The Australian National University ECON2021: Microeconomics 2 Semester Two, 2019 Tutorial 5 Answers Dr Damien S. Eldridge 22 August 2019 A Note on Sources Some, and possibly even all, of these questions might have been drawn from other sources. Key Concepts Budget-Constrained Utility Maximisation, Uncompensated (or Marshallian, or Walrasian, or Ordinary) Demands, Compensated (or Hicksian) Demands, Comparative Statics, Slutsky Decomposition, Total Effect, Substitution Effect, Income Effect, Slutsky Equation, Normal Good, Inferior Good, Giffen Good, Equivalent Variation, Compensating Variation, Hicksian Consumer Surplus, Marshallian Consumer Surplus, The Wagstaff Model. Tutorial Questions Tutorial Question 1 Consider a price taking consumer that has rational, continuous, strongly monotone, and strictly convex preferences defined over consumption bundles of non-negative amounts of each of two goods (call them food and clothing). This consumer is endowed with an income of $y > 0. Suppose that the government is considering implementing a specific (per unit) sales tax on 1 food. The tax rate will be $t per unit of food that is purchased. Assume that the supply curve for food is perfectly elastic at the pre-tax price. 1. Illustrate the determination of the consumer’s optimal consumption bundle before the introduction of the tax in an “indifference curvebudget constraint” diagram. 2. Illustrate the impact that the tax would have on the consumer’s budget set. 3. Illustrate the impact that the tax would have on the consumer’s optimal consumption bundle in an “indifference curve-budget constraint” diagram. 4. Illustrate the equivalent variation measure of the impact of the tax on the consumer’s welfare in an “indifference curve-budget constraint” diagram. Be sure to indicate this measure both in units of food and in units of clothing. How would you calculate the dollar value of this measure? 5. Illustrate the compensating variation measure of the impact of the tax on the consumer’s welfare in an “indifference curve-budget constraint” diagram. Be sure to indicate this measure both in units of food and in units of clothing. How would you calculate the dollar value of this measure? 6. Illustrate the Marshallian consumer surplus measure of the impact of the tax on the consumer’s welfare in a demand curve diagram. In what units is Marshallian consumer surplus measured? 7. Compare the EV, CV, and CS measures of the impact of the tax on the consumer’s welfare. Tutorial Question 2 Using budget lines and indifference curves, analyse each of the following situations in separate diagrams. In each case, determine whether the given condition would lead to an income effect or a substitution effect, or both. Clearly show the effects in your diagrams. 1. Milton lives on a fixed income, and the price of food falls. 2 2. Paul must drive 100 kilometres to a city store in order to gain a 50 per cent discount on clothing. He is indifferent between buying his clothes this way and purchasing them without the discount from a local store. 3. James is swindled out of half of his income, while the prices of all goods are unchanged. 4. Karl earns $100 per month and makes 200 phone calls per month. He is charged a flat fee of $35 for the first 100 calls, and $0.05 per call thereafter. The telephone company is considering an alternative pricing system which would eliminate the $35 fee and simply charge $0.20 per call. Tutorial Question 3 The government is considering two alternative ways to ease the financial pressures on families associated with the cost of education. (To keep things simple, suppose rather unrealistically that education can be purchased in infinitely divisible units.) One involves a specific (per unit) subsidy for education. The other involves the allocation of a voucher for a fixed amount that can be used to contribute towards the purchase of education, but cannot be used for anything else. Suppose that, due to strict enforcement, there would be no “black market” for these vouchers, if that option was chosen by the government. Imagine a family that behaves like a price-taking consumer whose preferences are defined over bundles of non-negative amounts of education and some composite consumption commodity known as “all other goods”. This family is endowed with a fixed money income of $y > 0. 1. Illustrate the impact of the subsidy policy option on the family’s budget constraint. How might this policy affect the family’s optimal choice of consumption bundle? 2. Illustrate the impact of the voucher policy option on the family’s budget constraint. How might this policy affect the family’s optimal choice of consumption bundle? 3. Compare and contrast the impacts of the two policy options on this family. 4. Are there any other factors that you might want to consider when assessing these two policy options? 3 Tutorial Question 4 Jason allocates a given sum of money between just two goods, food and drink. 1. Explain and illustrate how you would derive Jason’s Marshallian (or ordinary) demand curve for drink. 2. Explain and illustrate how you would derive the Hicksian (or compensated) demand curve for drink corresponding to Jason’s initial level of utility. 3. If food is an inferior good for Jason, which of these two demand curves for drink will show the greater responsiveness to a change in the price of drink away from its initial level. Justify your answer. Additional Practice Questions Additional Practice Question 1 Suppose that you have an income of $100 per week and can buy meat at a price of $1 per kilogram. 1. Draw your budget constraint for bundles of meat and some composite commodity known as “all other goods”, placing “quantity of meat” on the horizontal axis. 2. Now suppose that your parents, in an attempt to deter you from becoming a vegetarian, decide to subsidise your meat purchases by agreeing to always pay one-half of your meat bill. Illustrate the impact this has on your budget constraint. Compare your new budget constraint with your old one. 3. Illustrate the determination of your new optimal consumption bundle (call it bundle A). (Assume that bundle A contains a strictly positive amount of meat.) 4. Show in your diagram the cost of the subsidy to your parents (call this amount $S). 5. Now suppose that your parents choose to give you an unconstrained subsidy of $S per week instead of the meat subsidy. Illustrate your new budget constraint. 4 6. Is bundle A above, below, or on your new budget line? Would you rather have the meat subsidy or the gift. Justify your answer. Additional Practice Question 2 Sometimes when one variety of a good is provided for free, consumption of that variety of the good makes it impossible (or at the very least, highly difficult) to consume any other variety of that particular good. One possible example of this is education. If an individual attends a free public school full time, then he or she cannot also attend an expensive private school full-time. This question attempts to explore such a situation. 1. Illustrate the budget set for choices over bundles of (non-negative amounts of each of) two commodities: “education” and “all other goods”. Assume that education comes in two varieties: public and private. Public education must be consumed in a standardised amount equal to E ∗ , while private education can be purchased in varying amounts. (Hint: The budget set will consist of both the familiar budget set over bundles of two commodities, and a single point that lies outside that budget set.) 2. Redraw your budget set diagram from part one of this question and include an indifference curve that would imply that an individual would not accept the offer of a free public education. 3. Redraw your budget set diagram from part one of this question and include an indifference curve that would imply that an individual would accept the offer of a free public education, and that this would involve the individual consuming more education than would have been the case if he or she had instead chosen to purchase some amount of private education. 4. Redraw your budget set diagram from part one of this question and include an indifference curve that would imply that an individual would accept the offer of a free public education, and that this would involve the individual consuming less education than would have been the case if he or she had instead chosen to purchase some amount of private education. 5. When would you expect an offer of a standardised amount of free public education to reduce, rather than raise, the educational attainment of the population? (You may assume that any such subsidies would be 5 funded by lump-sum taxes on some other, non-education consuming, group within the jurisdiction in question. You may also assume that the public education subsidy and the associated lump-sum tax do not alter the prices of either private education or ”all other goods”.) Additional Practice Question 3 This question is about the Wagstaff model of the demand for health care. We examined the derivation of the consumption possibilities set for this model earlier this semester. The model is described in Wagstaff, A (1986), “The demand for health: Theory and applications”, The Journal of Epidemiology and Community Health 40(1), March, pp. 1–11. 1. Describe and illustrate the derivation of the optimal choice of health status, health care, and the composite “all other consumption” commodity in the Wagstaff model. 2. Analyse the impact of an increase in the price of health care on the optimal choice of health status, health care, and the composite “all other consumption” commodity in the Wagstaff model. 3. Analyse the impact of an increase in the income of an individual on his or her optimal choice of health status, health care, and the composite “all other consumption” commodity in the Wagstaff model. 4. Analyse the impact of an improvement in the “health status production technology” on the optimal choice of health status, health care, and the composite “all other consumption” commodity in the Wagstaff model. (This improvement might be due to advances in medical technology, or it might be due to an increase in the consumer’s stock of health capital. In either case, you may assume that the improvement results in a higher health status being attainable at each strictly positive level of health-care than was previously the case.) Answers to Tutorial Questions Answer to Tutorial Question 1 See the attached handwritten notes. 6 Answer to Tutorial Question 2 See the attached extract from Sproul, M, and J Hirshleifer (1988), Study guide to “Price theory and applications (fourth edition)”, Prentice-Hall, USA. Answer to Tutorial Question 3 See the attached extract from Hirshleifer, J (with the assistance of M Sproul) (1988), Price theory and applications (fourth edition), Prentice-Hall, USA. Answer to Tutorial Question 4 Part (a) q2 U3 U1 The Derivation of an Inverse Marshallian Demand Curve U2 y/p2 p11 < p12 < p13 0 y/p13 p1 q13 y/p12 q12 q1 y/p11 q11 p13 p12 p11 0 q13 q12 q11 p1D ( q1 ; p2 , y ) = ( q1D )-1 ( p1 ; p2 , y ) q1 7 Part (b) q2 The Derivation of an Inverse Hicksian Demand Curve U Slope = - ( p13 / p2 ) p11 < p12 < p13 Slope = - ( p11 / p2 ) 0 p1 q13 q12 q1 q11 Slope = - ( p12 / p2 ) p13 p12 p1HD ( q1 ; p2 , U ) = ( q1HD )-1 ( p1 ; p2 , U ) = h1-1 ( p1 ; p2 , U ) p11 0 q13 q12 q1 q11 Part (c) We are told the following. • There are two commodities (food and drink). • Food is an inferior good. • The price of drink changes. Since food is an inferior good, we know that all else equal, an increase in income will result in a decrease in the amount of food that is purchased. Similarly, all else equal, a decrease in income result in an increase in the amount of food that is purchased. In other words, if food is an inferior good, D ,pF ,y) then ∆xF (p∆y < 0, wherexF (pD , pF , y) is the Marshallian demand function for food. 8 In a two commodity world, if one commodity is an inferior good, then the other commodity must be a normal good (assuming that preferences are locally non-satiated so that the consumer exhausts his or her budget). If income increases, without any changes in price, and the consumer purchases less food than before, then all of the increase in income, and the additional income that is freed up by the reduction in food consumption and therefore food expenditure (recall that there are no price changes in this thought experiment), must be spent on drink. Thus drink consumption must increase, since there have been no price changes. Thus we know that drink must be a normal good for this consumer, D ,pF ,y) > 0, where xD (pD , pF , y) is the consumer’s Marshallian so that ∆xD (p∆y demand function for drink. We know from the Slutsky decomposition of the impact of a drink-price change on the Marshallian demand for drink into an income effect and a substitution effect that, for a sufficiently small change in the price of drink, it will be the case that ∆hD (pD , pF , U ) ∆xD (pD , pF , y) ∆xD (pD , pF , y) ≈ − xD (pD , pF , y) . ∆pD ∆pD ∆y This is simply the difference (rather than differential) version of the Slutsky equation for the impact of a drink-price change. Note that the total effect is measured by an arc approximation of the slope of the Marshallian demand curve for drink in the neighbourhood of the range of drink prices covered by D ,pF ,y) ), while the substitution effect is measured by the price change ( ∆xD (p ∆pD an arc approximation of the slope of the Hicksian demand curve for drink in the neighbourhood of the range of drink prices covered by the price change D ,pF ,U ) ( ∆hD (p ). The difference between these two effects is the income effect ∆pD D ,pF ,y) (−xD (pD , pF , y) ∆xD (p∆y ). Recall that the substitution effect must be non-positive, and will usually D ,pF ,U ) 6 0. be negative. This means that ∆hD (p ∆pD Since drink is a normal good, we know that ∆xD (pD ,pF ,y) ∆y > 0. Note that ∆xD (pD , pF , y) ∆xD (pD , pF , y) > 0 =⇒ xD (pD , pF , y) > 0, ∆y ∆y so that 9 ∆xD (pD , pF , y) ∆pD ≈ ∆hD (pD , pF , U ) ∆xD (pD , pF , y) − xD (pD , pF , y) ∆pD ∆y < ∆hD (pD , pF , U ) . ∆pD Since drink is a normal good, we know that it cannot be a Giffen good. As such, both the Marshallian demand curve for drink and the Hicksian demand curve for drink are downward sloping (or, at least, not upward sloping). Thus we have ∆xD (pD , pF , y) ∆hD (pD , pF , U ) < 6 0. ∆pD ∆pD This means that ∆xD (pD , pF , y) ∆hD (pD , pF , U ) > > 0. ∆pD ∆pl In other words, while both demand curves are downward sloping (or at least, not upward sloping), the Marshallian demand curve is steeper than the Hicksian demand curve when they are graphed using the standard mathematical convention with the drink price on the horizontal axis and the quantity of drink demanded on the vertical axis. As such, when the inverse Marshallian demand curve for drink and the inverse Hicksian demand curve for drink are graphed, so that the drink price is on the vertical axis and the quantity of drinks demanded is on the horizontal axis, the inverse Hicksian demand curve will be steeper than the inverse Marshallian demand curve. This algebraic argument is illustrated diagrammatically in the following diagram (in which commodity one is drink and commodity two is food). 10 q2 U1 A Comparison of a Marshallian Demand Curves and Hicksian Demand Curves for commodity one when commodity one is a normal good and commodity two is an inferior good. U2 y/p2 q22 U1 > U2 p11 < p12 q21 0 y/p12 y/p11 q1 p1 p12 h1 ( p1 ; p2 , U1 ) a d c p11 b x1 ( p1 ; p2 , y ) h1 ( p1 ; p2 , U2 ) 0 q12 h11 h12 q11 11 q1 Answers to Additional Practice Questions Answer to Additional Practice Question 1 See the attached document by Michael Sproule for some guidance about answering this question. Answer to Additional Practice Question 2 See the attached extract from McCloskey, DN (1982), The applied theory of price, Macmillan Publishing, USA (pp. 39–40). Answer to Additional Practice Question 3 Please consult the Wagstaff paper for the answers to the various parts of this question. 12 Tutorial Question 1 from ECON2101 Tutorial 5 in Semester Two of 2019. Tutorial Question 2 from ECON2101 Tutorial 5 in Semester Two of 2019. Tutorial Question 2 from ECON2101 Tutorial 5 in Semester Two of 2019. Tutorial Question 2 Part (b) from ECON2101 Tutorial 5 in Semester Two of 2019. Tutorial Question 2 Part (d) from ECON2101 Tutorial 5 in Semester Two of 2019. Tutorial Question 3 from ECON2101 Tutorial 5 in Semester Two of 2019. Tutorial Question 3 from ECON2101 Tutorial 5 in Semester Two of 2019. Guidance for Answering Additional Practice Question 1 from ECON2101 Tutorial 5 in Semester Two of 2019. Downloaded from http://www.csun.edu/~hceco008/tablecontents.htm on Friday 29 September 2017. This website is edited by Mike Sproul. 6.g. A Subsidy Versus a Cash Gift Many goods receive government subsidies: education, health care, and rail transit, to name a few. An important question for any consumer to ask is “Would I rather have the subsidy? Or would I rather have my share of the money the government spent on the subsidy?” For example, if the government’s subsidy to your education were $20,000 per year, you might reasonably prefer to have the government give you the $20,000, which you could then use to buy an education on the private market, or to buy some other good that you might value more than education. We can use figure 6.g.1 to show that a rational consumer will always prefer a cash gift to an equal-valued subsidy to some good. A consumer’s initial optimum is at point A, consuming a combination of education x and other goods y. A subsidy to education would make education cheaper, causing the budget constraint to rotate outward as shown. The consumer’s new optimum would be at point B, along the budget line labeled as “$100 subsidy”. y Figure 6.g.1 $100 Cash gift C B yB $100 Subsidy Subsidy= cash gift=$100 A yD D xB x (Education) At the optimum point B, the consumer is buying XB units of x and YB units of y. We can judge the government’s spending on the subsidy by comparing points B and D. Point D shows us that if the consumer had tried to buy XB units of x without the subsidy, he could have afforded only YD units of y. But with the subsidy, he can afford YB units of y while still buying XB units of x. Thus the value of the subsidy is equal to (YB -YD) units of y. We will assume that this amounts to 100 units of y. If y is a numeraire good, so that Py=$1, then this means that the government is spending $100 on the subsidy. Now suppose that instead of spending $100 on the subsidy, the government gave $100 in cash to the consumer. Starting from the original optimum at point A, this $100 cash gift would shift the budget line up by $100, to the new budget line labeled as “$100 Cash gift”. Note that the “$100 cash gift” budget line passes through point B, since we already know that point B is $100 above point D. Because the “$100 cash gift” line cuts through the indifference curve at point B, a higher indifference curve is now reachable at point C. In other words, the $100 cash gift allows the consumer to reach point C, while the $100 subsidy only allows the consumer to reach point B. Since C is on a higher indifference curve than B, the $100 cash gift is preferred to the $100 subsidy. Additional Practice Question 2 from ECON2101 Tutorial 5 in Semester Two of 2019.
