 Hi, everyone. Thank you for coming to today's Marcus Seminar. Our speaker today is Professor Saed Ali Samir from Yale University. Before I introduce the speaker, let's talk about some of the basic features that you can use for this Zoom meeting. So first of all, everyone will be muted upon entry to reduce background noise, but you can unmute yourself at the end of the presentation in the Q&A section if you have any questions. So, and there are three features that I would like you to pay attention to. The first one is this chat feature. You can use that to communicate with us if you have any technical difficulties. So for example, you cannot see the slides. You can tell us or you cannot hear the speaker. The second feature is raise hand. If you have a clarification question, you can use that feature. So let's say you would like a definition to be clarified during the presentation, you can use this. And finally, there is this Q&A feature that you can use to ask in-depth questions. So these questions will be addressed and answered at the end of the presentation. And if you have follow-up questions, you can unmute yourself to ask your follow-up questions. I just want to remind everyone, our next seminar is next week on the 19th. Professor Su from Columbia University will give a presentation. Professor Elise Samir is an associate professor of operations management with the Yale School of Management. He received his PhD in business administration decision science from Duke University. His research interest lies in the area of social responsibility and sustainability as well as healthcare, especially on policy, public policy related problems that involve dynamic decision making and learning. He has worked on government subsidy instruments in the renewable energy industry and agriculture. Today, he is going to talk about the role of smart appliances in electricity pricing. Professor, I will let you share your slides. Okay. Is my slides visible now? Can everybody see my slides now? It is, and it doesn't look like you're in presentation mode. Okay, I think that I can fix. Okay, how about now? That didn't change it. Is it good now? We can still see it. I see a sidebar, but that's not a problem. Because on my screen, it's actually a full screen, but maybe we can proceed with this. Is it visible? If it's visible, I think we should be fine. It is visible. I have a note from someone who says if you stop sharing and start sharing again. Okay, we can do that. That's better. Okay. Okay, hello everyone. Thanks, Chimu and Wahilla for the introduction and for organizing this talk. It's a pleasure to present in this seminar series. I hope you are enjoying your beautiful sunny afternoon in Palo Alto, which I was supposed to enjoy by the way, but if it wasn't for this pandemic. So in any case, I'm coming to you from a very rainy and gloomy day in Yale campus in New Haven. And thanks for the invitation. It's very exciting to actually see that you have this seminar series with a special focus on smart grid, given that this is such an interdisciplinary topic and it attracts attention from various fields. So hopefully today I'm going to contribute to this discussion and more from an operations management perspective from a more like an economic point of view. And especially with more focus on the details of consumer decision-making process. And as you will see the talk is slightly technical, but I will try to use my limited time to mostly convey the main message and try to spend less time on the technical details. But hopefully it will, I mean, I can still leave you with the main takeaways and I have to say, even though I have been teaching on Zoom for a couple of semesters, I'm not still an expert on it. So I can see your names and if you have any questions, please feel free to stop me, raise your hand, I will see that and I'm happy to answer your questions. Otherwise at the end you also can have a live Q&A after I'm done. So this is a joint work with my colleague from University of Texas at Dallas, Shujong Wang, and also one of our PhD students, Faribah Farajbat. And we started working on this topic a couple of years ago when we heard about the recent changes which is happening in electricity markets. As all of you are probably aware, most of the electricity markets around the world are undergoing significant changes, partly due to the fact that automation and smart technologies are becoming an integral part of power grids. And this technological empowerment, together with the rapid expansion of electricity markets, which for example in the US reached close to half a trillion dollars in revenue in 2016, has led to some very heated policy debates about how these smart technologies can lead to some solution for enhancing efficiency and resiliency of our power system. So there are certain features of electricity market which makes them different from other commodity markets. I mean, as all of you probably are aware, so one distinct feature of electricity market is that we usually face fluctuating demand, fluctuating demand, which is very hard to predict in advance. And it only realizes very close to a consumption point with the emergence of renewable energy which introduces supply intermittency that this problem is only magnified because we also have a lot of variability in the amount of renewable energy that we can generate in the electricity grid. And up to now, there are very limited or scarce economically viable storage capabilities, unlike other commodities which can be stored. That is not the case for electricity. At least as of now, battery or storage solutions are still very, very expensive. And at the same time, we cannot afford to allow for demand exceeding supply because that may lead to some brownout or blackouts that has significant economic and financial losses and have very negative consequences. So this introduces this big challenge which electrical system face, which is the supply demand mismatch, and the idea is, okay, what are the best things we can do and how we can use these smart technologies to mitigate the supply demand mismatch. In this graph, you can see electricity consumption in Spain, for example, over this span of seven years, and you can see that there are some seasonality patterns, but even if you allow for seasonality, there is still a lot of fluctuation compared to a baseline that we expect. As you can see, in terms of thousands of megawatt hours, there is still a lot of fluctuation up and down with respect to that baseline, which is represented by zero on the left axis. So what are we doing to address this problem? Well, these smart technologies, such as smart grid, which allows utility firms to monitor consumption in real time, has empowered them to experiment with various forms of so-called demand response programs. So I'm sure all of you are familiar with what demand response programs are. Majority, why some small portion of them are focused on direct load control and cartel management based on some pre-specified contractual agreement, majority of them are price-based demand response programs. So these kind of price-based incentive mechanisms have received tremendous popularity, and research has shown that they can lead to significant increase in the efficiency of the power grid if designed and implemented well. But the recent implementation of these kind of price-based demand response programs have generated some mixed results. So why is that? First of all, the prevailing assumption in design price-based demand response programs is that consumers are sensitive to price signals for electricity, and they are fully aware of their consumption at any point in time. And they find their consumption decision as they receive price signals. That's the underlying idea. Basically, it proceeds that they know exactly how their demand is going to unfold in the future after price signals are observed or after prices are set, and they informatively adjust their consumption decision up or down as they receive price signals. So, but as I said, the recent implementation of demand response program have generated mixed results, and basically these results, which is kind of against the expected outcomes that the developers of these price mechanisms have in mind, has undermined the validity of this assumption that indeed consumers are sensitive to these price signals. So what is the main challenge? The main challenge is that, well, even after prices are set and even after consumers observe the price, they still face some uncertainty in terms of their demand and in terms of the amount of electricity that they consume. And if you assume that these demand response programs work perfectly, it implies that the consumers should constantly monitor their consumption after they observe the price, and then they react to incorporate that into their decision-making and they optimize or readjust their consumption decision accordingly, which is obviously not the case. So what are these kind of fluctuations? Well, there can be many different sources, but probably the most obvious one, which has a big impact on our consumption decision, especially at the residential level, is variation in the ambient environment in particular weather, temperature changes. So this is the same graph that I showed you in the previous slides for electricity consumption in Spain, and then at the bottom, the blue line represents the temperature, which obviously has a very obvious cyclical pattern, and as you can see when the outdoor temperature is too high or too low, we observe a peak in electricity consumption. So temperature plays a big role, obviously, in the amount of electricity consumption, and many of these temperature changes happen unexpectedly, or there are some fluctuations that cannot be predicted accurately. And as you can see on the right, according to EIA energy outlook, air conditioning actually makes a big part of electricity consumption for an average household, close to 20% on average. And on a hot summer day or on a very cold day in winter, depending on which region you're living, it can actually increase to 60% of your electricity consumption. So that's only air conditioning. But there are some other consumers of electricity, which again also have to do with weather, like refrigeration or water heating, space heating, as you can see described. So those all essentially have to do with temperature and outdoor condition. And according to a recent study by prominent energy economists, actually they concluded that 20% of electricity consumption can be saved if we only adjust our AC temperature by only four degrees during the very hot summer days. And that would have prevented some of the blackouts that we observed in California over the last summer, which probably some of you had experienced. So, but why do we have this lack of response from the consumers in terms of responding to these changes in the ambient environment? Well, this has been attributed to the limited cognitive ability and the attention that consumers pay to what happens in the outside, as I said, in the ambient environment. So as Ernest Orstein put it, as probably some of you know him, he's a prominent energy economist at Berkeley. He says, in reality, consumers make such, which refers to electricity consumption, this is just with limited information, attention and cognitive ability. So that is something that the developers of this price based incentive programs or demand response programs, that is what they essentially miss. That is what they don't take into account. So the idea of this talk, and for this talk, I mainly focus on just temperature and air conditioning, but it can be essentially extended to other setting as well. So the idea here is to incorporate informational and cognitive limitations of consumers and then see how it can, how it can be incorporated into having a better understanding of the consumption patterns faced by utility firms. So maybe for many of us, we don't really adjust our air conditioning that much because maybe it's not a big part of our electricity bill or maybe for you guys who are living in California, there are not that much fluctuation in outdoor temperature. But here I have three examples from three different apartments in New York City from some of the less affluent neighborhoods. And as you can see, indeed the indoor temperature has some patterns, which is the red line, and then you can see that the blue line is the outdoor temperature. So and some of you may say, well, this is variations, the red temperature, which represents the indoor temperature, it may be attributable to other things. For example, whether I'm at home or not, or whether I have a guest over or whether I feel cold because I'm sick or others, of course, that can be part of these variations. But when we study this in more detail, we see that there is actually some correlation between the outdoor and indoor temperature, meaning that for these less affluent communities, they are really careful about how much electricity they consume. And when they, for example, sometimes if they see that the outside is very cold, maybe they lower their thermostat setting to save on electricity. So then the question is to what extent that is the case. And when you look at it at an aggregate level, how can we incorporate that to reach some meaningful conclusion for utility firms in terms of estimating their demand. So as I said, the idea is that consumers usually do not have sufficient cognitive capability, or we incur extensive opportunity costs if they want to perfectly optimize or re-optimize their consumption decision over time in response to changes in their ambient environment. So what is our goal here? So for the purpose of this talk, given the limited time that we have, my goal is to just shed sunlight on this. As I said with a special focus on the residential sector and air conditioning in particular, and basically it improved to some extent, improved our understanding of how consumers make decisions in light of these random demand shocks, which happens very close to consumption point. Because utility firms, if they want to design meaningful and effective demand response programs, they need to know what to expect for a given price. So if price is set at a particular level, how we should expect consumers to react to it. And then that can be used as a building block. So that's exactly our goal here. That can be used as a building block to design better and more effective demand response programs. And then the other question is, okay, how these values, sorry, how these changes with having more smarter homes or more smart appliances, how that changes and how that affects the utility firms pricing decisions. So I don't expect to be able to get to all of these during this talk, but we'll see how far we can proceed. So what we do here is basically we formalize the household decision making process and provide a theoretical foundation for analyzing their limited capability in responding to such a random shocks in the external environment. So I'm going to skip the literature review by just saying that most of the existing papers in this area, they examine how the total consumption react to a change in unit price. But they generally overlook random factors such as changes in outdoor temperature, for example, and they also don't take into account for household's possible behavioral or cognitive barriers. So that's exactly what we are trying to achieve here. So as some of these papers, for example, this paper from an response study in 2009 or this paper by E2 in 2014 show empirically by studying some real data from households, I believe from California, they argue that consumers are not capable of accurately inferring the marginal price that they face, and they only react basically to average price. So that is exactly another confirmation of the fact that consumers in general are not perfect or perfectly rational decision makers. So they face some cognitive limitations, and we have to take that into account when we want to predict how they make their decisions. So that is exactly our goal in this research. We want to incorporate these behavioral aspects and see how it shapes the overall consumption that's faced by a typical utility company. So with that, let me go into the details of our model. So the model that we construct here consists of a utility fair and a population of households. For the sake of simplicity, we represents this population of households as a single representative consumer. So let's say we are facing a population of thousands of households, but we want to capture their behavior as a single representative consumer. And then obviously this can be extended to situations where we have heterogeneous households, but for the purpose of this talk, just to keep things simple, I just assume that we have a single representative consumer which represents the entire population of households. So let's say price is set, if you have a flat-race price, which is set at P, that is the price of unit price of electricity that these representative consumer faces. And then we are facing some, so the consumer has to make some decision about her appliances setting, let's say for example, what is going to be my thermostat setting, I represent that by X. It's going to be my decision at any point in time. And at the same time, there is some external random shocks, let's say outdoor temperature, which is realized, and I'll represent it by W. That W is a random variable at any point in time, let's say at any 10 minutes or 15 minutes, it can potentially take a different value. It's up to us to define what is our time unit, it can be an hour, it doesn't matter. So this is the price, X is the decision that I make, let's say my indoor temperature, my thermostat setting, and W is the random shock or the outdoor temperature that I'm facing. So what is the consumption coming from, let's say my air conditioning. It has been shown in the literature that the electricity consumption follows this quadratic pattern, meaning that as a distance between my indoor setting and outdoor temperature increases. The electricity consumption increases based on the quadratic fashion. So if X and W are very close to each other, obviously I have very little consumption. But as X and W deviate from each other in either direction, then my electricity consumption is going to increase. And this parameter alpha that I have here, which again, this has been configured empirically, it's the notion of the energy efficiency of the consumer appliance. The energy efficient my dwelling is in terms of do I have like energy efficient windows and so on. So that essentially dictates how much consumption I should expect for a given outdoor temperature and a given indoor thermostat setting. Okay, now if I'm a consumer, if I'm a typical consumer, I have an ideal temperature. Let's say my ideal temperature is theta. I want to set my indoor temperature at theta. Ideally, that's what the level of temperature that gives me the maximum comfort or the maximum pressure. So I'm not pressure pleasure maximum pleasure. So then what is the utility that I have. So the utility that I have is beta zero beta zero is the maximum utility that I can enjoy. But as the indoor temperature setting X deviates from my ideal, which is theta, I incur some discomfort, which is represented by this beta. So beta represents my sensitivity to deviation from ideal temperature. So if my ideal temperature is 72 degrees, but I said my indoor setting at 70, for example, then the difference is two degrees. And then that gives me some discomfort from the maximum pleasure that I can get, which is beta zero. So if electricity was free, what would I do? I would definitely set X equal to theta. I just set my indoor temperature to my ideal level. There is no reason for me to deviate. But the point is that electricity is not free, or alternatively one can say, or I have environmentally conscious. So I don't really want to set my indoor temperature necessarily equal to outdoor temperature, especially set to my ideal temperature, especially when I see that outdoor is too hot or too cold. So I want to save a little bit. Now we can account for that by, of course, penalizing for the amount of money that I have to pay per my consumption. And my overall utility, therefore, is the pleasure or the comfort that I get minus the price that I have to pay multiplied by the total quantity that I consume. So this function together overall represents the overall consumer utility that is faced by a typical consumer. And then, okay, now considering a very naive setting, let's assume that we are always rational and we are always available to just readjust our AC temperature at any point in time. We call it a perfectly responsive consumer, which is far, far from reality, but for the sake of argument, look for now, let's assume you have a perfectly responsive consumer. Then obviously that perfectly responsive consumer wants to optimize this utility function. So therefore, and what is the decision variable, the decision variable is X. X is my thermostat setting. I want to find the optimal thermostat setting which maximizes my utility. This is a very easy problem, and this is optimal solution. The optimal solution is a linear combination of theta, ideal temperature and W, the outdoor temperature at any point in time. So basically, you see that the weight is represented by this ratio, which is increasing in price. So as price increases, I give more weight to outdoor temperature and less weight to my ideal temperature, which is very obvious. So as electricity becomes more expensive, I get my indoor to be closer to outdoor. On the other hand, as electricity becomes cheaper, I really don't care about outdoor anymore, I just make my indoor setting closer and closer to my ideal. So this is very obvious. That's what you expect. And then as W changes, let's say from this hour to the next hour, I readjust my X star. Again, I readjust my thermostat setting and get a new optimal temperature. So this is what happens in an ideal world where customers are perfectly responsive and there's no limited cognitive ability or there's no behavioral barrier for customers to readjust. But in reality, as I said, that's not the case. So now let's go to a more realistic setting, and that's exactly the purpose of this talk, how consumers in reality make decisions. So our goal here is to propose some theoretical framework that captures consumer behavior at an aggregate level. And for that, we are going to use a notion called rational inattention, which has been developed in economic literature over the last couple of decades and is gaining more and more attention. So what does rational inattention means? Rational attention argues that even in situations where a decision maker has access to all possible options and all information needed to make the optimal decision, the decision maker does not necessarily optimize his decision frequency because, as I said, there is some cost associated with the optimizing the decision. So now we want to use this notion into our decision making for a household to see how that affect the optimal thermostat setting, for example. In this setting, we say consumer decisions essentially can be represented as a random variable x and we make that more clear. So let's say capital W is the external random shock or like that outdoor temperature which can be presented using a random variable with a given mean mu and a standard deviation sigma, and in the literature it has been shown that normal distribution would be a very good representation of outdoor temperature. So let's say W is outdoor temperature which follows a normal random variable and then x is my decision which again is captured using a random variable. What does it mean? It means that my decision can be represented as a decision rule, which is a mapping from the value to a conditional distribution. So in other words, unlike the perfectly responsive case in which once W is realized, there is a corresponding optimal x, here we say once W is realized, we have a distribution over x, we have a conditional distribution which determines how I react to x, meaning that if W happens to be let's say 40 degrees, with some probability my indoor setting is going to be 70, with some probability it's going to be 75, with some probability it's going to be 80 and so on. So I don't have a singleton anymore, I have a distribution which represents my behavior, represents my decision rule, and this particularly makes sense when we are facing a collection of households. So maybe each household has a simpler decision rule, but at an aggregate level where we put them all together, their collective decision rule can be represented using a random variable. And then the question is, what is this conditional probability, how this conditional probability can be determined, and other than notion of rational inattention, when we know this conditional probability, we can infer how much information should be processed by the consumer in order to make that particular decision. So let's say, as I said, if the distribution G corresponds to outdoor temperature W, which for example can be normal, then this term which is referred to as Shannon mutual information for those of you who are familiar with information theory, those of you who are from computer science background, probably you are familiar with this notion of mutual information, Shannon's mutual information. So this quantity represents how much information I have to process if the conditional distribution is given by F. So the higher this value is, it means that I'm more active, I'm reacting more, I'm more responsive to W. On the other hand, if this quantity is small, it means that I really don't care much about W. So in other words, this quantity measures the amount of information inherent in X corresponding to W. So the closer these two are, which means I'm more active to, I'm more reactive to W, then this quantity is going to be higher. So then I can recast the consumer decision problem as maximization over this conditional distribution. So what would be, what would be a typical consumer do? So again, as before, a typical consumer would try to maximize her expected utility, the same that we had before, minus some cost of processing information. So this term is added to the consumer problem to capture that cognitive limitation, to capture the fact that we are not willing to always optimize our thermostat setting or optimize the optimize our consumption decision as the, as our ambient environment changes or if there is new shock to our demand pattern. So this is exactly the core. This is what is going to make the difference. So this parameter lambda can be represented as the marginal cost of processing information. In other words, the higher the lambda, the more difficult it is for me or there's a higher opportunity cost for me to readjust my decision. So this is exactly the link to a smart me to a smart technologies. So ideally smart technologies are meant to reduce this problem to make it easier for us maybe if you can pre program our thermostat or our other smart appliances, then this the optimization can happen in the background without us actively getting involved. And as a result, incurring this processing costs or incurring this opportunity costs. So that is the problem that should be incorporated into decision making and into the design of the Maddox programs. And probably the idea is that as more and more people start to use a smart appliances programmable appliances, this cost of processing information is going to be reduced. But again, if you look at the entire market, let's say there is a lambda hat associated with the entire market, which is captured using a representative household, and then that representative household has to solve this optimization problem. And the optimal solution to that is a FS start is conditional property, which is, as I said, is a decision rule, which tells us for any realization of w, what is the conditional distribution for my action x, and then correspondingly, what is the, what is the decision, what is the random variable representing the consumers decision as a function of price P. Okay, so luckily we are able to. So let me skip that for the moment and get back to data. So luckily we are able to fully characterize the optimal solution of the consumer in the presence of this competitive limitation. So let me remind you of the perfectly responsive responsive case. This was optimal solution when essentially lambda is zero. I can be optimized at any moment. I observe w, I adjust my optimal decision. Again, five minutes later, I again observe a different w, I adjust my decision. But what happens in the presence of lambda? In the presence of lambda, which is represents our cost of processing information, the consumer setting for the appliance X star and the random variable w which represents ambient environment follows a jointly normal distribution with given a correlation coefficient, as you can see here. And this is going to be the mean of my setting. And this is going to be the standard division. So what does it all mean? First of all, if you look at the mean, it resembles what we had above, except that w is replaced by mu. So on average, my thermostat setting is essentially the same as before on average, but now I have some fluctuation. So here, essentially, the variance of X is the same as the variance of w because for any w, I have a new X and correlation is one. But here, as you can see, if lambda is too high, this correlation is going to be zero, meaning that if consumers have a high cost of processing information, they completely become non responsive to what's what's happening in the ambient environment. So that's probably the case for some of us, if, for example, we really value our time, or we really do not want to incorporate this, what's going, what is happening outside into our decision and so probably that means that, well, our correlation coefficient is essentially zero. But on the other hand, as lambda decreases, correlation between my decision and what is happening in the ambient environment increases and becomes large. Okay, what does it tell us? So I told you, I showed you before like three different apartments from New York City and their behavior in terms of outdoor temperature versus outdoor. So this is the aggregate for a few, I think they are about like 100 apartments. So this is the aggregate level of their consumption decision. You can see the random variable representing their decision that random variable X that I talked about before. And this is the random variable w. And as you can see, there is a correlation coefficient of point 18. So if I'm a utility company, this is really important. It matters that how their indoor setting varies in response to outdoor temperature, because obviously that has significant implications on how the overall consumption of electricity is going to unfold over time. Okay, so what is the effect of price? The effect of price is relatively obvious here. For any given lambda, what it can be shown is that for any given price, as electricity becomes more and more expensive, the consumer decision gets closer, sorry, becomes less expensive as price decreases. The consumer decision becomes closer to theta, the ideal level, that is my ideal temperature, and it becomes less variable and less correlated with w, which makes sense. If electricity is cheap, I really don't care much about w, there is less correlation between my decision and outdoor temperature. But on the other hand, as p increases and it becomes more expensive, I become more responsive. I deviate more from theta, my ideal temperature, and my decision becomes more and more correlated with w. In particular, if electricity is very cheap, meaning that below a certain threshold represented by p hat, then I become fully non responsive, which means that what's happening in my ambient environment really does not influence my overall decision. Okay, so now on the other hand, what is the effect of lambda? As I said, presumably lambda can be a measure of the level of, basically how much, or what fraction of consumers incorporate smaller appliances into their, for example, into their house. And as a result, it lowers the cost of processing information, it would help to give attention that they have to pay to adjust their consumption decision. So as you can see, as lambda increases, essentially, even though the consumer decision remains insensitive to lambda on average, but it becomes less variable and less correlated with w's. And that's exactly, sorry, that's exactly what's expected to happen here. So essentially, if the cost of processing information is too hard, you really don't act, you are not really responsive. But if lambda decreases, then it's not really costly for you to adjust your decision. And as a result, you become more responsive and you lower, for example, maybe you lower your thermostat setting when outside is too cold, or you increase your thermostat setting by a few degrees when you realize that outside is too high in order to save electricity or in order to actually help the environment. So now what is the implication of this for the utility firm? Let me skip this in the interest of time and get to the bottom of it, because at the end of the day, if you are a utility firm, what you care about is overall consumption quantity. Okay, now that we know how the consumers make decision in the presence of this cognitive limitation, what is its implication on the overall consumption that the utility firm is facing. In effect, what we are doing here, it allows us to fully characterize in close form the stochastic demand curve which is being faced by the utility firm. So we can show that the overall consumption quantity faced by the utility firm follows a non-central square distribution and we can fully characterize its mean and its variance. And, well, okay, without paying too much attention to this complicated mathematical equation, what does it mean? Well, it means that as price increases, okay, consumption quantity decreases on average. That's expected, more expensive electricity, less consumption, and also as lambda increases, consumption on average increases, which is a bad thing. Again, as cost of process information is higher, we know that consumers consume more on average. How about variance? Interestingly, same is true about the variance, meaning that as price increases, the variance in the consumer's behavior, which leads to the variability in the total consumption faced by utility firm is also decreasing. And similarly, if the cost of processing information is higher, which means that if lambda increases, we see that there is more variability in what the utility firm should observe in the total consumption. So in other words, we see the dual role played by both price and lambda in regulating demand. Not only a higher price or a lower, not only a higher price or a lower lambda leads to less consumption, which is preferred. It also leads to less valuable consumption, which obviously has a huge impact on reliability of the electrical grid, meaning that the possibility of facing blackout and brownout is also governed by the value of P and value of lambda. So in other words, if you are the utility firm to take this into account, and you can fully characterize what is the stochastic demand curve that you are facing, and not only how the expected demand changes with respect to P and lambda, but also how the variability of demand also reacts to these changes in P and lambda. And I realize that I have only a couple of minutes to wrap up. So, therefore, I do not have time to go over the risk of our derivations in terms of, well, if you are utility firm, how you should incorporate this into finding you or into setting your optimal price. The only thing that I want to mention, so this is the utility firm's optimization problem, which I'm going to skip, but the bottom line is that if you are a utility firm, and as I said, you are facing this population of consumers which have a limited cognitive ability or face a cost of processing information, when you want to set your price, well, we can essentially picture it in these two-dimensional graphs. On the horizontal line, we have lambda, which represents cost of processing information, and as I said, it's a kind of a rough measure of, like, the level of smart appliances which is being incorporated into the consumer population. And then on the left, on the horizontal line, on the vertical line, we have this parameter cap up, which is kind of a notion of the weather, so my ideal temperature minus the average temperature divided by variance, and for different seasons and different geographical areas, obviously this quantity is going to be different. So we can see that, in fact, under the firm's optimal pricing, what's going to happen is that there are these two thresholds on this parameter lambda, so that if we are between these two thresholds, it is optimal for the firm to set price in a way that it induces no response from the consumer. On the other hand, once we are outside this region, the firm has to set its optimal price in a way that it induces more responsive behavior from the customer. And then what happens if there is, as all of us know, obviously firms do not have full freedom in setting their optimal price because they are, especially in the regulated markets, they are regulated by public utility commissions which set an upper bound on the price that they can set. So if you have an upper bound on price that can potentially introduce this new region in this two dimensional graph, which, again, in this region, the price is so that consumers are non responsive to changes in the environment under the optimal price. Now what if we have an additional restriction for reliability of the system, which means that you want to make sure that the variability or the variance in overall consumption is below a threshold so that the probability of facing a blackout or brownout is a smaller than a very small epsilon, let's say 1 over 1000. So if you have this reliability constraint, the results are pretty much similar. The only difference is that this non responsive region shrinks. So if you remember from the previous slide, in the previous slide, this non responsive region was represented using this U shaped region. Now it becomes a smaller in the presence of a reliability constraint. So with that, let me just wrap up by summarizing essentially what we have done here so the main goal of this research is to provide a unified framework for analyzing how schools are supposed to make a physical consumption decision. And this is an alternative and more informed explanation for some of the inelasticities that we observe in practice as in response to variations in price or variations in weather. This provides a kind of a building block for let's say utility firms or policymakers to come up with a better demand response programs design more effective demand response programs because it allows them to better predict the consumers behavior and what kind of a statistical pattern should expect for any given price. What kind of a statistical pattern should expect from the consumer population and the dual role which is played by price, not only in reducing the average consumption, but also in regulating the variation in consumption which obviously plays a role in reliability of the electrical grid. So with that, I think I've already two minutes over time. Let me stop here. The only thing that I want to mention is that actually luckily recently we have also obtained some data from one of the producers of one of the manufacturers of smart thermostats. It's a very rich data set and now they are using that data to empirically test the validity of this theoretical framework that they have developed here and see what is the estimated parameter lambda that I just presented here, what is the estimated parameter lambda for different customer population, let's say different cities or in different regions and how that can be used in order to come up with something commendations for better pricing decisions for policymakers and utility firms. So that's all I have to say. Thanks for attention. With that, I'm going to stop and happy to happy to hear any comments or any questions that you might have. Thank you for the presentation. I see that there is one question. The question is, will this smart technology be able to reduce lambda to zero? So, if I understand the question, so the question was does smart technologies will be able to reduce lambda? Is that the question? Yeah, that's exactly the idea because again, what does at an aggregate level, what does lambda represent? Lambda is a measure of, well, how hard it is for me to react to changes in the demand, changes in my ambient environment. Now, if you have a smart appliances, which can be pre-programmed, then, for example, I do not need to actively engage in making those kind of decisions or adjusting my different thermostat settings, for example, my water heater. I don't need to constantly check the outdoor temperature, let's say every hour or every two hours or every five hours. And then if I say, okay, this is a very cold night or this is a very hot summer afternoon and as a result, I want to readjust my decision. All of them can be pre-programmed into smart appliances and they can automatically implement those decisions for me without me actively trying to re-optimize. And as a result, this lambda potentially becomes much smaller and then our goal is to see, okay, now, how smaller lambda can help utility firms and how system operators to essentially lower overall consumption and the variability in consumption and improve reliability of the product. I don't know if that answered the question. If there is a follow-up question, you can unmute yourself to ask the question. Are there any other questions? Yes, there's, are there programs that are focusing on trying to reduce lambda already? So, to some extent, yes. For example, I mentioned some of these apartments in New York City. I was presenting this paper at Georgia Tech and one of the faculty there told me that, especially for some of these underprivileged neighborhoods, there are some of these nonprofit organizations which try to nudge customers, for example, based on what's happening, for example, with outdoor temperature. They send text messages to customers, for example, to say, this is a good time for you to, let's say, if it's a hot summer afternoon, to lower your thermostat setting by a couple of degrees and that helps you to save significantly on your electricity bill, for example, or that really helps us to improve the reliability of the grid. So, these kind of actions, yes, they are already underway, especially for people who are very sensitive to the electricity bill, which probably does not include people like us, but there are people who are really sensitive to electricity bill, or in Europe, for example, where electricity is much more expensive than here, people who need, react to what's happening outdoor. But now, in terms of, again, in terms of responding to the question, before we get into a, hopefully, at some point we get to a level where there are more and more smart appliances and lambda is significantly reduced. What is happening right now, as I said, is to nudge customers or actively provide them with the kind of information that they need so that maybe hopefully they can incorporate those and potentially that can lead them to adjust their decisions. So, because lambda is overall captures two things, it costs of acquiring and processing information. So, at least these kind of nudging systems using, let's say using text messages, that helps the customers for the acquiring part. At least they don't need to acquire, they don't need to think about it. Now, if I receive a text message, then I know that, okay, it's a good time for me to react to it and, for example, change my thermostat setting to save on my electricity bill or help the power grid. So that is some of the kind of ad hoc measures that are happening already. But, again, overall, I assume that the best solution to achieve lower lambda is incorporation of smart technologies as it's already happening in some parts. By the way, there is a follow up question to the first question, will lambda ever be reduced to zero? Well, maybe not necessarily to zero, but again, the question is, the point is that the smaller lambda is, so obviously it has implications on the friend's optimal pricing, but overall, it also reduces the overall variation that we see in the consumption. Basically, it would make it much easier for us to predict demand. So whether it's reduced to zero or not, it depends whether we ever get to a point that we have smart appliances which can, again, as I said, be pre-programmed fully to the extent that they react to what's happening outside. So if not only the presence of a smart appliances, it's also how we use them in the sense that, let's say, if I really don't, if I'm completely insensitive to price, and I only care about my comfort, and I really don't care about the environment, then incorporation of smart appliances is not going to really help. But the only thing that the smart appliances are going to do is to empower me to lower my lambda, but then again, it's a trade-off between my comfort, between my price sensitivity, and this cost of processing information. So if I really put a high weight on my comfort, and I put a very small weight on price sensitivity, which meaning that I'm not really price sensitive, or I'm not really environmentally conscious, then no matter how smart my appliances are, if I don't use that toward that goal, then obviously lambda is not going to reduce. On an aggregate level, I can imagine that, as I said, with more incorporation of smart appliances, I expect the magnitude of this parameter to decrease over time. Any other comments? Yeah, any other questions? Yeah, there is one. What do you mean by price of information processing? The rest of my knowledge lambda has to reflect the cost of power delivery at the location of the customer. Does the information processing a new term on the price? Oh, no, no, lambda is not price. So in my notation, in my formulation, so if I go back to the consumer optimization problem. Essentially, lambda is the, let's call it inconvenience or the cost that I incur if I want to readjust my decision. So, so this is, this is exactly what lambda represents. So on the one hand, I want to maximize my expected utility, which is my comfort from, which is the comfort that I get from my temperature setting minus the price that I have to pay for the electricity that I consume. But there is a third part which represents the hassle or the opportunity cost that I have to incur to reoptimize my decision or readjust my consumption decision. So that's what lambda represents. So again, the more closely read this X and WR X being the random variable which represents my action, W being the random variable which represents my ambient environment, the more closely these two are tied together. It means that I'm really responsive, meaning that my X is much more dependent on W. Okay, that means that I'm really responsive and processing a lot of information to readjust my consumption decision in response to W. As a result, I'm processing more information. So lambda is the unit cost of processing information. So what is this overall information that is being processed for two random variables X and W, that is what I presented here using the Shannon's mutual information. So we have to measure how closely tied these two distributions are, and the higher this value is, it means that I really care a lot about W. I'm really responsive to W. On the other hand, if it's really small, it means that I ignore W. This can be as low as zero, which means I completely disregard W when I set my X. This means that I'm a non responsive consumer. And the other extreme is when this is really high, which means that, well, I'm a very responsive consumer when I observe W, I readjust my X, and then obviously all those readjustments are costly from the cognitive perspective, and I have to penalize for that. So that's exactly what I'm doing here. I'm penalizing for the fact that I'm re-optimizing my decisions over time. And that is captured using this parameter lambda. Okay. Does that answer the question? Does that answer the question? There is a follow-up question. What is the typical meaning of W? Okay, so very good. So again, in the context of this talk, overall W is any random shock which influences my demand. Okay. Here, to set, to fix the idea, I use W as a random variable representing, let's say, for example, outdoor temperature. So meaning that, because think about it, if you set your thermostat setting, let's say at 72, how much your AC is going to consume depends on how cold or how hot outdoor is. Let's say I can keep my indoor setting at 72 for the entire season, so let's say for the entire winter, but how much I consume during each, let's say hour, obviously depends on outdoor temperature. So as outdoor temperature fluctuates, which is a random variable, the amount of consumption, even though my X is constant, the amount of consumption varies. So then the question is, do I react to that or I don't? And yeah, again, in response to a question, the physical meaning of W in the context of the model that I presented today is, let's say, outdoor temperature. It's just a random variable which represents how cold or how hot outdoor is. And then obviously, because that has an obvious implication on my overall consumption. There is a new message. Yes, to incorporate decision making frequency or interval. And my second minute or how to say the fact now. Okay, if I understand the question correctly, how to include decision making frequency or interval. So remember here, I'm capturing a population of households, you are right. You are saying that if it's an individual household individual households are not like, they are not like observing the optimal, observing the outdoor temperature and then they say, okay, now I'm using this conditional probability distribution to determine what is going to be my indoor setting. Of course, that's not how we make our decision. For example, we say maybe every five hour or maybe every day or every half a day we set our AC setting. So I assume that's exactly what you mean by interval. Meaning that we have a decision rule that we follow. Maybe in the morning when we leave the house, we set our AC to some level. And then when we come back, we readjust it to another level and then when we go to sleep, maybe we change it to another level and so on. So I assume that's what you mean by interval. But the argument here is that when we are facing a population of customers. Okay. It is as if this aggregate level decision follows this conditional distribution. So again, this is not to say that this is how in reality consumers make decision, but it says the consumer decision making can be made. It can be represented or can be framed using this framework at an aggregate level. Okay, so you have your own interval or you have your own decision rule in terms of how you set your AC. I have my own, let's say Chinwoo has his own. And then but an aggregate level, when you look at the entire market, entire population, it is as if there is this conditional distribution which governs the overall decision, the overall consumption decision that this representative consumer makes. So that's the idea here. Any more questions? I have one question. When you say aggregate, are you looking at a feeder level? So when you say what? When you say aggregate, when you aggregate the consumers, are you looking at a feeder level? What I mean is the, let's say a zip code. It depends on what population utility firm is facing in terms of the aggregate demand function. So let's say I'm serving, I'm supplying electricity to a region or to a city. And the question is on a hot summer afternoon, what should I expect? What kind of consumption pattern is statistically, what kind of consumption pattern I should expect from this population of households. So this framework is a kind of aggregate, is a way of capturing that overall consumption at an aggregate level, if that makes sense. So you are running this every 15 minutes or so, you know, because the problem itself is static, it's a static problem, but you run it every 15 minutes. So if you make it more dynamic, capturing the dynamics of the network, the network will tell you exactly or roughly how often you should run these optimization problems. So the idea, you're right, maybe I should have mentioned that in terms of, okay, what is the time frame in this optimization problem? Obviously, we can, it's up to us in terms of how we want to set the time horizon. But let's say the time horizon is a block of time during which it is safe to assume that, let's say, for example, outdoor temperature follows this normal distribution. So let's say my time frame is from 2pm to 7pm. I know that between 2pm and 7pm, this normal distribution is a good estimation or good approximation for my outdoor temperature. And then the question is, during this five hour block, what is going to be the distribution of x, which dictates the distribution of consumption quantity. If I run this optimization problem and follow up consumption quantity problem that I showed you in subsequent slides, that tells me that, look, during this five hour time block, this is the probability distribution of your total consumption. So, of course, as you say, total consumption varies by moment, but the answer to this question gives me the distribution of that, that gives me the statistical pattern or the distribution of total consumption over this time block, over this, let's say five hour time block. Does that make sense? Okay, yeah. Do we have any more questions? Okay, so, Professor Alissa Mears, thank you for the presentation.