I'IRITIIBIH UI'IIVERSITY

OF SCIENCE Fll'lD TECHNOLOGY

FACULTY OF HEALTH AND APPLIED SCIENCES

DEPARTMENT OF MATHEMATICS AND STATISTICS

QUALIFICATION: BACHELOR OF ECONOMICS

QUALIFICATION CODE: 07BECO LEVEL: 5

COURSE CODE: MFESllS COURSE NAME: MATHEMATICS FOR ECONOMISTS

SESSION: JUNE 2017 PAPER: THEORY

DURATION: 3 HOURS MARKS: 100

FIRST OPPORTUNITY EXAMINATION QUESTION PAPER

EXAMINER MRS S. MWEWA

MODERATOR: DR A. S. EEGUNJOBI

INSTRUCTIONS

1. Answer ALL the questions in the booklet provided.

2. Show clearly all the steps used in the calculations.

3. All written work must be done in blue or black ink and sketches must

be done in pencil.

PERMISSIBLE MATERIALS

1. Non-programmable calculator without a cover.

THIS QUESTION PAPER CONSISTS OF 7 PAGES (Excluding this front page)

"ITEMATHEMATICS: FOR. ECONOM'STS ‘MFESMQ ' 1$T OPPORTU NiTY EXAMlNAflON': g

>

Question 1 (1 mark each = 5 marks)

For each of the following statements, indicate whether True or False in your answer

booklet

1.1

1.2

1.3

1.4

1.5

Ines/g =\/-6—

Fixed costs are dependent on the number of units produced

A horizontal line has gradient equal to zero

Given thatf(x,y) = 9ye2"y, then fw = fyx

Given the production function Q = AKO‘ZLO‘s, a 3% increase in the level of technology,

capital and labour will result in an overall increase of3% in the production function

Section 2(2 marks each = 20 marks)

For each of the following indicate the response ofyour choice in the answer booklet

2.1

2.2

2.3

Consider the relationship between total profit (7:) and output is given by the

function7r 210005 — 250. In this function, a is described as the

A. Dependent variable B. Independent variable

C. Slope D. Intercept

The expression 16a4 —81b8 can be factorised fully as

A. (4a2 — 9b“)(4a2 + W)

B. (4a2 — 9b4)(2a —3b2)(2a + 3192)

c. (4a2 + 9b4 )(2a — 3b2 )(2a + 3b2) D. All of these

Compute the value ofy if 10g,r y= 100 and log2 x =10

10002A. 322210 B. y=0 c. y: D. y=2‘°0

Page 1 of 7

MA'T'R'EMATics F‘ORZECONOMlSTS (MFESllS) 1$T oppom‘unm EXAMINATION;'

:‘

2mm: 2017";'

._

2.4 A small economy produces bananas and the domestic demand and domestic supplycurves for bananas in thlS economy are given as:

P : ZO—lQ P = 2+—1—Q

The calculated equilibrium price and quantity will be:

A. P = N$2;Q = 4 bananas B. P = N$5; Q = 30bananas

C- P = N$2.50;Q = 45 bananas 13- P = N$1.00;Q = 50 bananas

2.5 Solve forn if 2n+4 - 2"+2 = 3

A.n=1 B.n=2 C.n=—1D.n=-—2

2.6 12M+M—3M—4fi is equalto

A. 28d? —5J7 B. 5% c. 56 D. 7\/§—\/7

2.7 In the graph below what does the line segment BD represent for Jerry?

/ Jerry’s Consumption Function

A //YEl

l

l

l

l

l

~——-«——->Jerry’sconsumption

DY1 Y2 Y

Jerry’s Income

A. Consumption when income equals Y1

B. Saving when income equals zero

C. Saving when income equals Y1

D. Consumption when income equals zero

2.8 If Ruth received a N$2500 bonus and her MPS = 0.2 her consumption will rise by

N$X and her savings will rise byN$Y. Find X and Y

A. X = N$1250; Y = N$1250

B. X = N$2500; Y = N$200

C. X = N$2000; Y = N$500

X = N$2400; Y = N$100

Page 2 of 7

2.9

2.10

MATHEMATICS FORECONOMISTSlMFESllS) 915T OPPQRTUNlTY‘ EXAMlNATtON: JUNEZ

Suppose that your grade on this examination is described by the function G = 3A0'5E0'5

where G is your numerical score, A is your ability, and E is your effort measured in

terms of hours studied. Suppose thatA = 100. How many hours would you have to

study to earn a grade of precisely 90?

A. 12 B. 6 C. 9 D. none of the above

An integral may be interpreted as:

A. The inverse of the objective function

8. The Lagrangian multiplier

C. The reverse of differentiation

D. Monotonic function

Section 3 (3 marks each = 15 marks)

For each of the following indicate the response of your choice in the answer booklet

3.1

3.2

Dalco Manufacturing estimates that its weekly profit, P, in hundreds of dollars, can

be approximated by the formula 7r = —3x2 + 6x+10 ,where x is the number of units

produced per week, in thousands. The number of units the company should

produce per week to earn maximum profit and the maximum weekly profit will be

A. 1000 units; N$1300 B. 3000 units; N$100

C. 1000 units; N5600 D. 2000 units; NSllOO

2

Determine the expanded form of logfl(3x — 2)4

A. logx2 —4log3x—4log2+log\/4x—1

3- 2 logx + élogmx — 1) — 4 log(3x — 2)

l

C. log at2 \l4x —1—log(3x — 2)4

D. None of the above

Page 3 of 7

-

MATHEMATICS FOR ECONOMISTS (MFESllS) 151' OPPORTUNVITY‘ EXAMINATION cf JUNEZOl? I 5

3.3 Consider the revenue function R(q) that is gained from selling q products given as

16

profit function will be given as

R(q) = 64(iq2 —3i2q) and the cost function is given as 8C(q) = 32 ~56q ,then the

A. 7r(q)=4q2+5q—4 B. 7r(q)=4q2—9q—4

c. 7z'(q)=4q2+54q—32 D. 7r(q)=16q2+32q—4

3.4 fiz+2ex+545dxx

5

A. 2ex+4x5—3+C B. 2e"+4x3+31nx+Cx

5 5

C. 2e"+§x4 —-E+C D. 2ex+4x4 —§+C4 x x

3.5 The marginal cost function of producing x units of a product is given as

C'(x)=10.6x. If the fixed cost is given as N$50, the total cost function will be given

as

A. C(x) =10.6 B. C(x) =10.6x2 +50

c. C(x) =5.3x2 +50 D. C(x) =5.3x+50

Page 4 of 7

MATHEMATICS FOR ECONOMISTS (MFESllS) lST OPPORTUNITY-EXAMINATION. JUNEZOI .1};

Section 4160 marks)

Show all the necessary calculations and steps in your answer booklet for the following

questions:

4.1 In a two—sector economy, the income determination model is given by:

Y = C +IWhere: C=Co+bY and [:10

4.1.1 Determine the reduced form of the above model (3)

4.1.2 Determine the value of Y if CO = 990; I0 = 840; b = 0.85 (2)

4.2 The lS-LM model for a three sector economy is described by:

Y=C+I+G C=15+O.8Yd 1:75—100i

Where Y = income; C = consumption; I = investment;

G = government expenditure and Ya is disposable income

4.2.1 If Yd is defined bde = Y—T, where T is taxes and T is given by T = 5 +0.2Y and

G = 300 determine Ydand hence the IS schedule for this economy (6)

4.2.2 If the transaction —

precautionary demand for money is given by

L1 = 0.2Y

And the speculative demand for money by

L2 2 750—2601’,

Where Md = L1+L2 at equilibrium and the supply of money is fixed at (3)

MA. = 950 , determine the LM schedule for this economy

4.2.3 Determine the equilibrium values of Yand i from your results above (6)

Page 5 of 7

'

MATHEMATICS For; ECONOMlSTS (MFE511S)

4.3

4.3.1

4.3.2

4.3.3

4.3.4

4.3.5

4.4

4.4.1

4.4.2

4.4.3

4.4.4

1$T OPPORTUNITY EXAMINATION“?

Consider a Cobb-Douglas production function with three inputs:

1 1 I

Y = K3L3H3

Where K is the capital (the number of machines), L is labour (the number of

workers), and H is the human capital (the number of college degrees among the

workers).

What will be the overall % increase in the production function if capital, labour and

human capital are all increased by 2%? (2)

What will be the overall % increase in the production function if labour is increased

by 2%, human capital by 3% and capital is held constant? (4)

If there are 300 units of capital available, 150 units of labour and 100 units of human

capital, what will be the level of production for this function? (2)

Show that if capital is doubled, labour and human capital are both tripled, the new

level of income will be

YM = 2.62Y (5)

l 2

Consider the Cobb—Douglas production output function is given asQ = 3AK3L3 ,

determine 92 (2)6K

The total revenue function and the total cost function for a company producing cell

phones are given as

R(x)=56Ox—2x2 and C(x)=60x+150

What costs will this company incur if no cell phones are produced? (2)

What is the marginal revenue from producing the 100th cell phone? (3)

Derive the profit function for this company (2)

Determine how many cell phones need to be produced to maximize profit (confirm

profit function yields a maximum using derivatives) (6)

Page 6 of 7

MATHEMATICS FOR ECONOMISTS(MFE5115) 1$T OPPORTUNITY EXAMINATION JUNE 2017

4.4.5 What is the maximum profit? (2)

4.5.1 A manufacturer’s marginal profit is

n" =e2" +10x~1

Find the additional profit fl' earned by increasing production from 1 unit to 3 units.

(5)

4-5-2 Compute the value of (5)

1— —du whereu=1-x2 0r 03x31JJZ f

END OF EXAMINATION!

GOOD LUCK!!

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