so in these lecture videos in problem sets and even in quizzes you probably have noticed that sometimes we assume that quantities have to be whole whole integers discrete units right you can have one cup of coffee you can have two cups of coffee you can have three cups of coffee but you can't have 2.678 cups of coffee other times we assume that quantities are continuous that any value is fine so you could have 2.678 units of whatever's for sale and what i'd like to do in this lecture video is first try and convince you that it's not such a terrible idea to think that we could potentially have right these units such as 2.678 right and also give you some guidance about right how to interpret the questions i'm asking so that you make the right assumption about right whether quantities are discrete or quantities are continuous right so the first question i'm going to ask is is it ridiculous to use non-integer values and here i'm just going to say that quantity demanded as a function of price is five minus p which of course if the price is equal to two dollars and eighty cents right this is going to mean that the quantity is going to be equal to 2.2 right now i cannot go to the coffee shop and order 2.2 cups of coffee well i actually can order 2.2 cups of coffee but they would either give me two cups or they would give me three so the first thing that i'm going to note is while i can't go to the coffee shop and order 2.2 cups of coffee what i can do is order two cups of coffee on monday tuesday wednesday and thursday and order three cups on friday which means my average quantity demanded is in fact 2.2 cups of coffee so in that case right it's not so ridiculous to think about 2.2 as the quantity now the other thing is right i've been talking about it and you almost certainly assume that we are talking about single cups of coffee but maybe the units of measure aren't single cups of coffee they are thousands of cups of coffee per day right and in that case 2.2 thousands of cups of coffee a day i eat 2 200 cups of coffee a day well that makes a lot of sense right so the other thing is is the units of measure might be something like thousands or millions and it's just easier to deal with small numbers right rather than multiply all the numbers by a thousand or divide the um slope by a thousand that gets kind of complicated all right so the next thing that i want to say is like at the end of the day these are models and we don't want to overestimate the precision of our models and even when we like have a lot of data and a lot of computing power and we estimate a model at the end of the day there's going to be a lot of error right and so at the end of the day an economist is not going to stake a claim right that the value is going to be exactly equal to 37.47 right but instead right is going to say that the answer is somewhere near this and so once again you know having these partial units right might not be such a ridiculous concept after all and finally right assuming continuous units right well that's sometimes but not always just going to make the problem more simple right if i give you an equation for supply and equation for demand right as long as the numbers work out relatively nice it's relatively easy to solve for equilibrium quantity and equilibrium price the other thing which i'm going to say is assuming discrete units well that's often not always but that's often going to make it easier for me to write test and exam questions that test your understanding of the concept as opposed to your ability to do math there's some some things to think about so one place where it matters right is right when we're trying to calculate area under a curve right and so here let's start with a discrete uh with the with the table and we're going to assume that values are discrete and when i draw it when i give you a table like that i'm implicitly assuming that values are discrete and so unit number one price equal four tells us the willingness to pay is equal to four right when the price is equal to three quantity is equal to two right quantity three price two right quantity four price one right and recall right demand just gives us a measure of benefit and if i wanted to know the total benefit from the first four units well the benefit of the first unit is four the benefit of the second unit is three benefit of the third is two benefit the fourth is one four plus three seven oh look at that i get 10. so some of you might look at that table and what you might say to yourself is self that looks an awful lot like the function that we just had on the previous page right in which case right what we have is that price as a function of quantity is going to be equal to five right minus q when q is equal to one p is equal to four when q is equal to two price equal 3 excellent and if we were to do that well we could then draw a line and so let's draw the line that we just depicted right and we can find the area under that curve and look at that we have a triangle right and if we have a triangle the area under the curve is just going to be base times height over two well the base is equal to five the height is equal to five over two is equal to twenty-five right and notice i've got a little discrepancy here when i assume that integer when the quantity had to be integers we said the benefit from all from consuming whatever the consumer wants at a price equal to zero we said the total benefit was equal to 10. in the case where quantities are continuous right we said for this consumer if we add up the benefit from all items consumed when the price is equal to zero right the area under the curve we would get oh 25 over 2 is equal to 12.5 didn't think i noticed that did you right 10 is still different than 12.5 and you're going to be like hey why is there a difference and this graph clearly shows why there's a difference and the difference is pink pink pink oh let's choose blue this area here pink pink pink think big right so there's two different ways of calculating it depending on whether we assume that the quantities are continuous or the quantities are are discrete and the area under the demand curve is going to be slightly different depending on which assumption that we make which tells us it's going to be careful and it's going to be important especially when the answer is graded by machine to make the right assumption so some final thoughts right so on test exams and quizzes this course will sometimes use continuous sometimes assume continuous and other times right it's going we're going to assume discrete quantities and as i said you have to be careful on computer graded questions i am going to do my best to be explicit when i write test que and exam questions of saying whether you should assume right that quantity is discrete or whether you can assume that quantities are continuous that something like 2.768 coffees are possible right but if i fail to explicitly say whether quantities are discrete or continuous what i can tell you is if i give you a table you ought to assume right that quantities are discrete and if i give you an equation if i give you a formula right in that case you ought to assume that non-integer quantities such as 2.768 are in fact possible the other thing that i promise that i will do right is i will try my best if i am going to assume that quantities are discrete and i give you a table i am going to do my best right to give you numbers where it is impossible for you right to have a linear to estimate a linear demand uh function right and so in this case i change the four to a five and all of a sudden the demand function is no longer linear which hopefully is a signal to you ooh maybe i shouldn't try and figure out right the continuous function associated with this table that being said have a great day see you later