3 A consumer has a generic utility function, u(x1,22), where du dx, 0 and 0u 0x 0 for i 1,2 The prices of the two goods are p and p2, and the target utility level is (a) Formulate the Lagrangian for the expenditure minimisation problem Find the three first order necessary conditions Denote the decision variables that solve this system as hi (P1, P2 u), h(P1 P2, u), and 0 (p, P2, u) (b) The second order sufficiency condition for this constrained minimisation problem is that the bordered Hessian matrix have a negative determinant What is the bordered Hessian matrix for this problem (c) Substitute the optimal solutions from part(a) into the three first order necessary con ditions Then, partially differentiate each of the three first order necessary conditions with respect to u Solve for Oh ou and Oh ou using Cramer's Rule Does the Hicksian quantity demanded of each good increase as the target utility level, u, is increased, ceteris paribus (d) Substitute the optimal solutions from part (a) into the three first order necessary condi tions Then, partially differentiate each of the three first order necessary conditions with respect to p Solve for ohi api and Oh ap, using Cramer's Rule What happens to the Hicksian quantity demanded of each good as the price of good 1, p, is increased, ceteris paribus You may assume that 0 (P, P2, ) 0 (e) How would your answer to part (d) change if you were presented with the consumer's utility maximisation problem and asked to determine the sign of dri api Explain in tuitively why it is possible to determine the sign of the partial derivative of the Hicksian demand function, but not that of the Marshallian demand function, ai, for an own price change

Question: 3. A consumer has a generic utility function, u(x1,22), where du/dx, >0 and 0u/0x <0 for i=1,2. The prices of the two goods are

3. A consumer has a generic utility function, u(x1,22), where du/dx, >0 and 0u/0x

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a ANS WER The Lag rang ian for the expenditure min im isation problem is L h 1 h 2 u h 1 h 2 p 1 h 1 p 2 h 2 u The three first order necessary conditi... View full answer

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