EE290O / IEOR 290 Lecture 05

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1 EE290O / IEOR 290 Lecture 05 Roy Dong September 7, 2017 In this section, we ll cover one approach to modeling human behavior. In this approach, we assume that users pick actions that maximize some function, known as their utility function. Such methods form a general class of human models, and include rational agent models, consumer preference theory, choice theory, utility-maximizing models, and von Neumann-Morgenstern utility theory. Much of this content will be drawn from David Autor s notes for MIT s Microeconomic Theory and Public Policy notes from Fall 2010, which is available on MIT s OpenCourseWare. Most of his material is drawn from (?), which is considered to be a standard textbook in economics, and is quite comprehensive. This section will heavily feature assumptions on human s preferences and behaviors. Our goal in this presentation is to outline what assumptions are commonly made in the literature, and what their consequences are. For clarity, whenever a formal proposition or theorem is stated, we will explicitly specify which assumptions are invoked in such a claim. 0.1 Ordinal preferences First, we outline the axioms of consumer preference theory. Suppose A and B are two outcomes, in a general sense. For example, A can represent receiving a bundle with 3 apples and B can represent receiving a bundle with 3 oranges. We wish to formally discuss whether a consumer would rather have outcome A or outcome B. Thus, a person is basically modeled by a set of relations of the form x is preferred over y and x and y are equally preferred over the set of possible states of the world. To regularize all possible sets of relations, we assume some axioms. For now, we will think of outcomes as just a general set of different points, but when applying this theory to particular models, we will be specific about what set of outcomes we are considering. Additionally, we will now suppose there is only one consumer under consideration, so we won t need to explicitly mention them throughout this text. As a reference, we ll let A denote the set of possible outcomes. Definition 0.1 (Preferences). For any two outcomes A A and B A, we say A B to denote that the consumer prefers A over B. We say A B to denote that the consumer is indifferent between A and B. We ll sometimes allow the notation A B to denote the case where A B or A B, and similarly, A B simply means B A. Assumption 0.2 (Completeness). For any two outcomes A A and B A, exactly one of the following is the case: 1. A B 2. A B 3. B A Assumption 0.2 ensures that a consumer must have an opinion between every pair of outcomes, and that opinion must be consistent in the sense that they cannot simultaneously prefer A over B and prefer B over A. Assumption 0.3 (Transitivity). For any three outcomes A, B, C A, if A B and B C, then A C. Assumption 0.3 hopefully seems reasonable. Deviations from this assumption are often seen as irrational, and are difficult to analyze. For example, if A is a doll, B is a toy truck, and C is a yo-yo, then an agent who does not satisfy Assumption 0.3 would pick up one toy, say, the yo-yo (C), see the toy truck, and put down the yo-yo to pick up the truck (B C), see the doll, and put down the truck to pick up the doll (A B), and see the yo-yo and put down the doll for the yo-yo (the non-transitive C A), and so on, and so on. 1

2 Nevertheless, there have been studies that have shown that Assumption 0.3 does not always hold. When Assumptions 0.2 and 0.3 hold for a preference relation, then is said to be rational. Definition 0.4 (Indifference curves). For an outcome A A, we define the indifference curve associated with A as IC(A) = {B A : A B}. Proposition 0.5 (Indifference curves are an equivalence class). Under Assumptions 0.2 and 0.3, the relation gives an equivalence relation, and for any A A, IC(A) is the equivalence class of A. This is easy to show; one only needs to check that the relation is reflexive, symmetric, and transitive. Note that this implies every outcome A A belongs to some indifference curve, the set A can be partitioned into indifference curves, and for any two indifference curves I 1, I 2, either I 1 = I 2 or I 1 I 2 =. We can define a cardinal utility function in a special case from these two assumptions. Proposition 0.6 (Existence of a utility function with finitely many outcomes). Let A be a finite set. Assumptions 0.2 and 0.3, there exists a function u : A R such that x y if and only if u(x) > u(y). Under This can easily be seen by just ordering the finite set of outcomes. Assumptions 0.2 and 0.3 are the most commonly used assumptions, although we will add a few more in this section to allow the definition of cardinal utility functions across a continuum of outcomes. Now, we specialize to the case where A R n, so an outcome x A can be thought of as x = (x 1,..., x n ). Let s interpret x i as the amount of a good i given to the consumer, so x = (x 1,..., x n ) as an outcome means that x 1 of good 1 is given, x 2 of good 2 is given, and so on. We then add another assumption which is both intuitive and incredibly convenient from a technical perspective. Assumption 0.7 (Continuity). We assume A R n and x, y A are two outcomes such that x y, then there exists an ɛ > 0 such that, for all y (B ɛ (y) A), x y. In words, if x is preferred to y, then x is preferred over all outcomes infinitesimally close to y. If this is the case, it turns out there are enough regularity assumptions to define a cardinal utility function. Proposition 0.8 (Existence of a utility function). Under Assumptions 0.2, 0.3, and 0.7, there exists a function u : A R such that x y if and only if u(x) > u(y). Assumption 0.9 (Non-satiation). We assume A R n, and for any x, y A, if x i y i for all i [n] where the equality is strict for at least one i [n], then x y. 1 Assumption 0.9 states that people always strictly want more. In particular, at any outcome x A, there s always some x that is preferred over x. Thus, consumers will always chase the dragon. This assumption also gives us a nice shape to our indifference curves. Note that, for a point x A, if y x and y x, then x i > y i for some i and x j < y j for some j. Additionally, this means that our indifference curves can be seen as functions. This is visualized in Figure 1. Proposition 0.10 (Indifference curves as a function). Under Assumption 0.9, for any indifference curve I, any point x I, and any good i [n], {y I : y j = x j for all j i} = {x}. In other words, if we pick the consumption of the n 1 other goods, then either there is no point on the indifference curve matching this consumption or there is a unique level of consumption for the last good which is on the indifference curve I. In this sense, each indifference curve can be thought of as a function from a subset of R n 1 to R. This subset is exactly the projection of the indifference curve I onto the n 1 coordinates. Definition 0.11 (Marginal utility and marginal rate of substitution). Suppose we are given some utility function u : A R, and, furthermore, suppose u is continuously differentiable. We define the marginal utility of a good i at x in the interior of A as Mu i (x) = u x i (x). We define the marginal rate of substitution of i for j at x in the interior of A as MRS i,j (x) = Mu i (x)/mu j (x). The marginal utility of i is how much another unit of good i would increase utility. The marginal rate of substitution of i for j is how much more of good j is required to compensate for the loss of one unit of i, i.e. how many units of j are required when i is decreased by one unit to keep utility constant. Now, we make an assumption that consumers prefer diversity in their consumption patterns. 1 We re starting to see some of the complexities of notation arise here, so be mindful of when is used to indicate indifference between two outcomes, and = is used to indicate that they are the same point in R n, and similarly versus >. 2

3 Figure 1: By Assumption 0.9, if we consider the indifference curve associated with x, we know that it must be entirely contained within the light-blue squares. (The strictness of the non-satiation principle states that the indifference curves cannot even be located at the dark blue lines.) Assumption 0.12 (Diminishing marginal utility). We assume that there exists a utility function u whose marginal utilities are well defined, i.e. u is continuously differentiable. For any good i and initial outcome x Ω, we assume the function α Mu i (x + αe i ) is decreasing. Recall that e i denotes the unit vector pointing at coordinate i. Proposition 0.13 (Diminishing marginal utility and convexity). Assumption 0.12 holds if and only if the indifference curves of u are the graphs of a convex function. As we have seen, convexity makes a lot of things very nice, so Assumption 0.12 is usually made to help regularize some problem formulations. In general, if a result can be derived without this assumption, it is better to exclude it. However, it is very important to note that this assumption leads to convexity of the indifference curves. Oftentimes, u itself will be concave, resulting in convex indifference curves. This is shown in Figure 2. Example 1 (Cobb-Douglass utility function). A commonly used utility function is one of the form u(x, y) = x a y 1 a for a (0, 1). We make one final note on cardinal versus ordinal preferences before we move almost entirely to cardinal utility functions. It is entirely possible that two different cardinal utility functions u and v give rise to the exact same set of ordinal preferences, i.e. for all x, y A, u(x) > u(y) if and only if v(x) > v(y). Under these different representations, we have that the marginal utilities may change, i.e. Mu i (x) may not equal Mv i (x). However, we have that the marginal rate of substitution is the same, i.e. MRS u i,j (x) = MRSv i,j (x). 0.2 Utility functions and demand From this point on, we will suppose we are given some utility function u : A R. Whereas ordinal preferences were easy to talk about, the value of the utility function is a little more abstract. For example, what exactly does it mean when u(x) = 5? What are the units of it? 3

4 Figure 2: By Assumption 0.9 and 0.12, the indifference curves of the utility function will always have this downward sloping convex shape. The common practice is to measure utility in terms of a unit called utils, and, usually, acknowledge that these are fundamentally different between people, i.e. we cannot compare the value of 5 utils to Jane with the value of 5 utils to John. Now, in this section, let us suppose that A = {x R n : x 0}, and u is continuously differentiable. Our consumer wants to maximize their utility subject to some budget constraint, i.e. we are given a price vector p R n, where p i is the price of good i, and some overall budget I. (We ll assume p i > 0 for each i.) We wish to solve: max x 0 u(x) subject to p x I Note that, in this formulation, the money spent doesn t affect the utility directly. In many settings, the utility of a consumer will also be a function of the amount of money spent, i.e. I ll be happier with the same goods but an extra $50 in my pocket. However, in this formulation, the utility is entirely a function of what goods the consumer has, and the consumer wants to totally spend their predetermined budget. Recalling some necessary conditions, we have that, if x opt is a local minima and regular and u is continuously differentiable, for any good i [n] such that x opt i > 0: This implies, for any pair of consumed goods i, j [n]: u x i (x opt ) λ opt p i = 0 MRS i,j (x opt ) = p i p j Thus, at optimum, the marginal rate of substitution is equal to the ratio of the prices, which is intuitive from an economic perspective. Another rearrangement of optimality conditions yields for all consumed goods i [n]: Mu i (x opt ) p i 4 = λ opt (1)

5 Thus, λ opt is a shadow price, which indicates how many more utils the consumer could get with the next dollar of budget. (It might be useful to think of p i as the derivative between amount of good i and money spent.) Indirect utility functions Take optimization (1). We can look at an optimizer x opt view it as a function of the price and budget, i.e. x opt = x opt (p, I). In this way: u(x opt (p, I)) = max x 0 u(x) subject to p x I Definition 0.14 (Indirect utility function). We define the indirect utility function as V (p, I) = u(x opt (p, I)). In contrast, we will often call u the direct utility function. Thus, the indirect utility function takes in a price and budget, implicitly optimizes the consumer s consumption profile, and outputs the utils of that price and budget pair. It s also interesting to note that there s a result in economics arising from the assumption of (1), known as the carte blanche principle. In words, it states that, since a consumer is always optimizing their utility subject to the price and budget constraints, they are always weakly better off with a cash donation than any in-kind transfer of identical value. Proposition 0.15 (Carte blanche principle). For any amount of cash c > 0 and gift y 0 such that p y = c: V (p, I + c) max x 0 u(x + y) subject to p x I Furthermore, the indirect utility function is positively homogenous with degree 0. Proposition 0.16 (Homogeneity of the indirect utility function). For any t > 0: V (tp, ti) = V (p, I) This is pretty easy to see by the definition. For this reason, much of the literature will state the indirect utility function as solely a function of prices, with the implicit assumption that the budget has been normalized to 1. Definition 0.17 (Normalized indirect utility function). We define the normalized utility function as: V (p) = V (p, 1) We also note that we can recover the utility function from the indirect utility function. Proposition 0.18 (Recovery of direct utility function). Suppose u is strictly concave and twice continuously differentiable. Furthermore, suppose Assumption 0.9 holds. Then for any x > 0: Proof. For any p 0 such that p x 1: u(x) = min p 0 V (p) subject to p x 1 u(x) max y 0 u(y) subject to p y 1 Note that the right-hand side is V (p); minimizing across p 0 such that p x 1 yields: u(x) min p 0 V (p) subject to p x 1 To prove the other direction, we ll have to use the sufficient optimality conditions for inequality constrained optimization. (To do so, we will think of the problem as minimizing u(x).) Fix x and we will provide a p such that x is optimal. Note that, by Assumption 0.9 and the fact x > 0, u x i (x) > 0 for all i. 5

6 Thus, if we define λ opt = u(x), x, we can see λ opt > 0. If we define p i = u x i (x)/λ opt, we can see that p > 0 and p x = 1. Additionally, x satisfies the conditions of Proposition?? using this p, with 2 xxu 0 by the strict concavity assumption. Thus, we have found a p 0 such that p x 1 and V (p) = u(x). It follows that, if we were to minimize a set including this p: u(x) min p 0 V (p) subject to p x 1 We have shown both directions of the inequality, proving equality Expenditure functions It s been established that I love duality. Economics is rife with duality as well, and the dual optimizations generally have nice interpretations. For example, if the primal optimization is profit maximization, then the dual optimization is cost minimization subject to an output constraint. In this setting, the same thing applies: maximizing utility subject to a budget constraint is dual to minimizing expenditures subject to a utility constraint. 2 Recall optimization (1), which is parameterized by p, I: max x 0 u(x) subject to p x I The dual optimization is as follows, parameterized by p, v: min x 0 p x subject to u(x) v For optimization (2), we can define the indirect function as well; in this setting it is called the expenditure function. We let x opt = x opt (p, v) be the solution to (2) as a function of the parameters p and v. Definition 0.19 (Expenditure function). The expenditure function is given by E(p, v) = p x opt (p, v). The expenditure function tracks how much money is spent when prices are p and the consumer has utility v. In theory, the expenditure function allows us to avoid directly addressing and comparing utils. For example, suppose there are two goods in the market and currently, the prices are p = (p 1, p 2 ) and the consumer is experiencing utility v from their purchases, which costs them E(p, v). If we introduce a $50 tax on good 2, then the consumer will have to spend E((p 1, p ), v) to attain the same level of utility. This means that this tax would cost the consumer E((p 1, p ), v) E(p, v) if they were to maintain the same utility. The expenditures and indirect utility functions are inverses of each other, in the following sense. Proposition 0.20 (Expenditures and indirect utility functions). We have for any prices p > 0, budget I, and utility level U: V (p, I) = U if and only if E(p, U) = I In other words: V (p, E(p, U)) = U E(p, V (p, I)) = I (2) Demand In economics, there are two types of demand functions: Marshallian (uncompensated) demand, and Hicksian (compensated) demand. Both take as inputs their prices, but the former holds income constant while the latter holds utility constant. Formal definitions follow. Definition The Marshallian (uncompensated) demand function is the mapping D M : p, I x opt (p, I), where x opt is the parameterized solution for the optimization (1). The Hicksian (compensated) demand function is the mapping D H : p, v x opt (p, v), where x opt is the parameterized solution for the optimization (2). 3 2 It s worth noting that this duality is, strictly speaking, not the same as duality in the optimization community. 3 In the economics literature, the Marshallian demand D M is often denoted d and the Hicksian demand D H is often denoted as h, but, to emphasize the fact both are demands, we use a slightly different convention. 6

7 Hopefully, the distinction between uncompensated and compensated is intuitive. Namely, if some policy were to affect the prices p, then the Marshallian (uncompensated) demand function states the amount of consumption at the same budget, i.e. consumers have the same income and must reallocate their spending to best handle the policy change. On the other hand, the Hicksian (compensated) demand is how the ratios of consumption change, when there is some compensation that allows users to attain the same level of utility as before. We ll quickly go over some economic concepts of goods now. For a demand function D, we ll write D[i](p, I) to denote the consumption of the ith good when the price is p and income is I. Definition 0.22 (Demand effects). The substitution effect (sometimes price effect) of a change in price from p 1 to p 2 (at income I) is defined as D H (p 2, U) D H (p 1, U) where U = V (p 1, I) is the utility from the original price and income. This is Hicksian in nature. The income effect of a change in price from p 1 to p 2 is D M (p 2, I) D H (p 2, U), where U = V (p 1, I). This is Marshallian in nature. Thus, the substitution effect tracks the change in the ratio between goods due to a change in their relative costs. It holds utility constant and compensates income so that the consumer stays on the same indifference curve. On the other hand, when prices increase, consumers become effectively poorer; the income effect isolates the effect of that change. This is visualized in Figure 3. Figure 3: This graph represents the income and substitution effects due to a change in the price. Initially, at price p and income I, the user chooses x init with utility U init. When the price of good 1 increases from p 1 to p 1, the feasible set of consumptions shrinks, moving the consumer s consumption to x m, on indifference curve U m. By increasing the user s income until the new budget line hits U init, we can find the compensated demand x h. The substitution effect is x init x h and the income effect is x m x h. Now, if we fix a good i and consider D M [i] (p, I). Note that this is how the demand of good i will change with a change in income; if this partial derivative is positive, then an increase in income increases the demand in the good. The income effect is equal to D M [i] (p, I)D M [i](p, I). Alternatively, if we consider D H[i] (p, V (p, I)), this represents the substitution effect due to a price change. If this partial derivative is negative, then when the price of good i increases, people will consume less of it as other goods are substituted for it. The term D H[i] (p, V (p, I)) actually equals the substitution effect. 7

8 Putting these two together, we recover the Slutsky equation: D H [i] (p, V (p, I)) D M [i] (p, I)D M [i](p, I) = D M [i] (p, I) There are also two very famous identities which allow us to recover the demand functions from our previously defined functions. Shepard s lemma allows us to find the Hicksian demand through the expenditures: D H [i](p, U) = E (p, U) Roy s identity allows us to find the Marshallian demand through the indirect utility function: Finally, we discuss good types. V p D M [i](p, I) = i (p, I) (p, I) V Definition 0.23 (Good types). Suppose both the Marshallian and Hicksian demands are continuously differentiable. Then: 1. A good i is normal at p, I if D M [i] (p, I) > 0 and D H[i] (p, V (p, I)) < 0. Thus a price increase (which essentially reduces income) causes consumption to go down both through the income and substitution effects. D M [i] 2. A good i is inferior at p, I if (p, I) < 0 and D H[i] (p, V (p, I)) < 0. Thus a price increase (which essentially reduces income) has indeterminate effect on consumption; through the income effect, consumption is increased, while through the substitution effect, it is decreased. 3. A good i is strongly inferior (Giffen) at p, I if D M [i] (p, I) < 0 and D H[i] (p, V (p, I)) < 0. Thus a price increase (which essentially reduces income) increases consumption. 0.3 Revealed preferences We quickly note that sometimes, economists operate under a weaker set of assumptions than those outlined in Section 0.1. The theory of revealed preferences is a weaker set of assumptions used when possible. Assumption 0.24 (Weak axiom of revealed preference). If A and B are feasible outcomes and A is chosen, then whenever A and B are feasible, the consumer will choose A over B. Simply put, consumers choose what they prefer, and they are consistent on their pairwise preferences. In practice, this means that if we ever observe a consumer choosing A over B, we know they will prefer A over B forever. The strong axiom adds transitivity. Assumption 0.25 (Strong axiom of revealed preference). If A i and A i+1 are feasible outcomes and A i is chosen, for i = 1,..., n 1, then whenever A 1 and A n are feasible, the consumer will choose A 1 over A n. 0.4 Expected utility theory All that we ve discussed thus far has assumed that all the outcomes can deterministically be chosen by the consumer. In a lot of real world settings, the outcomes are stochastic after the customer makes a decision. For example, if a consumer buys a lottery ticket for a $1, then the outcome for the consumer is not determined until after the Friday draw. In this section, we present some of the main results in expected utility theory; in particular, we show the von Neumann-Morgenstern utility theorem von Neumann-Morgenstern utility theorem In this section, we will let A denote the decision of the user, and Y denote the possible outcomes. Definition 0.26 (Probability distributions). Let (A) denote the set of probability distributions over the set A. When the set A is finite, a probability distribution p (A) is sometimes called a lottery. 8

9 Note that, even when A is finite, (A) is infinite, since it is the space of probability distributions on A. In this case, we can think of the user s decision as inducing a probability distribution over Y. In other words, the user picks an action a A, which induces a distribution over outcomes. We ll let this mapping be denoted f : A (Y). To go back to the lottery example, the user can decide to either buy or not buy a lottery ticket. Thus, A = {buy, not buy}. We have that f(not buy) yields the distribution that is identically $0, and f(buy) yields the distribution that is $1 with probability 1 ɛ and $1000 with probability ɛ. 4 Assumption 0.27 (Consequentialist agents). If two actions a 1, a 2 A induce the same probability distribution, i.e. f(a 1 ) = f(a 2 ), then the user is indifferent between a 1 and a 2. This essentially means that the agent s decision is independent of how the problem is framed, and only depends on the probability distribution over what happens in the end. For example, I can organize a free lottery which awards $1000 at the end with probability ɛ and takes $1 away from you with probability 1 ɛ. Alternatively, I can organize a $1 lottery which awards $999 with probability ɛ and otherwise does nothing. From most technical perspectives, these lotteries look totally equivalent; however, when people interact with these lotteries, they often make different decisions. Thus, some economists spend a lot of care to formalize lotteries in a fashion where these lotteries are distinct mathematical objects; we will not take such care in this section. Time permitting, we may discuss some results in behavioral economics which play with the difference between seemly equivalent lotteries. Now, we look at a preference relation over lotteries (Y). Again, we will use Assumptions 0.2 and 0.3, which are completeness and transitivity. Additionally, we will use a continuity assumption, which is slightly different than Assumption 0.7 due to the structural change in the formulation. Assumption 0.28 (Continuity). For any two lotteries L 1, L 2, if L 1 L 2, then there exists an ɛ > 0 such that for all L 1 B ɛ (L 1 ) (Y) and L 2 B ɛ (L 2 ) (Y), we have L 1 L 2. Another assumption is that mixing in a third lottery does not affect the preferences. Assumption 0.29 (Independence). For any three lotteries L 1, L 2, L, we assume that L 1 L 2 if and only if αl 1 + (1 α)l αl 2 + (1 α)l for all α (0, 1). Definition 0.30 (Expected utility form). Suppose Y = [n]. We say u : (Y) R has expected utility form if there exists (u 1,..., u n ) such that: n u((p 1,..., p n )) = p i u i This is sometimes called a von Neumann-Morgenstern (VNM) expected utility function. The nice thing about the expected utility form is that, like its name suggests, we can treat agents is simply maximizing their expected utility. Proposition A utility function has the expected utility form if and only if, for any set of lotteries {L k } and positive weights {α k } such that α k = 1: k=1 i=1 ( K ) K u α k L k = α k u(l k ) k=1 Proposition 0.32 (von Neumann-Morgenstern utility theorem). If a preference relation satisfies Assumptions 0.2, 0.3, 0.28, and 0.29, then it can be represented with a VNM expected utility function. The key property, often used, of VNM expected utility functions is: u(αl + (1 α)l ) = αu(l) + (1 α)u(l ) (3) This holds for any lotteries L, L and α (0, 1). In this sense, u is linear with respect to α, but it is not linear in general, i.e. u(2l) 2u(L). We ll investigate this in more detail in the next section. 4 Note that money in a lot of this theory is treated like any other good, with users having some level of preference of money over other goods. 9

10 0.4.2 Risk sensitivity In the previous section, we generally thought of each outcome of a lottery as some abstract entity: it could be a million dollars, a Grammy, or a Ford Fiesta. 5 When we talk about risk, we will assume all outcomes are basically some allocation of cash. Thus, a lottery must be a distribution between different cash allocations. In this section, suppose we have a VNM expected utility function u. Next, suppose we have some lottery L = (p 1,..., p n ) which distributes over outcomes Y = {y 1,..., y n }, where each y i R is some cash allocation. By assumption: n u(l) = p i u(y i ) i=1 Now, suppose we have some giant insurance company who can law-of-large-numbers out any randomness. They are willing to buy your lottery from you and give you the lottery s expected value. Let Y be a random variable with distribution L. Then: n E[Y ] = p i y i This is sometimes called the certainty equivalence lottery. Similarly, we will write u(l) = E[u(Y )]. Basically, when we analyze the risk sensitivity of an agent, we re asking: Would they accept this buyout of their lottery? That is, would they rather take the safe thing or roll the dice? Definition 0.33 (Risk sensitivity). An agent is risk averse if u(e[y ]) > E[u(Y )]. An agent is risk seeking if u(e[y ]) < E[u(Y )]. An agent is risk neutral if u(e[y ]) = E[u(Y )]. We note that these definitions are deeply reminiscent of Jensen s inequality. Proposition If an agent is risk averse, then u is concave. If an agent is risk seeking, then u is convex. If an agent is risk neutral, then u is affine. As far as indifference curves go, the opposite holds. The indifference curves are convex if u is concave and vice u versa. (Under the assumption of non-satiation, i.e. x i > 0.) We can quickly convince ourselves of this in a simple case. We have that u is twice continuously differentiable and convex. Suppose we have two goods, x and y, and we are considering the indifference curve {(x, y) : u(x, y) = U}. Non-satiation already guaranteed us that this function will be invertible. The implicit function theorem gives us that: ( ) 1 dy u dx = u x y We can see that dy dx < 0 by the non-satiation assumption; i.e. to maintain the same utility when the consumption of x is increased, we must decrease y. Let f(x, y) = dy dx (x, y), and let f x, f y denote the partial derivatives of f, and similarly f xx, f xy, f yy the second partial derivatives. Similarly, u, so f = u x u 1 y. If we take the next derivative: i=1 Breaking it up: f x = u xx u 1 y f y = u xy u 1 y d 2 y dx 2 = f dy x + f y dx = f x + f y f + u x u 2 y u xy = u 2 y ( u xx u y + u x u xy ) + u x u 2 y u yy = u 2 y (u yy u x u xy u y ) 5 Full disclaimer: I hate Ford Fiestas. 10

11 Putting it together: d 2 y dx 2 = u 2 y ( u xx u y + u x u xy ) + fu 2 y (u yy u x u xy u y ) = u 2 y ( u xx u y + u x u xy ) u x u 3 y (u yy u x u xy u y ) = u 2 y [ u xx u y + u x u xy u 2 xu 1 y u yy + u x u xy ] = u 2 y [u xx u y 2u x u xy + u 2 xu 1 y u yy ] = u 2 y [u xx u y 2u xy u x + u yy u 2 xu 1 y ] [ ] [ ] uy = u 2 uy y u x u 1 2 u y u x u 1 y By the assumed convexity, we have 2 u 0 and u y > 0, which lets us conclude that this function is actually concave. We part with one final comment. In the previous section, we showed that diminishing marginal utility would imply the indifference curve is convex. In this section, we showed that risk aversion leads to convex indifference curves. The key thing to note is that the axes have different interpretations in these two sections. In this section, each axis corresponded to one outcome, which, here, was an allocation of cash. In the previous section, each axis corresponded to an amount of good consumed. 11