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

 

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3. A consumer has a generic utility function, u(x1,22), where du/dx, >0 and 0²u/0x² <0 for i=1,2. The prices of the two goods are p₁ and p2, and the target utility level is ū. (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), h₂(P1.P2, u), and 0 (p₁, 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 p₁. Solve for ohi/api and Oh₂/ap, using Cramer's Rule. What happens to the Hicksian quantity demanded of each good as the price of good 1, p₁, is increased, ceteris paribus? You may assume that 0* (P₁, P2, ū) ≥ 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. 3. A consumer has a generic utility function, u(x1,22), where du/dx, >0 and 0²u/0x² <0 for i=1,2. The prices of the two goods are p₁ and p2, and the target utility level is ū. (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), h₂(P1.P2, u), and 0 (p₁, 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 p₁. Solve for ohi/api and Oh₂/ap, using Cramer's Rule. What happens to the Hicksian quantity demanded of each good as the price of good 1, p₁, is increased, ceteris paribus? You may assume that 0* (P₁, P2, ū) ≥ 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.

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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 the full answer