#LyX 2.0 created this file. For more info see http://www.lyx.org/ \lyxformat 413 \begin_document \begin_header \textclass amsart \begin_preamble \usepackage{graphicx} \usepackage{amsfonts} \end_preamble \use_default_options false \begin_modules theorems-ams eqs-within-sections figs-within-sections \end_modules \maintain_unincluded_children false \language english \language_package default \inputencoding auto \fontencoding global \font_roman ae \font_sans default \font_typewriter default \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \default_output_format default \output_sync 0 \bibtex_command default \index_command default \paperfontsize 12 \spacing single \use_hyperref false \papersize default \use_geometry false \use_amsmath 2 \use_esint 0 \use_mhchem 0 \use_mathdots 1 \cite_engine basic \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \use_refstyle 0 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle plain \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Title Expected Utility and Risk Aversion \end_layout \begin_layout Author Michael Peters \end_layout \begin_layout Date \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash today{} \end_layout \end_inset \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard This reading describes how people's aversion to risk affects the decisions they make about investment. Basically, the concepts used to do this analysis emerge naturally when people have expected utility preferences, but not otherwise. So, it illustrates one important way that expected utility is applied. \end_layout \begin_layout Standard This analysis also makes it possible to illustrate how to do \emph on comparative statics \emph default . Comparative statics typically involve calculations designed to show the direction in which changes in the environment move peoples' optimal decisions. Convincing comparative static results are ones that hold even if you only impose weak restrictions on preferences. So, for instance, to explain how an increase in price affects a buyer's demand when he or she has Cobb-Douglas preferences is not very convincing because Cobb Douglas preferences are a very special case. In fact, as we have already shown, an (uncompensated) increase in price will only reduce demand under very special conditions. That is why the method for doing comparative statics tend to be a little more sophisticated, and the questions asked tend not to be the most obvious ones. \end_layout \begin_layout Subsection Lotteries with a Continuum of Outcomes \end_layout \begin_layout Standard In portfolio theory, outcomes are all monetary. We looked at monetary lotteries in some of the example above. Yet it doesn't make sense when thinking about stocks or bonds to restrict to only three or four outcomes, there are really an infinity of possible outcomes. The way to think about this is to think of the set of potential outcomes \begin_inset Formula $\mathcal{X}$ \end_inset as an interval of \begin_inset Formula $\mathbb{R}$ \end_inset . Of course, we can't assign a positive probability to each of these outcomes the way we have so far. So instead we describe a lottery as a \emph on probability distribution function, \emph default \begin_inset Formula $F(x)$ \end_inset that takes each possible value \begin_inset Formula $x$ \end_inset in the set of \emph on potential \emph default outcomes and gives the probability that the \emph on actual \emph default outcome is less than or equal to \begin_inset Formula $x$ \end_inset . \end_layout \begin_layout Standard We could actually describe the lotteries we have considered so far this way. For example, in the last chapter we thought about lotteries with three outcomes, in particular, the outcomes were \begin_inset Formula $\{1000,500,0\}$ \end_inset , where each of these outcomes is supposed to be a monetary outcome. We assigned probability \begin_inset Formula $p_{1}$ \end_inset to the first outcomes, \begin_inset Formula $p_{2}$ \end_inset to the second and \begin_inset Formula $p_{3}=1-p_{1}-p_{2}$ \end_inset to the third. \end_layout \begin_layout Standard Suppose we think of \begin_inset Formula $\mathcal{X}=[-100,+2000]$ \end_inset as the set of possible outcomes. Then the following diagram shows you what the corresponding probability distribution function looks like for this lottery. The function \begin_inset Formula $F(x)$ \end_inset starts out at 0 when \begin_inset Formula $x$ \end_inset is equal to \begin_inset Formula $-100$ \end_inset which is the lowest possible monetary payoff. \begin_inset Newline newline \end_inset \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Graphics filename risk_aversion_fig2.eps \end_inset \end_layout \end_inset For every value of \begin_inset Formula $x$ \end_inset between \begin_inset Formula $-100$ \end_inset and \begin_inset Formula $0$ \end_inset the probability that the outcome of the lottery is less than or equal to these values is constantly zero. However, the probability that the outcome of the lottery is less than or equal to \begin_inset Formula $0$ \end_inset is exactly equal to the probability that \begin_inset Formula $0$ \end_inset occurs, in other words, \begin_inset Formula $p_{3}$ \end_inset , the probability that we have been assigning to the worst outcome. As the monetary payoff travels between \begin_inset Formula $0$ \end_inset and \begin_inset Formula $500$ \end_inset , say a value like \begin_inset Formula $x$ \end_inset in the diagram, the probability that the outcome is less than or equal to \begin_inset Formula $x$ \end_inset is then constant at \begin_inset Formula $p_{3}$ \end_inset . Suddenly it jumps up to \begin_inset Formula $p_{2}+p_{3}$ \end_inset when \begin_inset Formula $x$ \end_inset is \begin_inset Formula $500$ \end_inset , and so on, until it reaches 1 and stays there after \begin_inset Formula $x$ \end_inset is \begin_inset Formula $1000$ \end_inset . \end_layout \begin_layout Standard The previous chapter showed how the independence axiom makes it possible to compare different lotteries by computing the \emph on expected utility \emph default associated with those lotteries. The expected utility associated with the lottery \begin_inset Formula $\{p_{1},p_{2},p_{3}\}$ \end_inset for example, is given by \begin_inset Formula \[ p_{1}u(1000)+p_{2}u(500)+p_{3}u(0) \] \end_inset The lottery \begin_inset Formula $\{p_{1},p_{2},p_{3}\}$ \end_inset is the same as the probability distribution described in the Figure above. \end_layout \begin_layout Standard Now suppose that we want to describe a lottery with an infinite number of outcomes, for example, a share that has a return that could be anything between \begin_inset Formula $-100$ \end_inset and \begin_inset Formula $2000$ \end_inset . We can't describe such a lottery by assigning probabilities to each possible outcome in this case. There are just too many of them. We could, however, still describe the lottery by giving the probability distribution function associated with it. An infinite lottery might have a probability distribution that looked like the one in the next figure. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Graphics filename risk_aversion_fig3.eps \end_inset \end_layout \end_inset This lottery (i.e., the probability distribution function associated with this lottery) has the same values at \begin_inset Formula $0$ \end_inset , \begin_inset Formula $500$ \end_inset , and \begin_inset Formula $1000$ \end_inset as the previous lottery - \begin_inset Formula $p_{3},$ \end_inset \begin_inset Formula $p_{2}+p_{3}$ \end_inset , and \begin_inset Formula $1$ \end_inset . However, at a point like \begin_inset Formula $x$ \end_inset , the probability that the outcome in this lottery is less than or equal to \begin_inset Formula $x$ \end_inset is strictly higher than \begin_inset Formula $p_{3}$ \end_inset which was its value in the previous lottery. \end_layout \begin_layout Standard The expected utility theorem suggests that if the independence axiom holds for compound lotteries described as distribution functions, then we should be able to assign a utility value to \emph on all \emph default the outcomes between \begin_inset Formula $-100$ \end_inset and \begin_inset Formula $2000$ \end_inset . Then if we could somehow multiply these utility values by something like probabilities associated with each outcome we would have a single utility value for the lottery. The way to do this is to \emph on integrate \emph default the utility values multiplied by the amount by which the probability distributi on function increases at each possible outcome. This expression looks like \begin_inset Formula \[ \int_{-100}^{2000}u(x)dF(x) \] \end_inset where \begin_inset Formula $dF(x)$ \end_inset means the \emph on change \emph default in the probability distribution function. For a lottery like the first of the two above, \begin_inset Formula $dF$ \end_inset is equal to zero at every point except for the three special outcomes, \begin_inset Formula $0$ \end_inset , \begin_inset Formula $500$ \end_inset , and \begin_inset Formula $1000$ \end_inset . Then, \begin_inset Formula $dF$ \end_inset takes the values \begin_inset Formula $p_{3}$ \end_inset , \begin_inset Formula $p_{2}$ \end_inset and \begin_inset Formula $p_{1}$ \end_inset respectively. Then if we add up by integrating as above we just get the sum \begin_inset Formula \[ p_{1}u(1000)+p_{2}u(500)+p_{3}u(0) \] \end_inset \end_layout \begin_layout Standard In a case like the second lottery, \begin_inset Formula $dF$ \end_inset is the \emph on derivative \emph default of the probability distribution (sometimes called its density). In that case you would write \begin_inset Formula $dF$ \end_inset as \begin_inset Formula $F^{\prime}(x)dx$ \end_inset where \begin_inset Formula $F^{\prime}(x)=\frac{dF(x)}{dx}$ \end_inset . That is the formulation you have probably seen before. \end_layout \begin_layout Standard As an example, one special probability distribution function is the \emph on uniform \emph default distribution function. The idea behind it is that every possible outcome is equally likely. So if you take a point midway through the set of outcomes, i.e. 1050, then the probability the outcome is less than or equal to 1050 should be equal to \begin_inset Formula $\frac{1}{2}$ \end_inset . The probability distribution that does this is \begin_inset Formula \[ F(x)=\frac{x-(-100)}{2000-(-100)} \] \end_inset The expected utility associated with this lottery is \begin_inset Formula \[ \frac{1}{2100}\int u(x)dx \] \end_inset because of the fact that \begin_inset Formula $F^{\prime}(x)=\frac{1}{2100}$ \end_inset . Then using your high school algebra, the expected utility associated with the lottery whose probability distribution function is uniform is just the ratio of the area under the utility function between \begin_inset Formula $-100$ \end_inset and \begin_inset Formula $2000$ \end_inset , to the area in the rectangle whose sides have length 1 and 2100. \end_layout \begin_layout Subsection Risk Aversion \end_layout \begin_layout Standard As described above, the independence axiom implies that there is a \emph on utility for wealth \emph default function that people use to evaluate lotteries. One reasonable question to ask is whether we could guess something about the shape of this function from other properties of behavior. For example, the St. Petersberg bet that we discussed in the last chapter has an infinite expected value, but no one would pay an infinite amount to engage in the bet. \end_layout \begin_layout Standard An even simpler example might be the following - I propose a new grading scheme for the exam. You can choose either of the following options - either take the mark you get, or I will flip a coin - heads I raise your mark by 10 points, tails I lower your mark by 10 points. Your expected grade is the same under either scheme. You probably wouldn't want the second scheme because it is risky \emph on . \emph default An A grade could turn into a B, a pass could turn into a fail (or conversely). \end_layout \begin_layout Standard The second grading scheme involves what is called a \emph on fair \emph default gamble in the sense that the expected gain to you of taking the bet is exactly zero. It seems plausible that most people simply wouldn't be interested in taking a fair bet. \begin_inset Foot status open \begin_layout Plain Layout However, it should be added that the bets that people take in a casino are not fair. For example, if you fed coins into a slot machine for the rest of your life you would lose a lot of money. This doesn't seem to stop people from going to casinos. \end_layout \end_inset \begin_inset Foot status open \begin_layout Plain Layout The lotteries that you buy into at the corner store (like the Lotto 6-49) represent a strange variant on what happens in casinos. The way the lotteries work is roughly as follows - the lottery sells tickets. From the total revenue they earn by ticket sales, they take out a big chunk to cover operating costs and to give themselves some profit. The rest is put in a pool. A number is then drawn randomly by dropping balls from an urn (or some other method that is independent of the number of tickets). If one or more people has the winning number they split all the money in the pool. The interesting event occurs when no one has the ticket. Then, when the lottery is run the next period, a portion of ticket sales is added to the pool which already contains the pool from the last lottery. If no one wins for a long time, the pool can become so large that the lottery ticket will be a more than fair bet. For example, as this is written the chance of winning the 6-49 is about 1 in 18 million, but the current pool of money to be won is about 40 million. So the lottery ticket appears to have an expected value of a little over $2. It only costs $2 to buy a ticket. So it appears that everyone should buy a ticket. Many people do, leading to enormous revenues and profits for the lottery corporation. This is a bit misleading because the odds of winning are independent of the number of bidders. That means that if many people bid, more than one person is likely to have a winning ticket. The average payout to a winner will be considerably smaller than $40 million for this reason, which makes the expected value of the ticket smaller than $2. \end_layout \end_inset This kind of behavior does have some implications for the shape of the utility function. You can what the implications are in the next figure. \begin_inset Newline newline \end_inset \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Graphics filename risk_aversion_fig4.eps \end_inset \end_layout \end_inset Suppose our consumer has some initial level of wealth \begin_inset Formula $W$ \end_inset . A fair bet is one whose expected return is zero. One example would be a bet that pays \begin_inset Formula $x$ \end_inset with probability \begin_inset Formula $\frac{1}{2}$ \end_inset , but which costs you \begin_inset Formula $x$ \end_inset with the same probability. Doing nothing leaves you with \begin_inset Formula $W$ \end_inset for sure. The independence axiom implies that there is a utility level \begin_inset Formula $u(W)$ \end_inset associated with this outcome. The outcomes \begin_inset Formula $W-x$ \end_inset and \begin_inset Formula $W+x$ \end_inset also have associated utility values \begin_inset Formula $u(W-x)$ \end_inset and \begin_inset Formula $u(W+x)$ \end_inset . These are marked in the diagram. The expected utility of the lottery we just described is then \begin_inset Formula \[ \frac{1}{2}u(W-x)+\frac{1}{2}u(W+x) \] \end_inset In the diagram, this utility level is the distance from the point \begin_inset Formula $W$ \end_inset on the horizontal axis up to the point \begin_inset Formula $c$ \end_inset in the diagram. \begin_inset Foot status open \begin_layout Plain Layout To see this observe that the vertical distance between the points d and e is \begin_inset Formula $u(W+x)-u(W-x)$ \end_inset . By similar triangles, the ratio of the vertical distance between a and c (call it \begin_inset Formula $|c-a|$ \end_inset ) to \begin_inset Formula $u(W+x)-u(W-x)$ \end_inset is the same as the ratio of the horizontal distance between a and d to \begin_inset Formula $W+x-(W-x)$ \end_inset . This latter ratio is equal to \begin_inset Formula $\frac{1}{2}$ \end_inset . Therefore \begin_inset Formula \[ |c-a|=\frac{1}{2}(u(W+x)-u(W-x)) \] \end_inset Now adding \begin_inset Formula $u(W-x)$ \end_inset to both sides gives \begin_inset Formula \[ |c-d|+u(W-x)=\frac{1}{2}u(W-x)+\frac{1}{2}u(W+x) \] \end_inset \end_layout \end_inset The assertion that our consumer would rather not engage in this bet means that the utility value of \begin_inset Formula $W$ \end_inset (for sure) must be above c at a point like b. Since this preference not to take fair bets is likely to be true no matter what \begin_inset Formula $x$ \end_inset is and no matter what \begin_inset Formula $W$ \end_inset is, means that the utility for wealth function must lie everywhere above the line segment from d to e. In other words, the utility for wealth function must be concave. \end_layout \begin_layout Subsection Measuring Aversion to Risk \end_layout \begin_layout Standard Stocks and bonds are more complex than lotteries because the monetary outcomes usually aren't fair. Most traded stocks for example have a positive expected return. The thing that makes stock trading interesting is that it involves considerable risk. Whether or not an investor will want an initial public offering of some stock depends partly on the risk characteristics of the stock, and partly on the investors own attitudes toward risk. The economics literature has suggested some interesting ways of separating these two things. \end_layout \begin_layout Standard Let \begin_inset Formula $F$ \end_inset be the probability distribution over outcomes that is associated with some lottery. The expected payoff associated with the lottery is \begin_inset Formula \[ \mathbb{E}x\mathbb{=}\int xdF(x) \] \end_inset where \begin_inset Formula $dF(x)$ \end_inset depends on the nature of the lottery as discussed above. Lets assume for now that the probability distribution function \begin_inset Formula $F$ \end_inset has a density so that we can write the expected payoff (and all the expected utility calculations) using the better known manner \begin_inset Formula \[ \mathbb{E}x=\int xF^{\prime}(x)dx \] \end_inset \end_layout \begin_layout Standard Let \begin_inset Formula $W$ \end_inset be any level of wealth. Lets try to calculate the maximum amount our consumer would be willing to pay to play this lottery \begin_inset Formula $F$ \end_inset . Using the independence axiom we would find this price \begin_inset Formula $p$ \end_inset by solving \begin_inset Formula \[ u\left(W\right)=\int u\left(W-p+x\right)F^{\prime}\left(x\right)dx \] \end_inset It is going to tough to solve this exactly without more precise information about \begin_inset Formula $u$ \end_inset so lets solve it 'approximately'. \end_layout \begin_layout Standard First take a second order Taylor approximation to the utility function inside the integral sign on the right hand side of the equation. A first order Taylor approximation won't work very well here because it only gives a good approximation when \begin_inset Formula $x$ \end_inset is small. However, there will typically be some risk that \begin_inset Formula $x$ \end_inset could be quite large, so some correction for the curvature in \begin_inset Formula $u$ \end_inset will help. The second order approximation is given by \begin_inset Formula \[ \int\left\{ u\left(W\right)+u^{\prime}\left(W\right)(x-p)+u^{\prime\prime}\left(W\right)\frac{(x-p)^{2}}{2}\right\} F^{\prime}\left(x\right)dx \] \end_inset If this is a good approximation, then it is approximately equal to \begin_inset Formula $U(W)$ \end_inset when we choose the right \begin_inset Formula $p$ \end_inset , so simplify the equation \begin_inset Formula \[ \int\left\{ u\left(W\right)+u^{\prime}\left(W\right)(x-p)+u^{\prime\prime}\left(W\right)\frac{(x-p)^{2}}{2}\right\} F^{\prime}\left(x\right)dx=u(W) \] \end_inset to get \begin_inset Formula \[ u^{\prime}\left(W\right)\int(x-p)F'(x)dx+u^{\prime\prime}\left(W\right)\int\frac{(x-p)^{2}}{2}F^{\prime}\left(x\right)dx=0 \] \end_inset or \begin_inset Formula \[ p=\int xF^{\prime}(x)dx+\frac{u^{\prime\prime}(w)}{u^{\prime}(w)}\int\frac{(x-p)^{2}}{2}F^{\prime}(x)dx \] \end_inset The second term in this expression has a \begin_inset Formula $u^{\prime\prime}(w)$ \end_inset which is negative if the utility for wealth function is concave. So the expression says that the maximum amount the consumer will be willing to pay for the lottery is equal to its mean return less a term consisting of two parts. One part is the integral which is related to the variance of the lottery or the degree to which it is spread out and unpredictable. The other part depends only on the consumer's utility for wealth function. The larger in absolute value is the ratio \begin_inset Formula $\frac{u^{\prime\prime}(w)}{u^{\prime}(w)}$ \end_inset , the less the consumer will be willing to pay. \end_layout \begin_layout Standard This particular decomposition has proved to be very useful in thinking about portfolio theory. The term \begin_inset Formula \[ -\frac{u^{\prime\prime}\left(W\right)}{u^{\prime}\left(W\right)} \] \end_inset is referred to as the Arrow-Pratt measure of absolute risk aversion. \end_layout \begin_layout Standard We will shortly see how this measure can be used to get some insight into the way that investment decisions work. For the moment, the main thing to note is that the Arrow-Pratt measure is a conceptual device that emerges with the help of the expected utility theorem. One natural assumption would seem to be that the risk premium that a consumer would be willing to pay to avoid risk would be lower if the consumer were more wealthy. The formulation so far shows exactly how to formalize this idea - the function \begin_inset Formula $-\frac{u^{\prime\prime}\left(W\right)}{u^{\prime}\left(W\right)}$ \end_inset should be a decreasing function. We will come back to this idea momentarily. \end_layout \begin_layout Subsection The Portfolio Problem \end_layout \begin_layout Standard A fairly simple version of the portfolio problem can be studied by assuming that there are exactly two different securities that an investor can buy. A \emph on security \emph default is a lottery. The consequences of the lottery are possible rates of return on investment. If the consequence of the lottery is a rate of return \begin_inset Formula $s$ \end_inset , then the security pays \begin_inset Formula $1+s$ \end_inset dollars tomorrow for each dollar that is invested in the lottery today. In the problem that we are going to analyze, one of the two securities is safe in the sense that the lottery that produces the rate of return gives a rate of return of zero for sure. So, each dollar invested today gives back exactly one dollar tomorrow, no more and no less. There is also a risky security where the probability distribution function for the random rate of return is \begin_inset Formula $F$ \end_inset . \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash \end_layout \end_inset Sometimes this rate of return will be positive and the security will pay back more than one dollar for each dollar invested. Other times the rate of return will actually be negative, and a dollar invested will return less than one dollar tomorrow. \end_layout \begin_layout Standard The investor has \begin_inset Formula $w$ \end_inset dollars to invest in these two securities. His \emph on ex post \emph default income (i.e., his income tomorrow after the rate of return on the lottery is realized) when he invests \begin_inset Formula $i_{s}$ \end_inset in the safe security and \begin_inset Formula $i_{r}$ \end_inset in the risky security, is given by \begin_inset Formula \[ i_{s}+i_{r}\left(1+s\right) \] \end_inset \end_layout \begin_layout Standard A pair \begin_inset Formula $\left(i_{s},i_{r}\right)$ \end_inset is called a \emph on portfolio \emph default . Each portfolio generates a different lottery over monetary outcomes. Then relying on the independence axiom, we can invoke the expected utility theorem and conclude that there is a utility for wealth function \begin_inset Formula $u$ \end_inset such that one portfolio (and its associated lottery) \begin_inset Formula $\left(i_{s},i_{r}\right)$ \end_inset is preferred to another \begin_inset Formula $\left(i_{s}^{\prime},i_{r}^{\prime}\right)$ \end_inset if \begin_inset Formula \[ \int u\left(i_{s}+i_{r}\left(1+s\right)\right)F^{\prime}\left(s\right)ds\geq \] \end_inset \begin_inset Formula \[ \int u\left(i_{s}^{\prime}+i_{r}^{\prime}\left(1+s\right)\right)F^{\prime}\left(s\right)ds. \] \end_inset \end_layout \begin_layout Standard If that is true, then the investor should choose the portfolio that maximizes \begin_inset Formula \[ \int u\left(i_{s}+i_{r}\left(1+s\right)\right)F^{\prime}\left(s\right)ds \] \end_inset subject to the constraints that \begin_inset Formula \[ i_{s}+i_{r}\leq W \] \end_inset \begin_inset Formula \[ i_{r}\geq0 \] \end_inset \begin_inset Formula \[ i_{s}\geq0 \] \end_inset \end_layout \begin_layout Standard We could solve this using Lagrangian methods, recalling that before you do so, you need to tease out some properties of the solution in order to guess which constraints are likely to be important. To see a way to do this, simply imagine that the integral above is just a particular function \begin_inset Formula \[ U\left(i_{s},i_{r}\right)\equiv \] \end_inset \begin_inset Formula \[ \int u\left(i_{s}+i_{r}\left(1+s\right)\right)F^{\prime}\left(s\right)ds \] \end_inset and that the first constraint above is just a budget constraint where the prices of both securities are just equal to \begin_inset Formula $1$ \end_inset . So, we should be able to find the optimal portfolio by finding the highest indifference curve that touches the budget constraint, as the indifference curve \begin_inset Formula $II$ \end_inset does in Figure \begin_inset Formula $1$ \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash includegraphics[ \end_layout \begin_layout Plain Layout height=1.6302in, \end_layout \begin_layout Plain Layout width=2.0263in \end_layout \begin_layout Plain Layout ]{risk_aversion_fig1.eps} \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The slope of the indifference curve is given by the marginal utility of the good on the horizontal axis (i.e., \begin_inset Formula $i_{s}$ \end_inset ) divided by the marginal utility associated with the good on the vertical axis (i.e., \begin_inset Formula $i_{r}$ \end_inset ). Differentiating gives \begin_inset Formula \[ -\frac{\frac{\partial U\left(i_{s},i_{r}\right)}{\partial i_{s}}}{\frac{\partial U\left(i_{s},i_{r}\right)}{\partial i_{r}}}= \] \end_inset \begin_inset Formula \[ -\frac{\int u^{\prime}\left(i_{s}+i_{r}\left(1+s\right)\right)F^{\prime}\left(s\right)ds}{\int u^{\prime}\left(i_{s}+i_{r}\left(1+s\right)\right)\left(1+s\right)F^{\prime}\left(s\right)ds} \] \end_inset \end_layout \begin_layout Standard Now, following our usual procedure, we can try to check the conditions under which we might find a solution at the corner where all wealth is invested in the riskless asset. This would occur if the indifference curve happened to be steeper than the budget line. So, let's evaluate the slope of the indifference curve at the bundle \begin_inset Formula $\left(w,0\right)$ \end_inset . Substituting into the formula above gives \begin_inset Formula \[ -\frac{\int u^{\prime}\left(w\right)F^{\prime}\left(s\right)ds}{\int u^{\prime}\left(w\right)\left(1+s\right)F^{\prime}\left(s\right)ds}= \] \end_inset \begin_inset Formula \[ -\frac{1}{1+\int sF^{\prime}\left(s\right)ds} \] \end_inset This means that whether or not the investor optimally invests in the risky asset depends entirely on \begin_inset Formula $\int sF^{\prime}\left(s\right)ds$ \end_inset . If this is positive, the (absolute value of the) slope of the indifference curve is smaller than \begin_inset Formula $1$ \end_inset and the tangency has to occur somewhere up the budget line to the left of \begin_inset Formula $\left(w,0\right)$ \end_inset . On the other hand, if the mean value of \begin_inset Formula $s$ \end_inset is less than zero, then the indifference curve will be steeper than the budget line, and the optimal solution will be at the corner of the budget set. \end_layout \begin_layout Standard This is called the \emph on diversification theorem \emph default . If there is a risky asset whose expected return exceeds the expected return on the safe asset, then the optimal portfolio will always involve some investment in the risky asset. \end_layout \begin_layout Subsubsection Comparative Statics and Wealth \end_layout \begin_layout Standard We might assume that wealthy people invest more in risky assets. This assertion seems that it must be true. Surprisingly, this is not always the case. It is difficult to see why this might be by using intuition alone. As I have often mentioned, this is when mathematics can be very useful. Intuition is rarely wrong: it just doesn't give the whole story. Math can often help you think out the parts that you tend to gloss over when you think intuitively. Often the biggest insights in economics come by understanding the complexities that intuition can't see. \end_layout \begin_layout Standard The problem that is discussed in this section also provides an opportunity to see the way that comparative statics is often done in applications. We are interested in what the effect of a change in wealth \begin_inset Formula $w$ \end_inset will be on the optimal portfolio. One way to figure this out is to try to find out how a change in wealth will affect the position of the tangency between the indifference curve and budget line. To see how this line of argument works, start by substituting the fact that \begin_inset Formula $i_{s}=w-i_{r}$ \end_inset into the tangency condition to get \begin_inset Formula \[ \frac{\int u^{\prime}\left(w-i_{r}+i_{r}\left(1+s\right)\right)F^{\prime}\left(s\right)ds}{\int u^{\prime}\left(w-i_{r}+i_{r}\left(1+s\right)\right)\left(1+s\right)F^{\prime}\left(s\right)ds}=1 \] \end_inset Simplifying the arguments of the functions gives \begin_inset Formula \[ \frac{\int u^{\prime}\left(w+i_{r}s\right)F^{\prime}\left(s\right)ds}{\int u^{\prime}\left(w+i_{r}s\right)\left(1+s\right)F^{\prime}\left(s\right)ds}=1 \] \end_inset Multiplying both sides by the denominator on the left gives \begin_inset Formula \[ \int u^{\prime}\left(w+i_{r}s\right)F^{\prime}\left(s\right)ds-\int u^{\prime}\left(w+i_{r}s\right)\left(1+s\right)F^{\prime}\left(s\right)ds=0 \] \end_inset Canceling the common terms gives the equation \begin_inset Formula \begin{equation} \int u^{\prime}\left(w+i_{r}s\right)sF^{\prime}\left(s\right)ds=0\label{FOC} \end{equation} \end_inset \end_layout \begin_layout Standard The trick at this point is to assume that \begin_inset Formula $i_{r}$ \end_inset is actually a function of \begin_inset Formula $w$ \end_inset that adjusts in such a way that the equation above is always satisfied. Then, in fact, \begin_inset Formula \[ \int u^{\prime}\left(w+i_{r}\left[w\right]s\right)sF^{\prime}\left(s\right)ds\equiv0 \] \end_inset Since this holds uniformly under this definition, we can differentiate both sides of the expression with respect to \begin_inset Formula $w$ \end_inset and the derivatives will also be equal. In other words \begin_inset Formula \[ \int u^{\prime\prime}\left(w+i_{r}\left[w\right]s\right)sF^{\prime}\left(s\right)ds+\int u^{\prime\prime}\left(w+i_{r}\left[w\right]s\right)\frac{di_{r}\left[w\right]}{dw}s^{2}F^{\prime}\left(s\right)ds=0 \] \end_inset Solving for the derivative gives \begin_inset Formula \[ \frac{di_{r}\left[w\right]}{dw}=-\frac{\int u^{\prime\prime}\left(w+i_{r}\left[w\right]s\right)sF^{\prime}\left(s\right)ds}{\int u^{\prime\prime}\left(w+i_{r}\left[w\right]s\right)s^{2}F^{\prime}\left(s\right)ds} \] \end_inset We want to know how an increase in \begin_inset Formula $w$ \end_inset will change the amount invested in the risky asset. In other words, we want to know whether the derivative of the optimal value of \begin_inset Formula $i_{r}$ \end_inset with respect to a change in wealth will be positive. This expression almost gives the answer. The denominator is an integral. Each term in the integrand is negative provided the investor is risk averse (which gives \begin_inset Formula $u^{\prime\prime}<0$ \end_inset ). There is a minus sign in front of the fraction, and minus times minus is positive. So, we could conclude that the derivative is positive if we could show that the numerator is positive. Unfortunately, this is not obvious if it is true. The derivative \begin_inset Formula $u^{\prime\prime}$ \end_inset is certainly negative, but \begin_inset Formula $s$ \end_inset can be either positive or negative. The sign of the integral will depend on how big \begin_inset Formula $u^{\prime\prime}$ \end_inset is when \begin_inset Formula $s$ \end_inset is negative compared with how big it is when \begin_inset Formula $s$ \end_inset is positive. \end_layout \begin_layout Standard The lesson of the comparative statics has now been discovered. To conclude that increases in wealth raise investment, we need more information. Or, we need to restrict the set of preferences that we think are plausible. Fortunately, there is a fairly easy restriction that will do this trick. We have described the Arrow-Pratt measure of absolute risk aversion \begin_inset Formula $\frac{u^{\prime\prime}\left(w\right)}{u^{\prime}\left(w\right)}$ \end_inset . It is proportional to the size of the risk premium associated with fair gambles. Suppose we assumed that this measure of risk aversion is decreasing with wealth. That would mean that we would be assuming (or restricting attention to) investors whose risk premium falls as they become more wealthy. Surely these investors must raise investment as their wealth increases. So, let's check this out. \end_layout \begin_layout Standard Write out the Arrow-Pratt measure as it appears in our comparative static equation. Then we would have \begin_inset Formula \[ -\frac{u^{\prime\prime}\left(w+i_{r}s\right)}{u^{\prime}\left(w+i_{r}s\right)}\leq-\frac{u^{\prime\prime}\left(w\right)}{u^{\prime}\left(w\right)} \] \end_inset whenever \begin_inset Formula $s>0$ \end_inset by the assumption that the Arrow-Pratt measure is decreasing. Well the whole problem arises from the fact that we don't know that \begin_inset Formula $s$ \end_inset is positive. So, let's try a trick. If \begin_inset Formula $s$ \end_inset is positive, it must also be true that \begin_inset Formula \[ -\frac{u^{\prime\prime}\left(w+i_{r}s\right)}{u^{\prime}\left(w+i_{r}s\right)}s\leq-\frac{u^{\prime\prime}\left(w\right)}{u^{\prime}\left(w\right)}s \] \end_inset The nice thing about this expression is that if we change the sign of \begin_inset Formula $s$ \end_inset to negative (which of course means that \begin_inset Formula $-\frac{u^{\prime\prime}\left(w+i_{r}s\right)}{u^{\prime}\left(w+i_{r}s\right)}\geq-\frac{u^{\prime\prime}\left(w\right)}{u^{\prime}\left(w\right)}$ \end_inset ) then it would still be true that \begin_inset Formula \[ -\frac{u^{\prime\prime}\left(w+i_{r}s\right)}{u^{\prime}\left(w+i_{r}s\right)}s\leq-\frac{u^{\prime\prime}\left(w\right)}{u^{\prime}\left(w\right)}s \] \end_inset So, this last expression is actually correct no matter what the sign of \begin_inset Formula $s$ \end_inset . So, let's multiply both sides of this inequality by \begin_inset Formula $u^{\prime}\left(w+i_{r}s\right)$ \end_inset then integrate the result over \begin_inset Formula $s$ \end_inset to get \begin_inset Formula \[ -\int u^{\prime\prime}\left(w+i_{r}s\right)sF^{\prime}\left(s\right)ds\leq-\frac{u^{\prime\prime}\left(w\right)}{u^{\prime}\left(w\right)}\int u^{\prime}\left(w+i_{r}s\right)sF^{\prime}\left(s\right)ds \] \end_inset If you look back, you will see that the right hand side of this equation is proportional to the left hand side of ( \begin_inset CommandInset ref LatexCommand ref reference "FOC" \end_inset ) which is zero. So, we have \begin_inset Formula \[ -\int u^{\prime\prime}\left(w+i_{r}s\right)sF^{\prime}\left(s\right)ds\leq0 \] \end_inset which is just the result we wanted (since it shows that \begin_inset Formula $\int u^{\prime\prime}\left(w+i_{r}s\right)sF^{\prime}\left(s\right)ds\geq0$ \end_inset ). \end_layout \begin_layout Standard After all this work, what we have discovered is that an investor will increase his investment in the risky asset as his wealth rises provided his Arrow-Pratt measure of absolute risk aversion is decreasing as his wealth increases. \end_layout \end_body \end_document