MATHEMATICS FOR ECONOMICS II

EES 200: MATHEMATICS FOR ECONOMICS II

DATE: Tuesday, 29th December, 2009 TIME: 11.00 a.m. – 1.00 p.m.
————————————————————————————————————
INSTRUCTIONS:
Answer question ONE and any other TWO questions.
a) Find the product of matrix a and B given below:
? ? ? ?
?
?
? ? ? ?
?
?
=
1 4 6
9 6 3
8 8 7
5 2 1
A
? ? ?
?
?
? ? ?
?
?
=
7 5
3 4
9 0
B (3 marks)
b) z =5×2 -4xy+ y2
i) Get the 1st order and 2nd order total differentials of function z.
(3 marks)
ii) Comment on whether the function has a minima or a maxima point
(2 marks)
c) Find the consumer surplus for the function
P=35 – Q2 at price of 10 shillings (use graphs to illustrate your working)
(4 marks)

d) The following matrix A shows the input coefficient matrix of Jipange economy.
? ? ?
?
?
? ? ?
?
?
=
0.60 0.60 0.15
0.05 0.10 0.025
0.05 0.20 0.025
A
The final demand matrix is given as:
? ? ?
?
?
? ? ?
?
?
=
50
70
130
F
i) Find the total output requirements for the 3 sectors in Jipange economy.
(8 marks)
ii) Find the total primary input requirements for each of the 3 sectors
(2 marks)
e) An economy is represented by the following system of equations:
Y=C + I +G+ X -M
Where
Yd
o C b b1 = +
o I = I bo. > 0
o G=G 0 1 1 <b <
o X = X
M M M Y o 1 = + 0 0 1 1 M > < M < o
T t t Y o 1 = + 0 1 1 t >o <t < o
i) Find the equilibrium valve of Y (5 marks)
ii) If the government wants to increase the national income, should it adopt
an increase in tax rate or a decrease in tax rate. (Make your comment
based on the tax rate multiplier) (3 marks)

Q.2 a) Solve for X, Y and Z in the following system of equations:
3 2 4 3
2 4 0
4 3 9
– + =
+ + =
+ =
x y z
x y z
x y
(6 marks)
b) The utility function of a consumer is given as
2
1 2 2
2
1 U =Q +3Q Q -5Q
While his budget constraint is given as:
2 3 6. 1 2 Q + Q =
i) Find the level of Q1 and Q2 that will maximize the utility of the
consumer. (7 marks)
ii) Show that at the utility maximization point for this problem, the
slope of the indifference curve will be equal to the slope of the
budget line. (2 marks)
c) Find the degree of homogeneity of the following functions:-
i)
XY
Q XY x Y
3
4 2 -2 2
= (2 marks)
ii) 2 2
2 2
3
5 10 9
Z W
Y X Z XZW XW
+ –
= (3 marks)
Q.3 a) Differentiate the following non-algebraic functions:
i) x
ne
Y = L X (2 marks)
ii) Y =[1+3 X 2 ][e5x ] (3 marks)
iii) y L ( x ) n =2 1+ (3 marks)
b) The supply function of a firm is given as:
Q2 =4P-6
Find the producer surplus when P=10.5 (6 marks)
c) The total cost function for a firm is given as:
2 1 3 2 10 1 2
TC = Q 3 + Q 2 -Q Q +

Find:
i) The average cost for good 1. (1 mark)
ii) The marginal cost for good 1 and good 2 (3 marks)
iii) If function display increasing or decreasing marginal cost with
respect to:
i) Good 1 (1 mark)
ii) Good 2 (1 mark)
Q.4 a) Solve for X1 and X2 in the following linear programming problem. Using
graphical method
Maximize: 1 2 TT =5X +8X
Subject to:
i) 8 200 1 2 X + X =
ii) 2 100 1 2 X + X =
, 0 1 2 X X =
(8 marks)
Clearly label all your graphs.
b) Given a production function:
5
3
4
1
Q=5K L
i) Proof Eulers theorem for this function (3 marks)
ii) Find the nature of returns to scale for the functions (1 mark)
c) The demand and supply functions for 2 products are given by the
following functions:
1 1 2 2
Qd =4-P + 1 P 1 2 Qd =8+ 2P -2P
1 2 2
3
2
Qs = 5 + P 2 2 Qs =- 3+6P
Find:
i) Partial elasticity of demand for good 1 with respect to its own price
at equilibrium (4 marks)

ii) Cross partial elasticity of demand of good 2 with respect to price of
good 1 at the equilibrium. (3 marks)
iii) Are the 2 goods substitutes or complements? Give reasons for
your answer. (1 mark)
Q.5 a) Briefly explain the importance of comparative statics (2 marks)
b) Integrate the following functions with respect to dx :
i) X (X 2)dx
3
2
? 2 3 – (4 marks)
ii) dx X X . 5 3 8 7 3 ?? ?
?? ?
? – (4 marks)
c) The production function for a firm is given as:
Q= AKBL6 while the Isucost function is given as C =rK +wL show that
at the point where the firm minimizes its cost, the slope of the isuquart
will be equal to the slope of Isucost function. (8 marks)
d) Differentiate between consumer surplus and producer surplus
(2 marks)