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RANGEL: So far we have looked at very general forms of the maximization problem. One

useful thing, however, is that when one looks at maximization problems within the

context of economics, the problems have a specific structure that allows the

solutions to have a form that has a lot of economic intuition. So what we're going to

do for the next few minutes is add this economic structure into the problem to bring

out that economic intuition.

Let's take a look at what this additional structure look like. Remember that the

maximization problem that we have been studying so far it looks like this. Maximize

a function T of x over the level x of the action that you can take over its entire

domain from minus infinity to plus infinity. The additional economic structure that

we're going to impose has three key properties.

The first one is this one, which states that the objective function T of x can always

be written or can always be split into two components, a benefit function and a cost

function. The idea is extremely simple and very intuitive. Which is that, every time

you take an action x it is going to generate a benefit and it is going to generate a

cost. And that the total payoff for the economic actor-- a consumer, a firm, whoever

it is-- of taking the action is going to be the total benefits at level x minus the total

cost at level x. And that adds to the total payoff, T of x.

I want to give you a couple of examples to convince you that this structure is really

at the core of many, many economic examples of interest. But I won't actually argue

to most of the economic applications that we're going to do in this course.

So consider first an example of a firm. In this firm-- produces some sort of productand the level x just denotes how much to produce. The benefit is just the revenue of

selling x units. The cost is just the cost of producing these units-- x units-- basically,

the cost of production. And T of x is equal to the profit of selling those x units, which

is going to be just the revenue of x minus the cost of x-- I'm sorry-- minus the cost of

x which has this structure benefit of x minus cost of x that is assumed here.

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Now consider another prototypical problem that we're going to encounter time and

time again in this course, which is the problem of a consumer that has to decide

how many units of a good to buy in a particular market. And let's say, for example,

how many units of computing power to buy. How big a computer to buy. In this

case, x is the size of the computer-- how many units of computing he's buying. And

think of x equals 0 as no computer.

And B of x is going to be the benefit for the consumer-- the utility of usage-- for that

level of computing measured in dollars. C of x is the cost, or the market price, of

that amount of computing. Again, measured in dollars. And you can think of T of x

as the net utility for the consumer of buying x. Again, given by the benefit minus the

cost and measured in dollars.

So once more we see that the problem has a very natural structure that divides the

total well being that the agent is trying to maximize into a benefit minus a cost. Just

as is assumed in the structure that we have imposed.

The second assumption, or a structure that we're going impose, is very simple. It

just says that the action has always to be non-negative. In other words, the level of

the action x cannot take a negative value. Normally, if x is equal to 0, this is

interpreted as take no action. And then, as x increases, that is interpreted as take

more of the action.

Now, this should be fairly intuitive why this assumption is part of many economic

problems given the previous two examples. A firm cannot produce a negative

number. A consumer cannot buy a negative level of computing. The smallest action

that they can take, in both cases, is 0.

Now, the third assumption is perhaps the most valuable of all from a mathematical

point of view, but also the less intuitive. So let's take some time thinking about it.

First, let me just describe it to you.

Graphically, what it requires is that the function T-- which, of course, is given by the

sum of a benefit minus the cost-- but that the function T over the domain of the

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function that is relevant, which is x is greater or equal to 0, be a strictly concave.

And that basically says that somehow it looks like an inverted ball. Think of this as a

ball, think of the water being there, and the ball is inverted because it's facing down.

If you care about a precise, mathematical definition, it means for a differential

function that T prime-- the second derivative, I'm sorry, T double prime-- at x is lessthan 0 for all x greater or equal than 0.

Now, you may be wondering, why in God's name does this total benefit function T of

x is strictly concave in many economic problems? What does the economy has to

do with an inverse ball or with a second derivative. Bear with me a second. It's

actually pretty intuitive once you see what's going on.

Remember that by the first property of the structure that we have imposed-- the netpayoff function T of x-- is equal to a benefit function minus a cost function. Now let's

look at each of them in turn, the benefit and the cost. It turns out, that in many

economic problems, the benefit function looks like this. So if this is x and this is 0, it

has a shape that looks like this. Approximately.

The key properties is that the derivative is greater than 0, and the second derivative

is less than 0. In other words, every additional unit of x that individual x problem

gets or does, provides some additional benefit but at an ever decreasing rate,

because the second derivative is less than 0.

Now many of these economic problems also have the following cost function.

Something that looks like this. Now, the key properties here-- remember, this is x,

this is the cost function-- is that C prime of x is greater than 0, and C double prime

of x is also greater than 0. The key idea here is that every additional unit that gets

done increases the cost, and that every year unit increases the cost at an

increasingly rate. So buy more of the previous unit increases the cost.

For example, here the first unit increases the cost by this much, but the second unit

increases the costs by a little bit more, the third unit by a little bit more, et cetera.

That comes from the properties of the second derivative.

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Now look at what happens. Suppose that you are in a situation in which the natural

economic structure has benefits and cost functions that looks like this. And in

particular, it is the case that the second derivative of the benefit function is less or

equal than 0, the second derivative of the cost function is greater or equal than 0.

And it is the case that not both of them are 0. So either this one is strictly less than0. Or this one is greater than 0. Or both do not hold with equality.

If that is the case, notice what happens. We know that strict concavity requires the

second derivative to be less than 0. But we know that the second derivative is going

to be the sum of the second derivatives of B and C because of the additive structure

of the objective function. But we know that the second derivative is going to be less

or equal than 0. By this assumption, we know that the second derivative of the cost

is going to be greater or equal than 0 by this assumption. But there is a negative

sign there, so that means given that not both of them are 0-- this is less than 0-- ie,

it is strictly concave.

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