MICROECONOMICS 1 MODULE

TEACHING ASSISTANTS OF MICROECONOMICS AND MACROECONOMICS

ECONOMICS AND DEVELOPMENT STUDIES

FACULTY OF ECONOMICS AND BUSINESS

PADJADJARAN UNIVERSITY

2012

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ACKNOWLEDGEMENTIn 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. Iqbal Dawam Wibisono 1202101001562. Nedia Nurani 1202101100413. Rahma 1202101101244. Citra Kumala 1202101101555. Fierera Devi Febiosa 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 1REVIEW OF DIFFERENTIAL CALCULUS AND CONSTRAINED

OPTIMIZATION

1. Differentiate y=(x3+7 x−1)(5 x+3).

2. Differentiate y=x−2 ( 4+3 x−3 ) .

3. Differentiate y=x3 ln x .

4. Differentiate f ( x )=6 x2 /3 tan x .

5. Differentiate y=5 x2+sin xcos x .

6. Differentiate g ( x )=ex (7−√x ) .

7. Differentiate y=7 x ez2

.

8. Differentiate f ( x )=(x+8)4 sec (3 x).

9. Differentiate y=23 x+1 ln (5 x−11) .

10. Differentiate y=x2 sin3 (5x ) .

11. Differentiate y=(x3−7 x2)4 (1+9 x)1 /2 .

12. Differentiate y=sec2 ( x4 ) tan3 ( x4 ) .

13. Differentiate y= 2x+1

.

14. Differentiate y= x2

3 x−1.

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15. Differentiate y= 4 x3−7 x5 x2+2

.

16. Differentiate y= 4 sin x2 x+cos x

.

17. Differentiate y= 7 x2

4 ex−x.

18. Differentiate g ( x )= 1+ ln x

x2−ln x.

19. Differentiate g ( x )= 2x

2x−3x .

20. Differentiate f ( x )=(x2−1)3

(x2+1).

21. Differentiate f ( x )= 5 e−x

x+e−2 x .

22. Differentiate y= x3 ln xx+2

.

23. Differentiate f ( x )= x2(2 x−1)3

(x2+3)4 .

24. Differentiate g ( x )= 1

x √ x2+1.

25. Differentiate f ( x )=√ 3 x+22 x−1

.

26. Differentiate y=3 x4+ tan( xx−1 ) .

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27. Differentiate y=x2 e−xx+1 .

28. Find an equation of the line tangent to the graph of y= x3

x2−2 at x=1.

29. Find an equation of the line tangent to the graph of y=sin (2 x )

cos (3 x )+sec x at

x=Φ6

.

30. Consider the function f ( x )= x2

e2 x . Solve f ' ( x )=0 for x . Solve f ' ' ( x )=0

for x .

31. Find all points (x , y ) on the graph of f ( x )= x−12−x

where tangent lines are

perpendicular to the line 8 x+2 y=1.

32. Differentiate y=(3 x+1)2 .

33. Differentiate y=√13 x2−5 x+8.

34. Differentiate y=(1−4 x+7 x5)30 .

35. Differentiate y=(4 x+x−5)1/3 .36. Differentiate y=¿

37. Differentiate y=sin (5 x).

38. Differentiate y=e5 x2+7 x−13 .

39. Differentiate y=2cos x .

40. Differentiate y=3 tan√x .

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41. Differentiate y=ln (17−x ) .

42. Differentiate y=log ¿

43. Differentiate y=cos2 ( x3 ).

44. Differentiate y=( 15 )sec−4 (4+x3 ).

45. Differentiate y=ln (cos5 (3 x4 ) ) .

46. Differentiate y=√sin (7 x+ln (5 x )) .

47. Differentiate y=10¿

48. Differentiate y=4 ln ¿

49. Differentiate y=tan3 √cos (7 x) .

50. Assume that h ( x )=f (g ( x ) ) , where both f and g are differentiable

functions. If g (−1 )=2 , g' (−1 )=3 ,∧f ' (2 )=−3 ,what is the value of

h' (−1 )?

51. Assume that h ( x )=(f ( x ))3 ,where f is a differentiable function. If

f (0 )=−12

∧f ' (0 )=83

determine an equation of the line tangent to the

graph of 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. 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.

2. Transitivity: the individual’s choice are internally consistent, “A is preferred to B” ; “B is preferred to C” ; so “A is preferred to C”.

3. 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 = - dydx

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

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CHAPTER 2PREFERENCE, 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 )=√x2− y2

b. U ( x , y )= xyx+ y

2. Show that U ( x , y )=ln x+ ln y has a diminishing MRS!

3. A consumer has a utility function u ( x1 , x2 )=max ( x1 , x2 ). 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 x and y .

a. Graph is initial amounts on this person’s indifference curve map!b. 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, y) ?

5. A consumer has an indirect utility function of the form

v ( p1 , p2, m )= mmin ( p1 , p2)

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

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?

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7. A consumer has a direct utility function of the formU ( x1 , x2)=u ( x1 )+ x2

Good 1 is a discrete good; the only possible levels of consumption of good 1are x1=0 and x1=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 x1=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 e ( p , u)?

9. Show that the CES Function

αxδ

δ+ β

yδ

δ

is homotetic. How does the MRS depend on the rasio y/x?

10. Two goods have independent marginal utility if

∂2 U∂ y∂ x

= ∂2U∂ y∂ x

=0

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.

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CHAPTER 3UTILITY 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 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

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maximumutility=U (x1¿ , x2

¿ , …, xn¿ )

¿V ( p1 , p2 ,…, pn , 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( p1 , p2 , ……, pn ,U )

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

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CHAPTER 3UTILITY 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.

2. A consumer has a utility function u ( x1, x2 )=max {x1 , x2}. 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, p3 )= mmin {p1, p2 }

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 )= mp1+ p2

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

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6. A young connoisseur has $300 to spend to build a small wine cellar. She enjoys two vintages in particular : a 1997 French Bordeux (WF) at $20 per bottle and a less expensive 2002 California varietal wine (WC) 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. Make the budget constraint equationb. Suppose that income increase 50%. Make the new budget constraintc. What happen if price x decrease until 20% (with the first income given).

Make a new budget constraintd. Continuing from part c, now price y increase 25%. Make a new budget

constraint.e. Graph them !

8. A consumer has a direct utility function of the form

U ( x1 , x2 )=u ( x1 )+x2

Good 1 is a discrete good; the only possible levels of consumption of good 1 are x1=0 and x1=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 x1=1 if p1is 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) = b1/2n1 /2

10. A person has utility function U (x,y) = x0.4y0.8 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

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b. Calculate optimum combination of good x and good y to maximize utility

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 δx

∂ py is ambiguous, the substitution

effect is positive, the income effect is negative. In cases of more than two goods, demand relationships can be specified in two

ways

two goods are gross substitutes if δxi∂ pj

> 0 and gross complements if δxi∂ pj

<

0 because these price effects include income effects, they may not be

symmetric; it is possible that δxi∂ pj

≠ δxj∂ pi

If a group of goods has prices that always move in unison, expenditures on these goods can be treated as a “composite commodity” whose “price” is given by the size of the proportional change in the composite goods’ prices.

An alternative way to develop the theory of choice among market goods is to focus on the ways in which market goods are used in household production to yield utility-providing attributes.

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A composite comodity theorem applies to any group of commodities whose relative price all move together. It is possible to have more than one such commodity if there are several groupings that obey that theorem.

Slutsky-type Equation :

or, in elasticity term

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

1. The demand function for a particular good is x=a+bp. What are the associated direct and indirect utility functions?

2. Find the demanded bundle for a consumer whose utility function is

u ( x1, x2 )=x1

23 , x2

and her budget constraint is 3 x1+4 x2=100.

3. Calculate the substitution matrix for the Cobb-Douglas demand system with two goods. Verify that the diagonal terms are negative and the crossprice effects are symmetric.

4. Ellsworth's utility function is U (x , y) = min (x , y ). Ellsworth has $150 and the price of x and the price of y are both 1. Ellsworth's boss is thinking of sending him to another town where the price of x is 1 and the price of y is 2. The boss offers no raise in pay. Ellsworth, who understands compensating and equivalent variation perfectly, complains bitterly. He says that although he doesn't mind moving for its own sake and the new town is just as pleasant as the old, having to move is as bad as a cut in pay of $A. He also says he wouldn't mind moving if when he moved he got a raise of $B. What are A and B equal to?

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5. Suppose that utility is quasilinear. Show that the indirect utility function is a convex function of prices!

6. Consider a two-period model with Dave's utility given by u(x1 , x2)where

x1 represents his consumption during the first period and x2 is his second

period's consumption. Dave is endowed with (x1 , x2) which he could consume in each period, but he could also trade present consumption for future consumption and vice versa. Thus, his budget constraint is

p1 x1+ p2 x2=p1 x1+ p2 x2

where p1 and p2 are the first and second period prices respectively. Derive the Slutsky equation in this model. (Note that now Dave's income depends on the value of his endowment which, in turn, depends on prices: m=p1 x1+ p2 x2)

7. Draw two different diagrams, one illustrating the Slutsky version of income and substitution effects and the other illustrating the Hicks version of income and substitution effects. How do these two notions differ?

8. Two goods are available, x and y. The consumer's demand function for the x-good is given by lnx=a−bp+cm, where p is the price of the x-good relative to the y-good, and m is money income divided by the price of the y-good. What equation would you solve to determine the indirect utility function that would generate this demand behavior?

9. A consumer has a utility function u(x , y , z )=min(x , y)+z. The prices

of the three goods are given by ( px , py , p) and the money the consumer has to spend is given by m. What are the demand functions and the indirect utility function for the three goods.

10. Let (q ,m) be prices and income, and let p=q/m. Use Roy's identity to derive the formula

x i ( P )=

∂ v (P)∂ pi

∑j=1

k ∂ v(P)∂ p j

p j

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CHAPTER 5

UNCERTAINTY AND INFORMATION

The most common way to model behavior under uncertainty is to assume that individuals seek to maximize the expected utility of their actions.

A “fair game” is a random game with a specified set of prizes and associated probabilities that have an expected value of zero.

Individuals who exhibit a diminishing marginal utility of wealth are risk averse. That is, they generally refuse fair bets. Risk-averse individuals will wish to insure themselves completely against uncertain events if insurance premiums are actuarially fair.

If the utility-of-wealth function is concave (i.e., exhibits a diminishing marginal utility of wealth), then this person will refuse fair bets. A 50–50 bet of winning or losing h dollars, for example, yields less utility [Uh(W*)] than does refusing the bet. The reason for this is that winning h dollars means less to this individual than does losing h dollars.

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Risk aversion measure r(W), is defined as

r(W) = U ' ' (W )U ' (W )

The amount that a risk-averse individual is willing to pay to avoid a fair bet is approximately proportional to Pratt’ s risk aversion measure.

Whether risk aversion increases or decreases with wealth depends on the precise shape of the utility function. If utility is quadratic in wealth, risk aversion increases as wealth increases. On the other hand, if utility is logarithmic in wealth, risk aversion decreases as wealth increases.

Two utility functions have been extensively used in the study of behavior under uncertainty: the constant absolute risk aversion (CARA) function and the constant relative risk aversion (CRRA) function.

One of the most extensively studied issues in the economics of uncertainty is the “portfolio problem,” which asks how an investor will split his or her wealth between risky and risk-free assets. In some cases it is possible to obtain precise solutions to this problem, depending on the nature of the risky assets that are available.

A conceptual idea that can be developed concurrently with the notion of states of the world is that of contingent commodities. Examining utility-maximizing choices among contingent commodities proceeds formally in much the same way we analyzed choices previously. The principal difference is that, after the fact, a person will have obtained only one contingent good (depending on whether it turns out to be good or bad times).

Information is valuable because it permits individuals to make better decisions in uncertain situations. Information can be most valuable when individuals have some flexibility in their decision-making.

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CHAPTER 5UNCERTAINTY AND INFORMATION

1. Show that the willingness-to-pay to avoid a small gamble with variance v is approximately r(w)v/2.

2. What will the form of the expected utility function be if risk aversion is constant? What if relative risk aversion is constant?

3. Consider the case of a quadratic expected utility function. Show that at some level of wealth marginal utility is decreasing. More importantly, show that absolute risk aversion is increasing at any level of wealth.

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4. George is seen to place an even-money $100,000 bet on the Bulls to win the NBA Finals. If George has a logarithmic utility-of-wealth function and if his current wealth is $1,000,000, what must he believe is the minimum probability that the Bulls will win?

5. An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all of the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: (1) take all 12 eggs in one trip; or (2) take two trips with 6 eggs in each trip.

a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on average, 6 eggs will remain unbroken after the trip home under either strategy.

b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable?

c. Could utility be improved further by taking more than two trips? How would this possibility be affected if additional trips were costly?

6. Suppose the current wealth of Mr. Michael is $100,000. He faces the prospect of a 25 percent of losing his $20000 automobile through theft during the next year.a. Calculate the expected utility of him without insurance.b. Assuming that the insurance company only claims costs and

administrative costs are $0, how much a fair insurance premium will be? Regardless of whether the car is stolen, calculate the expected utility of him if he completely insures the car.

c. How much the maximum premium that he will be willing to pay?

7. Ms. Fogg is planning an around-the-world trip on which she plans to spend $10,000. The utility from the trip is a function of how much she actually spends on it (Y), given by U (Y) = ln Ya. If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her

cash on the trip, what is the trip’s expected utility?b. Suppose that Ms. Fogg can buy insurance against losing the $1,000 (say,

by purchasing traveler’s checks) at an “actuarially fair” premium of $250. Show that her expected utility is higher if she purchases this insurance than if she faces the chance of losing the $1,000 without insurance.

c. What is the maximum amount that Ms. Fogg would be willing to pay to insure her $1,000?

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8. A coin has probability p of landing heads. You are offered a bet in which you will be paid $21 if the first head occurs on the jth flip.

a. What is the expected value of this bet when p = 1/2?b. Suppose that your expected utility function is u(x) = lnx. Express the utility

of this game to you as a sum.

9. A farmer believes there is a 50–50 chance that the next growing season will be abnormally rainy. His expected utility function has the form

where YNR and YR represent the farmer’s income in the states of “normal rain” and “rainy,” respectively.

a. Suppose the farmer must choose between two crops that promise the following income prospects:

Which of the crops will he plant?b. Suppose the farmer can plant half his field with each crop. Would he choose

to do so? Explain your result.c. What mix of wheat and corn would provide maximum expected utility to

this farmer?

10. Let R1 and R2 be the random returns on two assets. Assume that R1 and R2 are independently and identically distributed. Show that an expected utility maximizer will divide her wealth between both assets provided she is risk averse; and invest all her wealth in one of the assets if she's risk loving.

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CHAPTER 6

PRODUCTION FUNCTION

The firm’s production function for a particular good, q, q=f (k , l)

shows the maximum amount of the good that can be produced using alternatives combinations of capital (k) and labor (l).

Marginal physical product of an input is the additional output that can be produced by employing one more unit of that input while holding all other inputs constant.

Marginal physical product of capital=MPk=∂ q∂ k

Marginal physical product of labor=MPl=∂ q∂l

Average product of labor (APl)

APl= outputlabor input

=ql=

f ( k ,l )l

The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant.

RTS ( l for k )=−dkdl

The return to scale exhibited by a production function record how output responds to proportionate increases in all inputs. If output increases proportionately with input use, there are constant return to scale. If there are greater than proportionate increases in output, there are increasing returns to scale, whereas if there are less than proportionate increases in output, there are decreasing returns to scale.

The elasticity of substitution (σ) provides a measure of how easy it is to substitute one input for another in production.

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σ=% Δ( k

l )% Δ RTS

Technical progress shifts the entire production function an its related isoquant map. Technical improvements may arise from the use of improved, more-productive inputs or from better methods of economic organization.

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CHAPTER 6PRODUCTION FUNCTION

1. Suppose the production function isQ = f (k, l) = 300 k2 l2 - k3 l3

Assume that k=10Calculate:a. Average product of labor when it reaches the maximum valueb. Optimum labor unit that should be hired

2. Please answer T if the statement is true, and answer F and correct the statement if the statement is false. a. Marginal physical product of capital is the additional output that can be

produced by employing one more unit of labor. b. Marginal rate of technical substitution shows the rate which labor can be

substituted for capital while holding output constant along an isoquant. c. Isoquant curve shows the combinations of k and l that can produce

different level of output. d. The elasticity of substitution provides a measure of how easy it is to

substitute one input for another in production. High elasticity of substitution implies that isoquants are nearly L-shaped.

3. Explain the term of marginal rate of technical substitution (RTS)! What does RTS=3 mean?

4. Explain and draw the curve! a. Linear production functionb. Fixed proportion production functionc. Cobb-Douglas production function

5. Why labor can’t be added indefinitely to a given amount of capital (when keeping amount of machine, land, etc) ? What concept that explains it?

6. Suppose that the production function of Wayne Enterprises is Q = 12 K0,4 L0,8

a. What is the type of return to scale (RTS) of this production function? Prove it!

b. Write the cost function if the price of L is 5 and the price of K is 2!c. Draw the isocost if the cost of Wayne Enterprises is $2000!

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7. Calculate the least cost combination of K and L with informations below! Draw the curve! C = 5L + 10 KQ = 100 K0,5 + 100 L0,5 and Q=3000

8. Fill in the blank!a. The slope of isoquant is termed as ....b. A production function measures the relation between .... & ....c. When increasing inputs by ¼ leads to an increase in 1/3 output, it is

called .... return to scale

9. Oliver Queen is considering producing Queen Consolidated High-Tech Computer. The production function is given by Q = 0,1 k 0,2 l 0,8

Where q is the number of Queen Consolidated High-Tech Computer produced in a week. K represent capital used and l represent the number s of labor employed. Oliver Queen would like to produce 10 Queen Consolidated High-Tech Computers and he allocated $1.000.000 for the production process.

a. Oliver Queen would like to buy and hire these two inputs in equal amounts because capital and labor both cost the same amount ($5000). How much of each input will he hire and how much the total cost?

b. Oliver Queen is recently study microeconomics. He wants to produce 10 Queen Consolidated High-Tech Computer by the least possible cost. How much labor will he hire and how much capital will he use? How much the total cost?

c. Now Oliver Queen is considering maximizing all of his budget. If he apply this method, how much labor will he hire and how much capital will he use? How much Queen Consolidated High-Tech Computer will he produce?

10. Based on the question number 9 above, what concept is the best used by Oliver Queen (b, or c)? Give your argument!

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28

CHAPTER 7

COST MINIMIZATION

We must differentiate between: Accounting cost: the accountant’s view of cost stresses out-of-pocket

expenses, historical costs, depreciation, and other bookkeeping entries. Economic cost: is that the cost of any input is given by the size of the

payment necessary to keep the resources in its present employment. The Lagrangian expression for cost minimization of producing q0 (Cobb-

Douglas) isL = vk + wl + (q0 - k a l b)

A firm that wishes to minimize the economic costs of producing a particular level of output should choose that input combination for which the rate of technical substitution (RTS) is equal to the ratio of the inputs’ rental prices.

The firm’s average cost (AC = C/q) and marginal cost (MC = C/q) can be derived directly from the total-cost function if the total cost curve has a general cubic shape, the AC and MC curves

will be u-shaped

The firm’s expansion path is the locus of cost-minimizing tangencies. Assuming fixed input prices, the curve shows how inputs increase as output increases. if the use of an input falls as output expands, that input is an inferior

input In the short run, the firm may not be able to vary some inputs

it can then alter its level of production only by changing the employment of its variable inputs

it may have to use nonoptimal, higher-cost input combinations than it would choose if it were possible to vary all inputs

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The long run average cost is the envelope of the firm’s short run average cost curves, and it reflects the presence or absence of returns to scale.

30

CHAPTER 7COST MINIMIZATION

1. For each cost function determine if it is homogeneous of degree one, monotonic, concave, and/or continuous. If it is, derive the associated production function.

a. C ( w , y )= y1 /2(w1 w2)3 /4

b. C ( w , y )= y ¿c. C ( w , y )= y (w1 e−w1+w2)d. C ( w , y )= y (w1−√w1w2+w2)

e. C ( w , y )=( y+ 1y )√w1w2

2. A firm producing hockey sticks has a production function given by

q=2√k .l

In the short run, the firm’s amount of capital equipment is fixed at k = 100. The rental rate for k is y = $1, and, the wage rate for l is w = $4.

a. Calculate the firm’s short-run total cost curve. Calculate the short-run average cost curve.

b. What is the firm’s short-run marginal cost function? What are the SC, SAC, and SMC for the firm if it produces 25 hockey sticks? Fifty hockey sticks? One hundred hockey sticks? Two hundred hockey sticks?

c. Graph the SAC and the SMC curves for the firm. Indicate the points found in part (b).

d. Where does the SMC curve intersect the SAC curve? Explain why the SMC curve will always intersect the SAC curve at its lowest point.

Suppose now that capital used for producing hockey sticks is fixed at k in the short run.

e. Calculate the firm’s total costs as a function of q, w, v, and k .f. Given q, w, and v, how should the capital stock be chosen to

minimize total cost?g. Use your results from part (f) to calculate the long-run total cost of

hockey stick production.h. For w = $4, v = $1, graph the long-run total cost curve for hockey

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stick production. Show that this is an envelope for the short-run curves computed in part (a) by examining values of k of 100, 200, and 400.

3. Suppose that a firm’s fixed proportion production function is given byq = min (5k, 10l).

a. Calculate the firm’s long-run total, average, and marginal cost functions.

b. Suppose that k is fixed at 10 in the short run. Calculate the firm’s short-run total, average, and marginal cost functions.

c. Suppose v = 1 and w = 3. Calculate this firm’s long-run and short-run average and marginal cost curves.

4. A firm’s production process can be represented by the following production functionQ = A Ka Lb

Where Q is the level of output produced, A>0 is technological parameter, K is the level of capital used, L is the number of labor used, and a>0, b>0 are parameters. The firm minimizes cost of production : C = wL + rKWhere w is the wage rate, and r is the rental rate of capital.

a. Calculate number of labor demand and capital demand .b. How to effect of technology change for input demand.

5. Calculate the number of labor (L1 & L2) and capital (K1 & K2) that solve the minimization problem below :

(i) Minimize wL1 +rK1 subject to Q = min {K1

1/3,

L1

2/3}

And(ii) Minimize wL2 +rK2 subject to Q = min {4K2 , 5L2}

Where w = 25 is the wage rate, and r = 10 is the rental rate of capital. The target level of output is Q= 100. Show the both of function with the relevant graph!

6. A firm has production function Q = 4√ K+2√L, where Q is the level of output produced, K is the level of capital used, L is the number of labor used.

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The target level of output is Q= 120 with the wage rate is w= $5 and the rent rate is r = $4.

a. What kind of production function above?b. Calculate number of Labor and Capital if the firm want to minimize

cost.c. Calculate the firm’s minimum cost.

7. Calculate the number of labor (L) and capital (K) that solve the minimization problem below :

minimize wL + rK subject to Q = 25

K+ 35

L

where w = 25 is the wage rate and r = 10 is the rent rate. The target output is Q = 100 units. Show with the relevant graph !

8. A firm has a production function given by f(x1,x2) = min(2x1+x2 , x1+2x2). What is the cost function for this technology? What is the conditional demand function for factors 1 and 2 as a function of factor prices (w1, w2) and output y?

9. Suppose the total-cost function for a firm is given byC = qw2/3 v1/3

a. Use Shephard’s lemma to compute the constant output demand functions for inputs l and k.

b. Use your results from part (a) to calculate the underlying production function for q.

10. A chair manufacturer hires its assembly-line labor for $22 an hour and calculate that the rental cost of its machinery is $110 per hour. Suppose that a chair can be produced using 4 hours of labor or machinery in any combination . if the firm is currently using 3 hours of labor for each hour of machine time, is it minimizing its cost of production? If so, why? If not, how can it improve the situation?

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CHAPTER 8PROFIT MAXIMIZATION AND PARTIAL EQUILIBRIUM

COMPETITIVE MODEL

A profit-maximizing firm chooses both its inputs and its outputs with the sole goal of achieving maximum economic profits is seeks to maximize the difference between total revenue and total economic costs

Total revenue for a firm is given by: R(q) = p(q)q In the production of q, certain economic costs are incurred [C(q)] Economic profits () are the difference between total revenue and total costs

(q) = R(q) – C(q) = p(q)q –C(q) To maximize economic profits, the firm should choose the output for which

marginal revenue is equal to marginal cost.

Profit Maximization MR=dRdq

=dCdq

=MC

“marginal” profit must be decreasing at the optimal level of q Because MR = MC when the firm maximizes profit, we can see that

MC=p(1+1

eq , p) p−MC

p=− 1

eq ,p The gap between price and marginal cost will fall as the demand curve facing

the firm becomes more elastic A firm’s economic profit can be expressed as a function of inputs:

= pq - C(q) = pf(k,l) - vk - wl Only the variables k and l are under the firm’s control: the firm chooses

levels of these inputs in order to maximize profits. Treats p, v, and w as fixed parameters in its decisions

We can apply the envelope theorem to see how profits respond to changes in output and input prices

∂ Π ( p , v , w )∂ p

=q( p , v , w )

∂ Π ( p , v , w )∂ v

=−k ( p , v , w )

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∂ Π ( p , v , w )∂w

=−l( p , v ,w )

Differentiation with respect to w yields

∂ l( p , v , w )∂ w

=∂ lc(v ,w , q )

∂w+

∂ lc(v , w , q )∂ q

⋅∂ q∂w

Short-run equilibrium prices are determined by the interaction of what demanders are willing to pay (demand) and what existing firms are willing to produce (supply). Both demanders and suppliers act as price takers in making their respective decisions.

In the long run, the number of firms may vary in response to profit opportunities. If free entry is assumed then firms will earn zero economic profits over the long run. Because firms also maximize profits, the long-run equilibrium condition is therefore P ¼ MC ¼ AC.

The shape of the long-run supply curve depends on how the entry of new firms affects input prices. If entry has no impact on input prices, the long-run supply curve will be horizontal (infinitely elastic). If entry raises input prices, the long-run supply curve will have a positive slope.

If shifts in long-run equilibrium affect input prices, this will also affect the welfare of input suppliers. Such welfare changes can be measured by changes in long-run producer surplus.

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CHAPTER 8PROFIT MAXIMIZATION AND PARTIAL EQUILIBRIUM

COMPETITIVE MODEL.

1. Let f ( x1 , x2 )be a production function with two factors and let w1 and w z be

their respective prices. Show that the elasticity of the factor share

(w2 x2/w1 x1) with respect to (x1/ x2) is given by 1σ−1.

2. Show that the elasticity of the factor share with respect to (w¿¿2/w1)¿ is

1−a

3. Let( pt , y t) for t=1 , … .., T be a set of observed choices that satisfy WAPM, and let YI and YO be the inner and outer bounds to the true production set Y. Let π+¿( p)¿, be the profit function associated with YO and

π−¿( p)¿ be the profit function associated with YI , and ( p) be the profit

function associated with Y. Show that for all p ,π+¿( p)≥ π ( p)≥ π−¿(p )¿ ¿.

4. The production function is f ( x )=20 x−x2and the price of output is

normalized to 1. Let w be the price of the x-input. We must have x≥ 0.(a) What is the first-order condition for profit maximization if x>0?(b) For what values of w will the optimal x be zero?(c) For what values of w will the optimal x be 10?(d) What is the factor demand function?(e) What is the profit function?(f) What is the derivative of the profit function with respect to w?

5. John’s Lawn Moving Service is a small business that acts as a price taker (i.e., MR = P). The prevailing market price of lawn mowing is $20 per acre. John’s costs are given by

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total cost=0.1 q2+10 q+50where q ¼ the number of acres John chooses to cut a day.a. How many acres should John choose to cut in order to maximize profit?b. Calculate John’s maximum daily profit.c. Graph these results and label John’s supply curve.

6. La Belle Boutique is a small business that acts as a price taker. The prevailing market price of La Belle Boutique is $30 per dress. La Belle’s costs are given by: C (Q )=0,1Q 2+10 Q+60a. How many quantity produced when the firm maximizing profit?b. How much its profit?

7.

Explain and show which is the Short-run Supply Curve? Give the detail label on the graph!

8. Suppose a perfectly competitive market has 2000 firms. In the very shor run, each of the firms has fixed supply of 200 units. The market demand is given by: Q=320.000 – 20.000P

a. Calculate the equilibrium price in the very short run!b. Calculate the demand schedule facing any one firm in the industry!

9. Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost function of the form

C (q )= 1300

q3+0,2 q2+4 q+10

a. Calculate the firm’s short-run supply curve with q as a function of market price (P).b. On the assumption that there are no interaction effects among costs of the firms in the industry,

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calculate the short-run industry supply curve.c. Suppose market demand is given by Q = -200P + 8,000. What will be the short-run equilibriumprice-quantity combination?

10. A perfectly competitive market has 1,000 firms. In the very short run, each of the firms has a fixed supply of 100 units. The market demand is given by

Q=160,000−10,000 P

a. Calculate the equilibrium price in the very short run.b. Calculate the demand schedule facing any one firm in this industryc. Calculate what the equilibrium price would be if one of the sellers decided to sell nothing or if one seller decided to sell 200 units.d. At the original equilibrium point, calculate the elasticity of the industry demand curve and theelasticity of the demand curve facing any one seller.

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CHAPTER 9

PARTIAL EQUILIBRIUM COMPETITIVE MODEL

Market demands curve is the ’’horizontal sum’’ of each individual’s demand curve at a price the quantity demanded in the market is the sum of the amount each individual demand for example at p* the demand in the market is x∗¿1+x∗¿2=¿ x∗¿¿¿¿ .

Timing of the Demand Response

In the analysis of competitive pricing, the time period under consideration is importantvery short run

no supply response (quantity supplied is fixed)short run

existing firms can alter their quantity supplied, but no new firms can

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enter the industry long run

new firms may enter an industry

Short-Run Market Supply CurveTo derive the market supply curve, we sum the quantities supplied at every price . q1

A + q1B = Q1

Long-Run Competitive Equilibrium A perfectly competitive industry is in long-run equilibrium if there are

no incentives for profit-maximizing firms to enter or to leave the industry

o this will occur when the number of firms is such that P = MC = AC and each firm operates at minimum AC

We will assume that all firms in an industry have identical cost curveso no firm controls any special resources or technology

The equilibrium long-run position requires that each firm earn zero economic profit

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• The shape of the long-run supply curve depends on how entry and exit affect firms’ input costs

a. in the constant-cost case, input prices do not change and the long-run supply curve is horizontal

b. if entry raises input costs, the long-run supply curve will have a positive slope

c. if entry reduces input costs, the long-run supply curve will have negative slope

41

CHAPTER 9PARTIAL EQUILIBRIUM COMPETITIVE MODEL

1. Suppose the market for widgets can be described by the following equations :Demand : P = 10 - QSupply : P = Q - 4

Where P is the price in dollars per unit and Q is the quantity in thousands of units. Then,

a. What is the equilibrium price and quantityb. Suppose the government imposes a tax 0f $1 per unit to reduce

widget consumption and raise government revenues. What will the new equilibrium quantity be? What price will the buyer pay? What amount per unit will the seller receive?

c. Suppose the government has a change of heart about the importance of widgets to the happiness of the American public. The tax is removed and a subsidy of $1 per unit granted to widget producers. What will the equilibrium quantity be? What price will the buyer pay? What amount per unit (including the subsidy) will the seller receive? What will be the total cost to the government ?

2. A vegetable fiber traded in a competitive world market and imported into the United States at a world price of $9 per pound U.S. domestic supply and demand for various price levels are shown in the following table :

PRICE U.S. SUPPLY(MILLION POUNDS)

U.S. DEMAND(MILLION POUNDS)

3 2 346 4 289 6 2212 8 1615 10 1018 12 4

Answer the following about the U.S. market :a. Confirm that the demand curve is given by Qd = 40 – 2P, and that

supply curve is given by Qs = 2/3 P.b. Confirm that if there no restriction on trade, the United States would

import 16 million pounds.

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3. Suppose that total cost of producing pizzas for the typical firm in a local town is given by C(q) = 2q + 2q2. In turn, marginal cost is given by MC = 2 + 4q. (if you know calculus, you should be able to derive this expression for marginal cost.)

a. Show that the competitive supply behavior of the typical pizza firm

is described by q = P4

−12

.

b. If there are 100 firms in the industry each acting as a perfect competitor , show that the market supply curve is, in inverse form, given by P=2 + Q/25.

4. We mentioned PT.TAMIMA and its control of plastic hanger market in the chapter. Suppose that the inverse demand for hanger is given by P= 6 -

Q8000

. Suppose further that the marginal cost of producing hangers is

constant at $2.a. What is the equilibrium price and quantity of hangers if the market

is competitive?b. What is the equilibrium price and quantity of hangers if the market

is monopolized? What is deadweight loss of monopoly in this market ? Show with graph!

5. Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost function of the form

C (q )= 1300

q3+ 0.2 q2 + 4 q + 10

a. Calculate the firm’s short-run supply curve with q as a function of market price (P) !

b. On the assumption that there are no interaction effects among costs of the firms in the industry, calculate the short-run industry supply curve !

c. Suppose market demand is given by Q = -200 P + 8000 . What will be the short-run equilibrium price-quantity combination?

6. Below is the inverse market demand curve :P = α−βQ

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Where P is price, Q is quantity of market demand α > 0, β>0 are demand

parameter. A firm has the following cost function :

C = η q2

Where C is cost of production , q is output supplied by the firm , and η > 0 is

parameter.a. Write down the profit function if the firm produce under a

competitive marketb. Show the first order condition for its profit maximizationc. What is the level of profit maximizing level of output.

7. Based on the above data (number 6)a. Write down the profit function if the firm is a monopolist !b. Show the first order condition for its profit maximization !c. What is the level of profit maximizing level of output !

8. Assume that the manufacturing of cellular phones is a perfectly competitive industry.

The market demand for cellular phones is described by a linear demand

function Qd = 6000−50 P

9 .

The inverse demand can easily be worked out, therefore, to be

P = 120 9

50Qd.

There are fifty manufactures of cellular phones. Each manufacture has the same production costs. These are described by the long-run total and marginal cost functions TC(q) = 100 + q2 + 10q, and MC (q) = 2q + 10, respectively.

a. Show that firm in this industry maximizes profit by producing q=P−10

2 !

b. Derive the industry supply curve and show that it is Qs = 25P – 250 !

c. Find the market price and aggregate quantity traded in equilibrium !d. How much output does each firm produce ? show that each firm

earns zero profit in equilibrium !

9. The perfectly competitive videotape copying industry is composed of many firms that can copy five tapes per day at an average cost of $10 per tape.

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Each firm must also pay a royalty to film studios, and the per-firm royalty rate (r) is an increasing function of total industry output (Q ):

r = 0.002Q

Demand is given by

Q = 1050 – 50P

a. Assuming the industry is in long-run equilibrium, what will be the equilibrium price and quantity of copied tapes? How many tape firms will there be? What will the per-film royalty rate be?

b. Suppose that demand for copied tapes increases toQ = 1600 – 50P

In this case, what is the long-run equilibrium price and quantity for copied tapes? How many tape firms are there? What is the per-film royalty rate?

10. You know that if a tax is imposed on a particular products , the burden of the tax is shared by producers and consumers. You also know that the demand for automobiles is characterized by a stock adjustment process. Suppose a special 20-percent sales tax is suddenly imposed on automobiles . will the share of the tax paid by consumers rise, fall or stay the same over time? Explain briefly ! Repeat for a 50-cents-per-gallon gasoline tax.

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