we've already explored the concept of utility Theory utility is the ability of something to satisfy our needs and wants and we use utility functions to represent people's preferences those utility functions are a mathematical expression of what is going on in people's heads when they compare and rank the different options available to them it may sound far-fetched but we mentioned in chapter one that there's a growing body of evidence from the field of neuro economics that this is pretty close to what our brains are actually doing when making decisions the purpose of our utility function is to rank the possibilities we face if these three bundles are our only options the utility function tells us which ones we like best and therefore which ones we will choose remember one of our principles in economics is optimization a belief that people will try and Achieve outcomes ranked highly in their preferences but we can reimagine our utility functions as a guide to what possibilities we wouldn't be able to choose between have you ever been to a restaurant and looked at the menu and you just couldn't design because so many things look good to eat it is as if your mind is crunching the numbers and telling you that it's a tie for the number one spot we call this indifference because you're equally happy with either choice and so you are indifferent to which one you get utility functions let us Define that indifference explicitly instead of thinking of utility as something that changes with our choices we can hold utility constant in math when we hold a variable constant we often put this little bar over it then we can rearrange our equation to see what the relationship between pizza and burgers is here we see that there's an inverse relationship between them if we were to hold utility constant at say 20 units there are lots of different combinations of pizza and burgers that would give us that much utility one slice of pizza and 20 Burgers would give us 20 units of utility but so would four slices of pizza and Five Burgers every one of these options here gives us the same amount of utility which means that if we try to rank them they would all be tied we don't prefer any of these options over the others we're totally indifferent between them give me two slices of pizza and ten burgers and I will be just as happy as if you gave me five slices of pizza and four Burgers it's all the same to me this wouldn't be economics if we didn't put this idea on a graph we can graph this function where we put Pizza on the vertical axis and burgers on the horizontal axis it doesn't really matter which one you put where when we graph the function it would look like this and it reflects the trade-off between pizza and burgers in our preferences is we get less Pizza we need more and more Burgers to make up for it and keep us at the same level of utility remember every point on this curve has the exact same result for utility so what would happen if we increased our utility as utility Rises our curve has to shift outwards away from the origin of the graph the further out we go the more utility we are getting so shifts in this direction represent increases in utility but remember for each curve individually every Point represents some combination of pizza and burgers that we are indifferent between but we would certainly prefer any combination on this line over any combination on this one so this point over here would be preferred to this point over here or even this point over here because those points are on a lower indifference curve note that R and difference curves never cross because that would violate our assumptions about people's preferences different kinds of preferences yield different in difference curves instead of pizza and burgers I'm just going to have any two goods X1 and X2 so this could still be pizza and burgers or it could be train tickets and airplane tickets or iPhones and Samsungs or Pepsis and coca-colas or jackets and sweaters it could be anything we want but I've also changed our utility function so to instead of multiplying our two goods we're going to add them together if we were to graph this utility function as an indifference curve we would get a straight line it looks a little like a budget constraint but instead this is representing our preferences instead of our budget in difference curves like this represent substitutes which are Goods we choose between like the ones we just listed train tickets versus airplane tickets iPhones or Samsungs Pepsi versus Coca-Cola Jackets versus sweaters those are things we tend to choose one or the other not both and so those are substitutes alternatively we could have a weird utility function like this one where we're taking the minimum of X1 and X2 think of something like left and right shoes if we have five pairs of shoes we would have five left shoes and five right ones if we plug those values into this utility function it would say that we have five units of utility because that is the minimum number between five and five but what if we found a sixth left shoe with no matching right shoe we got more stuff but our utility wouldn't increase the minimum value between five right shoes and six left shoes is still five we need that sixth right shoe to actually boost our utility up one if we graph this as an indifference curve we would get this L-shaped curve if this point is 5 for x 1 and 5 for X2 then we can see that increasing one variable without increasing the other doesn't change our utility we're indifferent between all of these options this is the most extreme example of what we call compliments or Goods that we consume together left and right shoes are an example of perfect complements things we want exactly the same number of but there are plenty of things we tend to consume together like peanut butter and jelly milk and cereal socks and shoes and more those two indifference curves show the extreme ends of a spectrum of possibilities which lie in between the sharper the bend in the indifference curve the more the relationship between those goods leans towards complements the lesser the bend the more the relationship looks like substitutes we don't really ever want a left shoe but not a right one and so those would be perfect complements but we might want peanut butter without jelly even though lots of times we want them together those are just regular compliments we tend to want them together but not always and then there's Pepsi and Coke most of the time we choose between those options but sometimes perhaps for throwing a party we might want both of them at the same time but when you're looking for pain relief Tylenol and generic brand acetaminophen are pretty much perfectly interchangeable they are perfect substitutes indifference curves are great because they already let us theorize about our consumption patterns and preferences but their real power comes from helping us solve the buyer's problem consumers you and I and everyone else are all trying to optimize we want to try to choose the best feasible option given all the available information and now we have some tools to help us model that decision-making process indifference curves can help us find the best option and budget constraints help us limit it to only the feasible ones when we put this in the language of our economic model of consumer Behavior what we assume people do when optimizing is that they try to maximize their utility subject to their budget constraint they want the most utility they can afford let's see what this looks like mathematically we will have some utility function and for now we don't need to Define it we can just leave it as a function and we want to choose values for X1 and X2 the goods or services we want to consume that gives us the highest value for utility but which are also affordable when considering our income why and the prices of X1 and X2 it could be a lot to look at this mathematically and I think it is more intuitive on the graph our graph places X1 and X2 on each axis first we will plot the budget constraint we saw previously that this is a straight line that connects the most X2 we can afford to the most X1 we can afford everything on this line is Affordable and so is everything beneath the line but go past our budget line and we find only bundles of X1 and X2 that we cannot afford they are not feasible next let's draw an indifference curve now we know what the we don't know what the utility function is so we don't know exactly how to draw but for now let's just draw the typical case in difference curves tend to look something like this what we will see is that our for our most basic conclusions we don't need to define utility explicitly we don't really want to we want a theory that applies to everyone and everyone has their own unique utility function but take a look at our placement of the indifference curve every point on this curve gives us the same set amount of utility let's say it's 10 units of utility if we want those 10 units of utility we can pick any point on the indifference curve but not all of them are affordable these points are not feasible they're too expensive we can't afford them but these ones fall below or on the budget line meaning they are feasible we can afford them so we're obviously going to need to pick one of the ones we can afford but is it possible for us to do better can we afford a bundle that gives us more than 10 units of utility for sure all of these bundles above our indifference curve but on or below the budget line are affordable to us and give us more than 10 units of utility as we increase our utility the indifference curve moves out like this and we want to keep going until there's only one affordable bundle left that will happen when our indifference curve is tangent to the budget line sharing just a single point which is the best choice we can make given our budget constraint remember a few videos ago when I said that our little assumption that people are able to rank the options available to them will let us unlock an entire theory of consumer Behavior well here we are we have an economic model that simulates one of those consumers and as we can see it shows us which option is the best for the consumer it tells us what decision consumers are going to make of course to predict what a specific consumer will do we would need their exact utility function and budget constraint and that's probably too much to ask but the power of this theory of the consumer is that it lets us simulate changes in the world and see how our representative consumer reacts to them it lets us make predictions about how people might change their decisions when conditions change and that prediction might apply generally to all consumers comparative Statics is the method of analyzing the impact of a change in the parameters of a model by comparing the equilibrium that results from the change with the original equilibrium our equilibrium is found where the indifference curve is tangent to the budget line they cross only at this one point and that point is the option the consumer will choose so how will that choice change when we alter the conditions of the decision let's start by thinking about what happens when we increase a person's income we saw before how higher incomes shift the budget line out keeping it parallel to the original line because the ratio of prices between X1 and X2 have not changed well now there are feasible options which would increase our utility we push our indifference curve out until it is again tangent to the budget line and this becomes our new choice at this new equilibrium we can see that we are consuming more X1 and more X2 it's a pretty good result if I pay you more money will you buy more of everything you like that matches reality pretty well as people's incomes go up they go on more vacations they buy bigger houses nicer cars and overall consume more stuff our prediction while not groundbreaking manages to State the obvious if it didn't well then it wouldn't be a very good theory okay what if instead we increase the price of X1 meaning it will be more expensive than before we saw previously how this rotates our budget line inwards changing the slope of the budget line because the ratio of prices has changed unfortunately with this new constraint our old choice is no longer feasible we can't afford it anymore because X1 got more expensive we need to move our indifference curve down until it touches a point on the new budget line in this case we get a pretty interesting result we end up with less X1 and we are not happy about it our utility is lower but what happens to X2 depends on whether or not X1 and X2 are complements or substitutes if we usually choose between the two goods like Pepsi and Coke that a higher price on X1 will probably mean we buy more X2 than before because we will substitute X2 for X1 alternatively if we like to have X1 and X2 together like milk and cereal then a higher price on one of them means a higher price on the combination of both of them together will buy less X2 on our graph these two goods appear to be unrelated they are neither complements or substitutes and our change in X1 didn't have any impact on x2 what if we decrease the price of X1 instead in this case the budget line rotates outward to reflect the lower price for X1 now we can increase our utility Shifting the indifference curve out to a new point where we buy more of X1 than before but here too what happens to X2 depends on whether or not the two goods are complements or substitutes cheaper X1 is bad news for X2 if it is a substitute but good news for X2 if it is a complement this model of consumer behavior is pretty powerful and already it has allowed us to make important predictions about consumer behavior that will lay the foundation of market demand but it also contains some great advice for how to live your life which we'll see next