn the one when the GB constrint is not binding, but the IR constrint is still binding only t ω =1. As the government budget gets more stringent, consumption of good x is more distorted nd the slope of the trget level utility gets steeper. Eventully, the trget utility becomes s steep s the right hnd side (RHS) of the IR constrint t ω =1. Then, consumption of good x cnnot be lowered without violting the IR constrint for high 14

15 ω. In this cse, the IR constrint becomes binding not only t ω =1but lso for lrge ω such tht ω ω < 1, where ω is threshold vlue. Nmely, for ω ω < 1, the trnsction schedule is chosen so s to mke the trget utility s steep s the RHS of the IR constrint: u (x (ω)+ω) =p + δ (p p ) (see Figure 3). Letting x b (ω) (b for binding ) denote the trnsction schedule stisfying this equlity, we hve x b (ω) = p b δ (p p ) b ω for ω ω < 1. (13) Since the trget utility is U (ω)+δ (U (ω) U (ω)), the trnsction schedule for ω ω < 1 is given by δz b (ω) U (ω)+δ (U (ω) U (ω)) U(x b (ω),p, ω). (14) On the other hnd, for ω < ω, the trnsction schedule is the sme s (11), while the trget utility is modified to U ( ω)+δ (U ( ω) U ( ω)) R ω ω u (x n (t)+t) dt. Therefore, the trnsction schedule for ω < ω is Z ω δ z n (ω) U ( ω)+δ (U ( ω) U ( ω)) u (x n (t)+t) dt U (x n (ω),p, ω). (15) ω The threshold vlue ω is determined by equting x n (ω) nd x b (ω). Thtis, ω solves the equlity p b (m +1) ω = p b δ (p p ) b ω, (16) tht is, ω =(1+δ)(p p ) /bm. (17) The more stringent the government budget constrint becomes, m rises, nd ω flls, mening tht the more individuls re mde just indifferent between utrky nd free trde. Then, nturlly, the following question rises. No mtter how smll the budget B is,sy,evenwhenb =, cn some individuls be mde strictly better off by free trde? Proposition 1 clims tht the nswer is no: when B =, the gins from free trde 15

16 cn be distributed to mke someone strictly better off nd no one worse off only when δ is sufficiently smll. If δ is lrge enough nd B is smll enough, there is no solution to problem (7). See Figure 6 for grphicl presenttion. Proposition 1 There is vlue δ (, 1) such tht for δ δ problem (7) hs solution for ny B. On the other hnd, for δ > δ, there is threshold function B (δ) > such tht B (δ) > nd for B< B (δ) problem (7) hs no solution. The proof is given in the ppendix. Here, we provide intuitive explntion in the following three steps. First, we show tht in problem (7), even when mking everyone just indifferent between utrky nd free trde, the government needs positive budget. With the trnsctiontrnsfer pir x b (ω),z b (ω) ª for ll ω, the government expenditure for individul ω is written s δz b (ω) (p p ) x b (ω) =U (ω)+δ (U (ω) U (ω)) U(x b,p ) Adding nd subtrcting U (ω), wehve ³ δz b (ω) (p p ) x b (ω) =(1+δ)(U (ω) U (ω)) + U (ω) U(x b,p ) The first term is negtive when it is ggregted, while the second term is positive since U (ω) =U (x,p ). By decomposing the first term by dding nd subtrcting U (x,p ), we hve δz b (ω) (p p ) x b (ω) ³ = (1+δ)(U (ω) U (x,p )+U (x,p ) U (ω)) + U (ω) U(x b,p ) (18) ³ = (1 + δ)(p p ) x (1 + δ)(u (x,p ) U (x,p )) + U (x,p ) U(x b,p ). The first term disppers when ggregted over ω 1. The second term is negtive: it represents, in terms of per cpit, the gins from free trde llowed in the both periods 16

17 (with the negtive sign in front of it). By giving everyone the IR-binding level of utility insted of the free-trde level of utility, the government cn collect this mount of tx revenue. The third term is positive, which is interpreted s the loss of the individuls induced to consume distortedly smll mount of x b +ω due to the IC constrint, insted of consuming the free trde level of x + ω. So, the government hs to compenste this loss to the individuls. Grphiclly (see Figure 5), the third term is big tringle representing the chnge in the consumer surplus by consuming x b + ω t p rther thn consuming x + ω t p.for the second term, (U (ω) U (x,p )) is smll tringle representing the chnge in the consumer surplus by consuming x +ω t p rther thn x +ω t p. Multiplied by (1 + δ), it becomes the second term. Since (x + ω) x b + ω =(1+δ)((x + ω) (x + ω)), the big tringle representing the third term is lwys lrger thn the tringles representing the second term. Therefore, expression (18) is positive, mening tht it requires positive government budget to mke everyone just indifferent between utrky nd free trde (i.e., mking the IR constrint binding for ll ω). Of course, mking the IR constrint binding for ll ω my not minimize the government expenditure. In fct, strting from the IR constrint binding for everyone (i.e., strting from ω =), the government cn improve its budget by giving someone strictly positive gins from trde (i.e., by incresing ω). So, second, we explin this. Tke prticulr individul with ω very smll. As ω increses from zero so tht his ω becomes below ω, his consumption is now less distorted, nd thus his utility level without trnsfer increses. By this, the government cn collect more txes from him (or needs to give less subsidy to him), with the government expenditure getting smller. On the other hnd, however, n increse in ω rises his trget utility level. This hppens becuse due to the less distorted (incresed) consumption, the slope of the trget utility of, not only him, but lso ll individuls between his ω nd ω, getsfltter. By this, the government cn collect less txes from him (or needs to give more subsidy to him), with the government expenditure getting lrger. Since the ltter effect is smll when ˆω is smll, the former effect is dominting so tht the government expenditure flls by n increse in ω from ω =. Eventully, 17

18 however, s ω is incresed sufficiently lrge, the government expenditure hits the bottom, since the ltter effect is getting bigger nd comes to outweigh the former. Third, we explin tht (18) is lrge when δ is lrge. This is seen by inspecting Figure 5. When δ is lrge, the distnce between p + δ(p p ) nd p is lrge, nd so is the distnce between x + ω nd x b + ω. Therefore, the re of the big tringle not covered by the smll tringles gets lrger. This mens tht (18) is lrge. Then, lthough the government expenditure decreses s ω increses, the lowest government expenditure is still positive if δ is lrge. So, when B is smller thn the lowest government expenditure, the government cnnot implement the trnsfer policy to redistribute the gins from free trde. 4.3 Should the trnsfers be dependent on trnsctions under utrky or under free trde? So fr, we hve considered trnsfer rules under which the mount of trnsfer is dependent on the trnsction mde in period. Tht is, the government offers trnsction-trnsfer menu x (ω),z (ω) ª,wherex (ω) is the trnsction in period nd z (ω) is the trnsfer given in period 1. In our nlysis, the resulting intertemporl utility cn be written s U x (ω),p, ω + δz (ω)+δu (ω). (19) On the other hnd, in order to redistribute the gins from free trde, we could consider trnsfer rules dependent on trnsctions mde in period 1. Nmely, the government could offer trnsction-trnsfer menu x 1 (ω),z 1 (ω) ª,wherex 1 (ω) is the trnsction in period 1 nd z 1 (ω) is the trnsfer given in period 1. In fct, this trnsfer rule is the one studied by FL (1991). Formlly, menu x 1 (ω),z 1 (ω) ª is solution to the 18

19 welfre-mximiztion problem formulted s follows: mx x(ω),z(ω) [U (x (ω),p, ω)+z (ω)] dω + B z (ω) dω (2) s.t. U (x (ω),p, ω)+z (ω) U x ω,p, ω + z ω for ll ω, ω, U (x (ω),p, ω)+z (ω) U (ω) for ll ω. z (ω) dω B. The solution to this problem, denoted by x 1 (ω),z 1 (ω) ª, is qulittively similr to x (ω),z (ω) ª, the solution to problem (7) we derived in the previous section. Nmely, when B is sufficiently lrge, the GB constrint is not binding nd the trnsction schedule is efficient: x 1 (ω) =x (ω). As B flls, the GB constrint becomes binding nd the trnsction is distorted so tht consumption is smller for lrge ω, whiletheirconstrint is binding only t ω =1. Then, s B flls further, eventully the IR becomes binding for some ω < 1. However, different from problem (7), there is lwys solution to problem (2), s shown by FL (1991). This is becuse, in problem (2), mking the IR constrint binding for ll ω (tht is, forcing everyone consume x + ω) cn be implemented by blnced budget (i.e., with B =). Therefore, strting from the IR binding for everyone, the government cn let some individuls with smll ω consume more thn x + ω to mke them strictly better off by trde, nd still the GB constrint is stisfied even when B = (See Proposition 4 of FL (1991)). By the trnsfer rule x 1 (ω),z 1 (ω) ª, the intertemporl utility is given by U (ω)+δz 1 (ω)+δu x 1 (ω),p, ω. (21) Compring (19) nd (21), we cn see tht in (19) the government does not intervene trnsctions in period 1, leving the period-1 utility U (ω) intct, while in (21) the government does not intervene trnsctions in period with leving the period- utility U (ω) intct. Then, we sk the following question: by which trnsfer rules cn the welfre be mde higher? In other words, to redistribute the gins from free trde to everyone 19

20 with stisfying the IC constrint, which is less hrmful, distorting period- trnsctions, or distorting period-1 trnsctions? The proposition below shows tht the nswer depends onthesizeofb. If the government hs lrge budget for trnsfer, the former is better, nd if it hs smll budget, the ltter is better. Proposition 2 If B (1 + δ)(u (1) U (1)), the trnsfer rule dependent on trnsctions in period gives higher welfre thn the trnsfer rule dependent on trnsctions in period 1. If B< B (δ), the trnsfer rule dependent on trnsctions in period 1 gives higher welfre thn the trnsfer rule depending on trnsctions in period. Proof. When B (1 + δ)(u (1) U (1)), the GB constrint is not binding for problem (7) nd x = x. Letting W denote the resulting intertemporl welfre of problem (7), W = = U x (ω),p, ω + δz (ω)+δu (ω) δz (ω)+(p p ) x (ω)+δb dω (U (ω)+δu (ω)+δb) dω. For problem (2), note tht the GB constrint is not binding if B U (1) U (1) (The reson comes from the sme logic discussed in Section 4.2). Therefore, when B (1 + δ)(u (1) U (1)), the government cn implement x 1 = x. Let W 1 denote the resulting welfre for problem (2). We hve W 1 = = U (ω)+δz 1 (ω)+δu x 1 (ω),p, ω δz 1 (ω)+δb dω (U (ω)+δu (ω)+δb) dω. Since R 1 U (ω) dω > R 1 U (ω) dω, it holds tht W >W 1. If B< B (δ), there is no solution to problem (7). So, the best thing the government cn do is to sty utrky in both periods, thus W = R 1 (U (ω)+δu (ω)+δb) dω. On the other hnd, for problem (2), the government cn give some individuls strictly 2

21 positive gins from trde. Therefore, W 1 = U (ω)+δu x 1 (ω),p, ω + δb dω > (U (ω)+δu (ω)+δb) dω = W. Moreover, since W nd W 1 re continuous in B B (δ),evenwhenb<(1 + δ)(u (1) U (1)), s long s B is lrge enough, W stys bove W 1. This cn be seen in Figure 6, where W is higher thn W 1 even when B is fr below (1 + δ)(u (1) U (1)). 5 Concluding remrks We hve considered two-period model where in period the utrky price is mintined ndinperiod1freetrdeisllowed. Tochieve Preto gins from trde, the government designs trnsfer rule to collect txes from giners nd give subsidies to losers from free trde. By considering trnsfer rules tht mke the mount of trnsfer dependent on trnsctions in the utrky period, we hve shown tht there is no incentive comptible trnsfer rules tht successfully redistribute the gins from free trde if the discount fctor is sufficiently lrge or if the government budget vilble for redistribution is limited. Nmely, our finding tells tht redistributing the gins from free trde is often very costly under incomplete informtion; redistribution uses up ll gins from free trde nd still not ble to mke everyone better off. 21

22 Appendix A. Solution to problem (7). (i) When the IR constrint is binding only t ω =1: From eqution (11) nd (12), the government expenditure is clculted s follows: δz (ω) dω (p p ) x (ω) dω =(1+δ)(U (1) U (1)) mb (2 m) 6 where U (1) U (1) = µ 1 2 (p p ) (p p ) 2b nd m = λ/ (1 + λ). The GB constrint is thus (1 + δ)(u (1) U (1)) mb (2 m) 6 = δb. Solving this for m, weget m = b p b 2 6b ((1 + δ)(u (1) U (1)) δb). b Note tht m is decresing in B. So, s B flls, m increses, x (ω) decreses, nd therefore the trget utility gets steeper. As B flls enough tht the trget utility is s steep s the RHS of the IR constrint t ω =1, the IR constrint becomes binding other thn ω =1.ThebordervlueofB below which this hppens is derived by u x (1) + 1 = p + δ (p p ) p + mb = p + δ (p p ). Solving this for B, we hve the border vlue of B: µ B = 1 (2 δ) (p p ) (1 + δ)(p p ). b 6δ From our ssumption tht ( p ) /b =1/2 < ( p ) /b < 1, wehve(p p ) /b < 1/2. So, the B bove is lwys positive, mening tht for ny prmeter vlues, if B is 22

23 smll enough, the IR constrint becomes binding for ω ω 1. (ii) When the IR constrint is binding for ω ω 1: For ω ω 1, the trnsction schedule nd the trnsfer schedule re respectively given by eqution (13) nd (14). For ω < ω, since the IR constrint is not binding, nd γ () = by trnsverslity condition, the trnsction schedule is given by (11), nd thus the trnsfer schedule is by (15). From these, the government expenditure is clculted s ³ δz b (ω) (p p ) x b (ω) ω µ ω = (1+δ)(p p 2 ) Z ω dω + 2 ((1 + δ) ω δ)(p p ) 2b (δ z n (ω) (p p ) x n (ω)) dω µ bm 3 bm2 ω 3. (22) 6 The threshold vlue ω is determined s follows. For ω ω 1, x b (ω) should stisfy (1): u (x b (ω)+ω) p = R ω ω μ (t) dt + λω b. 1+λ Here, we used tht γ () = nd μ (ω) =for ω < ω (the IR constrint is not binding for this rnge of ω). At the sme time, by the IC constrint, we hve u (x b (ω) +ω) = p + δ (p p ).Thus, Z ω ω μ (t) dt = λω (1 + λ) (1 + δ)(p p ). b Especilly, when ω = ω, it holds tht Z ω ω μ (t) dt = λ ω (1 + λ) (1 + δ)(p p ) b =. Solving this for ω gives ω =(1+δ)(p p ) /bm. Substituting this into (22), nd rrnging, we hve the following GB constrint: (1 + δ)(p p ) ω 2 (1 + δ)2 (p p ) 2 ω + δ (1 + δ)(p p ) 2 6 3b 2b = δb. (23) Combined with ω =(1+δ)(p p ) /bm, thisissolvedfor ω nd m. 23

24 B.ProofofProposition1. Consider the determinnt of the qudrtic eqution (23). It is nonnegtive if B (1 + δ)(p p ) 2 6bδ Ã! 3δ (1 + δ)2 (p p ). (24) b Define the RHS of (24) s B (δ). If B< B (δ), there is no solution to problem (7). Note tht B (δ) is negtive if 3δ (1 + δ)2 (p p ) b <. (25) Since the inequlity (25) holds when δ =, violted when δ =1, nd the LHS of (25) is incresing in δ, there is unique δ (cll it δ) belowwhich B (δ) <. Thus, for δ δ, there is solution to problem (7) for ny B. Forδ > δ, B (δ) > nd B (δ) = B (δ) (1 + δ) + (1 δ)(1+δ)2 (p p ) 3 6b 2 δ 2 >. 24

25 References [1] Che, Jihu nd Giovnni Fcchini, Dul Trck Reforms: with nd without Losers, Journl of Public Economics 91 (27): [2] Dixit, Avinsh nd Victor Normn, Gins from Trde without Lump-Sum Compenstion, Journl of Interntionl Economics 21 (1986): [3] Dong, Bomin nd Lsheng Yun, The Loss from Trde under Interntionl Cournot Oligopoly with Cost Asymmetry, Review of Interntionl Economics 18 (21): [4] Feenstr, Robert C. nd Trcy R. Lewis, Distributing the Gins from Trde with Incomplete Informtion, Economics nd Politics 3 (1991): [5] Grndmont, Jen-Michel nd Dniel McFdden, A Technicl Note on Clssicl Gins from Trde, Journl of Interntionl Economics 2 (1972): [6] Ichino, Ysukzu, Trnsction Flsifiction nd the Difficulty of Achieving Pretoimproving Gins from Trde through Lump-sum Trnsfers, Review of Interntionl Economics 2 (212): [7] Kemp, Murry C., The Gin from Interntionl Trde, Economic Journl 72 (1962): [8] Kemp, Murry C., The Gins from Trde in Cournot-Nsh Trding Equilibrium, Review of Interntionl Economics 18 (21): [9] Kemp, Murry C. nd Henry Y. Wn, Jr., The Gins from Free Trde, Interntionl Economic Review 13 (1972): [1] Wong, Kr-yiu, Interntionl Trde in Goods nd Fctor Mobility, Cmbridge: MIT Press, [11] Wong, Kr-yiu, Gins from Trde with Lump-Sum Compenstion. Jpnese Economic Review 48 (1997):

26 U U ( ) when the GB is not binding. U ( ) U *( ) (slope = p ) (slope = p* ) U ( x*( ), p, ) U ( ) ( U ( ) U * ( )) (slope = p ( p p*) ) 1 Figure 1: U (ω) when the GB constrint is not binding 26

27 U U ( U ( ) when the GB is binding nd the IR is binding only t 1. ) when the GB is not binding. U ( ) (slope = p ) U *( ) (slope = p* ) U ( ) ( U ( ) U * ( )) (slope = p ( p p*) ) 1 Figure 2: U (ω) when the GB constrint is binding nd the IR constrint is binding only t ω =1 27

28 U U ( ) when the IR is binding for some 1. U ( ) U *( ) (slope = p ) (slope = p* ) U ( ) ( U ( ) U * ( )) (slope = p ( p p*) ) 1 Figure 3: U (ω) when the IR constrint is binding for some ω < 1 28

29 x( ) x * ( ) x ( ) b x ( ) x ( ) when the IR is binding only t 1. x ( ) when the IR is binding for some 1. 1 Figure 4: Consumption of good x 29

30 p b U ( x*, p*) U ( x, p*) p ( p p*) p ( U( x*, p*) U( x, p*)) U( x*, p*) U( x, p*) p * u ( x ) b x x x * x Figure 5: 3

31 B.3.25 * B (1 )( U (1) U (1)) : bove this curve, the GB constrint is not binding Below this curve, the IR constrint is binding for some 1. Above this curve, W >W 1 : the trnsfer rule in our model gives higher welfre thn FL(1991) s..5 B( ): no solution below this curve A numericl exmple: when =2, b=1, p*=5/4 Figure 6: 31

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

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

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

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

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

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

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

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

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

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

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

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

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

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

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

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 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

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

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep. Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

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

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

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

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

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

Algorithms Chapter 4 Recurrences

Algorithms Chapter 4 Recurrences Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering

More information

Generalized Inverses: How to Invert a Non-Invertible Matrix

Generalized Inverses: How to Invert a Non-Invertible Matrix Generlized Inverses: How to Invert Non-Invertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax

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

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

1. The leves re either lbeled with sentences in ;, or with sentences of the form All X re X. 2. The interior leves hve two children drwn bove them) if

1. The leves re either lbeled with sentences in ;, or with sentences of the form All X re X. 2. The interior leves hve two children drwn bove them) if Q520 Notes on Nturl Logic Lrry Moss We hve seen exmples of wht re trditionlly clled syllogisms lredy: All men re mortl. Socrtes is mn. Socrtes is mortl. The ide gin is tht the sentences bove the line should

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

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

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

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

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

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified

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

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

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

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

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

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

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

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

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

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists. Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix

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

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

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

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

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

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

Exponents base exponent power exponentiation

Exponents base exponent power exponentiation Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily

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

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

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

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

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

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

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

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

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

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Health insurance exchanges What to expect in 2014

Health insurance exchanges What to expect in 2014 Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount

More information

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

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

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

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

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

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

Health insurance marketplace What to expect in 2014

Health insurance marketplace What to expect in 2014 Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

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

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

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

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

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

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

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

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

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

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999 Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,

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

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

N Mean SD Mean SD Shelf # Shelf # Shelf #

N Mean SD Mean SD Shelf # Shelf # Shelf # NOV xercises smple of 0 different types of cerels ws tken from ech of three grocery store shelves (1,, nd, counting from the floor). summry of the sugr content (grms per serving) nd dietry fiber (grms

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

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

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Solving BAMO Problems

Solving BAMO Problems Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only

More information

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

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

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

Worksheet 24: Optimization

Worksheet 24: Optimization Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I

More information

CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numerical Differentiation and Integration CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

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

Finite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh

Finite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is

More information