microeconomics 1 module -...
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MICROECONOMICS 1 MODULE
TEACHING ASSISTANTS OF MICROECONOMICSAND 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 ofMicroeconomics I. Shalawat and Salam be upon our Prophet Muhammad SAW, whohas brought us from the darkness into the brightness and guided us into the right wayof life.
In this opportunity, we also like to express our deep thanks to Dr. KodratWibowo, S.E. as the Head Department of Economics, Dr. Mohamad Fahmi, SE., MTas the Head of Undergraduate Program of Department of Economics, lecturers, andthose 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, weare 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 todeepen 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 COMPETITIVEMODEL...................................................................................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 theirunderstanding 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 upany confusion.
10. After successfully finishing the problems, the students can leave thelaboratory 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 theReview 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+3x−3 ) .
3. Differentiate y=x3 ln x .
4. Differentiate f ( x )=6 x2 /3 tan x .
5. Differentiate y=5 x2+sin x cos 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 (5 x ) .
11. Differentiate y=(x3−7 x2
)4(1+9x )
1/2.
12. Differentiate y=sec2 (x4 ) tan3 ( x4 ) .
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13. Differentiate y=2
x+1.
14. Differentiate y=x2
3 x−1.
15. Differentiate y=4 x3
−7x5 x2
+2.
16. Differentiate y=4sin x
2 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
.
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23. Differentiate f ( x )=x2
(2x−1)3
(x2+3)
4 .
24. Differentiate g (x )=
1
x √x2+1
.
25. Differentiate f ( x )=√ 3x+22 x−1
.
26. Differentiate y=3 x4+ tan( x
x−1 ) .
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 (2x )
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 .
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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 x2−5x+8.
34. Differentiate y=(1−4 x+7 x5)30 .
35. Differentiate y=(4 x+x−5)1 /3 .
36. Differentiate
8x−x6
x3
¿¿
y=¿
37. Differentiate y=sin (5 x) .
38. Differentiate y=e5x2+7x−13 .
39. Differentiate y=2cos x .
40. Differentiate y=3 tan√x .
41. Differentiate y= ln (17−x ) .
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42. Differentiate
x4+cos¿ .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 (5x )).
47. Differentiate
2−(6+7 x4)
¿1+¿¿
y=10¿
48. Differentiate
xln (ln (sec ¿)) .
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 ) ?
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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 thingsconsidered, he or she feels better off under situation A than situation B. There arethree basic properties of preference relation assumption:
1 Completeness: if A and B are any two situation, the person can chose threepossibilities: “A is preferred to B”; “B is preferred to A” ; or “A=B.
2 Transitivity: the individual’s choice are internally consistent, “A is preferredto B” ; “B is preferred to C” ; so “A is preferred to C”.
3 Continuity: If an individual reports “A is preferred to B” , then situationsuitably “close to” A must also be preferred to B. individual’s preferences areassumed 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 leastdesirable 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 effectbehavior (other things being equal).
Indifferent curve represents those combination of x and y from which theindividual derives the same utility. The slope of this curve represents the rate ofwhich individual is willing to trade x for y while remaining equally well off. Thenegative of the slope of an indifferent curve at the same point is termed themarginal 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 )=xy
x+ 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 )=m
min ( 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?
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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 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
y δ
δ
is homotetic. How does the MRS depend on the rasio y/x?
10 Two goods have independent marginal utility if
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∂2U∂ 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.
CHAPTER 3UTILITY MAXIMIZATION AND CHOICE I & II
To maximize utility, given a fixed amount of income to spend, an individualwill buy those quantities of goods that exhaust his or her total income and forwhich the psychic rate of trade-off between any two goods (the MRS) isequal to the rate at which the goods can be traded one for the other in themarketplace.
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
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the individual will equate the ratios of the marginal utility to pricefor every good that is actually consumed
The marginal rate of subsitution (MRS) of goods X and Y is the maximumamount of goods X that a person is willing to give up to obtain 1 additionalunit of Y. The MRS diminishes as we move down along an indifferencecurves. When there is a diminishing MRS, indifference curves are convex.
Consumers maximize satisfaction subject to budget constraint. When aconsumer 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 twogoods being purchased.
Maximization is sometimes achieved at a corner solution in which one goodis not consumed. In such cases, the marginal rate of substitution need to equalthe ratio of the prices.
The individual’s optimal choices implicitly depend on the parameters of hisbudget 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
maximumutility=U (x1¿ , x2
¿ ,…,xn¿)
¿V ( p1 , p2 ,…, pn , I)
The dual problem to the constrained utility-maximization problem is tominimize 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 theminimal expenditures necessary to achieve a given utility level for aparticular set of prices.
minimal expenditure=E (p1 , p2 ,……, pn ,U )
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Properties of expenditure functions : Homogeneity Expenditure functions are nondecreasing in prices Expenditure functions are concave in prices
CHAPTER 3UTILITY MAXIMIZATION AND CHOICE I & II
1 What is utility maximization? Graph and show where is the optimal quantityof x and y that maximize utility.
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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 utilityfunction? What is his expenditure function?
3 A consumer has an indirect utility function of the form
v ( p1 , p2 , p3 )=m
min {p1 , p2}
What is the form of the expenditure function for this consumer? What is theform of a (quasiconcave) utility function for this consumer? What is the formof the demand function for good l?
4 Explain mathematically first order condition for a maximum utility (for twogoods)
5 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?
6 A young connoisseur has $300 to spend to build a small wine cellar. Sheenjoys two vintages in particular : a 1997 French Bordeux (WF) at $20 perbottle and a less expensive 2002 California varietal wine (WC) priced at $4.How much of each wine should she purchase with Langrangian expression ifher 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. Priceof 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 constraint
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c What happen if price x decrease until 20% (with the first income given).Make a new budget constraint
d Continuing from part c, now price y increase 25%. Make a new budgetconstraint.
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 p1 is strictly less
than what?(c) What is the algebraic form of the indirect utility function associated withthis direct utilityfunction?
9 George has $300 to spend to buy book and novel. Price of book is $ 4 andprice of novel is $12.How much the MRS between book and novel? How much of each book andnovel 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) = x0.4y0.8 for good x and y. Assume hehas an income $100. Price of good x is $ 4 and price of good y is $12.a Show MRS between good x and good yb 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 examinerelationship among consumer goods. Although these relationship may becomplex, the analysis presented here provided a number of ways of categorizingand simplyfying them.
When there are only two goods, the income and substitution effects from thechange 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 thesegoods can be treated as a “composite commodity” whose “price” is given by thesize of the proportional change in the composite goods’ prices.
An alternative way to develop the theory of choice among market goods is tofocus on the ways in which market goods are used in household production toyield utility-providing attributes.
A composite comodity theorem applies to any group of commodities whoserelative price all move together. It is possible to have more than one suchcommodity if there are several groupings that obey that theorem.
Slutsky-type Equation :
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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 withtwo goods. Verify that the diagonal terms are negative and the crosspriceeffects 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 isthinking of sending him to another town where the price of x is 1 and theprice of y is 2. The boss offers no raise in pay. Ellsworth, who understandscompensating and equivalent variation perfectly, complains bitterly. He saysthat although he doesn't mind moving for its own sake and the new town isjust as pleasant as the old, having to move is as bad as a cut in pay of $A. Healso says he wouldn't mind moving if when he moved he got a raise of $B.What are A and B equal to?
5. Suppose that utility is quasilinear. Show that the indirect utility function is aconvex function of prices!
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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 presentconsumption for future consumption and vice versa. Thus, his budgetconstraint 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 incomedepends 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 incomeand substitution effects and the other illustrating the Hicks version of incomeand 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 utilityfunction 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 , p y , p) and the money the
consumer has to spend is given by m. What are the demand functions and theindirect 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
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x i ( P )=
∂ v(P)
∂ p i
∑j=1
k ∂ v (P)
∂ p j
p j
CHAPTER 5
UNCERTAINTY AND INFORMATION
The most common way to model behavior under uncertainty is to assume thatindividuals seek to maximize the expected utility of their actions.
A “fair game” is a random game with a specified set of prizes and associatedprobabilities 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
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themselves completely against uncertain events if insurance premiums areactuarially fair.
If the utility-of-wealth function is concave (i.e., exhibits a diminishing marginalutility of wealth), then this person will refuse fair bets. A 50–50 bet of winning orlosing h dollars, for example, yields less utility [Uh(W*)] than does refusing thebet. The reason for this is that winning h dollars means less to this individual thandoes losing h dollars.
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 isapproximately proportional to Pratt’ s risk aversion measure.
Whether risk aversion increases or decreases with wealth depends on the preciseshape of the utility function. If utility is quadratic in wealth, risk aversionincreases as wealth increases. On the other hand, if utility is logarithmic inwealth, risk aversion decreases as wealth increases.
Two utility functions have been extensively used in the study of behavior underuncertainty: the constant absolute risk aversion (CARA) function and the constantrelative 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 wealthbetween risky and risk-free assets. In some cases it is possible to obtain precisesolutions to this problem, depending on the nature of the risky assets that areavailable.
A conceptual idea that can be developed concurrently with the notion of states ofthe world is that of contingent commodities. Examining utility-maximizingchoices among contingent commodities proceeds formally in much the same waywe analyzed choices previously. The principal difference is that, after the fact, aperson will have obtained only one contingent good (depending on whether itturns out to be good or bad times).
Information is valuable because it permits individuals to make better decisions inuncertain situations. Information can be most valuable when individuals havesome 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 isapproximately r(w)v/2.
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2 What will the form of the expected utility function be if risk aversion isconstant? What if relative risk aversion is constant?
3 Consider the case of a quadratic expected utility function. Show that at somelevel of wealth marginal utility is decreasing. More importantly, show thatabsolute risk aversion is increasing at any level of wealth.
4 George is seen to place an even-money $100,000 bet on the Bulls to win theNBA Finals. If George has a logarithmic utility-of-wealth function and if hiscurrent wealth is $1,000,000, what must he believe is the minimumprobability that the Bulls will win?
5 An individual purchases a dozen eggs and must take them home. Althoughmaking trips home is costless, there is a 50 percent chance that all of the eggscarried on any one trip will be broken during the trip. The individualconsiders two strategies: (1) take all 12 eggs in one trip; or (2) take two tripswith 6 eggs in each trip.
a List the possible outcomes of each strategy and the probabilities ofthese outcomes. Show that, on average, 6 eggs will remain unbrokenafter 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 werecostly?
6 Suppose the current wealth of Mr. Michael is $100,000. He faces the prospectof a 25 percent of losing his $20000 automobile through theft during the nextyear.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 ofhim 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 actuallyspends 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
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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 thisinsurance than if she faces the chance of losing the $1,000 withoutinsurance.
c What is the maximum amount that Ms. Fogg would be willing to pay toinsure her $1,000?
8 A coin has probability p of landing heads. You are offered a bet in which youwill 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 willbe abnormally rainy. His expected utility function has the form
where YNR and YR represent the farmer’s income in the states of “normalrain” and “rainy,” respectively.
a Suppose the farmer must choose between two crops that promise thefollowing 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 R2are independently and identically distributed. Show that an expected utilitymaximizer will divide her wealth between both assets provided she is riskaverse; 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 alternativescombinations of capital (k) and labor (l).
Marginal physical product of an input is the additional output that can beproduced by employing one more unit of that input while holding all other inputsconstant.
Marginal physical product of capital=MPk=∂q∂k
Marginal physical product of labor=MPl=∂q∂l
Average product of labor (APl)
APl=output
labor input=
ql=
f (k , l )
l
The marginal rate of technical substitution (RTS) shows the rate at which laborcan be substituted for capital while holding output constant along an isoquant.
RTS (l for k )=−dk
dl
The return to scale exhibited by a production function record how outputresponds to proportionate increases in all inputs. If output increasesproportionately with input use, there are constant return to scale. If there aregreater than proportionate increases in output, there are increasing returns toscale, whereas if there are less than proportionate increases in output, there aredecreasing returns to scale.
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The elasticity of substitution (σ) provides a measure of how easy it is tosubstitute one input for another in production.
σ=
Δ ( kl )
Δ RTS
Technical progress shifts the entire production function an its related isoquantmap. 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 thestatement 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 ofsubstitution implies that isoquants are nearly L-shaped.
3 Explain the term of marginal rate of technical substitution (RTS)! What doesRTS=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 (whenkeeping 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!
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c Draw the isocost if the cost of Wayne Enterprises is $2000!
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-TechComputer. 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 Computerproduced in a week. K represent capital used and l represent the number s oflabor employed. Oliver Queen would like to produce 10 Queen ConsolidatedHigh-Tech Computers and he allocated $1.000.000 for the productionprocess.
a Oliver Queen would like to buy and hire these two inputs in equalamounts because capital and labor both cost the same amount($5000). How much of each input will he hire and how much thetotal cost?
b Oliver Queen is recently study microeconomics. He wants toproduce 10 Queen Consolidated High-Tech Computer by the leastpossible cost. How much labor will he hire and how much capitalwill he use? How much the total cost?
c Now Oliver Queen is considering maximizing all of his budget. Ifhe apply this method, how much labor will he hire and how muchcapital will he use? How much Queen Consolidated High-TechComputer will he produce?
10 Based on the question number 9 above, what concept is the best used byOliver Queen (b, or c)? Give your argument!
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32
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 particularlevel of output should choose that input combination for which the rate oftechnical 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 bederived 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 outputincreases. 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 employmentof its variable inputs
it may have to use nonoptimal, higher-cost input combinations than it
33
would choose if it were possible to vary all inputs
The long run average cost is the envelope of the firm’s short run average costcurves, and it reflects the presence or absence of returns to scale.
CHAPTER 7COST MINIMIZATION
34
1 For each cost function determine if it is homogeneous of degree one,monotonic, concave, and/or continuous. If it is, derive the associatedproduction function.
a C (w , y )= y1/ 2(w1 w2)
3/4
bw1+√w1 w2+w2
C ( w , y )= y ¿
c C (w , y )= y (w1 e−w 1+w2)
d C (w , y )= y (w1−√w1 w2+w2)
e C (w , y )=( y+1y )√w1 w2
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 theSC, SAC, and SMC for the firm if it produces 25 hockey sticks?Fifty hockey sticks? One hundred hockey sticks? Two hundredhockey sticks?
c Graph the SAC and the SMC curves for the firm. Indicate the pointsfound in part (b).
d Where does the SMC curve intersect the SAC curve? Explain whythe SMC curve will always intersect the SAC curve at its lowestpoint.
Suppose now that capital used for producing hockey sticks is fixed at k in theshort run.
e Calculate the firm’s total costs as a function of q, w, v, and k .
35
f Given q, w, and v, how should the capital stock be chosen tominimize total cost?
g Use your results from part (f) to calculate the long-run total cost ofhockey stick production.
h For w = $4, v = $1, graph the long-run total cost curve for hockeystick 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 costfunctions.
b Suppose that k is fixed at 10 in the short run. Calculate the firm’sshort-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 productionfunctionQ = A Ka Lb
Where Q is the level of output produced, A>0 is technological parameter, Kis the level of capital used, L is the number of labor used, and a>0, b>0 areparameters. 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 theminimization 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. Thetarget level of output is Q= 100. Show the both of function with the relevantgraph!
36
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 laborused. The target level of output is Q= 120 with the wage rate is w= $5 and therent 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 minimizationproblem 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 isQ = 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 conditionaldemand function for factors 1 and 2 as a function of factor prices (w1, w2) andoutput 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 demandfunctions for inputs l and k.
b Use your results from part (a) to calculate the underlying productionfunction for q.
10 A chair manufacturer hires its assembly-line labor for $22 an hour andcalculate that the rental cost of its machinery is $110 per hour. Suppose that achair can be produced using 4 hours of labor or machinery in anycombination . if the firm is currently using 3 hours of labor for each hour ofmachine time, is it minimizing its cost of production? If so, why? If not, howcan 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 solegoal of achieving maximum economic profits is seeks to maximize thedifference 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
pqepMC
,
11
pqep
MCp
,
1
The gap between price and marginal cost will fall as the demand curve facingthe 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 chooseslevels of these inputs in order to maximize profits. Treats p, v, and w as fixedparameters in its decisions
We can apply the envelope theorem to see how profits respond to changes inoutput and input prices
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),,(),,(
wvpqp
wvp
),,(),,(
wvpkv
wvp
),,(),,(
wvpw
wvpl
Differentiation with respect to w yields
w
q
q
qwv
w
qwv
w
wvp cc
),,(),,(),,( lll
Short-run equilibrium prices are determined by the interaction of whatdemanders are willing to pay (demand) and what existing firms are willingto produce (supply). Both demanders and suppliers act as price takers inmaking their respective decisions.
In the long run, the number of firms may vary in response to profitopportunities. If free entry is assumed then firms will earn zero economicprofits over the long run. Because firms also maximize profits, the long-runequilibrium condition is therefore P ¼ MC ¼ AC.
The shape of the long-run supply curve depends on how the entry of newfirms affects input prices. If entry has no impact on input prices, the long-run supply curve will be horizontal (infinitely elastic). If entry raises inputprices, the long-run supply curve will have a positive slope.
If shifts in long-run equilibrium affect input prices, this will also affect thewelfare of input suppliers. Such welfare changes can be measured bychanges in long-run producer surplus.
39
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
40
production set Y. Let +¿( p)
π ¿ , be the profit function associated with YO
and −¿( p)
π¿ be the profit function associated with YI , and ( p) be
the profit function associated with Y. Show that for all−¿( p)
+¿( p)≥π ( p)≥ π¿
p ,π¿.
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
totalcost=0.1q2+10q+50
where 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 prevailingmarket price of La Belle Boutique is $30 per dress. La Belle’s costs are given
by: C (Q )=0,1Q2+10Q+60
a How many quantity produced when the firm maximizing profit?
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b How much its profit?
7
Explain and show which is the Short-run Supply Curve? Give the detail labelon 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 givenby: 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 )=1
300q3
+0,2q2+4 q+10
a. Calculate the firm’s short-run supply curve with q as a function of marketprice (P).b. On the assumption that there are no interaction effects among costs of thefirms in the industry,calculate the short-run industry supply curve.c. Suppose market demand is given by Q = -200P + 8,000. What will be theshort-run equilibriumprice-quantity combination?
10 A perfectly competitive market has 1,000 firms. In the very short run, each ofthe firms has a fixed supply of 100 units. The market demand is given by
Q=160,000−10,000P
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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 decidedto sell nothing or if one seller decided to sell 200 units.d. At the original equilibrium point, calculate the elasticity of the industrydemand 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 demandcurve at a price the quantity demanded in the market is the sum of the amounteach individual demand for example at p* the demand in the market is
2=¿ x∗¿x∗¿¿
x∗¿1+¿¿
.
Timing of the Demand ResponseIn the analysis of competitive pricing, the time period under consideration isimportantvery short run
no supply response (quantity supplied is fixed)short run
existing firms can alter their quantity supplied, but no new firms can enter the industry
long runnew firms may enter an industry
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Short-Run Market Supply CurveTo derive the market supply curve, we sum the quantities supplied at everyprice . 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 theindustry
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 zeroeconomic profit
45
• The shape of the long-run supply curve depends on how entry and exit affectfirms’ 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 apositive slope
c if entry reduces input costs, the long-run supply curve will havenegative slope
46
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 ofunits. 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 thenew equilibrium quantity be? What price will the buyer pay? Whatamount per unit will the seller receive?
c Suppose the government has a change of heart about the importanceof widgets to the happiness of the American public. The tax isremoved and a subsidy of $1 per unit granted to widget producers.What will the equilibrium quantity be? What price will the buyerpay? What amount per unit (including the subsidy) will the sellerreceive? What will be the total cost to the government ?
2 A vegetable fiber traded in a competitive world market and imported into theUnited States at a world price of $9 per pound U.S. domestic supply anddemand 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 localtown 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 formarginal 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 perfectcompetitor , 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 thechapter. 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 thismarket ? 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 )=1
300q3
+ 0.2 q2 + 4 q + 10
a Calculate the firm’s short-run supply curve with q as a function ofmarket price (P) !
b On the assumption that there are no interaction effects among costsof the firms in the industry, calculate the short-run industry supplycurve !
c Suppose market demand is given by Q = -200 P + 8000 . What willbe the short-run equilibrium price-quantity combination?
6 Below is the inverse market demand curve :
48
P = α−βQ
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 = ηq2
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 competitiveindustry.
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 = 120 950
Qd .
There are fifty manufactures of cellular phones. Each manufacture has thesame production costs. These are described by the long-run total andmarginal 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 !
49
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 manyfirms that can copy five tapes per day at an average cost of $10 per tape.Each firm must also pay a royalty to film studios, and the per-firm royaltyrate (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 theequilibrium price and quantity of copied tapes? How many tapefirms 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 forcopied tapes? How many tape firms are there? What is the per-filmroyalty rate?
10 You know that if a tax is imposed on a particular products , the burden of thetax is shared by producers and consumers. You also know that the demand forautomobiles is characterized by a stock adjustment process. Suppose aspecial 20-percent sales tax is suddenly imposed on automobiles . will theshare 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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