THE UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF ECONOMICS

ECON1202 QUANTITATIVE ANALYSIS FOR

BUSINESS AND ECONOMICS

SEMESTER 1 2011

HPW 13th edition

Tutorial Booklet

2

General Guidelines

You must prepare the tutorial questions each week and take them to your tutorial. Additional discussion questions will be provided at the tutorial. Full references to textbooks are listed in the course outline and are identified here by the initials of their authors. HPW = Haeussler, Paul and Wood and KZB = Knox, Zima and Brown.

“WEEK X” at the beginning of each problem set indicates the week in which the problem set is due.

Week 2 tutorial is designed to review some Algebra and Calculus concepts. You should make every effort to attend tutorials and practice all problems.

3

WEEK 2

Remember that you must complete the “Lines and Inequalities” Adaptive Tutorial, which will give you access to the In-Tutorial Test in Week 4, worth 8% of your final mark.

Remember to register your assignment group members at this tutorial. TUTORIAL QUESTIONS

1. Is 2)125()( xxg a one-to-one function?

2. Find x in 2)46(log 2 xxx .

3. KZB 1.1 Problem 12, p.5 - Find the present value of $100 due in 3 months if the

rate is 11% p.a. simple interest.

4. HPW 5.2 Problem 8, p.216

5. HPW 5.2 Problem 12, p.216

6. HPW 5.2 Problem 21, p.217

7. Find the present value of $6500 due in four years at a bank rate of 5.8% compounded daily. Assume that the bank uses 360 days in determining the daily rate and that there are 365 days in a year; that is, compounding occurs 365 times in a year.

8. A student has to make four payments of a debt. The first is due in 1 month, the

second and third in 6 months and the fourth in 12 months respectively. The first is half the size of the second and twice the size of the third payment, while the first and the last payments are the same. If a student chooses to make repayments of $4,000 in 12 months and $8,000 immediately, find the value of the first repayment using 6 months as your focal date. Interest rate is 7% per annum compounded daily for the first 2 months, then 8% per annum compounded monthly for the following 4 months, and nominal rate of 10% continuously compounded for the rest of the year. Assume one month is equal to 30 days.

SELF STUDY QUESTIONS

4

1. Find the interest earned over 5 years on an investment of $2000 paying simple interest of 7.5% per annum (p.a.).

2. (Adapted from September 1998 exam) Which investment will have the greatest future value after 5 years?

(a) $2,000 invested for 2 years at 5.4% p.a. compounded quarterly, then for 3 years at 6.0% compounded monthly.

(b) $2,000 invested for 5 years at 5.85 % p.a. compounded daily. (c) $800 invested at 4.9% compounded weekly and $1,200 invested at 6.2% p.a.

compounded annually for 5 years. 3. Suppose a principal of $500 is placed in a bank account paying interest of 6% p.a.

compounded weekly. If no money is withdrawn from the account, how long will it take for the account balance to reach $700?

4. Find the effective annual rate of interest for the investments in Self Study Question

2, parts (a) and (b) above. 6. You are offered a choice of three types of bank account. These accounts earn,

respectively: (a) 4.1% per annum compounded monthly. (b) 4.05% per annum compounded daily. (c) 4.0% per annum compounded continuously. Which account would you choose? Explain your answer.

5

WEEK 3

Remember to complete online Quiz 1 this week (14-18 March 2011).

TUTORIAL QUESTIONS

1. KZB 8.1 Problem 6, p.224 (Attempt this problem using your calculator first, then check the result using Excel.) Which of the following projects should a company choose if each proposal costs

$50 000 and the cost of capital is 10% p.a.?

End-of-year cash flow Year 1 Year 2 Year 3 Year 4 Year5 Project A $20 000 $10 000 $5 000 $10 000 $20 000 Project B $5 000 $20 000 $20 000 $20 000 $5 000 Set up a spreadsheet for your calculations. Check your answers using the inbuilt NPV function.

2. (Adapted from QMA final exam, June 2000) Sam, a 20 year old graduate, has just found his first full-time job. An investment analyst suggests that he should try to have a substantial amount saved by the time of his planned retirement on his 65th birthday. The analyst presents a comparison between two savings plans.

Plan A: Sam should invest $5,000 on his 21st birthday then $2,000 on each birthday

up to and including his 31st . No further deposits are made but the money invested continues to earn interest until his 65th birthday.

Plan B: Sam should make the first deposit of $5,000 on his 31st birthday then

continue to deposit $2,000 per birthday with the last deposit being made on his 65th birthday.

If the effective annual rate of interest is 7.5% p.a., how much should Sam have in savings at age 65 under each plan?

3. KZB 2.6 Problem 12, p.50 A piece of land can be purchased by paying $50 000 cash or $20 000 deposit and two equal payments of $20 000 at the end of 2 years and 4 years respectively. To pay cash, the buyer would have to withdraw the money from an investment earning interest at j2 = 8% (i.e. 8% p.a. compounded twice per year). Which option is better and by how much, in present value terms?

4. HPW 5.4 Problem 6, p.227

6

5. HPW 5.4 Problem 10, p.227 6. HPW 5.4 Problem 12, p.227 7. HPW 5.4 Problem 16, p.227

8. HPW 5.4 Problem 22, p.227

SELF STUDY QUESTIONS

1. A farmer is planting a crop of potatoes using seed potatoes at a cost of $200 and has a choice of two harvest time options. If he harvests them at “cocktail size”, 12 weeks after planting, the yield will be 1,000 kg and he will receive a price of $1.25 per kilogram. However if they are left to grow to maturity at 24 weeks, the yield will be 2,500 kg and they will sell for 50c per kilogram. At an interest rate of 9% p.a. compounded weekly which option gives the higher net present value?

2. An investment is made at an initial cost of $50,000. It returns $15,000 after one

year and $45,000 after two years and then has no further returns. What is the internal rate of return for this investment?

3. A debt was originally due to be repaid with one payment of $10,000 in 4 years' time. Under new arrangements it will be repaid by installments of $x at the end of 1 year, $2x at the end of 3 years and $4x at the end of 5 years. Find x if interest is charged at a rate of 6% p.a. compounded quarterly.

7

WEEK 4

In-tutorial Test 1 is held during tutorial hour of this week. The test can be taken by all students who completed the “Lines and Inequalities” Adaptive Tutorial in Week 2. Make sure you attend your tutorial for the test. Material covers lectures 1-4.

TUTORIAL QUESTIONS

1. If 3 5 2 10 0.5 2 1

4 7 3 6 3 0.8 4A B C

Find (a) AB (b) BC (c) CA (d) BA if possible. If it is not possible to find any of these products give the reason why. 2. Find the determinants of the following matrices

(a) A =2 3

5 7

(b) B =

3 1 4

2 1 2

7 3 9

3. HPW 6.6, Problem 16, p. 283 4. Suppose that when 3 products (1, 2 and 3) are priced at p p p1 2 3, and , respectively,

the quantities demanded are q q qD D D1 2 3, , and respectively, and the quantities

supplied are q q qS S S1 2 3, , and respectively. The demand and supply functions are

interrelated and given by:

q p p p

q p p p

q p p

D

S

D

1 1 2 3

1 1 2 3

2 2 3

10 3 4 2

7 2 2

3 2 3

q p p

q p p p

q p p p

S

D

S

2 2 3

3 1 2 3

3 1 2 3

3 4

6 3 3

3 5 4

Write as three equilibrium equations then convert to matrix form, Ap = b. Find the

inverse of A then multiply to find the equilibrium prices (1p A b ). Finally find

8

the equilibrium quantities. Check your matrix inversion and multiplication using Excel. Remember to use array formulas as shown in the computing guide.

5. HPW 6.6, Problem 37, p. 283 6. If D, E , F and X are all matrices which have inverses, find X if 1 1DE XF D E SELF STUDY QUESTIONS 1. For the matrices

X Y

5

1

4

0

2

1

1 2 3

5 4 2

calculate XY and YX. What implication does this result have for solving matrix

equations? Show that XY Y XT T T . 2. HPW 6.3, Problem 66, p.249

3.

20 4 4 5 0 1

6 8 2 3 2 5A B C

In the matrix BAC what is the element 21

BAC ?

4. A, B, C, D, and X are all matrices which have inverses. Find X, if

CAXCABD 11 .

5. 4 6 3 2

1 9 5 7A B

.

Show that 1 1 1( )AB B A

6. Write the following linear systems in the form Ax = b. In which cases is it possible

to find 1A ? Where the inverse can be found, use this to solve for x. Where no inverse can be found give a reason.

9

136

52(a)

21

21

xx

xx

1 2 3

2 3

1 2 3

(b) 5 5 4 8

3 5

2 2 5

x x x

x x

x x x

1952

33

73(c)

21

21

21

xx

xx

xx

7. Solve the following system of equations by using Excel to find the inverse of the

coefficient matrix then multiplying.

1 2 3 4

1 3 4

1 2 3 4

1 2 3 4

4 5 2 5 108

2 6 250

2 2 190

4 3 5 9 400

x x x x

x x x

x x x x

x x x x

8. Calculate the determinants of the following matrices:

(a)

232

523

142

(b)

100

920

321

10

WEEK 5

Remember to complete online Quiz 2 this week (28 March -1 April). TUTORIAL QUESTIONS 1. HPW 6.6, Problem 16, p. 263 2. HPW 8.2, Problem 25, p. 374

3. HPW 8.1, Problem 8, p.363

4. HPW 8.1, Problem 21, p.363

5. A television station must schedule four programs for a particular night and has to decide which programs to show and the order in which they will run. The station has eight programs from which to choose. How many possible schedules are there?

6. Tram Loading. At a tourist attraction, two trams carry sightseers up a picturesque

mountain. One tram can accommodate seven people and the other eight. A busload of 18 tourists arrives, and both the trams are at the bottom of the mountain. Obviously, only 15 tourists can initially go up the mountain. In how many ways can the attendant load 15 tourists onto the two trams?

11

WEEK 6 In-tutorial Test 2 is held during tutorial hour of this week. Make sure you

attend your tutorial for the test. Material covers lectures 5-8. TUTORIAL QUESTIONS 1. (Adapted from a past exam question in QMB). Draw a probability tree to solve this problem. A qualifying test for a professional organisation was given at four locations. One thousand students sat for the test at each of locations A and B and 500 students sat for the test at each of locations C and D. Seventy percent of those who sat at location A passed the test. The percentages of students from locations B, C and D who passed were 75%, 65% and 72% respectively. If one student is selected at random from those who sat for the test, (a) what is the probability that the selected student passed the test? (b) if the selected student passed the test, what is the probability that the student sat at location C? 2. HPW 8.6, Problem 7, p.414

3. HPW 8.6, Problem 19, p.415

4. HPW 8.7, Problem 11, p.423

5. If is a transition matrix for a Markov chain, determine the values of a, b, and c.

6. is an initial state vector for the transition matrix

. Compute the state vector.

7. Find the steady state vector for the transition matrix

.

12

8. , find the probability of going from state 2 to state 1 after two steps.

SELF STUDY QUESTIONS 1. If a person watches a certain TV daily evening news program on one evening, then

the probability that the person watches that program the next evening is 0.7. However, if the person does not watch the program one evening, then the probability that the person watches the program the next evening is 0.2. (a) If the person watches the program on Monday, what is the probability that the person watches the program on Wednesday? (b) If 20% of the population watches the program on Thursday, what percentage can be expected to watch on Friday?

2. A college dining hall has available two fruit juices for breakfast: orange and

grapefruit. One hundred students drink juice on a daily basis. It is found that a student will not drink grapefruit juice on two successive days, and if a student drinks orange juice one day, then the following day the student is equally likely to drink orange juice as to drink grapefruit juice. If 40 of the regular juice drinkers drink orange juice on Monday, how many can be expected to drink orange juice on Wednesday?

3. is an initial state vector for the transition matrix

. Compute the state vector.

13

WEEK 7

You should work on Assignment Part A this week (11-17 April 2011).

TUTORIAL QUESTIONS 1. HPW 7.2, Problem 9, p.305 2. HPW 7.2, Problem 14, p.306 3. From QMA final exam June 2000 Excitement is spreading through the music industry as two legendary groups of the past have announced that they are simultaneously releasing new CDs. Supplies of the ADDA and Beetle albums will be limited due to manufacturing capacity. The owner of the only music store in a country town has had the following conditions imposed on her order for the first month after the release.

i. The minimum order for each group’s recording is 50 CDs. ii. No more than 400 Beetle albums will be available to each store.

iii. The total order for both albums cannot exceed 900. The owner also knows that past sales indicate that the number of ADDA fans in town is at least one third the number of Beetle fans so she will maintain this ratio in the number of albums ordered. The store will sell the ADDA CD for $30 and the Beetle double CD album for $45. It can be assumed that, due to the enormous publicity surrounding the releases, all albums ordered will be sold. How many of each album should be ordered to maximise the value of sales? (a) Show the objective function and all constraints.

(b) Draw a graph clearly marking the feasible region and at least one iso-objective line.

Find the number of albums from each group which should be ordered to maximise the value of sales.

(c) Suppose that the record company decided that instead of a limit of 400 Beetle albums, the store could order one extra album for each $5,000 of orders in the previous year. What would be the extra sales that would result from each extra album ordered and up to what limit would this value apply.

(d) If the store owner decided to use the Excel Solver function to solve this problem she would first need to provide an arbitrary value of the objective function. If the values in cells A2 and B2 below reflect arbitrary order numbers, what Excel formula (with cell addresses) should be used in C2 to calculate sales?

14

A B C 1 ADDA Beetles Sales 2 100 200 3

4. HPW 7.3, Problem 3, p.309

5. HPW 7.3, Problem 4, p.309

SELF STUDY QUESTIONS 1. (Part of Question 5 June, 1997 Exam) The Humongous Hamburger chain employs staff at two wage rates, $12 per hour

for seniors and $8 per hour for juniors. A store manager wishes to minimise the cost of wages over the next month. She must, however, satisfy the following conditions in setting staff levels.

The number of junior staff hours must be at least three times the number of senior staff hours. In any one month the total number of hours the staff are employed must be no less than 1% of the number of hamburgers sold. The expected number of hamburgers to be sold next month is 165,000. The store has contracted to employ seniors for at least 120 hours next month.

(a) Write down the objective function and constraints for this problem.

(b) Draw a graph, showing junior hours on the horizontal axis, which clearly

shows the feasible region and at least one iso-objective line. (c) Determine the combination of staff hours which will minimise the store’s

wages cost for the coming month.

15

WEEK 8 Assignment Part A should be submitted during this week (18-22 April).

TUTORIAL QUESTIONS 1. SOLVER “Transport authorities have promoted the use of buses and trains to those who will attend an event at a new showground. While this is desirable to avoid traffic problems, it offers the transport authority an opportunity to gain revenue from public transport. A maximum of 170,000 people are expected to arrive at the showground in any one day, but at least 36,000 will use private transport. The number of buses that can arrive in one hour is 60, each with a capacity of 70 persons. The number of trains that can arrive per hour is 20, each carrying up to 500 persons. Buses and trains arrive at the showground for 10 hours per day. The number of people travelling by bus is at least 25% of the number coming by train. Bus tickets cost $6 and train tickets cost $4 per person. Determine how many bus travellers and how many train travellers per day will maximise the transport authority’s revenue and state the maximum revenue.” The above linear programming problem is being solved using Excel Solver. Try to write down the objective function and constraints on paper then look at the screenshots below and answer the questions. (a)

16

Guess values for x (no. bus travellers) and y (no. train travellers) have been entered in A2 and B2. Write down the formulae that have been used in cells C2 and B4:B9. (Hint: Constraint 1 relates to the total number of public transport passengers, Constraint 2 relates to bus capacity, Constraint 3 relates to train capacity, ...) (b)

Show how this dialog box should be completed for each of the six constraints. (c)

17

Use the answer report to explain how many people must travel by bus and by train to produce maximum revenue and what is the maximum revenue. (d) Look at the Status column in the Answer Report and determine for what form of transport there is extra capacity when revenue is optimised. 2. HPW 10.2, Problem 57, p.476

3. HPW 10.3, Problem 34, p.482

4. HPW 10.4, Problem 23, p.486

5. HPW 11.4, Problem 72, p.526

6. HPW 11.5, Problem 80, p.533

7. HPW 12.3, Problem 12, p.554

SELF STUDY QUESTIONS

1. (from Final exam June 2000) (a) If revenue is given by R = pq where p = price and q = quantity, show using the

product rule that 11 where is point elasticity of demand.

dRp

dq

(b) Demand for haircuts in a certain chain of salons is given by the function 200

for 1110

q pp

Where q is the number of haircuts per day in one salon and p is the price of each haircut. Calculate the elasticity of demand when p = 15. (c) After a 10% GST is introduced, the price of haircuts will rise by the full 10% as there are no offsetting decreases in wholesale sales tax. Assuming that there is no change in the demand function, show whether turnover (i.e. revenue including the tax) will rise or fall when the price increases.

2. A surfboard manufacturer knows that the weekly cost of production, C, and revenue, R, are functions of output, q, given by:

qqR

qC

40058.0

8001550

2

Evaluate: (a) profit; (b marginal cost; (c) marginal revenue; (d) average cost when 30 surfboards are produced per week.

18

WEEK 9

Remember to complete online Quiz 3 and work on Assignment Part B this week (2-6 May).

Feedback for Assignment Part A will be given during tutorial.

TUTORIAL QUESTIONS 1. The supply of q units of a product at a price of p dollars per unit is given by

)12ln(*1025)( ppq . Find the rate of change of supply w.r.t. price, dq/dp. 2. HPW 12.2, Problem 26, p.549

3. HPW 12.4, Problem 11, p.560

4. HPW 12.4, Problem 23, p.560 5. HPW 12.7, Problem 34, p.561 6. Marginal cost: if 85023.0 2 qqc is a cost function, how fast is the marginal

cost changing when q=100?

19

WEEK 10 Assignment Part B should be submitted during this week (9-13 May). Remember that you must complete the “Linear Programming”

Adaptive Tutorial, which will give you access to the In-Tutorial Test in Week 4, worth 8% of your final mark.

TUTORIAL QUESTIONS 1. HPW 13.3, Problem 70, p.597 2. HPW 14.1, Problem 46, p.631: Review material

3. (From June 1998 exam)

A company has a single product which sells for a price of $(50-0.01q) per unit, where q is the quantity produced and sold. It has fixed costs of $35,000 per month and

variable costs of $

2

1

20 q per unit. How many units should it sell per month to

maximise profit? (Check the second order conditions for a maximum.) 4. HPW 14.5 Problem 68, p.652

5. A certain country’s marginal propensity to save is given by 2^3

8

1

2

13 IdI

dS where S

and I represent total national savings and income, respectively and are measured in billions of dollars.

a) Determine the marginal propensity to consume when total national income is $81 billion.

b) Determine the consumption function, given that savings are $3 billion when total national income is $24 billion.

c) Use the result in part (b) to show that consumption is $54.9 billion when total national income is $81 billion.

d) Use differentials and the results in parts (a) and (c) to approximate consumption when total national income is $78 billion.

6. (From June 1998 exam)

Suppose a company’s marginal cost function is given by qeMC 05.0616 ,

where q is the number of units of output produced. If the company is known to have fixed costs of $15,000, what is the average cost of producing100 units of output?

20

SELF STUDY QUESTIONS 1. The Springhill Spa company can sell 3000 litres of mineral water per day if it charges

80c per litre. For each extra 5c rise in price it sells 100 fewer litres of water per day. What selling price will maximise the company's income? What is the maximum daily income?

2. A fast food chain has been petitioned by environmentalists to reduce the amount of polystyrene used in packaging. The company wishes to design a new chicken container that will minimise the area of polystyrene needed. The box must have a volume of 800 cubic cm. It will be made from a single T shaped piece of material which can be folded to form a top, bottom, four sides and a tuck-in flap. The flap has the same dimensions as one of the sides. The top and bottom are equal sized squares. What are the dimensions of the box that minimise the area of polystyrene used?

3. A monopolist producing a single commodity has a production function given by

q x 1 2/ , where q is the quantity produced and x is the amount of the sole input used. The monopolist can buy any amount of the input at the per unit price of 2, and faces an industry (inverse) demand function given by p q 30 3 , where p is the price per unit of the output. Assume there are no fixed costs. If all output is sold and the production level is set to maximise profits, what will the market price be? Also derive an expression for the marginal revenue product of the input.

(From final exam Session 1, 1993) 4. Suppose that the sales revenue from a particular cultured pearl harvest is (6 + 2.4t)

thousand dollars if the pearls are harvested t years after implantation in the oysters. If the discount rate is 6% per annum compounded continuously, and the present value of sales revenue is to be maximised, when should the pearls be harvested and sold (expressed as number of years after implantation). Do not bother to check the second order condition for a maximum.

5. Evaluate the definite integrals

(a) x

x xdx

1

2 3230

1 (b) 3

12

1t

tdt

e

(c) te dtt 2

0

1

21

WEEK 11

In-tutorial Test 3 is held during tutorial hour of this week. The test can be taken by all students who completed the “Linear Programming” Adaptive Tutorial in Week 10. Make sure you attend your tutorial for the test. Material covers lectures 9-18.

TUTORIAL QUESTIONS 1. Find the area of the region bounded by the graphs of the given two equations.

.1523,3 yxxy 2. HPW 15.5, Problem 16, p.713 3. HPW 15.5, Problem 36, p.714 4. HPW 15.6, Problem 5, p.720

5. (From November 1998 exam) (a) In a country experiencing financial instability a rumour circulates that there is to be a

large devaluation of the currency. Those who hear the rumour rush to buy imported goods or US dollars. The number of people who have heard the rumour grows at a rate proportional to the product of the number who have heard it and the number who have not heard it. The population of the country is 50 million. Half a million have heard the rumour after one day and 2.5 million have heard it after 2 days. How long will it take before half the population is informed? (It is not necessary to show integration steps in part (a)).

(b) Due to the rumour and the panic buying it sets off, stocks of canned food available for

sale in the country fall at a rate proportional to their current level at any time. Originally 20 million tonnes of canned food were held. After two days the level had fallen to 15 million tonnes. Using integration derive a formula for the number of million tonnes, T, available at any time. How long will it take for the stock available to fall to 1 million cans?

SELF STUDY QUESTIONS 1. (Part of Problem 7 from June 1995 exam) Last year Desert Bloom sold its harvest of 300 tonnes of Nice’N’Salty wheat at a

price of $1800 per tonne. A market study undertaken last year by the firm of accountants Arthur Sanderson estimated that the price elasticity of annual demand for Nice’N’Salty wheat is given by

22

p

p( )2400 for p < 2400

where p is the price per tonne. Assuming the price elasticity expression above is correct, show that the annual

demand for Nice’N’Salty wheat as a function of p is given by

Qp

12002

where Q is the annual production in ones.

2. For each of the following differential equations, find the general solution and the

particular solution corresponding to the given initial condition.

(i) dy

dxx y y 3 0

1

22 ,

(ii) dy

dx

e

ey n

x

y , 0 2

(iii)

dy

dx

x

yy

1

2

25

0 7,

3. A country town, affected by the withdrawal of key manufacturing industries, has had a declining population for a considerable period. The rate of change at any time has been proportional to the population at that time. At 30th June, 1990 the population was 24,500 and four years later it was 18,000. At the current rate of change how long will it take for the town’s population to decline to 6,000 people?

23

WEEK 12

Remember to complete online Quiz 4 this week (23-27 May). Feedback for Assignment Part B will be given during tutorial.

TUTORIAL QUESTIONS 1. Determine the indicated function values for the given functions.

),(;),(

)4,5,3,(;0,,,(

00

22

yhxfeyxf

hut

rsutsrh

yx

2. HPW 17.1, Problem 21, p.754

3. HPW 17.1, Problem 39, p.754 4. HPW 17.2, Problem 9, p.758 5. HPW 17.2, Problem 13, p.759

6. HPW 17.5, Problem 17, p.768

SELF STUDY QUESTIONS

1. Find the local extrema of the function f x x x x x x1 2 1

3

2

3

1 28, .

24

WEEK 13 TUTORIAL QUESTIONS 1. HPW 17.6, Problem 35, p.776 2. HPW 17.7, Problem 5, p.783 3. Consider a problem very much related to the output-maximizing manager discussed

in the lecture. Again, production requires capital (K) and labour (L). The output level is given by the following function.

As before, capital costs r per unit, and labour costs w per unit, implying that total expenditure on inputs is:

As a manager, you are told to produce an output of . Your problem is to choose K and L so as to produce this output at the lowest possible cost.

a. Set up the problem, writing out the objective function, the constraint, and the Lagrangian function.

b. Write the system of first-order conditions. Is this a linear system?

c. Calculate the solution. Verify it makes sense.

d. Verify the second-order condition is satisfied when r=w= =1. e. Interpret the value of calculated in part 3.

4. An entrepreneur has 5 hours each day to devote to her business. Her business involves two types of tasks. If she devotes x hours to the first and y hours to the second, then her profit is

Her problem is to choose x and y in order to maximize subject to her time constraint.

a. Write out the entrepreneur’s time constraint. b. Set up the Lagrangian function c. Write out the first-order conditions in matrix form. d. Derive the optimal time allocation, and verify that the second-order conditions hold. e. Interpret the value of the Lagrange multiplier.

SELF STUDY QUESTIONS 1. A certain postal service only accepts rectangular packages having the sum of the

length and girth (i.e. perimeter of side with the smallest area) no greater than 120cm.

25

Find the dimensions (length width height) of the greatest volume rectangular package satisfying this requirement. Try to solve this problem without using the Lagrange multiplier method.

2. (Part of Problem 6, June 1994 exam) The Shiver family, which lives in a relatively cold region of Oz, has found from

experience that its home must be artificially heated for 1600 hours during the three months of winter. The Shivers’ home is equipped with both a gas heating system and an electrically powered reverse-cycle air conditioner (which can also be used for home heating).

Let x1 and x2 be the number of hours of home heating using gas and electricity,

respectively, for the Shivers’ home during winter. Also assume that there is no advantage to be gained from operating the two heating systems simultaneously.

The new resource pricing schemes are such that the prices paid per hour by the

Shivers for home heating during the winter months are given by 01 0 0005 1. . x (price of heating per hour of gas heating) 0 2 0 0003 2. . x (price per hour of heating with air conditioner) Determine the combination of x1 and x2 which minimizes the cost of heating the Shivers’ home 1600 hours during winter. What is this minimum cost? You need to check the second order conditions as well.