n -dimensional vectors follow naturally from the one

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1 B. Vectors ad sets B. Vectors Ecoomsts study ecoomc pheomea by buldg hghly stylzed models. Uderstadg ad makg use of almost all such models requres a hgh comfort level wth some key mathematcal sklls. I these otes we explore the cocepts that are absolutely essetal. Ad usg applcatos to the theory of the frm ad cosumer we llustrate ther value ecoomc aalyss. We beg wth the algebra of arrays. Ecoomsts typcally study arrays of umbers, both buldg models ad aalyzg data. The smplest such array s a sgle row or colum. Ths s called a vector. If the array has compoets t s sad to be a vector of dmeso or a -vector. For example suppose a frm uses z uts of put where,..., m to produce commodtes. The put vector s z ( z, z2,..., z ). The rules of addto ad subtracto of two dmesoal case. -dmesoal vectors follow aturally from the oe a b a b a b (,..., ) ad a b ( a b,..., a b ). Smlarly multplyg a vector by a umber scales all the compoets of the vector. (,..., ). c c c Cosder the vector a ( a, a2) depcted below. If we thk of ths fgure as a geographcal map, the compoets of the vector a are the coordates of a pot the plae. The frst compoet a s the dstace East from the org O, ad the secod compoet a 2 s the dstace North.

2 Legth of a vector It s ofte useful to cosder the dstace betwee a vector ad the org. Ths s called the legth of the vector ad wrtte as a. Appealg to Pythagoras Theorem, the square of the hypoteuse s the sum of the squares of the other two sdes. 2 a a a ad so a ( a a ). 2 2 /2 2 The three dmesoal case s depcted below. Applyg the Pythagoras Theorem, the square of the legth of the vector ( a, a 2,) s Applyg the Pythagoras Theorem aga, h a a a h a a a a 2

3 For vectors of hgher dmeso we have o physcal aalogy. We smply use the same formula called the Eucldea dstace. Defto: Eucldea legth of a vector of dmeso The square of the Eucldea legth of a vector a ( a,..., a ) s a a... a Dstace betwee two vectors Cosder the 2-dmesoal vectors a ad b depcted below. The dstace betwee the two vectors, wrtte as b a s the legth of the le jog a ad b. Appealg to Pythagoras Theorem, the square of the hypoteuse s the sum of the squares of the other two sdes. ( ) ( 2 2) b a b a b a ad so b a (( b a ) ( b a ) ). 2 2 /2 2 2 Now cosder the dfferece vector d b a,.e. ( d, d2) ( b a, b2 a2). Ths s depcted below. The square of the legth of ths dfferece vector s 2 ( ) ( 2 2) b a d d d b a b a 3

4 Thus the dstace betwee the vectors a ad b s the legth of the dstace vector b a. Remark: As you ca cofrm, the dstace betwee the vectors a ad b s also the legth of the dstace vector a b. The same argumet holds for vectors of hgh dmeso. The dstace betwee two vectors s the legth of the dfferece vector. Iequaltes For weak ad strct equaltes we make the followg dstctos. b a b s greater tha a. Ths s the statemet that b a,,..., b a b s strctly greater tha a. Ths s the statemet that b a,,..., ad at least oe equalty s strct. b a every compoet of b s greater tha the correspodg compoet of a. 4

5 Sumproduct of two vectors Let r ad z be two product) of the two vectors s r z rz... rmz m m -dmesoal vectors. The sumproduct (also called the dot product or er I words, the sumproduct of the two vectors s the sum of the term-by-term products of the compoets of each vector. Example : Cost of producto for a frm Let z ( z,..., z m ) be the vector of puts purchased by the frm ad r ( r,..., r m ) s the vector of put prces. The the total cost of these puts s TC r z rz... rmz m. Example 2: Reveue of a frm producg commodtes Let q ( q,..., q ) be the vector of outputs produced by the frm ad let p ( p,..., p ) be the vector of output prces. The the total reveue s TR p q p q... p q p q. j j j Lear combatos of vectors Let x, x,..., x be vectors of dmeso m. A lear combato of these vectors s ay learly weghted sum y x x... Covex combatos of vectors Let x, x,..., x be vectors of dmeso m. A covex combato of these vectors s a lear combato where the weghts are strctly postve ad sum to oe. follows: For the case of two vectors we wll ofte fd t coveet to wrte the covex combatos as 5

6 where t x( t) ( t) x tx Ths s llustrated the fgure below. Note that y () s the vector y ad y () s the vector y. As t creases from zero to oe the mappg from t to the vector yt () maps all the covex combatos of the two vectors. The ext fgure also depcts the dfferece vector x x. Note that x t t x tx x t x x ( ) ( ) ( ) Thus a covex combato s the sum of the vector x ad a fracto t of the dfferece vector, x x 6