so now let's think about the ricardian model we just did an example of it but let's think about uh developing it

generally so it's actually very simple we're going to have a country that we're going to call home okay and for that

country we're going to think about something that we're going to call the unit labor requirements so our unit

labor requirements we'll use these in a couple of different

models but our unit labor requirements we're going to designate this way a l our our good that home is going that

we're going to have two goods we're going to be producing wine and cheese so we're going to have a unit labor

requirement for both wine and cheese for home so let's call this one alw that's going to be our unit labor requirement

for home to make wine okay so this is our unit labor requirement for wine the L here stands for labor the W

stands for wine we're also going to have a unit labor requirement for cheese

okay these are just like the numbers I gave you in the first table with the farmer Rancher example so if we were

thinking about the farmer and we thought about the number of minutes that are required to produce meat and the number

of minutes required to produce potatoes all I was giving you were the unit labor requirements now I know that it it for a

lot of students it feels much more comfortable to have 60 Minutes or one hour in there instead of this thing but

I don't want you to be confused or get uncomfortable with this because what I would do if I give you a problem is i'

just tell you that this is equal to two or it might be equal equal to 1/2 so this simply tells you the number of

hours needed to make a unit of the good okay so it's not don't don't be confused or intimidated by it we're also going to

designate the amount of Labor that we've got by L this is going to be the total labor labor

Supply now I also gave you that in the farmer Rancher example by telling you the number of hours that they were going

to work so in our second table we had the amount that they could produce in an eight hour workday so there the amount

of Labor that both the farmer and the Rancher had was eight hours it's not the case that the two

people or the two countries have to have the same amount of Labor one country could have more labor than the other so

what we'll see is the we'll talk about what happens when they're different and it's it's not hard at all so again don't

don't be intimidated by any of this but that's L is going to represent the total labor Supply we're going to think about

the total production of wine and cheese we're going to designate those as qw this will be total production of wine

production of wine and QC is going to be production of

cheese so now let's think about how much wine and cheese our country here home can produce if that's

all they produce so the total amount of wine that they produce is going to be qw if we think about the amount of Labor

used to produce that wine it's going to be the unit labor requirement for wine multiplied by qw so the amount of Labor

used let's put it up here amount of Labor

used to produce say wine is going to be the unit labor requirement multiplied by the number of

units of wine produced so this is for wine so let's think about what that means let's suppose the unit labor

requirement is one that means that home needs one hour or one unit of Labor to produce a unit of wine and if

they produce five units of wine then 1 * 5 would be five they need five units of Labor to produce that much wine the

amount of Labor used to produce cheese is going to be the unit labor requirement for cheese multiplied by the

amount of cheese produced so that's the amount of Labor used to produce cheese okay and again this is just going to be

one number multiplied by another number now we can't use more labor than we've got home has a certain amount of Labor

given by L they can't use more than that here's the amount of Labor used to produce wine and the amount of Labor

used to produce cheese so if we add these two up we get the total amount of Labor used so if we

take the amount of Labor used to produce cheese plus the amount of Labor used to produce

wine that has to be less than or equal to the amount of Labor that home has okay so

this equation this is actually the equation for the production

possibilities Frontier okay so this is let's write that this is the equation

for the ppf and you can easily do this for the farmer Rancher example as a matter of

fact I'll do it in uh a little bit later on we'll just write out exactly what we're doing here for the farmer Rancher

example and you'll see that it's actually very simple it looks more complicated here than than it actually

is what we want to do is we want to graph this production possibilities Frontier now in order to graph it what

we need to do is we need to take this equation and we need to put it into slope intercept

form but the way that we're going to do that is we need to decide which good we're going to put on the horizontal

axis and which good we're going to put on the vertical axis so let's draw this I'm going to draw it right up here let's

put the quantity of cheese on our horizontal axis we'll put the quantity of wine on our vertical axis

now if we're going to put something into slope intercept form what we need to do is we need to solve for whatever we've

got up here on our vertical axis we need to solve for qw so I need to do a little bit of

algebra here to solve this thing for that okay so what I'm going to do is I'm going to move this thing to the other

side so let's put here it's going to leave me with uh the amount of Labor used to produce wine

has to be less than or equal to l minus this term what I need is that by itself so

I'm going to divide both sides by the unit labor requirement for wine so that's going to leave me with this qw

less than or equal to l divided by the unit labor requirement for WI minus the unit labor requirement for

cheese divided by the unit labor requirement for wine multiplied by the quantity of

cheese so now I've solved this equation for the production possibilities Frontier for what I have on the vertical

axis okay and if it's been a while since you've done this just think about this it what we've got here is qw and QC

but if you had say something you're probably more comfortable with y and x and you had

some function if you wanted to graph that function what you would need is you

would need y = mx + b that's the slope intercept of the for intercept form of this line and you need to have y by

itself okay then here here's your slope there's your vertical intercept and so all we're doing is we've replaced our Y

and our x with q's but I still need to solve for whatever variable I've got on the vertical axis there so that's why I

solved it for qw but now we can easily graph this if we graph it here is our

slope it's the thing the term that's multiplied by the other variable and here's our intercept our vertical

intercept our vertical intercept is going to be L divided by the unit labor requirement for

wine now that end point tells us the amount of wine that we can produce in home if that's all we produce so let's

do a quick example let's suppose that that we've got um say 8 hours for our total labor Supply suppose

l is equal to 8 and let's suppose that it takes 1 hour to produce a unit of wine so a LW is one then if we've got

eight hours and it takes an hour to produce each unit of wine if that's all we produce we could make eight units of

wine it would be 8 divided by one or if we had eight units of Labor and it took two hours to make each unit of wine then

we'd be able to make four 8 divided two would be four okay so we did the same thing with the farmer Rancher example

except we just had a number there here this is what that number looks like though if you actually do it

algebraically we're going to have a line This is linear and the slope of that line is

right there it's going to be equal to our slope is going to be equal to the ratio of the unit labor requirements

it's the unit labor requirement for for cheese divided by the unit labor requirement for

wine you can verify in the farmer Rancher example we'll do it a little bit later that all you have to do is take

the ratio of those numbers that I gave you in the first table and you'll get the slope of that production

possibilities Frontier this end point down here is going to be the amount of Labor we've

got divided by however much time it takes to produce each unit of cheese so it's the unit labor requirement for

cheese so right there is a general version of what the production possibilities Frontier looks like it's

not hard to derive we're just using this statement that tells us that we can't use any more labor than we've actually

got okay but this is exactly what we did with our farmer Rancher example our opportunity cost

then the opportunity cost of the good on the horizontal axis is given by this term it's going to be equal to the unit

labor requirement of the good on the horizontal axis that's in our numerator

divided by the unit labor requirement for the other good there's our opportunity cost so you

can start with the unit labor requirements for home and you can immediately our unit labor requirements

are right here you can immediately figure out the opportunity cost you don't necessarily have to graph the

production possibilities Frontier although I would if I were working on a problem so this production possibility

Frontier tells us that they're capable capable of producing any good on the ppf or inside

the ppf that's what this inequality tells us you can produce anything less than or equal to a quantity that lies on

that line what we need to do now is we need to think about um what I'm going to call

an ISO value line and that's probably something that you've never seen before at least you you may not have seen it

referred to as an ISO value line you probably in in a class or two have talked about that so it it won't be

something completely new but the terminology is going to be new so I need to erase this and then we'll think about

what an ISO value line looks like so let's think about what an ISO value line is if you think about the word ISO value

ISO means same and then value same value so what this is going to be is going to be a

line that represents combinations of output with the same value now the production possibilities Frontier is

also an ISO value line we don't call it that early on because it it would just be too confusing but remember that

a production possibilities Frontier represents all of the possible combinations of output that you can

produce using the same amount of Labor so what I just had up here was an ISO value line that represented all

combinations of wine and cheese that used L units of Labor so it's the exact same idea if you've seen a let's say

you've had your intermediate micro class and you talked about a budget constraint a budget constraint remember represents

all the combinations of the two goods that you can buy that costs whatever income you've got so that's also an ISO

value line so you've seen it before you just may not have heard this term so let's talk about the value of

production so what we have so far is we've talked about a production possibilities

Frontier and that shows all the possible combinations of the two goods that a country or a person can produce but then

the next question is which bundle do you actually produce now in the farmer Rancher example the way we solved that

was that we just said okay let's suppose the Rancher proposes a particular bundle to the farmer and then the farmer agrees

to it in the real world that's not typic Al how it's going to work in the real world we know that things are determined

in markets and so the bundle that you actually choose to produce is going to

be dependent upon how much you can sell each good for in the market and so you want to when you're making that decision

you want to choose the bundle that maximizes the value of what you're selling okay so you're going to maximize

the value of what you're producing so let's write this out the value we're going to designate that as V the value

of what you produce is going to be equal to the number of units of say wine that you produce times the price of

wine plus the number of units of cheese times the price of cheese so here's the amount of money you make off of the wine

you produce and here's the amount of money you make off of the cheese you produce and if you add those together

that tells you the value of what you've produced and your goal is to maximize that amount

that you get from selling the wine and the cheese now what we want to do is we want

to graph this and so we're going to be graphing it in this space I'm going to uh let's put it right over here I'm

going to graph it with qw on our vertical axis that's just like the picture that we drew with

our production possibilities Frontier we're going to to have qw on our vertical axis quantity of cheese on our

horizontal axis we want to draw it in that space okay so we're going to do the same thing that we did for that ppf if I

want to draw this it's not in slope intercept form I need to put it in slope intercept form so that I can draw it on

here which means I need to solve for that I need to solve for the quantity of wine that term right

there so let's do that I'm going to do a little bit of ranging I'm essentially going to move this term over here I'm

going to subtract it but I'm going to reverse the terms so what I'm going to get is

qw divided by the price or excuse me multiplied by the price is going to be equal to V minus the quantity of cheese

times the price of cheese so all I've done is subtract this term from both sides okay now I want this by itself so

I'm going to divide by the price of uh wine that leaves me with qw on the left side I get V divided by the price of

wine minus QC excuse me minus let's put our price there minus

PC over PW * QC so now I've

got quantity of line by itself and I've got my intercept and I've got my slope and you should see

that that in a lot of ways is very very similar to um the production possibilities Frontier that we just

graphed it's also very similar to what the equation of a budget constraint would look like in that case you would

also have prices but that V would be income but other than that it looks very similar hopefully to something you've

already seen so let's graph this our vertical intercept is V divided by the price of

wine it's downward sloping the inner or excuse me the slope here is the relative price it's p the

price of cheese divided by the price of wine we call that the relative price of chees

and then we don't have it here but if you were to solve for our horizontal intercept it would be V divided by the

price of cheese would look like that so there's an ISO value line that represents all combinations of

wine and cheese that are of the same value okay this bundle has the same value as that bundle the same value as

any bundle on here now a bundle that's back here would have a smaller value and a bundle out here would have a higher

value but all of the L all of the bundles on that line have the same value so it's an isov value

line now what we're going to be doing is we're going to be using this idea of an ISO value line to figure out which of

the possible bundles that the country can produce is the one bundle that they're going to choose so let's remind

ourselves here that an ISO value line isov value line farther from the origin has a higher value that's

important okay so any ISO value line out here here for example if I were to draw an isov value line right out there all

of the bundles on that isov value line would have a higher value than these bundles would on this ISO value line and

you can see that easily if we were to increase V then both of these intercepts would move out and the iso value line

would shift out in to the Northeast direction if we were to decrease V then this both intercepts would move towards

the or and our ISO value line would move back in this direction okay so the country or the person wants to get to

the iso value line that's farthest from the origin that's going to be the goal so let's do a quick example let's

suppose that I occasionally will use symbols in this class some of you will have seen them before that symbol of an

S with a little line through the bottom is the symbol that that I'm going to use for the word suppose and I'm going to

use that um frequently so let's suppose that V is equal to 20 um let's suppose the price of cheese is equal to four and

the price of wine is equal to two Okay so we're just going to pick some prices and we're going to think about a value

and then let's put that into this function so our value is equal to 20 that's equal to um let's do price of

wine first it's going to be equal to 2 * qw + 4 * QC so all I've done is plug these numbers into there now I want to

graph this so what I need to do is I need to solve for qw so let's go ahead and do that 2 qw is equal to 20 - 4 q q

c and then I'm going to divide by two so qw is equal to 10 - 2 QC so there's an ISO value line that

represents all of the bundles of wine and cheese that have a value of $20 and if I were to draw this then it

would look be very easy to draw let's put our quantity of wine again on our vertical

axis quantity of cheese on the horizontal axis I'm going to have I'm going to be careful about my units here

one two three four five six so there's 10 which is my intercept my vertical

intercept so right up there's where this ISO value line starts it has a slope of -2 so I go down two over one down two

over one by the the time I've gone down 10 I will have gone over five so there's the other end of that

ISO value line it's linear and it has a slope of two or -2 right which is the ratio of our two

prices it's the price of cheese divided by the price of wine four divid two is two so there's an ISO value line if I

were to increase V from say 20 to 30 30 and then draw that isov value line it would be somewhere to the northeast of

this one or if I were to decrease the value to say 10 and draw that isov value line it would shift down in this

direction to the southwest of that one so that line represents all of the possible bundles of wine and cheese that

have a value of 10 let's think about how each of these or excuse me have a value of 20 let's think about how each of

these end points works so this endpoint right where the country is producing 10 units of wine and no cheese so 10 units

of wine times the price of wine which is two is equal to $20 so the value of what they produce is 20 if we look at this

end point this endpoint they're only producing five units of cheese but cheese has a price of $4

which is equal to 20 so both of these end points have a value of $20 you could choose any other point on here and if

you take that quantity multiply it by the appropriate price add to it the other quantity multiplied by the

appropriate price you'd see that any bundle on that line also has a value of $20 it's a line of equal value

okay now what we want to do is we want to take this concept of an ISO value line and we want to figure out how that

helps us determine which production point the country is going to choose so in order to do that I need to clear the

board here and uh start with another picture so let me do that so now what we want to do is we want to use the output

prices and the idea of this ISO value line to figure out the bundle that our country is going to choose so let's

think about a production possibilities Frontier so let's suppose we have a production possibilities Frontier that

looks like this let's have our quantity of cheese down here quantity of wine up here I'm going to uh be a little bit

careful about my units here so three four five six let's have units up to 10 same thing

here one two three four so I'm going to use that nine and one two three four five six six let's

suppose we had a production possibilities Frontier that looked like that okay so this this is a production

possibilities Frontier so what this tells us is is that this country or this person can

produce any bundle on or inside of this Frontier they could produce nine units of cheese and no wine or they could

produce six units of wine and no cheese or they could produce any bundle on that production possibilities Frontier but

what we want to do is figure out well what's the bundle they are going to produce now the bundle that they're

going to choose to produce is going to be dependent upon the price of cheese and the price of wine and we're going to

use this idea of an ISO value line to help us figure that out out so let's suppose that the price of cheese is

equal to four and let's suppose the price of wine is equal to two

Okay so what we want to do is we want to consider a couple of different points let's just pick a point on this

production possibilities front here let's suppose that they we start with this point where they can produce two

excuse me four units of wine and two units of cheese that would be a point on that production

possibilities Frontier okay let's let's call that point a now what we want to do is we want to

look at the iso value line through that point now our ISO value line we just saw is going to have a slope equal to the

price of cheese divided by the price of wine so the slope of the iso value line is going to be the

price of cheese divided by the price of wine it's going to be equal to 4 over 2 which is two the slope of our ISO value

line is two now the slope of this production possibilities Frontier is going to be

six over 9 right it's going to be 2/3 so our ISO value line that represents the relative price of cheese is going to

have a steeper slope than that production possibilities Frontier let's suppose that we

draw the slope of the iso value line through point a it's going to have a slope of -2 so it's going to be

something similar to that that slope of this little line segment right here represents the slope

of the iso value line now now let's consider a different point let's suppose we think about a

point down here where they produce two units of wine and it's going to be what six

units six units of cheese that would be another point on that production possibilities

Frontier let's call that point B if we were to draw the iso value line through point B it would also have a

slope of two or -2 it would look something like this so these two segments right here

are going to be parallel to each other both with a slope of -2 and what we see is that this isov value line is farther

away from the origin than this isov value line meaning that if this country has a choice between a and b and it

wants to Max maximize the value of what it's selling they would rather produce and sell this bundle than they would

that bundle bundle B has a higher value than bundle a okay we could calculate the value of these two bundles and you

can verify that producing four units of wine and two units of cheese at those prices has a lower value than this

bundle B so they would rather be at bundle B than bundle a but it turns out there's an even better bundle isn't

there you can see that there would be a bundle right down here with the same slope it's not quite parallel but

pretend that it is that bundle right down there where they produce only cheese has a higher value than either of

these two bundles A or B so what we see is that the two prices here are output

prices that are going to be determined in the World Market we'll talk about that here in a little bit those two

prices are going to be the way that each country decides what they're going to produce now this shouldn't be surprising

to you because you already know from the farmer Rancher example that you're going to tend to specialize you're going to

tend to produce one good or the other in our farmer Rancher example the Rancher didn't we'll talk a little bit about

that later actually in example the gains from trade would have been even bigger had the Rancher specialized completely

in meat so what we've got here is now a way of figuring out which bundle the country each country is going to produce

as long as we know the output prices that they face what we need to do is figure out how these prices get

determined but that will be easy as with most things in economics it's going to Bo oil down to demand and Supply so

um it's it'll be something you've seen before so we'll erase this now and get back to our uh example of our our

production possibilities Frontier so let's think about now um how much

each input or how much the input labor is going to be paid in this model that we're developing so our only only input

in this particular model is going to be labor we'll have other models where we have a couple of inputs later but for

now all we've got is labor and labor is going to move to whichever sector either the wine sector or the cheese sector in

which it's paid the most so we've got here PC which we're St letting designate the price of

cheese and then we've got PW which we're going to designate as the price of wine now one of the basic things that we

know about microeconomics is that the hourly wage for an input is going to be equal to the value of what that input

can produce in an hour so we need to think about in this model what that looks like so we've got our unit labor

requirements remember that um our unit labor requirement for cheese this represents the number of hours needed to

produce a pound of cheese or a unit of cheese so this is number of hours to produce one

unit of cheese so if we take the inverse of that that's going to be the number of pounds

that can be produced in an hour or the number of units that can be produced in an hour one over that unit labor

requirement that's the number of units produced in an hour and let's think about why that

makes sense okay let's suppose that the unit labor requirement for cheese is two so that means that it takes 2 hours to

produce a unit of cheese well if we want to know how much cheese can be produced in 1 hour then if we take the reciprocal

of that the reciprocal of two would be 1/2 that means in each hour they can produce a half unit of cheese okay so

it's a little confusing when you think about the unit labor requirement and it's reciprocal but if you just pick an

example um here's the simplest example if this was one it takes one hour to produce a unit of cheese then each hour

they could produce one unit of cheese okay because the reciprocal of one is one so now if we wanted to know the

value of what this input can produce in one hour here's the amount that they can produce in one hour here's the price of

it so if we take the price and multiply it by the amount they produce p * one over the unit labor

requirement this product is going to be the value of what a worker can produce in one hour this is

the value of what a worker can produce in one

hour now this is important because we're going to use this a lot I'm going to write it this way I'm just going to

write p over the unit labor requirement okay so that's that's going to be the

value of what a worker can produce in one hour if they're producing cheese this is going to be let's make

this PC because we need to multiply it by the price of cheese if we wanted to know the value in

terms of wine of what a worker could produce in an hour th this would be the hourly wage in the cheese sector hourly

wage in cheese sector and if we wanted the hourly wage in the wine sector it would be the price of

wine divided by the unit labor requirement for wine that would be the hourly wage in the wine

sector so now let's consider a situation where wages in the cheese sector are higher than wages in the wine

sector so we just want to think about what would happen in that case so let's suppose wages in the cheese

sector wages in cheese sector are higher than the wine

sector let's think about what that would imply let's write out that statement so wages in the cheese sector here's the

hourly wage in the cheese sector are greater than wages in the wine sector now let's do a little bit of

rearranging there I'm going to divide both sides I'm going to move this PW down here and I'm going move that unit

labor requirement for cheese right up there so since this is an inequality I can do that without changing inequality

or anything like that so so I can rewrite this

equivalently so there's an equivalent statement to this one okay so if wages in the cheese sector are higher than

wages in the wine sector then that tells us that the relative price of cheese is higher than this slope right here now

remember that this is the slope of the production possibilities Frontier and this is the slope of that ISO value line

that we just talked about okay so this tells us that if wages in the cheese sector are higher than the wine sector

then workers are going to move to the cheese sector and we would specialize in cheese specialize in cheese now that's

exactly the what we saw when when we were talking about that example with the iso value line if the slope of that ISO

value line is steeper than the slope of the production possibilities Frontier then the bundle that produces the

highest value is going to be down there in that bottom right hand corner where they produce all cheese so this isn't

anything new all we're doing is we're using what we know about Labor and the fact that labor will move to whichever

sector is paid the most in to understand why the economy would specialize in cheese if the value relative value of

cheese is higher than the opportunity cost of producing cheese okay so this is a key

result and it's something we're going to use a lot what it tells us is that if the

relative price of cheese is higher than the opportunity cost of cheese that economy will speci specialize in cheese

the other side of that coin is if the relative price of wine is higher than the opportunity cost of producing wine

then the economy will specialize in wine and we didn't draw that picture that would be a picture where the iso value

line has a smaller slope than the opportunity cost of the um of producing cheese so we could go through that and

we will in a later example but you're going to specialize in cheesee if the relative price of cheese is higher

than the opportunity cost of producing it here's another thing that we need to to kind of finish this section with and

that is in the absence of trade the relative price of cheese is going to be equal to the opportunity

cost of cheese because there's no other place to purchase cheese from in the absence of trade in the autarchy

situation there's no other place to buy cheese or wine and so the relative price that

you're going to pay for them is going to be equal to your opportunity cost of producing

them what we need to do now is is go back to that General trade model and we've we've set this this model up for a

single country for home now let's add another country which we'll call foreign and then we'll kind of work through what

the trade model looks like we'll see where these prices are uh determined and then once we see that we'll have kind of

the whole model to work with