Natural Resource Economics

Transcription

1 Naura Rsourc Economics Acamic ar: Prof. Luca Savaici Lsson 14: Opima conro sufficin coniions Naura Rsourc Economics - Luca Savaici

2 FOCs Saic probm: Dnamic probm: max x, f x, V F,, x h x, 0 0. L f x,, x. h Max x0 x T A fr H, x,, F,, x -, x, Naura Rsourc Economics - Luca Savaici

3 FOCs Laranian rivaion: Hamionian rivaion: n boh cass, w h sam souion H x x x F x H F H 0 0,,. F L x x L Naura Rsourc Economics - Luca Savaici L x F x δ λ x ሶ λ 0

4 FOCs 1 maximum princip 2 3 canonica quaions 3 sa quaion 2 cosa quaion xprssin h prciaion of h sock -/ as h sum of h conribuions of h sa variab o h objciv funcion: boh irc F/x an hrouh h vau of h sock chan */x To sov h FOCs ssm w n ihr o know h iniia an fina vau of h sa variab or o us h ransvrsai coniion. Th souion of h ssm of iffrnia quaions is rprsn b: x*., *., *. Th anaica propris of h FOCs ssm rmins h shap of h phas iaram Naura Rsourc Economics - Luca Savaici

5 Opima conro: xamp Maximiz profis B T A p Q p Max T k p Q p H k Naura Rsourc Economics - Luca Savaici

6 FOCs B T A H Q p p H k Naura Rsourc Economics - Luca Savaici

7 Sufficin coniions Ncssar vs. sufficin coniions Sufficin coniions: ncssar coniions shap curvaur of h funcions invov in h probm Dnamic sufficin coniions: concavi convxi coniions of h hamionian w.r.. sa an conro variab Manasarian concavi convxi of h opima Hamionian H*. w.r.. sa variab Naura Rsourc Economics - Luca Savaici

8 Princip of opimai Naura Rsourc Economics - Luca Savaici

9 Wha is h vau of h sock? Onc i is assum ha h cision makr bhavs opima, h vau of a projc pns on on h sarin vau of h sa variab, x0. For his rason, h im pah of h cision variab is no an arumn in h opima vau funcion: V*[x*] max V[x,] Th assumpion ha h cision makr bhavs opima ransforms h funciona V ino h funcion V* n h jaron of h opima conro iraur, h im pah of h cision variab has bn 'maximiz ou'. Th opima vau funcion shows h raionship bwn h vau of a firm an is phsica capia sock. Diffrniain h opima vau funcion is h vau of a sin capia uni: λ 0 V X 0 X 0 Th shaow pric of capia, λ0, masurs h ffc of an xra capia uni on h vau of a firm. Naura Rsourc Economics - Luca Savaici

10 Anohr ook a h FOCs A an im, h vau of a firm quas h shaow pric of capia muipi b h firm s capia sock: λ * x. A ach poin in im, h proucion cision chans h vau of h firm b λ x λx ሶ λx ሶ Th firm s vau chans bcaus h amoun of phsica capia chans, an h vau of his capia chans. L f x, b h n rvnu a im : h cision makr chooss h im pah of oupu ha maximizs h sum of h momnar n rvnu an h chan in h vau of h firm a vr insan f x, λx ሶ λx ሶ Naura Rsourc Economics - Luca Savaici

11 Anohr ook a h FOCs Th im pah of h sock mus fufi h consrain x ሶ x, his ivs ris o an opima conro probm bcaus h rowh ra of h sa variab pns boh on h conro variab an h sa variab. Subsiuin h namic consrain is: f x, λ x, λx ሶ Paria iffrniaion proucs h wo coniions ha h opima im pahs of h cision variab an sa variab mus fufi: f λ 0 Th maximum princip sas ha a ach insan aon h opima im pah, h ffc of a sma chan in oupu xracion/harvs on n rvnu mus qua h marina usr cos of naura capia: i is worhwhi o rais oupu uni h incras in n rvnu is counrbaanc b h usr cos from usin h sock mor innsiv. f x λ x λ ሶ 0 Th oa bnfi of an xra capia uni is h sum of h immia incras in n rvnu an h chan in h vau of h capia; h cos of hoin capia is h fa in is shaow pric. Thus, h cosa quaion shows ha aon h opima im pah of capia h firm mainains a sufficin capia sock so ha h oa marina bnfi of capia quas is marina cos. Naura Rsourc Economics - Luca Savaici

12 Trasvrsai Opima fina sock Rvan for rnwab rsourcs Examp: pamn accorin o h v of sock a h n of h prio nsa of XT a T 0 Raiona: if h shaow pric of h sock was posiiv, h v cou no b opima b finiion BXT sava vau f B. is posiiv, how os ransvrsai coniion chan? B T X T Opima prio of uraion: Rvan for non-rnwab rsourcs H*x*,*,*,T 0 >F*T,*,x* - * *T,x*,* Naura Rsourc Economics - Luca Savaici