to maximize a function

Transcription

1 John Riley F Maimization with a sinle constraint F Constrained Maimization Many models in economics share the ollowin characteristics An economic aent chooses a non-neative bundle constraint o the orm () n bˆ to maimize a unction ( ) The bundle must also satisy a We write this succinctly as ollows: Ma{ ( ) ( ) bˆ } 0 Suppose that solves this problem We set i i and consider the ollowin two- variable problem Ma{ (,, ) (,, ) bˆ } 0 i i i i The solution to this problem is thereore (, ) We use this simpler problem to obtain necessary conditions or a maimum i To minimize notation we re-label commodities i and as commodities and Ma{ (, ) (, ) bˆ } 0 We assume that ( ) is strictly increasin and that the ollowin constraint qualiication holds Constraint qualiication I satisies the constraint () bˆ with equality, then or some, 0 In practice economic models always satisy this constraint qualiication and it is rarely mentioned So you can as well!

2 9 Auust 07 This ensures that or every such there is a neihborin point in the interior o the easible set Since ( ) is strictly increasin, the constraint must be bindin at the maimum Then consider any point ˆ, satisyin the constraint with equality This is depicted below Fiure -: Choice Appealin to the Implicit unction rule, we can compute the marinal rates o substitution or the level set ( ) ( ˆ) uˆ The slope o the raph o the level set is MRS ( ˆ) ( ˆ) / ( ˆ ) By the same arument, the slope o the raph o the level set ( ) ( ˆ) uˆ is ( ˆ ) ( ˆ ) / MRS ( ˆ ) As depicted in Fi -, at ˆ the decision-maker s marinal willinness to substitute or eceeds the easible marinal rate o substitution at ˆ Thereore the decision-maker is strictly better o increasin and decreasin And at ˆ the decision-maker s marinal willinness to substitute or is strictly less then the easible marinal rate o substitution Thereore the decision-maker is strictly better o decreasin and increasin The only bundle at which neither o these statements are true is where

3 9 Auust 07 MRS ( ) MRS ( ) A second possibility is depicted below At all points satisyin the constraint with equality MRS ( ) MRS ( ) Fiure -: Corner solution Thereore the decision-maker s willinness to substitute or strictly eceeds the easible marinal rate o substitution It ollows that as lon as ˆ 0, the aent is strictly better o substitutin more or Then the solution is consumption only o commodity Remark: I one or more o the non-neativity constraints is bindin at the optimum, the solution is called a corner solution Eample : Consumer choice with a budet constraint The consumer s utility unction is u ( ) (3 ) The price vector is p (,0) Income is 0 The budet constraint is thereore p p I Step : Write down the marinal utilities and hence MRS( ) 3

4 9 Auust 07 3, Thereore 3 MRS ( ) / () Step : Check to see i there is a solution 0 For this to be true 3 MRS MRS ( ) ( ) 0 Thereore Also the budet constraint must be satisied with equality Thereore But this violates the non-neativity constraint Consider then the corner I (,0) (0,0) p Alon the budet line the marinal rate o substitution MRS( ) (3 ) / alls as rises and alls It ollows that at or all bundles that satisy the budet constraint with equality, p MRS ( ) 0 0 p Thereore it is always optimal to increase utility maimizin bundle and reduce Thereore depicted below, is the 4

5 9 Auust 07 Fiure -3: Corner solution Eample : Robinson Crusoe s private island Robinson Crusoe is marooned on an island in the South Paciic He can row bananas and coconuts I he uses z acres to produce bananas and z acres o land to produce coconuts, the outputs o bananas and coconuts are A z and / A z / There are ˆb acres o usable land Robinson Crusoe s preerences are represented by the utility unction U ( ) At irst siht this eample looks more complicated than Eample However, with a little work we can reduce it to a simple constrained maimization problem Method : Substitute outputs into the utility unction so that utility becomes a unction o the inputs u A A z z The input allocation z ( z, z) must satisy the constraint / / () z z bˆ Method : Invert the mappin rom input to output as ollows: 5

6 9 Auust 07 A z and z A Then substitute into the input constraint We take the irst approach Note that we can square both sides o each production equation and rewrite them as ollows: A z and A z () Thereore A z z A Robinson Crusoe s consumption bundle must thereore satisy the ollowin constraint ( ) z ˆ z b A A The easible set is elliptical in shape and is depicted below 6

7 9 Auust 07 Appealin to the Implicit unction rule A A MRS ( ) ( ) A A and MRS ( ) Since utility is zero i consumption o at least one commodity is zero and strictly positive or 0 there can be no corner solution Thereore at the optimum, A MRS MRS ( ) ( ) ( ) A Hence A ( ) ( ) ie A A A Hence A MRS MRS (3) ( ) ( ) A Also ( ) ( ) ( ) A A bˆ Thereore ( ) ( ) bˆ A A It ollows that A bˆ / ( ) (4) 7