UNIVERSITY OF NAIROBI
UNIVERSITY EXAMINATIONS 2020/2021
EXAMINATIONS FOR THE DEGREE OF BACHELOR OF ECONOMICS AND
BACHELOR OF ECONOMICS AND STATISTICS
XEQ 301: MATHEMATICS FOR ECONOMISTS II
DATE: JANUARY 22, 2021
TIME: 2.30 P.M.- 5.00 P.M
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INSTRUCTIONS:
Answer any THREE questions
All Question Carry Equal Marks
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Question One (23 Marks)
(a) Differentiate between the following terms
(i) Homogenous and non-homogenous differential equations
(ii) Complementary and particular integral
(b) You are given the following demand and supply functions
( , β, , δ > 0)
i.
ii.
iii.
iv.
Assuming that the rate of change of price overtime is directly proportional to the excess
demand find the time path p
general solution and definite solution
What is the intertemporal equilibrium price
What is the market clearing equilibrium price
State the role of the complementary function and particular integral in relation to the equilibrium.
Question Two (23 Marks)
(a) Find the time path of capital K(t) given the following rates of net investment flow functions
(i) I(t) = 20t1/3
K(0) = 200
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(ii) I(t) = 8t3/4
K(0) = 60
(iii) I(t) = 60t1/5
K(0) = 25
(iv) For each of (i) to (iii), find the amount of capital formation over the time interval {1,3]
(b) The marginal cost of two firms are given by the following
1) C’(Q) = 100e0.1Q
2) C’(Q) = 25 +40Q -10Q2
TC= 3000 when Q = 0
TC =200 when Q = 0
(i) Find the total cost function C(Q) for each firm
(ii) Find the average cost function for each firm
(iii) What is the fixed cost for each firm
Question Three (23 Marks)
a) Find the general and definite solutions of the differential equations
i) dy/dt + 9y=0
y(0) =12
ii) dy/dt - 15y=17
y(0) =25
iii) dy/dt + 0.1y= 5
y(0) =1/5
iv) dy/dt + 16y= 2/5
y(0) =0.8
v) 7dy/dt + 3y=0
y(0) =9
b) Given the following differential equations, verify that they are exact differential equations and find
the solution to the equations.
i)
ii)
iii)
2yt3dy + 3y2t2dt = 0
3y2tdy + (52 + y3t)dt = 0
4ytdy + (4t +2y2t)dt =0
ɛ
c) Given the following price elasticity of demand of QP = -1/2, P > 0
i.
Form a differential equation from the information
ii.
Solve the equation to find to obtain the required demand function.
Question Four (23 Marks)
(a) Given the following demand and supply functions for the cobweb model, find the intertemporal
equilibrium price and determine whether the equilibrium is stable.
(i) Qdt =14 – 3P1
(ii) Qdt =11 – 2P1
(iii) Qdt =19 – 6P1
Qst = -5 + 3Pt-1
Qst = -3 + Pt-1
Qst = -7 + 5Pt-1
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(b) In period analysis, the dynamic stability of equilibrium depends on
(the
complementary function). Whether the equilibrium is dynamically stable or not is a
question of whether the complementary function will tend to zero as t
.
i) Given the range (-ꝏ to +ꝏ), show using illustrations significance of the value of b
in determining the dynamic stability of equilibrium - disregarding the coefficient
A.
ii) Also, explain the role of A in
Question Five (23 Marks)
a) Find the definite integrals of the following functions
(i)
ʆ x(x2 +10)49 dx
(ii)
ʆ xlnx dx
(iii)
ʆ 20x5/(x6 +9) dx
(iv)
ʆ (lnx)3 dx
(v)
ʆ x2(x3 +9)1/4 dx
b) You are given the following demand and supply functions
Qd = 12 - 0.4P
Qs = -2 + 0.3P
(i) Find the consumer surplus
(ii) Find the producer surplus
demand function
Supply function
(c) Find the general and definite solutions of the difference equations
i)
Yt=1 + 3Yt= 17
y0 =5
ii)
Yt=1 + 14Yt= 1
y0 =18
iii)
Yt=1 + 0.3Yt= 3
y0 =7
iv)
Yt=1 = -0.6Yt
y0 =15
v)
Yt=1 - 6Yt= 18
y0 =12
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