Final Exam Advanced Mathematics for Economics and Finance

Transcription

1 Final Exam Advanced Mathematics for Economics and Finance Dr. Stefanie Flotho Winter Term /5 March 5 General Remarks: ˆ There are four questions in total. ˆ All problems are equally weighed. ˆ This is an open-book exam. You are allowed to use the book by Sydsæter, Further Mathematics for Economic Analysis or hardcopies thereof, your hand-written lecture notes, hardcopies of the lecture slides no electronic versions on laptops or other electronic devices, hardcopies of all problem sets plus solutions to the problem sets no electronic versions on laptops or other electronic devices, a non-programmable pocket calculator. ˆ Good luck!

2 Dr. Stefanie Flotho Final Exam Advanced Mathematics /5 Problem Consider the matrix A 3 a Find the rank of A, show that AA exists, and find this inverse. Note: A denotes the transpose of A. b Compute the matrix C A AA. c Show that ACb b for every -dimensional column vector b. d Use the results above to find a solution of the system of equations Problem Consider the following maximizing problem x + x + x 3 x + x + 3x 3 max fx, y x + y s.t. gx, y 5x + 6xy + 5y a Write down the necessary Kuhn-Tucker conditions of the problem. b Find all points that satisfy these conditions. c Which of the points found in b solve the maximization problem? d Determinate the approximate changes in the maximum value of fx, y if the constraint is replaced by gx, y. Problem 3 Sketch the domain A {x, y : y, x y } of integration and compute the following integral: J ye x dx dy Problem Consider the following system of differential equations A ẋ x + y ẏ x y a Find the general solution of the system using the method of eigenvalues. b Is the equilibrium point a stable point?

3 Final Exam Advanced Mathematics for Economics and Finance Sketch of Solutions Dr. Stefanie Flotho Winter Term /5 March 5 Problem a The matrix A 3 has rank, as there are two linearly independent rows. The matrix AA has got the determinant A 6, i.e. the matrix has got an inverse. AA b C A AA c To show that ACb b for every -dimensional column vector b we look at the left-hand side and compute ACb AA AA b Ib b where I denotes the identity matrix. d The system of equations x + x + x 3 x + x + 3x 3

4 Dr. Stefanie Flotho Final Exam Solutions Advanced Mathematics /5 can be written as Ax b where x x, x, x 3, and b,. Comparing with the result in c we get Problem Ax b and ACb b x Cb 6 Consider the following maximizing problem 5 max fx, y x + y s.t. gx, y 5x + 6xy + 5y a The Kuhn-Tucker conditions for a point x, y to solve the problem are the following: A. Lagrangian function: B. First-order conditions: C. Complementary slackness: D. The constraint must hold: Lx, y fx, y λgx, y x x x 3 x + y λ 5x + 6xy + 5y L xx, y x λx + 6y L yx, y y λ6x + y λ, λ if gx, y < gx, y 5x + 6xy + 5y b First, we start assuming that the constraint is binding: gx, y 5x +6xy +5y. Solving both FOCs and for λ and equating both expressions results in λ x x + 6y y 6x + y x y x y or x y ˆ First case: x y. This implies λ 8 values for x y ±. >. Using the constraint we get the following ˆ Second case: x y. This implies λ >. Using the constraint we get the following values for x ± which implies y.

5 Dr. Stefanie Flotho Final Exam Solutions Advanced Mathematics /5 Second, we assume that the constraint is not binding: Complementary slackness implies that λ. The FOCs and imply x y. To sum up, we have five candidates for a solution: x, y,, λ 8 x, y,, λ 8 x 3, y 3 x, y,,, λ, λ x 5, y 5,, λ c To find out which of these five candidates solve the maximization problem compute the value of the function at these points: f, 8 f, f, f, f, x 3, y 3, solve the maximization problem. and x, y,, with λ d As the maximum is attained with a binding constraint we can use one of the envelope theorem results: df db λ where b is the right-hand side of the constraint. This implies for the approximate changes in the maximum value of fx, y if the constraint is replaced by gx, y. : df λdb. i.e., the maximum value of f will increase by. 3

6 Dr. Stefanie Flotho Final Exam Solutions Advanced Mathematics /5 Problem 3 Problem J A e ye x dx dy [ye x ] y dy y ye y y dy [e y y ] a To find a general solution of the system ye x dx dy ye y ye dy e e + ẋ x + y ẏ x y ye y y dy first, we have to compute the equilibrium point: Setting ẋ ẏ and solving the system of linear equations yields: x, y, 3. Second, we introduce new variables: a x and b y 3 Third, we express the system in these new variables: Fourth, rewrite this into matrix form: ȧ ḃ ȧ a + b ḃ a b 3 a b a A b Fifth, compute the eigenvalues of the matrix A setting up the characteristic polynomial: pλ λ λ and λ 3 Sixth, compute the eigenvectors corresponding to the eigenvalues: λ : v v v v v v λ 3 : u u u 3 u u u

7 Dr. Stefanie Flotho Final Exam Solutions Advanced Mathematics /5 Seventh, the general solution for the system 3 is given by with C and D constants: a Ce t + De 3t b Finally, this implies for the solution for the problem: x Ce t + De 3t y + b To determine the stability of the equilibrium point we have to inspect the matrix A which has a trace of tra < and a determinant of deta A 3 <. These two conditions imply that the equilibrium point is a saddle point. 3 5