Budget Constraints and Consumer Optimization

let's talk about what a budget constraint represents so the objective of the of the consumer is going to be to maximize utility now that we understand what an indifference curve is we know that the objective of the consumer is going to be to reach the indifference curve that is farthest away from the origin in other words to reach the highest elevation of utility that they possibly can but the constraint that they have is that they can only spend the amount of income that they've got so we're going to be thinking about um let's talk about the assumptions we're going to make here in terms of our budget constraint we're going to be assuming that the prices are not going to be changing for the consumer as they as they try to decide which bundle to choose the prices are going to stay fixed for this short period of time that they're trying to figure this out we're not assuming that prices never change but for right now for this point in time the consumer knows what the prices are and they're not going to change the consumer is also going to have a particular amount of income the income that they have isn't going to change at this point in time it could change in the future but not right now and then we're going to think about the consumer not saving or borrowing we could come up with complicated models where the consumer can save some of their income for later but we're not going to have them do that at least to begin with okay so the consumer has no reason to save any of their income so their budget constraint let's think about what the budget constraint looks like the budget constraint simply says that you can't spend more money than you have okay and there are only two things to spend money on good X and good y so the budget constraint is going to look like this I which I'll use to designate income your income the amount of income that you have has to be spread between the amount you spend on good X and the amount you spend on good y so it's going to be PX times QX the number of units of x that you buy multiplied by however much each of those units Cost Plus p y q y so this is the total amount you spend on good X and this is the total amount you spend on good Y and those two amounts have to add up to your income remember it's going to add up to exactly your income because you're going to spend all of your income you're not going to save and you can't borrow what we need to do is we want to graph this and we're going to graph it in that picture of the indifference curves that we just talked about and remember that the thing on our vertical axis when we drew the indifference curves we had this picture it was QX down here I think back there in that video I was just calling it X but let's be more precise we'll call it qy back then I just called it y I think and we had an indifference curve U1 if we want to graph our budget constraint in this picture then we need to solve for whatever is on that vertical axis we need to solve for q y so I'm going to do that I'm going to get qy by itself that means I need to move that term over so income minus PX QX is equal to p y q y I'm going to flip everything around because I like to have the thing I'm solving for on the left side of the equal the equal sign so I'm going to have p y q y equal to income minus p x q X so I just flipped it around now I'm going to divide through by P Y that's going to give me over here just q y I get income divided by P Y minus p x divided by P Y oops multiplied by QX there's my budget constraint all I did was I solved it for q y and now I can graph that thing it's in slope intercept form so here's QX down here here's q y up here The Intercept is income divided by the price of good y it has a negative slope and that slope is PX divided by P Y and you can't see it from that version of it but I can tell you this horizontal intercept is the amount of income divided by the price of good X so there's what our budget constraint looks like it's linear we've got our n intercept and we've got our slope let's just do a quick example so let's suppose that income is equal to a hundred dollars let's suppose the price of good X is equal to 10 and the price of good Y is equal to 5. so let's plug in those numbers into this version of it and then we'll solve it for q y and then we'll graph it so 100 is equal to 10 QX plus 5 q y there's what my budget constraint looks like I'm going to move my q y stuff to one side so I get 5 q y is equal to 100 minus 10 QX and then I'm going to divide through by 5 I get q y is equal to 20 minus 2 QX there's our budget constraint and that's easy to graph if we graph that here's QX there's q y our intercept our vertical intercept is 20. this thing has a slope of negative 2 so it goes down two over one by the time it's gone down 20 it's gone over 10 so this thing looks just like that it has a slope of negative two and you can see that if we look at what each of these things represent that it makes perfect sense this term right up here is income divided by the price of good y this tells us how many units of good y you can buy if you buy only good y well if you've got a hundred dollars of income and each of those units cost you five dollars then you can buy 20 units if you spent all your money on good why this number down here tells you how many units of good X you could buy if you spent all of your income on good X well if you've got a hundred dollars and good X cost you ten dollars then you can buy 10 units the slope here is the price of good x divided by the price of good Y which is 10 divided by 5 so the slope is indeed two so you can see that it's very simple especially with the numbers to graph your budget constraint now here's what the budget constraint tells us graphically it represents all of the bundles the consumer can afford the consumer could spend all of their money on good Y and buy 20 of them or all of their money on good X and buy 10 or they could spend half their money on one good and half their money on the other good but any point on on this budget constraint represents an affordable bundle of course the consumer could buy a bundle right there but that bundle doesn't exhaust their income they wouldn't want to be at that bundle because they can afford bundles to the Northeast they can have more of both Goods so the consumer is going to end up at some point on that budget constraint question is which point and that's how we're going we're going to figure out the answer to that by combining this with our indifference curves first we want to talk about how the budget constraint changes if any of these three things change so we'll think about what happens if income changes we'll think about what happens if either of the price chain prices change let's start by thinking about what happens if income changes so let's talk about a change in income let's suppose income Falls to fifty dollars well since the income now is going to get cut in half the consumer will be able to afford fewer bundles than before what that means is that this budget constraint is going to shift in they're not going to be able to afford any of the bundles on that original budget constraint so if we think about how remember this endpoint is their income divided by the price of good why well if their income is a is 50 and the price of good Y is five now they can only afford 10 units of good y they will only be able to afford five units of good X so the budget constraint moves parallel to itself it's going to move from this one where it's oops where it's 20 and 10 is going to move down to this one where it's 10 and 5. the slope will not change because we didn't change either of the prices price of good X is still 10 and the price of good Y is still 5. so the slope is still 2. if we were to increase income then the budget constraint would shift out in this direction again the slope would not change it would simply move out the consumer would be able to afford more bundles than before so that's what happens if we change income let's think about what happens if we change prices price changes let's suppose that the price of good y increases to 10. suppose price of good y increases to ten dollars that's going to be bad for the consumer right a price went up they will be able to afford fewer bundles now if the price of good y goes to 10 they can't buy 20 units anymore since the price of good X didn't change they will still be able to buy 10 units of good X but now with a hundred dollars of income if the price of good y goes to 10 they will only be able to buy 10 units of good why so can only buy 10 units of y in other words when the budget constraint excuse me when the price of good y Rises the budget constraint pivots down like that it goes from having a slope of two to having a slope of one because the price of each good is now ten dollars and ten over ten is one so if you have an increase in the price of good y let's make sure that we remember that this is good y up here here's good X down here an increase in the price of good y will pivot this thing down that direction so let's put here let's let's draw a few pictures I'm going to draw actually two pictures where we change the price of good Y and then we'll draw two pictures where we change the price of good X and I'm going to start with essentially the same budget constraint in all four of my pictures and then in this red color we'll draw the new budget constraint so if we increase the price of good y that means the consumer can afford less good y so an increase in the price of Y will pivot that budget constraint down that direction this is an increase in the price of good y a decrease in the price of good Y is good for the consumer that will pivot the budget constraint up in that direction here it goes down like that this is a decrease in the price of good y if we think about an increase in the price of good X that will pivot the budget constraint in like that there's an increase in the price of good X a decrease in the price of good X will pivot the budget constraint out in that direction there is a decrease in the price of good X so now we know how the budget constraint will change depending upon whether or not income changes depending upon whether or not the price of good y changes or the price of good X changes and it's important that you remember how these things work but remember that as a as a last resort you can always just plug in the new prices or the new income and just graph the new budget constraint okay what we want to do now I'm going to clear this off and we'll talk about what happens if we have what we refer to as a non-standard budget constraint so the budget constraint represents all of the affordable bundles that the consumer can choose from we don't yet we haven't yet gone through how they're going to pick the particular bundle we've got we have to combine it with the indifference curves to do that but first let's talk about different ways that the budget constraint can can be affected by things like a quantity discount or maybe a quantity limit so let's start by thinking about a quantity discount this is very common out there the idea here is that if you buy more than some particular number then you get a discount so let's suppose you buy if you buy the five up to five of these you pay this price but if you buy more than five six or seven then for those additional units we'll give them to you for half price okay so there are lots of places where you see quantity discounts and let's just work through an example doing adjusting the budget constraint for something like this is very straightforward but there's a process you need to go through to do it and it's it's fairly easy let's suppose that we have two two goods suppose our two goods are going to be pizzas and tickets to a baseball game let's suppose that the price of tickets is ten dollars each but if you buy more than 10 you get them for half price more than 10 they're half price they only cost five dollars let's suppose pizzas are twenty dollars and let's suppose that income is two hundred dollars so you've got two hundred dollars to spend on pizzas and tickets those are the only two goods there are the first 10 tickets cost you ten dollars but if you buy 11 or 12 or more than that you get the additional ones the 11th and the 12th and the 13th for half price you can buy those for five dollars so the first step is graph the budget constraint without the discount graph budget constraint without the discount so let's do that first that means that our price of P of tickets is 10 our price of pizzas is 20 and we've got two hundred dollars to spend so 200 is equal to 10 the price of uh tickets multiplied by the quantity of tickets plus 20 the price of pizzas multiplied by the quantity of pizzas now we can draw this any way we want I'm going to put tickets on the vertical axis I'm going to draw this in this space we're going to put the quantity of pizzas down here and we'll put the quantity of tickets up here so in order to graph this I need to solve for that okay so this is going to be 10 QT is equal to 200 minus 20 QP and divide through by 10 QT is equal to 20 minus 2 QP there's my budget constraint it's in slope intercept form I'm going to put now I know what this is going to end up looking like so I'm going to put 10 here and 20 and 30 and then let's down here let's put five and ten so if I graph this it's going to start here at 20 that's my intercept it's got a slope of negative 2 so by the time I've gone down 20 I will have gone over 10. here's what the initial budget constraint looks like I could spend all of my money on pizzas I've got two hundred dollars and a pizza cost twenty dollars so I could buy 10 pizzas ignoring the discount I could spend all of my money on tickets I've got two hundred dollars and each ticket cost ten dollars so I could buy twenty now we've got to figure out what the discount does okay so let's think about where the consumer gets the discount the consumer gets the discount at 10 units so let's look at that point that point of 10 units is right here I'm going to move my five over there there's the point that we get the discount 10 tickets well if we spend our money on 10 tickets that's a hundred dollars and then if we spend twenty dollars on five pizzas that's another hundred dollars so our income is exhausted right there but now let's think about what happens if we buy more than 10 tickets so if we if we buy the first 10 tickets at ten dollars each that takes up a hundred of our dollars and then we've got another hundred dollars left over if we spent all of that hundred dollars on additional tickets we could buy 20 of them so that means in addition to the first 10 we could buy 20 more that means if we spent all of our money on tickets we could buy a total of 30 tickets what that means is we get an additional portion on the budget constraint that comes down here and looks just like that the budget constraint actually ends up being trace it out in a little bit different color the budget constraint ends up being this portion right here it comes on down it makes a little turn right here there's a kink in it and then it's this portion right down there once we've got the full budget constraint drawn this portion of the budget constraint that we drew just to kind of work our way through it this doesn't matter so this little portion right in here I'm going to kind of just mark it out that portion ignore it the actual budget constraint is not linear it's piecewise linear it's got a kink right there at that point so you can see that the budget constraint changes when you've got a quantity discount and we could add an additional discount it could be that the 11th through 15th tickets cost five dollars or ten dollars no half price was five and we could have we could make any tickets past uh fifteen two dollars and fifty cents and then there'd be another kink in it you can do that let's finish up our special budget constraints with a quantity limit let's think about what the budget constraint looks like if there's some limit on the amount of something you can buy and it's pretty straightforward let's suppose that we draw a budget constraint and let's just put QX and Qi down here and let's suppose we have a budget constraint that looks like this we don't need to worry about numbers for this one but let's suppose there's a limit on the amount of good y that you can buy and that limit I'm going to use a bar over it let's suppose that limit is right there you can't buy more than this many units of good y well what that means is that the budget constraint simply is this portion of the original budget constraint and then it just gets cut off right there we get this flat segment this portion is irrelevant because the consumer can't buy any of these units of good y so in that case the final budget constraint ends up looking like this so you get a different type of kink in the budget constraint okay and we could have a a quantity limit on good y we could have a quantity limit on good X typically we would just have a quantity limit on one of the goods not both of them at the same time now that we understand the budget constraint what we want to do is think about consumer optimization so let's think about the consumer's problem the consumer's problem so the consumer's problem we've started this uh chapter's worth of material on that the consumer's problem is to maximize their well-being to maximize their utility given their income so what that means is the consumer can choose any bundle on their budget constraint any bundle that's affordable but what they want to do is choose the one bundle that provides them with the most utility the one bundle that gets them as high up on their utility function as they can another way to say that is they want to choose the one bundle that allows them to reach the indifference curve that is farthest from the origin because the indifference curves farther from the origin correspond to higher levels of utility so to solve that problem we need to combine the indifference curves with the budget constraints so let's draw kind of a big picture here and think about how this is going to work thank you Let's uh draw a budget constraint let's go ahead and put QX down here and q y up here and let's draw a budget constraint now the consumer can choose any bundle on that budget constraint there they will exhaust all of their income so we know they're not going to choose any bundle down in this range but they can't afford any of the bundles out here so let's just pick a bundle and let's think about starting there let's pick a bundle like bundle a so consider bundle a in order to understand something about how the consumer prefers that bundle a to other bundles on the picture we need to draw the indifference curve through bundle a so let's do that let's suppose that we draw the indifference curve through bundle a and suppose that that indifference curve looks like this here's the indifference curve through bundle a we'll call it U1 the consumer is indifferent between bundle a and any other bundle on that indifference curve so now that we have the indifference curve we can think about whether or not bundle a would maximize utility and we see real quickly that it won't because what could happen is the consumer can afford bundle B undle B is an affordable bundle and we know that bundle B would be preferred to bundle a because bundle B has more of both Goods than this bundle right down here which the consumer is indifferent to a so we know that given that budget constraint and given that indifference curve the consumer would never choose bundle a they would rather have bundle B so now let's draw the indifference curve through bundle B it's going to be a higher indifference curve than U1 it's going to look something like this it's going to come down through here suppose it does this there's indifference curve U2 bundle B provides higher utility than bundle a but we also know that bundle B is not the best bundle for the consumer because the consumer could afford this bundle right there what we'll call bundle C bundle C is Affordable and bundle C would provide higher levels of utility or a higher level of utility than bundle B so let's draw the indifference curve through bundle C and let's suppose that it looks like this it comes down right through here and it's just tangent right there at Point C and then it curves on around like this and we'll call that indifference curve u3 indifference curve u3 is farther from the origin than U1 or U2 so we know that the consumer at Point C is getting more utility than either of those other two points and we also see that there's no other bundle on this picture that the consumer can afford that would allow the consumer to get to a higher indifference curve so what we can say is that point c ends up being the consumer's optimal Choice that's their best bundle okay so bundle C foreign is the optimal bundle there's no other way that the consumer could change how much good X and how much good y they buy and still afford it and reach a higher indifference curve bundle C allows the consumer to achieve the highest level of utility now the coordinates of that bundle C tell us how much good X and good y the consumer would consume what this tells us is that the consumer would buy that much good X and this much good y q X star q y star that's the optimal bundle the optimal amount of good X and good Y for the consumer to choose now notice that the outcome of what we've just done here is that we end up at a tangency between the indifference curve and the budget constraint right so at Point C the indifference curve and the budget constraint are tangent to each other which means they have the same slope so we know that the slope of the indifference curve is the marginal rate of substitution and I'm going to put that negative sign out there so at the optimal point the marginal rate of substitution of good X for good y which we know is equal to the marginal utility of the good on the horizontal axis divided by the marginal utility of the good on the vertical axis it's the ratio of the marginal utilities that's equal to the slope of the budget constraint and if you think back just a little bit ago we talked about what the slope of the budget constraint was and it was the price of good x divided by the price of good y this is important really it's this portion of it that is most important at the optimal point the ratio of the marginal utilities will be equal to the ratio of the two prices we call this the relative price of good X so now what I want to do is clear this half of this off and then we're going to go back and take another quick look at what's true at Point a compared to point C let's go back and take another look at what's true at Point a remember point a is right here and make it a little bit bigger there's point a at Point a let's think about the relationship between the slope of the indifference curve and the slope of the budget constraint now both of the slopes are negative so I'm not going to worry about the negative signs at this point but what we see is that this at Point a the slope of the indifference curve is bigger than the slope of the budget constraint so at a the slope of the indifference curve slope of the indifference curve is bigger than the slope of the budget constraint now let's think about what that means slope of the indifference curve is bigger than the slope of the budget constraint the slope of the indifference curve is the ratio of the marginal utilities so the slope of the indifference curve looks like this it's the marginal utility of good x divided by the marginal utility of good y that slope of the indifference curve is greater than the slope of the budget constraint which is p x over p y so we know that that is true at Point a now I'm going to manipulate this just a little bit I'm just going to move my PX down here and I'm going to move this marginal utility up there I don't need to change the sign I'm not going to multiply by a negative or anything like that this statement is equivalent to this statement the marginal utility of X over the price of good X is greater than the marginal utility of Y divided by the price of good y now let's think about what that is of course this term tells us how much additional utility you get out of another unit of good X and then this tells us how much that other unit of good X is going to cost you this is like uh additional utility per dollar from good X it's like to bang for your buck out of good X this is the additional utility you get from consuming another unit of Y and this is how much another unit of Y costs you this is dollar or utility per dollar out of good why it's essentially the bang for your buck out of good why what this tells us is that at Point a you get more bang for your buck out of good X than you get out of good y well what that means is you're not spending your dollars in the best way possible you're getting more bang for your buck out of good X that means you should buy more good X you should spend more dollars on good X because you get more out of them so that's why starting at Point a the consumer buys more good X and less good why that's why B is better than a but we also see that at point B this is still true they're still getting more bang for their Buck out of good X than they get out of good y and so they should buy more good X and less good Y and so they move to point C and at Point C this is true at Point C the slope of the indifference curve is equal to the slope of the budget constraint which means that at Point C the marginal utility of good x divided by the price of good X is exactly equal to the marginal utility of Y divided by the price of good y at Point C they get exactly the same bang for their Buck out of good X as what they get out of good why there's no reason to readjust that's the definition of being at the optimal Point there's no other adjustment that you can make that would make you better off so we can see that in terms of of the bang for their buck that they get we can see it in terms graphically of the fact that they have reached the highest indifference curve the indifference curve that is farthest from the origin so at the optimal point this will be true at any other point there will be an inequality and they will need to readjust now let's talk about an important implication of this all consumers in a particular Market are going to face the same prices that means that all consumers are going to face the same relative price of good X that means at the optimal outcome the slope of everybody's indifference curve is going to be equal to each other and your book contains a picture of this and here's the picture let's let's draw two pictures and then I'll combine them into one picture like your book does I don't like initially combining them because I think it's misleading for students let's suppose we have two consumers this is consumer one this is consumer two each of them has their own indifference curve their own their own utility function with their own sets of indifference curves and exactly what their indifference curve picture looks like depends on their preferences and this persons are going to be different from that person's but they're both going to be facing the same exact set of prices so the slope of their budget constraint will be the same now where are their budget constraint ends up being depends on how much income each of them has but they're both going to have the same slope I tried to draw those so that they're both parallel to each other so they're both going to have a different map of their indifference curves but at their optimal point they're going to end up at a point of tangency between an indifference curve and their budget constraint let's suppose that that person ends up at a point of tangency there and this person ends up at a point of tangency somewhere down here they don't have to consume the same amounts of good X and good y because they have different incomes and they have different preferences but the important point is for both of these people the marginal rate of substitution will be equalized between the two because they both have the exact same slope to their budget constraint now here's what your picture does so so let's say that that this is person one and this is person two what we will be able to say is that the marginal rate of substitution for person one will be equal to the marginal rate of substitution for person two and this will be Goods X and Y x y so when both of them maximize their utility they'll make different decisions in terms of how much they consume but because they face the same prices they're going to have the same marginal rates of substitution now the picture that your book draws is this picture they put it on one picture they have they give the consumers the same income and of course they face the same prices so the consumers two budget constraints will lie right on top of each other and then they have one consumer optimizing by choosing this bundle and another consumer optimizing by choosing this bundle so one person's right there and one person's right there and I don't like that picture because it shows two indifference curves intersecting now it's fine because these are indifference curves for different people a single person's indifference curves will never intersect but two different peoples indifference curves can intersect so as long as you're careful about understanding the picture there's nothing wrong with it but I feel like it's better to draw this picture first and then combine them so that you get a better understanding of what's going on in that picture what we need to do now is clear this off and then we'll finish up by talking about what's known as a corner solution let's talk about what a corner solution is the best way to understand what a corner solution is is to understand what an interior solution is so when we think about a consumer maximizing utility let's put the picture back up here we've got quantity of good X quantity of good y we've got a budget constraint here and the consumer has to choose any of these infinite number of bundles that are on that budget constraint and we know that the bundle that they're going to end up choosing ends up being the bundle where one of their indifference curves is just tangent to that budget constraint it's going to look something like that indifference curve U1 we end up at a point right there and we call that an interior solution it's an interior solution because it's on the interior of this picture the coordinates of that point tell us how much good X and how much good y the consumer chooses and the optimal point is on the inside of this picture rather than being down here at one of these corners so once you understand what an interior solution is then it's much easier to think about what a the solution has one think about what a corner solution looks like so let's think about a corner solution so a corner solution will only happen when you have unusually shaped indifference curves so either those indifference curves have to be extremely steep or extremely flat or they have to be linear they have to be perfect substitutes and that's really pretty rare real usually we're not going to be dealing with Goods that have perfect that are perfect substitutes for each other or we're typically not going to be dealing with Goods where the indifference curves are are unusually steep or unusually flat but let's think about what happens if we did let's suppose that we start with a budget constraint that looks something like this there's Q we'll say in my notes here I've used uh pizza and burritos for some reason let's go with that so the consumer can afford any bundle on this budget constraint let's consider a bundle like this a bundle like a now let's go through the same process we went through just a little bit over there we're going to draw the indifference curves through bundle a and let's suppose when we draw the indifference curves through bundle a they're kind of unusually shaped and let's suppose they're really Steep and they're just always really steep let's suppose they look something like this they've got a little bit of curvature to them but they're really steep we'll call that indifference curve U1 well we know that the consumer would never choose U1 because there are bundles that are affordable out here that would be on a higher indifference curve so we can think about a bundle like bundle B undle B would be preferred to any bundle on this indifference curve it would be on a higher indifference curve and so if we draw the indifference curve through bundle B let's suppose that it looks something like this difference curve U2 that's still not the optimal bundle because the consumer would prefer any of these affordable bundles to bundle B what you can see is that we're going to end up given the shape of these indifference curves we're going to end up at this point right down here where the consumer ends up right at that bundle right there which we'll call bundle C and it all has to do with the shape of the indifference curves it leads this consumer to buy all of one good and none of the other and that we call a corner solution so this is a corner solution where the consumer spends all of their money on Pizza they buy no burritos at all and it's all due to this kind of unusual shape of their um indifference curves and remember that the steepness of the indifference curves tells us how much the consumer likes the one good relative to the other and here we can see that the consumers willing to give up a whole lot of burritos for a little bit more pizza they really like pizza compared to burritos so we get this unusual outcome but most of the time we have interior Solutions it's relatively rare that we have a corner solution let's finish up this discussion by talking about what we're going to call expenditure minimization expenditure minimization so we've been thinking about an optimization problem a maximization problem so our problem that we've been thinking about is to maximize utility so our problem maximize utility given income it's a constrained optimization problem constrained optimization it's actually a very simple calculus problem but we're not going to work through the calculus of it the constraint is income and the thing that we're trying to optimize is utility so what we get of course is this picture where we draw the constraint and then we end up with this optimization where we're trying to move to the highest indifference curve so we look at some indifference curves and then we move push those indifference curves as far as we can to the Northeast and we end up right there at that point so this curved thing that represents utility that's what we call the objective function the objective function and this budget constraint right down here the line that's our constraint we can think about this problem in a different way we can think about it in terms of a minimization problem so in order to do that let's rephrase the problem let's think about a new problem we'll see that the two problems end up being mirror images of each other we mathematically we say that they are dual problems which just means that there is two ways of looking at the same problem you're just looking at it from a different point of view so here's a new problem what's the smallest amount of money we can spend to reach a particular utility level smallest amount of money we can spend to reach a particular utility level now notice that this changes the role of the objective function and the constraint so now our constraint is that we have to achieve a particular level of utility and our objective is to minimize the amount of money that we spend to reach that level of utility so our objective function now is going to be not the curved line or the curved thing it's going to be the line and our constraint is not going to be the line our constraint will be the indifference curve so here's what this problem looks like now here's the constraint we need to reach that level of utility QX qy this becomes our constraint and what we need to do is figure out which of these bundles on this indifference curve is going to be the cheapest bundle is it going to be this bundle or is it going to be that bundle or some bundle down in here and so what we have to do is consider a bundle let's pick a bundle like bundle a and then we can draw essentially the budget constraint that would go through that bundle now we don't have a particular amount of income we've got to decide how much income we have to spend so it's it's technically not correct to call it a budget constraint but it still represents a line or it still is a line that would represent all of the bundles that cost the same so let's draw that line through here it's going to look like a budget constraint but we want to move that thing as close to the origin as possible so we can see that this bundle is not the cheapest way to do it we could buy this bundle and if we drew the line through that bundle we would see that that bundle actually costs less because it's closer to the origin remember that the less income you have the closer your budget constraint was to the origin of course that's not the cheapest bundle it turns out that we get the situation where we have a tangency right there at the optimal point and you can see that this now the Line is now our objective we want to move it as close to the origin as possible the indifference curve is our constraint so compared to up here the role of the objective function in the constraint has switched but you can see that we still end up at a point of tangency between a line and a curve the outcome of the two problems looks the same now if you've had any background in calculus you know that an optimization problem simply a constrained optimization problem you can do dual problems you can reverse the role of the constraint one of them is a maximization one of them is minimization the way that you do the two problems is the same you take derivatives the partial derivatives set those equal to zero and work through it it's a system of equations and here we're not going to do any of that we're just going to think about it graphically but you can see that this expenditure minimization problem is just like looking at the problem kind of backwards now we'll come back to something like that in our next video what we're going to do is we're going to going to work more with the utility maximization problem in a chapter after that we're going to come back to something that looks a lot like this only what we'll be thinking about will be cost minimization for a firm but it's going to look exactly like that so I'll see in a future video

let&#39;s talk about what a budget constraint represents so the objective of the of the consumer is going to be to maximize utility now that we understand what an indifference curve is we know that the objective of the consumer is going to be to reach the indifference curve that is farthest away from the origin in other words to reach the highest elevation of utility that they possibly can but the constraint that they have is that they can only spend the amount of income that they&#39;ve got so we&#39;re going to be thinking about um let&#39;s talk about the assumptions we&#39;re going to make here in terms of our budget constraint we&#39;re going to be assuming that the prices are not going to be changing for the consumer as they as they try to decide which bundle to choose the prices are going to stay fixed for this short period of time that they&#39;re trying to figure this out we&#39;re not assuming that prices never change but for right now for this point in time the consumer knows what the prices are and they&#39;re not going to change the consumer is also going to have a particular amount of income the income that they have isn&#39;t going to change at this point in time it could change in the future but not right now and then we&#39;re going to think about the consumer not saving or borrowing we could come up with complicated models where the consumer can save some of their income for later but we&#39;re not going to have them do that at least to begin with okay so the consumer has no reason to save any of their income so their budget constraint let&#39;s think about what the budget constraint looks like the budget constraint simply says that you can&#39;t spend more money than you have okay and there are only two things to spend money on good X and good y so the budget constraint is going to look like this I which I&#39;ll use to designate income your income the amount of income that you have has to be spread between the amount you spend on good X and the amount you spend on good y so it&#39;s going to be PX times QX the number of units of x that you buy multiplied by however much each of those units Cost Plus p y q y so this is the total amount you spend on good X and this is the total amount you spend on good Y and those two amounts have to add up to your income remember it&#39;s going to add up to exactly your income because you&#39;re going to spend all of your income you&#39;re not going to save and you can&#39;t borrow what we need to do is we want to graph this and we&#39;re going to graph it in that picture of the indifference curves that we just talked about and remember that the thing on our vertical axis when we drew the indifference curves we had this picture it was QX down here I think back there in that video I was just calling it X but let&#39;s be more precise we&#39;ll call it qy back then I just called it y I think and we had an indifference curve U1 if we want to graph our budget constraint in this picture then we need to solve for whatever is on that vertical axis we need to solve for q y so I&#39;m going to do that I&#39;m going to get qy by itself that means I need to move that term over so income minus PX QX is equal to p y q y I&#39;m going to flip everything around because I like to have the thing I&#39;m solving for on the left side of the equal the equal sign so I&#39;m going to have p y q y equal to income minus p x q X so I just flipped it around now I&#39;m going to divide through by P Y that&#39;s going to give me over here just q y I get income divided by P Y minus p x divided by P Y oops multiplied by QX there&#39;s my budget constraint all I did was I solved it for q y and now I can graph that thing it&#39;s in slope intercept form so here&#39;s QX down here here&#39;s q y up here The Intercept is income divided by the price of good y it has a negative slope and that slope is PX divided by P Y and you can&#39;t see it from that version of it but I can tell you this horizontal intercept is the amount of income divided by the price of good X so there&#39;s what our budget constraint looks like it&#39;s linear we&#39;ve got our n intercept and we&#39;ve got our slope let&#39;s just do a quick example so let&#39;s suppose that income is equal to a hundred dollars let&#39;s suppose the price of good X is equal to 10 and the price of good Y is equal to 5. so let&#39;s plug in those numbers into this version of it and then we&#39;ll solve it for q y and then we&#39;ll graph it so 100 is equal to 10 QX plus 5 q y there&#39;s what my budget constraint looks like I&#39;m going to move my q y stuff to one side so I get 5 q y is equal to 100 minus 10 QX and then I&#39;m going to divide through by 5 I get q y is equal to 20 minus 2 QX there&#39;s our budget constraint and that&#39;s easy to graph if we graph that here&#39;s QX there&#39;s q y our intercept our vertical intercept is 20. this thing has a slope of negative 2 so it goes down two over one by the time it&#39;s gone down 20 it&#39;s gone over 10 so this thing looks just like that it has a slope of negative two and you can see that if we look at what each of these things represent that it makes perfect sense this term right up here is income divided by the price of good y this tells us how many units of good y you can buy if you buy only good y well if you&#39;ve got a hundred dollars of income and each of those units cost you five dollars then you can buy 20 units if you spent all your money on good why this number down here tells you how many units of good X you could buy if you spent all of your income on good X well if you&#39;ve got a hundred dollars and good X cost you ten dollars then you can buy 10 units the slope here is the price of good x divided by the price of good Y which is 10 divided by 5 so the slope is indeed two so you can see that it&#39;s very simple especially with the numbers to graph your budget constraint now here&#39;s what the budget constraint tells us graphically it represents all of the bundles the consumer can afford the consumer could spend all of their money on good Y and buy 20 of them or all of their money on good X and buy 10 or they could spend half their money on one good and half their money on the other good but any point on on this budget constraint represents an affordable bundle of course the consumer could buy a bundle right there but that bundle doesn&#39;t exhaust their income they wouldn&#39;t want to be at that bundle because they can afford bundles to the Northeast they can have more of both Goods so the consumer is going to end up at some point on that budget constraint question is which point and that&#39;s how we&#39;re going we&#39;re going to figure out the answer to that by combining this with our indifference curves first we want to talk about how the budget constraint changes if any of these three things change so we&#39;ll think about what happens if income changes we&#39;ll think about what happens if either of the price chain prices change let&#39;s start by thinking about what happens if income changes so let&#39;s talk about a change in income let&#39;s suppose income Falls to fifty dollars well since the income now is going to get cut in half the consumer will be able to afford fewer bundles than before what that means is that this budget constraint is going to shift in they&#39;re not going to be able to afford any of the bundles on that original budget constraint so if we think about how remember this endpoint is their income divided by the price of good why well if their income is a is 50 and the price of good Y is five now they can only afford 10 units of good y they will only be able to afford five units of good X so the budget constraint moves parallel to itself it&#39;s going to move from this one where it&#39;s oops where it&#39;s 20 and 10 is going to move down to this one where it&#39;s 10 and 5. the slope will not change because we didn&#39;t change either of the prices price of good X is still 10 and the price of good Y is still 5. so the slope is still 2. if we were to increase income then the budget constraint would shift out in this direction again the slope would not change it would simply move out the consumer would be able to afford more bundles than before so that&#39;s what happens if we change income let&#39;s think about what happens if we change prices price changes let&#39;s suppose that the price of good y increases to 10. suppose price of good y increases to ten dollars that&#39;s going to be bad for the consumer right a price went up they will be able to afford fewer bundles now if the price of good y goes to 10 they can&#39;t buy 20 units anymore since the price of good X didn&#39;t change they will still be able to buy 10 units of good X but now with a hundred dollars of income if the price of good y goes to 10 they will only be able to buy 10 units of good why so can only buy 10 units of y in other words when the budget constraint excuse me when the price of good y Rises the budget constraint pivots down like that it goes from having a slope of two to having a slope of one because the price of each good is now ten dollars and ten over ten is one so if you have an increase in the price of good y let&#39;s make sure that we remember that this is good y up here here&#39;s good X down here an increase in the price of good y will pivot this thing down that direction so let&#39;s put here let&#39;s let&#39;s draw a few pictures I&#39;m going to draw actually two pictures where we change the price of good Y and then we&#39;ll draw two pictures where we change the price of good X and I&#39;m going to start with essentially the same budget constraint in all four of my pictures and then in this red color we&#39;ll draw the new budget constraint so if we increase the price of good y that means the consumer can afford less good y so an increase in the price of Y will pivot that budget constraint down that direction this is an increase in the price of good y a decrease in the price of good Y is good for the consumer that will pivot the budget constraint up in that direction here it goes down like that this is a decrease in the price of good y if we think about an increase in the price of good X that will pivot the budget constraint in like that there&#39;s an increase in the price of good X a decrease in the price of good X will pivot the budget constraint out in that direction there is a decrease in the price of good X so now we know how the budget constraint will change depending upon whether or not income changes depending upon whether or not the price of good y changes or the price of good X changes and it&#39;s important that you remember how these things work but remember that as a as a last resort you can always just plug in the new prices or the new income and just graph the new budget constraint okay what we want to do now I&#39;m going to clear this off and we&#39;ll talk about what happens if we have what we refer to as a non-standard budget constraint so the budget constraint represents all of the affordable bundles that the consumer can choose from we don&#39;t yet we haven&#39;t yet gone through how they&#39;re going to pick the particular bundle we&#39;ve got we have to combine it with the indifference curves to do that but first let&#39;s talk about different ways that the budget constraint can can be affected by things like a quantity discount or maybe a quantity limit so let&#39;s start by thinking about a quantity discount this is very common out there the idea here is that if you buy more than some particular number then you get a discount so let&#39;s suppose you buy if you buy the five up to five of these you pay this price but if you buy more than five six or seven then for those additional units we&#39;ll give them to you for half price okay so there are lots of places where you see quantity discounts and let&#39;s just work through an example doing adjusting the budget constraint for something like this is very straightforward but there&#39;s a process you need to go through to do it and it&#39;s it&#39;s fairly easy let&#39;s suppose that we have two two goods suppose our two goods are going to be pizzas and tickets to a baseball game let&#39;s suppose that the price of tickets is ten dollars each but if you buy more than 10 you get them for half price more than 10 they&#39;re half price they only cost five dollars let&#39;s suppose pizzas are twenty dollars and let&#39;s suppose that income is two hundred dollars so you&#39;ve got two hundred dollars to spend on pizzas and tickets those are the only two goods there are the first 10 tickets cost you ten dollars but if you buy 11 or 12 or more than that you get the additional ones the 11th and the 12th and the 13th for half price you can buy those for five dollars so the first step is graph the budget constraint without the discount graph budget constraint without the discount so let&#39;s do that first that means that our price of P of tickets is 10 our price of pizzas is 20 and we&#39;ve got two hundred dollars to spend so 200 is equal to 10 the price of uh tickets multiplied by the quantity of tickets plus 20 the price of pizzas multiplied by the quantity of pizzas now we can draw this any way we want I&#39;m going to put tickets on the vertical axis I&#39;m going to draw this in this space we&#39;re going to put the quantity of pizzas down here and we&#39;ll put the quantity of tickets up here so in order to graph this I need to solve for that okay so this is going to be 10 QT is equal to 200 minus 20 QP and divide through by 10 QT is equal to 20 minus 2 QP there&#39;s my budget constraint it&#39;s in slope intercept form I&#39;m going to put now I know what this is going to end up looking like so I&#39;m going to put 10 here and 20 and 30 and then let&#39;s down here let&#39;s put five and ten so if I graph this it&#39;s going to start here at 20 that&#39;s my intercept it&#39;s got a slope of negative 2 so by the time I&#39;ve gone down 20 I will have gone over 10. here&#39;s what the initial budget constraint looks like I could spend all of my money on pizzas I&#39;ve got two hundred dollars and a pizza cost twenty dollars so I could buy 10 pizzas ignoring the discount I could spend all of my money on tickets I&#39;ve got two hundred dollars and each ticket cost ten dollars so I could buy twenty now we&#39;ve got to figure out what the discount does okay so let&#39;s think about where the consumer gets the discount the consumer gets the discount at 10 units so let&#39;s look at that point that point of 10 units is right here I&#39;m going to move my five over there there&#39;s the point that we get the discount 10 tickets well if we spend our money on 10 tickets that&#39;s a hundred dollars and then if we spend twenty dollars on five pizzas that&#39;s another hundred dollars so our income is exhausted right there but now let&#39;s think about what happens if we buy more than 10 tickets so if we if we buy the first 10 tickets at ten dollars each that takes up a hundred of our dollars and then we&#39;ve got another hundred dollars left over if we spent all of that hundred dollars on additional tickets we could buy 20 of them so that means in addition to the first 10 we could buy 20 more that means if we spent all of our money on tickets we could buy a total of 30 tickets what that means is we get an additional portion on the budget constraint that comes down here and looks just like that the budget constraint actually ends up being trace it out in a little bit different color the budget constraint ends up being this portion right here it comes on down it makes a little turn right here there&#39;s a kink in it and then it&#39;s this portion right down there once we&#39;ve got the full budget constraint drawn this portion of the budget constraint that we drew just to kind of work our way through it this doesn&#39;t matter so this little portion right in here I&#39;m going to kind of just mark it out that portion ignore it the actual budget constraint is not linear it&#39;s piecewise linear it&#39;s got a kink right there at that point so you can see that the budget constraint changes when you&#39;ve got a quantity discount and we could add an additional discount it could be that the 11th through 15th tickets cost five dollars or ten dollars no half price was five and we could have we could make any tickets past uh fifteen two dollars and fifty cents and then there&#39;d be another kink in it you can do that let&#39;s finish up our special budget constraints with a quantity limit let&#39;s think about what the budget constraint looks like if there&#39;s some limit on the amount of something you can buy and it&#39;s pretty straightforward let&#39;s suppose that we draw a budget constraint and let&#39;s just put QX and Qi down here and let&#39;s suppose we have a budget constraint that looks like this we don&#39;t need to worry about numbers for this one but let&#39;s suppose there&#39;s a limit on the amount of good y that you can buy and that limit I&#39;m going to use a bar over it let&#39;s suppose that limit is right there you can&#39;t buy more than this many units of good y well what that means is that the budget constraint simply is this portion of the original budget constraint and then it just gets cut off right there we get this flat segment this portion is irrelevant because the consumer can&#39;t buy any of these units of good y so in that case the final budget constraint ends up looking like this so you get a different type of kink in the budget constraint okay and we could have a a quantity limit on good y we could have a quantity limit on good X typically we would just have a quantity limit on one of the goods not both of them at the same time now that we understand the budget constraint what we want to do is think about consumer optimization so let&#39;s think about the consumer&#39;s problem the consumer&#39;s problem so the consumer&#39;s problem we&#39;ve started this uh chapter&#39;s worth of material on that the consumer&#39;s problem is to maximize their well-being to maximize their utility given their income so what that means is the consumer can choose any bundle on their budget constraint any bundle that&#39;s affordable but what they want to do is choose the one bundle that provides them with the most utility the one bundle that gets them as high up on their utility function as they can another way to say that is they want to choose the one bundle that allows them to reach the indifference curve that is farthest from the origin because the indifference curves farther from the origin correspond to higher levels of utility so to solve that problem we need to combine the indifference curves with the budget constraints so let&#39;s draw kind of a big picture here and think about how this is going to work thank you Let&#39;s uh draw a budget constraint let&#39;s go ahead and put QX down here and q y up here and let&#39;s draw a budget constraint now the consumer can choose any bundle on that budget constraint there they will exhaust all of their income so we know they&#39;re not going to choose any bundle down in this range but they can&#39;t afford any of the bundles out here so let&#39;s just pick a bundle and let&#39;s think about starting there let&#39;s pick a bundle like bundle a so consider bundle a in order to understand something about how the consumer prefers that bundle a to other bundles on the picture we need to draw the indifference curve through bundle a so let&#39;s do that let&#39;s suppose that we draw the indifference curve through bundle a and suppose that that indifference curve looks like this here&#39;s the indifference curve through bundle a we&#39;ll call it U1 the consumer is indifferent between bundle a and any other bundle on that indifference curve so now that we have the indifference curve we can think about whether or not bundle a would maximize utility and we see real quickly that it won&#39;t because what could happen is the consumer can afford bundle B undle B is an affordable bundle and we know that bundle B would be preferred to bundle a because bundle B has more of both Goods than this bundle right down here which the consumer is indifferent to a so we know that given that budget constraint and given that indifference curve the consumer would never choose bundle a they would rather have bundle B so now let&#39;s draw the indifference curve through bundle B it&#39;s going to be a higher indifference curve than U1 it&#39;s going to look something like this it&#39;s going to come down through here suppose it does this there&#39;s indifference curve U2 bundle B provides higher utility than bundle a but we also know that bundle B is not the best bundle for the consumer because the consumer could afford this bundle right there what we&#39;ll call bundle C bundle C is Affordable and bundle C would provide higher levels of utility or a higher level of utility than bundle B so let&#39;s draw the indifference curve through bundle C and let&#39;s suppose that it looks like this it comes down right through here and it&#39;s just tangent right there at Point C and then it curves on around like this and we&#39;ll call that indifference curve u3 indifference curve u3 is farther from the origin than U1 or U2 so we know that the consumer at Point C is getting more utility than either of those other two points and we also see that there&#39;s no other bundle on this picture that the consumer can afford that would allow the consumer to get to a higher indifference curve so what we can say is that point c ends up being the consumer&#39;s optimal Choice that&#39;s their best bundle okay so bundle C foreign is the optimal bundle there&#39;s no other way that the consumer could change how much good X and how much good y they buy and still afford it and reach a higher indifference curve bundle C allows the consumer to achieve the highest level of utility now the coordinates of that bundle C tell us how much good X and good y the consumer would consume what this tells us is that the consumer would buy that much good X and this much good y q X star q y star that&#39;s the optimal bundle the optimal amount of good X and good Y for the consumer to choose now notice that the outcome of what we&#39;ve just done here is that we end up at a tangency between the indifference curve and the budget constraint right so at Point C the indifference curve and the budget constraint are tangent to each other which means they have the same slope so we know that the slope of the indifference curve is the marginal rate of substitution and I&#39;m going to put that negative sign out there so at the optimal point the marginal rate of substitution of good X for good y which we know is equal to the marginal utility of the good on the horizontal axis divided by the marginal utility of the good on the vertical axis it&#39;s the ratio of the marginal utilities that&#39;s equal to the slope of the budget constraint and if you think back just a little bit ago we talked about what the slope of the budget constraint was and it was the price of good x divided by the price of good y this is important really it&#39;s this portion of it that is most important at the optimal point the ratio of the marginal utilities will be equal to the ratio of the two prices we call this the relative price of good X so now what I want to do is clear this half of this off and then we&#39;re going to go back and take another quick look at what&#39;s true at Point a compared to point C let&#39;s go back and take another look at what&#39;s true at Point a remember point a is right here and make it a little bit bigger there&#39;s point a at Point a let&#39;s think about the relationship between the slope of the indifference curve and the slope of the budget constraint now both of the slopes are negative so I&#39;m not going to worry about the negative signs at this point but what we see is that this at Point a the slope of the indifference curve is bigger than the slope of the budget constraint so at a the slope of the indifference curve slope of the indifference curve is bigger than the slope of the budget constraint now let&#39;s think about what that means slope of the indifference curve is bigger than the slope of the budget constraint the slope of the indifference curve is the ratio of the marginal utilities so the slope of the indifference curve looks like this it&#39;s the marginal utility of good x divided by the marginal utility of good y that slope of the indifference curve is greater than the slope of the budget constraint which is p x over p y so we know that that is true at Point a now I&#39;m going to manipulate this just a little bit I&#39;m just going to move my PX down here and I&#39;m going to move this marginal utility up there I don&#39;t need to change the sign I&#39;m not going to multiply by a negative or anything like that this statement is equivalent to this statement the marginal utility of X over the price of good X is greater than the marginal utility of Y divided by the price of good y now let&#39;s think about what that is of course this term tells us how much additional utility you get out of another unit of good X and then this tells us how much that other unit of good X is going to cost you this is like uh additional utility per dollar from good X it&#39;s like to bang for your buck out of good X this is the additional utility you get from consuming another unit of Y and this is how much another unit of Y costs you this is dollar or utility per dollar out of good why it&#39;s essentially the bang for your buck out of good why what this tells us is that at Point a you get more bang for your buck out of good X than you get out of good y well what that means is you&#39;re not spending your dollars in the best way possible you&#39;re getting more bang for your buck out of good X that means you should buy more good X you should spend more dollars on good X because you get more out of them so that&#39;s why starting at Point a the consumer buys more good X and less good why that&#39;s why B is better than a but we also see that at point B this is still true they&#39;re still getting more bang for their Buck out of good X than they get out of good y and so they should buy more good X and less good Y and so they move to point C and at Point C this is true at Point C the slope of the indifference curve is equal to the slope of the budget constraint which means that at Point C the marginal utility of good x divided by the price of good X is exactly equal to the marginal utility of Y divided by the price of good y at Point C they get exactly the same bang for their Buck out of good X as what they get out of good why there&#39;s no reason to readjust that&#39;s the definition of being at the optimal Point there&#39;s no other adjustment that you can make that would make you better off so we can see that in terms of of the bang for their buck that they get we can see it in terms graphically of the fact that they have reached the highest indifference curve the indifference curve that is farthest from the origin so at the optimal point this will be true at any other point there will be an inequality and they will need to readjust now let&#39;s talk about an important implication of this all consumers in a particular Market are going to face the same prices that means that all consumers are going to face the same relative price of good X that means at the optimal outcome the slope of everybody&#39;s indifference curve is going to be equal to each other and your book contains a picture of this and here&#39;s the picture let&#39;s let&#39;s draw two pictures and then I&#39;ll combine them into one picture like your book does I don&#39;t like initially combining them because I think it&#39;s misleading for students let&#39;s suppose we have two consumers this is consumer one this is consumer two each of them has their own indifference curve their own their own utility function with their own sets of indifference curves and exactly what their indifference curve picture looks like depends on their preferences and this persons are going to be different from that person&#39;s but they&#39;re both going to be facing the same exact set of prices so the slope of their budget constraint will be the same now where are their budget constraint ends up being depends on how much income each of them has but they&#39;re both going to have the same slope I tried to draw those so that they&#39;re both parallel to each other so they&#39;re both going to have a different map of their indifference curves but at their optimal point they&#39;re going to end up at a point of tangency between an indifference curve and their budget constraint let&#39;s suppose that that person ends up at a point of tangency there and this person ends up at a point of tangency somewhere down here they don&#39;t have to consume the same amounts of good X and good y because they have different incomes and they have different preferences but the important point is for both of these people the marginal rate of substitution will be equalized between the two because they both have the exact same slope to their budget constraint now here&#39;s what your picture does so so let&#39;s say that that this is person one and this is person two what we will be able to say is that the marginal rate of substitution for person one will be equal to the marginal rate of substitution for person two and this will be Goods X and Y x y so when both of them maximize their utility they&#39;ll make different decisions in terms of how much they consume but because they face the same prices they&#39;re going to have the same marginal rates of substitution now the picture that your book draws is this picture they put it on one picture they have they give the consumers the same income and of course they face the same prices so the consumers two budget constraints will lie right on top of each other and then they have one consumer optimizing by choosing this bundle and another consumer optimizing by choosing this bundle so one person&#39;s right there and one person&#39;s right there and I don&#39;t like that picture because it shows two indifference curves intersecting now it&#39;s fine because these are indifference curves for different people a single person&#39;s indifference curves will never intersect but two different peoples indifference curves can intersect so as long as you&#39;re careful about understanding the picture there&#39;s nothing wrong with it but I feel like it&#39;s better to draw this picture first and then combine them so that you get a better understanding of what&#39;s going on in that picture what we need to do now is clear this off and then we&#39;ll finish up by talking about what&#39;s known as a corner solution let&#39;s talk about what a corner solution is the best way to understand what a corner solution is is to understand what an interior solution is so when we think about a consumer maximizing utility let&#39;s put the picture back up here we&#39;ve got quantity of good X quantity of good y we&#39;ve got a budget constraint here and the consumer has to choose any of these infinite number of bundles that are on that budget constraint and we know that the bundle that they&#39;re going to end up choosing ends up being the bundle where one of their indifference curves is just tangent to that budget constraint it&#39;s going to look something like that indifference curve U1 we end up at a point right there and we call that an interior solution it&#39;s an interior solution because it&#39;s on the interior of this picture the coordinates of that point tell us how much good X and how much good y the consumer chooses and the optimal point is on the inside of this picture rather than being down here at one of these corners so once you understand what an interior solution is then it&#39;s much easier to think about what a the solution has one think about what a corner solution looks like so let&#39;s think about a corner solution so a corner solution will only happen when you have unusually shaped indifference curves so either those indifference curves have to be extremely steep or extremely flat or they have to be linear they have to be perfect substitutes and that&#39;s really pretty rare real usually we&#39;re not going to be dealing with Goods that have perfect that are perfect substitutes for each other or we&#39;re typically not going to be dealing with Goods where the indifference curves are are unusually steep or unusually flat but let&#39;s think about what happens if we did let&#39;s suppose that we start with a budget constraint that looks something like this there&#39;s Q we&#39;ll say in my notes here I&#39;ve used uh pizza and burritos for some reason let&#39;s go with that so the consumer can afford any bundle on this budget constraint let&#39;s consider a bundle like this a bundle like a now let&#39;s go through the same process we went through just a little bit over there we&#39;re going to draw the indifference curves through bundle a and let&#39;s suppose when we draw the indifference curves through bundle a they&#39;re kind of unusually shaped and let&#39;s suppose they&#39;re really Steep and they&#39;re just always really steep let&#39;s suppose they look something like this they&#39;ve got a little bit of curvature to them but they&#39;re really steep we&#39;ll call that indifference curve U1 well we know that the consumer would never choose U1 because there are bundles that are affordable out here that would be on a higher indifference curve so we can think about a bundle like bundle B undle B would be preferred to any bundle on this indifference curve it would be on a higher indifference curve and so if we draw the indifference curve through bundle B let&#39;s suppose that it looks something like this difference curve U2 that&#39;s still not the optimal bundle because the consumer would prefer any of these affordable bundles to bundle B what you can see is that we&#39;re going to end up given the shape of these indifference curves we&#39;re going to end up at this point right down here where the consumer ends up right at that bundle right there which we&#39;ll call bundle C and it all has to do with the shape of the indifference curves it leads this consumer to buy all of one good and none of the other and that we call a corner solution so this is a corner solution where the consumer spends all of their money on Pizza they buy no burritos at all and it&#39;s all due to this kind of unusual shape of their um indifference curves and remember that the steepness of the indifference curves tells us how much the consumer likes the one good relative to the other and here we can see that the consumers willing to give up a whole lot of burritos for a little bit more pizza they really like pizza compared to burritos so we get this unusual outcome but most of the time we have interior Solutions it&#39;s relatively rare that we have a corner solution let&#39;s finish up this discussion by talking about what we&#39;re going to call expenditure minimization expenditure minimization so we&#39;ve been thinking about an optimization problem a maximization problem so our problem that we&#39;ve been thinking about is to maximize utility so our problem maximize utility given income it&#39;s a constrained optimization problem constrained optimization it&#39;s actually a very simple calculus problem but we&#39;re not going to work through the calculus of it the constraint is income and the thing that we&#39;re trying to optimize is utility so what we get of course is this picture where we draw the constraint and then we end up with this optimization where we&#39;re trying to move to the highest indifference curve so we look at some indifference curves and then we move push those indifference curves as far as we can to the Northeast and we end up right there at that point so this curved thing that represents utility that&#39;s what we call the objective function the objective function and this budget constraint right down here the line that&#39;s our constraint we can think about this problem in a different way we can think about it in terms of a minimization problem so in order to do that let&#39;s rephrase the problem let&#39;s think about a new problem we&#39;ll see that the two problems end up being mirror images of each other we mathematically we say that they are dual problems which just means that there is two ways of looking at the same problem you&#39;re just looking at it from a different point of view so here&#39;s a new problem what&#39;s the smallest amount of money we can spend to reach a particular utility level smallest amount of money we can spend to reach a particular utility level now notice that this changes the role of the objective function and the constraint so now our constraint is that we have to achieve a particular level of utility and our objective is to minimize the amount of money that we spend to reach that level of utility so our objective function now is going to be not the curved line or the curved thing it&#39;s going to be the line and our constraint is not going to be the line our constraint will be the indifference curve so here&#39;s what this problem looks like now here&#39;s the constraint we need to reach that level of utility QX qy this becomes our constraint and what we need to do is figure out which of these bundles on this indifference curve is going to be the cheapest bundle is it going to be this bundle or is it going to be that bundle or some bundle down in here and so what we have to do is consider a bundle let&#39;s pick a bundle like bundle a and then we can draw essentially the budget constraint that would go through that bundle now we don&#39;t have a particular amount of income we&#39;ve got to decide how much income we have to spend so it&#39;s it&#39;s technically not correct to call it a budget constraint but it still represents a line or it still is a line that would represent all of the bundles that cost the same so let&#39;s draw that line through here it&#39;s going to look like a budget constraint but we want to move that thing as close to the origin as possible so we can see that this bundle is not the cheapest way to do it we could buy this bundle and if we drew the line through that bundle we would see that that bundle actually costs less because it&#39;s closer to the origin remember that the less income you have the closer your budget constraint was to the origin of course that&#39;s not the cheapest bundle it turns out that we get the situation where we have a tangency right there at the optimal point and you can see that this now the Line is now our objective we want to move it as close to the origin as possible the indifference curve is our constraint so compared to up here the role of the objective function in the constraint has switched but you can see that we still end up at a point of tangency between a line and a curve the outcome of the two problems looks the same now if you&#39;ve had any background in calculus you know that an optimization problem simply a constrained optimization problem you can do dual problems you can reverse the role of the constraint one of them is a maximization one of them is minimization the way that you do the two problems is the same you take derivatives the partial derivatives set those equal to zero and work through it it&#39;s a system of equations and here we&#39;re not going to do any of that we&#39;re just going to think about it graphically but you can see that this expenditure minimization problem is just like looking at the problem kind of backwards now we&#39;ll come back to something like that in our next video what we&#39;re going to do is we&#39;re going to going to work more with the utility maximization problem in a chapter after that we&#39;re going to come back to something that looks a lot like this only what we&#39;ll be thinking about will be cost minimization for a firm but it&#39;s going to look exactly like that so I&#39;ll see in a future video

Transcript for:Budget Constraints and Consumer Optimization

Transcript for:
Budget Constraints and Consumer Optimization