Practice Problems Module 2 PDF

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Intermediate Microeconomics and Mathematical Economics - JMMK14

Practice problems Module 2: The Consumer & The Market

This document includes problems from Varian’s Intermediate Microeconomics with Calculus, former exercise, former exams etc. We acknowledge the effort from our colleagues creating these problems. Please, report error in the problems and/or answers to Paul Nystedt ( [email protected])

PROBLEM 1. A. Calculate the MRS for the following utility functions: U= (a) X1+X2 (b) 3X1+4X2 (c) X10.5+X2 (d) 2lnX1+4X2 (e) X1X2 (f) aX1bX2 (g) X12X22 (h) X1a+X2b (i) lnX1+lnX2 (j) X1aX2b B. Calculate the Marshallian demand functions for the utility functions in problem A: e,f,g,i,j above. Are the two goods substitutes or complements (check the sign of the cross-price elasticites of demand)? Are the two goods normal or inferior (check the income elasticities of demand)? C. Calculate the indirect utility function for e and j in problem B above. D. For utility e in problem B above, if one of the prices of the two commodities is doubled, how much more income does the individual need to have in order for the utility level to be intact (compensating variation)?

PROBLEM 2. Explain what is “wrong” with the following statements, i.e. what assumptions do they violate? Why? (a) Two indifference curves can cross each other. (b) Bundle A (6 apples and 8 bananas) and bundle B (7 apples and 9 bananas) are equally good.

PROBLEM 3. A consumer’s indifference curve has been constructed using data from questionnaire surveys. A line of best fit which traces the relationship between the two commodities Qx and Qy shows a downward sloping indifference curve, convex to the origin and with an equation of Qy = 36/Qx. The equation for the consumer’s budget line is Qy = 12 – Qx. (a) Calculate how much Qx and Qy the consumer should choose in order to maximize utility. (b) Prove that the slopes of the indifference curve and the budget line are tangential at these values of Qx and Qy.

PROBLEM 4. Suppose Jean thinks it takes one slice of turkey and two slices of bread to make a “perfect” sandwich. Two slices of turkey or three slices of bread to a sandwich do not make her happier, but she always prefers more perfect sandwiches to less: in fact she gets one unit of utility from each perfect sandwich. (a) Write down the formula for Jean’s indifference curves and plot them in a diagram. (b) Assuming that Jean has $10 to spend on sandwiches and bread costs $1 a slice, write down the formula for Jean’s demand for turkey slices against turkey prices.

PROBLEM 5. Suppose Albert always uses exactly two pats of butter on each piece of toast. If toast costs $0.1/slice and butter costs $0.2/pat, (a) Find Albert’s best affordable bundle if he has $12 /month to spend on toast and butter. Suppose Albert starts to watch his cholesterol and therefore alters his preferences to using exactly one pat of butter on each piece of toast. (b) How much toast and butter would Albert then consume each month?

PROBLEM 6. Consider two goods that are perfect substitutes. What is likely to be true about their relative prices? Can you confirm your hypothesis with examples? PROBLEM 7. What information is contained in the slope of an indifference curve? Why are these curves typically convex to the origin? PROBLEM 8. For each of the utility functions below, draw a set of indifference curves showing utility levels U = 12, U =16, and U = 24. (a) U = XY (b) U = X + Y (c) U = X -Y (d) What is true about the commodities in (b)? (e) What about the commodities in (c)?

PROBLEM 9. What happens to the budget line if the government increases tax on cigarettes but nothing else?

PROBLEM 10. The estimated demand function for processed pork in Canada; Q = 171 − 20p + 20𝑝𝑏 + 3𝑝𝑐 + 2Y ,where Q is the quantity of pork demanded in million kilograms of dressed cold pork carcass weight per year, p is the price of pork in Canadian dollars per kilogram, pb is the price of beef (a substitute good) in dollars per kilogram, pc is the price of chicken (another substitute good in dollars per kilogram, and Y is the income of consumers in thousands of dollars per year ceteris paribus. Show how the quantity demanded, Q, at a given price changes as per capita income Y, increases slightly.

PROBLEM 11. Suppose a consumer has income of $120 per period and faces prices pX = 2 and pZ = 3. Her goal is to maximize her utility, described by the function U = 10X 0.5Z 0.5. Calculate the utility maximizing bundle (X *, Z *) using the Lagrangian method.

PROBLEM 12. Suppose a consumer has a utility function constants.

U (x, y) = AY  X  where

A, , and  are

(a) If the price of X, px is 2 and the price of Y, py is 1, what is the consumer’s optimal bundle? (b) Suppose that M is the consumer’s income and that you don’t have any information about the prices. Derive the consumer’s demand curve. PROBLEM 13. Jöran likes both apples and bananas. He consumes nothing else. The consumption bundle where Jöran consumes 𝑥𝐴 bushels of apples per year and 𝑥𝐵 bushels of bananas per year is written as (𝑥𝐴 , 𝑥𝐵 ). Last year, Jöran consumed 20 bushels of apples and 5 bushels of bananas. It happens that the set of consumption bundles (𝑥𝐴 , 𝑥𝐵 ) such that Jöran is indifferent between (𝑥𝐴 , 𝑥𝐵 ) and (20, 5) is the set of all bundles such that 𝑥𝐵 = 100/𝑥𝐴 .

The set of bundles (𝑥𝐴 , 𝑥𝐵 ) such that Jöran is just indifferent between (𝑥𝐴 , 𝑥𝐵 ) and the bundle (10, 15) is the set of bundles such that 𝑥𝐵 = 150/ 𝑥𝐴 . (a) Draw a graph and plot the indifference curve that passes through the point (20, 5). Do the same for the indifference curve passing through the point (10, 15). (b) In your graph shade in the set of commodity bundles that Agnes weakly prefers to the bundle (10, 15). Shade in the set of commodity bundles such that Agnes weakly prefers (20, 5) to these bundles. For each of the following statements about Jöran’s preferences, write “true” or “false.” (c) (30, 5) ∼ (10, 15) (d) (10, 15) ≻ (20, 5) (e) (20, 5) ≽ (10, 10) (f) (24, 4) ≽ (11, 9.1) (g) (11, 14) ≻ (2, 49) (h) A set is convex if for any two points in the set, the line segment between them is also in the set. Is the set of bundles that Jöran weakly prefers to (20, 5) a convex set? (i) Is the set of bundles that Jöran considers inferior to (20, 5) a convex set? (j) The slope of Jöran’s indifference curve through a point, (𝑥𝐴 , 𝑥𝐵 ), is known as her marginal _______ of ________at that point. (k) Jöran’s indifference curve through the point (10, 10) has the equation 𝑥𝐵 = 100/𝑥𝐴 . The slope of a curve is just its derivative, which in this case is −100/𝑥𝐴2 . Find Jöran’s marginal rate of substitution at the point, (10, 10). (l) Do the indifference curves you have drawn for Jöran exhibit diminishing marginal rate of substitution? PROBLEM 14. Jöran has a utility function given by u(𝑥1 , 𝑥2 ) = 𝑥1 ^2 + 2𝑥1 𝑥2 + 𝑥2 ^2  (a) Compute Jöran’s marginal rate of substitution. (b) Again, Jöran’s cousin Pontus, has a utility function v(𝑥1 , 𝑥2 ) = 𝑥1 + 𝑥2 . Compute Pontus’ marginal rate of substitution. (c) Do u(𝑥1 , 𝑥2 ) and v(𝑥1 , 𝑥2 ) represent the same preferences? (d) Can you show that Jöran’s utility function is a monotonic transformation of Pontus’?

PROBLEM 15. (a) Jöran now consumes apples and bananas. His utility function is U(𝑥𝐴 , 𝑥𝐵 )= 𝑥𝐴 𝑥𝐵 . Jöran has 40 apples and 5 bananas. What is his utility for the bundle (40,5)? The indifference curve through (40,5) includes all commodity bundles (𝑥𝐴 , 𝑥𝐵 ) such that 𝑥𝐴 𝑥𝐵 = _______. What is the equation for 𝑥𝐵 when the indifference curve goes through (40,5)? In a graph, draw the indifference curve showing all of the bundles that Jöran likes exactly as well as the bundle (40, 5). (b) Pontus, Jörans cousin, offers to give Jöran 15 bananas if he will give him 25 apples. Would Jöran have a bundle that he likes better than (40,5) if he makes this trade? What is the largest number of apples that Pontus could demand from Jörgen in return for 15 bananas if he expects him to be willing to trade or at least indifferent about trading? (Hint: If Pontus gives Jöran 15 bananas, he will have a total of 20 bananas. If he has 20 bananas, how many apples does he need in order to be as well-off as he would be without trade?) PROBLEM 16. (a) What does it mean that preferences are reflexive, complete and transitive? (b) What characterises convex preferences? (c) Explain what the assumption of monotonicity of preferences imply. (d) If the five properties above (reflexive, complete transitive, convexity and monotonicity) are satisfied, preferences are said to be “well-behaved”. Explain, which of these properties that make sure that: i. indifference curves cannot slope upwards, ii. indifference curves (for an individual) cannot cross each other, iii. indifference curves are bowed into the origin.

PROBLEM 17. In a pure exchange economy with two goods, G and H, the two traders have Cobb-Douglas utility functions. Amos’s utility is Ua = (Ga) (Ha)1- and Elise’s is Ue = (Ge) (He)1- . What are their marginal rates of substitution? PROBLEM 18. Jöran’s friend Malin consumes meat (𝑥𝑀 ) and vegetables (𝑥𝑉 ) and her utility function is u(𝑥𝑀 , 𝑥𝑉 ) = 𝑥𝑀 𝑥𝑉 . Suppose that the price of meat is 1, the price of vegetables is 2, and Malin’s income is 40. (a) Draw in a graph Malin’s budget line. Plot a few points on the indifference curve that gives Malin a utility of 150 and sketch this curve. Now plot a few points on the indifference curve that gives Malin a utility of 300 and sketch this curve. (b) Can Malin afford any bundles that gives her a utility of 150? (c) Can Malin afford any bundles that gives her a utility of 300?

(d) On your graph, mark a point that Malin can afford and that gives her a higher utility than 150. (e) Neither of the indifference curves that you drew is tangent to Malin’s budget line. At any point, (𝑥𝑀 , 𝑥𝑉 ), Malin’s marginal rate of substitution (MRS) is a function of 𝑥𝑀 and 𝑥𝑉 . Calculate Malin’s MRS. Calculate the slope of Malin’s budget line. Write an equation that implies that the budget line is tangent to an indifference curve at (𝑥𝑀 , 𝑥𝑉 )  [There are many Answers to this equation. Each of these Answers corresponds to a point on a different indifference curve.] Calculate the optimized amount of bundle consumed. (f) What is Malin’s utility if she consumes the optimized bundle? On a graph draw her budget line and the indifference curve at the optimized point.

PROBLEM 19. Malin’s preferences are represented by the utility function u(𝑥1 ,𝑥2 ) = 𝑥1 ^2𝑥2 ^3. The prices of 𝑥1 and 𝑥2 are 𝑝1 and 𝑝2 . (a) Calculate the slope of Malin’s indifference curve at the point (𝑥1 , 𝑥2 ). (b) If Malin’s budget line is tangent to her indifference curve at (𝑥1 , 𝑥2 ), calculate the 𝑝 𝑥 ratio 1 1 . [Hint: Look at the equation that equates the slope of his indifference 𝑝 𝑥 2 2

curve with the slope of his budget line.] When she is consuming the best bundle she can afford, what fraction of her income does Malin spend on 𝑥1 ? (c) Other members of Malin’s family have similar utility functions, but the exponents may be different, or their utilities may be multiplied by a positive constant. If a family member has a utility function v(𝑥1 , 𝑥2 ) = c𝑥1𝑎 𝑥2𝑎 where a, b, and c are positive numbers, what fraction of his or her income will that family member spends on 𝑥1 ?

PROBLEM 20. Malin now consumes coffee and brownies. Her utility function is now u(𝑥1 ,𝑥2 ) = 4√{𝑥1 } + 𝑥2 where 𝑥1 is his consumption of coffee and 𝑥2 is her consumption of brownies. (a) What is the slope Malin’s indifference curve at (𝑥1 ,𝑥2 )? Solve her demand of

coffee (𝑥1 ).

(b) Calculate her demand for brownies. (c) You are given the information that 𝑝1 = 1, 𝑝2 = 2, and M = 9 . Calculate 𝑥1 and

𝑥2 . What does your result for 𝑥2 imply? What amount must income (M) be for Malin to demand a positive amount of both goods?

PROBLEM 21 U=3( Coke )+6( Pepsi ) The price of Pepsi is 2, and the price of Coke is 0.8 . What is the Optimal Consumption Bundle for the individual? First assume, Coke to be on y-axis MRS=-(3/6) =-(1/2) , slope of budget line=-(2/5)=-(10/4)=-2.5 We know that the slope of the budget line is greater than the MRS, the slope of the indifference curve. So what good will the individual consume and why?

PROBLEM 22 Find the Optimal Consumption Bundles of x and y for the following utility functions: 2 4

U(𝑥, 𝑦) = 𝑥3 𝑦 5 U(𝑥, 𝑦) = 𝑥 2 + y Where the price of good x is 2$ and the price of good y is 1$ and income is 10$.

PROBLEM 23. Nancy Lerner is taking Professor Stern’s economics course. She will take two examinations in the course, and her score for the course is the minimum of the scores that she gets on the two exams. Nancy wants to get the highest possible score for the course. Write a utility function that represents Nancy’s preferences over alternative combinations of test scores x1 and x2 on tests 1 and 2 respectively.

PROBLEM 24. Shirley thinks a 16-ounce can of beer is just as good as two 8-ounce cans. Lorraine only drinks 8 ounces at a time and hates stale beer, so she thinks a 16-ounce can is no better or worse than an 8-ounce can. (a) Write a utility function that represents Shirley’s preferences between commodity

bundles comprised of 8-ounce cans and 16-ounce cans of beer. Let X stand for the number of 8-ounce cans and Y stand for the number of 16-ounce cans. (b) Now write a utility function that represents Lorraine’s preferences. (c) Would the function utility U(X, Y ) = 100X+200Y represent Shirley’s preferences? Would the utility function U(x, y) = (5X + 10Y )2 represent her preferences? Would the utility function U(x, y) =X + 3Y represent her preferences?

(d) Give an example of two commodity bundles such that Shirley likes the first

bundle better than the second bundle, while Lorraine likes the second bundle better than the first bundle.

PROBLEM 25. Jöran has an income of $40 to spend on two commodities. Commodity 1 costs $10 per unit, and commodity 2 costs $5 per unit. Write down Jöran’s budget equation. What is the slope of the budget line? If you spent all your income on commodity 1, how much could Jöran buy? If you spent all of your income on commodity 2, how much could Jöran buy? Suppose that the price of commodity 1 falls to $5 while everything else stays the same. Write down Jöran’s new budget equation. (f) What is the slope of the new budget line? (g) Suppose that the amount you are allowed to spend falls to $30, while the prices of both commodities remain at $5. Write down Jöran’s budget equation. (h) Draw a diagram and shade in the area representing commodity bundles that Jöran can afford with the budget in Part (e) but could not afford to buy with the budget in Part (a). Shade in the area representing commodity bundles that Jöran could afford with the budget in Part (a) but cannot afford with the budget in Part (e). (a) (b) (c) (d) (e)

PROBLEM 26. An Individual has a weekly income of 400$ and spends half of it on good x and half on good y (assume x is food and y is “all other goods”). The individual is also receiving social allowances in form of food stamps worth 10$ that he buys for 5$ each. Show the budget line with and without the food stamps.

PROBLEM 27. Individual 1: U(x ,x )=x^2 y^3 Individual 2: U(x ,x ) =3x +2y Individual 3: U(x, y)=min{x , 2y} (a) What type of utility functions does the individuals above have? (b) Find the MRS of the utility functions. (c) Find demand functions for each individual, the budget constraint is pxx+pyy=m

PROBLEM 28. David can consume two goods, good 1 and good 2 where 𝑥1 and 𝑥2 denote the quantity consumed of each good. These goods sell at prices 𝑝1 and 𝑝2 , respectively. David's preferences are represented by the following utility function: u(𝑥1 , 𝑥2 ) = √𝑥1 𝑥2 . David has an income of m. (a) Derive David's Marshallian demand functions for the two goods. (b) Assume that 𝑝1 = 5,  𝑝2 = 5 and m = 100 .What are David's demands for good 1 and good 2? What share of his income is spent on good 1? (c) Are the goods i) normal? ii) ordinary? Explain your answer! (d) Suddenly, 𝑝1 increases to 10. What are the income and substitution effects? How much does david consume of goods 1 and 2 after the price increase? What share of income is spent on good 1?

PROBLEM 29 Jöran’s friend Malin consumes meat (𝑥𝑀 ) and vegetables (𝑥𝑉 ) and her utility function is u(𝑥𝑀 , 𝑥𝑉 ) = 𝑥𝑀 𝑥𝑉 Suppose that the price of meat is 1, the price of vegetable is 2, and Malin’s income is 40. Calculate Malin’s choice of the two goods that maximises her utility. What is the demand for the two goods as a function of income and prices?

PROBLEM 30. Suppose an individual’s (A) utility function is given by: 𝑢 = 𝑥0.5𝑦0.5 (a) Compute the demand functions for the two goods, and the income elasticities of demand. (b) Draw the Income offer curve and Engel Curves that relate to this individual´s demand if the prices of the two goods are equal to 1.

Another individual´s (B) utility function is given by: U= x + ln y. The prices of both of these goods are 1. (a) What are the income elasticities of the demand for the two goods? (b) Draw the Income offer curve and the Engel curves that relate to this individual’s demand.

PROBLEM 31. A Giffen good must be inferior, but an inferior good is not necessarily a Giffen good. Is this correct? Explain. PROBLEM 32. True, False or Uncertain; explain your answer. When income rises and the price of x falls, the consumer will always buy more units of x.

PROBLEM 33. A consumer faces prices for hot dogs and hamburgers of $1 each. Consumption of the two commodities at various weekly income levels are shown below. (a) Use the information to sketch the income consumption curve on a graph. (b) Draw the Engel curves for hot dogs and hamburgers. Income $10 15 20

Hot Dogs 3 6 10

Hamburgers 7 9 10

(c) What is the income elasticity of hot dogs for this consumer as income

increases from $10 to $15?

PROBLEM 34. Bob is a drug addict and his demand for cocaine is Q = 100-3p where Q is the quantity demanded and p is the price paid. The market price for cocaine is $10 a bag. At this price, what is Bob’s consumer surplus of consuming cocaine?

PROBLEM 35. Demand in Market 1 for X is Qd = 80 - p. Demand in Market 2 is Qd =120 - 2p. At a price of $20, which has a larger consumer surplus?

PROBLEM 36. Elin owns a small chocolate factory, located close to a river that occasionally floods in the spring, with disastrous consequences. Next summer, Elin plans to sell the factory and retire. The only income she will have is the proceeds of the sale of his factory. If there is no flood, the factory will be worth $500,000. If there is a flood, then what is left of the factory will be worth only $50,000. Elin can buy flood insurance at a cost of $0.10 for

each $1 worth of coverage. She thinks that the probability that there will be a flood this spring is 1/10. Let 𝑐𝐹 denote the contingent commodity dollars if there is a flood and 𝑐𝑁𝐹 denote dollars if there is no flood. Elin’s von Neumann-Morgenstern utility function is u(𝑐𝐹 , 𝑐𝑁𝐹) = 0.1√𝑐𝐹 + 0.9√𝑐𝑁𝐹 . (a) Calculate Elin’s contingent commodity bundle if he buys no insurance. (b) Calculate Elin’s contingent commodity bundle if he does buy insurance. (c) Write down Elin’s budget equation when then “price” of 𝑐𝑁𝐹 is 1. (d) Calculate Elin’s marginal rate of substitution between the two contingent

commodities – dollars if there is no floor and dollars if there is a floor. (e) Calculate the optimal consumption bundle. How much will his insurance premium

pay her if there is a floor? How much is the amount of insurance premium she has to pay? PROBLEM 37. Explain carefully the relationship between marginal utility of income and 1) risk neutrality, 2) risk aversion and 3) risk preference. Use graphs in your answers! Furthermore, explain why ordinal utility is ...