ECON 323 HW 1, 2, 3 questions

The cost of producing n units of output in a certain factory is given by the expression C(x)=2x+x^2. Then the expression of the derivative of the average C is? 3x 2x 2 1 2+2x

1 C(x)/x=2+x. The derivative of this expression is 1.

The cost of producing n pairs of shoes is C(n)=n^2. What is the equation of the line that is tangent to the cost function C at n=3? C=9n+3 C=3n+3 C=6n-9 C=9n-3 C=3n+9

C=6n-9 The tangent passes through (n,C(n)) i.e., (3,9). The slope of this tangent is the derivative of C at n=3. Since C'(3)=6, the equation of the tangent is C-9=6(n-3), i.e., C=6n-9

Alexandra is maximizing her utility over goods x and y subject to her budget constraint. Her preferences are smooth (maximizers are always interior). At her optimal consumption bundle, her MRS of goof y for good x is equal to 1. The price of good x is $4. What is the price of good y? $1 $3 We need to know Alexandra's income to find the price. $2 $4

$4 Let (x*,y*) be Alexandra's optimal consumption. We know that at an interior maximizer MRSxy(x*,y*)=px/py. Now, we also know that MRSxy(x*,y*)= 1/MRSyx(x*,y*). Thus, 1= px/py. Thus, py=$4.

Consider the function F given by the following expression: F(n,k)=2n+k where n and k are numbers. Which of the following pairs (n,k) belongs to the same iso-level set as (1,3)? (5,0) (0,3) (0,5) (3,0) (5,5)

(0,5) The value of the function at (1,3) is F(1,3)=2+3=5. From the given options the only one whose F value is 5 is (0,5), because F(0,5)=2*0+5=5.

Jane consumes bundles of apples and pears. In order to understand her behavior we use a graph of her consumption space in which we draw two axes. We measure apples in the horizontal axis and pears in the vertical axis. We write (x,y) to represent a bundle of x apples and y pears. Consider the following list of consumption bundles: (1,2); (1,3); (2,6); (1/2,1/3); and (3,7). Which of these bundles is located the most to the south west in our graph? (2,6) (3,7) (1,2) (1/2,1/3) (1,3)

(1/2,1/3) The bundle that is most to the south west is the one that has the least of each commodity, i.e., (1/2,1/3).

Elena consumes bundles of food and transportation. In order to understand her behavior we use a graph of her consumption space in which we draw two axes. We measure food in the horizontal axis and transportation in the vertical axis. Which of the following bundles has the largest amount of food (if the bundles are written consistently with the way we graph consumption)? (100,80) (99,150) (95,95) (80, 100) (98,98)

(100,80) With two commodities, say x and y, in which we draw x in the horizontal axis and y in the vertical axis, a bundle (x,y) contains x of the first commodity and y of the second commodity. Thus, bundle (100,80) has 100 units of food, which is the largest amount of food among these bundles; (99,150) has 100 units of transportation, which the maximum among these bundles.

Chuck's utility function on the number of cookies he eats, denoted by c, and the number of glasses of milk he drinks, denoted by m, is given by: U(c,m)=min{2c,3m}. What is the point that is most to the south-west in Chuck's indifference curve that passes through (2,8)? 8,8). (2,4/3). (5,5). (2,2). (4/3,8).

(2,4/3). U(2,8)=4. U(2,4/3)=2. For any point to the south west of (2,4/3) the utility is below 4. Thus, (2,4/3) is the point most to the south-west in Chuck's indifference curve through (2,8).

Chuck's utility function on the number of cookies he eats, denoted by c, and the number of glasses of milk he drinks, denoted by m, is given by: U(c,m)=min{2c,m3}. What is the point that is most to the south-west in Chuck's indifference curve that passes through (4,8)? (2,4). (4,2). (5,5). (8,8). (2,2).

(4,2). U(4,8)=8. U(4,2)=8. For any point to the south west of (4,2) the utility is below 8. Thus, (4,2) is the point most to the south-west in Chuck's indifference curve through (4,8).

Elena consumes bundles of food and transportation. In order to understand her behavior we use a graph of her consumption space in which we draw two axes. We measure food in the horizontal axis and transportation in the vertical axis. Which of the following bundles has the largest amount of transportation (if the bundles are written consistently with the way we graph consumption)? (98,98) (95,95) (100,80) (80, 100) (99,150)

(99,150) With two commodities, say x and y, in which we draw x in the horizontal axis and y in the vertical axis, a bundle (x,y) contains x of the first commodity and y of the second commodity. Thus, bundle (100,80) has 100 units of food, which is the largest amount of food among these bundles; (99,150) has 100 units of transportation, which the maximum among these bundles.

What is the slope of the line that passes through (0,10) and (10,9)? -8/10 9/10 -9/10 0 -1/10

-1/10 m=(y2-y1)/(x2-x1), then m=(9-10)/(10-0)=-1/10.

Otis consumes two goods: x and y. If the price of x is $2 and the price of y is $5, then the marginal rate of transformation of y into x is: -1/5 -2/5 1/4 -1/4 2/5

-2/5 MRTxy=-px/py=-2/5

Linda consumes two goods: x and y, has preferences that are smooth and maximizers are always interior, and income W=$150. If px=$25 and py=$5, what is the marginal rate of transformation of y into x (MRTxy) at her optimal bundle? -5 7/2 -3 -1/2 1/5

-5 MRTxy is actually independent of where it is measured. It only depends on prices, MRTxy=-px/py=-25/5=-5.

Consider the utility function U(x,y)=4x+2y. Someone who has this utility function is indifferent between consuming 4 units of x and 0 units of y and which of the following? 3 units of x and 3 unit of y 2 units of x and 3 units of y 0 units of x and 8 units of y 3 units of x and 1 unit of y 2 units of x and 1 unit of y

0 units of x and 8 units of y Calculate the utility of each of the alternatives and only one has the same utility as (4,0).

Jane's utility function on bundles of two commotities (x,y) is given by: U(x,y)=4x2y3. What is the value of MRSxy(x,y) when x=2 and y=1? 4/3 3 1 1/3 2/3

1/3 Recall that MRSxy(x,y)=MUx(x,y)/ MUy(x,y). Then, MRSxy(x,y)=8xy3/12x2y2=2y/3x.

Juno has preferences that are represented by the utility function U(x,y)=x1/2y1/2 (this preference is smooth, which implies that maximizers are interior). If Juno has income W=$1000 and px=$50 and py=$25, what is Juno's consumption of good x (assuming Juno maximizes her utility given her budget constraint?) 20 units 10 units 25 units 50 units 80 units

10 units Let (x*,y*) be Juno's optimal consumption. We know that when maximization happens at an interior point, it must be the case that MRSxy(x*,y*)=-MTRxy. Thus, MRSxy(x*,y*)=px/py=50/25=2. We also know that MRSxy(x,y)=MUx(x,y)/MUy(x,y)=(1/2)x-1/2y1/2/(1/2)x-1/2y1/2=y/x. Thus, y*/x*=2 and y*=2x*. Now, we know that June will consume all her income, so 50x*+25y*=1000. Thus, 100x*=1000 and x*=10

Leonid consumes bundles of food and transportation. His utility from bundle (10,10) has a value of 4. If for Leonid more is better, then among the following options, which can be the value of utility of bundle (20,20)? 3 1 2 1000 0

1000 Since more is better for Leonid, the utility of (20,20) has to be greater than the utility of (10,10). The only possible value is 1000. All other values are lower than 4.

Linda has an income of $100 which she spends completely on two goods: x and y. The price of x is $20 and the price of y is $12. What is the equation of Linda's budget constraint? 4x+100y=12 6x+10y=50 10x-6y=50 10x+6y=50 20x - 12y=100

10x+6y=50 The budget constraint is pxX+pyY=Income. Here, 20x+12y=100. Simplifying we get 10x+6y=50.

Jane's utility function on bundles of two commotities (x,y) is given by: U(x,y)=4x2y3. What is the expression for MUy(x,y)? 8x2y3 12x2y2 8xy3 4xy3 8x3y3

12x2y2 Recall that MUy(x,y) is the partial derivative of U with respect to y, which is the derivative of the function when we assume x is a constant. That is, dU/dy(x,y)=4x2(3)y(3-1)=12x2y2.

Kenneth consumes bundles of food and clothing. He has constant MRS of food for clothing equal to 5/7. Then, what is the maximal amount of clothing that he is willing to give up in order to consume 3 units more of food. 15/7 5/21 10/7 3 9

15/7 Since MRS food for clothing is 5/7, the maximal amount of clothing that he is willing to give up in order to consume 1 unit more of food is 5/7. Then, the maximal amount of clothing that he is willing to give up in order to consume 3 units more of food is 3*5/7=15/7.

Consider the utility function U(x,y)=2x+3y. The marginal utility of good x is given by: 3/2 2/3 2 3 1/3

2 MUx(x,y) is the partial derivative with respect to x, i.e., 2.

Consider the utility function given by U(x,y)=min{x,y}. Someone who has this utility function is indifferent between consuming 4 units of x and 2 units of y and which of the following? 20 units of x and 0 units of y. 2 units of x and 2 units of y. 1 unit of x and 4 units of y. 1 unit of x and 1 unit of y. 4 units of x and 1 unit of y.

2 units of x and 2 units of y. Calculate the utility of each of the alternatives and only one has the same utility as (4,2).

Jan Claude has the following utility function on the number of homework assignments, denoted by h, and the number of exams, denoted by e: U(h,e)=-5h-10e. If Jan Claude's instructor increases the number of exams by one, how many hw assignments should he/she reduce to maintain Jan Claude's welfare unaffected? 1. 3. 2. 0. We do not have enough information to solve the question. We would need to know the initial number of exams and hw assignments.

2. Since utility is linear an increase in e of one unit causes utility to drop by 10 units. So h has to decrease by 2 to compensate.

Suppose a cup of coffee at the campus coffee shop is $2.50 and a cup of hot tea is $1.25. Suppose a student's beverage budget is $20 per week. What is the algebraic expression of his/her budget constraint (denote cups of coffee by C and cups of tea by T)? 2.5C + 1.25T=20 2.5C + 2.5T = 20 1.25T = 20 2.5C = 20 1.25C + 2.5T = 20

2.5C + 1.25T=20

Charlie consumes apples and bananas. Charlie's utility function is U(A,B)=A*B where A is the number of applies and B is the number of bananas. Which of the following bundles would lie on the same indifference curve as the bundle 40 apples and 5 bananas? 10 apples, 10 bananas 20 apples, 10 bananas 20 apples, 11 bananas 6 apples, 35 bananas 80 apples, 4 bananas

20 apples, 10 bananas Calculate the utility of each of the alternatives and only one has the same utility as (40,5).

What is the solution for y of the following system (for positive values of both variables): x2/y2=9 and x+y=100. 15 18 75 30 25

25 Since x2/y2=9, then for positive values of both x and y, we have that x/y=3. Then, x=3y. Replacing on the other equation we get that 3y+y=100, which is y=25. Then, x=3y=75.

What is the horizontal (x) intercept for the following budget constraint: 2x+3y=25? 15 0 25/2 5 25/3

25/2 Is x such that 2x+3*0=25, i.e., 25/2.

Suppose the price of good x is $10, the price of good y is $15, and Katrina's income is $30. What is the equation of Katrina's budget constraint? 3x + 2y = 6 2x + 3y = 6 15x + 10y = 30 3x + 2y = 12 2x + 3y = 12

2x + 3y = 6 The budget constraint is pxX+pyY=Income. Here it is: 10x+15y=30. Simplifying, 2x+3y=6.

Jane's utility function on bundles of two commotities (x,y) is given by: U(x,y)=4x2y3. What is the expression for MRSxy(x,y)? x/y 2x/3y 2y/3x 2x/3y2 3x/2y

2y/3x ecall that MRSxy(x,y)=MUx(x,y)/ MUx(x,y). Then, MRSxy(x,y)=8xy3/12x2y2=2y/3x.

Consider the utility function U(x,y)=4x+2y. An agent who has this utility function prefers which of the following baskets: 1 unit of x and 3 units of y 2 units of x and 2 units of y 2 units of x and 1 unit of y 0 units of x and 1 unit of y 3 units of x and 1 unit of y

3 units of x and 1 unit of y Evaluate the utility of each option and the one with greatest utility will be the preferred basket.

Linda consumes two goods: x and y, has preferences that are smooth and maximizers are always interior, and income W=$150. If px=$25 and py=$5, what is the marginal rate of substitution of good x for good y for Linda at her optimal bundle? 5 -3 7/2 1/5 1/2

5 Let (x*,y*) be Linda's optimal consumption. We know that when maximization happens at an interior point, it must be the case that MRSxy(x*,y*)=-MTRxy. Thus, MRSxy(x*,y*)=px/py=25/5=5.

Suppose that initially the price of good x is $10, the price of good y is $15, and Ellie's income is $30. Budget cuts at work reduce Ellie's income to $25. What is the new vertical (y) intercept of Ellie's budget constraint? 3 2 5/3 There is not enough information to calculate it. 5/2

5/3 The y intercept is where Ellie consumes all her income on good y, i.e., Income/py=25/15=5/3. One can also find the equation of the budget constraint and find its intercept.

Jane's utility function on bundles of two commotities (x,y) is given by: U(x,y)=4x2y3. What is the value of MUx(x,y) when x=1 and y=2? 48 64 12 32 128

64 Recall that MUx(x,y) is the partial derivative of U with respect to x, which is the derivative of the function when we assume y is a constant. That is, dU/dx(x,y)=4(2)x(2-1)y3=8xy3=8(1)(23)=8(8)=64.

What is the solution for x of the following system (for positive values of both variables): x2/y2=9 and x+y=100 15 30 18 25 75

75 Since x2/y2=9, then for positive values of both x and y, we have that x/y=3. Then, x=3y. Replacing on the other equation we get that 3y+y=100, which is y=25. Then, x=3y=75.

Napoleon has preferences on bundles of two goods (x,y). Napoleon finds (x,y) is at least as good as (x',y') whenever the following is true: either x>x', or x=x' and y≥y'. What is the shape of an indifference set of Napoleon's preferences? A U shaped curve. A curved line that satisfies more is better. A single point. An L shaped curve. A straight line.

A single point. Given (x,y) the only other bundle that is indifferent to (x,y) is (x,y). To see this, just take (x',y') that is different from (x,y) and such that (x,y) is at least as good as (x',y') and (x',y') is at least as good as (x,y). If x'>x, then it is not true that (x,y) is at least as good as (x',y'). If x>x', then it is not true that (x',y') is at least as good as (x,y). Thus, it must be the case that x=x'. Now, if y>y', we have that it is not true that (x',y') is at least as good as (x,y). If y'>y, we have that it is not true that (x,y) is at least as good as (x',y'). Thus, it must be the case that y=y'. We concluded that (x,y)=(x',y'). So the indifference set of Napoleon's preference contains a single point.

Suppose that the cost of producing n pairs of shoes is given by the expression C(n)=30+5n. The average cost of producing a pair of shoes is given by A(n)=C(n)/n. What is the derivative of the average cost function at n=10? 1. A'(10)=-35 2. A'(10)=-0.35 3. A'(10)=350 4. A'(10)=-5 5. A'(10)=-0.30

A'(10)=-0.30 Since A(n)=30/n+5, then A'(n)=-30/n2. Thus, A'(10)=-0.30

Suppose that the cost of producing n pairs of shoes is given by the expression C(n)=30+5n. The average cost of producing a pair of shoes is given by A(n)=C(n)/n. What is the average cost at n=30? 1. A(n)=5 2. A(n)=6 3. A(n)=8 4. A(n)=350 5. A(n)=35

A(n)=6 Since A(n)=30/n+5, then A(30)=30/30+5=6.

A student gets the following partial grades in this course. Hw1: 100; Hw2: 95; Hw3: 87.5; Hw4: 95; Hw5: 70.625; Hw6: 84.5833; Hw7: 88; Exam 1: 90; Exam2: 72.5; Final exam: 88.33. What is the final letter grade of this student in the course, if he or she misses three participation quizzes with no University accepted excuse? B F D C A

B The final exam replaces exam 2 so the student final score without the participation points is 88.42653. Since the student missed three quizzes, his or her participation grade bump is 1. Thus, this student final score is 89.42653. The rounded score is 89. The grade is B.

A preference is complete if: Between any two alternatives a and b, if a has more of each commodity, then a is preferred to b. Between any two alternatives a and b, a is indifferent to b. Between any two alternatives a and b, if a is preferred to b, then b is preferred to a. Between any two alternatives a and b, at least one of the following holds: a is at least as good as b, or b is at least as good as a. Between any two alternatives a and b, at least one of the following holds: a is preferred to b, or b is preferred to a.

Between any two alternatives a and b, at least one of the following holds: a is at least as good as b, or b is at least as good as a.

Assume that the cost of producing n pairs of shoes is given by C(n)=50+3n. The marginal cost of producing shoes at n is the derivative of C at n. What is the expression of the marginal cost of producing shoes at n? C'(n)=3 C'(n)=50n C'(n)=3n C'(n)=53 C'(n)=53

C'(n)=3 Recall that the derivative of a constant is zero, the derivative of 3n is 3, and the derivative of a sum is the sum of the derivatives. Then, C'(n)=3.

Suppose that the depreciation of a car, which is measured in dollars, is determined by the miles it is driven and its age. Which of the following expressions may represent the depreciation of a car if depreciation increases with both miles driven and age? (D is depreciation, m is miles driven, and a is age of the car in years). D(m,a)=0.1*m+100*a D(a)=100*a D(m,a)=0.1*m-100*a D(m)=2m. D(m,a)=100*a-0.1*m

D(m,a)=0.1*m+100*a D(m,a)=0.1*m+100*a. All the other options represent either a function of a single variable, or are functions that do not increase with both miles and age.

Aggeliki works for a multinational corporation. They relocate her to a city in which housing and food is double as expensive as in her original city, but all the other goods, like transportation, entertainment, education, etc. are half the price. The company does not know how Aggeliki spends her money. If they want to make sure that Aggeliki is not worse off with the change, what is the minimal change in salary that they need to give her? increase her salary by 50%. Double her salary. Triple her salary. Decrease her salary by 25%. Decrease her salary by half.

Double her salary. They have to double her salary. If she is spending all her money in housing and food, this is the only option with which she can afford her consumption in the first city.

The following is the graph of function C. This function assigns an amount of money to each q. The graph of C passes through the origin. Then it is increasing. There are three reference levels of q in the graph. They are denoted from left to right by q1, q2, and q3. There is an auxiliary line shown in the graph, which interpolates the origin and (q1,C(q1)). This line has a slope of 1. If one traces a line passing through the origin and (q2,C(q2)), this line's slope will be lower than 1. If one traces a line passing through the origin and (q3,C(q3)), this line's slope will be lower than 1, but will be higher than the slope of the line passing through the origin and (q2,C(q2)). (line curves above slope until (Cq1,q1), below slope at (Cq2,q2) and (Cq3,q3), goes above slope a little after We learn from this graph that AC is always decreasing True. False.

False. False. AC is not decreasing. It decreases from 0 to q2. After q2 it increases again. You can see this by drawing the line that interpolates the curve and seeing that after q2 the line becomes steeper.

A marketing firm is interested in learning about millennials' preferences on learning experiences. In order to do this they survey students at a university. They ask them six questions as follows. They consider the following leaning experiences: a traditional class, a flipped course (half of the content is delivered online), and an online class. For each pair of these alternatives, say a and b, they ask the students "is a at least as good as b." In order to make this operational, they provide the students with a picture in which the three alternatives are depicted as circles with their description next to it. The students are asked to draw arrows between the alternatives whenever the answer to a question is affirmative. That is, if a student finds alternative a is at least as good as alternative b, the student is asked to draw an arrow from the circle with label a to the circle with label b. (No question is a at least as good as a was asked; assume that the answer to each of these trivial questions is affirmative.) No matter what the answers of a student look like, they cannot violate completeness and transitivity at the same time. True. False.

False. False. Suppose that the answers are the following two arrows: Traditional class->Online class, and Online class ->Flipped course. There is no arrow between traditional class and flipped course or vice versa. Thus, the answers are incomplete. There is no arrow from Traditional class to Flipped course, thus the answers are not transitive.

Jon will get a $1000 tax refund this year. John files his taxes with a software company that offers him two options to get his tax refund. First, he can get the deposit back from the government. Alternatively, he can get a $1050 gift card from Amazon, which can be sold at a 6% discount online. Then, Jon will for sure be better off by choosing the Amazon card than the refund. False. True.

False. Jon may be constrained to buy only products available at amazon. Even though his income would increase with the card, his unrestricted utility maximization may require he buys part of the $1000 in commodities not available in Amazon. For instance, he may need this money to help him pay school. Note that getting the card and selling it would give him less than $1000.

Ronald consumes apples and bananas. The value that Ronald's utility assigns to a bundle of one apple and one banana is U(1,1)=-100. Is it necessarily true that Ronald prefers zero apples and zero bananas, i.e., (0,0) to (1,1)? True. False.

False. Utility has no meaning on its own. The meaning of utility is only to compare between bundles. It is possible that U(0,0)<U(1,1). For instance U(0,0)=-1000. So Ronald may prefer (1,1) to (0,0).

Natalia and her sister, Gina, have the following utility functions on the number of slices of pizza (x) and cans of soda (y) they consume in the semester. Natalia's is U(x,y)=3x+2y and Gina's is V(x,y)=4x+2y. If we plot their indifference curves with x measured in the horizontal axis and y measured in the vertical axis, we find that: Natalia's indifference curves are steeper than Gina's. Natalia's indifference curve through (3,2) crosses Gina's indifference curve through (0,0) The indifference curves of both sisters are parallel. Gina's indifference curves are steeper than Natalia's. Natalia's indifference curves are L shaped and Gina's are lines.

Gina's indifference curves are steeper than Natalia's. Both sisters have linear preferences. Thus their indifference curves are line. Take for instance the point (1,1). The equation of Natalia's indifference curve through (1,1) is 5=3x+2y, and the equation of Gina's indifference curve through (1,1) is 6=4x+2y. Thus for Natalia y=5/2-3x/2 and for Gina y=3-2x. Thus, Gina's indifference curves are steeper: the slope is more negative. The only point in Gina's indifference curve through (0,0) is (0,0). The utility of Natalia in her indifference curve through (3,2) is 13. Thus, (0,0) is not on Natalia's indifference curve through (0,0).

Victor is the manager of a local bank branch in College Station where he consumes bundles of two commodities x and y. Prices in College Station are px=1 and py=5. He is offered a transfer to Dallas where prices are px=2 and py=8; Victor is guaranteed a salary in Dallas with which he would be able to buy exactly what he buys in College Station. Victor's utility function is U(x,y)=xy and his income in College Station is $5000. What happens to Victor's utility if he accepts the change (Victor's utility maximization is always characterized by the tangency rule)? Increases 1.00% Increases 1.50% Increases 1.75% Increases 1.25% Decreases 1.00%

Increases 1.25% Solving for utility maximization we find that in College Station victor consumes (2500,500). Thus his salary in Dallas is 9000. Solving for utility maximization he consumes (2250, 565.5) in Dallas with the new salary. Thus his utility changes from 1250000 to 1265625. Increases 1.25%.

James works for a delivery company in College Station where he consumes bundles of two commodities x and y. Prices in College Station are px=1 and py=5. He is offered a transfer to Dallas where prices are px=2 and py=5; James is guaranteed a salary in Dallas with which he would be able to buy exactly what he buys in College Station. James's utility function is U(x,y)=x1/3y2/3 and his income in College Station is $2500. What happens to James' utility if he accepts the transfer (James' utility maximization is always characterized by the tangency rule)? Increases 1.275% Increases 5.827% Increases 4.236% Decreases 3.245% Increases 6.234%

Increases 5.827% Solving for utility maximization we find that in College Station James consumes (833.33,333.33). Thus his salary in Dallas is $3333.33. Solving for utility maximization he consumes (555.55,444.44) in Dallas with the new salary. Thus his utility changes from 452.402936 to 478.763264. Increases 5.827%.

A decreasing MRS of x for y means: The tangent to an indifference curve is flatter as x decreases. It has no behavioral meaning. It is more and more difficult to substitute x for y as the consumption of x is higher. It is more and more easy to substitute x for y as the consumption of x is higher. It is more and more easy to substitute y for x as the consumption of y is higher.

It is more and more difficult to substitute x for y as the consumption of x is higher.

Linda receives a raise at work and her income increases. How does her budget line change? It makes a parallel shift outward. It does not change. It makes a parallel shift inward. It gets flatter. It gets steeper.

It makes a parallel shift outward.

Jon has complete and transitive preferences on bundles of sodas and pizza. Jon finds one soda and two slices of pizza at least as good as two sodas and a slice of pizza. Moreover, two sodas and a slice of pizza is not at least as good as one soda and two slices of pizza. Then, we know that: Jon's preferences satisfy more is better. Jon's utility function is differentiable. Jon is indifferent between two sodas and one slice of pizza and one soda and two slices of pizza. Jon prefers two sodas and one slice of pizza to one soda and two slices of pizza. Jon prefers one soda and two slices of pizza to two sodas and one slice of pizza.

Jon prefers one soda and two slices of pizza to two sodas and one slice of pizza. Think of sodas a being represented in a horizontal axis and slices of pizza on the vertical. The two bundles in the statement are (1,2) and (2,1). Since (1,2) is at least as good as (2,1) but the opposite is not true, then (1,2) is preferred to (2,1). Since these bundles are not related by order, i.e., one is not greater-greater than the other, there is no implication for more is better here. Indeed, even if the bundles were related by order and there was no violation of more is better, we would not be able to say the property is satisfied in general, because we would not know if somewhere else there is a violation of it.

Phillip's utility function is given by U(x,y)=min{x,2y}. Then his indifference curves are: Smooth curves Horizontal lines L shaped curves Vertical lines Straight lines that never cross

L shaped curves

When asked about her preferences between a and b, Lisa answered that a is not at least as good as b. If Lisa has complete preferences, then: Lisa prefers b to a Lisa is indifferent between a and b It is impossible to know what alternative between a and b is better for Lisa. Lisa's preferences are not transitive Lisa prefers a to b

Lisa prefers b to a Completeness means that at least one of the following is true: a is at least as good as b or b is at least as good as a. Since a is not at least as good as b, then b must be at least as good as a. Thus Lisa prefers b to a.

Adam has complete and transitive preferences on cars. We know that he finds a Hyundai Sonata is at least as good as a Mazda 3, and that a Mazda 3 at least as good as a Honda Civic. Can we conclude that he prefers a Hyundai Sonata to a Honda Civic? Yes. No.

No. No, we cannot conclude this. We can only conclude that Hyundai Sonata is at least as good as the Honda Civic. For instance, it is possible that he is indifferent among the three cars.

Victor is the manager of a local bank branch in College Station where he consumes bundles of two commodities x and y. Prices in College Station are px=1 and py=5. He is offered a transfer to Dallas where prices are px=2 and py=8; Victor's utility function is U(x,y)=xy and his income in College Station is $5000. (Victor's utility maximization is always characterized by the tangency rule). Will Victor be able to afford what he was buying in College Station if he is offered a salary in Dallas that guarantees his welfare is the same with the transfer? Yes. No.

No. Solving for utility maximization we find that in College Station victor consumes (2500,500). His utility in College Station is 1250000. Thus a salary in Dallas that would make this affordable is at least 9000. Solving for utility maximization when he has a salary 9000 and the prices are those in Dallas, he consumes (2250, 565.5). His utility in Dallas with income 9000 would be 1265625. Thus, the new salary that makes him indifferent with the transfer is less than 9000. Thus, he will get a salary that is less than 9000 in Dallas.

Napoleon has preferences on bundles of two goods (x,y). Napoleon finds (x,y) is at least as good as (x',y') whenever the following is true: either x>x', or x=x' and y≥y'. Do Napoleon's preferences violate completeness? No. Yes.

No. The preferences are complete. Take (x,y) and (x',y'). There are only three possibilities: x>x', x=x', or x<x'. In the first, (x,y) is at least as good as (x',y'); in the third (x',y') is at least as good as (x,y); in the second, if y≥y', then (x,y) is at least as good as (x',y'), and if y'>y, (x',y') is at least as good as (x,y).

Leon has complete and transitive preferences on greek sweets. He finds a piece of Bougatsa is at least as good as a portion of Melomakarona, and he prefers a portion of Melomakarona to a piece of Halva. Can the utility of a piece of Halva be greater than the utility of a piece of Bougatsa? Yes. No.

No. If a piece of Halva has greater utility than a piece of Bougatsa, then a portion of Melomakarona needs to have a higher utility than a piece of Halva and then a piece of Bougatsa. But since a piece of Bougatsa is at least as good as a portion of Melomakarona, the utility of the first has to be greater than or equal than the utility of the second. Thus the utility of a piece of Halva cannot be greater than the utility of a piece of Bougatsa.

Two people have identical preferences when for each pair of alternatives, say a and b, their answers to the questions "is a at least as good as b" is the same. Suppose that Evdekia has utility function, on bundles (x,y), U(x,y)=6x+8y, and Georgos, her fiance, has utility function U(x,y)=3xy. Are Evdekia and her boyfriend's preferences identical? No. Yes.

No. For instance Georgos is indifferent between (0,1) and (1,0). Thus, Georgos finds (1,0) at least as good as (0,1). Evdekia, on the other hand, finds (1,0) is NOT at least as good as (0,1).

Jane has complete and transitive preferences. Then: Her marginal rate of transformation is always positive. She can rank only alternatives that she likes. If she prefers a to b and c to d, then she prefers c to b. Her preferences are linear. She can rank any pair of alternatives.

She can rank any pair of alternatives.

A marketing firm is interested in eliciting Silvia's tastes on a set of ten automobiles. In order to do this they ask Silvia, for each pair of alternatives (x,y), whether she finds x at least as good as y. Silvia is interested in both the reliability of the car and its safety record. She answers that an alternative x is at least as good as y if and only if x is at least as reliable as y and x has at least the safety ranking as y. Then, Silvia's answers necessarily satisfy completeness. Silvia's answers may violate transitivity. Silvia's answers may violate completeness. Silvia's answers will have a cycle. Silvia's answers will always violate completeness.

Silvia's answers may violate completeness. Imagine that there are two cars x and y as follows: x is more reliable than y and y is safer than x. Silvia would not be able to say that x is at least as good as y, nor that y is at least as good as x.

A marketing firm is interested in eliciting Silvia's tastes on a set of ten automobiles. In order to do this they ask Silvia, for each pair of alternatives (x,y), whether she finds x at least as good as y. Silvia is interested in both the reliability of the car and its safety record. She looks at the ratings of both reliability and safety and constructs a combined index by taking the average of the ratings. She answers that an alternative x is at least as good as y if and only if the average index of x is greater than or equal than the average index of y. Silvia's answers may violate transitivity. Silvia's answers will always violate completeness. Silvia's answers will have a cycle. Silvia's answers necessarily satisfy completeness. Silvia's answers may violate completeness.

Silvia's answers necessarily satisfy completeness. In between any two numbers there is always one that is greater than or equal than the other. Thus, given two alternatives x and y, the average rating of one must be greater than or equal than the other. Thus, at least one car will be at least as good as the other. Thus, Silvia's preferences will always satisfy completeness.

Suppose that the cost of producing n pairs of shoes is given by the expression C(n)=30+5n. The average cost of producing a pair of shoes is given by A(n)=C(n)/n. Is the average cost a function of n? True False

True True. Given n we have all information to know how much A(n) is

Linda's utility function is given by U(x,y)=2x+y. Then Linda's indifference curves are: Straight upward slopping lines. L shaped curves. Smooth curves that exhibit increasing marginal rate of substitution. Straight downward slopping lines. Vertical lines.

Straight downward slopping lines.

A marketing firm is interested in learning about millennials' preferences on learning experiences. In order to do this they survey students at a university. They ask them six questions as follows. They consider the following leaning experiences: a traditional class, a flipped course (half of the content is delivered online), and an online class. For each pair of these alternatives, say a and b, they ask the students "is a at least as good as b." In order to make this operational, they provide the students with a picture in which the three alternatives are depicted as circles with their description next to it. The students are asked to draw arrows between the alternatives whenever the answer to a question is affirmative. That is, if a student finds alternative a is at least as good as alternative b, the student is asked to draw an arrow from the circle with label a to the circle with label b. (No question is a at least as good as a was asked; assume that the answer to each of these trivial questions is affirmative.) Suppose that a student's answers are as follows: one arrow from the traditional class to the online class, one arrow from the traditional class to the flipped course, one arrow from the online class to the flipped course, and no more arrows. The student's statements violate completeness, but are transitive. The student's statements violate transitivity, but are complete. The slope of the Marginal Rate of Transformation for the student's statements is tangent to the indifference curve. The student's statements satisfy completeness and transitivity. The student's statements violate transitivity and completeness.

The student's statements satisfy completeness and transitivity. Completeness is not violated because there is at least one arrow in between any two alternatives. Transitivity is not violated. The only triple of arrows among three different alternatives is Traditional Class->online class->Flipped course. Since Traditional class->Flipped course, transitivity is not violated.

A marketing firm is interested in learning about millennials' preferences on learning experiences. In order to do this they survey students at a university. They ask them six questions as follows. They consider the following leaning experiences: a traditional class, a flipped course (half of the content is delivered online), and an online class. For each pair of these alternatives, say a and b, they ask the students "is a at least as good as b." In order to make this operational, they provide the students with a picture in which the three alternatives are depicted as circles with their description next to it. The students are asked to draw arrows between the alternatives whenever the answer to a question is affirmative. That is, if a student finds alternative a is at least as good as alternative b, the student is asked to draw an arrow from the circle with label a to the circle with label b. (No question is a at least as good as a was asked; assume that the answer to each of these trivial questions is affirmative.) Suppose that a student's answers are as follows: one arrow from the traditional class to the online class, one arrow from the traditional class to the flipped course, and no more arrows. The slope of the Marginal Rate of Transformation for the student's statements is tangent to the indifference curve. The student's statements violate transitivity and completeness. The student's statements violate completeness, but are transitive. The student's statements violate transitivity, but are complete. The student's statements satisfy completeness and transitivity.

The student's statements violate completeness, but are transitive. Completeness is violated because there is no arrow from the online class to the flipped course. Transitivity is not violated because this would need two consecutive arrows between three different alternatives.

The following is the graph of function C. This function assigns an amount of money to each q. The graph of C passes through the origin. Then it is increasing. There are three reference levels of q in the graph. They are denoted from left to right by q1, q2, and q3. There is an auxiliary line shown in the graph, which interpolates the origin and (q1,C(q1)). This line has a slope of 1. If one traces a line passing through the origin and (q2,C(q2)), this line's slope will be lower than 1. If one traces a line passing through the origin and (q3,C(q3)), this line's slope will be lower than 1, but will be higher than the slope of the line passing through the origin and (q2,C(q2)). (line curves above slope until (Cq1,q1), below slope at (Cq2,q2) and (Cq3,q3), goes above slope a little after We learn from this graph that AC(q1)>AC(q2). False. True.

True. True. AC(q1) is the slope of the line that interpolates the origin and (q1,C(q1)). AC(q2) is the slope of the line that interpolates the origin and (q2,C(q2)). The first line, which is showed in the figure is steeper than the second, which is not shown in the figure.

Jan Claude has the following utility function on the number of homework assignments, denoted by h, and the number of exams, denoted by e: U(h,e)=-5h-10e. Then, Jan Claude's indifference curves are thin lines that never cross? False. True.

True. More-is-worse for Jan Claude. In the same way that we saw in class that more-is-better implies indifference curves are thin and never cross, one can see that more-is-worse implies the same.

Suppose a utility function assigns 3 to bundle a, 2 to bundle b, and 1 to bundle c. Which of the following utility functions represent the same preferences? U(a)=16, U(b)=16, U(c)=16 U(a)=-1, U(b)=-2, U(c)=-3 U(a)=-1, U(b)=-2, U(c)=0 U(a)=1/10, U(b)=1, U(c)=1/2 U(a)=100, U(b)=-100, U(c)=0

U(a)=-1, U(b)=-2, U(c)=-3 This utility function says that a is preferred to b and b is preferred to c. There is only one utility function in the options provided that says the same.

Roman loves corn bread. He buys corn meal and wheat flour in order to make corn bread. His recipe calls for three cups of corn meal and two cups of flour for each batch that he bakes. More corn bread is better for Roman. He can bake fractions of a batch, but has no use for corn meal or flour that is left over. What of the following is Roman's utility function on cups of corn meal, denoted by c, and cups of flour, denoted by f? U(c,f)=min{c,2f} U(c,f)=min{2c,3f} U(c,f)=2c+3f U(c,f)=min{2c,f} U(c,f)=min{3c,5f}

U(c,f)=min{2c,3f} Roman's utility is U(c,f)=min{2c,3f}. Note that U(3,2)=6. If Roman gets extra of only one of the items, his utility does not increase. For instance, U(3,3)=6 and U(4,2)=6. The other options do not satisfy this. For instance, if U(c,f)=min{2c,f}, U(3,2)=2. If this were Roman's utility, he would be indifferent between (3,2) and (2,2). But he can only bake a fraction of a batch with (2,2) so this cannot be true. All the other options have similar problems.

Napoleon has preferences on bundles of two goods (x,y). Napoleon finds (x,y) is at least as good as (x',y') whenever the following is true: either x>x', or x=x' and y≥y'. Are Napoleon's preferences complete? Yes. No.

Yes, the preferences are complete. Take (x,y) and (x',y'). There are only three possibilities: x>x', x=x', or x<x'. In the first, (x,y) is at least as good as (x',y'); in the third (x',y') is at least as good as (x,y); in the second, if y≥y', then (x,y) is at least as good as (x',y'), and if y'>y, (x',y') is at least as good as (x,y).

Vilfredo has complete and transitive preferences on cars. We know that he prefers a Hyundai Sonata to a Mazda 3, and that he finds a Mazda 3 at least as good as a Honda Civic. Can we conclude that he prefers a Hyundai Sonata to a Honda Civic? Yes. No.

Yes, we can conclude this. First, since he prefers a Hyundai Sonata to a Mazda 3, he finds a Honda Sonata is at least as good as a Mazda 3. Since his preferences are transitive, he must find Hyundai Sonata is at least as good as a Honda Civic. Now, it cannot be the case that he finds a Honda Civic is at least as good as a Hyunday Sonata. If this were true, then since his preferences are transitive, a Mazda 3 would be at least as good as a Hyundai Sonata. But this cannot be true because a Hyundai Sonata is preferred to a Mazda 3. Thus, he prefers the Hyundai Sonata to the Honda Civic.

Two people have identical preferences when for each pair of alternatives, say a and b, their answers to the questions "is a at least as good as b" is the same. Suppose that Elena has utility function on bundles of two goods (x,y), U(x,y)=6x+8y, and Tuna, her husband, has utility function V(x,y)=3x+4y. Are Elena and her husband's preferences identical? Yes. No.

Yes. Yes, they are identical. Suppose that (x1,y1) is at least as good as (x2,y2) for Elena. Then 6x1+8y1=6x2+8y2. Then, 3x1+4y1=3x2+4y2. This means that Tuna finds (x1,y1) is at least as good as (x2,y2). One can see that the symmetric statement is also true. If Tuna finds (x1,y1) is at least as good as (x2,y2), then Elena finds the same. Thus, Elena and Tuna will have the same answers to all "at least as good" questions.

Napoleon has preferences on bundles of two goods (x,y). Napoleon finds (x,y) is at least as good as (x',y') whenever the following is true: either x>x', or x=x' and y≥y'. Are Napoleon's preferences transitive? No. Yes.

Yes. The preferences are transitive. Suppose that (x,y) is at least as good as (x',y') and that (x',y') is at least as good as (x'',y''). Then, x≥x' and x'≥x''. Thus, x≥x''. If x>x'', (x,y) is at least as good as (x'',y''). Now, if x=x'', it must be that x=x'=x''. Since (x,y) is at least as good as (x',y') and x=x', it must be that y≥y'. Since (x',y') is at least as good as (x'',y'') and x'=x'', it must be that y'≥y''. Thus, y≥y'≥y''. Thus, x=x'', and y≥y''. Thus, (x,y) is at least as good as (x'',y'').

Andrea's grandparents gave her a $50 gift card from the local bookstore during the holidays. Is it possible that Andrea would have been better off with $50 cash? Yes. No.

Yes. Andrea may allocate her income in a different way if she had no restriction on where to spend it. That may make her better off.

A marketing firm is interested in learning about the dynamics of viral video dissemination on the web. In order to do this they show three videos to students at a university and ask them to fill out a survey related to the videos. The videos are the epic splits of van Dame (a), the baby (b), and Chuck Norris (c). The survey contains six questions as follows. For each pair of videos, say a and b, they ask the students "is a at least as good as b." In order to make this operational, they provide the students with a picture in which the three videos are depicted as circles with their description next to it. The students are asked to draw arrows between the videos whenever the answer to a question is affirmative. That is, if a student finds video a is at least as good as video b, the student is asked to draw an arrow from the circle with label a to the circle with label b. (No question of the type, "is a at least as good as a" was asked; assume that the answer to each of these trivial questions is affirmative.) In what follows we write a->b when the student draw an arrow from a to b, and so on. The following are the answers to the survey of five different students. Which student finds Chuck's split is preferred to the other two? a-> b, b->a, c->a, c->b, a->c. a-> b, b->a, c->a, c->b. c->a, a->b. a-> b, b->a, c->a. a-> b, b->a, c->b.

a-> b, b->a, c->a, c->b. c is preferred to a when c->a and it is not true that a->c; c is preferred to b when c->b and it is not true that b->c. These two happen only for a-> b, b->a, c->a, c->b.

A marketing firm is interested in learning about the dynamics of viral video dissemination on the web. In order to do this they show three videos to students at a university and ask them to fill a survey related to the videos. The videos are the epic splits of van Dame (a), the baby (b), and Chuck Norris (c). The survey contains six questions as follows. For each pair of videos, say a and b, they ask the students "is a at least as good as b." In order to make this operational, they provide the students with a picture in which the three videos are depicted as circles with their description next to it. The students are asked to draw arrows between the videos whenever the answer to a question is affirmative. That is, if a student finds video a is at least as good as video b, the student is asked to draw an arrow from the circle with label a to the circle with label b. (No question of the type, "is a at least as good as a" was asked; assume that the answer to each of these trivial questions is affirmative.) In what follows we write a->b when the student draw an arrow from a to b, and so on. The following are the answers to the survey of five different students. Which student's preferences are not transitive? a-> b, b->a, c->b, b->c, a->c, c->a. a-> b, b->a, c->a. c->a, c->b. c->a, b->a. a-> b, b->c, a->c.

a-> b, b->a, c->a. Preferences are not transitive when transitivity is violated. This is so when there are three alternatives, say a, b, and c such that a is at least as good as b, b is at least as good as c, but a is not at least as good as c. This happens for answers [a-> b, b->a, c->a]. Note that here c is at least as good as a, a is at least as good as b, but c is not at least as good as b. The other survey answers have no violation of transitivity.

A marketing firm is interested in learning about the dynamics of viral video dissemination on the web. In order to do this they show three videos to students at a university and ask them to fill a survey related to the videos. The videos are the epic splits of van Dame (a), the baby (b), and Chuck Norris (c). The survey contains six questions as follows. For each pair of videos, say a and b, they ask the students "is a at least as good as b." In order to make this operational, they provide the students with a picture in which the three videos are depicted as circles with their description next to it. The students are asked to draw arrows between the videos whenever the answer to a question is affirmative. That is, if a student finds video a is at least as good as video b, the student is asked to draw an arrow from the circle with label a to the circle with label b. (No question of the type, "is a at least as good as a" was asked; assume that the answer to each of these trivial questions is affirmative.) In what follows we write a->b when the student draw an arrow from a to b, and so on. The following are the answers to the survey of five different students. Which student has complete preferences? c->a, c->b. a-> b, b->a, c->b. a-> b, b->c, c->a. c->a, a->b. a-> b, b->a, c->a.

a-> b, b->c, c->a. Completeness means that for any two alternatives at least one has to be "at least as good as the other". Here this means that between two alternatives, say a and b, either a->b or b->a (or both). The only survey answer satisfying this is a-> b, b->c, c->a.

Suppose that initially the prices of x and y are the same. If the price of good x doubles and the price of good y triples, while income is held constant, the budget line (x is measured in the horizontal axis and y in the vertical axis): Does not change. becomes flatter. Shifts inward (and remains parallel to the initial budget line). becomes steeper. Stays the same.

becomes flatter. The slope of the budget constraint is initially -px/py=-1. Then it becomes -2px/3py. That means is less negative and the line is flatter.

Catherine has complete and transitive preferences on three alternatives a, b, and c. We know that for her: a is not at least as good as b and b is not at least as good as c. Then we know that for Catherine: a is preferred to b. c is preferred to a. a is preferred to c. b is preferred to c. a is indifferent to c.

c is preferred to a. Since Catherine has complete preferences and a is not at least as good as b, then b must be at least as good as a. For the same reason c must be at least as good as b. By transitivity, c must be at least as good as a. Now, it is impossible that a is at least as good as c, for otherwise by transitivity, a would be at least as good as b. Thus, c is preferred to a.

The amount of a commodity that a firm can produce efficiently is a function of the numbers of hours of labor employed, L, and the stock of capital, K, and is given by the expression F(L,K)=4L2K3. What is the partial derivative of F with respect to L? dF/dL(L,K)= 12L2K2 dF/dL(L,K)= 8L2K3 dF/dL(L,K)= 8L3K3 dF/dL(L,K)= 8LK3 dF/dL(L,K)= 4LK3

dF/dL(L,K)= 8LK3 Recall that the partial derivative of F with respect to L is the derivative of the function when we assume K is a constant. That is, dF/dL(L,K)=4(2)L(2-1)K3=8LK3.

The following is the graph of two iso-level sets of a function of two variables, x and y. The graph is a two-axis graph in which the variable x is measured in the horizontal axis and the variable y is measured in the vertical axis. There are two red linear curves that represent the iso-level sets of the function. These lines are parallel and have negative slope. The x intercept of the curve that is closest to the origin is closer to the origin than the y intercept of this curve. Both x and y are measured in the same scale in the graph. Which of the following is the expression of the function whose iso-level sets are shown in the graph? (the expression min{a,b} is the minimum between a and b; for instance min{1,2}=1). (two parallel budget lines sloping downward) f(x,y)=2x+3y. f(x,y)=xy2 f(x,y)=min{x,y} f(x,y)=3x+2y. f(x,y)=x2y

f(x,y)=3x+2y. The graph of the iso-level sets of functions f(x,y)=x2y and f(x,y)=xy2 are smooth curves that are not lines. The iso-level sets of f(x,y)=min{x,y} are L shaped curves. The iso-level sets of f(x,y)=3x+2y and f(x,y)=2x+3y are lines that are parallel to each other and have negative slope. The information about the intercepts of the iso-level sets in the graph tells us that the function is f(x,y)=3x+2y because the x intercept closer to the origin than the y intercept. For f(x,y)=2x+3y the x intercept of an iso-level set is further away from the origin than the y intercept of the same iso-level set.

Jane prefers a to b, b to c, and c to a. Based on this information, what properties are violated by Jane's preferences (only one answer is correct)? cannot be determined with the information provided. completeness and transitivity completeness more is better transitivity

transitivity Since Jane prefers a to b and b to c, then she must find a at least as good as b and b at least as good as c. Transitivity would imply that a is at least as good as c. But she prefers c to a, then a cannot be at least as good as c. So Lisa's statements violate transitivity. Completeness is satisfied because between any to alternatives one is at least as good as the other.

Jane has preferences that are represented by the utility function U(x,y)=x2y3 (this preference is smooth, which implies that maximizers are interior). If she has income W=$1000, px=$50, and py=$25, what is her preferred consumption bundle? x = 14 and y = 12 x = 10 and y = 20 x = 3 and y = 34 x = 12 and y = 16 x = 8 and y = 24

x = 8 and y = 24 Let (x*,y*) be Jane's optimal consumption. We know that when maximization happens at an interior point, it must be the case that MRSxy(x*,y*)=-MTRxy. Thus, MRSxy(x*,y*)=px/py=50/25=2. We also know that MRSxy(x,y)=MUx(x,y)/MUy(x,y)= 2xy3/3x2y2=2y/3x Thus, 2y*/3x*=2 and y*=3x*. Now, we know that June will consume all her income, so 50x*+25y*=1000. Thus, 125x*=1000 and x*=8. Thus, 400+25y*=1000. Thus, y*=24.

Mark has utility function U(x,y)=x3y. Then, the equation of Mark's indifference curve through (2,1) is: y=8/x. y=x3. y=3x/8y. y=3xy. y=8/x3.

y=8/x3. Mark's utility at (2,1) is 8. This his indifference curve has equation 8=x3y. Thus, y=8/x3.