Page 1 of 5 DEPARTMENT OF MANAGEMENT AND ECONOMICS Royal Military College of Canada ECE 256 - Modelling in Economics Instructor: Lenin Arango-Castillo Final Examination 13:00-16:00, December 11, 2017 INSTRUCTIONS 1. The exam is THREE HOURS in length. 2. This examination paper contains THREE sections. Section A includes Question 1 to Question 4, Section B includes Question 5 and Question 6, and Section C includes Question 7 and Question 8. 3. Answer THREE questions from Section A. Answer ONE question from Section B. Answer ONE question from Section C. Only FIVE questions will be graded. Each question is worth 20 marks. 4. Write all your answers in the answers booklet. 5. Read the questions carefully and make sure you fully understand them before giving your answers. Always show your work or explain how you got your answer. Good luck! Aids permitted Non-programmable calculator Formula sheet (2 pages) * Please note: Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer exam question as written. This material is copyrighted and is for the sole use of students registered in ECE 256. This material shall not be distributed or disseminated. Failure to abide by these conditions is a breach of copyright, and may also constitute a breach of academic integrity.
Page 2 of 5 Section A. Question 1 a) (10 marks) Show that if a + b > 0, then a > b implies a 2 > b 2. b) (10 marks) Show that (A B) C A (B C) for any sets A, B, and C. Question 2 In the macroeconomic model below, Y is (aggregate) output, C is consumption, I is investment, r is interest rate, G 0 is government spending, M 0 is supply of money, and t is tax rate. The variables Y, I, C, and r are endogenous, G 0, M 0, and t are exogenous, and a, b, c, d, k, and m are parameters. Y = C + I + G 0 C = a + b(1 t)y I = c dr M 0 = ky mr a) (8 marks) Write the system of equations in the form Ax = b where A is a matrix of coefficients, x is a vector of endogenous variables, b is the solution vector. b) (8 marks) Find a condition on the parameters and on the tax rate that guarantees a unique solution to this system of equations. c) (4 marks) Using the Cramer s rule, solve for Y. Question 3 (20 marks) Using the definitions of concavity and convexity check whether z = x 2 + y 2 is concave, convex, strictly concave, strictly convex, or neither. Question 4 Consider the matrix X and Y given by 1 1 1 2 X = 1 3 and Y = 1 4 1 5 1 2 4 4 6 a) (5 marks) Find (X X). Is this matrix symmetric? b) (10 marks) Find (X X) 1 and X Y. Is (X X) singular? c) (3 marks) Find the least square parameters such as [ ] β0 β = = (X X) 1 X Y β 1
Page 3 of 5 d) (2 marks) Consider the parameters γ 1 and γ 2 given by and γ 1 = n n i=1 x i2y i n i=1 x n i2 i=1 y i n n i=1 x2 i2 ( n i=1 x i2) 2 γ 0 = ȳ γ 1 n i x i2 n where x i2 denotes the element in the i-th row and the second column of X, y i denotes the i-th row in the matrix Y, ȳ = (1/n) n i y i denotes the sample mean, and n denotes the number of rows of X. Find γ 1. Using γ 1 find γ 0. Are γ 0 and β 0 equal? Are γ 1 and β 1 equal? Is any relationship between the denominator of γ 1 and the determinant of (X X)?
Page 4 of 5 Section B. Question 5 A production possibilities (transformation) curve in two goods, x and y, is given by T (x, y) = k, where T (x, y) = x 2 + y. A social utility function is given by u(x, y) = α log x+β log y. The efficient choices for x and y are given at the point where the marginal rate of substitution equals the marginal rate of transformation MRS = MRT or (u x /u y ) = (T x /T y ). Thus, there are two equations determining the solution and u x u y = T x T y (1) x 2 + y = k. (2) a) (10 marks) For the functions u(x, y) and T (x, y), find u x, u y, T x and T y and give (u x /u y ) = (T x /T y ) in terms of x and y. b) (10 marks) Use the implicit function theorem to determine x α, y α, x k, y k. Question 6 Consider the matrix A = 3 0 0 0 16 4 0 4 1 a) (5 marks) Write down the quadratic form in (x, y, w) associated with matrix A. b) (5 marks) Find all of the characteristic roots (eigenvalues) of matrix A. c) (5 marks) Determine whether the quadratic form (and matrix A) is positive definite, positive semi-definite, negative definite, negative semi-definite, or indefinite. d) (5 marks) Write down all the principal minors of order one of matrix A.
Section C. Page 5 of 5 ( ) Question 7 Mark s utility function for goods c and m is U(c, m) = ln c 1 3 m 1 2. He must pay p c for a unit of good c and p m for a unit of good m. Mark s income is I. a) (4 marks) Write down Mark s constrained utility maximization problem and his Lagrangian function. Use λ as the Lagrange multiplier. b) (4 marks) Derive the first order conditions and find candidate solution (c, m ). c) (4 marks) Show that the demand function c = c (p c, p m, I) is homogeneous of degree zero in princes and income, i.e. c (tp c, tp m, ti) = t 0 c (p c, p m, I) for t > 0. d) (4 marks) Determine the value of λ associated with this problem. Interpret λ. e) (4 marks) Using the bordered Hessian, H, check whether the point (c, m ) satisfies the second-order conditions for a local maximum of this optimization problem. Question 8 Consider the following minimization problem min p 1x + p 2 y x>0,y>0 subject to: 6x 1 3 y 1 2 = ū a) (5 marks) Write down the Lagrangian function and determine the optimal amounts of goods x and y Mark should purchase. Use γ as the Lagrange multiplier. b) (5 marks) Find e(p 1, p 2, ū) defined by e(p 1, p 2, ū) p 1 x + p 2 y Show that e(p 1, p 2, ū) is homogeneous of degree one in prices, i.e. e(tp 1, tp 2, ū) = te(p 1, p 2, ū) for t > 0. c) (5 marks) Determine the optimal values of γ and provide a written interpretation of this value as it specifically applies to this problem. d) (5 marks) Write down the bordered Hessian, H, for this optimization problem. Write down all the leading principal minors of H? END OF PAPER