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1 2004multivrilefunctions University of Sydney Deprtment of Economics Mthemticl Methods of Economic Anlysis MULTIVARIABLE FUNCTIONS SUMMARY 1. Functions 1) Given two sets X nd Y, function from X to Y, written f: X Y, is rule tht ssocites with ech element X unique element f() Y. The set X is clled the domin of the function. The set Y goes under vrious nmes including codomin, trget nd trget spce. The element f() is clled the vlue of the function t or the imge of under f. The set of ll the f() vlues is clled the imge or the rnge of the function, denoted f(x), i.e., f(x) = { f() Y X }. The rnge my e proper suset of the codomin, e.g., if X = R, Y = R nd f() = 2, the rnge of the function is R + which is proper suset of the codomin R. If the rnge nd codomin re equl, then the function is sid to mp X onto Y (the generl sttement is tht the function mps X into Y onto is specil cse of into). If f mps X onto Y, then f() = y hs solution for for ech y Y. 1) In working with functions, it is importnt to e cler on the nottionl conventions tht govern them. Suppose, for emple, tht we hve function f: X Y. In the epression f(), the letter is simply ny element elonging to the domin X nd f() is the element of the codomin Y tht results from pplying the function rule f to tht element of the domin. The fct tht the letter is used hs no significnce. Suppose, for emple, tht we hve function f with domin given y X = { R 0 10 } nd tht f() = 2 for ll in the domin. Then it is eqully true tht f(y) = y 2 for ll y X, f(z) = z 2 for ll z X nd f( 1) = ( 1) 2 for ll ( 1) X. Note tht in this lst emple ( 1) X iff More generlly, given function f: X Y, it is permissile to replce the in f() y ny oject whtsoever provided X. Assuming this condition is met, f( ) is given y the following condition: if =, then f( ) = f(). If f() is given y formul, then this mens tht f( ) is clculted y replcing y wherever ppers in the formul for f(), e.g., if then For emple, if = ( c), then f() = for ll 2, 2, f( ) = for ll 2, 2. f( c) = [2 2 + c] 2 3 [ c] [ c] 2 for ll c 2, 2.

2 2 1c) A function of n vriles cn e represented y f: X Y where X is suset of R n nd Y is suset of R. Economic emples include the production function Q = F (K, L) (where K is the quntity of cpitl nd L is the quntity of lour) nd the utility function U = U( 1, 2,..., n ) (where 1, 2,..., n re quntities of goods 1,..., n). Using vector nottion, the utility function my e written more succinctly s U = U(), where = ( 1, 2,..., n ). 1d) Just s with functions of single vrile, it is importnt to understnd the nottion for functions of severl vriles. Suppose, for emple, tht we hve function f with domin given y X = { R 3 1, 2, 3 0 }. Then the domin is to e interpreted s consisting of ll 3-dimensionl vectors with non-negtive components, not just ll vectors with non-negtive components. function s domin. Thus if y = [7, 0, 4], then tht y vector elongs to the Similrly, in the epression f(), the letter is simply ny element elonging to the domin X, nd f() is the element of the codomin Y tht results from pplying the function rule f to tht element of the domin. For ny X, we my clculte f( ) y pplying the rule: if =, then f( ) = f(). If f() is given y formul, then this mens tht f( ) is clculted y replcing the ith component of (i.e., i ) y the ith component of wherever i ppers in the formul for f(), e.g., if then f( 1, 2, 3 ) = e 1 for ll ( 1, 2, 3 ) R 3 f(7, 2, y) = y + e 7 for ll (7, 2, y) R 3 1e) As further illustrtion of the generlity of the function concept, consider the following. Suppose tht we hve two functions, ech of three vriles: y 1 = f 1 ( 1, 2, 3 ) for ll ( 1, 2, 3 ) R 3 (1) y 2 = f 2 ( 1, 2, 3 ) for ll ( 1, 2, 3 ) R 3. (2) Then, s n lterntive to considering these s two functions, we cn think of them s just one function f: R 3 R 2 which ssocites with ech element ( 1, 2, 3 ) in the domin of R 3 unique element (y 1, y 2 ) in the codomin of R 2. Such functions re sometimes termed vector-vlued. Using vector nottion, we my write (1) nd (2) s y = f(), where f is vector-vlued function. 1f) In economics we frequently encounter composite functions or functions of functions. These re formlly defined s follows: Given f: X Y nd g: Y Z, where Y includes f(x) (i.e., the domin of g includes the rnge of f), we my define the composite function, h: X Z, where ( ) h() g f() for ll X.

3 3 This is illustrted in Figure 1 (the domin of g hs not een shown eplicitly, ut is to e understood s including ll f() vlues): Z g Y f X g( f( )) f() h Figure 1 e.g., if X = Y = Y = Z = R, f() = 3 nd g(y) = 2 + y 2, then we my define the composite function, h: X Z, where ( ) h() = g f() = g(3) = 2 + (3) 2. Note tht, rther thn introduce new letter such s h to denote the composite function, n lterntive is to use g f, which mkes clerer how the vlues of the composite function re clculted. Thus, insted of h: X Z, we cn write g f: X Z. As n economic emple, suppose tht utility is function of the quntity consumed of ech of n consumption goods, i.e., we hve U = g( 1,..., n ). Further suppose tht the demnd for good i depends on the prices of goods 1 to n nd on money income M, i.e., 1 = f 1 (p 1,..., p n, M) 2 = f 2 (p 1,..., p n, M).. n = f n (p 1,..., p n, M). These demnd functions cn e regrded s single vector-vlued function f with domin of R n+1 +, codomin of R n + nd vlues given y = f(p, M). Utility is determined directly y the quntities of goods consumed ut, since these consumption quntities re determined y prices nd income, it follows tht utility is determined indirectly y prices nd income. We represent this mthemticlly y the use of composite function. In effect this involves sustituting the demnd functions into the utility function.

4 4 Formlly, we hve demnd function f: R n+1 + R n + nd utility function g: R n + R. From these two we form the composite function h: R n+1 + R, where ( ) h(p 1,..., p n, M) = g f 1 (p 1,..., p n, M),..., f n (p 1,..., p n, M) }{{}}{{} 1 n or, more succinctly, h(p, M) = g(f(p, M)). This gives utility s function of prices nd income. The composite function h(p, M) is known s the indirect utility function. 1g) Note tht composite function my involve more thn two functions. One could hve, sy, four functions,,, c nd d nd form the composite function: h() = ((c(d()))). 1h) Let f nd g e two sclr-vlued functions. Then the sum, difference, product, quotient, minimum nd mimum of f nd g re defined s follows: Sum: Difference: Product: Quotient: Minimum: Mimum: (f + g)() = f() + g() (f g)() = f() g() (f g)() = f() g() (f/g)() = f()/g() min{f, g}() = min{f(), g()} m{f, g}() = m{f(), g()}. The domin in ll cses is the intersection of the domins of f nd g, suject to the qulifiction tht the quotient function is only defined for those vlues of for which g() 0. 1i) The function f ssocites with ech X unique y Y, ut it need not do the reverse, i.e., ssocited with y Y my e more thn one vlue of X, e.g., if f() = 2, then ech positive y is ssocited with two vlues, ± y. The inverse imge of y Y is defined s the set of ll X such tht f() = y: f 1 (y) = { X, f() = y } In the emple descried ove, f 1 (y) = { R, 2 = y } Note tht the inverse imge my e n empty set. In the set just defined, f 1 (y) = { ± y } for y 0, ut f 1 (y) = for y < 0. The inverse imge of suset of the codomin, Y Y, cn lso e defined s follows: f 1 (Y ) = { X, f() Y }, e.g., if Y = { y 1 < y < 4 } nd f() = 2, then f 1 (Y ) = { 1 < < 2 or 2 < < 1 }.

5 5 1j) If for ech y Y, the inverse imge consists of t most one point (recll tht it my e n empty set), then the function is sid to e one to one. Another wy of sying this is tht, if f is one to one, then the eqution f() = y (where y Y ) hs t most one solution for. Yet nother wy of defining one to one function is to sy tht f( 1 ) = f( 2 ) 1 = 2 for ll 1, 2 X. 1k) If f is oth one to one nd onto, then there eists n inverse function f 1 : Y X, i.e., f 1 ssocites with ech y Y unique vlue f 1 (y) = X, where f 1 (y) is the inverse imge defined erlier, i.e., 1l) If y is in the rnge of f, then f 1 (y) = { X, f() = y } ( ) f f 1 (y) = y. However, it is not necessrily true tht for X: f 1( ) f() =. It is true if f is one to one. In the generl cse, we hve f 1( ) f(), ut f 1( ) f() my include other elements in ddition to. 2. Limits nd Continuity 2) Consider function f: X Y, where X R n nd Y R m. Let 0 R n nd s R m. Note tht it is possile ut not required tht 0 X nd s Y. We do ssume, however, tht 0 is n ccumultion point of X. 1 If the it of f s pproches 0 equls s, then we denote this with the following nottionl forms: f() = s or f() s s 0. This it reltion holds iff the following condition is met: For every ɛ > 0, there eists δ > 0 such tht f() s < ɛ for ll X stisfying 0 < 0 < δ. (3) For emple, if X = Y = R nd f() = 2, then f() = The point 0 R n is n ccumultion point of X if every ɛ-ll out 0 contins t lest one point X, where 0.

6 6 since, if we set δ = ɛ/2, then 2 6 < ɛ for ll R stisfying 0 < 3 < δ = ɛ/2, so condition (3) is stisfied. 2) Note tht the condition (3) specifies tht 0 > 0. Thus the vlue of f() t = 0 plys no role in the definition of the it s noted erlier, it is not even necessry tht 0 e in the domin of f or tht s e in its codomin. 2c) The it of function my or my not eist. It my fil to eist if the function jumps t 0 or if it oscilltes infinitely rpidly t 0. 2d) An lterntive ut equivlent definition of the it of function my e given in terms of sequences. Let f, 0 nd s e s defined ove. Then f() = s iff, for every sequence { k } in X stisfying k 0 (ll k) nd k 0, the sequence {f( k )} hs it of s, i.e., f( k) = s or f( k ) s s k. k Note tht, s in the erlier definition, the vlue of f t 0 is irrelevnt. 2e) As with its, we cn give two definitions of continuous function. Define f s ove nd ssume 0 X. Then f is sid to e continuous t 0 iff, for every ɛ > 0, there eists δ > 0 such tht f() f( 0 ) < ɛ for ll X stisfying 0 < δ. (4) Plinly, unlike in the definition of the it of function given ove, it is necessry for continuity t 0 tht f() e defined t = 0 (to e defined in this contet, f( 0 ) must e finite) nd the vlue of f( 0 ) is fundmentl to the definition. 2f) The sequence definition of continuity is s follows: Define f s ove nd ssume 0 X. Then f is sid to e continuous t 0 iff, for every sequence { k } in X with it of 0, the sequence {f( k )} hs it of f( 0 ), i.e., f( k) = f( 0 ) or f( k ) f( 0 ) s k. k Note tht, unlike in the it of function definition, we do not require tht k 0, i.e., we dmit wider clss of sequences. 2g) As the ove definitions will suggest, the continuity of function is closely relted to the concept of function s it. We my formlise this s follows. Defining f s ove, if 0 is n ccumultion point of X, then f is sid to e continuous t 0 iff f() = f( 0 ).

7 7 If 0 X ut 0 is not n ccumultion point (i.e., it is n isolted point), then f is utomticlly continuous t 0. Note tht this reltionship cn e used in two wys: if we know the it, then we cn test for continuity; on the other hnd, if we know tht function is continuous, then the reltionship provides n esy wy to find the it. 0 X. 2h) The function f: X Y is sid to e continuous iff it is continuous t every point 2i) As efore, consider function f: X Y, where X R n nd Y R m. Let Y i e the set of possile vlues of the ith component of vector elonging to Y, i.e., Y i = { y R y Y, y = y i }. Then f is continuous iff its component functions f i : X Y i re continuous for i = 1... m. ssume: Then 2j) Let f nd g e e two functions from X to Y, where X R n nd Y R m, nd f() = nd g() =. Further, if m = 1, then (f + g)() = + (f g)() = (f g)() =. (f/g)() = / for 0 min{f, g}() = min{, } m{f, g}() = m{, }. 2k) Let f nd g e s defined ove. If f nd g re oth continuous, then the functions formed y tking the sum, difference, inner product, nd (for m = 1) quotient, minimum nd mimum of these functions re lso continuous on the domin for which the new function is defined (in the quotient cse, this domin will eclude ny points t which the function in the denomintor is zero). 2l) Given f: X Y nd g: V Z, where f(x) V, we my define the composite function, h: X Z, where ( ) h() g f() for ll X, e.g., if f() = 2 nd g(y) = y 2, then ( ) h() = g f() = (2) 2. If f nd g re oth continuous functions, then the composite function h is lso continuous.

8 8 3. Continuity nd Sets 3) Let f: R n R m e function from R n to R m. Then f is continuous iff, for every open set V R m, the inverse imge f 1 (V ) is n open set. Similrly, f is continuous iff, for every closed set Y R m, the inverse imge f 1 (Y ) is closed set. 3) Let f: R n R m e function from R n to R m. If f is continuous nd X R n is compct, then f(x) is compct. 3c) Intermedite Vlue Theorem. Let f e rel sclr-vlued function which is continuous t ech point on the closed intervl [, ]. Choose two ritrry points 1 < 2 in [, ] such tht f( 1 ) f( 2 ). Then f tkes on every vlue etween f( 1 ) nd f( 2 ) somewhere on the intervl ( 1, 2 ). 3d) Weierstrss s Theorem. Let S e compct suset of R n nd let f e rel sclr-vlued function which is continuous t ech point on S. Then f hs oth minimum nd mimum vlue on S, i.e., there eist nd elonging to S such tht f( ) f() f( ) for ll S. 4. Homogeneous nd Homothetic Functions 4) A set X R n is cone iff X implies tht t X for ll t R ++. 4) A function f: X I (where X R n, X is cone nd I R) is homogeneous of degree k iff f(t 1, t 2,..., t n ) = t k f( 1, 2,..., n ) for ll X nd t R ++. 4c) Let I, J R. Then function g: I J is monotonic trnsformtion iff g is strictly incresing function, i.e., iff > y implies g() > g(y). 4d) A function is homothetic iff it is monotonic trnsformtion of homogeneous function. Formlly, ssume X R n, X is cone nd I, J, K R. Then h: X J is homothetic iff there eists homogenous function f: X I nd monotonic trnsformtion g: K J (where f(i) K) such tht h() = g f() for ll X. This is of interest ecuse monotonic trnsformtion of function hs the sme level curves (indifference curves, isoqunts or whtever) s the originl function. 5. Conve Sets 5) A set C R n is conve iff for every C nd y C, the vector z = λ+(1 λ)y lso elongs to C for every λ stisfying 0 < λ < 1. Geometriclly, this mens tht, for every pir of elements of C, stright line etween the points lies wholly within C.

9 9 An epression of the form λ + (1 λ)y, where 0 λ 1, is known s conve comintion of nd y. Note. A set which is not conve is simply clled non-conve. The opposite of conve set is not concve set: there is no such thing s concve set. 5) By convention, the empty set is conve. 5c) The intersection of finite or infinite numer of conve sets is lso conve. 5d) The sum of finite numer of conve sets is conve set. 5e) Any sclr multiple of conve set is conve set. 5f) The crtesin product of finite numer of conve sets is conve set. 5g) The union of two (or more) conve sets is not necessrily conve. 5h) A conve set C is conve cone iff, in ddition to stisfying the requirements for conve set, it hs the property tht C implies tht t C for ll t > 0. A set C is strictly conve iff, for every C nd y C, where y, the element z = λ + (1 λ)y elongs to interior of C for ll λ stisfying 0 < λ < Concve/Conve nd Qusi-Concve/Qusi-Conve Functions 6) Definitions of the four function types re given elow. These definitions ll depend upon the domins of the functions eing conve sets (if the domins re not conve, then the definitions cnnot e pplied). Accordingly, it is to e understood in ll of the definitions to follow tht the functions domins re conve sets. 1. A function y = f() with domin X is concve if nd only if for every, X: f(λ + (1 λ) ) λf( ) + (1 λ)f( ) for ll 0 < λ < A function y = f() with domin X is conve if nd only if for every, X: f(λ + (1 λ) ) λf( ) + (1 λ)f( ) for ll 0 < λ < A function y = f() with domin X is qusi-concve if nd only if for every, X: { } f(λ + (1 λ) ) min f( ), f( ) for ll 0 < λ < A function y = f() with domin X is qusi-conve if nd only if for every, X: { } f(λ + (1 λ) ) m f( ), f( ) for ll 0 < λ < 1. If the wek inequlity in the ove definitions is replced y strict inequlity nd if we dd the condition tht, then we get the definition of strict concvity, strict conveity, strict qusi-concvity, nd strict qusi-conveity respectively.

10 10 f ( ) Concve Function f ( λ + (1 λ) ) f ( ) λf ( ) + (1- λ) f( ) f ( ) 0 λ + (1 λ) Figure 2 f ( ) Conve Function f ( ) λf ( ) + (1- λ) f( ) f ( ) f( λ + (1 λ) ) 0 λ + (1 λ) Figure 3

11 11 f ( ) Qusi-Concve Function f ( λ + (1 λ) ) f ( ) min {f ( ), f ( )} = f ( ) 0 λ + (1 λ) Figure 4 f ( ) Qusi-Conve Function m {f ( ), f ( )} = f ( ) f( λ + (1 λ) ) f ( ) 0 λ + (1 λ) Figure 5

12 12 6) Every concve (conve) function is qusi-concve (qusi-conve), ut the converse is not true, i.e., concvity/conveity is stronger (i.e., more restrictive) ssumption thn qusi-concvity/qusi-conveity, e.g., f() = is qusi-concve ut not concve. { 2 for ll 0 0 for ll > 0 6c) If f is (strictly) concve function, then f is (strictly) conve function nd vice vers. Similrly, if f is (strictly) qusi-concve function, then f is (strictly) qusi-conve function nd vice vers. 6d) A liner function is oth concve nd conve. 6e) A monotonic trnsformtion of qusi-concve(conve) function is itself qusiconcve(conve). By contrst, monotonic trnsformtion of concve(conve) function need not e concve(conve). 6f) Any non-negtive liner comintion of concve (conve) functions is concve (conve), i.e., if F () = n λ i f i () where λ i 0 for ll i, i=1 then F () is concve (conve) if f i () is concve (conve) for ll i. By contrst, non-negtive liner comintion of qusi-concve (qusi-conve) functions is not necessrily qusi-concve (qusi-conve), e.g. the two functions f 1 () = { 2 for ll 0 0 for ll > 0 nd f 2 () = { 0 for ll 0 2 for ll > 0 re oth qusi-concve, ut their sum F () = f 1 () + f 2 () = 2 is not qusi-concve. 6g) If f is concve/conve function, then it is continuous on the interior of its domin. 7. The Reltionship Between Function Type nd Conve Sets Another wy of chrcterising function s type i.e., whether it is concve, conve, qusi-concve or qusi-conve involves the use of conve sets. This cn e little confusing since people tend to feel tht conve functions should e relted to conve sets nd concve functions should e relted to concve sets. However, since there is no such thing s concve set, this is oviously impossile. Insted, oth conve nd concve functions (nd the corresponding qusi- functions) re defined in terms of conve sets.

13 13 Specificlly, function f() is conve/concve/qusi-conve/qusi-concve if nd only if prticulr set defined in terms of f() is conve. Four differently defined sets re used in this test one for ech function type. These set definitions re given elow. It is to e understood in ech cse tht the function hs conve domin, denoted y X. 1. A function f() is concve if nd only if the set T 1 = { (, y ) X, y f() } is conve. 2. A function f() is conve if nd only if the set T 2 = { (, y ) X, y f() } is conve. 3. A function f() is qusi-concve if nd only if the set T 3 = { X, f() k } is conve for ech rel numer k. 4. A function f() is qusi-conve if nd only if the set T 4 = { X, f() k } is conve for ech rel numer k. Note. While it is possile to relte strict concvity etc. to strictly conve sets, it is rther tricky nd is therefore omitted here.