EC7088
All Candidates
January Examinations 2016
DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY
THE CHIEF INVIGILATOR
Department
Economics
Module Code
EC7088
Module Title
Mathematical Methods for Economics
Exam Duration (in words)
Two Hours
CHECK YOU HAVE THE CORRECT QUESTION PAPER
Number of Pages
4
Number of Questions
4
Answer three questions. All questions carry equal marks.
Instructions to Candidates
FOR THIS EXAM YOU ARE ALLOWED TO USE THE FOLLOWING:
Calculators
Permitted calculators are the Casio FX83 and FX85 models
Books/Statutes provided by No
the University
Are students permitted to
bring their own
Books/Statutes/Notes?
Additional Stationery
Version 1
No
No
Page 1 of 4
EC7088
All Candidates
1.
(a) The determinant of a square real matrix is defined to be an antisymmetric multilinear real valued function of its rows whose value for the identity matrix is 1. Prove
the following:
(i)
(ii)
(iii)
(iv)
If a row of a matrix is 0 or if two rows of a matrix are identical, then its
determinant is 0.
[15 marks]
Adding to one row of a matrix a linear combination of other rows does not
change the value of its determinant.
[15 marks]
The determinant of a diagonal matrix is the product of its diagonal
elements.
[15 marks]
The determinant of an upper triangular matrix is the product of its diagonal
elements.
[15 marks]
(bi) Use Gaussian elimination to solve the following system of simultaneous linear
algebraic equations [30 marks]:
x1  3x2  2x3 
2 x1  4 x2  3x3 
6
8
 3x1  6 x2  8x3   5
(bii) Hence, or otherwise, find the determinant of the following matrix [10 marks]:
 1  3  2
 2  4  3


 3 6
8 
Version 1
Page 2 of 4
EC7088
All Candidates
2.
(a) Suppose that, for some   0 , q  maximizes the function Lq   u q   bq  , where
 
 
b q  0 and b q   0 . Prove that q maximizes the function u q  subject to the
[15 marks]
constraint bq   0 .
(b) (The linear expenditure system). A price taking consumer with exogenously given
income y  0 maximizes her utility function u q  subject to her budget constraint
 in1 pi qi  y , where pi  0 is the price of good i and q i is the quantity of good i
demanded by the consumer, i=1,2,…,n. Let u q    in1 ai ln qi  bi  , where ai and bi are
constants satisfying: ai  0 , bi ≥ 0, i=1,2,…,n.  in1 ai  1 ,  in1 pi bi  y . Derive the
consumer's demand function for good i . Take care to justify your method. [70 marks]
(c) A price taking consumer with exogenously given income y  0 maximizes her utility
function u q1 , q2   q12  q22 subject to her budget constraint p1q1  p2 q2  y , where pi is
the price of good i and qi is the quantity of good i demanded by the consumer.
Assuming 0  p1  p2 , derive the consumer's demand functions for goods 1 and 2. Take
care to justify your method.
[15 marks].
3.
(a)
(b)
(c)
Version 1
dx
dy
 x1  x  y  ,
 y x  y  .
dt
dt
Find the positive equilibrium and analyze its stability.
Sketch the phase diagram.
Give an economic or biological interpretation of this model.
Consider the dynamic system:
[35 marks]
[50 marks]
[15 marks]
Page 3 of 4
EC7088
All Candidates
4.
Consider a decision maker who aims to choose path for the control variable ct  so as
to maximize the objective function
 u k , c, t  dt ,
T
t 0
subject to the dynamic constraint
dk
 f k , c, t ; 0  t  T ,
dt
the initial condition
and the following terminal condition
k 0  is given,
k T   k min .
(a) Formulate, clearly, the principle of optimality (Bellman's equation).
[10 marks]
(b) Give an outline derivation of the Hamilton-Jacobi equation.
[60 marks]
(c) Define the Hamiltonian, Maximized Hamiltonian and the costate variable. Give their
economic interpretations.
[20 marks]
(d) Clearly state the transversality conditions for this problem.
[10 marks]
END OF PAPER
Version 1
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