[MUSIC]. Our next task is to formulate and solve the consumer's optimization problem using utility theory. That sounds pretty technical and complicated, but all we're saying is that given the money they have in their pockets, consumers are going to find the best way to spend it. That's optimizing behavior. And all economic agents are assumed to optimize something as a way of explaining their behavior. For example, firms typically are assumed to maximize profits. In the consumers case, we assume that consumers maximize utility subject to a budget or income constraint. In plain terms this means that consumers have a certain amount of income to spend. And subject to their budget constraint and given a menu of prices, they will choose a market basket of goods that will provide them with the greatest utility or satisfaction. The question is how do consumers do this? And the answer lies in something called the utility maximizing rule or equimarginal principle. A consumer with a fixed income facing market prices will achieve maximum satisfaction when the marginal utility of the last dollar spent on each good, is exactly the same as the marginal utility of the last dollar spent on any other good. Let's take a few minutes now to prove this. By so doing, we can derive the downward sloping demand curve and gain further insight into market equilibrium on the consumer's side of the market. Assume then that health food nut Greg [INAUDIBLE] is trying to decide how many Big Macs and Dove Bars he should buy with his fixed daily income of $10. This table summarizes the marginal utility that Greg will get from the consumption of the first, second, third and so on units of each of these goods. Note that for both products, marginal utility is declining. Now in order to make this table really work for us, we have to take into account the different prices of each of the two products. We can do so by converting the marginal utilities in the table to a per dollar spent basis. We do this simply by dividing each marginal utility by the product price. This allows us to compare apples and oranges, as the saying goes, or in this case Big Macs and Dove Bars. Adding these columns to the table, here's what this looks like. Take a minute now to try and fill in the empty boxes and columns three and five. Did you get it? You can see that since the price of Dove Bars is one, the marginal utility per dollar is the same as the marginal utility. However, for Big Macs, a marginal utility of, say, 18 converts to a marginal utility per dollar of nine because it's price is $2. Now this next table illustrates the sequence of purchases that Greg will make to maximize his utility. With his first potential choice, he'll start by buying one Big Mac which will yield him a marginal utility per dollar of 12 and leave him with a dollar's income. This is clearly superior to buying the first Dove Bar, which only yields an MU of ten per dollar. Next, Greg will buy his first Dove Bar for $1.00 and a second Big Mac for $2.00, because both yield an MU per dollar of ten. So what will Greg do next with his remaining $5.00? Try filling in the remaining boxes. That's right, in his third potential choice, Greg opts for a third Big Mac, and with his fourth potential choice, he spends his remaining income of $3 on a second Dove Bar and fourth Big Mac. What's really interesting about this, besides the fact that Greg is a heart attack waiting to happen, is that Greg runs out of money exactly where the marginal utility per dollar of the two goods are equal. In this case, equal to eight. This proves more generally that utility maximized when the marginal utility of the last dollar spent on each good is exactly the same as the marginal utility of the last dollar spent on any other good. This is the utility-maximizing rule or the equimarginal principle. What's really neat about the equimarginal principle, is that it perfectly explains why demand curves slope downward. Let me show you how. Suppose that at the equilibrium point in our last example, we hold the marginal utility per dollar constant for the two goods. In this case, it would be equal to eight. Now further suppose that the price of Dove Bars increases. What happens to the marginal utility per dollar of Dove Bars, and how do you think that our consumer Greg will respond? Because price is in the denominator. The marginal utility per dollar of Dove Bars falls below the same ratio for Big Macs. Therefore, in order to maximize his utility, Greg will have to decrease his Dove Bar consumption and increase that of Big Macs. This clearly implies a downward sloping demand curve. As the price of Dove Bars rises, quantity demanded falls. This example can also help us understand the income and substitution effects that we introduced in the previous lecture. In this case, the substitution effect is obvious. As we've shown when the price of Dove Bars rises Greg increases his consumption of Big Macs. This is because the last dollars spent on Dove Bars yields less utility than the last dollars spent Big Macs. But what about the income effect? To understand this we have to understand the difference between nominal income and real income. Nominal income is the face value of what we have in our pocket or bank account. For Greg it's 10 bucks. Note however that when inflation increases, in this case when the price of Dove Bars rises, it reduces Greg's actual purchasing power. But simply, he can no longer afford to buy the same combination of Dove Bars and Big Macs that he once could, so real income is nominal income adjusted for inflation. And here's the punchline [SOUND] ; the portion of the increase in Greg's purchase of Big Macs due to his reduction in real income is the income effect. Now to complete these thoughts, let's use a little algebra to generalize the utility-maximizing or equimarginal principle to the case of many goods. It looks like this, study it carefully. And here's the second part of the equimarginal principle rule. Try to fill the empty boxes correctly with either an equals sign, a greater than sign, or a less than sign. Did you get it right?