ecg765-oldmidterms2

ecg765-oldmidterms2 - Atsushi Inoue ECG 765 Mathematical...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Sp Midterm Exam ♯ 2 1. (a) Compute the derivative of f ( x ) = ln( x 2 + 4 x + 3) (5 points). (b) Find the first three terms in the Taylor series ap- proximation of f ( x ) = x 1 + x around the point x = 0 (5 points). 2. Consider the following equation F ( x, y, z ) = x 2 + 3 xy + 2 yz + y 2 + z 2 − 11 = 0 . (a) Verify that the assumption of the implicit function theorem is satisfied around the point ( x, y, z ) = (1 , 2 , 0) (5 points). (b) Find ∂z/∂x and ∂z/∂y by the implicit function theorem and evaluate them at the point ( x, y, z ) = (1 , 2 , 0) (5 points). 3. Determine whether each of the following functions is concave, convex, strictly concave, strictly convex, or neither. (a) f ( x ) = exp( x ) (5 points). (b) g ( x, y ) = x 2 − y 2 (5 points). 1 4. Consider f ( x, y ) = x 2 + xy + 2 y 2 + 3 . (a) Compute the gradient of f , i.e., ∇ f ( x, y ) = f x ( x, y ) f y ( x, y ) . (5 points) (b) Compute the Hessian of f , i.e., ∇ 2 f ( x, y ) = f xx ( x, y ) f yx ( x, y ) f xy ( x, y ) f yy ( x, y ) . (5 points) (c) Find (¯ x, ¯ y ) which satisfies ∇ f (¯ x, ¯ y ) = . (5 points) (d) Determine whether f (¯ x, ¯ y ) is a maximum, mini- mum or neither. Explain why. (5 points) 5. Consider the following utility maximization problem: max α ln x + β ln y subject to px + qy = I where α > 0, β > 0, p > 0, q > 0 and I > 0. (a) Write the Lagrangian function (5 points). (b) Write the first order conditions (5 points). 2 (c) Find (¯ x, ¯ y ) which satisfies the first order conditions (5 points). (d) Verify that the (¯ x, ¯ y ) satisfies the second-order con- dition (5 points). 6. Consider the following utility maximization problem: max α ln x + β ln y + γ ln z subject to px + qy + rz ≤ I where α > 0, β > 0, γ > 0, p > 0, q > 0, r > 0 and I > 0. (a) Write the Lagrangian function (5 points). (b) Write the Kuhn Tucker conditions including the nonnegativity constraint on the Lagrange multi- plier and the complementary slackness condition (5 points). (c) Find (¯ x, ¯ y, ¯ z ) which satisfies the Kuhn Tucker con- ditions (5 points). (d) Prove that α ln x + β ln y + γ ln z is concave and that px + qy + rz is convex (5 points). (e) Are the Kuhn Tucker conditions necessary for the optimal solution? Explain why. (5 points) (f) Are the Kuhn Tucker conditions sufficient for the optimal solution? Explain why. (5 points) 3 Atsushi Inoue ECG 765: Mathematical Methods for Economics Sp Midterm Exam ♯ 2 1. (a) i. Find the Hessian matrix of f ( x, y ) = ln( x 2 + y 2 ) . (5 points) ii. Find the Jacobian matrix of g ( x, y ) = exp( x + y ) ( x + y ) 2 x + y ....
View Full Document

This note was uploaded on 10/22/2009 for the course ECG 765 taught by Professor Fackler during the Fall '07 term at N.C. State.

Page1 / 19

ecg765-oldmidterms2 - Atsushi Inoue ECG 765 Mathematical...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online