Chapter 4: Utility PDF

Chapter 4: Utility 1 Overview 1. Utility functions 2. Constructing a utility function 3. Some examples of utility functions 4. Marginal utility 5. Marginal utility and MRS 6. Utility for commuting Source: Varian, chapter 4 2 1...

Chapter 4: Utility 1 Overview 1. Utility functions 2. Constructing a utility function 3. Some examples of utility functions 4. Marginal utility 5. Marginal utility and MRS 6. Utility for commuting Source: Varian, chapter 4 2 1. Utility functions A utility function assigns a number to every possible consumption bundle such that more-preferred bundles get assigned larger numbers than less-preferred bundles. Consider a utility function u(X ). This function will assign numbers to bundles such that: (x1 , x2 )  (y1 ; y2 ) if and only if u(x1 , x2 ) > u(y1 , y2 ) The only important feature of the numbers a utility function assigns to bundles is that it preserves the ranks. Because of this emphasis on ordering bundles of goods, this kind of utility is referred to as ordinal utility. 3 Suppose we have three bundles A, B, and C , where A  B  C. All of the following utility functions represent the same preferences: u1 (A) = 3 u2 (A) = 17 u3 (A) = −1 u1 (B) = 2 u2 (B) = 10 u3 (B) = −2 u1 (C ) = 1 u2 (C ) = 0.002 u3 (C ) = −3 Since only the ranking of the bundles matters, there can be no unique way to assign utilities to bundles of goods. Any transformation of a set of numbers that preserves the order of the numbers is called a monotonic transformation. A monotonic transformation is a function f (u) that transforms each number u into some other number f (u), in a way that u1 > u2 implies f (u1 ) > f (u2 ). 4 A transformation is monotonic if it has a positive slope: df (u) >0 du Proposition: If f (u) is any monotonic transformation of a utility function that represents some particular preferences, then f (u(x1 , x2 )) is also a utility function that represents those same preferences. Proof: 1. If u(x1 , x2 ) represents a set of preferences, then u(x1 , x2 ) > u(y1 , y2 ) ⇔ (x1 , x2 )  (y1 , y2 ). 2. If f (u) is a monotonic transformation, then u(x1 , x2 ) > u(y1 , y2 ) ⇔ f (u(x1 , x2 )) > f (u(y1 , y2 )). 3. Therefore, f (u(x1 , x2 )) > f (u(y1 , y2 )) ⇔ (x1 , x2 )  (y1 , y2 ). A monotonic transformation of a utility function is a utility function that represents the same preferences as the original utility function. 5 In a cardinal theory of utility, the size of the utility difference between two bundles of goods is supposed to have some sort of significance. It tells us how much more a consumer prefers bundle A over bundle B. Cardinal utility isn’t needed to describe choice behavior. 6 2. Constructing a utility function Well-behaved preferences (satisfying the assumptions of completeness, transitivity, monotonicity, convexity) can be represented by a utility function. A utility function is a way to label indifference curves. All bundles on the same indifference curve have the same utility. Higher indifference curves get assigned a larger value. 7 Figure 1. Constructing a utility function from indifference curves (Fig 4.2 in Varian) 8 3. Some examples of utility functions 3.1. Indifference curves from utility To draw indifference curves from a given utility function u(x1 , x2 ), plot all the points (x1 , x2 ) such that u(x1 , x2 ) equals a constant k. Suppose utility is given by u(x1 , x2 ) = x1 x2. Each indifference curve has the formula: x2 = k x1. Consider now the utility function v (x1 , x2 ) = x12 x22 v (x1 , x2 ) = x12 x22 = (x1 x2 )2 = u(x1 , x2 )2 v (x1 , x2 ) is a monotonic transformation of u(x1 , x2 ) and has the same shaped indifference curves. Only the labels change. v (x1 , x2 ) describes exactly the same preferences as u(x1 , x2 ) since it orders all of the bundles in the same way. 9 Figure 2. Indifference curves k = x1 x2 for different values of k (Fig 4.3 in Varian) 10 3.2. Perfect substitutes Preferences for perfect substitutes can be represented by a utility function of the form u(x1 , x2 ) = ax1 + bx2 The slope of the indifference curve is −a/b and indicates the rate at which good 1 is traded off against good 2. Examples: Good 1 is 0.25l Stella and good 2 0.25l Jupiler u(x1 , x2 ) = x1 + x2 Good 1 is 0.25l Stella and good 2 0.5l Jupiler u(x1 , x2 ) = x1 + 2x2 11 3.3. Perfect complements The utility function for perfect complements takes the form u(x1 , x2 ) = min(ax1 , bx2 ), where a and b are positive numbers indicating the proportions in which the goods are consumed. Examples: x1 is the number of right shoes and x2 is the number of left shoes u(x1 , x2 ) = min(x1 , x2 ) A consumer uses 2 teaspoons of sugar with each cup of tea. If x1 is the number of cups of tea and x2 is the number of teaspoons of sugar:   1 u(x1 , x2 ) = min x1 , x2 2 12 3.4. Quasilinear preferences Quasilinear preferences can be represented by the following utility function u(x1 , x2 ) = v (x1 ) + x2 In this case, the utility is linear in good 2 but possibly nonlinear in good 1. All indifference curves are vertically shifted versions of one indifference curve. The equation for an indifference curve takes the form x2 = k − v (x1 ) with k a different constant for each indifference curve. √ Examples: u(x1 , x2 ) = x1 + x2 or u(x1 , x2 ) = ln(x1 ) + x2 13 Figure 3. Quasilinear preferences (Fig 4.4 in Varian) 14 3.5. Cobb-Douglas preferences Another commonly used utility function is the Cobb-Douglas utility function u(x1 , x2 ) = x1c x2d , with c and d positive numbers. Cobb-Douglas indifference curves are convex monotonic indifference curves. Cobb-Douglas preferences are the standard example of indifference curves that look well-behaved. As with other utility functions, we can represent the exact same preferences with any monotonic transformation of the Cobb-Douglas utility function. 15 Figure 4. Cobb-Douglas indifference curves (Fig 4.5 in Varian) 16 Two examples of monotonic transformations of a Cobb-Douglas utility function: 1. v (x1 , x2 ) = ln(x1c x2d ) = c ln x1 + d ln x2 These indifference curves will look exactly the same as the Cobb-Douglas indifference curves. 1 c d 2. v (x1 , x2 ) = (x1c x2d ) c+d = x1c+d x2c+d (1−a) Define a = c c+d. Then we can rewrite v (x1 , x2 ) = x1a x2. This means we can always use a monotonic transformation to ensure that the exponents in a Cobb-Douglas utility function sum to 1. 17 4. Marginal utility How does a consumer’s utility change when we increase the quantity of good 1? The rate of change of the utility function as we add a little more of good 1 is called the marginal utility with respect to good 1: ∆U u(x1 + ∆x1 , x2 ) − u(x1 , x2 ) MU1 = = ∆x1 ∆x1 This shows how utility changes when the quantity of good 1 is changed, for a given quantity of good 2. The marginal utility of good 1 can be calculated by the partial derivative of the utility function w.r.t. good 1: u(x1 + ∆x1 , x2 ) − u(x1 , x2 ) ∂u(x1 , x2 ) MU1 = lim = ∆x1 →0 ∆x1 ∂x1 We take the partial derivative because we hold the quantity of good 2 constant. 18 The marginal utility w.r.t. good 2 is defined in a similar way: u(x1 , x2 + ∆x2 ) − u(x1 , x2 ) ∂u(x1 , x2 ) MU2 = lim = ∆x2 →0 ∆x2 ∂x2 Note that the magnitude of the marginal utility depends on the magnitude of utility. Taking monotonic transformations of the utility function will change the magnitude of the marginal utility. Marginal utility itself has no behavioral content, but we care only about the ordering of bundles implied by any utility function. 19 5. Marginal utility and MRS The MRS (marginal rate of substitution) measures the slope of the indifference curve at a given bundle of goods. It can be interpreted as the rate at which a consumer is just willing to substitute a small amount of good 2 for good 1. Consider a change in the consumption of each good (∆x1 , ∆x2 ) that keeps utility constant. MU1 ∆x1 + MU2 ∆x2 = ∆U = 0 Solving for the slope of the indifference curve we have ∆x2 MU1 MRS = =−. ∆x1 MU2 The MRS is negative, because to stay on the same indifference curve, an increase in one good implies a decrease in the other. 20 The general equation for an indifference curve is u(x1 , x2 ) = k. Totally differentiating this identity gives: ∂u(x1 , x2 ) ∂u(x1 , x2 ) du = dx1 + dx2 = 0 ∂x1 ∂x2 The first term measures the change in utility as we change x1 , and the second term measures the change in utility as we change x2. We want to pick these changes so that the total change in utility, du, is zero. dx2 We can now solve for dx1 , which is the MRS: dx2 ∂u(x1 , x2 )/∂x1 =− dx1 ∂u(x1 , x2 )/∂x2 This is analogous to the expression for the MRS on the previous slide. 21 Suppose we take a monotonic transformation v (x1 , x2 ) = f (u(x1 , x2 )). We can calculate the MRS by using the chain rule ∂v /∂x1 ∂f /∂u ∂u/∂x1 ∂u/∂x1 MRS = − =− =− ∂v /∂x2 ∂f /∂u ∂u/∂x2 ∂u/∂x2 The MRS is independent of the utility representation. If two utility functions have the same MRS, then they represent the same preferences. While the utility function and the marginal utility function are not uniquely determined, the ratio of marginal utilities gives us an observable magnitude: the MRS. If you multiply utility by 2, the MRS becomes: 2MU1 MU1 MRS = − =− 2MU2 MU2 22 Example: Cobb-Douglas preferences Cobb-Douglas preferences are of the form: u(x1 , x2 ) = x1c x2d The MRS is: MRS = − ∂u/∂x ∂u/∂x2 1 cx1c−1 x2d =− dx1c x2d−1 cx2 = − dx 1 23 We can also compute the MRS by taking the log transformation of the Cobb-Douglas utility function: v (x1 , x2 ) = c ln x1 + d ln x2 The MRS is: ∂v /∂x1 MRS = − ∂v /∂x2 c/x1 = − d/x 2 cx2 = − dx 1 In both cases, the MRS depends only on the parameters c, d and the quantities of the two goods currently in the bundle. 24 5. Utility for commuting Utility functions are a way of describing choice behavior. In the field of transportation economics, economists have estimated utility functions to study consumers’ commuting behavior: Commuters can choose between taking public transport or driving to work. Each of these alternatives represents a bundle of characteristics: travel time, waiting time, out-of-pocket costs,... Let (x1 , x2 ,..., xn ) be the characteristics of driving and (y1 , y2 ,..., yn ) of taking public transport. Average consumer’s preferences for characteristics can be represented by a utility function of the following form where the coefficients β1 , β2 ,...,βn are unknown parameters: U(x1 , x2 ,..., xn ) = β1 x1 + β2 x2 +... + βn xn 25 Domenich and McFadden (1975) estimated such a utility function by observing commuter behavior. U(TW , TT , C ) = −0.147TW − 0.0411TT − 2.24C TW = walking time, TT = travel time and C = $ costs The coefficients describe the weights that an average household places on the various characteristics (marginal utility of each characteristic). The ratio of one coefficient to another measures the marginal rate of substitution. The average commuter would be willing to subsitute 1 minute of walking time for about 3 minutes of travel time. The average commuter valued a minute of commute time at 0.0411/2.24 = 0.0183 $ per minute 26 We can use the marginal rate of substitution to estimate the value that each commuter places on reduced travel time or improvements in other characteristics of public transport. Estimated utility functions can be very valuable for determining whether or not it is worthwhile to make some investments in the public transportation system. We can also answer questions like, what happens to commuting decisions when the price of gasoline goes up, or when the price of transit passes goes up? 27