#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{amssymb} \usepackage{graphicx} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \usepackage{amsmath,amssymb,epsfig,bbm,ifthen,capt-of,calc} \usepackage{babel} \usepackage{babel} \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 latin1 \fontencoding global \font_roman palatino \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 1 \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 Applications of Choice Theory: The Theory of Demand \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 Preference Theory tells us that individuals who can express opinions about the various alternatives available to them will act as if they are maximizing a utility function. It is important to remember that this isn't meant to be a description of what people actually do when they make decisions. Obviously people don't consciously maximize anything when they make choices. Consumers who never got the hang of finding \begin_inset Formula $x$ \end_inset in high school, nonetheless seem perfectly capable of deciding how to spend their money. The basic presumption in economics is that there is no way to know what people are actually thinking when they make decisions. \begin_inset Foot status collapsed \begin_layout Plain Layout Not everyone agrees with this. For example, polling at elections is done under the assumption that respondents will truthfully reveal who they want to vote for - they are often pretty close. Psychologists often run experiments in which they simply ask people what they would do if ... . Neuroeconomists believe that new brain scanning technology will make it possible to observe preferences directly. \end_layout \end_inset \end_layout \begin_layout Standard The fact that individuals' choices will look just like solutions to maximization problems allows us to use methods and concepts from mathematics to help describe behavior. This mathematical representation of behavior ultimately leads to the greatest contribution of economics, the concept of \emph on equilibrium \emph default behavior. We will begin the description of equilibrium behavior later on in this course. \end_layout \begin_layout Standard To make use of the method that the utility theorem provides, we have to add something to what we have so far. Suppose we are trying to figure out how people will react to a price change. At the initial price, we can use the theorem that we proved in the chapter on preferences to show that there is a utility function and that the choices our consumer makes maximize this utility function subject to whatever constrain ts she faces. The construction of this utility function depends on the alternatives over which the consumer has to decide. There is nothing in the theorem that says that the consumers preferences won't change when the choice set does. Our consumer might believe that a higher price means that the good she is buying has a higher quality than she initially thought. After the price change, she might `want' the good more than before. \begin_inset Foot status collapsed \begin_layout Plain Layout My favourite example of this is Adobe Acrobat Software used for making pdf files (like the file you are currently reading). Many free programs will produce pdf files. However, Adobe had the idea to offer an expensive software package to do the same thing, so that people would incorrectly believe that it was higher quality software. This strategy worked brilliantly, at least among my colleagues who have jointly shelled out thousands of dollars from their research grants to Adobe for free software - thousands of dollars they could have paid to graduate students. \end_layout \end_inset Perhaps more importantly, the price change might affect what other people do. Some goods are more desirable when other people like and use them, for example. \begin_inset Foot status collapsed \begin_layout Plain Layout Telephones are an obvious example. The fashion industry seems to work on this principle, as well. Companies advertizes their brand heavily (for example, product placement in a popular movie like \shape slanted I, Robot \shape default , or \shape slanted The Italian Job \shape default ), then raise the product price to make it exclusive. Suddenly, everyone wants the product and is willing to pay a lot for it. \end_layout \end_inset \end_layout \begin_layout Standard To make use of the maximization approach, we need to make assumptions about utility and how it changes when we change the environment. These assumptions are called an economic \emph on model \emph default . We use our economic model to make a prediction. We will start with one of the oldest and perhaps simplest economic models in the next section, and I will explain these extra assumptions and how to use the maximization approach to understand it. \end_layout \begin_layout Standard You might wonder about this. Does this mean that economic predictions are just elaborate assumptions about the way people behave? Why should I believe these assumptions? If you are thinking this way, you are on the right track. Economist spend an enormous amount of time and effort collecting and analysing data - often with the purpose of \emph on testing \emph default some economic model. You'll be learning how to use models in this course, so we won't say much more about testing, but we might find that our prediction is inconsistent with what appears to be going on in the data we have collected. This may require that we go back and revise the assumptions of our model to try to get things to work out. So, the assumptions evolve with our knowledge of how people behave. \end_layout \begin_layout Standard Perhaps this leads you to a second, closely related question. If models are just elaborate guesses about preferences designed to generate predictions, why not just start off with the predictions? For example, suppose we are interested in the impact of an increase in price. It seems perfectly reasonable to guess that if the price of a good rises, then people will buy less. \begin_inset Foot status collapsed \begin_layout Plain Layout This is called the Law of Demand. In October 1981, American Senator William Proxmire gave his Golden Fleece Award to the National Science Foundation for funding an empirical test of the Law of Demand. Pigeons in a laboratory would receive food by pecking a lever. Once the scientists had trained the pigeons to peck on the lever to get food (the first ten years of the project), they changed the rules so that the pigeons had to peck twice on the lever to get food, instead of only once. The idea was that if the law of demand holds, then pigeons should eat less when they have to peck twice than they would if they only had to peck once. \end_layout \end_inset Then why bother to write down a maximization model, find Lagrange multipliers, take derivatives, and do all that other tedious stuff? After all, we can always test our guess, and refine it if we are wrong. \end_layout \begin_layout Standard There are basically two answers. Part of the answer is that mathematics is universal: everyone, no matter what their field of study, knows math. Formal mathematical models can, in principal, be understood by everyone, not just specialists in economics. Apart from the obvious connection with math and statistics, the modeling approach in economics is similar to that used in some branches of computer science, theoretical biology, and zoology. In an odd way, formal modeling makes economic theory more inclusive. \end_layout \begin_layout Standard The real benefit of formal modeling (to all these fields) is that it helps make up for the deficiencies in our own intuition. Our intuition is rarely wrong, but it is almost always incomplete. It is also lazy. It wants to push every new and challenging fact into an existing `intuitive' box, which makes us very conservative intellectually. Careful mathematical analysis of well-defined models makes up for this. It helps us to see parts of the story that we might otherwise have missed. Often, those insights-gained through painstaking mathematical analysis-lead to the most fundamental changes in thinking. So, don't despair if you spend hours thinking through the logic of one of the problems in the problem set without actually getting the answer. You are often laying the groundwork for important leaps in your understanding that will often transcend the particular problem you are working on. \end_layout \begin_layout Standard At a more practical level, mathematical analysis of a model will often reveal implications that your intuition would never have imagined. These implications can often be critical. For example, it isn't hard to show that the law of demand mentioned above need not be true. Nothing in the nature of preferences or the characteristics of markets requires it to be true. If our model doesn't tell us anything about demand curves, what use is it? Rational behavior does impose restrictions on demand that are amenable to econometric test. I will show you enough of the argument below for you to see that the real implications of rational behavior in a market like environment can not be understood using intuition, you need formal analysis. \end_layout \begin_layout Standard Bear in mind as we go along, that the content of economics is not the particular models we study, but the method of using models like this to generate predictio ns, then modifying these until the predictions match the information we have in our data. \end_layout \begin_layout Section Consumer Theory \end_layout \begin_layout Standard A \emph on consumer \emph default is an individual who wants to buy some stuff. The `'stuff \begin_inset Quotes erd \end_inset will be a list of quantities of the goods that she wants. We express this list as a \emph on vector \emph default , that is, an ordered list of real numbers \begin_inset Formula $x_{1},x_{2},\ldots,x_{n}$ \end_inset where \begin_inset Formula $x_{1}$ \end_inset is the total units of good 1 she wants, and so on. We refer to a generic \emph on bundle \emph default of goods as \begin_inset Formula $x\in\mathbb{R}^{n}$ \end_inset , where this latter notation means that \begin_inset Formula $x$ \end_inset is an ordered list consisting of exactly \begin_inset Formula $n$ \end_inset real numbers. \end_layout \begin_layout Standard For the moment, let \begin_inset Formula $\mathbb{B}$ \end_inset be the set of bundles that our consumer can afford to buy. If we propose different alternatives in \begin_inset Formula $\mathbb{B}$ \end_inset to our consumer, she will be able to tell us which one she prefers. If these preferences are transitive, along with an appropriate continuity assumption (see the previous chapter), then there will be a utility function \begin_inset Formula $u$ \end_inset which converts bundles in \begin_inset Formula $\mathbb{R}^{n}$ \end_inset into real numbers, and our consumer will look just like she is maximizing \begin_inset Formula $u$ \end_inset when she chooses a bundle from \begin_inset Formula $x$ \end_inset . \end_layout \begin_layout Standard Now, let \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset be a pair of alternatives in \begin_inset Formula $\mathbb{B}$ \end_inset . For the sake of argument, suppose that \begin_inset Formula $x\succeq y$ \end_inset (which means that the consumer prefers \begin_inset Formula $x$ \end_inset to \begin_inset Formula $y$ \end_inset ). Classical consumer theory makes two very strong assumptions. First, the preferences of our consumer are independent of the preferences and choices of all other consumers. Second, the preferences are independent of the budget set that the consumer faces. The first assumption just means that we can think about one consumer in isolation. No one really believes this is a good assumption, and we will begin to relax it later on. It does make it much easier to explain the approach. \end_layout \begin_layout Standard The second assumption can be stated more formally given the notation we have developed. If the consumer prefers \begin_inset Formula $x$ \end_inset to \begin_inset Formula $y$ \end_inset when these are offered as elements of \begin_inset Formula $\mathbb{B}$ \end_inset , then the consumer will still prefer \begin_inset Formula $x$ \end_inset to \begin_inset Formula $y$ \end_inset if these are offered as choices from any other budget set \begin_inset Formula $\mathbb{B}'$ \end_inset . \end_layout \begin_layout Standard What does this mean in words? Well as a good Canadian, you no doubt drink foreign beer like Molson (Coors, USA), or Labatt (Interbrew, Belgium). Suppose you would prefer a Molson to a Labatt if you are given a choice \begin_inset Foot status collapsed \begin_layout Plain Layout The presidents of Molson, Labatt and Big Rock Brewery (Calgary) once went for a beer after attending a conference together. The waiter asked the president of Molson what he wanted to drink. He said proudly, \begin_inset Quotes eld \end_inset I'll have a Canadian. \begin_inset Quotes erd \end_inset \begin_inset Quotes eld \end_inset Fine, \begin_inset Quotes erd \end_inset said the waiter. Then, he asked the president of Labatt, who said he would like a Labatt Blue. \begin_inset Quotes eld \end_inset Fine, \begin_inset Quotes erd \end_inset said the waiter, \begin_inset Quotes eld \end_inset good choice. \begin_inset Quotes erd \end_inset Then, he asked the president of Big Rock. \begin_inset Quotes eld \end_inset I'll have a Coke, \begin_inset Quotes erd \end_inset she said. \begin_inset Quotes eld \end_inset Pardon? \begin_inset Quotes erd \end_inset asked the waiter. \begin_inset Quotes eld \end_inset They aren't drinking beer so I don't think I will either, \begin_inset Quotes erd \end_inset she replied. \end_layout \end_inset . If you suddenly won a lottery that gave you $ 1 million for life, would you still prefer Molson to Labatt? Probably. You might not want a Molson or Labatt-because you could then afford to buy champagne or something-but, if you are given a choice between those two only, you would probably still choose Molson. \end_layout \begin_layout Standard Whatever you think of these two assumptions, let us accept them for the moment and try to show how to draw out their implications. \end_layout \begin_layout Subsection The Budget Set \end_layout \begin_layout Standard The \emph on budget set \emph default refers to the set of consumption bundles that the consumer can afford. We can provide a mathematical characterization of this set fairly easily. Let's assume that the consumer knows the prices of each of the goods, and that these prices can be represented as a vector \begin_inset Formula $p\in\mathbb{R}^{n}$ \end_inset , where \begin_inset Formula $p$ \end_inset is an ordered list \begin_inset Formula $\{p_{1},\ldots p_{n}\}$ \end_inset . Let's assume further that the consumer has a fixed amount of money \begin_inset Formula $W$ \end_inset to spend on stuff. The set of consumption bundles that the consumer can afford to buy is the set \begin_inset Formula \begin{equation} \left\{ x:x_{i}\geqslant0\forall i;\sum_{i=1}^{n}p_{i}x_{i}\leqslant W\right\} \label{budget} \end{equation} \end_inset The brackets around the expression are used to describe the set. The notation inside the bracket means the set of \begin_inset Formula $x$ \end_inset such that ( \begin_inset Formula $:$ \end_inset ) each component of \begin_inset Formula $x$ \end_inset is at least as big as zero, and such that ( \begin_inset Formula $;)$ \end_inset if you sum up the product of the price and quantity across all components you end up with something less than, or equal to, the amount of money you have to spend in the first place. Hopefully, you find the mathematical expression a lot more compact. However, the real benefit of using the math is yet to come. \end_layout \begin_layout Standard It helps to mix formal arguments together with pictures like the ones you saw in your first-year course. To do this, imagine that there are only two goods. Call them good \begin_inset Formula $x$ \end_inset and good \begin_inset Formula $y$ \end_inset . The price of good \begin_inset Formula $x$ \end_inset will be \begin_inset Formula $p_{x}$ \end_inset and the price of \begin_inset Formula $y$ \end_inset will be \begin_inset Formula $p_{y}$ \end_inset . The amount of money you spend buying good \begin_inset Formula $x$ \end_inset is \begin_inset Formula $p_{x}x$ \end_inset . The amount you spend on \begin_inset Formula $y$ \end_inset is \begin_inset Formula $p_{y}y$ \end_inset . Total spending is \begin_inset Formula $p_{x}x+p_{y}y$ \end_inset , which can be no larger than the money you have, \begin_inset Formula $W$ \end_inset . That is exactly what the math says in equation ( \begin_inset CommandInset ref LatexCommand ref reference "budget" \end_inset ). \end_layout \begin_layout Standard To help you think about this, let's draw the following picture. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Graphics filename undergrad_demand_fig1.eps \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption \begin_layout Plain Layout Figure 1 \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "udfig1" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard In the picture above, our consumer has $ \begin_inset Formula $W$ \end_inset to spend on two different goods called \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset . If she spends her entire income on good \begin_inset Formula $x$ \end_inset , she can actually purchase \begin_inset Formula $\frac{W}{p_{x}}$ \end_inset units in all. This point is labeled on the horizontal axis, and represents one feasible consumption bundle; i.e., \begin_inset Formula $\frac{W}{p_{x}}$ \end_inset units of good \begin_inset Formula $x$ \end_inset , and no units of good \begin_inset Formula $y$ \end_inset . By the same token, she could spend all her money on good \begin_inset Formula $y$ \end_inset , and purchase \begin_inset Formula $\frac{W}{p_{y}}$ \end_inset units of good \begin_inset Formula $y$ \end_inset , and no \begin_inset Formula $x$ \end_inset . This point is labeled on the vertical axis as another feasible consumption bundle. \end_layout \begin_layout Standard Any combination of these two would also work. For example, spending half her income on each good would yield the consumption bundle \begin_inset Formula $(\frac{\frac{1}{2}W}{p_{x}},\frac{\frac{1}{2}W}{p_{y}})$ \end_inset . This bundle lies halfway along the line segment that joins the points \begin_inset Formula $(\frac{W}{p_{x}},0)$ \end_inset on the horizontal axis, and \begin_inset Formula $(0,\frac{W}{p_{y}})$ \end_inset on the vertical axis. \end_layout \begin_layout Standard She doesn't really have to spend all her money either. Since she doesn't have any good \begin_inset Formula $x$ \end_inset or \begin_inset Formula $y$ \end_inset to sell, the set of feasible consumption bundles consists of all the points in the triangle formed by the axis and the line segment joining the point \begin_inset Formula $(\frac{W}{p_{x}},0)$ \end_inset to the point \begin_inset Formula $(0,\frac{W}{p_{y}})$ \end_inset . \end_layout \begin_layout Standard The \emph on budget line \emph default is the upper right face of the triangle. The slope of this line (rise over run) is \begin_inset Formula $-\frac{p_{x}}{p_{y}}$ \end_inset . The \emph on relative price of good \begin_inset Formula $x$ \end_inset \emph default is the ratio of the price of good \begin_inset Formula $x$ \end_inset to the price of good \begin_inset Formula $y$ \end_inset ( \begin_inset Formula $-1$ \end_inset times the slope of the budget line). \end_layout \begin_layout Subsection Using The Utility Theorem \end_layout \begin_layout Standard Implicitly, when we say that the bundle \begin_inset Formula $(x,y)$ \end_inset is at least as good as \begin_inset Formula $(x',y')$ \end_inset , we interpret this to mean that, given the choice between the bundles \begin_inset Formula $\text{\ensuremath{(x,y)}}$ \end_inset and \begin_inset Formula $\text{\ensuremath{(x',y')}}$ \end_inset , our consumer would \emph on choose \emph default \begin_inset Formula $\text{\ensuremath{(x,y)}}$ \end_inset . If that is true, then once we describe the budget set, we must expect the consumer to choose a point in the budget set that is at least as good as every other point in the budget set. Our `utility function' theorem says that-as preferences are complete, transitiv e and, continuous-there will be a function \begin_inset Formula $u$ \end_inset such that a bundle \begin_inset Formula $(x,y)$ \end_inset will be at least as good as every other bundle in the budget set if and only if \begin_inset Formula $u(x,y)$ \end_inset is at least as large as \begin_inset Formula $u(x',y')$ \end_inset for every other bundle \begin_inset Formula $(x',y')$ \end_inset in the budget set. If we knew the function \begin_inset Formula $u$ \end_inset , then we could find the bundle by solving the problem \begin_inset Formula \begin{equation} \max u(x,y)\label{objective} \end{equation} \end_inset subject to the constraints \begin_inset Formula \begin{equation} p_{x}x+p_{y}y\leqslant W\label{budget-constraint} \end{equation} \end_inset \begin_inset Formula \begin{equation} x\geqslant0\label{x-positive} \end{equation} \end_inset \begin_inset Formula \begin{equation} y\geqslant0\label{y-positive} \end{equation} \end_inset Now, before we try to use the mathematical formulation, let's go back for a moment to the characterization you learned in first-year economics. \end_layout \begin_layout Standard As we have assumed that our consumer's preferences are independent from the budget set he faces, we can construct a useful conceptual device. Take any bundle \begin_inset Formula $(x,y)$ \end_inset . Form the set \begin_inset Formula \[ \{\text{\ensuremath{(x',y'):\text{\ensuremath{(x,y)\succeq\text{\ensuremath{(x',y')\textrm{ and }(x',y')\succeq\text{\ensuremath{(x,y)}\}}}}}}}} \] \end_inset In words, this is the set of all bundles \begin_inset Formula $(x',y')$ \end_inset such that the consumer is \emph on indifferent \emph default between \begin_inset Formula $(x',y')$ \end_inset and \begin_inset Formula $(x,y)$ \end_inset . This set is referred to as an \emph on indifference curve \emph default . If the bundle \begin_inset Formula $(x,y)$ \end_inset is preferred to the bundle \begin_inset Formula $(x',y')$ \end_inset , then every bundle in the indifference curve associated with \begin_inset Formula $(x,y)$ \end_inset will be preferred to every bundle in the indifference curve associated with \begin_inset Formula $(x',y')$ \end_inset . This follows by the \emph on transitivity \emph default of preferences (remember that preferences are transitive if \begin_inset Formula $x\succeq y$ \end_inset and \begin_inset Formula $y\succeq z$ \end_inset implies that \begin_inset Formula $x\succeq z$ \end_inset ). So, the consumer's choice problem outlined above is equivalent to choosing the highest indifference curve that touches his or her budget set. This gives the tangency condition that you are familiar with, as in Figure \begin_inset CommandInset ref LatexCommand ref reference "tangency" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Graphics filename undergrad_demand_fig2.eps \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption \begin_layout Plain Layout Figure 2 \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "tangency" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The two bundles \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset and \begin_inset Formula $(x^{\ast}+dx,y^{\ast}-dy)$ \end_inset both lie on the same indifference curve. The vertical distance \begin_inset Formula $dy$ \end_inset is the amount of good \begin_inset Formula $y$ \end_inset that this consumer is willing to give up in order to get \begin_inset Formula $dx$ \end_inset additional units of good \begin_inset Formula $x$ \end_inset . When \begin_inset Formula $dx$ \end_inset is very small, the ratio of \begin_inset Formula $dy$ \end_inset to \begin_inset Formula $dx$ \end_inset is referred to as the \emph on marginal rate of substitution of \begin_inset Formula $y$ \end_inset for \begin_inset Formula $x$ \end_inset \emph default . Using your elementary calculus, notice that this marginal rate of substitution is the same thing as the slope of the consumer's indifference curve. \end_layout \begin_layout Standard Now, we can bring our utility theorem to bear. Assuming that the consumer's preferences are complete, transitive, and continuous, they must be represented by some utility function: let's call it \begin_inset Formula $u(x,y)$ \end_inset . Then, the indifference curve must be the set of solutions to the equation \begin_inset Formula \[ u(x',y')=u(x^{\ast},y^{\ast}) \] \end_inset We could then calculate the slope of the indifference curve (that is, the marginal rate of substitution) from the total differential \begin_inset Formula \[ u_{x}(x,y)dx+u_{y}(x,y)dy=0 \] \end_inset or \begin_inset Formula \[ \frac{dy}{dx}=-\frac{u_{x}(x,y)}{u_{y}(x,y)} \] \end_inset where \begin_inset Formula $u_{x}(x,y)$ \end_inset means the \emph on partial derivative \emph default of our utility function \begin_inset Formula $u$ \end_inset with respect to \begin_inset Formula $x$ \end_inset evaluated at the point \begin_inset Formula $(x,y)$ \end_inset . \end_layout \begin_layout Standard Since the highest indifference curve touching the budget set is the one that is just tangent to it, the marginal rate of substitution of \begin_inset Formula $y$ \end_inset for \begin_inset Formula $x$ \end_inset must be equal to the slope of the budget line, \begin_inset Formula $-\frac{p_{x}}{p_{y}}$ \end_inset . \end_layout \begin_layout Standard Now, let's take the utility function that we know exists, go back to the purely mathematical formulation and maximize ( \begin_inset CommandInset ref LatexCommand ref reference "objective" \end_inset ) subject to the constraints ( \begin_inset CommandInset ref LatexCommand ref reference "budget-constraint" \end_inset ) through ( \begin_inset CommandInset ref LatexCommand ref reference "y-positive" \end_inset ). By the Lagrangian theorem, there are three multipliers (one for each of the three constraints) \begin_inset Formula $\lambda_{1},\lambda_{2},$ \end_inset and \begin_inset Formula $\lambda_{3}$ \end_inset such that the Lagrangian function can be written as \begin_inset Formula \[ u(x,y)+\lambda_{1}(p_{x}x+p_{y}y-W)-\lambda_{2}x-\lambda_{3}y \] \end_inset At the optimal solution to the problem, the following first order conditions must hold \begin_inset Formula \begin{equation} u_{x}(x,y)+\lambda_{1}p_{x}-\lambda_{2}=0\label{x} \end{equation} \end_inset \begin_inset Formula \begin{equation} u_{y}(x,y)+\lambda_{1}p_{y}-\lambda_{3}=0\label{y} \end{equation} \end_inset \begin_inset Formula \begin{equation} p_{x}x+p_{y}y-W\leqslant0;\lambda_{1}\leqslant0 \end{equation} \end_inset \begin_inset Formula \begin{equation} -x\leqslant0;\lambda_{2}\leqslant0 \end{equation} \end_inset \begin_inset Formula \begin{equation} -y\leqslant0;\lambda_{3}\leqslant0 \end{equation} \end_inset where the last three conditions holding with complementary slackness. \end_layout \begin_layout Standard Suppose that we knew for some reason that the solution must involve positive amounts of both \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset (you will see an example like this below). Then by complementary slackness, the multipliers associated with both of these variables would have to be zero. Then ( \begin_inset CommandInset ref LatexCommand ref reference "x" \end_inset ) and ( \begin_inset CommandInset ref LatexCommand ref reference "y" \end_inset ) would simplify to \begin_inset Formula \[ u_{x}(x,y)=-\lambda_{1}p_{x} \] \end_inset and \begin_inset Formula \[ u_{y}(x,y)=-\lambda_{1}p_{y} \] \end_inset Dividing the first condition by the second gives exactly the same result that we deduced from the picture \begin_inset Formula \[ \frac{u_{x}(x,y)}{u_{y}(x,y)}=\frac{p_{x}}{p_{y}} \] \end_inset \end_layout \begin_layout Section A Simple Example \end_layout \begin_layout Standard If we know more about the utility function, then the mathematical approach can be quite helpful. For example, in the section on Lagrangian theory it was assumed that the utility function had the form \begin_inset Formula \begin{equation} u(x,y)=x^{\alpha}y^{(1-\alpha)}\label{cobb-douglas} \end{equation} \end_inset Then the first order conditions becAme \begin_inset Formula \begin{equation} \alpha x^{(\alpha-1)}y^{(1-\alpha)}+\lambda_{1}p_{x}-\lambda_{2}=0 \end{equation} \end_inset \begin_inset Formula \begin{equation} (1-\alpha)x^{\alpha}y^{-\alpha}+\lambda_{1}p_{y}-\lambda_{3}=0 \end{equation} \end_inset \begin_inset Formula \begin{equation} p_{x}x+p_{y}y-W\leqslant0;\lambda_{1}\leqslant0\label{cs3} \end{equation} \end_inset \begin_inset Formula \begin{equation} -x\leqslant0;\lambda_{2}\leqslant0\label{cs1} \end{equation} \end_inset \begin_inset Formula \begin{equation} -y\leqslant0;\lambda_{3}\leqslant0\label{cs2} \end{equation} \end_inset where ( \begin_inset CommandInset ref LatexCommand ref reference "cs3" \end_inset ), ( \begin_inset CommandInset ref LatexCommand ref reference "cs1" \end_inset ), and ( \begin_inset CommandInset ref LatexCommand ref reference "cs2" \end_inset ) hold with complementary slackness. At first glance, this mess doesn't look particularly useful. However, notice that if either \begin_inset Formula $x$ \end_inset or \begin_inset Formula $y$ \end_inset are zero, then utility is zero on the right hand side of ( \begin_inset CommandInset ref LatexCommand ref reference "cobb-douglas" \end_inset ). If the consumer has any income at all, then she can do strictly better than this by purchasing any bundle where both \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are positive. As a consequence, we can be sure that, in any solution to the consumer's maximization problem, both \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are positive. Then, by the complementary slackness conditions ( \begin_inset CommandInset ref LatexCommand ref reference "cs1" \end_inset ) and ( \begin_inset CommandInset ref LatexCommand ref reference "cs2" \end_inset ), \begin_inset Formula $\lambda_{2}$ \end_inset and \begin_inset Formula $\lambda_{3}$ \end_inset must both be zero. \end_layout \begin_layout Standard In addition, the solution will also require that the consumer use up her whole budget since the right hand side of ( \begin_inset CommandInset ref LatexCommand ref reference "cobb-douglas" \end_inset ) is strictly increasing in both its arguments. Complementary slackness in ( \begin_inset CommandInset ref LatexCommand ref reference "cs3" \end_inset ) unfortunately doesn't tell us that \begin_inset Formula $\lambda_{1}$ \end_inset is positive, it is possible, but unlikely that both the constraint and its multiplier could be zero. \end_layout \begin_layout Standard Let's continue. The logic of the Lagrange theorem is that the first order conditions have to hold at a solution to the problem. Remember that the converse is not true: a solution to the first order condition s may not give a solution to the maximization problem. Now, as long as both prices are strictly positive and both \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset must also be so, a solution to the maximization problem (if it exists) must satisfy \begin_inset Formula \begin{equation} \alpha x^{(\alpha-1)}y^{(1-\alpha)}=-\lambda_{1}p_{x}\label{simple-1} \end{equation} \end_inset and \begin_inset Formula \begin{equation} (1-\alpha)x^{\alpha}y^{-\alpha}=-\lambda_{1}p_{y}\label{simple-2} \end{equation} \end_inset Now, divide ( \begin_inset CommandInset ref LatexCommand ref reference "simple-1" \end_inset ) by ( \begin_inset CommandInset ref LatexCommand ref reference "simple-2" \end_inset ) (which means divide the left hand side of ( \begin_inset CommandInset ref LatexCommand ref reference "simple-1" \end_inset ) by the left hand side of ( \begin_inset CommandInset ref LatexCommand ref reference "simple-2" \end_inset ) and the same for the right hand sides). You will get \begin_inset Formula \begin{equation} \frac{\alpha}{1-\alpha}\frac{y}{x}=\frac{p_{x}}{p_{y}} \end{equation} \end_inset or \begin_inset Formula $p_{x}x=p_{y}y\frac{\alpha}{1-\alpha}$ \end_inset . Again, this last equation has to be true at any solution to the maximization problem. Since it also has to be true that \begin_inset Formula $p_{x}x+p_{y}y=W$ \end_inset , then \begin_inset Formula $p_{y}y\frac{\alpha}{1-\alpha}+p_{y}y=W$ \end_inset . This means that is has to be true that \begin_inset Formula \begin{equation} y=W(1-\alpha)/p_{y}\label{demand} \end{equation} \end_inset Similarly, \begin_inset Formula $x=W\alpha/p_{x}$ \end_inset . These two equations are great because they tell us the solution to the maximization problem for all different values of \begin_inset Formula $p_{x}$ \end_inset , \begin_inset Formula $p_{y}$ \end_inset , and \begin_inset Formula $W$ \end_inset . These last two equations are `demand curves,' just like the ones you saw in your first-year economics course. You can easily see that the `law of demand' holds for this utility function: an increase in price lowers demand. \end_layout \begin_layout Standard This simple example takes us a long way along the road to understanding what it is that economists do differently from many other social scientists. We started with some very plausible assertions about behavior; in particular, given any pair of choices, consumers could always make one, and these choices would be transitive. This showed us that we could `represent' these preferences with a utility function. Using this utility function, we can conclude that the consumer's choice from any set of alternatives will be the solution to a maximization problem. \end_layout \begin_layout Standard By itself, this seems to say very little - if you give a consumer a set of choices, she will make one. However, we now have the wherewithal to formulate models - additional assumptio ns that we can add to hone our predictions. We added two of them. The first is basic to all the old-fashioned consumer theory - the way the consumer ranks any two bundles does not depend on the particular budget set in which the alternatives are offered. The second assumption was that the the utility function has a particular form as given by ( \begin_inset CommandInset ref LatexCommand ref reference "cobb-douglas" \end_inset ). \end_layout \begin_layout Standard Putting these together we were able to apply some simple mathematics to predict what the consumer would do in all the different budget sets that we could imagine the consumer facing. This is the demand function ( \begin_inset CommandInset ref LatexCommand ref reference "demand" \end_inset ) that we derived above. As promised above, the mathematics has delivered \emph on all \emph default the implications of our model. The demand function shows that there are a \emph on lot \emph default of implications, so it shouldn't be too hard for us to check whether the model is right. \begin_inset Foot status collapsed \begin_layout Plain Layout This is both good and bad when a model has lots of implications. This is good because the model is easy to test. That may make it a bad model, as well, if its predictions are obviously wrong. The utility function in ( \begin_inset CommandInset ref LatexCommand ref reference "cobb-douglas" \end_inset ) is like this. It predicts that the consumer will consume positive amounts of every good - no sensible consumer would pay for Microsoft Windows, or buy an SUV. \end_layout \end_inset \end_layout \begin_layout Standard The utility function theorem allows us to unify our approach (though not our model) to virtually all behavioral problems. We don't even need to confine ourselves to human behavior. For instance, animals make both behavioral and genetic choices. Transitivity is arguably plausible and we can assume that they are always able to make some choice (completeness). So, we could also represent their choices as solutions to utility maximization problems. Genetics involves choices made by biological systems in response to changes in environmental conditions. Completeness and transitivity of these choices are both compelling. Completeness is immediate. The idea that organisms evolve seems to rule out the kind of cyclic choices implied by intransitivity (which would require that one evolves then eventually reverts back again). So we could try to model genetic behavior using the maximization approach. \begin_inset Foot status collapsed \begin_layout Plain Layout I can't resist suggesting one of my favourite arguments by Arthur Robson (http://www.sfu.ca/ \begin_inset space ~ \end_inset robson/wwgo.pdf). The formal title is \begin_inset Quotes eld \end_inset Why we grow Large and then grow old: Biology, Economics and Mortality \begin_inset Quotes erd \end_inset , the informal title of his talk was \begin_inset Quotes eld \end_inset Why we Die \begin_inset Quotes erd \end_inset . Yes, it is the solution to a maximization problem. \end_layout \end_inset \end_layout \begin_layout Standard This unified approach is nice, but not necessarily better. After all, we need to add a model (assumptions about utility, for example) that could quite well be wrong. Fortunately, the econometricians have taught us how to test our models and reject the ones that are wrong, so that we can refine them. If you are taking econometrics, you might want to learn how. If you take logs of equation ( \begin_inset CommandInset ref LatexCommand ref reference "demand" \end_inset ) you will get \begin_inset Formula \begin{equation} \log(y)=\log(1-\alpha)+\log(W)-\log(p_{y})\label{linear-equation} \end{equation} \end_inset If you add an error term to this, you get a simple linear regression equation in which the coefficient associated with the log of price is supposed to be 1. That is very easy to test (and reject). \end_layout \begin_layout Section How to Test Demand Theory \end_layout \begin_layout Standard If we make assumptions about the utility function, we can say a lot about how consumers behave. As with the formulation given by ( \begin_inset CommandInset ref LatexCommand ref reference "cobb-douglas" \end_inset ), these strong predictions often won't be borne out in whatever data we have. For example, an econometric test of ( \begin_inset CommandInset ref LatexCommand ref reference "linear-equation" \end_inset ) will almost surely fail. Then we can reject our model. However, we will most likely be rejecting our assumption that the utility function has the form given in ( \begin_inset CommandInset ref LatexCommand ref reference "cobb-douglas" \end_inset ). What if we wanted to test the assertion that preferences are independent of the budget set the consumer faces? To do that, we need to find a prediction that will be true no matter what form the utility function has, then find a situation where the consumer doesn't obey that prediction. \end_layout \begin_layout Standard This creates a bit of a problem. Suppose our consumer simply doesn't care what consumption bundle she gets. Then our model is consistent with any pattern of behavior at all, and we could never reject it. Neither would we find such a model useful, because it doesn't really make any predictions. So a useful and testable economic model will inevitably involve some assumption s about the utility function. \end_layout \begin_layout Standard Fortunately, if we simply add the assumption that consumers always prefer more of a good to less of it, we get a prediction that is true no matter what other properties the consumer's preferences have. It goes the following way - suppose we observe at particular array of prices, a level of income, and the choice the consumer makes under those circumstances. Then, suppose that, at another time, we observe a new array of prices, and a new level of income such that the consumer could just afford to buy the consumption bundle that she purchased in the first case. Of course, along this new budget line we will get to observe another choice by the consumer. Along this new budget line there will be some consumption bundles that would have been inside (strictly) the budget set at the old prices and level of income. If the consumer picks one of these then she is not acting as predicted by our model, and we can reject our model. \end_layout \begin_layout Standard Let me illustrate this in the simple case where there are only two goods. The basic idea is depicted in Figure \begin_inset CommandInset ref LatexCommand ref reference "revealed" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Graphics filename undergrad_demand_fig3.eps \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption \begin_layout Plain Layout Figure 3 \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "revealed" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The point \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset is the solution to the consumer's problem at the initial set of prices. Here we simplify a bit by assuming that at the initial situation, the price of good \begin_inset Formula $x$ \end_inset is \begin_inset Formula $p$ \end_inset while the price of good \begin_inset Formula $y$ \end_inset is just \begin_inset Formula $1$ \end_inset . The budget set for the consumer is the triangle formed by the axis and the line between the points \begin_inset Formula $(0,W)$ \end_inset and \begin_inset Formula $(\frac{W}{p})$ \end_inset . \end_layout \begin_layout Standard Now we present the consumer with a new higher price for good \begin_inset Formula $x$ \end_inset . The new price is \begin_inset Formula $p'$ \end_inset . At this new price, good \begin_inset Formula $x$ \end_inset is more expensive than it was before, so our consumer could not afford to buy the bundle \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset unless there is some change in her income. So, let's suppose we can give her just enough income to buy the bundle \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset that she bought before the change in prices. The compensated income is denoted \begin_inset Formula $W'$ \end_inset . The new income, along with the new price \begin_inset Formula $p'$ \end_inset , gives her the blue budget line. By construction, this budget line just passes through the point \begin_inset Formula $(x^{\ast},y^{\ast}).$ \end_inset \end_layout \begin_layout Standard This is all reasoning from your first-year economics course. Along the new budget line, the consumer should pick a point like \begin_inset Formula $(x',y')$ \end_inset . If she picks a point like \begin_inset Formula $(x^{0},y^{0})$ \end_inset instead, then she would be choosing a point that she could have afforded to buy at the initial price \begin_inset Formula $p$ \end_inset before her income changed. \end_layout \begin_layout Standard What would be wrong with that? Well, remember, we are trying to figure out whether our model is true. The model consists of three kinds of assumptions. The first are our most basic axioms - completeness, transitivity and continuity of preferences. The second is our assumption that preferences are independent of the budget set that is presented to the consumer. The third is the assumption that the consumer prefers more of each good. \end_layout \begin_layout Standard Since \begin_inset Formula $(x^{0},y^{0})$ \end_inset is inside the budget set when the price of \begin_inset Formula $x$ \end_inset is \begin_inset Formula $p$ \end_inset (we leave out the additional qualifier \begin_inset Quotes eld \end_inset and when income is \begin_inset Formula $W$ \end_inset \begin_inset Quotes erd \end_inset to make the argument is little shorter), then whatever the consumer's indiffere nce curves actually look like, there must be other bundles in the initial budget set that are strictly preferred to \begin_inset Formula $(x^{0},y^{0})$ \end_inset . We have no idea what all these bundles are, but suppose that one such bundle is \begin_inset Formula $(x'',y'')$ \end_inset (which isn't marked in the picture). Since the consumer chose \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset from that budget set, it must be that \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset is at least as good from the consumer's point of view as \begin_inset Formula $(x'',y'')$ \end_inset . Yet \begin_inset Formula $(x'',y'')$ \end_inset is strictly better than \begin_inset Formula $(x^{0},y^{0})$ \end_inset . By transitivity, \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset is strictly better for the consumer than \begin_inset Formula $(x^{0},y^{0}).$ \end_inset Then, if preferences are the same in every budget set, the consumer could do strictly better in the new budget set at prices \begin_inset Formula $p'$ \end_inset by choosing \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset . If our consumer chooses a bundle like \begin_inset Formula $(x^{0},y^{0})$ \end_inset then there must be something wrong with our story. \end_layout \begin_layout Standard So, if our model of the consumer is correct, we should observe that an \emph on income compensated \emph default increase in the price of any commodity will result in a fall in demand for that commodity. I will leave it to your econometrics courses to tell you how the tests of consumer demand theory have worked out. \end_layout \begin_layout Section Comparative Statics and The Envelope Theorem \end_layout \begin_layout Standard To appreciate most modern economic theory, you need to understand that the consumer's choice depends on the constraint set she faces. If we characterize the choice as the solution to a maximization problem, then the consumer's choice could be thought of as a \emph on function \emph default of the parameters of the constraint set she faces. In general, we refer to this as a \emph on best reply \emph default function. In consumer theory, the best reply function is called a demand function. More generally, the parameters that affect the choice sets may not be prices. In game theory, the parameters that affect the individual's choice behavior are the actions that she thinks others will take. \end_layout \begin_layout Standard You have seen a best reply function already. When preferences are given by ( \begin_inset CommandInset ref LatexCommand ref reference "cobb-douglas" \end_inset ) then the amount of good \begin_inset Formula $y$ \end_inset the consumer will buy for \emph on any \emph default pair of prices \begin_inset Formula $(p_{x},p_{y})$ \end_inset and \emph on any \emph default level of income \begin_inset Formula $W$ \end_inset is given by ( \begin_inset CommandInset ref LatexCommand ref reference "demand" \end_inset ). The demand for good \begin_inset Formula $y$ \end_inset is a function of its price and the consumer's income. \end_layout \begin_layout Standard It is actually pretty unusual to have the demand function in such a complete form. To get such a thing, you actually need to be able to find a complete solution to the first order conditions. That requires assumptions about utility that are unlikely to pass any kind of empirical test. However, it is often possible to use mathematical methods to say useful things. \end_layout \begin_layout Standard Let's go back to the case where preferences are represented by a function \begin_inset Formula $u(x,y)$ \end_inset and assume there is a demand function, \begin_inset Formula $D(p_{x},p_{y},W)$ \end_inset , that tells us for each possible argument what quantity of good \begin_inset Formula $x$ \end_inset the consumer will choose to buy. This function probably looks something like ( \begin_inset CommandInset ref LatexCommand ref reference "demand" \end_inset ), but we can't really say exactly what it is like. Let's make the heroic assumption that this function looks like ( \begin_inset CommandInset ref LatexCommand ref reference "demand" \end_inset ) in the sense that it is differentiable; that is, \begin_inset Formula $D(p_{x},p_{y},W)$ \end_inset has exactly three partial derivatives, one for each of its arguments. \end_layout \begin_layout Standard In particular, for preferences given by ( \begin_inset CommandInset ref LatexCommand ref reference "cobb-douglas" \end_inset ), the demand function for good \begin_inset Formula $y$ \end_inset is \begin_inset Formula \[ D(p_{x},p_{y},W)=\frac{(1-\alpha)W}{p_{y}} \] \end_inset The three partial derivatives are given by \begin_inset Formula \[ \frac{\partial D(p_{x},p_{y},W)}{\partial p_{x}}\equiv D_{p_{x}}(p_{x},p_{y},W)=0 \] \end_inset \begin_inset Formula \[ \frac{\partial D(p_{x},p_{y},W)}{\partial p_{y}}\equiv D_{p_{y}}(p_{x},p_{y},W)=-(1-\alpha)W\left(\frac{1}{p_{y}}\right)^{2} \] \end_inset \begin_inset Formula \[ \frac{\partial D(p_{x},p_{y},W)}{\partial W}\equiv D_{W}(p_{x},p_{y},W)=\frac{1-\alpha}{p_{y}} \] \end_inset More generally, we can just refer to the partial derivatives as \begin_inset Formula $D_{p_{x}}$ \end_inset , \begin_inset Formula $D_{p_{y}}$ \end_inset and \begin_inset Formula $D_{W}$ \end_inset as long as you remember that these derivatives depend on their arguments. \end_layout \begin_layout Subsection Implicit Differentiation \end_layout \begin_layout Standard The method of implicit differentiation will sometimes give you a lot of information about a best reply function. To be honest, it doesn't really work very well in demand theory, but I will explain it anyway. We will use this method in our discussion of portfolio theory below. \end_layout \begin_layout Standard Let's simplify things a bit and hold the price of good \begin_inset Formula $y$ \end_inset constant at 1 and vary only the price \begin_inset Formula $p$ \end_inset of good \begin_inset Formula $x$ \end_inset , and the level of income \begin_inset Formula $W$ \end_inset . Let's suppose as well that for some price \begin_inset Formula $p$ \end_inset and level of income \begin_inset Formula $W$ \end_inset , the solution to the consumer's maximization problem involves strictly positive amounts of both goods \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset . Then by the Lagrangian theorem, there must be a multiplier \begin_inset Formula $\lambda$ \end_inset such that the first order conditions \begin_inset Formula \begin{equation} u_{x}(x,y)+\lambda p=0\label{foc21} \end{equation} \end_inset \begin_inset Formula \begin{equation} u_{y}(x,y)+\lambda=0\label{foc22} \end{equation} \end_inset \begin_inset Formula \begin{equation} px+y=W\label{foc23} \end{equation} \end_inset hold. \end_layout \begin_layout Standard As we vary \begin_inset Formula $p$ \end_inset slightly, the values of \begin_inset Formula $x$ \end_inset , \begin_inset Formula $y$ \end_inset , and \begin_inset Formula $\lambda$ \end_inset will change so that ( \begin_inset CommandInset ref LatexCommand ref reference "foc21" \end_inset ) to ( \begin_inset CommandInset ref LatexCommand ref reference "foc23" \end_inset ) continue to hold. Then, by the chain rule of calculus, \begin_inset Formula \begin{equation} u_{xx}(x,y)\frac{dx}{dp}+u_{xy}(x,y)\frac{dy}{dp}+\lambda+p\frac{d\lambda}{dp}=0 \end{equation} \end_inset \begin_inset Formula \begin{equation} u_{yx}(x,y)\frac{dx}{dp}+u_{yy}(x,y)\frac{dy}{dp}+\frac{d\lambda}{dp}=0 \end{equation} \end_inset \begin_inset Formula \begin{equation} x+p\frac{dx}{dp}+\frac{dy}{dp}=0 \end{equation} \end_inset In this notation, the terms like \begin_inset Formula $u_{xx}(x,y)$ \end_inset are second derivatives. For example, when preferences are given by ( \begin_inset CommandInset ref LatexCommand ref reference "cobb-douglas" \end_inset ), \begin_inset Formula $u_{xx}(x,y)=\alpha(\alpha-1)x^{\alpha-2}y^{1-\alpha}$ \end_inset . The terms like \begin_inset Formula $\frac{dx}{dp}$ \end_inset are the derivatives of the implicit functions that satisfy the first order conditions ( \begin_inset CommandInset ref LatexCommand ref reference "foc21" \end_inset ) to ( \begin_inset CommandInset ref LatexCommand ref reference "foc23" \end_inset ) as \begin_inset Formula $p$ \end_inset changes a little. \end_layout \begin_layout Standard We are interested in trying to figure out properties of \begin_inset Formula $\frac{dx}{dp}$ \end_inset . In principle, we could use these last three equations to learn about it. There are three equations and three unknowns. They are non-linear, so there is no guarantee they will have a solution, but they probably will. The complication is that this solution is complicated and won't actually say much. For what it is worth, pure brute force gives the following \begin_inset Formula \begin{equation} \frac{dx}{dp}=\frac{(u_{xy}-pu_{yy})x-\lambda}{u_{xx}-2pu_{xy}-p^{2}u_{yy}}\label{cs} \end{equation} \end_inset This is pretty bleak, because there is too much in the expression that we don't know. The sign of the expression could be either positive or negative depending on the sizes of the cross derivatives. Then, there is the mysterious multiplier term \begin_inset Formula $\lambda$ \end_inset . \end_layout \begin_layout Standard The one advantage of this approach is that it will often tell you what you need to \emph on assume \emph default in order to get the result that you want. Since the irritating terms are the cross derivatives, suppose that we make the utility function \emph on separable \emph default . For example, it might have the form \begin_inset Formula $u(x,y)=v(x)+w(y)$ \end_inset where \begin_inset Formula $v$ \end_inset and \begin_inset Formula $w$ \end_inset are concave functions (which means that their derivatives get smaller as their arguments get larger). Then \begin_inset Formula $u_{xy}=u_{yx}=0$ \end_inset and ( \begin_inset CommandInset ref LatexCommand ref reference "cs" \end_inset ) reduces to \begin_inset Formula \begin{equation} \frac{dx}{dp}=\frac{-pw_{yy}x-\lambda}{v_{xx}-p^{2}w_{yy}} \end{equation} \end_inset This still not enough. If we assume that the function \begin_inset Formula $w$ \end_inset and \begin_inset Formula $v$ \end_inset are both concave, then their second derivatives can't be positive. The multiplier is less than or equal to zero by the complementary slackness conditions, so the numerator is non-negative. The denominator can be either positive or negative depending on the magnitudes of the second derivatives. \end_layout \begin_layout Standard This leads us to the second most famous special functional form in economics. If we assume that \begin_inset Formula $w(y)=y$ \end_inset , we get something called a \emph on quasi-linear \emph default utility function. Then \begin_inset Formula $w_{yy}=0$ \end_inset and we know that the demand function is at least downward sloping. Quasi-linear utility functions are widely used in the theory of mechanism design and auctions. \end_layout \begin_layout Subsection Graphical Methods \end_layout \begin_layout Standard The arguments above are a bit obscure. Graphical methods will often provide some more insight. The methods in the previous section are also \emph on local \emph default methods, since they assume that all the changes that are occurring are small. Graphical analysis won't really give you a full solution to the problem you are trying to solve, you will ultimately need to return to the math for a full solution. Yet graphical analysis will often point in the right direction. \end_layout \begin_layout Standard If you simply want to understand why the demand function doesn't slope downward, a graphical trick will show you. Go back to Figure \begin_inset CommandInset ref LatexCommand ref reference "revealed" \end_inset where the consumer was faced with an increase in the price of good \begin_inset Formula $x$ \end_inset , but was given enough income to allow her to afford her initial consumption bundle. We concluded that this combination of changes in her budget set would induce her to lower her demand for good \begin_inset Formula $x$ \end_inset . We can decompose these changes into their constituent parts - an increase in price, followed by an increase in income. The two changes together appear in Figure \begin_inset CommandInset ref LatexCommand ref reference "slutsky" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Graphics filename undergrad_demand_fig4.eps \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption \begin_layout Plain Layout Figure 4 \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "slutsky" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The picture shows a problem similar to the one in Figure \begin_inset CommandInset ref LatexCommand ref reference "revealed" \end_inset . The initial price for good \begin_inset Formula $x$ \end_inset is \begin_inset Formula $p$ \end_inset . At the price and her initial income, the consumer selects the bundle \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset . As we saw before, if we raise the price of good \begin_inset Formula $x$ \end_inset to \begin_inset Formula $p'$ \end_inset but give the consumer enough extra income that she could just purchase the original bundle \begin_inset Formula $(x^{\ast},y^{\ast})$ \end_inset , then she must respond by purchasing more good \begin_inset Formula $y$ \end_inset . In other words, her compensated demand for good \begin_inset Formula $x$ \end_inset must fall. For example, she might choose the new bundle \begin_inset Formula $(x',y')$ \end_inset as in the Figure. \end_layout \begin_layout Standard If we want to know how the impact of the price increase by itself will influence her demand, we need to take away the extra income we gave her so that she could afford her initial bundle. In the picture we do this by shifting the budget line downward (toward the origin) from the red line to the blue line. Since we are holding both prices constant as we take away this income, the slope of the budget line doesn't change as we shift it in. (Make sure you understand why the blue line goes through the point \begin_inset Formula $(0,W)$ \end_inset ). \end_layout \begin_layout Standard As the picture is drawn, our consumer chooses the bundle \begin_inset Formula $(x'',y'')$ \end_inset . The remarkable thing about this bundle is that it actually involves more good \begin_inset Formula $x$ \end_inset than there is in the initial bundle \begin_inset Formula $(x^{\ast},y^{\ast}$ \end_inset ). An increase in the price of good \begin_inset Formula $x$ \end_inset has actually caused an increase in demand for good \begin_inset Formula $x$ \end_inset . The diagram illustrates why. As our consumers income rises (shifting the budget line up from the blue line to the red line, her demand for good \begin_inset Formula $x$ \end_inset actually falls. Goods that have this property are called \emph on inferior goods \emph default as you may recall from your first-year course. \end_layout \begin_layout Subsection The Envelope Theorem \end_layout \begin_layout Standard There is one special theorem associated with the Lagrangian that is sometimes quite useful. Suppose that we are trying to solve the problem \begin_inset Formula \begin{equation} \max_{x}u(x) \end{equation} \end_inset subject to \begin_inset Formula \begin{equation} G_{1}(x,y)\leqslant0 \end{equation} \end_inset \begin_inset Formula \[ \vdots \] \end_inset \begin_inset Formula \begin{equation} G_{m}(x,y)\leqslant0 \end{equation} \end_inset where \begin_inset Formula $x\in\mathbb{R}^{n}$ \end_inset , \begin_inset Formula $m\geqslant1$ \end_inset , and \begin_inset Formula $y$ \end_inset is some parameter that affects our constraints, for example, the price of one of the goods, or the consumer's income. If we could find a solution to this problem, the we could call the \emph on value \emph default of the solution \begin_inset Formula $V(y)$ \end_inset . This value is a function of the parameter \begin_inset Formula $y$ \end_inset . If \begin_inset Formula $y$ \end_inset were a price, for instance, then the maximum value of utility would be a decreasing function of price. Suppose we are interested in finding out how a change in \begin_inset Formula $y$ \end_inset will change this maximum value - i.e., we want to know something about \begin_inset Formula $\frac{dV(y)}{dy}$ \end_inset . \end_layout \begin_layout Standard One way to do this to use implicit differentiation as we did above. The vector \begin_inset Formula $x^{\ast}$ \end_inset that solves the problem is an implicit function of \begin_inset Formula $y$ \end_inset . Imagine that \begin_inset Formula $x^{\ast}[y]$ \end_inset is the function that gives us the solution to the problem. For example, in the consumer's problem, if we think of \begin_inset Formula $y$ \end_inset as the price of good \begin_inset Formula $x$ \end_inset , then \begin_inset Formula $x^{\ast}[y]$ \end_inset is the \emph on bundle \emph default that provides the maximum utility. Whatever the actual interpretation, it should be clear that \begin_inset Formula $V[y]=u[x^{\ast}[y]]$ \end_inset . We could then compute the impact of a change in \begin_inset Formula $y$ \end_inset by finding all the partial derivatives of \begin_inset Formula $u$ \end_inset with respect to each of the \begin_inset Formula $x$ \end_inset 's evaluated at the initial optimal solution, multiplying each of these by the total derivative of the corresponding solution with respect to a change in \begin_inset Formula $y$ \end_inset , then summing everything up. In math \begin_inset Formula \[ \frac{dV(y)}{dy}=\sum_{i=1}^{n}\frac{\partial u(x^{\ast}[y])}{\partial x_{i}}\frac{dx_{i}^{\ast}[y]}{dy} \] \end_inset This would require not only that we take a lot of partial derivatives, but also that we compute function \begin_inset Formula $x^{\ast}[y]$ \end_inset and find its total derivatives - a daunting amount of work. \end_layout \begin_layout Standard Fortunately, there is a very nice way around this. Recall that the Lagrangian function associated with this maximization problem is \begin_inset Formula \[ L(x,\lambda,y)=u(x)+\sum_{j=1}^{m}\lambda_{j}G_{j}(x,y) \] \end_inset Then the envelope theorem says that \end_layout \begin_layout Theorem \begin_inset Formula \begin{equation} \left.\frac{dV(y)}{dy}=\frac{\partial L(x,\lambda,y)}{\partial y}\right|_{x=x^{\ast};\lambda=\lambda^{\ast}} \end{equation} \end_inset \end_layout \begin_layout Standard This says that to compute the total derivative of the maximum value, then we only need to compute the \emph on partial \emph default derivative of the Lagrangian evaluated at the optimal solution. This is much easier. \end_layout \begin_layout Standard I am going to show you why this is true, and how nicely it works. Our consumer solves the problem \begin_inset Formula \[ \max u(x,y) \] \end_inset subject to \begin_inset Formula \[ px+y-W\leqslant0 \] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ -x\leqslant0 \] \end_inset \begin_inset Formula \[ -y\leqslant0 \] \end_inset The Lagrangian is \begin_inset Formula \[ u(x,y)+\lambda_{1}(px+y-W)-\lambda_{2}x-\lambda_{3}y \] \end_inset Suppose I want to find out the impact of an increase in wealth on the consumer' s optimal utility starting from an initial price \begin_inset Formula $p_{0}$ \end_inset and wealth level \begin_inset Formula $W_{0}$ \end_inset . The Envelope theorem says that we first need to solve the consumer's problem and find the utility maximizing demands, call them \begin_inset Formula $x^{0}$ \end_inset and \begin_inset Formula $y^{0}$ \end_inset , as well as the multipliers that satisfy the first order conditions at the optimal solution, \begin_inset Formula $\lambda_{1}^{0}$ \end_inset , \begin_inset Formula $\lambda_{2}^{0}$ \end_inset , and \begin_inset Formula $\lambda_{3}^{0}$ \end_inset . The Lagrangian is generally a complicated function of \begin_inset Formula $W$ \end_inset because all the multipliers and the optimal \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are changing with \begin_inset Formula $W$ \end_inset . Nonetheless the derivative of this optimal value is simply \begin_inset Formula \[ \frac{\partial L(x,y,\lambda_{1},\lambda_{2},\lambda_{3})}{\partial W}=-\lambda_{1} \] \end_inset The significance of the \begin_inset Formula $\partial L$ \end_inset instead of \begin_inset Formula $dL$ \end_inset is that we don't have to worry about all the implicit functions. \end_layout \begin_layout Standard Here is the proof of the envelope theorem: \end_layout \begin_layout Proof First observe that \begin_inset Formula \[ V(y)=u(x^{\ast})\equiv L(x^{\ast},\lambda^{\ast},y) \] \end_inset \begin_inset Formula \begin{equation} =u(x^{\ast})+\sum_{j=1}^{m}\lambda_{j}^{\ast}G_{j}(x^{\ast},y) \end{equation} \end_inset It might seem that this would be false because of the sum that we add to \begin_inset Formula $u(x^{\ast})$ \end_inset . However, by the complementary slackness conditions, the product of the multiplier and the constraint will always be zero at the solution to the first order conditions. So, the sum is exactly zero. \end_layout \begin_layout Proof As long as we think of \begin_inset Formula $x^{\ast}$ \end_inset and \begin_inset Formula $\lambda^{\ast}$ \end_inset as implicit functions of \begin_inset Formula $y$ \end_inset , then this is an identity, so we find the derivative using the chain rule. \begin_inset Formula \[ \frac{dV(y)}{dy}=\sum_{i=1}^{n}\frac{\partial u(x^{\ast})}{\partial x_{i}}\frac{dx_{i}^{\ast}}{dy}+\sum_{j=1}^{m}\left[\frac{d\lambda_{j}^{\ast}}{dy}G_{j}(x^{\ast},y)+\lambda_{j}^{\ast}\sum_{i=1}^{n}\frac{\partial G_{j}(x^{\ast},y)}{\partial x_{i}}\frac{dx_{i}^{\ast}}{dy}+\lambda_{j}^{\ast}\frac{dG_{j}(x^{\ast},y)}{dy}\right] \] \end_inset First consider the terms \begin_inset Formula $\frac{d\lambda_{j}^{\ast}}{dy}G_{j}(x^{\ast},y)$ \end_inset . By complementary slackness, either \begin_inset Formula $G_{j}(x^{\ast},y)$ \end_inset is zero, or \begin_inset Formula $\lambda_{j}^{\ast}$ \end_inset is zero, or both are zero. In the first case, and the last case, we can forget about the term \begin_inset Formula $\frac{d\lambda_{j}^{\ast}}{dy}G_{j}(x^{\ast},y)$ \end_inset because it will be zero. What happens when \begin_inset Formula $G_{j}(x^{\ast},y)<0$ \end_inset ? Then \begin_inset Formula $\lambda_{j}^{\ast}$ \end_inset is zero. In that event, changing \begin_inset Formula $y$ \end_inset , say by \begin_inset Formula $dy$ \end_inset , will not change the solution very much and we can rely on continuity to ensure that \begin_inset Formula $G_{j}(x^{\ast}[y+dy],y+dy)$ \end_inset is still negative. If that is the case, then again using complementary slackness, it must be that \begin_inset Formula $\lambda_{j}^{\ast}[y+dy]=0$ \end_inset , which means that \begin_inset Formula $\frac{d\lambda_{j}^{\ast}}{dy}$ \end_inset =0. \end_layout \begin_layout Proof Using this, we can rewrite the derivative as follows \begin_inset Formula \[ \frac{dV(y)}{dy}=\sum_{i=1}^{n}\left(\frac{\partial u(x^{\ast})}{\partial x_{i}}+\sum_{j=1}^{m}\lambda_{j}^{\ast}\frac{\partial G_{j}(x^{\ast},y)}{\partial x_{i}}\right)\frac{dx_{i}^{\ast}}{dy}+\sum_{j=1}^{m}\lambda_{j}^{\ast}\frac{dG_{j}(x^{\ast},y)}{dy} \] \end_inset Now notice that the terms in the first sum over \begin_inset Formula $i$ \end_inset are all derivatives of the Lagrangian with respect to some \begin_inset Formula $x_{i}$ \end_inset evaluated at the optimal solution. Of course the optimal solution has the property that the derivatives of the Lagrangian with respect to the \begin_inset Formula $x_{i}$ \end_inset are all equal to zero. Consequently the derivative reduces to \begin_inset Formula \[ \frac{dV(y)}{dy}=\sum_{j=1}^{m}\lambda_{j}^{\ast}\frac{\partial G_{j}(x^{\ast},y)}{\partial y} \] \end_inset which is just the partial derivative of the Lagrangian with respect to the parameter \begin_inset Formula $y$ \end_inset . \end_layout \end_body \end_document