Ok, now, let's focus on our Expected Utility Function. So, now, we know, for input '3', we can get a value for f(x) = 9. If we plug in the value '9' to this inverse function, we get 3. So, if we want to find out, for what value of x, we can get f(x) or y = 9, then this inverse function can answer the question. For various input x, you get various output y, right? The inverse function of f(x) is a function that will find out, for what input value 'x', you will get a given 'y' value.ДЇor example, if a function is, then, its inverse function is. If you are not familiar with Inverse Function, then here is the simplest explanation. Once you get the Expected Utility, you need to find the Inverse Function of your Utility Function. In that way, you will get a number for Expected Utility E, as shown here. So, whatever is your utility function, you pass that real-world value 400,000 and -50,000 to your utility function. The expected utility of a random variable is basically the weighted sum of the utility value, where the weight represents the probability, as depicted by the following expression. Say, your utility function is U(X), where x is the real-world monetary gain from a risky investment.Д«ased on that utility function, you can calculate the Expected Utility of the above investment. If you can model your utility function for monetary gain from a risky investment (like this one), then using that utility function, you can calculate your certainty equivalent. We have learned that utility function maps a real-world value to a satisfaction level. Perhaps, you should ask more than 5,000$? How can you figure out? Is there any way to find that number mathematically? But you do not know exactly what should be the right price to sell out your business plan. Say, you are not convinced by the 5,000$ selling price of your business plan that your friend proposed. Let's get back to your original business plan. If you agree to pay that 2000$ to buy the piece of mind that, you are covered for that worst-case scenario, then your certainty equivalent for this gamble is 2000$. But, if something breaks down that would cost 10,000$ to repair, then, your gain will be 8,000$ in that event. If it won't break down, then, your upfront payment of 2,000$ will be certainly lost. Well, you are not sure if the car will definitely break down or not. The dealer is asking that, if you pay just 2,000$ to buy an extended warranty, then in those worst cases, they will cover the cost.

Say, you bought a used car and you are worried that if something happens to the car (engine or transmission break down) in the next 5 years, it can cost you 10,000$. It is a guaranteed return that someone would accept, rather than taking a chance on a higher, but uncertain return.ДЄ practical example of certainty equivalent is your insurance payment or extended warranty of something very expensive to repair. Therefore, Certainty Equivalent is the amount of money that is equivalent in your mind to a given situation that involves uncertainty (or risk). What do you think? will you give up the business opportunity for 5,000$? If yes, then, in the economics term, this 5,000$ is the Certainty Equivalent for this uncertain opportunity, as depicted by the following decision tree.

You simply get a certain 5,000$ for giving up the business opportunity. Your friend will invest his money and take the risk. The condition is that, you won't execute that plan, rather your friend will do that. Your business plan is so solid that, one of your friends proposed you buy out the business plan from you for 5,000$.

But there is a 50/50 chance that you may lose all of your investment money and gain nothing. Within a year, you can expect a profit of 400,000$. According to your plan, you need to invest 50,000$ in your business. Thinking about a new startup and got an excellent idea.

Power utility certainty equivalent