TEACHING ASSISTANTS OF MICROECONOMICS
AND MACROECONOMICS
ECONOMICS AND DEVELOPMENT STUDIES
FACULTY OF ECONOMICS AND BUSINESS
ACKNOWLEDGEMENT
In the name of Allah, The Most Gracious, The Most Merciful
Alhamdulillah, all praises to Allah SWT, The Almighty, for giving belief, health, confidence and blessing for the writers to accomplish this Module of Microeconomics I. Shalawat and Salam be upon our Prophet Muhammad SAW, who has brought us from the darkness into the brightness and guided us into the right way of life.
In this opportunity, we also like to express our deep thanks to Head Department of Economics,Coordinator 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. Arjuni Rahmi Barasa 120210130104 2. Hygea Marwany 120210130091 3. Amatul Ghina 120210120130 4. Safira Kirami 120210120016 5. Citra Kumala 120210110155 6. Rahma 120210110124 7. Ridho Al-Izzati 120210110095
Acknowledge and Agree, Coordinator of Undergraduate Program of Department of Economics
Dr. Adiatma Yudistira Manogar Siregar, S.E., ME. conSt. NIP. 19801205 200812 1 001
CHAPTER 1
REVIEW OF DIFFERENTIAL CALCULUS AND CONSTRAINED
OPTIMIZATION
1. Differentiate 𝑦 = (𝑥3+ 1𝑥 − 7)(3𝑥 + 5).
2. Differentiate 𝑦 = 𝑥−2(5 + 3𝑥−3).
3. Differentiate 𝑦 = 𝑥3ln 𝑥.
4. Differentiate 𝑦 = 10𝑥𝑒𝑧2.
5. Differentiate𝑦 = 23𝑥+1ln(5𝑥 − 11).
6. Differentiate 𝑦 = (𝑥3− 7𝑥2)4 (1 + 9𝑥)1/2.
7. Differentiate 𝑦 = 4
𝑥+1.
8. Differentiate 𝑦 = 𝑥3
3𝑥−1.
9. Differentiate 𝑦 =4𝑥3−7𝑥
5𝑥2+2.
10. Differentiate 𝑦 = 6𝑥2
4𝑒𝑥−𝑥.
11. Differentiate 𝑔(𝑥) = 1+ln 𝑥
𝑥2−ln 𝑥.
12. Differentiate 𝑔(𝑥) = 2𝑥
2𝑥−3𝑥.
13. Differentiate 𝑓(𝑥) =(𝑥2−1)3
(𝑥2+1).
14. Differentiate 𝑓(𝑥) = 5𝑒−𝑥
𝑥+𝑒−2𝑥.
15. Differentiate 𝑦 =𝑥3ln 𝑥
𝑥+3 .
16. Differentiate 𝑓(𝑥) =𝑥2(2𝑥−1)3
(𝑥2+3)4 .
17. Differentiate 𝑔(𝑥) = 1
18. Differentiate 𝑓(𝑥) = √3𝑥+2
2𝑥−1.
19. Consider the function 𝑓(𝑥) = 𝑥2
𝑒2𝑥. Solve 𝑓′(𝑥)= 0 for 𝑥. Solve 𝑓′′(𝑥)= 0 for 𝑥.
20. Differentiate 𝑦 = (5𝑥 + 1)2.
21. Differentiate 𝑦 = √83𝑥2− 5𝑥 + 8.
22. Differentiate 𝑦 = (1 − 4𝑥 + 7𝑥5)30.
23. Differentiate 𝑦 = (4𝑥 + 𝑥−5)1/3.
24. Differentiate 𝑦 = (4𝑥−𝑥6
𝑥3 )− 4 5.
25. Differentiate 𝑦 = 𝑒5𝑥2+17𝑥−13.
26. Differentiate 𝑦 = ln(17 − 𝑥).
27. Differentiate 𝑦 = 10(1 + (2 − (6 + 7𝑥4)9)3)5.
28. Differentiate 𝑦 = ( 𝑥 + 1 ) −1
3𝑒𝑥
29. Diffferentiate 𝑦 = ln(6 − 𝑥2).
CHAPTER 2
PREFERENCE, UTILITY, AND UTILITY FUNCTION
Let’s begin with the concept of “preference”:
An individual who reports that “A is preffered to B” is taken to mean that all things considered, he or she feels better off under situation A than under situation B. There are three basic properties of preference relation assumption:
1. Completeness. If A and B are any two situations, the individual can always
specify exactly one of the following three possibilities: “A is preffered to B”;
“B is preffered to A”; or “A andB are equally attractive”.
2. Transitivity.The individual’s choice are internally consistent. If “A is
preffered to B” and “B is preffered to C,” then he or she must also report that “A is preffered to C.”
3. Continuity. If an individual reports “A is preffered to B,” then situations suitably “close to” A must also be preffered to B.
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.
The ceteris paribus (other things being equal) assumption is invoked in all economic analyses of utility-maximizing choices so as to make the analysis of choices manageable within a simplified setting.
Indifference curve represents all combinations of market baskets that provide a
consumer with the same level of utility.
The negative of the slope of an indifference curve (U1) at some point is termed
the Marginal Rate of Substitution (MRS) at that point.
𝑴𝑹𝑺 = − 𝒅𝒚𝒅𝒙 |
𝑼 = 𝑼𝟏′
Utility Functions
a. Cobb Douglas Utility, U(x,y) = xαyβ b. Perfect Substitutes, U(x,y) = αx + βy
c. Perfect Complement, U(x,y) = min(αx, βy)
MULTIPLE CHOICE
1. A typical indifference curve...
a. Shows that as a consumer has more of a good he or she is less willing to exchange it for one unit of another good
b. Shows all combinations of goods that give a consumer the same level of utility
c. Shifts out if income increases d. Both b and c
2. A utility function shows the relation between...
a. The amount of goods consumed and a consumer’s utility
b. Income and a consumer’s utility
c. Prices and a consumer’s utility
d. Maximum utility and the prices and income facing a consumer
3. Marginal utility is the...
a. Relative value of two goods when a utility-maximizing decision has been made
b. Change in total utility that results from increasing the amount of a good consumed by one unit
c. Change in the amount of a good consumed that increases total utility by one unit
c. Marginal rate of substitution d. Ratio of the prices of the goods
5. Along an indifference curve...
a. The ratio of the marginal utilities is constant b. The MRS is constant
c. The price ratio is constant d. None of the above
6. Which one of the following is CES utility function? a. b.
c. d.
7. Moving along an indifference curve the
a. Consumer does not prefer one consumption point to another b. Consumer prefers some of the consumption points to others
c. Marginal rate of substitution for a good increases as more of the good is consumed
d. Marginal rate of substittution is constant
8. We have ask Dave to rank his preferences between three market baskets, A, B, and C. If Dave prefers B to C but does not care if he gets A or B, then
a. A is on a higher indifference curve than B
b. B is on a higher indifference curve than C but it is not possible to determine wheter C is on a higher, lower, or the same indifference curve as A
c. C is on a higher indifference curve than A d. A and B are on the same indifference curve
9. As a consumer moves away from the origin onto higher indifference curves, what happens?
a. The consumer reaches less preffered combinations of goods b. The consumer reaches more affordable combinations of goods c. The consumer reaches more preffereed combinations of goods d. Nothing
10. Normally shaped indifference curves are bowed towards the origin of the graph. The reason for this shape is
a. The law of demand
b. The principle of diminishing marginal rate of relative price c. Diminishing marginal rate of substitution
ESSAY
1. Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether the MRS declines as x increases).
a. 𝑈(𝑥, 𝑦) = 3𝑥 + 𝑦 b. 𝑈(𝑥, 𝑦) = √𝑥 + 𝑦 c. 𝑈(𝑥, 𝑦) = √𝑥2− 𝑦2
2. Consider the following utility functions: a. 𝑈(𝑥, 𝑦) = 𝑥𝑦
b. 𝑈(𝑥, 𝑦) = 𝑥2𝑦2 c. 𝑈(𝑥, 𝑦) = ln 𝑥 + ln 𝑦
3. Many advertising slogans seem to be asserting something about people’s
preferences. How would you capture the following slogans with a mathematical utility function?
a. Promise margarine is just as good as butter. b. Things go better with Coke.
c. You can’t eat just one Pringle’s potato chip.
d. Krispy Kreme glazed doughnuts are just better than Dunkin’.
4. A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve?
5. A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle (8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consuming bundle (4, 4). She is indifferent between these two bundles. Assuming that the utility function is Cobb-Douglas of the form U(x,y) = xαyβ, where α and β are positive constants, what is the utility function for this consumer?
6. There are a few standard assumptions about what an indifference map can and cannot look like. Which are these assumptions, and what reasoning lies behind them?
7. a. What is the marginal rate of substitution, MRS? State the definition and explain, in words, what it means.
8. Often, we assume that consumers have diminishing MRS. Explain what that means and how it is reflected in indifference curves.
9.
a. In figure, we have drawn an indifference curve for a certain consumer. Calculate an estimate of her marginal rate of substitution, MRS, in point A. b. Can we say anything about whethe point B is better or worse for the
consumer, as compared to point A?
CHAPTER 3
UTILITY MAXIMIZATION AND CHOICE
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 substitution (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.
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, … . . , 𝑝𝑛𝑎𝑛𝑑 𝐼
𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 = 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
𝑚𝑖𝑛𝑖𝑚𝑎𝑙 𝑒𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑢𝑟𝑒 = 𝐸(𝑝1, 𝑝2, … … , 𝑝𝑛, 𝑈)
Properties of expenditure functions :
Homogeneity
Expenditure functions are nondecreasing in prices
MULTIPLE CHOICES
1. There is slope of budget constraint:
a. 𝑀𝑈𝑦
2. Consumers maximize satisfaction subject to: a. Expenditure.
b. Indifferent function.
c. Marginal Utility. d. Budget constraint.
3. Which point does maximize individual’s utility?
a. D. b. C.
c. C,B. d. A, B,C.
4. In the case of utility maximization, dual problem concern:
a. Allocating income in such way as to achieve maximum utility with a given level of price.
b. Allocating combination of goods to achieve maximum revenue.
c. Allocating income in such a way as to achieve a given utility level with the minimal expenditure.
d. Allocating income in such a way as to achieve maximum utility with a given budget constraint.
5. Expenditure function equals the inverse of: a. Budget constraint.
b. Indirect utility function.
c. Direct utility function. d. Income function.
6. The concern in Consumer’s primal problem is:
a. Minimizing expenditure subject to a utility constraint. b. Maximizing expenditure subject to a utility constraint.
d. Maximizing utility subject to budget constraint.
7. To reach maximum utility, each individual should choose combination of commodity which:
a. The MRS between any two of goods is equal to ratio of those goods’ utilities.
b. The MRS between any two of goods is greater than ratio of those goods’ market prices.
c. The MRS between any two of goods is equal to ratio of those goods’ market
prices.
d. The MRS between any two of goods is equal to ratio of those goods’ marginal
utilities.
8. A consumer is making purchases of product A and product B such have marginal utility of product A is 40 and the marginal utility of product B is 30. The price of product A is $10 and $20 for product B. for maximizing his utility the consumer should:
a. Increase consumption of product A and decrease consumption of product B. b. Decrease consumption of product A and increase consumption of product B. c. Switch product B to the cheaper one.
d. Make no change in the consumption of product A or B.
9. Which of the following map illustrate Cobb-Doughlas utility function?
a.
b.
c.
10. The second order should be required for maximum utility is:
a. The marginal rate of substitution need to equal the ratio of the prices. b. MRS is assumed to be diminishing.
c. The second order derivative of utility function equal to zero. d. Expenditure functions are nondecreasing in prices
ESSAY
1. What is utility maximization? Graph and show where is the optimal quantity of x and y that maximize utility.
2. Suppose that Miranti only eats breads (b) and meats (m) for her supper. Which provide her utility of
𝑢𝑡𝑖𝑙𝑖𝑡𝑦 = 𝑈(𝑏, 𝑚) = √𝑏𝑚
If breads cost $10 per bundle and $25 per kg for meat, how should Miranti spends $100 in order to maximize his utility?
3. Assumed the utility function of individual is given by 𝑈(𝑥, 𝑦) = 𝑥𝛼𝑦𝛽 where
𝛼 + 𝛽 = 1. 𝑃𝑥 , 𝑃𝑦 are price of good x, good y and I is income. Calculate the value of good x that maximize individual’s utility!
4. 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
b. Calculate optimum combination of good x and good y to maximize utility!
5. On a given evening, Zahara enjoys the consumption of seblak (s) and green tea latte (g) according to the function
𝑈(𝑑, 𝑔) = 20𝑠 − 𝑠2+ 18𝑔 − 3𝑔2
a. How many seblak and glasses of latte does she consume during evening? (cost is no object to Zahara)
b. If Zahara should limit the sum of glasses of latte and seblak consumed to 5. How many glasses of latte and seblak will she consume under these circumstances?
6. Mr. Gumelar enjoys commodities x and y according to the utility function
What is Mr. Gumelar’s maximum utility if 𝑃𝑥 = $3, 𝑃𝑦= $4 , and he has
$50 to spend.
7. Consider the indirect utility function given by
𝑣(𝑝1, 𝑝2, 𝑚) = 𝑝1𝑚+𝑝2
a. What are the demand functions? b. What is the expenditure function? c. What is the direct utility function?
8. A person has an income $100. He uses 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 equation
b. Suppose that income increase 50%. Make the new budget constraint 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 budget constraint.
9. Aditya 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) = 𝑏12 𝑛12
CHAPTER 4
THE THEORY OF OPTIMUM CONSUMER’S CHOICE
Demand function: A representation of how quantity demanded depends on prices, income, and preferences.
Homogeneity: Individual demand functions are homogeneous of degree zero in all prices and income. Changing all prices and income in the same proportions will not affect the physical quantities of goods demanded.
Substitution and Income Effect are due to change in good’s price.
Substitution Effect: The part of the change in quantity demanded that is caused by substitution of one good for another. A movement along an indifference curve.
Income Effect: The part of the change in quantity demanded that is caused by a change in real income. A movement to a new indifference curve.
Demand characteristic which caused by change in income.
Giffen’s Paradox: A situation in which an increase in a good’s price leads people to consume more of the good.
Relation among good is due to changes in the price of another good Complements: Two goods such that when the price of one increases, the quantity demanded of the other falls.
Substitutes: Two goods such that if the price of one increases, the quantity demanded of the other rises.
Consumer surplus: The extra value individuals receive from consuming a good over what they pay for it. What people would be willing to pay for the right to consume a good at its current price.
MULTIPLE CHOICE
1. The figure below can derived if increasing in income happen (ratio Px/Py stays constant). It indicate that...
a. The utility-maximizing conditions get the MRS stay constant as the individual
b. Individual moves to higher levels of satisfaction.
c. The MRS is therefore the same at point (X3, Y3) as at (X1, Y1). d. All of above
2. Given expenditure function at initial price level (Px), Eo = E(Px, Py, Uo). And if at any time the price increase to the new level (Pxi), so the expenditure function would be Ei = E(Pxi, Py, Uo). Thus, welfare change is Eo – Ei. So, it’s mean that…
3. Based on figure above, what is good Z and Y, respectively? a. Substitute Good, Normal Good
b. Inferior Good, Normal Good
c. Complement Good, Substitute Good d. Normal Good, Inferior Good
Following figures are for number 3 and 4:
4. What is kind of relationship between good X and Y (Figure A)? a. Substitute good
b. Complement Good c. Normal and Inferior Good
d. Giffen’s Paradox
5. What is kind of relationship between good X and Y (Figure B)? a. Substitute good
b. Complement Good c. Normal and Inferior Good
d. Giffen’s Paradox
6. Two goods, X and Y, are said gross substitutes if…
a. Changes in price Y toward quantity demand of good X = 0 b. Changes in price Y toward quantity demand of good X > 0 c. Changes in quantity demand Y toward price of good X < 0 d. Changes in price Y toward quantity demand of good X < 0
7. Two goods, X and Y, are said gross complement if…
a. Changes in price Y toward quantity demand of good X = 0 b. Changes in price Y toward quantity demand of good X > 0 c. Changes in quantity demand Y toward price of good X < 0 d. Changes in price Y toward quantity demand of good X < 0
8. If the income effect of a price change is strong enough, the change in price and the resulting change in the quantity demanded could actually move in the same direction. For example: An increase in the price of potatoes therefore reduced real income substantially. People were forced to cut back on other luxury food consumption in order to buy more potatoes. This phenomena usually call..
a. Composite Good Phenomena
b. Giffen’s Paradox
9. Based on figure below, please show the substitution and income effect…
a. Subtitution Effect: Y* to Y** and income effect: Y** to I/Py b. Subtitution Effect: X* to XB and income effect: XB to X** c. Subtitution Effect: XB to X** and income effect: X* to XB d. Subtitution Effect: XB to B and income effect: X* to X**
10. If the demand of good Z increase caused by increasing price of good Y, what can you conclude?
ESSAY
1. Graph demand curve for good X from indifference and budget constraint curve, show the quantity of demand at any level of price!
2. What is difference between Marshallian Demand and Hicksian Demand (compensated demand)? Please explain briefly!
3. Following equation is indirect utility function:
𝑣 = (4𝑃1
𝑥+
3 4𝑃𝑦) 𝑚
With Roy’s Identity, please define Marshallian demand for good x and y.
4. What is clear distinction between Roy’s Identity and Sheppard Lemma?
5. Graph Substitution and Income Effect for inferior good when the income increase! (hint: let the horizontal axis is inferior good noted by X and vertical axis is normal good noted by Y)
6. Tell with graph Substitution and Income Effect for Giffen’s Paradox (of course
when the price of good X increase)! Complete your answer with explanation. (hint: let the horizontal axis is inferior good noted by X and vertical axis is normal good noted by Y)
7. Given utility fuction for food (X) and coffee (Y), ( , ) = 0.5 0.5. Use the simple
income equation, lets find the compensated demand function!
8. Show that if there are only two goods (X and Y) to choose. If X is inferior, how do changes in income affect the demand for Y? (Use graph)
9. Please graph substitution and income effect for good Y as a normal good when its price is decrease! (hint: good Y stand on vertical axis)
CHAPTER 5
UNCERTAINTY AND INFORMATION
In the St. Petersburg paradox, this game has an infinite number of outcomes. The expected value of the St. Petersburg paradox is:
The von Neumann-Morgenstern Theorem could be derived from more basic axioms of “rational” behavior that represent an attempt by the authors to generalize the foundations of the theory of individual choice to cover uncertain situations. The formula is U(xi) = i · U(xn) + (1 - i) · U(x1).
A rational individual will choose among gambles based on their expected utilities, they will act as if they choose the option that maximizes the expected value of their von Neumann-Morgenstern utility index.
An individual who always refuses fair bets is said to be risk averse.They will exhibit diminishing marginal utility of income and will to pay to avoid taking fair bets through insurance.
The most commonly used risk aversion measure was developed by Pratt, is defined as
For risk averse individuals, U”(W) < 0.
Diminishing marginal utility would make potential losses less serious for high-wealth individuals as well as makes the gains from winning gambles less attractive.
If utility is quadratic in wealth, Pratt’s risk aversion measure is
If utility is logarithmic in wealth, Pratt’s risk aversion measure is
If utility is exponential in wealth, Pratt’s risk aversion measure is
The portfolio problem seems that the fraction invested in risky assets should be smaller for more risk-averse investors. To get started, assume that an investor has a certain amount of wealth, W0, to invest in one of two assets. The first asset
yields a certain return of rf , whereas the second asset’s return is a random
variable, . If we let the amount invested in the risky asset be denoted by k, then this person’s wealth at the end of one period will be
.
In the State-Preference Approach, there is a need to develop new techniques to incorporate the standard choice-theoretic framework. Outcomes of any random event can be categorized into a number of states of the world. Contingent commodities are goods delivered only if a particular state of the world occurs.
The expected utility associated with these two contingent goods is V(Wg,Wb) =
U(Wg) + (1 - )U(Wb).
Assume that the person can buy $1 of wealth in good times for pg and $1 of wealth
in bad times for pb.. His budget constraint is W = pgWg + pbWb.
If markets for contingent wealth claims are well-developed and there is general agreement about , prices for these goods will be actuarially fair
pg = and pb = (1- ).
If contingent claims markets are fair, a utility-maximizing individual will opt for a situation in which Wg = Wb.. Maximization of utility subject to a budget
constraint requires that .
Consider two people, each of whom starts with an initial wealth of W*. Each seeks to maximize an expected utility function of the form
To illustrate why information has value, assume that an individual faces an uncertain situation involving “good” and “bad” times and that he or she can invest in a “message” that will yield some information about the probabilities of these outcomes. Hence, when this person is considering purchasing the message, expected utility is given by: Ewith m = pV1 + (1-p)V2 and without purchasing the
MULTIPLE CHOICE
1. Suppose that Smith and Jones decide to flip a coin. Heads (x1) where Jones will
pay Smith $1, tails (x2) where Smith will pay Jones $1. From Smith’s point of
2. A coin is flipped until a head appears. If a head appears on the nth flip, the player is paid $2n
.The probability of getting of getting a head on the ith trial is a. (1/4)i
b. (1/2)i
c. (1)i
d. (2)i
3. The theory that define the utility of xi as the expected utility of the gamble that
the individual considers equally desirable to xi shown by formula U(xi) = i ·
U(xn) + (1 - i) · U(x1) is …
a. St. Petersburg Paradox Game b. Expected Utility
c. The von Neumann-Morgenstern Theorem d. The State-Preference Approach
4. A consumer has an income for the next year of $14400, but faces a 0.5 probability of a monetary loss of $ 4400 due to illness. There are no other losses in utility cause by the illness. Her utility function is U(W) = W 0.5 where W is the amount of income, net of any loss she suffers. The amount of expected dollar loss is … a. $8800
b. $4400
c. $2200 d. $1100
5. Lala is buying an insurance for her house. The land costs $20 and the house costs $80, so her total wealth is 100. Probability of fire is 0.2. If fire occurs, her wealth is only $20. Given the data above, the expected wealth is …
a. 20 b. 36
c. 68 d. 84
a. U(W*) > Uh
(W*) > U2h
(W*) b. U(W*) > U2h(W*) > Uh(W*)
c. Uh(W*) > U(W*) > U2h
(W*) d. Uh(W*) > U2h
(W*) > U(W*)
7. Let h be the winnings from a fair bet, so E(h) = 0. Let p be the size of the insurance premium that would make the individual exactly indifferent between taking the fair bet h and paying p with certainty to avoid the gamble. So, the proper equation to measure risk aversion is
a. E[U(W )] = U(W - p)
b. E[U(W + h)] = U(W - p)
c. E[U(W)] = U(W)
d. E[U(W + h)] = U(W )
8. If utility is quadratic in wealth, U(W) = a + bW + cW 2 where b > 0 and c < 0. Due
to Pratt’s risk aversion measurement, risk aversion … as wealth…
a. constant, constant b. increases, increases c. declines, declines d. increases, declines
9. If we assume that individuals exhibit a diminishing marginal utility of wealth, they will also be…
a. risk taker b. risk neutral c. risk averse d. price taker
10. Suppose that Jeff’s utility function is given by U(W) = W 0.5, where W represents
total wealth. So, Jeff is a. risk taker
ESSAY
1. Suppose a person must accept one of three bets:
Bet 1:Win $100 with probability 1/2; lose $100 with probability ½ Bet 2:Win $100 with probability 3/4; lose $300 with probability ¼ Bet 3:Win $100 with probability 9/10; lose $900 with probability 1/10
a. Show that all of these are fair bets.
b. Graph each bet on a utility of income curve similar to Figure 5.1. c. Explain carefully which bet will be preferred and why.
2. An individual purchases a dozen wine glasses and must take them home. Although making trips home is costless, there is a 50% chance that all of the wine glasses carried on any one trip will be broken during the trip. The individual considers two strategies:
Strategy 1: Take all 12 wine glasses in one trip. Strategy 2: Take two trips with 6 in each trip
a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that on the average, 6 wine glasses 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?
3. A local politician has Bernoulli utility function u(x) = √ x. A construction entrepreneur approaches the politician to offer him a bribe in exchange for being selected to perform a public works project. If the politician accepts the bribe, there is some probability p ∈ (0, 1) that this will be made public before the next election, in which case the politician will have to pay back the value of the bribe in fines and will not be reelected. The politician places monetary value b > 0 on being reelected.
a. What is the minimum bribe the entrepreneur can offer to the politician in exchange for the public works contract?
b. Suppose that a number of scandals involving local politicians and public works contracts have surfaced recently, so that media scrutiny increases the likelihood that a bribe-taking politician is found out. What happens to the minimum bribe the politician accepts?
c. Suppose that local politicians vote themselves a salary increase, so that the value of holding this office increases. What happens to the minimum bribe the politician accepts?
expected utility = (1/2) ln YNR + (1/2) ln YR, where YNR and YR represent the
flower farmer’s income in the states of “normal rain” and “rainy,” respectively. a. Suppose the flower farmer must choose between two flowers that promise
the following income prospects:
Flower YNR YR
Rose $28,000 $10,000
Jasmine $19,000 $15,000
Which of the flower will he plant?
b. Suppose the flower farmer can plant half his field with each flowers. Would he choose to do so? Explain your result.
c. What mix of what and flower would provide maximum expected utility to this farmer?
d. Would rose crop insurance, available to flower farmers who grow only rose, which costs $4000 and pays off $8000 in the event of a rainy growing season, cause this farmer to change what he plants?
5. Explain Utility of Wealth from Two Fair Bets of Differing Variability using the graph!
6. Consider a person with a current wealth of $100,000 who faces a 25% chance of losing his automobile worth $20,000. Suppose also that the person’s von Neumann-Morgenstern utility index is U(W) = ln (W).
a. The person’s expected utility.
b. The amount of a fair insurance premium.
c. If the individual will likely be willing to pay more than $5,000 to avoid the gamble. How much will he pay?
7. Suppose there is a 50-50 chance that a risk-averse individual with a current wealth of $20,000 will contract a debilitating disease and suffer a loss of $10,000.
a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-income graph to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured.
b. Suppose two types of insurance policies were available:
A fair policy covering the complete loss.
A fair policy covering only half of any loss incurred.
Calculate the cost of the second type of policy and show that the individual will generally regard it as inferior to the first.
8. Ms. Ria 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 (Z), given by U(Z) = ln(Z)
a. If there is a 25 percent probability that Ms. Ria will lose $1000 of her cash on the trip, what is the trip’s expected utility?
b. Suppose that Ms. Ria can buy insurance against losing the $1000 (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 $1000 without insurance.
c. What is the maximum amount that Ms. Ria would be willing to pay to insure her $1000?
9. Consider a person with wealth of $100,000 who faces a 25% chance of losing his automobile worth $20,000
wealth with no theft (Wg) = $100,000 and probability of no theft = 0.75
wealth with a theft (Wb) = $80,000 and probability of a theft = 0.25
If we assume logarithmic utility, how much is the expected utility?
