If there are k binding constraints at x then re-label these constraints so that they are the first k constraints.

Transcription

1 Mathematcal Foundatons -1- Constaned Optmzaton Constaned Optmzaton Ma{ f ( ) X} whee X {, h ( ), 1,, m} Necessay condtons fo to be a soluton to ths mamzaton poblem Mathematcally, f ag Ma{ f ( ) X}, then what condtons must be satsfed? Constant Qualfcatons If thee ae k bndng constants at then e-label these constants so that they ae the fst k constants The constant qualfcatons holds at X f h () the assocated gadent vecto ( ), 1,, k h () X {, ( ) ( ), 1,, k} has a non-empty nteo (ths s the set of non-negatve vectos satsfyng the lneazed bndng constants)

2 Mathematcal Foundatons -- Constaned Optmzaton Defne the Lagangan L (, ) f ( ) h( ) Poposton: Fst Ode Condtons fo a Constaned Mamum Suppose solves Ma{ f ( ) X} whee X {, h( ) } If the constant qualfcatons hold at then thee ests a vecto of shadow pces such that and L (, ), 1,, n wth equalty f L (, ), 1,, m wth equalty f Kuhn-Tucke condtons Restatement of FOC () L f h and L, 1,, n L L () h( ) and, 1,, m

3 Mathematcal Foundatons -3- Constaned Optmzaton In vecto notaton () L f h and L L L () h ( ) and

4 Mathematcal Foundatons -4- Constaned Optmzaton f Case (): ( ) Then the Kuhn-Tucke condtons hold at f Case (): ( ) Bndng constants Suppose that k constants ae bndng at Re-label constants to that these ae the fst k constants Fo k set Replace each bndng constant by ts tangent hypeplane at h h ( ), 1,, k eplaced by ( ) ( ) h To be a constant we eque that the gadent vecto ( )

5 Mathematcal Foundatons -5- Constaned Optmzaton The lneazed feasble set s then h X {, ( ) ( ), 1,, k} Uppe contou set of f f Replace U {, f ( ) f ( )} by U { ( ) ( ) } f Poposton: If the Constant Qualfcatons hold at and ( ), then s a soluton to the lneazed poblem: f Ma{ ( ) X} h whee X {, ( ) ( ), 1,, k}

6 Mathematcal Foundatons -6- Constaned Optmzaton We appeal epeatedly to the followng esult 1 f Lemma: Suppose that f( ) and that Then fo (,1) and suffcently small, les n the nteo of U { f ( ) f ( )}, whee (1 ) les n the nteo of the set U { ( ) ( ) }

7 Mathematcal Foundatons -7- Constaned Optmzaton Consde the -th bndng constant h ( ) h ( ) The lneazed constant s h ( ) ( ) Consde any vecto ˆ satsfyng all the lneazed constants, that s h ( ) ( ˆ ), 1,, k By the constant qualfcaton, thee s some such that h ( ) ( ) Fo any conve combnatons of these two vectos ˆ ˆ (1 ) h ˆ h ( ) ( ) ( ) ((1 ) ) ˆ h h ˆ (1 ) ( ) ( ) ( ) ( )

8 Mathematcal Foundatons -8- Constaned Optmzaton Net consde conve combnatons of ˆ and, that s (1 ) ˆ h Agung as above ( ) ( ) Appealng to the Lemma, fo all suffcently small les n the nteo of U { h ( ) h ( )} Repeatng ths agument fo each bndng constant fo all suffcently small les n the nteo of {, h ( ) h ( )} Fo non-bndng constants ths must also be tue f s suffcently small Thus fo all suffcently small, les n the nteo of X {, h ( ) h ( ), 1,, m} Suppose that the Poposton s false Then fo some ˆ X t must be the case that f f f ( ) ˆ ( ), that s ( ) ( ˆ )

9 Mathematcal Foundatons -9- Constaned Optmzaton We defned the conve combnaton ˆ (1 ) ˆ Then f ( ) ( ˆ ) fo suffcently close to zeo Agung as above, t follows that fo suffcently close to zeo and all (,1) f ( ) ( ) Appealng to the Lemma once moe, fo all suffcently small f ( ) f ( ) But we have aleady shown that X But ths cannot be tue snce solves Ma{ f ( ) X} Then the Poposton s tue QED

10 Mathematcal Foundatons -1- Constaned Optmzaton Gven ths esult, ag Ma{ f ( ) X} whee X {, h ( ), 1,, m} only f f h ag Ma{ ( ) X} whee X {, ( ) ( ), 1,, m} Thus we can obtan necessay condtons by consdeng the lneazed poblem Snce t s a lnea optmzaton poblem (lnea pogammng poblem) and hence concave, we can to appeal to the sepaatng hypeplane theoem to pove estence of a vecto of shadow pces (See EM Chapte1)

11 Mathematcal Foundatons -11- Constaned Optmzaton Eample 1: Constant qualfcaton holds Ma{ f ( ) ln, h( ) } 1 1 As s eadly confmed, the mamzng value of s (1,1) The feasble set and contou set fo f though (1,1) ae depcted The Lagangan s L (, ) ln 1 ln ( 1 ) The fst ode condtons ae theefoe Fg 1-1: Constaned mamum () L 1, wth equalty f >, =1, L () 1, wth equalty f > As s eadly checked, the necessay condtons ae all satsfed at (, ) (1,1,1)

12 Mathematcal Foundatons -1- Constaned Optmzaton Eample : Constant qualfcaton does not hold Ma f h 3 { ( ) ln 1 ln, ( ) ( 1 ) } Snce the feasble set and mamand ae eactly the same as n eample 1, the soluton s agan (1,1) The Lagangan s 3 L (, ) ln 1 ln ( 1 ) Fg 1-1: Constaned mamum Dffeentatng by, L 1 3 ( 1 ) 1 at (1,1)

13 Mathematcal Foundatons -13- Constaned Optmzaton Thus the fst ode condton does not hold at the mamum Ou ntutve agument beaks down because h, the gadent of the constant functon, s zeo at the mamum Thus the patal devatves of h no longe eflect the oppotunty cost of the scace esouce Moe fomally, at the mamum, the lnea appomaton of the constant s L h h ( ) h( ) ( ) ( ) Then, as long as the constant s bndng ( h ( ) ) and the gadent vecto s not zeo, the lneazed constant s h ( ) ( ), 1,, m

14 Mathematcal Foundatons -14- Constaned Optmzaton Thee s a second (though unlkely) stuaton n whch the lnea appomatons fal Consde the followng poblem Ma{ f ( ) 1 h( ) ( ), } The feasble set s the shaded egon n the fgue Also depcted s the contou set fo f though (,) Fom the fgue t s clea that f takes on ts mamum at (,) Fg 1-a: Ognal poblem Howeve, L ( ) 1 3 ( 1 ) 1 1 Thus agan the fst ode condtons do not hold at the mamum

15 Mathematcal Foundatons -15- Constaned Optmzaton Ths tme the poblem occus because the feasble set, afte takng a lnea appomaton of the constant functon, looks nothng lke the ognal feasble set h At, the gadent vecto ( ) (, 1) Thus the lnea appomaton of the constant h ( ) though s Lneazed feasble set Fg 1-b: Lneazed poblem h h h ( ) ( ) ( )( ) ( ) 1 1 Snce must be non-negatve, the only feasble value of s In Fgue 1-b the lneazed feasble set s theefoe the hozontal as Then the soluton to the lneazed poblem s not the soluton to the ognal poblem

16 Mathematcal Foundatons -16- Constaned Optmzaton Poposton: Suffcent Condtons fo a Mamum Suppose that f and h, 1,, m ae quas-concave and the feasble set X {, h( ) } has a non-empty nteo h ( ) f ( ) f ( ) f If the Kuhn-Tucke condtons hold at, ( ) h and fo each bndng constant, ( ), X then solves Ma{ f ( ), h ( ), 1,, m} h( ) 1 1 Intuton: Unde these assumptons the feasble set s conve and hence the lneazed feasble set X contans the ognal feasble set X Then t s suffcent to show that X has a non-empty nteo

17 Mathematcal Foundatons -17- Constaned Optmzaton Techncal Remak: Satsfyng the second constant qualfcaton Poposton: Suppose that we e-label the constants so that t s the fst I that ae bndng at, that s h I Suppose also that thee ae J vaables fo whch the non-negatvty constant ( ), 1,, s bndng Then consde the I lneazed bndng constants and J bndng non-negatvty constants Ths s a system of I J lnea constants the second constant qualfcaton holds at A ( ) If the ows of A ae lnealy ndependent then As a pactcal matte, ths esult s almost neve employed n economc analyss The eason s that economsts typcally make assumptons that mply that the feasble set X s conve and has a nonempty nteo As we shall see, when these assumptons hold, checkng the fst constant qualfcaton s enough Poof: Suppose that at thee ae I bndng constants We e-label these so that h I and ( ), 1,, h I ( ), The lneazed feasble set s theefoe h {, ( ) ( ), 1,, } X I

18 Mathematcal Foundatons -18- Constaned Optmzaton Suppose that thee ae J vaables fo whch, I 1,, I J We e-label these vaables so that Thus thee ae K I J bndng constants n the lneazed feasble set Suppose we choose K (1), Then the K bndng constants can be e-wtten as a system of K lnea equatons n K unknowns h1 h1 h1 h1 ( ) ( ) ( ) ( ) 1 I I 1 K 1 1 hi hi hi h I I I ( ) ( ) ( ) ( ) I I I 1 K I1 I K K 1 () Choose b, k 1,, K and consde the followng lnea equaton system k

19 Mathematcal Foundatons -19- Constaned Optmzaton h h h h ( ) ( ) ( ) ( ) b h h h h b b 1 1 b K K I I 1 n I I I I I I I ( ) ( ) ( ) ( ) I I I 1 I I 1 I 1 I 1 K (3) If the columns (o ows) of the mat ae ndependent, then the mat s nvetble Then fo each b thee s a unque soluton ( b), k 1,, K Fo k K defne Consde any b Fom (3) t follows that h b b I k () b n ( )( ( ) ), 1,, } and ( ), 1,, 1 k k b b I K Remembe that b k K Also (), k 1,, I k( ) k, k Then fo b and suffcently small ( b), k 1,, I Thus fo b and suffcently small h b I n ( )( ( ) ), 1,, and ( ) 1 k b Thus X has a non-empty nteo QED

20 Mathematcal Foundatons -- Constaned Optmzaton Eecses 1 Utlty mamzaton u( ) 1 4 The pce vecto s p (1,1) 1 1 Solve fo the utlty mamzng consumpton vecto f () I 6 () I 1 Utlty mamzaton u( ) 1ln 4 The pce vecto s p (1,1) 1 (a) Solve fo the utlty mamzng consumpton vecto f I 7 (b) Fo what ncome levels (f any) s consumpton of commodty equal to zeo? 3 Cost mnmzaton The mamum output that a fm can poduce wth nput vecto z ( z1, z) s mappng s called a poducton functon) q ( z z ) (Ths 1/ 1 (a) Confm that the constant qualfcatons hold fo any feasble nput vecto (b) If the nput pce vecto s, solve fo the cost mnmzng nputs and hence the mnmum cost unde the assumpton that q ( 1/ ) (c) Show that magnal cost s an nceasng functon fo q ( 1/ )

21 Mathematcal Foundatons -1- Constaned Optmzaton (d) Show also that an ncease n one of the two nput pces has no effect on magnal cost Do you fnd ths puzzlng? (e) Solve also fo mnmzed cost f q ( 1/ ) (f) Is magnal cost MC( q ) contnuous on? 4 Peak and off-peak pcng An electc powe company dvded the day nto thee equal peods The demand pces n the thee peods ae as follows p a q, p q, p q The unt cost of poducng electcty n each peod s 1 If the capacty of the fm s q (so that the mamum output n each peod s q) the nteest cost pe day s 4q That s, the unt cost of capacty s 4 (a) If a1 14 solve fo the poft mamzng outputs and pces n each peod (b) Solve agan f a1 16 (c) Retunng to the fst case, show that the total (e goss) beneft of the poducton plan ( q1, q, q 3) s B 14q q q q q q (d) Show that socal suplus (SS = B-C) s mamzed wth a plan twce as lage as the poft mamzng plan

22 Mathematcal Foundatons -- Constaned Optmzaton Answe to Eecse 3 The poducton functon q F() z s the mamum output that can be poduced usng nput vecto z Thus the set Z { z F( z) q } s the set of feasble nput vectos As depcted, cost s mnmzed at z If you check the slopes at A and B you wll see why one of these ponts wll be cost mnmzng fo some ange of nput pce atos whle the othe s neve cost mnmzng Note that f q = the cost mnmzng nput vecto s z = Hencefoth we consde q > Then fo feasblty, z Defne h( z) ( z z ) q Then 1/ 1 h 1/ 1/ 1/ (( z1 z ),( z1 z ) z ) z

23 Mathematcal Foundatons -3- Constaned Optmzaton Constant Qualfcatons Note that h ( z ) z 1 f z > Then the fst CQ s satsfed Note net that z z s concave snce t s the sum of two concave functons Snce 1/ 1 y s stctly nceasng on t follows that 1/ h( z) ( z z ) q s stctly quas-concave Then we olny need to 1 check that the feasble set Z { z F( z) q } has a non-empty nteo Note that f 1/ z ( q, q) then h( z) ( z z ) q 8q Thus the nteo of Z s ndeed non-empty 1/ 1 Mnmzaton poblem Mn{ z z ( z z ) q } z 1/ Note that f the constant s not bndng, then cost can be educed by educng z Thus at z the constant must be bndng Mamzaton poblem Ma{ z z ( z z ) q } z 1/ 1 1 1

24 Mathematcal Foundatons -4- Constaned Optmzaton The Lagangan fo ths optmzaton poblem s L z z (( z z ) q) 1/ Kuhn-Tucke condtons L 1/ 1 ( z 1 z ) z 1 wth equalty f z1 (1) L z ( z z ) z 1/ 1/ 1 wth equalty f z () L 1/ ( z 1 z ) q wth equalty f (3) Snce q, z If z1 (1) s an equalty and so If z () s an equalty and so agan

25 Mathematcal Foundatons -5- Constaned Optmzaton We can ewte (1) and () as follows: ( z z ) wth equalty f z1 (1) 1/ 1 1 ( z z ) z wth equalty f z () 1/ 1/ 1 Case () Ty z Then both (1) and () ae equaltes and so z 1/ 1 Then z and so 1/ 1 z 1 ( ) Fom (3) ( z z ) q Theefoe 1/ 1 z 1 1/ 1 q ( ) Note that snce we assumed that z1, q ( ), that s 1/ 1 q 1 4( ) Total cost s C( q) z 1 q 1/ 1

26 Mathematcal Foundatons -6- Constaned Optmzaton Case () Ty z1 Then fom (3) 4z Then q C( q) q 4 z fo q 1 4( ) Note: Fo a complete soluton you should check that equalty (1) and nequalty () ae both satsfed Magnal cost MC MC dc 1, q 4( ) dq 4 dc 1, q 4( ) dq q 1 1 1/ If you check you wll fnd that MC s contnuous