discuss how the above-solved equilibrium levels Ye, Ce, Ge and Te will change (i.e., increase or decrease) if the income tax rate (t) is going to be increased.

Linear Algebra for Economics and Finance (2021-22Sem1) –– Special Assignment (80 marks, contributing 10% to the course’s overall assessment) Submitted to TA (Miss Tina Kong at T1-301-R5-H10) or the Lecturer before 5:00pm, Tuesday, 7 December 2021
Consider the following simplified national-income model, Y = C + G + I0 (1) C =  +  (Y – T) (  > 0, 0 <  < 1) (2) G =  +  Y (  > 0, 0 <  < 1) (3) T = d + tY (d > 0, 0 < t < 1) (4) where Y, C, G and T represent the endogenous variables of national income, private consumption expenditure, government consumption expenditure, and income tax, I0 stands for the exogenous variable of investment expenditure,  ,  ,  ,  , and d are independent structural parameters, and t is the (private) income tax rate which is considered as a policy or control parameter, respectively. It is also understood that  and  , standing for private and government consumption rates, respectively, satisfy the reasonable conditions of  –  d > 0 and  +  < 1.
Denote x = [
𝑌 𝐶 𝐺 𝑇
] and write equations (1)-(4) in matrix format as:
x = Ax + b (5)
Question 1 (20 marks) (i) What is A and what is b? (5 marks) (ii) Find the column space and null space of A, each written as a subspace spanned from a set of some independent vectors as usual. What is the rank of A? (11 marks)
(iii) Determine if v1 = [
2  2𝑡 2  𝑡 ] and v2 = [
 2𝑡 2  2𝑡 ] belong to the column space of A or not. (4 marks)
Question 2 (20 marks) Just for this Question 2 only, assume  = t 2 [i.e., government consumption rate equals to (private) income tax rate squared]. (i) Find all eigenvalues of A. (10 marks) (ii) Discuss A is diagonalizable or not. (10 marks)
Question 3 (40 marks) (i) Solve for Y, C, G and T from equations (1)-(4) [or from equation (5)] in terms of the exogenous variable (I0) and parameters (  ,  ,  ,  , d and t), formally called their equilibrium values which are denoted as Ye, Ce, Ge and Te, respectively. Indicate the conditions or restrictions (if any) needed in order to find the equilibrium values. (20 marks) (ii) Discuss how the above-solved equilibrium levels Ye, Ce, Ge and Te will change (i.e., increase or decrease) if the income tax rate (t) is going to be increased. (20 marks)
—- End of Special Assignment —-