Chapter 1 1 Chapter 1 The Market This chapter was written so I would have something to talk about on the first day of class. I wanted to give students an idea of what economics was all about, and what my lectures would be like and yet not have anything that was really first day, and a good number of them won't nec ily be at the lecture. way t I chose to discuss a housing market since it gives a way to describe a number gives a of economic ideas in very simple language and gives a good guide to what lies ahead. In this chap sing results-analytic insights that would thinking"about a problem. The two most surprising results that ndominium example and the tax example in Secti sizing in class just why these results are true, anc nodeling. It also makes tions. Suppose that every con- dominium conve and creating two apart- ments. Then wha condor Suppose that the condominiums a consider renting an apartment. In ea ng apartments would rise when cond The point of a si red here is to focus our thoughts on wha come to a once-and-for-all conclusion about t is offered by these examples is that you ha the demand side of the apartment market when t of this particular policy. The only concept that le with in this chapter is the idea of Pareto efficiency idea a little more than is in the book and rephrase it a few times. But then I tell them not to worry about it too much, since we'll look at it in great detail later in the course The workbook problems here are pretty straightforwa. The biggest problem is getting the students to draw the true(discontinuous demand curve, as in Figure 1.1, rather than just to sketch in a downward-slopir curve as in Figure 1.2. This is a good time to emphasize to the students that when they are given n numbers describing a curve, they have to use the numbers-they can't just sketch in any old shape
Chapter 1 1 Chapter 1 The Market This chapter was written so I would have something to talk about on the first day of class. I wanted to give students an idea of what economics was all about, and what my lectures would be like, and yet not have anything that was really critical for the course. (At Michigan, students are still shopping around on the first day, and a good number of them won’t necessarily be at the lecture.) I chose to discuss a housing market since it gives a way to describe a number of economic ideas in very simple language and gives a good guide to what lies ahead. In this chapter I was deliberately looking for surprising results—analytic insights that wouldn’t arise from “just thinking” about a problem. The two most surprising results that I presented are the condominium example and the tax example in Section 1.6. It is worth emphasizing in class just why these results are true, and how they illustrate the power of economic modeling. It also makes sense to describe their limitations. Suppose that every condominium conversion involved knocking out the walls and creating two apartments. Then what would happen to the price of apartments? Suppose that the condominiums attracted suburbanites who wouldn’t otherwise consider renting an apartment. In each of these cases, the price of remaining apartments would rise when condominium conversion took place. The point of a simple economic model of the sort considered here is to focus our thoughts on what the relevant effects are, not to come to a once-and-for-all conclusion about the urban housing market. The real insight that is offered by these examples is that you have to consider both the supply and the demand side of the apartment market when you analyze the impact of this particular policy. The only concept that the students seem to have trouble with in this chapter is the idea of Pareto efficiency. I usually talk about the idea a little more than is in the book and rephrase it a few times. But then I tell them not to worry about it too much, since we’ll look at it in great detail later in the course. The workbook problems here are pretty straightforward. The biggest problem is getting the students to draw the true (discontinuous) demand curve, as in Figure 1.1, rather than just to sketch in a downward-sloping curve as in Figure 1.2. This is a good time to emphasize to the students that when they are given numbers describing a curve, they have to use the numbers—they can’t just sketch in any old shape
2 Chapter Highlights The Market A. Example of an economic model- the market for apartments 1. models are simplifications of reality 2. for example, assume all apartments are identical 3. some are close to the university, others are far away 4. price of outer-ring apartments is exogenous -determined outside the model 5. price of ring apartments is endogen determined within the B. Two principles of economics 1. optimization principle- people choose actions that are in their interest 2. equilibrium principle- people's actions must eventually be consistent with each othe C. Constructing the demand curve 1. line up the people by willingness-to-pay. See Figure 1.1 2. for large numbers of people, this is essentially a smooth curve as in Figure D. Supply curve 1. depends on time frame 2. but we'll look at the short run- when supply of apartments is fixed E. Equilibrium 1. when demand equals supply 2. price that clears the market Comparative statics 1. how does equilibrium adjust when economic conditions change? 2.“ comparative”- compare two equilibria 3.statics"- only look at equilibria, not at adjustment 4. example- increase in supply lowers price; see Figure 1.5 5. example - create condos which are purchased by renters: no effect on 1.6 G. Other ways to allocate apartments 1. discriminating monopolist 2. ordinary monopoli H. Comparing different institutions 1. need a criterion to compare how efficient these different allocation methods are 2. an allocation is Pareto efficient if there is no way to make some group of people better off without making someone else worse off 3. if something is not Pareto efficient, then there is some way to make some people better off without making someone else worse off 4. if something is not Pareto efficient, then there is some kind of " waste"in Checking efficiency of different methods 1. free market - efficient 2. discriminating monopolist -- efficient 3. ordinary monopolist- not efficient 4. rent control not efficient
2 Chapter Highlights The Market A. Example of an economic model — the market for apartments 1. models are simplifications of reality 2. for example, assume all apartments are identical 3. some are close to the university, others are far away 4. price of outer-ring apartments is exogenous — determined outside the model 5. price of inner-ring apartments is endogenous — determined within the model B. Two principles of economics 1. optimization principle — people choose actions that are in their interest 2. equilibrium principle — people’s actions must eventually be consistent with each other C. Constructing the demand curve 1. line up the people by willingness-to-pay. See Figure 1.1. 2. for large numbers of people, this is essentially a smooth curve as in Figure 1.2. D. Supply curve 1. depends on time frame 2. but we’ll look at the short run — when supply of apartments is fixed. E. Equilibrium 1. when demand equals supply 2. price that clears the market F. Comparative statics 1. how does equilibrium adjust when economic conditions change? 2. “comparative” — compare two equilibria 3. “statics” — only look at equilibria, not at adjustment 4. example — increase in supply lowers price; see Figure 1.5. 5. example — create condos which are purchased by renters; no effect on price; see Figure 1.6. G. Other ways to allocate apartments 1. discriminating monopolist 2. ordinary monopolist 3. rent control H. Comparing different institutions 1. need a criterion to compare how efficient these different allocation methods are. 2. an allocation is Pareto efficient if there is no way to make some group of people better off without making someone else worse off. 3. if something is not Pareto efficient, then there is some way to make some people better off without making someone else worse off. 4. if something is not Pareto efficient, then there is some kind of “waste” in the system. I. Checking efficiency of different methods 1. free market — efficient 2. discriminating monopolist — efficient 3. ordinary monopolist — not efficient 4. rent control — not efficient
J. Equilibrium in long run supply will 2. can examine efficiency in this context as well
Chapter 1 3 J. Equilibrium in long run 1. supply will change 2. can examine efficiency in this context as well
Chapter 2 Budget Constraint ost of the material here is pretty straightforward. Drive home the formula he slope of the budget line, emphasizing the derivation on page 23. Try some different notation to make sure that they see the idea of the budget line, and dont just memorize the formulas. In the workbook, we use a number of different choices of notation for precisely this It is also worth pointing out that the slope of a line depends on the(arbitrary)choice of which variable is plotted on the vertical axis. It is surprising how often confusion arises on this Students sometimes have problems with the idea of a numeraire good. They understand the algebra, but they don't understand when it would be used. One nice example is in foreign currency exchange. If you have English pounds and American dollars, then you can measure the total wealth that you have in either tw In the workbook, students sometimes get thrown in exercises where one of the goods has a negative price, so the budget line has a positive slope. This comes from trying to memorize formulas and figures rather than thinking about the problem. This is a good exercise to go over in order to warn students about the dangers of rote learning Budget Constraint A. Consumer theory: consumers choose the best bundles of goods they can afford 1. this is virtually the entire theory in a nutshell 2. but this theory has many surprising consequences B. Two parts 1."can afford"-budget constraint 2.“best”- according to consumers' preferences
4 Chapter Highlights Chapter 2 Budget Constraint Most of the material here is pretty straightforward. Drive home the formula for the slope of the budget line, emphasizing the derivation on page 23. Try some different notation to make sure that they see the idea of the budget line, and don’t just memorize the formulas. In the workbook, we use a number of different choices of notation for precisely this reason. It is also worth pointing out that the slope of a line depends on the (arbitrary) choice of which variable is plotted on the vertical axis. It is surprising how often confusion arises on this point. Students sometimes have problems with the idea of a numeraire good. They understand the algebra, but they don’t understand when it would be used. One nice example is in foreign currency exchange. If you have English pounds and American dollars, then you can measure the total wealth that you have in either dollars or pounds by choosing one or the other of the two goods as numeraire. In the workbook, students sometimes get thrown in exercises where one of the goods has a negative price, so the budget line has a positive slope. This comes from trying to memorize formulas and figures rather than thinking about the problem. This is a good exercise to go over in order to warn students about the dangers of rote learning! Budget Constraint A. Consumer theory: consumers choose the best bundles of goods they can afford. 1. this is virtually the entire theory in a nutshell 2. but this theory has many surprising consequences B. Two parts to theory 1. “can afford” — budget constraint 2. “best” — according to consumers’ preferences
C. What do we want to do with the theory 3. use observed behavior to estimate underlying value &5 is adequate to describe consumer behavior 2. predict how behavior changes as economic environ a)cost-benefit analysis b)predicting impact of some policy D. Consumption bundle 1.(a1, T2)-how much of each good is consumed es of the two goods 3. m- money the consumer has to spend 4. budget constraint: P101+P2T2 < m 5. all(r1, I2) that satisfy this constraint make up the budget set of the consumer. See Figure 2.1 1. theory works with more than two goods, but cant draw pictures 2. often think of good 2(say)as a composite good, representing money to spend on other goods 3. budget constraint becomes P1T1+a2< 4. money spent on good 1 (p1I1) plus the money spent on good 2(a2)has to be less than or equal to the amount available(m) 1.p11+p2x2=m 2. also written as T2=m/p2-(p1/p2)T 3. budget line has slope of -pi/p2 and vertical intercept of m/p 4. set a1=0 to find vertical intercept(m/p2);set r2=0 to find horizontal intercept(m/pi) 5. slope of budget line measures opportunity cost of good 1- how much of good 2 you must give up in order to consume more of go G. Changes in budget line 1. increasing m makes parallel shift out. See Figure 2 2. increasing Pi makes budget line steeper. See Figure 2.3 3. increasing p2 makes budget line fatter 4. just see how intercepts change 5. multiplying all prices by t is just like dividing Income 6. multiplying all prices and income by t doesnt change budget line a)"a perfectly balanced infation doesn't change consumption possibili- H. The numeraire 1. can arbitrarily assign one price a value of l and measure other price relative to that 2. useful when measuring relative prices; e. g, English pounds per dollar, 1987 dollars versus 1974 dollars. etc. axes, subsidies, and rationing 1. quantity tax - tax levied on units bought: pi +t 2. value tax -tax levied on dollars spent: P1 TPI. Also known as ad 3. subsidies -opposite of a tax a)
Chapter 2 5 C. What do we want to do with the theory? 1. test it — see if it is adequate to describe consumer behavior 2. predict how behavior changes as economic environment changes 3. use observed behavior to estimate underlying values a) cost-benefit analysis b) predicting impact of some policy D. Consumption bundle 1. (x1, x2) — how much of each good is consumed 2. (p1, p2) — prices of the two goods 3. m — money the consumer has to spend 4. budget constraint: p1x1 + p2x2 ≤ m 5. all (x1, x2) that satisfy this constraint make up the budget set of the consumer. See Figure 2.1. E. Two goods 1. theory works with more than two goods, but can’t draw pictures. 2. often think of good 2 (say) as a composite good, representing money to spend on other goods. 3. budget constraint becomes p1x1 + x2 ≤ m. 4. money spent on good 1 (p1x1) plus the money spent on good 2 (x2) has to be less than or equal to the amount available (m). F. Budget line 1. p1x1 + p2x2 = m 2. also written as x2 = m/p2 − (p1/p2)x1. 3. budget line has slope of −p1/p2 and vertical intercept of m/p2. 4. set x1 = 0 to find vertical intercept (m/p2); set x2 = 0 to find horizontal intercept (m/p1). 5. slope of budget line measures opportunity cost of good 1 — how much of good 2 you must give up in order to consume more of good 1. G. Changes in budget line 1. increasing m makes parallel shift out. See Figure 2.2. 2. increasing p1 makes budget line steeper. See Figure 2.3. 3. increasing p2 makes budget line flatter 4. just see how intercepts change 5. multiplying all prices by t is just like dividing income by t 6. multiplying all prices and income by t doesn’t change budget line a) “a perfectly balanced inflation doesn’t change consumption possibilities” H. The numeraire 1. can arbitrarily assign one price a value of 1 and measure other price relative to that 2. useful when measuring relative prices; e.g., English pounds per dollar, 1987 dollars versus 1974 dollars, etc. I. Taxes, subsidies, and rationing 1. quantity tax — tax levied on units bought: p1 + t 2. value tax — tax levied on dollars spent: p1 + τp1. Also known as ad valorem tax 3. subsidies — opposite of a tax a) p1 − s b) (1 − σ)p1
6 Chapter Highlights 4. lump sum tax or subsidy -amount of tax or subsidy is independent of the consumers choices. Also called a head tax or a poll tax 5. rationing -can't consume more than a certain amount of some good J. Example -food stamps 1. before 1979 was an ad valorem subsidy on food a)paid a certain amount of money to get food stamps which were worth more than they cost b)some rationing component- could only buy a maximum amount of 2. after 1979 got a straight lump-sum grant of food coupons. Not the san as a pure lump-sum grant since could only spend the coupons on food e
6 Chapter Highlights 4. lump sum tax or subsidy — amount of tax or subsidy is independent of the consumer’s choices. Also called a head tax or a poll tax 5. rationing — can’t consume more than a certain amount of some good J. Example — food stamps 1. before 1979 was an ad valorem subsidy on food a) paid a certain amount of money to get food stamps which were worth more than they cost b) some rationing component — could only buy a maximum amount of food stamps 2. after 1979 got a straight lump-sum grant of food coupons. Not the same as a pure lump-sum grant since could only spend the coupons on food
Chapter 3 Preferences This chapter is more abstract and therefore needs somewhat more motivation than the previous chapters. It might be a good idea to talk about relations in general before introducing the particular idea of preference relations. Try the relations of"taller, and "heavier. " and "taller and heavier. Point out that aller and heavier" isnt a complete relation, while the other two are. This general discussion can motivate the general idea of preference relations Make sure that the students learn the specific examples of preferences suck as perfect substitutes, perfect complements, etc. They will use these examples any, many times in the next few weeks When describing the ideas of perfect substitutes, emphasize that the defining characteristic is that the slope of the indifference curves is constant, not that it is-1. In the text, I always stick with the case where the slope is -l, but in the workbook, we often treat the general case. The warning goes with the perfect complements case. I work out the symmetric case in the text and try to get the students to do the asymmetric case in the workbook. The definition of the marginal rate of substitution is fraught with "sign confusion. "Should the mrs be defined as a negative or a positive number? I've chosen to give the MRS its natural sign in the book, but I warn the students that many economists tend to speak of the MRS in terms of absolute value. Example diminishing marginal rate of substitution refers to a situation where the absolute value of the mrs decreases as we move along an indifference curve. The actual value of the MRS (a negative number) is increasing in this movement Students often begin to have problems with the workbook exercises here The first confusion they have is that they get mixed up about the idea that indifference curves measure the directions where preferences are constant, and instead draw lines that indicate the directions that preferences are increasing The second problem that they have is in knowing when to draw just arbitrary curves that qualitatively depict some behavior or other, and when to draw exact Try asking your students to draw their indifference curves between five dollar ills and one dollar bills. Offer to trade with them based on what they draw. In ddition to getting them to think, this is a good way to supplement your facult salary
Chapter 3 7 Chapter 3 Preferences This chapter is more abstract and therefore needs somewhat more motivation than the previous chapters. It might be a good idea to talk about relations in general before introducing the particular idea of preference relations. Try the relations of “taller,” and “heavier,” and “taller and heavier.” Point out that “taller and heavier” isn’t a complete relation, while the other two are. This general discussion can motivate the general idea of preference relations. Make sure that the students learn the specific examples of preferences such as perfect substitutes, perfect complements, etc. They will use these examples many, many times in the next few weeks! When describing the ideas of perfect substitutes, emphasize that the defining characteristic is that the slope of the indifference curves is constant, not that it is −1. In the text, I always stick with the case where the slope is −1, but in the workbook, we often treat the general case. The same warning goes with the perfect complements case. I work out the symmetric case in the text and try to get the students to do the asymmetric case in the workbook. The definition of the marginal rate of substitution is fraught with “sign confusion.” Should the MRS be defined as a negative or a positive number? I’ve chosen to give the MRS its natural sign in the book, but I warn the students that many economists tend to speak of the MRS in terms of absolute value. Example: diminishing marginal rate of substitution refers to a situation where the absolute value of the MRS decreases as we move along an indifference curve. The actual value of the MRS (a negative number) is increasing in this movement! Students often begin to have problems with the workbook exercises here. The first confusion they have is that they get mixed up about the idea that indifference curves measure the directions where preferences are constant, and instead draw lines that indicate the directions that preferences are increasing. The second problem that they have is in knowing when to draw just arbitrary curves that qualitatively depict some behavior or other, and when to draw exact shapes. Try asking your students to draw their indifference curves between five dollar bills and one dollar bills. Offer to trade with them based on what they draw. In addition to getting them to think, this is a good way to supplement your faculty salary
8 Chapter Highlights Preferences A. Preferences are relationships between bundles 1. if a consumer would choose bundle(r1, r2)when(y1, y2)is available, then it is natural to say that bundle (a1, r2)is preferred to(y1, y2) by thi consumer 2. preferences have to do with the entire bundle of goods, not with individual B. Notation 1.(1, r2)>(y1, y2) means the x-bundle is strictly preferred to the y- bundle 2.(1, T2)N(y1, y2) means that the x-bundle is regarded as indifferent to 3.(31, 122(91, y2) means the x-bundle is at least as good as(preferred to or indifferent to) the y-bundle C. Assumptions about preferences 1. complete- any two bundles can be compared 2. reflexive any bundle is at least as good as itself 3. transitive-if X >Y and Y>Z, then X>Z a)transitivity necessary for theory of optimal choice D. Indifference curves 1. graph the set of bundles that are indifferent to some bundle. See Figure 2. indifference curves are like contour lines on a map 3. note that indifference curves describing two distinct levels of preference cannot cross. See Figure 3.2. a)proof- use transitivity E. Examples of preferences 1. perfect substitutes. Figure 3.3 a)red pencils and blue pencils; pints and quarts b) constant rate of trade-off between the two goods b)right shoes and left shoes; coffee and crean 3. bads. Figure 3.5. 4. neutrals. Figure 3.6 5. satiation or bliss point Figure 3.7. F. Well-behaved preference monotonicity more of either good is better a)implies indifference curves have negative slope. Figure 3.9 2. convexity -averages are preferred to extremes. Figure 3.10 a)slope gets Hatter as you move further to righ b) example of non-convex preferences G. Marginal rate of substitution l. slope of the indifference curve 2. MRS=A r2/ AT1 along an indifference curve. Figure 3.11 3. sign problem - natural sign is negative, since indifference curves will generally have negative sloy for consumption of good 2. Figure 3. 12 trade off consumption of good 1 4. measures how the consumer is willing to
8 Chapter Highlights Preferences A. Preferences are relationships between bundles. 1. if a consumer would choose bundle (x1, x2) when (y1, y2) is available, then it is natural to say that bundle (x1, x2) is preferred to (y1, y2) by this consumer. 2. preferences have to do with the entire bundle of goods, not with individual goods. B. Notation 1. (x1, x2) (y1, y2) means the x-bundle is strictly preferred to the ybundle 2. (x1, x2) ∼ (y1, y2) means that the x-bundle is regarded as indifferent to the y-bundle 3. (x1, x2) (y1, y2) means the x-bundle is at least as good as (preferred to or indifferent to) the y-bundle C. Assumptions about preferences 1. complete — any two bundles can be compared 2. reflexive — any bundle is at least as good as itself 3. transitive — if X Y and Y Z, then X Z a) transitivity necessary for theory of optimal choice D. Indifference curves 1. graph the set of bundles that are indifferent to some bundle. See Figure 3.1. 2. indifference curves are like contour lines on a map 3. note that indifference curves describing two distinct levels of preference cannot cross. See Figure 3.2. a) proof — use transitivity E. Examples of preferences 1. perfect substitutes. Figure 3.3. a) red pencils and blue pencils; pints and quarts b) constant rate of trade-off between the two goods 2. perfect complements. Figure 3.4. a) always consumed together b) right shoes and left shoes; coffee and cream 3. bads. Figure 3.5. 4. neutrals. Figure 3.6. 5. satiation or bliss point Figure 3.7. F. Well-behaved preferences 1. monotonicity — more of either good is better a) implies indifference curves have negative slope. Figure 3.9. 2. convexity — averages are preferred to extremes. Figure 3.10. a) slope gets flatter as you move further to right b) example of non-convex preferences G. Marginal rate of substitution 1. slope of the indifference curve 2. MRS = ∆x2/∆x1 along an indifference curve. Figure 3.11. 3. sign problem — natural sign is negative, since indifference curves will generally have negative slope 4. measures how the consumer is willing to trade off consumption of good 1 for consumption of good 2. Figure 3.12
5. measures marginal willingness to pay(give up) a)not the same as how much you have to pay b)but how much you would be willing to pay
Chapter 3 9 5. measures marginal willingness to pay (give up) a) not the same as how much you have to pay b) but how much you would be willing to pay
10 Chapter Highlights Chapter 4 Utility In this chapter, the level of abstraction kicks up another notch. Students often Lave trouble with the idea of utility. It is sometimes hard for trained economists to sympathize with them sufficiently, since it seems like such an obvious notion Here is a way to approach the subject. Suppose that we return to the idea of the"heavier than"relation discussed in the last chapter. Think of having a big palance scale with two trays. You can put someone on each side of the balance scale and see which person is heavier, but you don't have any standardized weights. Nevertheless you have a way to determine whether a is heavier than y Now suppose that you decide to establish a scale. You get a bunch of stones, check that they are all the same weight, and then measure the weight of individuals in stones. It is clear that z is heavier than y if z's weight in stones is heavier than ys weight in stones Somebody else might use different units of measurements--kilograms, pounds or whatever. It doesnt make any difference in terms of deciding who is heavier At this point it is easy to draw the analogy with utility-just as pounds give to represent the" heavier than"order numerically, utility gives a way to represent the preference order numerically. Just as the units of weight are the units of utility This analogy can also be used to explore the concept of a positive monotonic transformation, a concept that students have great trouble with. Tell them that a monotonic transformation is just like changing units of measurement in the weight example However, it is also important for students to understand that nonlinear changes of units are possible. Here is a nice example to illustrate this. Suppose hat wood is always sold in piles shaped like cubes. Think of the relation"one pile has more wood than another. Then you can represent this relation by looking at the measure of the sides of the piles, the surface area of the piles, or he volume of the piles. That is, a, x or a' gives exactly the same comparison between the piles. Each of these numbers is a different representation of the utility of a cube of wood Be sure to go over carefully the examples here. The Cobb-Douglas example is an important one, since we use it so much in the workbook. Emphasize that it is just a nice functional form that gives convenient expressions. Be sure to
10 Chapter Highlights Chapter 4 Utility In this chapter, the level of abstraction kicks up another notch. Students often have trouble with the idea of utility. It is sometimes hard for trained economists to sympathize with them sufficiently, since it seems like such an obvious notion to us. Here is a way to approach the subject. Suppose that we return to the idea of the “heavier than” relation discussed in the last chapter. Think of having a big balance scale with two trays. You can put someone on each side of the balance scale and see which person is heavier, but you don’t have any standardized weights. Nevertheless you have a way to determine whether x is heavier than y. Now suppose that you decide to establish a scale. You get a bunch of stones, check that they are all the same weight, and then measure the weight of individuals in stones. It is clear that x is heavier than y if x’s weight in stones is heavier than y’s weight in stones. Somebody else might use different units of measurements—kilograms, pounds, or whatever. It doesn’t make any difference in terms of deciding who is heavier. At this point it is easy to draw the analogy with utility—just as pounds give a way to represent the “heavier than” order numerically, utility gives a way to represent the preference order numerically. Just as the units of weight are arbitrary, so are the units of utility. This analogy can also be used to explore the concept of a positive monotonic transformation, a concept that students have great trouble with. Tell them that a monotonic transformation is just like changing units of measurement in the weight example. However, it is also important for students to understand that nonlinear changes of units are possible. Here is a nice example to illustrate this. Suppose that wood is always sold in piles shaped like cubes. Think of the relation “one pile has more wood than another.” Then you can represent this relation by looking at the measure of the sides of the piles, the surface area of the piles, or the volume of the piles. That is, x, x2, or x3 gives exactly the same comparison between the piles. Each of these numbers is a different representation of the utility of a cube of wood. Be sure to go over carefully the examples here. The Cobb-Douglas example is an important one, since we use it so much in the workbook. Emphasize that it is just a nice functional form that gives convenient expressions. Be sure to