. The firm makes different types of furniture. Let x ( x1,..., x n. If the firm produces nothing it rents out the entire space and so has a profit of

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1 Joh Riley F Maimizatio with a sigle costrait F3 The Ecoomic approach - - shadow prices Suppose that a firm has a log term retal of uits of factory space The firm ca ret additioal space at a retal rate of per uit or ret out some of the uits at this same rate If the firm produces othig it rets out the etire space ad so has a profit of The firm makes differet types of furiture Let ( 1,, ) be the quatities of each type the profit of the firm from furiture sales is f( ) The space required i the productio of is g ( ) Addig the reveue from retig out some of the space (or the cost of retig additioal space, the profit of the firm is L (, ) f ( ) ( bˆ g( )) For every shadow price 0 we ca solve the followig profit maimizatio problem Ma{ L (, ) f ( ) g( )} 0 Let ( ) be a solutio of the maimizatio problem The total demad for space is b ( ) g( ( )) Cosider furiture type If 0, the the margial profit must be zero If 0, the the margial profit caot be positive Therefore we have the followig ecessary coditios for a maimum L f g (, ) ( ) ( ) 0 with equality if 0 (11) The firm s demad for space is the mappig from the shadow price to demad b ( ) If we put b o the horizotal ais ad o the vertical price, the graph is of the demad price

2 fuctio This is the price that clears the market for ay supply of space Suppose that for all b there is some demad price () b 1 Figure 3-1: Demad price fuctio for The for ay there is a demad price firm chooses the productio vector b () ˆ ˆ Thus, with this market price, the profit maimizig ad uses the leased space b ˆ g( ( b ˆ )) Cosider ay that satisfies the space costrait g() bˆ Sice is profit maimizig, it follows that f ( ) ˆ ( bˆ g( )) f ( ˆ) ˆ ( bˆ g( ˆ)) Rearragig this iequality, f ( ) f ( ˆ) ˆ ( g( ) g( ˆ)) For ay satisfyig the space costrait, the right had side is egative Therefore f ( ) f ( ˆ ) 0 1 This will ecessarily be the case if the profit fuctio is cocave 2

3 We have therefore proved the followig result Propositio: Solutio to the costraied maimizatio problem Ma{ f ( ) bˆ g( ) 0} 0 Let ( ) solve the profit maimizatio problem whe it is possible to buy or sell the iput, ie Ma{ L (, ) f ( ) ( bˆ g( ))} 0 ad let b( ) g( ( )) be the implied demad for the iput If there is a shadow price that b( ˆ ) bˆ, the ( ˆ ) solves the costraied maimizatio problem ˆ such Remark: College courses o calculus, such as at UCLA, do ot cover maimizatio problems with o-egativity ad iequality costraits If a College level tet book cosiders a costraied maimizatio problem, the costrait holds with equality ad is ay vector of real umbers, ie the problem aalyzed is the closely related but simpler optimizatio problem Ma{ f ( ) bˆ g( ) 0} We have iterpreted L(, ) f ( ) ( bˆ g( )) as the profit of a firm To a mathematicia, this is the Lagragia of the costraied maimizatio problem The price (or shadow price), is called the Lagrage multiplier Eample 1: Divisio of a firm with a budget costrait A firm ca produce output q f () z usig the iput vector z The iput price vector is r The divisio maager is give b dollars ad istructed to maimize output, q f () z without violatig the budget costrait r z r ˆ z b 1 Eample 2: The cost fuctio of the firm A firm ca produce output q f () z usig the iput vector z The iput price vector is r The 3

4 divisio maager is asked to solve for the miimum cost of producig ay output ˆq, give that the iput price vector is r Remark: Suppose that you have already solved the maimum output problem for some productio fuctio f( ) Let z * ( b, r) be the output maimizig iputs so that q b * ( ) f ( ( b, r)) This mappig is depicted below Figure 3-2: Maimum output with a fied budget Thus for ay ˆq we ca choose so that qˆ q() bˆ Thus the miimum cost is o more tha Sice q * ( b ) is strictly icreasig, it follows that for ay bˆ bˆ, the maimum output is strictly less the ˆq Thus the miimized cost is Havig solved for the mappig qˆ q() bˆ, the miimized total cost is the iverse of this fuctio 1 C( r, qˆ) q ( qˆ) 4

5 Eample 3: Utility maimizatio with Cobb-Douglas prefereces Prefereces are represeted by the followig strictly icreasig fuctio U( ) l 1 i i The budget costrait is 1 p p I Sice utility is strictly icreasig the solutio equality The Lagragia of the problem is must satisfy the budget costrait with L (, ) l ( I p ) 1 1 FOC L (, ) p 0, 1,,, with equality if 0 (12) Sice p I, there is some k for which k 0 Therefore Therefore L k (, ) pk 0 k k L p k (, ) 0, 1,, p k k Note that U L as 0 so (, ) is strictly positive for sufficietly small Thus there is o corer solutio where oe or more compoets of the solutio is zero Hece 5

6 p 1 (13) 1 1 p Therefore p (14) Substitutig ito the budget costrait, p I Therefore 1 I `1 Substitutig for i (14), p I `1 Remark: if you ca see how to prove the followig rule for 2, the you might like to use it Ratio Rule: Equal ratios are also equal to the ratio of sums of the umerators ad deomiators a If r, 1,, b ad b 0 the a b r From (13) ad the Ratio Rule 6

7 p 1 p p I Oe ca the solve immediately for the demad fuctios 7

EVEN NUMBERED EXERCISES IN CHAPTER 4

Joh Riley 7 July EVEN NUMBERED EXERCISES IN CHAPTER 4 SECTION 4 Exercise 4-: Cost Fuctio of a Cobb-Douglas firm What is the cost fuctio of a firm with a Cobb-Douglas productio fuctio? Rather tha miimie