▁ ▁Opt im ality ▁properties ▁for ▁an ▁entire ▁market ▁system ▁may ▁be ▁stated ▁in ▁mathematical ▁terms , ▁as ▁in ▁form ulation ▁of ▁the ▁two ▁fundamental ▁the or ems ▁of ▁w elf are ▁econom ics ▁and ▁in ▁the ▁Ar row – Deb reu ▁model ▁of ▁general ▁equilibrium ▁( also ▁discussed ▁below ). ▁More ▁con cret ely , ▁many ▁problems ▁are ▁am enable ▁to ▁analyt ical ▁( form ula ic ) ▁solution . ▁Many ▁others ▁may ▁be ▁sufficiently ▁complex ▁to ▁require ▁numerical ▁methods ▁of ▁solution , ▁a ided ▁by ▁software . ▁Still ▁others ▁are ▁complex ▁but ▁tract able ▁enough ▁to ▁allow ▁comput able ▁methods ▁of ▁solution , ▁in ▁particular ▁comput able ▁general ▁equilibrium ▁models ▁for ▁the ▁entire ▁economy . ▁ ▁Linear ▁and ▁non linear ▁programming ▁have ▁prof ound ly ▁affected ▁micro e conom ics , ▁which ▁had ▁earlier ▁considered ▁only ▁equality ▁constraints . ▁Many ▁of ▁the ▁mathematical ▁econom ists ▁who ▁received ▁Nobel ▁P riz es ▁in ▁Econom ics ▁had ▁conducted ▁notable ▁research ▁using ▁linear ▁programming : ▁Leon id ▁Kant or ov ich , ▁Leon id ▁Hur w icz , ▁T j alling ▁Ko op mans , ▁Kenneth ▁J . ▁Ar row , ▁Robert ▁Dorf man , ▁Paul ▁Samuel son ▁and ▁Robert ▁Sol ow . ▁Both ▁Kant or ov ich ▁and ▁Ko op mans ▁acknowled ged ▁that ▁George ▁B . ▁D ant zig ▁des erved ▁to ▁share ▁their ▁Nobel ▁Prize ▁for ▁linear ▁programming . ▁Econom ists ▁who ▁conducted ▁research ▁in ▁non linear ▁programming ▁also ▁have ▁won ▁the ▁ ▁Nobel ▁prize , ▁not ably ▁R agn ar