ECON0006/0010:
Introduction to Mathematics for Economics
2024/2025 Sample Exam III
Problem Set II
Instructions
Exam Instructions
Exam:
You are expected to attempt all the questions in sections A and B.
◼ Keep your answers inside of the boxes. Solve the questions separately on scrap paper, only then decide which of the workings you are going to include in the boxes of the exam paper.
◼ Workings presented outside of the boxes will not be considered for marking.
◼ Each of the 5 questions carry 20 marks. Sub-questions within a question carry equal weight within that question.
Section A: Calculational questions
1] Matrix Algebra
◼ a] Given the vectors a1 = {1, 1, 1}, a2 = {2, -2, -2}, a3 = {1, -2, 1}, and a4 = {2, 0, 2}, calculate the matrices A = a1⊗a2 and B = a3⊗a4, and their commutator C = A.B - B.A.
◼ b] Find the characteristic equation and the eigenvalues for the matrix A + B.
◼ c] Find the eigenvectors of the matrix A + B.
2] Optimization problems
◼ a] For f[x] = 4/1 x(1 - x) y(5 - y), give the first-order conditions and find all the local optima.
◼ b] Determine the nature of all the local optima.
◼ c] Sketch a few isoquants of this function that highlight the local optima.
3] Difference equations
A firm is using 2 resources with quantities x1 and x2, a uses a technology that gives them a produc-tion function F[x] = 10 x1 x2. The resources come at a unit cost p1 = 1
2 , p2 = 3
2
while one unit of output generates a revenue of 1.
◼ a] Calculate the difference equation describing how the firm might ‘learn’ its optimal levels of inputs using gradient-learning, and comment on any unknown parameters left in the equation.
◼ b] Find all the stationary states.
◼ c] Use an approximation to find a linear difference equation for small deviations δx[t] from the stationary state, find a solution for δx[0] = {1, -1}, and sketch the gradient-flow close to the stationary state(s).
Section B: Exploratory questions
4] Matrix algebra exploration:
Consider a matrix A and its transpose AT ≠ A.
◼ a] How do we know that S = A . AT + AT . A must have real-valued eigenvalues and eigenvectors that form. an orthonormal basis?
Now it is given that this matrix A has the following two properties:
i) C[A, AT] ≡ A . AT - AT . A = and that ii) A . e1 = 0, where e1 = {1, 0, 0, 0, ...}.
and we define the vectors ϕn = (AT) n . e1.
◼ b] Show that A . ϕ1 = e1, and that ϕ1 . ϕ1 = 1.
◼ c] Show that A . ϕn = n ϕn-1 and that this implies ϕn . ϕn = n! for n = 2, 3, ... .
◼ d] Show that S . ϕn = f[n] ϕn, determine the eigenvalue f[n].
We now normalise these ϕn vectors by ϕn = 1n!ϕn.
◼ e] Give an argument without calculations why we can assume that ϕn . ϕm = 0 if m ≠ n and show that in the orthonormal basis of the ϕn the matrix A has the matrix-elements
Ajk = k j = k - 1, and 0 otherwise.
5] Minimizing production cost
Consider a firm which uses two inputs for its production process in quantities x1 and x2. The production technology they use gives them a production function given by F[x] = 16 x1 x2 . The cost of these resources is, using the price vector p = {4, 2}, simply C[x] = p . x . The revenue the firm gets from selling a single unit of output is 1. The firm management decides that they need to achieve an output Q and they seek a set of inputs x that will do so at minimum cost.
◼ a] Write down a Lagrangian for this constrained minimization problem and explain your answer.
◼ b] Give the first-order conditions and find a solution for Q = 36.
◼ c] Calculate the bordered Hessian at the local optimum.
◼ d] Find the normalised gradient of the constraint, at the optimum, calculate the projected, bordered Hessian, determine its eigenvalues and check whether it is a minimum.
◼ e] Sketch cost and output isoquants close to the optimum, and include vectors representing the cost and output gradients at the optimum as well as the eigenvectors on the projected, ordered Hessian. Comment on your sketch.