Department of Economics

ECON0006/0010:

Introduction to Mathematics for Economics

Problem Set 3, 2025

General Instructions

These are the instructions you will see on the final exam. They do not apply to this prob- lem set! But it is good to familiarize yourself with themanyway.

Time allowance: You have THREE (3) hours to complete this examination.

If you have been granted SoRA extra time and/or rest breaks, your individual examination duration will be extended pro-rata.

Calculators (UCL approved models only) are permitted in this examination.

Number of Questions Answered Policy: You are expected to attempt all questions.

Recording of answers:

Use the separate paper provided to do all the necessary rough work, calculations or drafting of answers. If you need more paper for rough work, alert the invigilators by raising your hand. Enter   your final answers, answer steps, and explanation/interpretation (as required) in the answer-boxes provided within this question-and-answer booklet. Only results, workings, diagrams, and writing   provided within the appropriate boxed answer-spaces will be marked. Comments, notes or any-thing else outside of the provided boxed answer-spaces will be ignored.

Do not remove any pages from this answer booklet!

If you inadvertently damage your answer booklet during the exam, alert the invigilators and request a new answer booklet. You will not get additional time!

Submission of answers:

Ensure that you have provided in the spaces above your correct student number (seeyour student card) and your correct name, in block capitals. Failure to provide this information may mean that  any marks awarded cannot be allocated to you correctly. Only submit your answer booklet!

Submitting a damaged or incomplete answer booklet may result in a mark of 0.

Instructions for this Exam

. This exam consists of 5 questions, 1]-5], and each question has 4 sub-questions, a]-d]. All sub- questions have the same weight.

. You are expected to attempt all questions, there is no selection of questions to be made.

. Use the provided ‘scrap-paper’ to do your workings and solve the questions.

. In each answer-box, complete the answer options at the top, and then complete the ‘workings’ or ‘comments’ or ‘sketch’ sections as appropriate.

. You will likely need to select the important steps in your workings on the scrap paper and

transcribe them into the answer boxes. Don’t use the boxes for initial, rough, workings and draft answers.

1] Using spectral methods in Matrix Algebra

Consider the following 3 ×3 matrix

. a] Compute the eigenvalues mj of M as well as the normalisedeigenvectors m(^)j.

. b] Verify that we have

. c] Find a matrix R = M1/2 for which R2 = M and which has only real-valued components.

. d] Consider the system differential equations given by

for some constant vectorf(-)0 = {1, -1, 1}. Find a solution withx(-)[0] = {0, 1, 0}?

2] Gradient-learning

Consider a firm that produces a service or good for which it can charge an exogenous price per   unit q = 5, and its production technology gives it an output off[K, L] = Log[K] + 3 Log[L] units as a function of the inputs of labour L and capital K. In order to acquire these inputs the firm faces exogenous labor and capital market prices of w = 2 and r = 3/2 respectively.

. a] Find the profit π[K, L] the company makes as a function of its labour and capital inputs.

. b] Determine the optimal inputs {L*, K*} and the corresponding optimal output f* = f[K*, L*], and demonstrate that it is indeed a profit maximum π* .

If we assume that the firm is ‘learning’ what their optimal inputs are via gradient-learning, then over time the firm’s will inputs will evolve according to

. c] Assume that α = 1. Show that the optimum {L*, K*} is a stationary state and use small

deviations {δL[t], δK[t]} from the optimum, to show this stationary state is stable under gradient- learning.

. d] Determine the rate-of-change of profit dw/d π* and dr/d π* at the current prices w = 2 and r = 3/2.

3] Making a Matrix from Vectors

Consider the vectors b(→)1 = {1, 1, 0} and b(→)2 = {1, -1, 1}.

. a] Find a third vector b(→)3 such that it is orthogonal to both b(→)1 and b(→)2, and then normalize all three into an orthonormal basis b(^)jwith j = 1, 2, 3.

. b] Consider the matrix

verify that it is an orthogonal matrix and find a similar expression for BT.

. c] Consider the matrix

Calculate R[0] and M = dy/d R[y] y=0 and demonstrate this means that R[y] is a solution to the initial value problem

. d] Consider quadratic functions of x ∈ [-1, 1] and the two linearly independent functions

Find a third function β3 [x] that forms an orthonormal set with the other two, calculate the function

and demonstrate that or an arbitrary quadratic function f[x] = a + bx + cx2 we get

4] Optimization problems

Consider the quadratic function

We want to find the local optimum, subject to the constraint

. a] Write down the Lagrangian for this problem, and the first-order conditions.

. b] Calculate the bordered Hessian HB.

. c] Determine the constrained, local, optima ford = -1 and c = 2/1 , find its type, and sketch a few

isoquants ofF[x(-)] and the constraint. Discuss whether there are restrictions on the value of d when we want to ensure that in generalx(-)* is not a saddle-point.

d] What changes when the constraint becomes an inequality constraint g[x(=)] ≤ 0?

5] A Quadratic flow

Consider the following vector-field

and the difference equation

. a] Determine all stationary states x(→)s .

Consider small deviations from the stationary states

. b] Find a linear approximation for the difference equation for δx[t] around each of the stationary states.

. c] Find the eigenvalues of the corresponding Jacobians, determine whether or not the stationary states are also asymptotic states for pathsx(-)[t] starting sufficiently close to the stationary state, and sketch the ‘flow’ close to each of the stationary points.

. d] Determine the eigenvectors of the Jacobians, sketch the overall ‘flow’ in a single sketch containing all stationary states.