ECON 381 SC ASSIGNMENT 2

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1 ECON 8 SC ASSIGNMENT 2 JOHN HILLAS UNIVERSITY OF AUCKLAND Problem Consider a consmer with wealth w who consmes two goods which we shall call goods and 2 Let the amont of good l that the consmer consmes be x l and the price of good l be p l Sppose that the consmer s preferences are described by the tility fnction ) ) 2) ) Set p the tility maximisation problem and write down the Lagrangian The maximisation problem is And the Lagrangian is max x 2 sbject to + w L λ w) + λw ) 2) Write down the first der necessary conditions f an interi maximm The first der conditions are λ 0 ) λ 0 ) λ w 0 ) Solve the first der conditions to obtain the Marshallian ncompensated) demand fnctions There are many ways to solve these eqations I ll give one From eqations and 2 we obtain 4) 5) λ ) λ λ ) x λ Sbstitting eqations 4 and 5 into eqation we obtain w + ) x λ 0 6) λ x 2 w Date: Second Semester 2002

2 2 JOHN HILLAS UNIVERSITY OF AUCKLAND and sbstitting sbstitting this vale back into into 4 and 5 we obtain w w )w )w w )w Ths the Marshallian demands are w) w/ and w) )w/ Now sppose that there are three goods which we shall call with eqal lack of imagination goods 2 and Let the amont of good l that the consmer consmes be x l and the price of good l be p l Sppose that the consmer s preferences are described by the tility fnction where + + ) x2 x 4) Again set p the tility maximisation problem and write down the Lagrangian write down the first der necessary conditions f an interi maximm and solve the first der conditions to obtain the Marshallian ncompensated) demand fnctions The idea was that yo se the two good case as a model to solve the three good problem The problem is not really that mch me difficlt It is a bit me difficlt to keep the notation straight if yo jmp straight into the three good problem In this case the maximisation problem is max x2 x And the Lagrangian is sbject to + + p w L λ w) x2 x + λw p p ) The first der conditions and the soltion follow exactly the model of the two good case The first der conditions are 7) 8) 9) 0) x λ p 0 x2 λ p2 0 x2 x λ p 0 λ w p 0

3 ECON 8 SC ASSIGNMENT 2 From eqations 7) 8) and 9) we obtain x2 x λ p x2 x λ p2 x2 x λ p ) 2) ) λ x2 x p λ x2 x p2 λ x2 2x p Sbstitting eqations ) 2) and ) into eqation 0) we obtain w + + ) x2 x λ 0 4) λ w and sbstitting this vale back into eqations ) 2) and ) we obtain x2 x w x2 x w x2 x w p w w w p Ths the Marshallian demands are w) w/ w) w/ and w) w/p 5) Consider the expenditre minimisation problem min + + p sbject to: x2 x Write down the Lagrangian f this problem The Lagrangian is L λ ) p + λ x2 x ) 6) Write down the first der necessary conditions f an interi minimm

4 4 JOHN HILLAS UNIVERSITY OF AUCKLAND 5) 6) 7) The first der conditions are λ x 0 λ x2 0 p λ x2 x 0 8) λ x2 2x 0 7) Solve the first der conditions to obtain the Hicksian compensated) demand fnctions From eqations 5) 6) and 7) we obtain 9) 20) 2) and from eqation 8) λ x2 x λ x2 x p λ x2 x 22) x2 x 2) 24) 25) Sbstitting eqation 22) into eqations 9) 20) and 2) gives λ λ p λ λ λ λ p Sbstitting these back into eqation 22) we obtain λ) + + p p p2 p λ And sbstitting this back into eqations 2) 24) and 25) we obtain 26) 27) 28) p p2 p h ) p p2 p h 2 ) p p2 h ) ) p

5 ECON 8 SC ASSIGNMENT 2 5 Problem 2 Take the Hicksian demand fnctions that yo fond in the first problem which we shall label h l ) and sbstitte them back into the objective fnction + + p to obtain the expenditre fnction e ) h ) + h 2 ) + p h ) ) Differentiate the fnction e that yo ve fond with respect to and confirm that this vale is eqal to the vale f h ) that yo fond in the previos problem First we find e ) e ) Ths e l p l l p l l ) ll p ) p p p2 p p p2 p l p p2 p p p2 p p2 p p which is what we fond in the previos qestion 2) Sbstitte v w) f and se the dal relationship p2 p e v w)) w to find the indirect tility fnction v w) F or particlar expenditre fnction the previos eqation is p p2 p v w) w which we solve to find v w) w p ) Rewrite the indirect tility fnction as a fnction ṽ of and where q i p i /w First we note that w w w + + w w w Ths v w) w w w ) w ) w w ṽ ) ) p p

6 6 JOHN HILLAS UNIVERSITY OF AUCKLAND 4) Consider the problem min ṽq ) sbject to: + + and write ot the Lagrangian and the first der conditions f minimisation The Lagrangian is L λ ) λ q x + + ) 29) 0) ) The first der conditions are ) + + λx 0 ) + + λx2 0 ) + + λx 0 2) λ ) Solve the first der conditions to find q q2 and q as fnctions of x x2 and Sbstitte these vales back into ṽ ) to obtain a tility fnction giving tility as a fnction of and and confirm that this is the tility fnction that yo started with in part 4) of the previos problem From eqations 29) 0) and ) we obtain ) 4) 5) λ λ λ and sbstitting these into eqation 2) we obtain + + λ 6) λ Sbstitting eqation 6) back into eqations ) 4) and 5) and solving we obtain ) 7) ) x ) ) 8) 9) ) x ) ) 2 ) x ) )

7 ECON 8 SC ASSIGNMENT 2 7 Now sbstitting eqations 7) 8) and 9) back into the fmla f ṽ we obtain the minimised tility fnction ũ ) ṽ ) ) )) / / / x x 2 x which is the tility fnction that we started with