Teori Mikroekonomi 1 (Microeconomics).docx

MICROECONOMICS 1 MODULE

TEACHING ASSISTANTS OF MICROECONOMICS AND MACROECONOMICS ECONOMICS AND DEVELOPMENT STUDIES FACULTY OF ECONOMICS AND BUSINESS PADJADJARAN UNIVERSITY 2012 1

ACKNOWLEDGEMENT In the name of Allah, The Most Gracious, The Most Merciful Alhamdulillah, all praises to Allah SWT, The Almighty, for giving belief, health, confidence and blessing for the writers to accomplish this Module of Microeconomics I. Shalawat and Salam be upon our Prophet Muhammad SAW, who has brought us from the darkness into the brightness and guided us into the right way of life. In this opportunity, we also like to express our deep thanks to Dr. Kodrat Wibowo, S.E. as the Head Department of Economics, Dr. Mohamad Fahmi, SE., MT as the Head of Undergraduate Program of Department of Economics, lecturers, and those who contributed and helped in the process of making this module. All of your \kindness and help means a lot to us. Thank you very much We realise that the contents in this module is not that perfect. Therefore, we are willing to receive and consider feedback, suggestions and constructive criticisms, and eager to implement improvements. Hopefully this module can be the short guide for the students in order to deepen the understanding and the analysis of Microeconomics I theory. Thank you. List of the Module Writers: 1. 2. 3. 4. 5.

Iqbal Dawam Wibisono Nedia Nurani Rahma Citra Kumala Fierera Devi Febiosa

120210100156 120210110041 120210110124 120210110155 120210120012 Acknowledge and Agree, Head of Undergraduate Program of Department of Economics

Dr. Mohamad Fahmi, SE., MT NIP19731230200012100

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TABLE OF CONTENTS MICROECONOMICS 1 MODULE.................................................1 ACKNOWLEDGEMENT...............................................................2 TABLE OF CONTENTS...............................................................3 MODULE AND LABORATORY GUIDANCE....................................4 REVIEW OF DIFFERENTIAL CALCULUS AND CONSTRAINED OPTIMIZATION..........................................................................5 PREFERENCE, UTILITY, AND UTILITY FUNCTION.........................9 UTILITY MAXIMIZATION AND CHOICE I & II..............................12 THE THEORY OF OPTIMUM CONSUMER’S CHOICE I & II.............16 UNCERTAINTY AND INFORMATION...........................................19 PRODUCTION FUNCTION..........................................................24 COST MINIMIZATION................................................................28 PROFIT MAXIMIZATION AND PARTIAL EQUILIBRIUM COMPETITIVE MODEL................................................................................... 33 PARTIAL EQUILIBRIUM COMPETITIVE MODEL...........................38

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MODULE AND LABORATORY GUIDANCE 1.

This module was arranged as a media to help the students deepen their understanding during the laboratory session of Microeconomics 1. 2. This module could only be used during the laboratory of Microeconomics 1. 3. The students are not allowed to bring and copy the module unless they get permission from the Team of Teaching Assistant. 4. For any reasons, the students are not allowed to write anything in the module unless they get permission from the Team of Teaching Assistant. 5. The answers are written on the answer sheet/other paper that has been provided by the Team of Teaching Assistant. 6. The materials in each laboratory meeting is adjusted based on the material that has been given by each of the lecturers in the class. 7. During the laboratory, all of the students should obey the rules that has been made by each of the Teaching Assistant. 8. The maximum duration for Laboratory is 2.5 hours (180 minutes) 9. For any incorrect or unclear questions that you found difficult, please re-read the appropriate question or ask directly to the Teaching Assistant to clear up any confusion. 10. After successfully finishing the problems, the students can leave the laboratory room with the permission from the Teaching Assistant. 11. Here below we kindly inform the general rule during the laboratory:  The laboratory has 10 (ten) meetings. The Teaching Assistant will take only 7 (seven) best mark and one other mark that comes from the Review in the 10th meeting.  The students are not allowed to change their laboratory schedule without any permission from their Teaching Assistant.  The students are not allowed to cheat, work together, and open the book/note while solving the problems in the laboratory.  Other rules that are agreed by the Teaching Assistant and the students in each laboratory.

Team of Teaching Assistant of Microeconomics 1

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CHAPTER 1 REVIEW OF DIFFERENTIAL CALCULUS AND CONSTRAINED OPTIMIZATION 3

1.

Differentiate

y=( x +7 x−1)(5 x+ 3) .

2.

Differentiate

y=x−2 ( 4+ 3 x −3 ) .

3.

Differentiate

y=x ln x .

4.

Differentiate

f ( x )=6 x 2 /3 tan x .

5.

Differentiate

y=5 x 2+ sin x cos x .

6.

Differentiate

x g ( x ) =e ( 7−√ x ) .

7.

Differentiate

y=7 x e z .

8.

Differentiate

f ( x )=(x+ 8) sec ⁡( 3 x ) .

9.

Differentiate

y=23 x+1 ln ( 5 x−11 ) .

3

2

4

10. Differentiate

y=x 2 sin3 ( 5 x ) .

11. Differentiate

y=( x3 −7 x 2)4 (1+ 9 x )1/ 2 .

12. Differentiate

y=sec 2 ( x 4 ) tan 3 ( x 4 ) .

5

13. Differentiate

y=

2 . x+ 1

14. Differentiate

y=

x2 . 3 x−1

15. Differentiate

y=

4 x 3−7 x . 2 5 x +2

16. Differentiate

y=

4 sin x . 2 x +cos x

17. Differentiate

y=

7 x2 . 4 e x −x

18. Differentiate

g (x )=

1+ ln x . x 2−ln x

2x ( ) g x = . 19. Differentiate 2 x −3 x (x 2−1)3 20. Differentiate f ( x )= (x 2+ 1) . 5 e−x ( ) f x = . 21. Differentiate x+ e−2 x

22. Differentiate

x 3 ln x y= . x +2 6

x 2(2 x−1)3 f ( x ) = . 23. Differentiate ( x 2 +3)4

24. Differentiate

25. Differentiate

26. Differentiate

g (x )=

f ( x )=

1 . x √ x 2 +1

√

3 x +2 . 2 x−1

y=3 x 4 + tan

x ( x−1 ).

−x

27. Differentiate

y=x 2 e x+1 .

28. Find an equation of the line tangent to the graph of

y=

x3 x 2−2

at

graph

of

x=1. 29. Find

an

equation

sin ⁡(2 x ) y= cos ( 3 x )+ sec x

of

the

line

tangent

to

the

Φ x= . at 6

x2 ( ) f x = f ' ( x )=0 for 2 x . Solve 30. Consider the function e f ' ' ( x )=0 for x .

7

x . Solve

31. Find all points

(x , y )

on the graph of

8 x+ 2 y =1.

lines are perpendicular to the line 32. Differentiate

y=(3 x+1)2 .

33. Differentiate

y=√ 13 x 2−5 x +8.

34. Differentiate

y=(1−4 x+7 x 5 )30 .

35. Differentiate

y=(4 x + x−5)1 /3 . 6

36. Differentiate

8 x−x 3 x ¿ ¿ y=¿

37. Differentiate

y=sin ⁡( 5 x) .

38. Differentiate

y=e5 x +7 x−13 .

39. Differentiate

y=2cos x .

40. Differentiate

y=3 tan √ x .

41. Differentiate

y=ln ( 17−x ) .

f ( x )=

2

8

x−1 2−x where tangent

x 4 +cos ¿. 42. Differentiate y=log ⁡¿ 43. Differentiate

y=cos 2 ( x 3 ) .

44. Differentiate

y=

45. Differentiate

y=ln ( cos5 ( 3 x 4 ) ) .

46. Differentiate

y=√ sin ⁡( 7 x+ ln ( 5 x ) ).

( 51 ) sec

−4

( 4+ x 3 ) .

2−(6+7 x 4 ) ¿ 1+ ¿ 47. Differentiate ¿ y=10 ¿ x ln ⁡ ( ln ⁡ ( sec ¿)) . 48. Differentiate y=4 ln ⁡¿ 49. Differentiate

50. Assume that

y=tan

3

√cos ⁡(7 x ).

h ( x )=f ( g ( x ) ) ,

differentiable functions. If the value of

where both

f and

g

are

g (−1 )=2, g' (−1 )=3,∧f ' ( 2 )=−3, what is

h' (−1 ) ? 9

51. Assume that

f ( 0 )= graph of

h ( x )=(f ( x ) )3 , where f

−1 ' 8 ∧f ( 0 ) = 2 3

is a differentiable function. If

determine an equation of the line tangent to the

h at x=0 .

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CHAPTER 2 PREFERENCE, UTILITY, AND UTILITY FUNCTION 

When individual reports that “A preferred to B” its taken to mean that all things considered, he or she feels better off under situation A than situation B. There are three basic properties of preference relation assumption: 1 2 3



Completeness: if A and B are any two situation, the person can chose three possibilities: “A is preferred to B”; “B is preferred to A” ; or “A=B. Transitivity: the individual’s choice are internally consistent, “A is preferred to B” ; “B is preferred to C” ; so “A is preferred to C”. Continuity: If an individual reports “A is preferred to B” , then situation suitably “close to” A must also be preferred to B. individual’s preferences are assumed to be represented by a utility function of the form: U (x1,x2,…,xn).

Utility, when people are able to rank in order all possible situations from the least desirable to the most. The situations offer more utility than the other. Utility = U (W).

 

The cateris paribus assumption is holding constant the other things that effect behavior (other things being equal). Indifferent curve represents those combination of x and y from which the individual derives the same utility. The slope of this curve represents the rate of which individual is willing to trade x for y while remaining equally well off. The negative of the slope of an indifferent curve at the same point is termed the marginal rate of substitution. MRS = -

dy dx

U = U1



Cobb-Douglas Utility,

U ( x , y ) = xα yβ



Perfect Substitution,

U ( x , y ) = αx+ βy



Perfect Complement,

U ( x , y ) = min (αx, βy)



CES Utility ,

U ( x , y )=ln x +ln y 11

CHAPTER 2 PREFERENCE, UTILITY, AND UTILITY FUNCTION 1

Graph a typical indifference curve for the following utility function and determine whether they have convex indifference curve (that is, whether the MRS declines as

x increses)!

a

U ( x , y )= √ x 2− y 2

b

U ( x , y )=

xy x+ y

U ( x , y )=ln x +ln y has a diminishing MRS!

2

Show that

3

A consumer has a utility function

u ( x 1, x 2 )=max ( x 1, x 2 ) . What is the

consumer's demand function for good l? What is his indirect utility function? What is his expenditure function? 4

Suppose that a person has initial amounts of the two goods that provide utility to him or her. This initial amounts are given by a b

´x and

Graph is initial amounts on this person’s indifference curve map! If this person can trade x for y (or vice versa) with other people, what kind of trade would he or she voluntarily make? How do these trades relate to this person’s MRS at the point ( ´x ,

5

´y .

´y ) ?

A consumer has an indirect utility function of the form

v ( p1 , p2 , m )=

m min ⁡( p 1 , p2)

What is the form of the expenditure function for this consumer? What is the form of a (quasiconcave) utility function for this consumer?

12

6

Consider the indirect utility function given by

v ( p1 , p2 , m )=

m ( p1 + p2 )

(a) What are the demand functions? (b) What is the expenditure function? (c) What is the direct utility function? 7

A consumer has a direct utility function of the form

U ( x 1, x 2 )=u ( x 1 ) + x 2 Good 1 is a discrete good; the only possible levels of consumption of good 1are

x 1=0 and x 1=1 . For convenience, assume that u ( 0 )=0

and

p2=1 .

(a) What kind of preferences does this consumer have? (b) The consumer will definitely choose

x 1=1 if

p1 is strictly less

than what? 8

A consumer has an indirect utility function of the form

v ( p , m ) =A ( p ) m (a) What kind of preferences does this consumer have? (b) What is the form of this consumer's expenditure function 9

Show that the CES Function δ

δ

x y α +β δ δ

is homotetic. How does the MRS depend on the rasio y/x? 10 Two goods have independent marginal utility if

13

e (p ,u) ?

∂2 U ∂2 U = =0 ∂y∂x ∂ y∂x Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing MRS. Provide an example to show that the converse of this statement is not true.

CHAPTER 3 UTILITY MAXIMIZATION AND CHOICE I & II 

To maximize utility, given a fixed amount of income to spend, an individual will buy those quantities of goods that exhaust his or her total income and for which the psychic rate of trade-off between any two goods (the MRS) is equal to the rate at which the goods can be traded one for the other in the marketplace.



To reach a constrained maximum, an individual should:  spend all available income  choose a commodity bundle such that the MRS between any two goods is equal to the ratio of the goods’ prices

14



the individual will equate the ratios of the marginal utility to price for every good that is actually consumed



The marginal rate of subsitution (MRS) of goods X and Y is the maximum amount of goods X that a person is willing to give up to obtain 1 additional unit of Y. The MRS diminishes as we move down along an indifference curves. When there is a diminishing MRS, indifference curves are convex.



Consumers maximize satisfaction subject to budget constraint. When a consumer maximizes satisfaction by consuming some of each of two goods, the marginal rate of substitution is equal to the ratio of the prices of the two goods being purchased.



Maximization is sometimes achieved at a corner solution in which one good is not consumed. In such cases, the marginal rate of substitution need to equal the ratio of the prices.



The individual’s optimal choices implicitly depend on the parameters of his budget constraint  choices observed will be implicit functions of prices and income  utility will also be an indirect function of prices and income



Demand functions show the dependence of the quantity of each goods demanded on

p1 , p2 , ….. , pn∧I

maximum utility=U ( x ¿1 , x ¿2 , … , x ¿n ) ¿ V ( p1 , p2 , … , p n , I ) 

The dual problem to the constrained utility-maximization problem is to minimize the expenditure required to reach a given utility target  yields the same optimal solution as the primary problem  leads to expenditure functions in which spending is a function of the utility target and prices



Expenditure function is the individual’s expenditure function shows the minimal expenditures necessary to achieve a given utility level for a particular set of prices.

minimal expenditure=E ( p 1 , p2 , … … , pn , U ) 15



Properties of expenditure functions :  Homogeneity  Expenditure functions are nondecreasing in prices  Expenditure functions are concave in prices

CHAPTER 3 UTILITY MAXIMIZATION AND CHOICE I & II

1

What is utility maximization? Graph and show where is the optimal quantity of x and y that maximize utility.

16

2

A consumer has a utility function

u ( x 1 , x 2) =max ⁡{x 1 , x 2 } . What is

the consumer's demand function for good l? What is his indirect utility function? What is his expenditure function? 3

A consumer has an indirect utility function of the form

v ( p1 , p2 , p 3 )=

m min ⁡{ p1 , p 2 }

What is the form of the expenditure function for this consumer? What is the form of a (quasiconcave) utility function for this consumer? What is the form of the demand function for good l?

4

Explain mathematically first order condition for a maximum utility (for two goods)

5

Consider the indirect utility function given by

v ( p1 , p2 , m )=

m p 1+ p 2

(a) What are the demand functions? (b) What is the expenditure function? (c) What is the direct utility function? 6

A young connoisseur has $300 to spend to build a small wine cellar. She enjoys two vintages in particular : a 1997 French Bordeux (W F) at $20 per bottle and a less expensive 2002 California varietal wine (W C) priced at $4. How much of each wine should she purchase with Langrangian expression if her utility is: U (WF, WC ) = WF 2/3 WC 1/3

7

A person has an income $100. His use his money to buy good x and y. Price of good x is $10 and price of good y is $20. a b

Make the budget constraint equation Suppose that income increase 50%. Make the new budget constraint

17

c

What happen if price x decrease until 20% (with the first income given). Make a new budget constraint Continuing from part c, now price y increase 25%. Make a new budget constraint. Graph them !

d e 8

A consumer has a direct utility function of the form

U ( x 1 , x 2) =u ( x 1 )+ x2 Good 1 is a discrete good; the only possible levels of consumption of good 1 are

x 1=0

and

x 1=1 .

For

convenience,

assume

that

u ( 0 )=0∧ p2=1 . (a) What kind of preferences does this consumer have? (b) The consumer will definitely choose

x 1=1

if

p1 is strictly less

than what? (c) What is the algebraic form of the indirect utility function associated with this direct utilityfunction?

9

George has $300 to spend to buy book and novel. Price of book is $ 4 and price of novel is $12. How much the MRS between book and novel? How much of each book and novel should he purchase with Langrangian expression if his utility is: U ( b,n ) =

b1 /2 n1 /2

10 A person has utility function U (x,y) = x 0.4y0.8 for good x and y. Assume he has an income $100. Price of good x is $ 4 and price of good y is $12. a Show MRS between good x and good y b Calculate optimum combination of good x and good y to maximize utility

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CHAPTER 4 THE THEORY OF OPTIMUM CONSUMER’S CHOICE I & II 



In this chapter we used the utility maximizing model of choice to examine relationship among consumer goods. Although these relationship may be complex, the analysis presented here provided a number of ways of categorizing and simplyfying them. When there are only two goods, the income and substitution effects from the change in the price of one good (py) on the demand for another good (x) usually work in opposite directions; the sign of



two goods are gross substitutes if

δxi ∂ pj 

 

δxi ∂ pj

> 0 and gross complements if