NationStates Jolt Archive

This thread will be the follow up of my ”Question for the lefties” (http://forums.jolt.co.uk/showthread.php?t=524833). In it I will, over the next few days or weeks, explain what a market is, what it can do and what it cannot do. It’ll basically be an explanation of “general equilibrium theory” and the welfare theorems associated with them.

I’m by no means a complete expert in this, I’m only an undergrad student in economics. The stuff I’m talking about is the subject of one of my courses at the moment. Nonetheless, I think I’ll be able to answer most questions that come up. So if anything isn’t entirely clear to you, or more importantly, if you think anything I say is bullshit, speak up. It would be a shame for me to go through all this effort and nobody caring.

Even if at the end of this you won’t have changed any of your opinions, even if you think this is a silly exercise in abstract nonsense, at least you’ll know a little bit about microeconomics, which can’t be a bad thing.

The model I’m building up is not of any particular school of thought in economics. It’s pretty much the mainstream, and you’ll notice that the assumptions are deliberately kept extremely general. The idea is to create a model of behaviour that is so general that one cannot attack it on a psychological level. Nonetheless, it’s the assumptions that this stands and falls with, so by all means question them.

So here we go.

PART 1 The consumer

In the beginning, there was darkness…

We begin in a world where there is no market, no prices, no politics, no nothing. There are only two things in the world: a person, and commodities. A commodity is defined as a set of attributes of a product or service, such as their physical characteristics, their time of delivery, the state of the world at the time of delivery, and so on.

This person, let’s call him Stewie, can consume commodities. More generally, Stewie chooses commodity bundles. These are groups of commodities…mathematically one would call them vectors where each possible type of commodity is associated with a quantity of that commodity. So if there were apples and oranges in the world, a commodity bundle might be 1 apple and 2 oranges, which we could write as x = (1,2) if we define the first number as apples and the second as oranges, and call the commodity bundle “x”.

Any consumption bundle in the world that Stewie can consume is part of the “consumption set”. That’s not all commodities in the world, because there are some which he cannot both consume. For example, Stewie cannot both choose to go on a holiday to Brazil and a holiday to New Zealand at the same time. So only those bundles which can logically be consumed are part of the consumption set. The rest will be ignored.

If Stewie chooses between commodity bundles, there are basically four options:
He can prefer bundle x to bundle y. That means that he will choose bundle x over bundle y, and we could write x > y.
He could prefer y to x. Same deal.
He could be indifferent between the two. That means that if asked to choose between them, he will say “I don’t think one is better than the other, I like both exactly the same.” We would write x ~ y.
Or maybe he’s not able to compare them at all. He might say “Blast, I cannot choose between them.” That’s not the same as being indifferent between the two.

Note that we know nothing yet about how Stewie makes the decision. We know simply that he does make a decision, and that it can take one of these four forms.

Of course, this also tells us pretty much nothing. If we would like to know more about how Stewie picks between options, we’ll have to introduce some assumptions.

The ones we’ll pick are the following:
Reflexivity
Reflexivity simply means that one commodity bundle cannot be preferred to itself. So x ~ x must always be true.
Transitivity
We’ll assume that there are no circular preferences in what Stewie picks. If he prefers x to y, and y to z, then it must be the case that he prefers x to z.
Anti-symmetry
This one's simple as well. If we say that x ≥ y and y ≥ x, then it must be that x ~ y.
Continuity
This one’s a bit tricky. It basically says that there can’t be sudden jumps in preferences. Mathematically speaking, if we had a bundle x that was preferred to y, then another bundle which approaches x to the limit must eventually be preferred to y as well. An example of a discontinous ordering can be found in an encyclopedia: the ordering of the words goes by the first letter, then the second and so on. Which means that if the first letter becomes the same, the ordering can suddenly switch around. If Ag comes before Ca, and then Ba, but then Aa is suddenly before Ag, then that’s not a continuous ordering.
Completeness
This one can be a bit controversial. We basically assume away the “I cannot pick” option. We say that Stewie can say “I prefer this to this” or “I am indifferent between the two” for any two commodity bundles in the world. There’s been some theoretical disagreement about it, but I personally think that in the real world it holds together quite well. My lecturer has assured me though that one can come to the same conclusions without this assumption, though it’s a whole lot more complicated.

So if we have these five assumptions…we can rank all of Stewie’s choices in a long list, for example. We can put one bundle at the top, and rank all the possible bundles down to the one he hates most.

The gag with a ranking that exhibits these characteristics will be covered in the next post.

Why are fruits that grow in Florida and Washington part of the same commodity bundle?

Why are fruits that grow in Florida and Washington part of the same commodity bundle?
I'm not entirely sure what you're getting at.

If you think there is a difference between fruits from Florida and fruits from Washington, then the two might be two different commodities.

Also, for mathematical purposes, every commodity bundle will have all the commodities in it, but for most of them the quantity will be 0. If there are n different types of commodities, a commodity bundle will be an n-dimensional vector.

So if there exist only apples, oranges and pet elephants, then every commodity bundle will have three dimensions and x will be (a, b, c). Even if Stewie doesn't want any pet elephanst, it will still have three dimensions, and simply say (a, b, 0).

I'm going to attack the concept of Transitivity. It's always erked me that economics in its pursuit of the scientific methology attempts to breaks everything down into a mathematical model, and a simple one at that. In this case Transitivity seems to be suggesting that a commodity bundle can be given an absolute value.

Take holiday's as an example (please bear with me as I try to figure this out).

It can be in one of three places - France, Wales, or Egypt.
It can be as long as a - weekend break, one week or two weeks
And it can either be in the summer or the winter.

In general Person A prefers; France, then Egypt, then Wales; longer holidays; and holidays in the summer.

A is given three choices
x) 2 weeks in the summer in Wales
y) 1 week in the summer in Egypt
z) 1 weekend in the winter in France

A isn't keen on Wales and would always prefer to go to France. (z>x)

However, x is preferable to y because Egypt isn't to pleasant in the summer (it gets too hot). So though A isn't keen on Wales, he'd prefer to spend a longer holiday in slightly cooler conditions in Wales. (x>y)

A prefers France to Egypt, but can't decide between France in winter and Egypt in summer. So he prefers the Egyptian holiday to the French holiday as it is longer. (y>z)

Does that makes sense? :confused:

You may well counter than if presented with three choices he will always make a choice - a holiday is better than no holiday, right? But this presumably means A would be choosing at random, and hence would A make the same choice every time?

I've got lost in my own argument... I'll just go have breakfast now...

I don't have any particular objections to this thus far.

In this case Transitivity seems to be suggesting that a commodity bundle can be given an absolute value.
Well, you'll see that that really isn't the point. Economists these days are absolutely paranoid about assigning any absolute values to anything.

Nonetheless, you make a good point.

Take holiday's as an example (please bear with me as I try to figure this out).
You did well to figure it out, it shows you thought about it.

What you described is intransitivity, which a microeconomist would call irrational, and which can then be used to squeeze money out of him or her.

Say you have your holiday in France. I offer to swap you a holiday to Egypt and you'll pay me a cent. You take the offer.

Then you have a holiday in Egypt. So I come along and offer you the trip to Wales in return for the Egypt holiday and a cent. Again, you take the offer.

And once you have the trip to Wales, I offer you the holiday in France and again ask you to pay a cent.

Everytime you make a choice that makes you happier, and everytime you act in your self-interest by paying me that cent. And I will continue to lead you around in circles until you have no cents left.

So, does that mean that intransitive preferences don't exist? Obviously not.

But knowing that intransitivity will be your ruin, we are more or less resigned to accepting that we won't be able to apply what follows to you as an individual. Hey, we might even assume that at some point you'll learn and force transitivity upon your own preference orderings.

But in short: you're correct. There are people who don't fit the model - but there are usually reasons for why economists have not compromised on the further stages of it.

Part 2 “Utility” Functions and Indifference Curves

The Utility Function

If we look again at the properties of this preference ordering described above, we can think of other orderings that exhibit the same characteristics. Or at least one. Think.

The number system, for one thing. The numbers are reflexive (2 must always be equal to 2), transitive (if 3 > 2 and 2 > 1, then 3 > 1), anti-symmetric, continuous and complete (any two numbers can be compared).

Which means that we can use the ordering system of the numbers to describe the ordering of preferences. Of course, note that we are only interested in the ordering, not any values. We are not assigning a value to every bundle, we are assigning a place in the ordering of the number system.

A “utility function” is a function that assigns every commodity bundle a number, which we will call “utility”. Utilities are ordinal, not cardinal, meaning that we are only interested in the order, not the value. Just because x has a utility of 2 (which we would write as U(x) = 2) and y has a utility of 10 does not mean that I would be indifferent between five x and one y. All it says is that y will be preferred to x.

As a side note, this is actually somewhat reverse-engineered by modern theorists. In the early days of economics, people actually thought we could measure happiness, assign values (ie utility) to any level of happiness and then calculate what is best for society based solely on that. This is different: the utility these men meant back then is very different from what we call utility today. That’s why some economists don’t like the word “utility function” and instead prefer “preference relation” or “representation function”.

The Indifference Curve

We can observe in the real world that Stewie ends up picking a commodity bundle to consume. In other words, Stewie makes a choice and picks only one bundle out of the entire consumption set.

Obviously if he picks just one, that has implications for the possibility of being indifferent between two choices. There must be a unique choice he can make that makes him happy (or, as we can say now, maximises his utility).

So what do we know about being indifferent? At the moment, not much.

For any utility, there exists an entire set of bundles within the consumption set between which Stewie is indifferent, correct? For any choice Stewie makes, we should be able to find another bundle between which and his choice Stewie is indifferent.

Let’s call that an “indifference set”. It’s not a curve yet, because it could have any shape, hell for all we know it could be the entire consumption set if Stewie gets the same happiness from any possible combination.
So again we have a problem. We have a very general model which must hold true for everyone. But it’s also so incredibly general that it doesn’t tell us anything. So again we have to pick up the pen and start writing down assumptions, perhaps slash away at the generality of it but also gaining a whole lot in usefulness. These are the assumptions we will need:

The more, the better
This one is always going to be disputed. There are two ways of expressing this mathematically: the first is “strict monotonicity”, which essentially means that any point in the positive orthant (http://en.wikipedia.org/wiki/Orthant) of another is preferred to it. In other words, if there is a bundle that has more of all commodities, or the same of all and more of at least one, then it will be preferred to the other. Note that this only works if there is no trade-off, if the increase in one commodity doesn’t come with a decrease in another.
The other is “local non-satiation”, which says that any bundle must have another bundle in its vicinity that is preferred to it.
Both mean that there can be no zones of indifference, that the set must always be a curve.
Strict Convexity
This assumption makes sure that if we draw that indifference curve, it must be convex. It works by assuming that people prefer to mix their bundles over any extremes. On the graph, you see that either point near the axes has a lot more of one commodity than of the other. The curve is convex because we assume that if you mix the commodities and come closer to the centre, relatively less of each will give you just as much utility. So the curve is bent.
This ties in with diminishing marginal returns to something, ie the idea that if you already had 20 chocolate eggs, you’d be willing to pay a little less for the 21st than what you would have paid for the 1st. There is some evidence gathered by psycho-economists that people indeed exhibit diminishing marginal utility.

http://instruct1.cit.cornell.edu/courses/econ101-dl/images/lecture-uses-demand-tax&transfer.gif

This is a bunch of the curves that you get when you assume these two things of indifference sets. Every curve stands for a different level of utility, and Stewie will be indifferent between any bundles on them. Of course every person has their own indifference curve.

In the next post I'll introduce prices into this model and we'll see what implications that has for Stewie's choices.

The Market used to be a human named James Heward, until a massive surge of gaydiation turned him into a being of godlike proportions. He can emit blasts of gaydiation strong enough to send people hurtling into the nearest clothing store, and can direct it at homophobes to use the reversing energy to lift himself into the air. He cannot, for some reason, get a girl. No one's quite sure why.

Part 3 How to make a decision

Prices

We still don’t actually have a market. We now say that there are indeed prices out there, but we aren’t specifying who sets them and how.

So let’s say Stewie stands in a big building with a big neon sign saying all the prices for all the commodities in the world.

What is a price? It’s an amount of money associated with a commodity.

What is a price system? It is a vector in which every commodity is associated with a price. If apples cost $2 and oranges cost $3, then a price system p might be p = (2, 3), if we say that the first number stands for apples and the second for oranges.

Stewie’s budget

Let’s say Stewie has some initial endowment of resources that we will call w. w is a commodity bundle like any other, except that Stewie starts out with it.

Stewie walks into this big building, sees the neon sign, puts his commodity bundle w on the ground, and receives some amount of money for it. That money is Stewie’s budget, with which he can now secure himself some other commodity bundle that makes him happy.

On a 2d graph, the budget can be expressed as a straight line between an extreme on one axis (the quantity of commodity 1 if he spent all his money on it) and the other extreme on the other (same deal with commodity 2).

Stewie can consume any combination of resources of which the total value is less than or equal to the total value of his initial endowment.

So p.x ≤ p.w, where p is the price system, x is the bundle Stewie chooses, and w is Stewie’s initial bundle. The dot represents the dot product (http://en.wikipedia.org/wiki/Dot_product) of the vectors (ie each number in the price system gets multiplied with the respective quantity in the commodity bundle).

Stewie’s decision

Given that in both the budget and the indifference curve we’re expressing things in terms of commodity bundles, we can put the two on the same graph, with the commodity quantities representing the axes.

If we assume that Stewie wants to maximise his utility with the choice he makes, it should become obvious that the higher the indifference curve, the better. So he will make the decision that allows him to pick a point on the highest indifference curve he can get onto.

And because our indifference curves are convex, we can see that there will be a single point at which the outer limit of the budget set (ie the budget line) will form the tangent of the indifference curve, at which the two will only just touch at one single point.

Mathematically, this will be where the slope of the indifference curve (ie its differentiatial) will be equal to the slope of the budget line.

http://cepa.newschool.edu/het/essays/paretian/image/pareto1.gif

This is the whole thing graphically. We know that Stewie won’t pick F, because he can get a greater utility than that, since we assumed strict monotonicity before: there are points to the upper right of it which he can still afford.

He won’t pick H either, because even though it uses up all his budget, by taking a less extreme bundle and mixing commodities more, he could make it onto a higher indifference curve.

As much as he’d want to, he can’t pick G because it is higher than his budget line and he can’t afford it.

Which leaves only E. It’s as high an indifference curve as he can make with his budget, and there are no points to the upper right of it that he can afford. So E is his utility maximising commodity bundle, given a budget constraint.

And that’s it. We now know how Stewie makes a decision. We haven’t assumed all that much about why he makes it, we haven’t assumed all that much about Stewie himself, and we haven’t assumed anything about the environment or economic system in which he lives.

There are lots of twists on this, particularly if you introduce uncertainty about future states of the world, or say that Stewie doesn’t pick from fixed bundles, but rather from uncertain lotteries, but that doesn’t really help me make my ultimate point, so I’ll leave it aside.

But now that we have a consumer who makes decisions about the resources he wants, we can start thinking about resource allocations and interactions between different people, the start of which will be covered by my next post.

Are you sure that you shouldn't call this thread microeconomics? This is really just my micro class being put on a forum.(well, without the supply and demand curves I suppose)

Are you sure that you shouldn't call this thread microeconomics? This is really just my micro class being put on a forum.(well, without the supply and demand curves I suppose)
Supply and demand curves carry with themselves more assumptions, which I'm trying to keep to a minimum.

You'll see where I'm going with this. At the moment I just established the framework, and next I'll start getting to the interesting stuff, ie I actually start talking about the market as a way of allocating resources.

Supply and demand curves carry with themselves more assumptions, which I'm trying to keep to a minimum.

You'll see where I'm going with this. At the moment I just established the framework, and next I'll start getting to the interesting stuff, ie I actually start talking about the market as a way of allocating resources.
Ah, yes, I suppose not having them makes sense.

Well, I suppose I will have to wait to see all of this unfold, but I still somehow get this feeling that this is going to be micro economics even if it does lack a few assumptions.

Ah, yes, I suppose not having them makes sense.

Well, I suppose I will have to wait to see all of this unfold, but I still somehow get this feeling that this is going to be micro economics even if it does lack a few assumptions.I'm just waiting for something to argue against. :)

i hated how much maths there was studying economics at uni. same reason i gave up reading the OP just now

Well, I suppose I will have to wait to see all of this unfold, but I still somehow get this feeling that this is going to be micro economics even if it does lack a few assumptions.
It is microeconomics. Just that rather than simply presenting you with graphs and curves like in first year, I'm building them from the ground up, to make sure that everyone can follow every step.

Sorry about the break, I was busy on other things for a while there. But since I've got an exam on this stuff tomorrow morning I figured I might as well get it going again.

As Holyawesomeness said the stuff I've been going on about so far isn't revolutionary, nor does it have great implications for the debates going on on NSG. Nonetheless, I think it was important to illustrate how we can model people's actions with maths. If I had just launched into the next topics people would have immediately questioned how you can use numbers or algebra to talk about something as complex as human behaviour.

Part 4 Allocations and how to rank them

Allocations and the ownership economy

Now we finally move beyond Stewie in his bubble and add more people. The assumption of our economy for the time being is there are many consumers, such that the decisions of any one of them have no impact on prices. Everyone is a price taker. This isn't necessarily absolutely vital for now, but it makes it easier. I'll also keep things simple by only adding one other person, but obviously the process is no different if you add millions.

So what is an allocation? It's basically a vector that associates with each person a commodity bundle. If we say that there are two people in the world, Stewie and Brian, and two commodities, apples and oranges, then an allocation x would be x = (Stewie's commodity bundle, Brian's commodity bundle) or x = ((2 apples, 3 oranges),(3 apples, 2 oranges)).

Of course we're assuming that apples and oranges are actually owned. This is an ownership economy, meaning that every commodity in the economy is owned by someone. It's obviously the basis for the market we'll construct soon - but if you're not a fan of property rights, you can still think of allocations just as the things people consume which then aren't available for others anymore. Even a communist utopia without property rights would exhibit the characteristic that if you eat an orange, no one else will be able to eat that orange.

Ranking allocations

Of course that's simply a descriptive statement. The real question is how do we tell whether one allocation is better than another one?

The short answer is that we can't. Some might think that everyone having the same would be perfect, someone else wants Stewie to get more (or all). However, there is a way to at least exclude some possible allocations.

The first is by excluding all the allocations that aren't feasible - meaning all allocations that have in them greater quantities of commodities than are available in the economy at this point. If there are only 2 oranges and 2 apples, then an allocation x = ((10, 5),(77,9)) isn't feasible and can be disregarded.

The second is through the concept of "Pareto Optimality".

Pareto Optimality

This concept was developed by a guy called Pareto, hence the name. There are many misunderstandings of it out there, so let me clear them up:

If we know that something is pareto optimal, that doesn't tell us a whole lot. If we know that something is not pareto optimal, that does!

An allocation is said to strictly pareto dominate another allocation if it makes everyone better off. So keeping our assumption of "more is better" from before, x = ((5,3),(4,2)) is better than y = ((4,1),(4,0)).

An allocation is said to pareto dominate another if it makes at least one person better off. So x = ((5,3),(4,2)) is better than y = ((5,3),(4,0)).

An allocation is pareto optimal if there exists no allocation that pareto dominates it.

You can tell that an important characteristic of such an allocation is that you cannot make anyone better off without making someone else worse off. Hence the fact that knowing that something is not pareto optimal is important: If it isn't, you could make someone (maybe even everyone) better off without making anyone worse off. Pareto optimality is therefore a measure of efficiency as much as anything else. If we can allocate the commodities in the economy in such a way that no one gets hurt but someone's situation is improved, that obviously tells us that the previous allocation wasn't an efficient use of the commodities. There must have been some waste of some sort.

Individually Rational Allocations

It's actually quite easy to come up with a pareto optimal allocation. How about "Stewie gets everything and Brian gets nothing"? In that case you couldn't make anyone better off without taking something away from Stewie, so it's pareto optimal.

Obviously that allocation isn't particularly desirable though, hence the fact that pareto optimality alone doesn't tell us much in terms of rankings.

The concept of individually rational allocations comes to help. Let's assume that there exists some initial allocation of resources when we "start" the economy, which we will call w. w consists of the initial endowment of Stewie and the initial endowment of Brian.

An allocation is said to be individually rational if no person in the economy would be better off by simply not taking part in the economy and just sticking with their initial endowment.

We can see that Stewie getting everything isn't individually rational. Brian would simply not want to be part of that economy and instead consume his initial endowment.

Experimental economics show preferences contain systematic and significant intransitivities. Classical microeconomics may fall over at step 1.

On the other hand, assuming the convexity of preferences may prove an adequate substitute for the transitivity assumption. The jury remains out; but very busy.

In the end, I think the seperation between market and non-market outcome are:
-Markets have efficient but arbitary outcomes
-Planned economic outcomes are not arbitary, but cannot be efficient

Markets add +25% income and +1 happiness if you have access to ivory or gems.
That's all I need to know.