Special Utility Functions PDF

4 Special Utility Functions The last notes showed how to approach a consumer’s maximization problem in general. In these notes, we will explore some specific utility functions that commonly appear in economics and discuss some of their properties. Concepts Covered Cobb...

4 Special Utility Functions The last notes showed how to approach a consumer’s maximization problem in general. In these notes, we will explore some specific utility functions that commonly appear in economics and discuss some of their properties. Concepts Covered Cobb-Douglas Perfect Substitutes Perfect Complements Properties of Utility Functions Corner Solutions 4.1 Cobb-Douglas The Cobb-Douglas utility function is one of the most widely used in all of economics. We have actually already seen the Cobb-Douglas utility function a few times so far. In general, a Cobb-Douglas utility function is one that has the form U (x, y) = xα y 1−α Where the parameter α will have an economic interpretation that we will discuss below. In past examples we used the utility function U (x, y) = x1/2 y 1/2 , which is a Cobb-Douglas function with α = 1/2. Let’s find the Marshallian demands (values of x and y that maximize the Cobb-Douglas function). First let’s find the MRS αxα−1 y 1−α α y M RS = α −α = (1 − α)x y 1−αx Note that the MRS is decreasing in x/y and is a function only of the ratio of x and y. Solving for Marshallian demands gives us I x∗ = α px I y ∗ = (1 − α) py Let’s take a look at these functions. First note that we could define the fraction of income spent on x as pxI x. Plugging in the Marshallian demand for x, we get that the fraction of expenditure is equal to α. Similarly, the fraction of expenditure on good y is given by 1 − α. 1 Example To see this property with some numbers plugged in, let’s assume U (x, y) = x1/3 y 2/3 px = 2 py = 4 I = 120 With these numbers, we get an optimal consumption of x∗ = 20 and y ∗ = 20. Since each unit of x costs 2 and y costs 4, the total spending on x is 40 (which is a third of their income), and the total spending on y is 80 (two thirds of their income). Notice that if we increased the price of one of the goods, then we change the quantity demanded of that good, but not the share. Let’s say we changed px = 4, then the quantity of x would fall to 10, but the consumer would still be spending 40 total on x. This property is specific to Cobb-Douglas and would not hold for a general utility function. 4.2 Perfect Substitutes Another type of utility that we’ve already seen in the class is called perfect substitutes. In these preferences, the consumer does not need to consume both goods to get utility. They can substitute one good with another. The general form for the perfect substitutes utility function is U (x, y) = Ax + By We add the coefficients A and B to allow for cases where the consumer is willing to substitute but not one for one. For example, if A = 4 and B = 1, then the consumer would need 4 units of y to substitute for 1 unit of x. If we take the MRS of this function we get A M RS = B Since A and B are fixed constants, the MRS does not change as we increase x and y, which means that indifference curves have constant slope. What if we try to set this MRS equal to the price ratio? A px = B py This expression doesn’t seem to be much help. There’s no x and y at all! To solve the problem, we need to think a bit differently. Let’s put in some numbers to make it more concrete. Return to the example above with A = 4 and B = 1. Then our MRS is 4. The consumer would be willing to trade one unit of x for 4 units of y. The price ratio tells us how many units they can actually trade. For example, if px = 2 and py = 1 then the consumer can trade 1 unit of x for 2 units of y. But 2 units of y is not enough to compensate for losing a unit of x. The consumer will always prefer to consume x over y and will therefore only want x at the optimum. 2  In general, the consumer will want to consume only x when the benefit of consuming x is greater than the cost. They will only consume y when the benefit of consuming x is less than the cost. This intuition gives us our result for Marshallian demand with perfect substitutes. We have 3 cases 1. If M RS > ppxy , the relative benefit of consuming x is always greater than the relative cost and the consumer will want to spend all of their income on x. Then the Marshallian demand for x is I x∗ = px ∗ y =0 2. If M RS < ppxy , , the relative benefit of consuming x is always less than the relative cost and the consumer will want to spend all of their income on y. Then the Marshallian demand for y is x∗ = 0 I y∗ = py 3. If M RS = ppxy , the consumer is totally indifferent between consuming x or y. Any combi- nation of x and y that is on the budget line maximizes consumption. Graphical Representation Since the MRS is constant for perfect substitutes preferences, indifference curves are linear. We can no longer find the point where the budget line is tangent to an indifference curve. Instead, indifference curves will always either be steeper or flatter than the budget line (or parallel). The graph on the left below shows the case where indifference curves (in blue) are steeper than the budget constraint. In this case, we have MRS is greater than the price ratio for any value and so the best the consumer can do is consume all x. This optimal point occurs at the lower right corner of the graph (see more details on “corner” solutions below). The graph on the right shows the opposite case. Indifference curves are flatter than the budget constraint so MRS is less than the price ratio and the optimal choice is to only consume y 10 10 8 8 6 6 y y 4 4 2 2 0 0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 x x 3 4.3 Perfect Complements On the other end of the preference spectrum from perfect substitutes, we have perfect com- plements. With these preferences, a consumer only gets utility if they consume both goods together. The general form for perfect complements is U (x, y) = min(Ax, By) For example x could be left shoes and y could be right shoes. One is not useful unless you also have the other. As with perfect substitutes, we cannot use the usual MRS equal to price ratio condition. In this case, it doesn’t work because min is not a differentiable function. Once again we have to be a bit more clever. Indifference Curves To plot an indifference curve, we again want to set Ū = min(Ax, By) It might not be immediately obvious how to plot such a function so let’s plug in some arbitrary numbers to make it more concrete. 50 = min(10x, 5y) The minimum function means that we will always take the minimum of the two values in the parentheses. Let’s start with x = 5, y = 10. Notice that with these numbers 10x = 5y = 50 so the consumer’s utility is exactly 50 as well. Now think about what happens as we increase x or y holding the other one fixed. If we increase x, the left-hand term in the min function increases, but the right hand stays fixed at 50 so the minimum is still 50. Similarly, if we increase y but hold x constant, utility also remains at 50. With other values for Ū the story is similar and the indifference curves end up looking like those pictured below. 20 15 Ū = 70 y 10 Ū = 50 Ū = 30 5 0 0 5 10 15 20 x 4 Marshallian Demand As mentioned above, we cannot use the MRS to find Marshallian demand in this case. However, we can use a similar process. Take a look at the graph below. Can you figure out which point represents the consumer’s Marshallian demand? 20 15 Ū = 70 y 10 Ū = 50 Ū = 30 5 0 0 5 10 15 20 x Although we cannot find the point tangent to the budget constraint since the curve is non- differentiable, we can find a similar point which occurs at the corner of the indifference curve that just barely touches the budget set. To find this point, we replace our MRS=price ratio condition with a new condition that Ax = By In other words, the two terms inside the minimum function need to give us the same value, which means we are not wasting money by buying more of one good without matching it with some of the other (i.e. we are not buying two left shoes when we only have one right shoe). We can then use this condition and the budget constraint to solve for Marshallian demand. Solving the condition above for y gives A y= x B Plug this into the budget constraint to get   A A I = p x x + py y = px x + py x = x px + p y B B Solve for x to get the Marshallian demand I I BI x∗ = A
= Bpx +Apy = px + B py B Bpx + Apy And plugging back into y gives AI y∗ = Bpx + Apy 5 4.4 Properties of Preferences While the previous sections outlined three possible forms of a utility function, it is also helpful to examine some properties of preferences and utility functions more broadly. This section outlines two important classes of preferences: convex preferences and homothetic preferences. Convex Preferences To this point, we have generally looked at utility functions that produce convex indifference curves. As we will see below, this convexity is important for our solution to the maximization problem. When a preference results in convex indifference curves, we say that the preference is convex. A consumer has convex preferences when they prefer a mix of different goods rather than all of one good or another. For example, in our Cobb-Douglas preferences above, U = x1/2 y 1/2 , we can see that a consumer gets more utility from a bundle like x = 10, y = 10 than they do from a more unbalanced bundle like x = 1, y = 25. Given a utility function, we can check for convexity by finding the MRS and checking that it is decreasing as we increase the ratio of x to y (an easy way to check this is to hold y constant and increase x - if the MRS decreases then we have convex preferences). If this property holds, then it means that the slope of an indifference curve is getting flatter as we move down and to the right (increase x and decrease y), which creates the convex shape. Homothetic Preferences A preference is homothetic if its MRS depends only on the ratio between goods and not the absolute amount. In other words, if we can write the MRS between x and y as a function of xy , then the preference is homothetic. Homotheticity is a useful property because it implies that the absolute amount of x and y are not important for determining the consumer’s choice. We only need to know the percentage of their income they spend on each good. A nice implication is that no matter what a consumer’s income is, they will still consume x and y in the same ratio, which allows us to avoid answering difficult questions about how to deal with consumers who have very different incomes. Of course, in reality, this property is unlikely to hold in many cases, but its importance for making problems easier to solve keeps it as an important piece of economic analysis. Examples: Show that the preferences represented by utility functions 1-4 are homothetic but 5 is not 1. U (x, y) = 3x2 + 4y 2 2. U (x, y) = ln(x) + ln(y) 3. U (x, y) = x + y 4. U (x, y) = x4 y 5. U (x, y) = x2 + x + y 2 6 4.5 Corner Solutions Concave Indifference Curves Imagine we had a utility function U (x, y) = x2 + y 2. Let’s try to solve for Marshallian demand using our usual strategy. For concreteness let’s take px = py = 1 and I = 100. Taking the MRS 2x x M RS = = 2y y Setting equal to the price ratio x = 1 =⇒ x = y y And plugging into the budget constraint 100 = x + y = 2x =⇒ x∗ = 50, y ∗ = 50 But is this our solution? This bundle gives U = 502 + 502 = 5000. What if we had instead spent all of our income on x? We could buy 100 units which gives U = 1002 = 10000. So the initial bundle is not the utility maximizing bundle. Let’s look at some indifference curves to see why. 120 100 80 y 60 40 20 0 0 20 40 60 80 100 120 x Looking at the curve that is tangent to the budget constraint, we can see a clear difference between the typical graphs we have drawn. Before, at the tangency point, there were no feasible bundles that gave as much utility as the tangent point. That is clearly not the case here. When we have concave indifference curves, instead of finding the maximum utility we can achieve by spending all income, we actually found the minimum. In this case, the maximum occurs at the corners of the graph (consuming all of one good or the other). How could we anticipate this result without drawing the graph? Let’s look back at the MRS. Remember that the definition of convex preferences we gave above required MRS to be decreasing as the ratio of x to y increased. But with M RS = x/y, it is clearly increasing as x/y increases. With these preferences, rather than preferring a mixture of x and y, the consumer prefers the extreme. The more x they receive, the more they like x to y and vice versa. Whenever we have concave preferences, we can no longer use the MRS equals price ratio condition and should instead check the corner points. 7 Corner Solutions with Convex Preferences While concave preferences always have corner solutions, convex preferences can potentially have corner solutions as well (particularly when they are convex but non-homothetic). To see an example, let’s take the utility function U (x, y) = ln(x) + y Taking the MRS, we get ∂U ∂x 1 M RS = ∂U = ∂y x Setting this equal to the price ratio 1 px py = =⇒ x∗ = x py px We can then plug this back into the budget constraint to get y ∗ I I = px x + py y = px (py /px ) + py y =⇒ y ∗ = −1 py For some parameter values, this solution does not present any new problems. Let’s assume px = py = 10 and I = 20. Then the consumer consumes 1 unit of x and 1 unit of y. Our solution works. But what if we only had I = 5? Plugging into the equations we found, we would get x=1 y = −1/2 So we get negative consumption for y!. Since we cannot consume a negative amount of the product, something must have gone wrong. Remember that when we set M RS = ppxy , we are looking for the point where the slope of an indifference curve is the same as the slope of the budget line. But there is no guarantee in general that this point will exist in the positive range. In this case, that point is at (1,-1/2). So what can we do? If we can’t have negative consumption, a logical next choice would be to make consumption of y as small as possible. So we set y = 0. That means all of our income is spent on x so we get x = pIx = 1/2. Let’s check the MRS at this point 1 M RS = =2 x The MRS is greater than the price ratio, which means we still want to consume more x if we can. We would be willing to trade some y to get additional units of x. But we have no y left to trade. Consuming 0 is the best we can do. 8  The graph below shows what is happening here 2 1 y A 0 B −1 −2 0 0.5 1 1.5 2 x While point B is the tangency point, it is not feasible because it would require a negative consumption of y. Instead, the consumer gets as close as they feasibly can to the tangency point by only purchasing x and ends up at point A. Corner Solutions Summary The two previous examples suggest the following steps for solving utility maximization problems in general 1. Calculate the MRS. If it is increasing as the ratio of x to y increases, the consumer has concave preferences. Check both corner points (consuming all x or all y) and choose whichever gives higher utility and we are done. 2. If we have convex preferences, calculate using MRS equal price ratio and the budget con- straint as usual. If both x and y are positive we are done (this is called an interior solution) 3. If one of the values for x or y is negative, we again have a corner solution. Set that value to 0 and the consumer will spend all of their income on the other good. 9 Optional: Constant Elasticity of Substitution The three types of utility function above are actually all special cases of a more general class of functions called constant elasticity of substitution (CES) preferences. The general form for CES preferences can be written as  1/r r r U (x, y) = αx + (1 − α)y It turns out that this function can be made equivalent to Cobb-Douglas (r = 0), perfect substi- tutes (r = 1), and perfect complements (r = −∞). You do not need to know how to show why that’s the case, but if you go further in economics, you may see them again. Once again let’s take the MRS  1/r−1 r r (1/r) αx + (1 − α)y rαxr−1 M RS =  1/r−1 r (1/r) αx + (1 − α)y r r(1 − α)y r−1 Which simplifies considerably to  r−1 α x M RS = 1−α y (Plug in r = 0 to see this is the same MRS as Cobb-Douglas and r = 1 to see perfect substitutes with A = α and B = 1 − α). CES preferences are both homothetic and convex and are widely used in economics. 10