2019-Midterm-Resit

Examination Information

  • Course: EC251-5-SP Mathematical Methods in Economics Resit Midterm Examination 2019

  • University: University of Essex

  • Time Allowed: 2 hours

  • Permitted Calculators: Casio FX-83GT PLUS or Casio FX-85GT PLUS

  • Instructions:

    • Do not leave your seat without permission from an invigilator.

    • No communication with other candidates.

    • Do not open the question paper until instructed.

    • Use provided answer books for all work including rough work, clearly marking rough work.

    • Remain seated until all answer books are collected and dismissal is permitted.

Question Set Overview

  • Question 1 [45 marks]

    • Matrix M: Given matrix M represented as:[ M = \begin{pmatrix} 0 & 1 & 0 \ a & b & c \ 1 & 0 & 1 \end{pmatrix} ]

    • Tasks: Compute determinant, check inverse existence, cofactors, find inverse, solve MV = B for variables.

  • Question 2 [40 marks]

    • Macroeconomic Model: National output represented as:[ Y = C + I + G_0 ]

      • Consumption ( C = bY - T_0 )

      • Investment ( I = 0.4Y + 1 )

    • Tasks: Represent in matrix form, compute determinant and rank, conditions for unique solutions, apply Cramer’s rule for equilibrium consumption.

  • Question 3 [15 marks]

    • Matrix H: Given matrix H represented as:[ H = \begin{pmatrix} 1 & 0 & a \ 1 & 0 & 0 \ 1 & 1 & 1 \ 1 & 1 & b \end{pmatrix} ]

    • Tasks: Conditions for rank (4, 3, 2), and determining bases with provided vectors.

Detailed Questions

Question 1: Analysis of Matrix M

(a) Determinant Calculation

  • Compute determinant |M|.

(b) Existence of Inverse

  • Check if M⁻¹ exists for all values of a, b, c.

(c) Matrix of Cofactors

  • Calculate the matrix of cofactors for M.

(d) Inverse of M

  • Compute the inverse M⁻¹.

(e) Solving System MV = B

  • Solve for variables x, y, z in the system given V and B.

Question 2: Macroeconomic Model

(a) Matrix Representation

  • Formulate system as AX = B;

    • Matrix A \

    • Vector X: ( X_0 = (Y, C, I) )

(b) Determinant and Rank

  • Calculate |A| and find the rank depending on b.

(c) Unique Solution Conditions

  • Identify values of b for unique solutions.

(d) Cramer's Rule for Consumption

  • Apply Cramer’s rule to find equilibrium consumption C∗.

(e) Conditions for Solutions

  • Establish conditions on G0 and T0 leading to infinite or no solutions.

Question 3: Matrix H and Vectors

(a) Rank Conditions

  • Identify values of a and b for Rank(H) = 4.

(b) Rank Set Values

  • Set values of a and b resulting in Rank(H) = 3 and Rank(H) = 2.

(c) Basis for R⁴

  • Examine vectors V1, V2, V3, V4 and determine values of b for them to form a basis for R⁴.

    • Vectors:

      • V1: ( \begin{pmatrix} 1 \ 1 \ 1 \ 1 \end{pmatrix} )

      • V2: ( \begin{pmatrix} 0 \ 0 \ 1 \ 1 \end{pmatrix} )

      • V3: ( \begin{pmatrix} a \ 1 \ 1 \ b \end{pmatrix} )

      • V4: ( \begin{pmatrix} 1 \ 0 \ 0 \ 0 \end{pmatrix} )