Lecture 4: Choice and Demand - 001 Microeconomics PDF

# Lecture 4: Choice and Demand ## 001 Microeconomics ### Ques 1. a) Jim consumes two goods and he has income m = $120. The prices of the two goods are p₁ =$2, p₂=$5. Write down his budget line and draw it in a graph. What is the slope of his budget line? b) If now the government decides to impos...

# Lecture 4: Choice and Demand ## 001 Microeconomics ### Ques 1. a) Jim consumes two goods and he has income m = $120. The prices of the two goods are p₁ =$2, p₂=$5. Write down his budget line and draw it in a graph. What is the slope of his budget line? b) If now the government decides to impose a quantity subsidy of s on good 2. What is the equation for Jim's new budget line? Suppose that s = $2. Draw his new budget line in the same graph in (1). What is the slope of his new budget line? 1. (1) Budget line: 2x₁+5x₂ = 120 slope = dx₁/dx₂ = p₁/p₂ = 2/5 2. Jim's new budget line: 2x₁+(5-s)x₂=120 s = $2. Jim's new budget line: 2x₁+3x₂ = 120 slope = dx₁/dx₂ = p₁/p₂ = 2/3 ### Ques 2. a) Laura's monthly income is $240 and she allocates her income among two goods: Burger Queen's hamburgers and PepsiCo's coke. Suppose Burger Queen's hamburgers cost $4 each and PepsiCo's coke costs $2 per bottle. Draw her budget line and calculate the slope of it. b) Suppose now that in this month (October) Burger Queen offers every customer 25 percent off the price of each hamburger up to 20 hamburgers. Draw Laura's budget line for October in the same graph of (1). You must label all the necessary numbers on the two axes. c) Suppose instead that Burger Queen announces that it will have a new promotion plan next month (November). The plan is that Burger Queen will give each customer a coupon. By using the coupon each customer can get up to 10 hamburgers for free. Draw Laura's budget line for November in a separate graph. You must label all the necessary numbers on the two axes. 2. a. Let x: hamburger, y:coke. Budget line: 4x+2y=240 slope = dy/dx = pₓ/pᵧ = 4/2 = -2 b. If x ≤ 20, p₁ =(1-0.25)×4=3 If x > 20, p = 4 → If. x ≤ 20, budget line: 3x+2y = 240 If x > 20, budget line: 4(x-20)+2y=240-3×20 → If x≥20, budget line: 3x + 2y = 240 If x > 20, budget line: 4x+2y = 260 ### Ques 3. a) Michelle eats cereal (C) and milk (M) for breakfast everyday. She always mixes cereal and milk in the following particular ratio: 3 spoons of cereal to 10 spoons of milk. Write down her utility function and draw some of her indifference curves. What is her marginal rate of substitution between cereal and milk? a) Michael Phelps enjoys crackers (C) but has no preference for ice cream (I). Write down his utility function and draw some of his indifference curves. What is his marginal rate of substitution between crackers and ice cream? b) Brad Pitt also enjoys crackers, but to look fit, he hats ice cream. Write down his utility function and draw some of his indifference curves. What is his marginal rate of substitution between crackers and ice cream? 3. u(C,M) min{10C,3M} Case1: 10C = 3M, MRSC= not defined Case2: 10C ≥ 3M, u(C,M)=10C, MRSCм = MUM/MUC = 0/10 = 0 Case3: 10C < 3M, u(C,M)=3M, MRSCм = MUM/MUC = 3/0 = ∞ 4. (1)Michael enjoys crackers (goods) but has no preference for ice cream (neutral good), u=u(C), ∂u/∂C=u'(C)>0. MRSC = MUC/MU₁ = u'(C)/0 = -∞ (2) Brad Pitt enjoys crackers (goods), but hates icecream (bads) u=u(C,I), ∂u/∂C=u(C)>0, ∂u/∂I=u'(I)<0. MRSC = MUC/MU₁ = u'(C)/u'(I) > 0 ### Ques 4. For Stacy, every 2 cups of tea (T) can always substitute 3 cups of coffee (C) and vice versa. Write Stacy's utility function for tea and coffee. What is her marginal utility from drinking tea, and from drinking coffee? Calculate her marginal rate of the substitution between tea and coffee and draw some of his indifference curves. Be sure to label all the necessary numbers on two axes. 4. u(T, C)=3T+2C MU₁ = ∂u/∂T = 3, MUC = ∂u/∂C = 2 MRSTC = MU₁/MUC = 3/2 ### Ques 5. Kate's preferences for good 1 and good 2 are described by the utility function u = x²x². Calculate her marginal utility from consuming x₁, her marginal utility from consuming x₂, and her marginal rate of substitution between x₁ and x₂. 5. 1) u = x²x² MU₁ = ∂u/∂x₁ = 2x₁x² MU₂ = ∂u/∂x₂ = 2x₁²x₂ MRS₁₂ = MU₁/MU₂ = 2x₁x²/2x₁²x₂ = x₂/x₁ ### Ques 6. Which of the following function(s) can also represent Kate's preferences? Why? Show your calculations. (а) u₁ =10+17x₁x₂ (b) u₂-3ln x₁-2lnx₂-100 (c) u=6log.x₁ +4log x₂ (d) u₁ = -2x₁x₂ (e) u₅ = 3x₁²/⁵x₂²/⁵ (a) u₁=10+17x₁x₂=10+17, ∂u₁/∂x₁ = 17>0 (b) u₂=-3ln x₁-2lnx₂-100=-lnx₁³x₂²-100 = -ln u -100, ∂u₂/∂u = - 1/u < 0 (c) u₁ = 6log x₁ + 4log x₂=logx₁⁶x₂⁴ = logu, ∂u₁/∂u = 2>0, ∂u/∂u = 2>0 (d) u₁ = -2x₁x₂ = -2u, ∂u₁/∂u = -2 <0 (e) u₅ = 3x₁²/⁵x₂²/⁵, ∂u/∂u = 3/5 > 0 So what is your answer? (a), (c) and (e) ### Ques 7. Pete's preferences for good 1 and good 2 are described by the utility function up = ln x₁ +5x₂. a. What is his marginal utility from consuming x₁? What is the change in his marginal utility from consuming x₁? What is his marginal utility from consuming x₂? What is his marginal rate of substitution between x₁ and x₂? b. Mary's preferences for good 1 and good 2 are described by the utility function u=(lnx₁+5x₂)^1/2. Do you think Pete and Mary can be a good couple in making the consumption plan for x₁ and x₂? Explain your answer. Ans (7)a. u = ln x₁ +5x₂ MU₁ = ∂u/∂x₁ = 1/x₁ MU₂ = ∂u/∂x₂ = 5 MRS₁₂ = MU₁/MU₂ = 1/5x₁ (2) u = (lnx₁+5x₂)^1/2 = u^1/2 ∂u/∂u = 1/2u^1/2 > 0 ### Ques 8. Larry's preferences for good 1 and good 2 are described by the utility function up = 3x₁ +5x₂. c. What is his marginal rate of substitution between x₁ and x₂? Write down Larry's demand function for x₁, and draw it in a graph. Make sure you label all the necessary numbers on the two axes. d. Suppose that the price of x₁ is $4 and the price of x₂ is also $4. Larry has income $120 to spend on the two goods. Find out Larry's demand bundle. Budget constraint: p₁x₁+p₂x₂ = m, slope = - p₁/p₂. Utility function: u=3x₁+5x₂, MRS₁₂ = MU₁/MU₂ = 3/5 Case1: x₁=0≤x₁≤m, if p₁/p₂ =MRS₁₂, or p₁ p₂ MRS₁₂ Case2: x₁=0, if p₁/p₂ >MRS₁₂, or p₁ > p₂ MRS₁₂ Case3: x₁ = m/p₁, if p₁/p₂ <MRS₁₂, or p₁ < p₂ MRS₁₂ (2) p₁=4p₂=4,m=120 p₁/p₂ =4/4 = 3/5 =>MRS₁₂ =(x₂/x₁) = (0, m/p₂) = (0,30) ### Ques 9. Beyonce only cares about the amount of Pepsi that she drinks. Pepsi either comes in a six pack (x6) or in a case (x24). A case has 24 cans of soda. a) Draw Beyonce's budget line when her soda budget is $30 and the price of a case is (p24) is $3 and the price of a six pack (p6) is $1. b) How many six packs and cases does Beyonce buy when m=30, p24=3, and p6=1? Beyonce is facing two perfect substitutes here packs of Pepsi (x6) and cases of Pepsi (x24). There are 6 cans in one x6 and 24 cans in one x24. Beyonce will therefore view, for example, 4x6 as good as one x24 since both give her 24 cans of Pepsi. Since a case costs $3, a can in a case costs $3/24 = $0.125. On the other hand, a pack costing $1 implies a can in a pack costs $1/6 = $0.167. Obviously, it is cheaper per can to buy a case rather than a pack of Pepsi. Since Beyonce only cares about cans of Pepsi regardless of whether they come from a case or a pack, to maximize her utility she will spend all $30 on cases, and this gives her 10 cases and zero pack. We already know that Beyonce's utility maximizing bundle is (x6; x24) (0; 10). Given she obtains 240 cans from this bundle, she will consider (x6; x24) = (40; 0) to be as good as (x6; x24) (0; 10) because it also gives her 240 cans. Connecting the two bundles give you her indifference curve through her utility maximizing bundle. ### Ques 10. Phoebe likes spending time at Central Park. While at Central Park she always wants two scones (xS) for every one cup of coffee (xC). a) Draw Phoebe's budget line when her Central Park budget is $20, and the price of a scone is $1 and the price of a cup of coffee is $2? b) How many scones and cups of coffee does she buy when m=20, pS=1, and pC=2? c) Draw a picture of the indifference curve where she maximizes utility Phoebe considers scones (xS) and coffee(xC) as perfect complements. In particular, her preferred mix of them is two xS for every one xC. The total cost of two xS and one xC is $1X2+$2 = $4. Given her budget of m = $20, she can buy 5 sets of such combination of scones and coffee, and this maximizes her utility. In these 5 sets, there are 5 X 2 = 10 scones and 5 cups of coffee. The general lesson drawn from Figure 2 is that, the optimum consumption bundles of two perfect complements are always in the preferred mix where the kink on the indifference curve touches the budget line. ### Ques 11. Suppose we observe an individual's consumption of x and y over three periods. In all periods his income is $100, in the first period px = 1 and py = 1, in the second period px = 2 and py = 1, and in the third period px = 1 and py = 2. In the first period he consumes (x, y)=(50,50), in the second period he consumes (x, y)=(33,33), and in the third period (x, y)=(33,33). What do you think his utility function might look like? The first thing you should be able to spot from the consumption bundles in the three periods is that x and y are always in a fixed proportion. What kind of goods do an individual always like to consume in a fixed proportion: perfect complements. So, you know that the utility function should be in the form of min (ax; by) for some positive constants a and b. Since the proportion of x and y is always one-to-one, we can further infer that a = b. Thus, an example of the utility function will be U(x; y) = min(ax,by) in which I set a = b = 1. Of course, for example ^U (x; y) = min(2x; 2y) will also work so long as a = b. ### Ques 12. Suppose we observe an individual's consumption of x and y over three periods. In all periods his income is $100, in the first period px = 1 and py = 1, in the second period px = 2 and py = 1, and in the third period px = 1 and py = 2. In the first period he consumes (x, y)=(100,0), in the second period he consumes (x, y)=(40,20), and in the third period (x, y)=(100,0). What do you think his utility function might look like? The consumption bundles in the first and third period, (x; y) = (100; 0), are boundary optimums. The presence of boundary optimums suggests that x and y should be perfect substitutes. The utility function of perfect substitutes are in the form of U(x; y) = ax+by for some positive constants a and b. Your task is to figure what they are, and the bundle in the second period helps. For perfect substitutes, we know that there could be non-boundary optimums only when the indifference curve has the same slope as the budget line and coincide with each other. Indeed, in this case, every consumption bundle on the indifference curve that coincides with the budget line is an optimal consumption bundle; it just happens in this case that the individual picks (x; y) = (40; 20) among them. The slope of the indifference curve is the MRSx;y, which is a/b in this case. The slope of the budget line is-Px/Py, which is -2 in the second period. Thus, any U(x; y) = ax + by with a/b = 2 can be a utility function. For example, U(x; y) = 2x + y will do, and so will ^U(x; y) = x + 1/2y. ### Ques 13. Suppose we observe an individual's consumption of x and y over three periods. In all periods his income is $100, in the first period px = 1 and py = 1, in the second period px = 2 and py = 1, and in the third period px = 1 and py = 2. In the first period he consumes (x, y)=(60,40), in the second period he consumes (x, y)=(30,40), and in the third period (x, y)=(60,20). What do you think his utility function might look like? Do Yourself! ### Ques 14. A schoolgirl spends her pocket money of $20 per week on movie tickets and sundaes. A movie ticket costs $5, and a sundae costs $2. Show graphically the change in her budget constraint for the following events (consider one event at a time): i. The price of a sundae increases to $2.50. ii. The schoolgirl finds a $5 bill on the pavement. iii. In addition to the pocket money, Dad buys her two movie tickets and three sundaes a week. The girl cannot sell what Dad gives her to somebody else. iv. Being busy, Dad does not go to see the movie with her, and she figures out she can sell the tickets Dad buys her to her classmate for $4. The sundaes Dad buys her still have no resale value. v. The movie theater introduces a discount program: for everyone buying more than two movie tickets per week each movie ticket starting with the third one gives the right to see another movie for free. vi. The movie theater changes the discount program to give everyone buying more than two tickets per week 50% discount on the tickets purchased afterwards.