2 Outline for this week’s tutorial Past exam question ( from last week) – 5 min. Question 1 – 2 min. Question 2 – 3 min. Question 4 – 5 min. Question 5 – 10 min. Question 7 – 20 min. Solutions for Q6 and Q8 are in the slides (and on Moodle) – please review them and me or come to office hours if you have questions, but we will not be going over them in tutorial.
3 Past exam question from last week From Week 3 tutorial (Question 6), we learned the following equations:The Incidence of a tax (or change in price with respect to a change in tax):𝑑𝑝 𝑑𝑡 = 𝜀 𝑆 /( 𝜀 𝑆 − 𝜀 𝐷 )Price Elasticity of Supply and Price Elasticity of Demand (this is the point elasticity method, which involves taking the derivative of the demand or supply curve):𝜀 𝑆 = 𝑃 𝑄 ∗(𝑑𝑆 𝑑𝑝 )𝜀 𝐷 = 𝑃 𝑄 ∗(𝑑𝐷 𝑑𝑝 )
4 Suppose D=120-4P. Find the price elasticity at a price of 10 and at a price of 20. Use the standard mathematical method, not the midpoint method.-0.2, -2, respectively-0.5, -2, respectively-0.5, -4, respectivelyNot possible to say without knowing what the corresponding level of demand is.p/q * dq/dp = 10/80 * -4 = -1/2 and 20/40 * -4 = -2
5 An Inferior Good: Is a Giffen good Has a positive income elasticity of demandHas a negative income elasticity of demandHas an upward sloping demand function
6 Suppose a demand curve is written D=60-3P Suppose a demand curve is written D=60-3P. Find the intercept and slope of the corresponding inverse demand curve.Slope of -20, intercept of 3Slope of -1/3, intercept of 20Slope of -3, intercept of 20Slope of 1/3, intercept of 60
7 Suppose D=200-2P and S=20+4P. What is the equilibrium price and quantity? P*=20, Q*=100P*=30, Q*=140P*=50, Q*=220P*=40, Q*=180
8 Question 1Assuming an indifference curve which is convex to the origin, what can this tell us about a consumer’s marginal rate of substitution between coffee and muffins?From Mankiw Pg. 443 (2nd Ed.)
9 Question 2Explain why the consumer’s optimal choice occurs where the marginal rate of substitution (MRS) is equal to the relative price of the two goods.The optimal choice is where one indifference curve is touching the budget constraint at exactly one point (where the indifference curve is tangent to the budget constraint).The MRS is the same thing as the slope of the budget constraint.If a line is tangent to another line, their slopes are equal at the point of tangency.Mankiw pg. 445 (2nd Ed.)
10 Question 3Distinguish between the concepts of marginal utility (MU) and the marginal rate of substitution (MRS). MRS is a ratio of marginal utilities. For example, the MRS of X and Y is 𝑀𝑈 𝑋 𝑀𝑈 𝑌 .Mankiw pg. 447 (2nd Ed.)
11 Question 4Would the assumption that goods are perfect substitutes be valid in a study of intertemporal food purchases?Food todayFood tomorrowPerishable foods in one time period are not a perfect substitute for food in another time period.When we talk about perfect substitutes, we have the quantities on both axes, for good 1 and good 2. In this question, we are considering that good 1 is food today, and good 2 is food in the future. Can we say that food today and food tomorrow are perfect substitutes?
12 Mankiw pg. 444 (2nd Ed.)Note: A nickel is a 5-cent piece and a dime is a 10-cent piece.
14 Question 5(a)Use a diagram to distinguish between the income and substitution effects of a change in the price of muffins when the price of coffee stays constant. Assume both goods are normal goods. So, what are a substitution effect and an income effect? First, we draw our original budget constraint, original indifference curve, and the new budget constraint. Next, we can find the substitution effect, the movement along the indifference curve, to a point whose MRS is equal to the slope of the new budget constraint. Finally, we can find the income effect – which will move us to a new indifference curve on the new budget constraint – this depends on whether our good whose price changed is a normal, inferior or giffen good.
15 Blank slide for in-class work using smartboard.
21 Question 6(a)Suppose a consumer only buys two goods: hot dogs and hamburgers. Suppose the price of hot dogs is £1, the price of hamburgers is £2, and the consumer's income is £20. Plot the consumer's budget constraint. Measure the quantity of hot dogs on the vertical axis and the quantity of hamburgers on the horizontal axis. Explicitly plot the points on the budget constraint associated with the even numbered quantities of hamburgers (0, 2, 4, ).
22 hot dogs: £1, hamburgers: £2; consumer's income: £20.
23 Question 6(b)Suppose the individual chooses to consume six hamburgers. What is the maximum amount of hot dogs that he can afford? Draw an indifference curve on your diagram that establishes this bundle of goods as the optimum. Answer: Eight.
24 Question 6(c) What is the slope of the budget constraint? Rise over run = 2/1. This is also the price ratio of price of hamburgers to price of hot dogs = £2/£1. The slope of the indifference curve is also 2/1. (Note: all of these slopes are negative.)What is the slope of the consumer's indifference curve at the optimum?What is the relationship between the slope of the budget constraint and the slope of the indifference curve at the optimum?At the optimum, the indifference curve is tangent to the budget constraint so their slopes are equal.What is the economic interpretation of this relationship?Thus, the trade-off between the goods that the individual is willing to undertake (MRS) is the same as the trade-off that the market requires (slope of budget constraint).
25 Question 6(d)Explain why any other point on the budget constraint must be inferior to the optimum.Because the highest indifference curve reachable is tangent to the budget constraint, any other point on the budget constraint must have an indifference curve running through it that is below the optimal indifference curve so that point must be inferior to the optimum.
26 Question 7(a)Suppose the price of a magazine is £2, the price of a book is £10, and the consumer's income is £100. Which point on the graph represents the consumer's optimum: X, Y, or Z? What are the optimal quantities of books and magazines this individual chooses to consume? Answer: Point Z. 25 magazines and 5 books.
27 Question 7(b)Suppose the price of books falls to £5. What are the two optimum points on the graph that represent the substitution effect (in sequence)?Answer: From point Z to point X.What is the change in the consumption of books due to the substitution effect? Answer: From 5 to 8 books.
28 Question 7(c)Again, suppose the price of books falls to £5. What are the two optimum points on the graph that represent the income effect (in sequence)? Answer: From point X to point Y. What is the change in the consumption of books due to the income effect? From 8 to 6 books.
29 Question 7(c) ctd.Is a book a normal good or an inferior good for this consumer? Explain. Books are inferior because an increase in income decreases the quantity demanded of books.
30 Question 7(d)For this consumer, what is the total change in the quantity of books purchased when the price of books fell from £10 to £5? Answer: The quantity demanded increased from five books to six books.
31 Question 7(e)Use the information in this problem to plot the consumer's demand curve for books on a diagram.
32 Question 8(a)Explain why if two goods (called 1 and 2) are perfect substitutes then their utility function is linear – for example, U=q1+q2. Answer: Rearrange this so q2 is on the left-hand-side and everything else is on the right. The slope is -1 (ie 45o downward sloping) and the intercept on the q2 axis is U.
33 Question 8(b)Show, in this case, that demand for good 1 is Y/p1 if p1<p2 and 0 otherwise (where Y is income) while the demand for good 2 is q2=Y/p2 if p2<p1 and 0 otherwise. Answer: The budget constraint says that income, Y, equals expenditure on good 1 plus expenditure on good 2. So: p1q1 + p2 q2 = Y. You can rearrange this to get q2 on the left hand side and everything else on the right. That is: q2=Y/p2 – (p1/p2)q1 . Since prices and income are fixed this is a linear equation and the slope of this budget constraint is just -(p1/p2). This is steeper than 45o if p1>p2 and shallower if p1<p2 . So if p1>p2 then the consumer only buys the cheaper good 2 and so q2=Y/p2 ; and if p2>p1 then the consumer only buys the cheaper good 1 and so q1=Y/p1.
34 Question 8(c)It is common in many economic applications to assume that a utility function is “quasi-linear”. An example of quasi-linear is the equation U = q2+q1. Show that this implies that indifference curves are parallel. HINT: The indifference curve can rewritten as q2=U-q1 and you can find the slope of this relationship by applying the rule on slide 9 of lecture 6 and noting that q1 = q1½. Answer: Applying the rule, the slope of q2 = U-q1½ is –(½) q1-½, or - 1/2q1, which depends only on q1, not on Y. So whatever the value of Y the slope is the same.
35 Question 8(d)The demand curve can be found by equating the slope of the budget constraint to the slope of the indifference curve. Do this and show that q1 does not depend on Y, but that q2 does. Answer: the demand curve is defined by: the price ratio = MRS -(p1/p2) = -1/2q1 so q1 = p2/2p1 or q1 = [p2/p1]2/4 - which does not depend on Y. Substitute this into the budget constraint and solve to get q2 on the left hand side to find that that the demand for good 2 is given by: q2=(Y/p2)-(p2/4p1), which does depend on Y.Note: The solution give on Moodle might have an error for q2: q2=(Y/p2)-(p22/4p1), the correct solution for q2 is shown above: q2=(Y/p2)-(p2/4p1).
36 Question 8(e)What happens to the demand for q2 if Y<= p22/4p1? And what happens to the demand for q1 when this happens? Answer: If Y= p22/4p1 then q2=0. This remains true for lower values of Y. If q2=0 then Y=p1q1 and so q1=Y/p1 – so everything is spend on good 1.Note: There was an error in the original question.The original question read: What happens to the demand for q2 if Y<= p2/4p1? But should instead say (as above): What happens to the demand for q2 if Y<= p22/4p1?
37 Next Week Tutorial 5 Worksheet Practice exam questions in tutorial. Access past exams here:You will need to select a year, then select ECON 100 or ECON 101. You will be looking at approximately the first 10 questions from each year’s exam.If the link doesn’t work, you can just do a google search for: “lancaster past papers”The first two or three links should send you to the student registry page. From there, click on the link on the right-hand-side that says “Past Papers”.The exams are all the final end-of-year exam, and they cover topics from all four exams during the year. Generally the first ten or eleven questions will be from the material on Exam 1. The next ten or eleven questions from Exam 2, and so on.