Utility Maximization PDF

3 Utility Maximization In the last notes we talked about how to describe consumers’ preferences in a mathematical framework. Now we will use that framework to discuss how consumers make choices given their preferences. We will approach this decision in a couple different...

3 Utility Maximization In the last notes we talked about how to describe consumers’ preferences in a mathematical framework. Now we will use that framework to discuss how consumers make choices given their preferences. We will approach this decision in a couple different ways. These notes lay out the idea of utility maximization, meaning consumers will choose to buy whichever bundle of goods provides them with the highest possible utility. Concepts Covered The Consumer’s Utility Maximization Problem The Relationship Between MRS and Price Ratio Graphical Representation of Utility Maximization Marshallian Demand 3.1 The Consumer’s Utility Maximization Problem Setup We assume that a consumer is deciding how much to consume of two goods x and y given the prices of each good px and py and a given income I (Note that this framework could relatively easily be extended to more than two goods, but we will stick to two goods throughout this class). This setup takes the form of a constrained maximization problem. Formally, we could say that the consumer’s objective function (what they are trying to maximize) is their utility function max U (x, y) x,y While their constraint says that their total expenditure cannot be more than their income. px x + py y ≤ I We call this constraint the budget constraint. Although we write the constraint as an inequality here, we will always assume that the consumer gets positive utility from consuming more of each good and don’t get any utility from leftover income, so they will never want to spend less than their total income. Therefore, we can always take the budget constraint as a binding constraint. Setting this maximization problem up as a Lagrangian, we have L = U (x, y) + λ(I − px x − py y) 1 Solving the Problem Let’s take first order conditions of the Lagrangian above ∂L ∂U ∂U = − px λ = 0 =⇒ = px λ ∂x ∂x ∂x ∂L ∂U ∂U = − py λ = 0 =⇒ = py λ ∂y ∂y ∂y ∂L = I − px x − py y = 0 =⇒ I = px x + py y ∂λ Solving the first two equations for λ gives us ∂U ∂U ∂U ∂x ∂y ∂x px λ= = =⇒ ∂U = px py ∂y py Let’s look at this equation we just derived. The left hand side should look familiar based on the discussion in the previous notes. It is none other than the marginal rate of substitution of x for y. On the right hand side we have the relative price between x and y (we usually call this the price ratio). Before continuing to solve the problem, let’s dig deeper into this condition we just derived. 3.2 The Relationship Between MRS and Price Ratio Solving the consumer’s utility maximization problem gave us the result that when they maximize their utility, it must be true that the marginal rate of substitution between two goods must be equal to the price ratio of those two goods. This relationship is one of the most fundamental concepts not only in this class, but in all of economics, so it is worth taking some time to understand. To understand the economic intuition behind the result, let’s think about the tradeoff the consumer faces between consuming x vs y. On the benefit side, the consumer receives utility from consuming each good. Other things equal, they would prefer to consume the good that gives them more utility. However, they also need to consider the cost of each good. If the good that provides more utility is also more expensive, then it may not be worth the additional cost. The condition we derived above navigates this tradeoff by setting the marginal utility per dollar spent equal to each other. ∂U ∂U ∂x ∂y λ= = px py Note that this equation also gives us an economic interpretation for the Lagrange multiplier λ. It is the additional utility that the consumer would receive from spending one additional dollar (we could think of this as the marginal utility of income). We won’t spend much time on this idea in this course, but it is worth pointing out. Why is this condition necessary to be at a maximum utility? Let’s think about what would happen if it did not hold by looking at some examples. 2 Examples Assume a utility function of U (x, y) = x1/2 y 1/2 For each of the following consumption choices and prices, determine whether the consumer should be consuming more or less of each good to maximize consumption. Hint: we calculated in the last notes that the MRS for this utility function is y/x. 1. x = 4, 2. x = 2, 3. x = 3, 4. x = 2, 5. x = 4, y = 4, y = 2, y = 3, y = 4, y = 2, px = 1, px = 2, px = 1, px = 2, px = 3, py = 2 py = 1 py = 1 py = 2 py = 3 Looking at the first example, we have M RS = y/x = 4/4 = 1. Other things equal, the consumer would be willing to trade 1 unit of x for exactly 1 unit of y. However, the price of y is higher than the price of x. The consumer would be better off selling some y, which would allow them to buy twice as much x. Since they value both goods the same, this transaction would increase their utility. In the second example, we have the reverse case. Now x is relatively expensive compared to y, but the relative benefit I receive is the same for both goods (MRS is still 1). Therefore, I would be better off trading some x for some y. The third example has both MRS and the price ratio are equal to 1. Therefore, this bundle is already the maximizing bundle and there is no way to trade one good for the other to increase utility. For the 4th and 5th examples, we have equal price ratios (both equal to 1), but the MRS is no longer equal to 1. Let’s look at number 4. This consumer has more of good y than good x. Plugging into the MRS, we can see that they have M RS = 2. An MRS of 2 means that the consumer would be willing to trade 1 unit of x for 2 units of y. In other words, an additional unit of x is worth more to them than an additional unit of y. Since prices are equal, the consumer would be better off trading some of their y for an equal amount more x. The 5th example is the same logic but reversed, the consumer would prefer more y Summary We can summarize the results we saw in the examples above as follows: ∂U px 1. If ∂x ∂U > py , then the relative benefit of consuming more of good x (MRS) is greater than ∂y the relative cost of consuming more of good x (price ratio). Then the consumer should consume more x (and less y) ∂U px 2. If ∂x ∂U < py , then the relative benefit of consuming more of good x (MRS) is less than ∂y the relative cost of consuming more of good x (price ratio). Then the consumer should consume less x (and more y) ∂U px 3. If ∂x ∂U = py , then the relative benefit of consuming more of good x (MRS) is equal to ∂y the relative cost of consuming more of good x (price ratio). Then the consumer cannot increase utility by consuming more or less of x or y. They are at their optimal point. 3 3.3 Graphical Representation of Utility Maximization In this section we will further expand on the intuition of the utility maximization problem by looking at a graphical representation of the problem. Plotting the Feasible Set Let’s start by showing on a graph all of the combinations of x and y that are feasible choices for this consumer. In other words, we want to show any consumption bundle that lies within the consumer’s budget constraint. We can plot the budget constraint as I px y= − x py py Which we can see is just a line with intercept I/py and slope − ppxy (it is worth pointing out that the intercepts of the line show us how much the consumer could purchase if they spent all their income on just one of the goods - the x-intercept is I/px ). For example, the line below shows the budget constraint and the feasible set for an income of 16, a price of x of 1 and a price of y of 4. 10 8 6 y 4 2 0 0 2 4 6 8 10 12 14 16 x The shaded area in the graph above shows the possible consumption choices this consumer could make given their income. Everything outside the shaded area is too expensive for this consumer given their income. As we argued earlier, the consumer will always want to spend all of their income, but in principle they could feasibly choose to spend less. It is important to note here that the slope of the budget constraint is equal to the (negative) price ratio between x and y. If the price of y increased relative to x, the budget constraint would get flatter and if the price of x increased relative to y, the constraint would get steeper. Increasing the consumer’s income would shift the curve out parallel to the initial line. Now that we have shown all of the feasible choices this consumer could make, we want to find which point within this set maximizes the consumer’s utility. To do that, we will return to the concept of indifference curves introduced last time. 4 Maximizing Utility In the last set of notes, we defined an indifference curve that shows all combinations of x and y that give the same utility. Since the consumer’s goal is to maximize utility, they will want to be on the highest possible indifference curve. If there were no constraint, this would mean consuming as much x and y as they possibly can. However, remember that now we also have to worry about the consumer’s budget constraint. The budget constraint shows us the set of bundles that are feasible to buy for the consumer. We can summarize the consumer’s problem graphically as: choose the point that lies within the budget set that lands on the highest indifference curve (gives the most utility) Where will this occur? Let’s add in some indifference curves to the budget constraint graph above (the indifference curves are for the utility function U (x, y) = x1/2 y 1/2 ) 10 8 6 y 4 2 5 4 3 0 0 2 4 6 8 10 12 14 16 x First, note that all the points that lie on the indifference curve that gives 5 utility are out of reach for the consumer. They are too expensive given the consumers budget constraint and so cannot be choices. The consumer would be allowed to choose many of the points that lie on the indifference curve that gives 3 utility. However, the consumer can do better than that. Hopefully you can see that the maximum utility the consumer can achieve is at the point where the indifference curve that gives 4 utility just touches the budget constraint. Any other point in the budget set will be on a lower indifference curve than that point. In mathematical terms, we would say that the indifference curve is tangent to the budget line at that utility maximizing point. All that means is that the derivative of the indifference curve at that point is equal to the slope of the budget line. What is the slope of the budget line? It’s just px py. And what is the derivative of an indifference curve? It’s the MRS. So once again we have derived the result that the consumer maximizes utility when px M RS = py 5 3.4 Marshallian Demand Definition So far we have derived a condition that must hold for the consumer to be at a maximum (with some exceptions that we will discuss next time), which tells us that the MRS between two goods should be equal to the ratio of their prices. The next step is to use this condition to solve for the actual choice of x and y. Our ultimate goal is to derive an expression for the optimal consumption of x and y as a function of prices and income x∗ = x(px , py , I) y ∗ = y(px , py , I) We will call these the Marshallian Demands for x and y (named after Alfred Marshall, who developed much of this theory in the late 1800s). Example Let’s return to the utility function we used above U (x, y) = x1/2 y 1/2 We could then set up their Lagrangian as L = x1/2 y 1/2 + λ(I − px x − py y) We already showed that the solution to this problem must set px y px M RS = =⇒ = py x py Solve this equation for y (could also solve for x) px y=x py Now we can use our budget constraint to solve for x   px 1 I I = px x + py y = px x + py x = 2px x =⇒ x∗ = py 2 px Plug this back in to find y px 1 I px 1I y ∗ = x∗ = = py 2 px py 2 py These are our Marshallian demands for x and y. 6 Discussion Let’s examine these demand functions 1 I x∗ = x(px , py , I) = 2 px 1I y ∗ = y(px , py , I) = 2 py First, let’s just restate what these functions are telling us. The consumer takes their income and the prices of each good as a given. They cannot control these parameters. What they can do is choose how much of each good to consume given income and prices. The Marshallian demand for each good tells us how much they would like to consume for any set of income and prices. For example, let’s assume a consumer has an income of I = 100, px = 5 and py = 10. Plugging these numbers in, we find 1 100 x∗ = = 10 2 5 1 100 y∗ = =5 2 10 To check that we did the problem correctly, we can always check to make sure this allocation is actually feasible by plugging back into our budget constraint px x + py y = 5(10) + 10(5) = 100 = I We can also try plugging in some other feasible allocations into the utility function to convince ourselves that this is actually the best the consumer can do. For example, the bundle we found gives U = (10)1/2 (5)1/2 ≈ 7.07 utility. Let’s pick another random feasible bundle like x = 4, y = 8, which only gives U = (4)1/2 (8)1/2 ≈ 5.66 utility. The next thing to notice is what happens when we change the parameters of the model. Since the equations are symmetric, let’s focus on the Marshallian demand for x 1 I x∗ = x(px , py , I) = 2 px We can see that if we increase income, the amount of x the consumer wants to consume will increase (you may remember from Econ 1 that this means we can call x a normal good - more on this later in the class). As we increase the price of good x, the quantity the consumer will consume falls. This result means that the demand curve for x is downward sloping (as price goes up, quantity demanded goes down). In general, the price of good y could also have an impact on the quantity of good x, but it does not in this case due to the specific utility function chosen (called a Cobb-Douglas function - we will see this in greater detail in the next set of notes) 7