ECON 255 Introduction to Mathematical Economics

Transcription

1 Page 1 of 5 FINAL EXAMINATION Winter 2017 Introduction to Mathematical Economics April 20, 2017 TIME ALLOWED: 3 HOURS NUMBER IN THE LIST: STUDENT NUMBER: NAME: SIGNATURE: INSTRUCTIONS 1. This examination paper contains TWO sections (Section A and Section B). Both sections have five questions. 2. Answer FOUR questions from section A. Answer FOUR questions from section B. Only EIGHT questions will be graded (FOUR for each section). The marks for each question are indicated at the beginning of each question. You must indicate the questions to be graded in the chart below. If you answer more questions per section we will grade only the first four. 3. 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. 4. This IS NOT an OPEN BOOK exam. 5. Students are allowed to use a Casio 991 CALCULATOR. Table 1: Cross with an X the questions to be graded (FOUR in each Section) Section A Q1 Q2 Q3 Q4 Q5 Section B Q6 Q7 Q8 Q9 Q10 1

2 Section A Question 1. (I) (15 marks) Let A, B, and U be sets. Assume A, B U (U denotes the universal set), A = 25, B = 10, and U = 100. ( denotes the cardinality of the set). (a) What is A c? What is B c? (b) What is A B? (c) What is the smallest A B can be? (d) What is the largest A B can be? (e) Suppose you also know that A B = 5. What must A B be? (II) (5 marks) For any events (sets) A, B defined on the sample space U (Universal set). Show that (A B) c = A c B c Question (15) Let A and B be square matrices of order 4 such that det(a) = 5 and det(b) = 3 (det denotes the determinant). Find (a) det(3b). (b) det((ab) ). (c) det(a 1 ). 2. (5 marks) If A 1, A 2, A 3,..., A n are invertible matrices of size n, prove that Hint. You should use induction. (A 1 A 2 A 3 A n ) 1 = A 1 n A 1 3 A 1 2 A 1 1. Question 3. Consider the matrix A = [ 1 k ]. 1. (5 marks) If A is the augmented matrix of a system of linear equations, determine the number of equations and the number of variables. 2

3 2. (5 marks) If A is the augmented matrix of a system of linear equations, find the value(s) of k such that the system is consistent. 3. (5 marks) If A is the coefficient matrix of a homogeneous system of linear equations, determine the number of equations and the number of variables. 4. (5 marks) If A is the coefficient matrix of a homogeneous system of linear equations, find the value(s) of k such that the system is consistent. Question 4. Using the definitions of concavity and convexity check whether z = x 2 y 2 is concave, convex, strictly concave, strictly convex, or neither. Question (15 marks) Consider the squared loss function L(α, β) = n (y i α βx i ) 2. i=1 (a) Compute the First Order Conditions and Second Order conditions. (b) Write the FOCs as a system of equations in the form Ax = b where A is a matrix of coefficients, x is a vector of endogenous variables (α, β), b is the solution vector. Find the critical points (ˆα, ˆβ). (c) Using the SOCs, find conditions such that the Hessian matrix is positive definite provided that n > (5 marks) The least squares estimator of β, b is defined as: b = (X X) 1 X y Let ŷ be defined as: ŷ = Xb. Show that X (y ŷ) = 0. 3

4 Section B Question 6. A firm pays tax on gross revenue R(Q) at a rate which depends on output volume t(q). Suppose that and that Then the total tax paid is R(Q) = k ln(q + 1) cq α, k > 0, c > 0, α > 1, t(q) = 1 e bq, b > 0. T (Q) = R(Q)t(Q) = [k ln(q + 1) cq α ] ( 1 e bq) a) (7 marks) Is R(Q) concave? Justify. b) (8 marks) Use the product rule to find T (Q). c) (5 marks) Suppose that Q maximizes R(Q). Show that T (Q ) > 0 Question 7. Consider the following utility function over goods x and y, u(x, y) = α ln x + (1 α) ln y, 0 < α < 1. The price of x is p x > 0 and the price of y is p y > 0. The consumer has a total income of m > (4 marks) Write down the consumer constrained utility maximization problem and his Lagrangian function. 2. (4 marks) Derive the first order conditions and find a candidate solution (x, y ). 3. (4 marks) Using the bordered Hessian, check whether the point (x, y ) satisfies the second-order conditions for a local maximum of this optimization problem. 4. (4 marks) Determine the value of λ (Lagrange multiplier) associated with this problem. Interpret λ. 5. (4 marks) Find v(p x, p y, m) defined by v(p x, p y, m) u(x, y ) Show that v(p x, p y, m) is homogeneous of degree zero in prices and income. 4

5 Question 8. Consider a profit maximizing monopoly. The demand for the monopoly s product is given by Q = ln(a bp ) and its cost function is C(Q) = ce Q, where the parameters a, b, c are all positive. Let Q be the monopoly s optimal (profit-maximizing) output. Derive an expression for Q / a and determine its sign. Question 9. Consider a firm that has a Cobb-Douglas technology. The firm wishes to minimize the cost of producing y units of output and has access to perfectly competitive factor markets. The firm s cost minimization problem is given by min k,l subject to: wl + rk k α l β = y. Let µ denote the Lagrange multiplier on the output constraint. 1. (4 marks) Write down the Lagrangian function. 2. (4 marks) Derive the first order conditions and find the candidate solution l (w, r, y) and k (w, r, y). 3. (4 marks) Using the bordered Hessian, check whether the point (l (w, r, y), k (w, r, y)) satisfies the second order conditions for a local minimum of this optimization problem. 4. (4 marks) Find the cost function defined such as C (w, r, y) wl (w, r, y) + rk (w, r, y) 5. (4 marks) Find µ. What is its interpretation? Question 10. Consider the following National-Income Model Y = C + I 0 + G 0 C = α + β(y T ) (α > 0, 0 < β < 1) T = γ + δy (γ > 0, 0 < δ < 1) where the endogenous variables are Y (national income), C (consumption), and T (taxes). The exogenous variables are I 0 (investment), G 0 (government expenditure) are nonnegative. The exogenous variables are nonegative. Find Y/ G 0 and T/ G 0. (Hint: You should consider the system of equations as a system of implicit functions). END OF PAPER 5