## A Few Scattered Notes, Observations, and Examples of Graphs and Diagrams in Teaching and Doing Economics/The Honest Broker

Perhaps starting a broader conversation...

**: In Defense of Funny Diagrams: "Brad DeLong asks a question about which of the various funny diagrams economists love...**

...should be taught in Econ 101. I say production possibilities yes, Edgeworth box no--which, strange to say, is how we deal with this issue in Krugman/Wells. But students who go on to major in economics should be exposed to the box--and those who go on to grad school really, really need to have seen it, and in general need more simple general-equilibrium analysis than, as far as I can tell, many of them get these days. There was, clearly, a time when economics had too many pictures. But now, I suspect, it doesn’t have enough....

This is partly a personal bias. My own mathematical intuition, and a lot of my economic intuition in general, is visual: I tend to start with a picture, then work out both the math and the verbal argument to make sense of that picture. (Sometimes I have to learn the math, as I did on target zones; the picture points me to the math I need.) I know that’s not true for everyone, but it’s true for a fair number of students, who should be given the chance to learn things that way.

Beyond that, pictures are often the best way to convey global insights... in the sense of thinking about all possibilities as opposed to small changes.... The production possibility frontier is that it gives students a way to think about what efficiency means--if you want to explain inefficiency in production, you put a point inside the PPF, if you want to explain inefficiency in allocation, you talk about choosing the wrong point on the PPF. The Edgeworth box is good for explaining what it takes to be efficient in production and also efficient in distribution--I learned all of this from the classic Francis Bator paper on welfare maximization--but is just too hard for freshpeople.

I also retain, even after all these years, a soft spot for at least some of the profusion of diagrams that characterized trade theory when I was a student....

I have the sense that too many majors and/or grad students were shortchanged.... They can do game theory, they can solve sets of equations, but their sense of how the pieces fit together is lacking... don’t have the technique to cut through what should be easily avoidable confusion. (I sometimes find myself wanting to shout ‘Use an offer curve, dammit!’)... [While] the real economy isn’t characterized by competitive general equilibrium. But it’s still a useful baseline — not so much an idealization as a description of how things should be, which helps to cast how they really are into much sharper relief.

Draw, baby, draw.

For graduate students, Paul Krugman is (of course) completely right: if you aren't very and roughly equally comfortable speaking verbal-narrative, drawing graphical-analytic-geometric, and writing down equation-based mathematical analyses and descriptions of an economic problem, you are doing it wrong and should fix it by learning your tools.

Of course, it may well be that attaining and maintaining a kind of balance that avoids, say, hopelessly confusing yourself with Neo-Fisherman idiocy is unattainable for you. In that case, you should remember the principle of comparative advantage, and go thou and do something else.

And, although each piece--verbal-narrative, graphical-analytic-geometric, and equation-based mathematical-descriptive--of the analysis is easy, somehow combining them is hard. And that was what led John Maynard Keynes to write in his obituary for Alfred Marshall:

The study of economics does not seem to require any specialised gifts of an unusually high order. Is it not, intellectually regarded, a very easy subject compared with the higher branches of philosophy and pure science? Yet good, or even competent, economists are the rarest of birds. An easy subject, at which very few excel!

The paradox finds its explanation, perhaps, in that the master-economist must possess a rare

combinationof gifts. He must reach a high standard in several different directions and must combine talents not often found together. He must be mathematician, historian, statesman, philosopher-in some degree. He must understand symbols and speak in words. He must contemplate the particular in terms of the general, and touch abstract and concrete in the same flight of thought. He must study the present in the light of the past for the purposes of the future.No part of man's nature or his institutions must lie entirely outside his regard. He must be purposeful and disinterested in a simultaneous mood; as aloof and incorruptible as an artist, yet sometimes as near the earth as a politician. Much, but not all, of this ideal many-sidedness Marshall possessed. But chiefly his mixed training and divided nature furnished him with the most essential and fundamental of the economist's necessary gifts-he was conspicuously historian and mathematician, a dealer in the particular and the general, the temporal and the eternal, at the same time...

And as Keynes wrote to Roy Harrod:

It seems to me that economics is a branch of logic, a way of thinking; and that you do not repel sufficiently firmly attempts à la Schultz to turn it into a pseudo-natural-science. One... cannot get very far except by devising new and improved models. This requires... 'a vigilant observation of the actual working of our system'. Progress in economics consists almost entirely in a progressive improvement in the choice of models.... The later classical school, exemplified by Pigou... overwork[s] a too-simple or out-of-date model... not seeing that progress lay in improving the model.... Marshall often confused his models, for the devising of which he had great genius, by wanting to be realistic and by being unnecessarily ashamed of lean and abstract outlines....

Economics is a science of thinking in terms of models joined to the art of choosing models which are relevant to the contemporary world. It is compelled to be this, because, unlike the typical natural science, the material to which it is applied is, in too many respects, not homogeneous through time. The object of a model is to segregate the semi-permanent or relatively constant factors from those which are transitory or fluctuating so as to develop a logical way of thinking about the latter, and of understanding the time sequences to which they give rise in particular cases. Good economists are scarce because the gift for using 'vigilant observation' to choose good models, although it does not require a highly specialised intellectual technique, appears to be a very rare one...

Nevertheless, when we move down from the graduate students to the freshpeople, I do want to disagree with Paul, I now definitely want to throw overboard the ludicrously-overcomplicated diagrams:

and also throw overboard many of the not-so-ludicrous ones, including the Edgeworth Boxes and the Production-Possibility Frontiers. I now want to get as quickly as possible to supply-and-demand and to producer and consumer surplus.

Some of this is peculiar to Berkeley.

Our introduction to economics is a one-semester course, which makes it extra-complicated to teach. The students can handle it: this is the best public university in the world, and these are the most capable public university students in the world. But it is a great rush.

This year I tried to make space for the PPF.

But, as evidenced by Problem Set 1, it did not go terribly well.

From the Problem Set 1 answer key:

## Problem Set 1

1) In the economy of the university town of Avicenna (if you wish, cf.: Peter Beagle (1986): The Folk of the Air http://amzn.to/1RxRFQJ (New York: Del Rey: 0345337824)) there are produced two and only two commodities: yoga lessons, and lattes. There are ten workers in the economy. They are able to produce the following amounts of lattes or teach the following amounts of yoga lessons each day:

1a) On a graph, draw the Production Possibility Frontier of Avicenna

**Answer: Begin with everybody teaching yoga lessons and nobody making lattes—at the point (2750, 0) on the graph (if you have chosen yoga for your x- and lattes for your y-axis). Now suppose the economy wants to produce lattes. Any of the ten people could produce 500 lattes. But in order to produce those lattes, you have to move them to the café and off of the mat and so sacrifice having some yoga lessons taught. What is the best thing to do? Well, looking at the table you see that Alfred (an Old English name meaning “wisdom of the elves”. The most prominent historical figure was King of the English realm of Wessex—shortened “Westseaxna Rice”, “Realm of the West-Saxons” or “western realm of the knife-guys” —from 871-899; he is the reason that we today think of England as more Germanic than Norse) has a comparative advantage at making lattes of 10:1: he can make 10 lattes for the time and resources he needs to teach one yoga lesson.**

**Pull Alfred off the mat and put him to work at the milk frothing machine and you only sacrifice 50 yoga lessons—fewer than anybody else. So we pull Alfred off of the yoga mat and put him to work at the milk frother, have him produce 500 lattes, and draw the next point (2700, 500) on the PPF.**

**What if we don’t want 500 but only 250 lattes? Well, if Alfred spends half his time at the café and half his time on the mat, he can make 250 lattes and teach 25 yoga lessons—and similarly for some other combination of shifts. Therefore draw a straight line between (2750, 0) and (2700, 500) for that part of the PPF.**

**Now suppose we are at (2700, 500) and want to produce more lattes—1000, say? who is the next person we should pull off of the yoga mat and put to work at the milk frother in order to have the most efficient economy? Who has the second greatest comparative advantage in making lattes? Well, from the table it’s Beatrice (Latin: “she who blesses” or “she who brings joy”. The most prominent figure in literature is the Beatrice is Dante’s patron and guide through the Afterlife in his Divine Comedy), with a 5:1 comparative advantage. She should shift next. So put Alfred and Beatrice to work in the café, and draw in the (2600, 1000) point on the PPF.**

**By a similar argument as with Alfred, Beatrice could produce 250 lattes and teach 50 yoga lessons if she split her shifts. So connect (2600, 1000) to (2700, 500) for the next segment of the PPF.**

**Note that along the straight-line segments of the PPF, the slope of the PPF is equal to the comparative advantage of the person who is, as you move along that straight line, switching from teaching on the mat to working the milk frothing machine.**

**Continuing to work down the table, note that I have already arranged the potential workers not just in alphabetical order but from most to least in their comparative advantage at making lattes. So, in order, move successively Cixi (Mandarin “Empress of the Western Palace”. The historical figure known as Cixi was born Yehenara, “Little Orchid”, and was Regent of Qing China from 1861-1908. At the end of her reign, China was one of the only ten or so Asian and African realms that had not been colonized by European powers) is next.**

**She is then followed, in order, by Dante, Earendil, Faramir, Gaius, Hrothgar, Indira, and Jenghiz off of the yoga mat and into the café. So draw in the points (2450, 1500), (2250, 2000), (2000, 2500), (1700, 3000), (1350, 3500), (950, 4000), (500, 4500), and (0, 5000)—the last corresponding to the situation in which all ten are working at the café and nobody is teaching yoga.**

**Connect those points with straight lines to complete the PPF.**

**And you are done.**

**Note that if we had simply moved away from producing 2750 lessons and zero lattes by having everybody work half-time at the café and half-time at the gym, we would be producing 1375 lessons and 2500 lattes—well inside the PPF. We could produce 1800 lessons and 3000 lattes if we just had Gaius, Hrothgar, Indira, and Jenghiz teaching yoga. Specializing in your comparative advantage in production is a very good thing to do!**

1b) On a graph, draw the supply curve for lattes if the price of yoga lessons is $10/lesson.

**Answer: You can start either from the table, or from the PPF…**

**From the PPF: We know that the price of yoga is $10/lesson. Suppose that the price of lattes were $0/latte. What would the value of production be? Well, the value of production would then be simply ten times the number of lessons taught. So start at the (2750, 0) point, calculate that (2750 yoga lessons) x ($10/lesson) = $27,500 worth of yoga taught, and draw a straight line upward from (2750, 0). That straight line tells you the combinations of yoga lessons and lattes that are worth $27,500 at a yoga price of $10/lesson and a latte price of $0/latte. Points to the right and above this line are worth more; points below and to the left of this line are worth less.**

**Is there any point on the PPF to the right of this line? No. That tells us that combinations of yoga lessons and lattes that produce more than $27500 of value at these prices are unattainable: they are outside the economy’s PPF.**

**Is there any point on this line other than (2750, 0)? No. That tells us that (2750, 0)—with all ten potential workers on the mat in the gym teaching yoga, and nobody working at the café—is the only attainable combination at which $27,500 of value is created, and is the most profitable way for workers to arrange their shifts. And that tells us that the supply of lattes is 0 when the price of lattes is 0.**

**From the table: At a price of $10/lesson for yoga and $0/latte for lattes, would any of the ten potential workers make more money working at the café than teaching yoga in the gym? No. All ten will therefore choose to work in the gym, none will choose to work in the café, and so the supply of lattes is zero when the price of lattes is $0/latte.**

**Draw the (0, $0) point on the supply curve, with the quantity of lattes produced on the x-axis and the price of lattes on the y-axis.**

**Now let’s gradually raise the price of lattes from $0/latte to just a hair under $1/latte. What happens?**

**On the PPF: As the price of lattes rises from $0/latte toward $1/latte, the line that tells us combinations of yoga lessons and lattes that are worth $27,500 rotates counterclockwise around the point (2750,0). At a price of $0/latte that line passed through (2750, 0) and (2750, 500). At a price of $0.50/latte that rotating line passed through (2750, 0) and (2700, 1000). At a price of $0.99/latte that rotating line passes through (2700, 505.0505…). **

**Is there any point on the PPF to the right of and above the rotating line? No. That tells us that combinations of yoga lessons and lattes that produce more than $27500 of value at these prices are unattainable: they are outside the economy’s PPF.**

**Is there any point on this line other than (2750, 0)? No. That tells us that (2750, 0)—with all ten potential workers on the mat in the gym teaching yoga, and nobody working at the café—is the only attainable combination at which $27,500 of value is created, and is the most profitable way for workers to arrange their shifts. And that tells us that the supply of lattes is 0 when the price of lattes is anything from $0/latte but less than $1/latte.**

**From the table: At a price of $10/lesson for yoga and more than $0/latte but less than $1/latte for lattes, would any of the ten potential workers make more money working at the café than teaching yoga in the gym? No. All ten will therefore choose to work in the gym, none will choose to work in the café, and so the supply of lattes is zero when the price of lattes is anything from $0 to just less than $1/latte.**

**Draw the line from (0, $0) up to a hair less than (0, $1) on the supply curve, with the quantity of lattes produced on the x-axis and the price of lattes on the y-axis.**

**Now let’s raise the price of lattes from a hair under to exactly $1/latte. What happens?**

**On the PPF, when the price of yoga is $10/lesson and the price of lattes is $1/latte, the rotating line corresponding to $27,500 of value produced now intersects the PPF along its whole length from (2750, 0) to (2700, 500): producing at any of those combinations—and they are all attainable—generates $27500 of value.**

**From the table: all of the potential producers except Alfred still make more money teaching yoga than pulling lattes. They remain at work in the gym rather than at the café. Alfred, however, can make $500 teaching yoga at a price of $10/lesson, can make $500 pulling lattes at a price of $1/latte, or split his shifts and produce any linear combination of 50 yoga lessons and 500 lattes and make $500. Alfred is indifferent. He will produce up to 500 lattes at a price of $1/latte depending on what demand at $1/latte is, and spend the rest of his time happily teaching yoga.**

**Draw a straight line on the supply curve from (0, $1) to (500, $1), to show that at a price of $1/latte the quantity of lattes supplied can and will be anything between 0 and 500, depending on what the demand for lattes is.**

**Now let’s gradually raise the price of lattes from $1/latte to just a hair under $2/latte. What happens?**

**From the PPF: As the price of lattes rises from $1/latte toward $2/latte, the line that tells us combinations of yoga lessons and lattes that are worth as much as is produced at the (2700 lessons, 500 lattes) combination rotates counterclockwise around the point (2700, 500). At a price of $1/latte, that line passed through (2700, 500) and (2650, 1000). At a price of $1.50/latte, that line passed through (2700, 500) and (2625, 1000). At a price of $1.99/latte, that line passes through (2700, 500) and (2600.5 , 1000).**

**Is there any point on the PPF to the right of and above the rotating line? No. That tells us that combinations of yoga lessons and lattes that produce more value than (2700 lessons, 500 lattes) at these prices are unattainable: they are outside the economy’s PPF.**

**Is there any point on this line other than (2700 lessons, 500 lattes)? No. That tells us that (2700 lessons, 500 lattes)—with Alfred working at the café, and all other nine potential workers on the mat in the gym teaching yoga—is the only attainable combination at which the most of value is created, and is the most profitable way for workers to arrange their shifts. And that tells us that the supply of lattes is 500 when the price of lattes is anything from $1/latte up to less than $2/latte.**

**From the table: At a price of $10/lesson for yoga and more than $1/latte but less than $2/latte for lattes, would any of the ten potential workers besides Alfred make more money working at the café than teaching yoga in the gym? No. All workers except Alfred will therefore choose to work in the gym, only Alfred will choose to work in the café, and so the supply of lattes is 500 when the price of lattes is anything from $1 to just less than $2/latte.**

**Draw a straight line from (500, $1) up but not quite to (500, $2) to show that the quantity supplied of lattes is 500 when the price is less than $2/latte but is $1/latte or greater.**

**Now let’s raise the price of lattes from a hair under to exactly $2/latte. What happens?**

**On the PPF, when the price of yoga is $10/lesson and the price of lattes is $2/latte, the rotating line corresponding to maximum value produced now intersects the PPF along its whole length from (2700, 500) to (2600, 1000): producing at any of those combinations—and they are all attainable—generates the maximum value.**

**From the table: all of the potential producers except Alfred and Beatrice still make more money teaching yoga than pulling lattes. They remain at work in the gym rather than at the café. Alfred, however, makes more at the café. And Beatrice can make $1000 pulling lattes at a price of $2/latte, or split her shifts and produce any linear combination of 100 yoga lessons and 500 lattes and make $1000. Beatrice is indifferent. She will produce up to 500 lattes at a price of $2/latte depending on what demand at $2/latte is, and spend the rest of her time happily teaching yoga. And Alfred will produce 500 lattes.**

**Draw a straight line on the supply curve from (500, $2) to (100, $2), to show that at a price of $2/latte the quantity of lattes supplied can and will be anything between 500 and 1000, depending on what the demand for lattes is.**

**Continue walking down the table and rotating the maximum-value line on the PPF. Find out that at $3/latte Cixi, $4 Dante, $5 Earendil, $6 Faramir, $7 Gaius, $8 Hrothgar, $9 Indira, and $10 Jenghiz are willing to move over to the café. Jump up the quantity of lattes supplied by 500, therefore, when the price hits $3, $4, $5, $6, $7, $8, $9, and $10. And you are done:**

**[...parts c-g omitted...]**

2) In the economy of the university town of Avicenna (if you wish, cf.: Peter Beagle (1986): The Folk of the Air http://amzn.to/1RxRFQJ (New York: Del Rey: 0345337824)) there are produced two and only two commodities: yoga lessons, and lattes. When the price of yoga lessons is $10/lesson, the supply curve for lattes is:

SUPPLY: P = 0 + Qs/500 up to a maximum quantity produced of 5000 lattes

2a) On a graph, draw the supply curve for lattes if the price of yoga is $20/lesson. Also draw the supply curve for lattes if the price of yoga is $40/lesson.

**Answer: Remember: behind your supply curve there is, lurking somewhere, a Production-Possibility Frontier and a table telling you what the resources and capabilities of the potential workers in the economy are. Workers are trying to choose what is best for themselves—and it is their behavior in response to the different incentives provided them by market prices that determines and is summarized in the supply curve.**

**At a value of yoga lessons of $20/lesson, the opportunity cost, measured in dollars, of producing lattes is twice as great as in the baseline case, where the price of yoga was $10/lesson and the supply curve was P = 0 + Qs/500. Workers will therefore demand twice as much in order to induce them to make lattes. The supply is thus:**

**P = 0 + Qs/250 when the price of yoga is $20/lesson.**

**Similarly, at a value of yoga lessons of $40/lesson, the opportunity cost, measured in dollars, of producing lattes is twice as great as in the baseline case, where the price of yoga was $10/lesson and the supply curve was P = 0 + Qs/500. Workers will therefore demand twice as much in order to induce them to make lattes. The supply is thus:**

**P = 0 + Qs/125 when the price of yoga is $40/lesson.**

**So we have:**

2b) What do you think the Production Possibility Frontier of Avicenna is? Draw what you think is the PPF. Explain why you think this is the PPF.

**Answer: This question was a disaster—and the reason why we are curving the grades on this problem set up.**

**What I had hoped that you would do was think: “In 1 we built a PPF and a supply curve out of a table. Now we have a supply curve. Surely we can reverse-engineer the table and the PPF lurking behind the supply curve.”**

**The next step would be to have noticed that the supply curve for a $10/lesson yoga price of:**

**P = 0 + Qs/500**

**starts at (0, $0) and ends at (5000, $10) and looks a lot like the supply curve built in 1b)—except that it is smooth and upward-sloping rather than stair-stepped. And the step after that would have been to say: if the supply curve is very similar, the underlying table should be very similar, and the PPF should be very similar—only in some sense “smoother” than the PPF in problem 1). And that would have been enough for full credit.**

**I expected the bulk of students to stop there.**

**I expected some to go on, and draw the supply curve for a $10/lesson price of yoga:**

**and to then start walking down the supply curve and using that walking-down process to draw the exact PPF.**

**We know that the maximum production point of the economy—when all resources are devoted to making lattes—is (5000 lattes, 0 yoga lessons). So we have a starting point: with a $10/lesson yoga price, and a $10/latte latte price, we are at the (5000 lattes, 0 lessons) point on the PPF and the (5000 lattes, $10/latte) point on the supply curve.**

**Now let us walk down the supply curve to (0 lattes, $0/latte). What has happened? Well, as the price of lattes fell, people decided that they could do better teaching yoga in the gym than pulling lattes in the café, and so they exited the latte-producing business and went and taught yoga lessons instead. How many yoga lessons can they teach? We see from the supply curve that as the price of lattes falls it induces people to switch from making lattes to teaching yoga falls smoothly and linearly. Thus the average person shifts from making lattes to teaching yoga when the price of lattes falls below $5.**

**The average person, therefore, then makes as much money teaching $10 yoga lessons as they make making $5 lattes.**

**Therefore the average person can teach half as many yoga lessons as they can make lattes.**

**Thus when everyone has switched from making lattes to teaching yoga, the economy has moved from making 5000 lattes (and teaching 0 lessons) to teaching 2500 yoga lessons (and making 0 lattes).**

**Now we have both ends of the PPF.**

**What about the middle? How many yoga lessons are taught when 2500 lattes are produced?**

**To walk down the supply curve from 5000 to 2500 lattes, the price has to fall from $10 to $5/latte to induce enough people to switch to yoga to reduce the quantity supplied to 2500. That means the average person who switches in this walk-down switches when the price is $7.50/latte. That means that the average person who switches in this walk-down can make as much money teaching yoga at $10/lesson as pulling lattes when the price of lattes is $7.50/latte. The average person who switches in this walk-down can thus teach ¾ as many yoga lessons as they can make lattes. So in this walk-down the economy gives up 2500 lattes to make 1875 yoga lessons.**

**The middle point on the PPF, with latte production cut in half, is thus: (2500 lattes, 1875 yoga).**

**Can you now see the pattern and calculate the PPF?**

**Yes, you can see the pattern: When the price of lattes moves from $10/latte to $P/latte, the average person who switches can teach (10+P)/20 as many yoga lessons as lattes. As the price drops from $10/latte to $P/latte, the economy gives up 500(10-P) lattes from its maximum latte production and thus makes:**

**L = 500 - 5000(10 - P) = 500P lattes—that’s the supply curve.**

**As the price drops from $10/latte to $P/latte, the economy makes (10+P)/20 yoga lessons for each latte it gives up, and thus makes:**

**Y = 500(10-P) x (10+P)/20 yoga lessons—that’s the number of yoga lessons.**

**Algebra 1 then tells us:**

**Y = 500(10-P)(10+P)/20 = 25(100 - P2) = 2500 - 25P2 yoga lessons. And if you let P: $0—>$10 the point (L, Y) traces out the PPF.**

**Or simply note that $P = L/500, and use Algebra 1 to substitute in for the equation for yoga lessons:**

**Y = 2500 - 25P2 = 2500 - 25(L2/250000) = 2500 - L2/10000**

**and we are done with the exact PPF—indeed, very similar to what we did in 1a), only “smoother” and having maximum yoga production of 2500 rather than 2750 yoga lessons:**

**And I expected a few people to draw the supply curve, and then to say: People switch from making lattes to teaching yoga when the value of the yoga lessons they can produce with those resources rises above the value of those resources making lattes. The height of the supply curve—the price—tells you the value of the latte that the economy is about to stop producing. The value of yoga lessons produced is thus simply the integral under the supply curve as the amount of lattes produced falls from 5000 toward zero:**

**With the value of a yoga lesson set at $10, the number of yoga lessons is then:**

**Y = 2500 - 25P ^{2}**

**And Algebra 1, plus L = 500P, gets you to:**

**Y = 2500 - L ^{2}/10000**

**And my question for those of us who did this third is: Why are you here in Econ 1? Why aren’t you in Econ 2?**

It seems to me that if you are going to teach the PPF, it is only worth doing if people then understand how it relates to the capabilities, resources, and incentives of producers on the one hand, and to the supply curve which summarizes how produces in aggregate respond to market incentives given their capabilities and resources on the other. I spent more time than I had to spare on the PPF, and yet... I was not pleased with how much they learned. It seems to me that I have to chalk this down as one of my mistakes...