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What is it defined over? \end_layout \begin_layout Enumerate What are the two properties of a \emph on rational \emph default preference relation? For each of these properties, describe a realistic situation in which the property would fail. \end_layout \begin_layout Enumerate What does it mean for a person to be `rational' in a particular economic environment? What does it not mean? \end_layout \begin_layout Enumerate Why is it \begin_inset Quotes eld \end_inset almost tautological \begin_inset Quotes erd \end_inset to \begin_inset Quotes eld \end_inset assume that a person chooses the alternative that he or she prefers most \begin_inset Quotes erd \end_inset ? Given this, where does the predictive content of economic theory come from? (i.e. what is the testable hypothesis?) \end_layout \begin_layout Enumerate Why is it \begin_inset Quotes eld \end_inset a waste of time to argue whether or not consumers are rational \begin_inset Quotes erd \end_inset ? What is the real objection that one might make? \end_layout \begin_layout Enumerate Explain why a \emph on rational \emph default preference relation implies that indifference curves can not have a point in common. \end_layout \begin_layout Enumerate Consider a game of tennis. If you are standing to the right, I prefer to hit the ball to the left. But, if I hit the ball to the left, you prefer to go to the left. But, if you are standing to the left, I prefer to hit the ball to the right. Thus, a consequence of my preference to hitting the ball to the left is that I prefer to hit the ball to the right (and vice versa). Does this mean my preferences are intransitive? \end_layout \begin_layout Enumerate What does it mean for a function, \begin_inset Formula $u$ \end_inset , to \emph on represent \emph default a preference relation on \begin_inset Formula $X$ \end_inset ? (i.e what properties must an arbitrary function possess in order for it to be called a `utility' function?) \end_layout \begin_layout Enumerate What are the benefits of using a utility function, as opposed to working with its underlying preference relation? \end_layout \begin_layout Enumerate What does it mean for a preference relation to be `monotonic'? Describe a scenario where the assumption of monotonicity would be reasonable, and one where it would not. \end_layout \begin_layout Enumerate Show that a utility function can not exist if the preference relation is not transitive. \emph on Hint \emph default : Try a `proof by contradiction.' \end_layout \begin_layout Enumerate Pick an arbitrary consumption bundle \begin_inset Formula $(x,y)\in\mathbb{R}_{+}^{2}$ \end_inset and draw the sets \begin_inset Formula $B$ \end_inset and \begin_inset Formula $W$ \end_inset (in \begin_inset Formula $\mathbb{R}_{+}^{2}$ \end_inset ) for the following situations: \end_layout \begin_deeper \begin_layout Enumerate I like consuming \begin_inset Formula $x$ \end_inset , and the more the better. I am completely indifferent to the amount of \begin_inset Formula $y$ \end_inset that I consume. \end_layout \begin_layout Enumerate As long as my consumption of \begin_inset Formula $x$ \end_inset is less than \begin_inset Formula $\bar{x}$ \end_inset , the more the better. After \begin_inset Formula $\bar{x}$ \end_inset the less the better. I am completely indifferent to the amount of \begin_inset Formula $y$ \end_inset that I consume. \end_layout \begin_layout Enumerate I like both \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset , but a unit of \begin_inset Formula $x$ \end_inset must be paired with a unit of \begin_inset Formula $y$ \end_inset in order for me to enjoy it (like left and right shoes). \end_layout \begin_layout Enumerate I prefer the bundle \begin_inset Formula $(\bar{x},\bar{y})$ \end_inset to all other bundles. If I have any other bundle, then I am indifferent between having this bundle and having nothing (0,0). \end_layout \end_deeper \begin_layout Enumerate \begin_inset CommandInset label LatexCommand label name "pq1" \end_inset This question is designed to take you through the steps of the proof of the Theorem presented in the text (the existence of a utility function) by using specific description of preferences. \end_layout \begin_deeper \begin_layout Standard Garrett likes owning shoes, but only gets use out of them if they are in a pair. Let \begin_inset Formula $L$ \end_inset be the number of left shoes he owns, and \begin_inset Formula $R$ \end_inset be the number of right shoes he owns. Now consider adding more shoes to the bundle \begin_inset Formula $(L,R)$ \end_inset . In particular, if Garrett were to add \begin_inset Formula $\ell\geq0$ \end_inset left shoes and \begin_inset Formula $r\geq0$ \end_inset right shoes, then he would (strictly) prefer the new bundle, \begin_inset Formula $\left(L+\ell,R+r\right)$ \end_inset , to the original bundle, \begin_inset Formula $\left(L,R\right)$ \end_inset , if and only if \emph on both \emph default \begin_inset Formula $\ell$ \end_inset and \begin_inset Formula $r$ \end_inset are strictly positive. For example, this implies that he is indifferent to having more left shoes but not more right shoes. \end_layout \end_deeper \begin_layout Enumerate Are these preferences rational (complete and transitive)? Are they monotonic? \end_layout \begin_deeper \begin_layout Enumerate Draw (with \begin_inset Formula $L$ \end_inset on the vertical axis, and \begin_inset Formula $R$ \end_inset on the horizontal axis) the following bundles and sets. \end_layout \begin_deeper \begin_layout Enumerate \begin_inset Formula $Z$ \end_inset . \end_layout \begin_layout Enumerate An arbitrary bundle, \begin_inset Formula $(L,R)\in{\mathbb{R}}_{+}^{2}$ \end_inset . \end_layout \begin_layout Enumerate \begin_inset Formula $B$ \end_inset and \begin_inset Formula $W$ \end_inset (relative to the above bundle). \end_layout \begin_layout Enumerate \begin_inset Formula $P^{+}$ \end_inset and \begin_inset Formula $P^{-}$ \end_inset (again, relative to the above bundle). \end_layout \begin_layout Enumerate The `indifference bundle', \begin_inset Formula $z\in Z$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Construct the utility function as suggested in the proof ( \emph on hint \emph default : Pythagoras' theorem might come in handy). \end_layout \begin_layout Enumerate Can you think of a simpler utility function? \end_layout \begin_layout Enumerate Check that both utility functions (the suggested one, and the simpler one) actually represent preferences. \end_layout \end_deeper \begin_layout Enumerate Repeat Question \begin_inset CommandInset ref LatexCommand ref reference "pq1" \end_inset using the following description of preferences. \end_layout \begin_deeper \begin_layout Standard Ken also like shoes, but unlike Garrett, he does not wear them. He makes `shoe sculptures' out of them. As far as he is concerned, a left shoe is equally as good as a right shoe. That is, suppose that we were to add more shoes to Ken's initial bundle of \begin_inset Formula $(L,R)$ \end_inset . In particular, if were to add \begin_inset Formula $\ell\geq0$ \end_inset left shoes and \begin_inset Formula $r\geq0$ \end_inset right shoes, then he would (strictly) prefer the new bundle, \begin_inset Formula $\left(L+\ell,R+r\right)$ \end_inset , to the original bundle, \begin_inset Formula $\left(L,R\right)$ \end_inset , if and only if \emph on either \emph default \begin_inset Formula $\ell$ \end_inset or \begin_inset Formula $r$ \end_inset is strictly positive \begin_inset Foot status collapsed \begin_layout Plain Layout This includes the case in which both are strictly positive. \end_layout \end_inset \end_layout \end_deeper \begin_layout Enumerate These questions are designed to highlight the role of the various assumptions used in the proof. \end_layout \begin_deeper \begin_layout Enumerate What part of the proof would fail if we removed each of the following assumption s? \end_layout \begin_deeper \begin_layout Enumerate Continuity. \end_layout \begin_layout Enumerate Monotonicity. \end_layout \begin_layout Enumerate Transitivity. \end_layout \begin_layout Enumerate Completeness. \end_layout \begin_layout Standard Which of these are `crucial' in the sense that a failure of the assumption does implies that a utility function will not exist? \end_layout \end_deeper \begin_layout Enumerate The proof was only concerned with consumption bundles in \begin_inset Formula ${\mathbb{R}}_{+}^{2}$ \end_inset (two-good consumption bundles). What problems, if any, would be introduced if we considered consumption bundles in \begin_inset Formula ${\mathbb{R}}_{+}^{k}$ \end_inset ( \begin_inset Formula $k$ \end_inset -good consumption bundles), where \begin_inset Formula $k>2$ \end_inset ? \end_layout \begin_layout Enumerate The proof was by `construction:' it showed that a utility function existed (under the stated assumptions) by actually building such a function. Is this suggested function unique? Prove your claim. \end_layout \end_deeper \begin_layout Enumerate Draw a graph showing the indifference curve associated with each of the following utility functions when the level of utility is equal to 1 \end_layout \begin_deeper \begin_layout Itemize \begin_inset Formula $u\left(x,y\right)=\max\left\{ x,y\right\} $ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $u\left(x,y\right)=x+2y$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $u\left(x,y\right)=x-2y$ \end_inset \end_layout \end_deeper \begin_layout Enumerate Simone has an endowment of $20 in period 1 and $40 in period 2. Her utility is \begin_inset Formula $u\left(c_{1},c_{2}\right)=c_{1}+c_{2}$ \end_inset . She can put money in a bank account in period 1, and get it back again in period 2 without interest. She can also borrow money from a loan shark. For each dollar she borrows in period 1, she must pay back \begin_inset Formula $\$1.50$ \end_inset in period 2. She can also buy a chair in period 1 for $10 that she knows that she will be able to resell in period \begin_inset Formula $2$ \end_inset for \begin_inset Formula $\$15$ \end_inset . \end_layout \begin_deeper \begin_layout Itemize Draw her budget line assuming that she does not buy the chair (labeling you diagram carefully). \end_layout \begin_layout Itemize Draw her budget line if she does buy the chair. \end_layout \begin_layout Itemize Should she buy the chair, or not? What consumption bundle should she pick in each case ? \end_layout \begin_layout Itemize Predict her consumption, and whether or not she will buy the chair if her utility function is instead given by \begin_inset Formula $u\left(x,y\right)=\min\left\{ x,y\right\} $ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Solve the problem \begin_inset Formula \[ \max2x^{\frac{1}{2}}+2y^{\frac{1}{2}}\] \end_inset subject to the constraints \begin_inset Formula $3x+y\leq10$ \end_inset ; \begin_inset Formula $x\geq0$ \end_inset ; \begin_inset Formula $y\geq0$ \end_inset using the method of Lagrangian multipliers. Write down the Lagrangian function, and each of the first order conditions before solving the problem. \end_layout \begin_layout Enumerate Solve these problems: \end_layout \begin_deeper \begin_layout Enumerate Solve for the optimal value for \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset in the problem \begin_inset Formula $max\, u(xy)=\frac{xy}{x+y}$ \end_inset subject to \begin_inset Formula $px+y\leq W$ \end_inset , \begin_inset Formula $x\geq0$ \end_inset , \begin_inset Formula $y\geq0$ \end_inset using the method of Lagrangians. Hint: the derivative of \begin_inset Formula $u(x,y)$ \end_inset with respect to \begin_inset Formula $x$ \end_inset is \begin_inset Formula $\frac{y^{2}}{(x+y)^{2}}$ \end_inset . \end_layout \begin_layout Enumerate Find the demand function associated with utility function \begin_inset Formula $u(x,y)=x+xy$ \end_inset (Hint: there are corner solutions to this problem - the Lagrangian method is too slow for this problem). \end_layout \begin_layout Enumerate Solve the problem maximize \begin_inset Formula $u(x,y)=x$ \end_inset subject to \begin_inset Formula $2x+y\leq10$ \end_inset , \begin_inset Formula $x\geq0$ \end_inset , and \begin_inset Formula $y\geq0$ \end_inset using the method of Lagrangians (i.e., find the optimal value for x and y and all the multipliers). \end_layout \end_deeper \begin_layout Enumerate A consumer has downward-sloping indifference curves for goods \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset . Below the 45 \begin_inset Formula $^{0}$ \end_inset line, the indifference curves are straight lines with slope \begin_inset Formula $-\frac{1}{2}$ \end_inset . When the curves hit the 45 \begin_inset Formula $^{0}$ \end_inset line, they become steeper. Above the 45 \begin_inset Formula $^{0}$ \end_inset line they are also straight lines but their slope is equal to \begin_inset Formula $-2$ \end_inset . Can you use the method we used to prove the existence of a utility function to provide a utility function that represents these preferences? \end_layout \begin_layout Enumerate Consider the following situation: \end_layout \begin_deeper \begin_layout Standard Geraldine prefers sports cars to hybrid cars because sports cars are faster and more powerful. She prefers SUVs to sports cars because SUVs are roomier. She prefers hybrid cars to SUVs, however, because they are more fuel-efficient and environmentally sustainable. \shape italic Honest Abe's Used Cars \shape default charges $1000 to trade in Geraldine's vehicle each time to \begin_inset Quotes eld \end_inset help her make up her mind. \begin_inset Quotes erd \end_inset \end_layout \begin_layout Standard Why is it that people \begin_inset Quotes eld \end_inset who exhibit intransitive preferences [...] quickly change their behavior when this is pointed out to them \begin_inset Quotes erd \end_inset ? \end_layout \end_deeper \begin_layout Section Demand Theory \end_layout \begin_layout Enumerate In what sense is the optimal consumption problem a special case of the more abstract choice problem introduced in the previous chapter? In particular, what is the special name given to the choice set \begin_inset Formula ${\mathcal{X}}$ \end_inset ? What is a typical element \begin_inset Formula $x\in{\mathcal{X}}$ \end_inset ? \end_layout \begin_layout Enumerate Explain how the `existence of a utility function' theorem allows us to make progress with understanding consumer demand. \end_layout \begin_layout Enumerate In order to make sharper predictions about economic behaviour, we will often need to add assumptions (e.g. on preferences) to the more fundamental axioms. The collection of such assumptions are known as a \emph on model \emph default . What is the point of analysing a model, when a model is simply a set of assumptions? \end_layout \begin_layout Enumerate What are the two `big' assumptions made by Classical consumer theory? Think of a scenario in which each would be inappropriate. \end_layout \begin_layout Enumerate Do the classical assumptions require that a consumer have an unchanging preference relation over \begin_inset Formula ${\mathbb{B}}$ \end_inset ? That is, suppose that we observed a consumer's choice of consumption bundle at two points in time, \begin_inset Formula $x_{t}^{*}$ \end_inset and \begin_inset Formula $x_{t'}^{*}$ \end_inset (where both of these are in \begin_inset Formula ${\mathbb{R}}^{n}$ \end_inset ). Further, suppose that prices and incomes were identical in the two time periods. What predictions could we make about \begin_inset Formula $x_{t}^{*}$ \end_inset and \begin_inset Formula $x_{t'}^{*}$ \end_inset if we allowed for the possibility that the consumer's preference relation over \begin_inset Formula ${\mathbb{B}}$ \end_inset changed over time? Could we ever find evidence that our model is wrong? \end_layout \begin_layout Enumerate There are two goods, \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset . Their prices are \begin_inset Formula $p_{x}$ \end_inset and \begin_inset Formula $p_{y}$ \end_inset , and income is \begin_inset Formula $W$ \end_inset . Draw \begin_inset Formula ${\mathbb{B}}$ \end_inset , the budget set. Show (graphically) the effect of \end_layout \begin_deeper \begin_layout Enumerate An increase in \begin_inset Formula $W$ \end_inset . \end_layout \begin_layout Enumerate An increase in \begin_inset Formula $p_{x}$ \end_inset . \end_layout \begin_layout Enumerate All prices changing by a factor of \begin_inset Formula $\alpha$ \end_inset . \end_layout \begin_layout Enumerate All prices and income changing by a factor of \begin_inset Formula $\alpha$ \end_inset . \end_layout \begin_layout Standard What is the implication of the last of these for the classical consumer theory assumptions? \end_layout \end_deeper \begin_layout Enumerate A consumer is growing tomatoes and has \begin_inset Formula $W$ \end_inset kg available for consumption in period 1. Whatever she doesn't eat in a period will grow by a factor of \begin_inset Formula $R$ \end_inset by the next period. Let \begin_inset Formula $x_{t}$ \end_inset represent the consumption of tomatoes in period \begin_inset Formula $t$ \end_inset . \end_layout \begin_deeper \begin_layout Enumerate Draw the budget set for the case when there are only two periods. \end_layout \begin_layout Enumerate Describe this budget set in formal notation. \end_layout \begin_layout Enumerate Describe this budget set in formal notation for the case in which there are \begin_inset Formula $T$ \end_inset periods. \end_layout \end_deeper \begin_layout Enumerate Derive the demand function when preferences are given by: \end_layout \begin_deeper \begin_layout Enumerate \begin_inset Formula $u(x,y)=x^{2}+y^{2}$ \end_inset . \emph on Hint \emph default : This highlights the fact that the FOCs are necessary but \series bold not \series default sufficient - draw the indifference curves. \end_layout \begin_layout Enumerate \begin_inset Formula $u(x,y)=x+y$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Suppose that a consumer's preferences are described by a utility function, \begin_inset Formula $u(x)$ \end_inset . Prove that it is also true that his preferences are represented by a new function, \begin_inset Formula $h(x)$ \end_inset , which is defined as \begin_inset Formula $h(x)\equiv g(u(x))$ \end_inset , where \begin_inset Formula $g$ \end_inset is an increasing function (e.g. an exponential or logarithmic function). \end_layout \begin_layout Enumerate Use the previous question to help you derive the demand function when preference s are given by \begin_inset Formula \begin{equation} u(x,y)=Z^{f(x,y)},\end{equation} \end_inset where \begin_inset Formula $Z>1$ \end_inset and \begin_inset Formula \begin{equation} f(x,y)\equiv A-\frac{1}{{\alpha}\ln(x)+(1-\alpha)\ln(y)}.\end{equation} \end_inset \end_layout \begin_layout Enumerate A quick inspection of the properties of the utility function often makes it much easier to derive the demand functions since the effects of the constraints are more transparent. For each of the utility functions below, identify the conditions under which positive quantities of both goods will be consumed at the optimum (so that the non-negativity constraints can be ignored when such conditions are met): \end_layout \begin_deeper \begin_layout Enumerate \begin_inset Formula $u(x,y)=x^{\alpha}y^{\beta}$ \end_inset , where \begin_inset Formula $\alpha,\beta>0$ \end_inset . \end_layout \begin_layout Enumerate \begin_inset Formula $u(x,y)=x+\ln(y)$ \end_inset . \end_layout \begin_layout Enumerate \begin_inset Formula $u(x,y)=x^{\alpha}+y^{\alpha}$ \end_inset , where \begin_inset Formula $\alpha\in(0,1)$ \end_inset . \end_layout \begin_layout Enumerate \begin_inset Formula $u(x,y)=x^{\alpha}+y^{\alpha}$ \end_inset , where \begin_inset Formula $\alpha\in[1,\infty)$ \end_inset . \end_layout \begin_layout Enumerate \begin_inset Formula $u(x,y)=(x+y)(1-x-y)$ \end_inset . \end_layout \begin_layout Enumerate \begin_inset Formula $u(x,y)=(x^{\alpha}+y^{\alpha})(1-x^{\alpha}-y^{\alpha})$ \end_inset , where \begin_inset Formula $\alpha\in(0,1)$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate For each of the functional forms in the previous question, identify the conditions under which the income constraint will be binding. \end_layout \begin_layout Enumerate Suppose we want to test the basic assumption that a consumer's preferences are independent of the budget set. Describe the test proposed in the text. What is the extra assumption on preferences that was required? Can you think of another approach to testing this basic assumption? \end_layout \begin_layout Enumerate It is oftenq thought that econometricians assist theorists by collecting and analysing data in order to determine whether a model is a plausible description of the world. This question shows that the reverse is sometimes true too - theorists may assist econometricians in terms of how they construct their econometric model. \end_layout \begin_deeper \begin_layout Standard Elliot the eager econometrician knows that consumer theorists talk about demand functions, and are especially interested in the effects of income and prices. Elliot thinks that theorists are full of hot wind and wants to test certain aspects of demand theory. To do this, each day he carefully records the prices of the \begin_inset Formula $n$ \end_inset goods that he consumes. Along with this, he also records daily his income and his chosen consumption bundle (a quantity for each of the \begin_inset Formula $n$ \end_inset goods). Thus, after a year or so he has a data set that contains observations on prices, incomes and demands. To facilitate his many probing tests, he needs to determine how prices and income influence his demand. To keep things simple, he decides to focus on his demand for pocket protectors. The most basic specification that comes to mind is the following econometric model: \begin_inset Formula \[ x_{1t}=\beta_{0}+\beta_{w}W_{t}+\alpha_{1}p_{1t}+\sum_{i=2}^{n}\alpha_{i}p_{it}+\varepsilon_{t}\] \end_inset where \begin_inset Formula $x_{1t}$ \end_inset is his demand for pocket protectors on day \begin_inset Formula $t$ \end_inset , \begin_inset Formula $W_{t}$ \end_inset is his income on day \begin_inset Formula $t$ \end_inset and \begin_inset Formula $p_{it}$ \end_inset is the price of good \begin_inset Formula $i$ \end_inset on day \begin_inset Formula $t$ \end_inset (pocket protectors are good 1). \end_layout \end_deeper \begin_layout Enumerate While waiting for his expensive statistical package to load, Elliot asks your opinion on his specification of the demand function. Do you see any problems? In particular think about: \end_layout \begin_deeper \begin_layout Enumerate If demand theory is taken seriously, what \emph on must \emph default \begin_inset Formula $x_{1t}$ \end_inset be when \begin_inset Formula $W_{t}=0$ \end_inset ? How does this restrict the possible values of \begin_inset Formula $\beta_{0}$ \end_inset and \begin_inset Formula $\alpha_{j}$ \end_inset ( \begin_inset Formula $j=1,...,n$ \end_inset )? Does demand theory itself impose these latter restrictions on the partial effects of prices on demands? \end_layout \begin_layout Enumerate Again, if demand theory is taken seriously, then how must \begin_inset Formula $x_{1t}$ \end_inset change when all prices and income are changed in the same proportion? How does this restrict what \begin_inset Formula $\beta_{w}$ \end_inset and \begin_inset Formula $\alpha_{j}$ \end_inset ( \begin_inset Formula $j=1,...,n$ \end_inset ) can be? Does demand theory itself impose these latter restrictions on the partial effects of income and prices on demands? \end_layout \end_deeper \begin_layout Enumerate The general mistake that Elliot has made was to forget that a demand function is \emph on derived \emph default from preferences. That is, he assumed a demand function (based on simplicity) rather than assuming a form on preferences. In other words, it may very well be the case that there are no preferences that would produce this particular demand function. Do you have any general advice to give Elliot before he decides on some other specification? \end_layout \begin_layout Enumerate Use graphical methods to describe preferences (draw indifference curves) for goods \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset for the following scenarios. \end_layout \begin_deeper \begin_layout Enumerate As income increases, the demand for good \begin_inset Formula $x$ \end_inset falls. \end_layout \begin_layout Enumerate As the price of good \begin_inset Formula $x$ \end_inset increases, so too does demand for \begin_inset Formula $x$ \end_inset . \end_layout \begin_layout Enumerate As income changes, the ratio of the demand for \begin_inset Formula $x$ \end_inset to the demand for \begin_inset Formula $y$ \end_inset is constant. \end_layout \begin_layout Enumerate As the demand for \begin_inset Formula $y$ \end_inset is unaffected by the price of good \begin_inset Formula $x$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Use the above question to answer the following. \end_layout \begin_deeper \begin_layout Enumerate Is it possible for the quantity demanded of \begin_inset Formula $x$ \end_inset to decrease as income increases, yet increase when it's price increases? \end_layout \begin_layout Enumerate Is it possible for the quantity demanded of \begin_inset Formula $x$ \end_inset to increase as it's price increases, yet increase when income increases? \end_layout \begin_layout Enumerate Is it possible for the quantity demanded of \emph on both \emph default \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset to fall as income increases? \end_layout \end_deeper \begin_layout Section Discontinuous Budget Sets \end_layout \begin_layout Standard 1. In the text, a non-linear pricing problem was analyzed in which the per unit price increased after some threshold demand was reached. In this problem, we do the same thing, but assume that the per unit price falls. Let the per-unit price of good \begin_inset Formula $y$ \end_inset be constant and equal to 1. The per unit price of good \begin_inset Formula $x$ \end_inset is assumed to be \begin_inset Formula $p$ \end_inset for each unit up to the \begin_inset Formula $n^{th}$ \end_inset . Then each additional unit costs \begin_inset Formula $p-dp$ \end_inset . In other words, each additional unit of output purchased above and beyond the first \begin_inset Formula $n$ \end_inset costs \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash tmem{less } \end_layout \end_inset than the first \begin_inset Formula $n$ \end_inset unit. Assume that preferences are given by \begin_inset Formula \[ u(x,y)=x^{\alpha}y^{1-\alpha}\] \end_inset Repeat the exercise given in text, and show how demand for good \begin_inset Formula $x$ \end_inset varies with \begin_inset Formula $\alpha.$ \end_inset \end_layout \begin_layout Standard 2. Tickets for the World Cup Quidditch match have been sold in bundles of three, and are now being resold by scalpers. The lowest priced scalper offers his three tickets at a unit price of $ 100. So if a consumer wants \begin_inset Formula $x\leqslant3$ \end_inset tickets, the price she will pay is \begin_inset Formula $100x$ \end_inset . The next lowest price scalper is offering her three tickets for $ 110, the cheapest after that is $ 120 and so on. A consumer has preferences given by \begin_inset Formula $u(x,y)=x^{\alpha}y^{1-\alpha}$ \end_inset where \begin_inset Formula $x$ \end_inset stands for the number of tickets she purchases, and \begin_inset Formula $y$ \end_inset is the amount of money she has left over for other stuff. She has $ 3000 to spend on tickets and other stuff. Write a computer program that takes as its input the value of \begin_inset Formula $\alpha$ \end_inset and outputs the number of tickets that the consumer with that value of \begin_inset Formula $\alpha$ \end_inset will buy. You can assume that tickets are infinitely divisible. \end_layout \begin_layout Standard Hint: If our consumer decides to buy \begin_inset Formula $x$ \end_inset tickets, then she will pay as follows: compute \begin_inset Formula $x/3$ \end_inset and take its integer part and call it \begin_inset Formula $j(x)$ \end_inset (e.g. the integer part of \begin_inset Formula $22/3$ \end_inset is \begin_inset Formula $7$ \end_inset , so \begin_inset Formula $j(22/3)=7$ \end_inset ); let \begin_inset Formula $p_{0}=90;$ \end_inset compute \begin_inset Formula \[ \sum_{i=1}^{j(x)}3(p_{0}+i10)+(p_{0}+j(x)10)(x-j(x)3)\] \end_inset This procedure gives you the slope and position of the budget line for different values of \begin_inset Formula $x$ \end_inset . Now generalize the method in the text to solve the problem. \end_layout \begin_layout Standard One way to do this would be to write a short computer program which takes as its input a value for \begin_inset Formula $\alpha$ \end_inset and outputs a demand and expenditure on tickets. Scripting on computers is a useful skill to develop as an economist. It doesn't really matter which language you use to write scripts (c, java, perl, bash, php etc) since they all use similar principals which you can apply in a variety of contexts, for example, statistics and econometrics packages, or computer algebra programs. \end_layout \begin_layout Standard 3. In the text, the response of a consumer with quasi-linear preferences to the imposition of a fixed fee is analyzed. Preferences in this problem are given by \begin_inset Formula $u(x,y)=y+\log(x)$ \end_inset . Consumer income is \begin_inset Formula $W$ \end_inset and the firm's pricing scheme requires that the consumer pay a fixed fee \begin_inset Formula $K$ \end_inset , which entitles her to \begin_inset Formula $n$ \end_inset 'free' units of good \begin_inset Formula $x$ \end_inset . Each unit in addition to the first \begin_inset Formula $n$ \end_inset costs the consumer \begin_inset Formula $p$ \end_inset . Hold the price \begin_inset Formula $p$ \end_inset and the initial number of units \begin_inset Formula $n$ \end_inset constant, and suppose that the firm using this scheme wants choose a fee \begin_inset Formula $K$ \end_inset to maximize the profit it receives from this consumer. What fee would it choose? If you were the competitor of this firm (i.e., you are the firm selling good \begin_inset Formula $y$ \end_inset what might you do to counteract this strategy by the firm? \end_layout \begin_layout Standard 3. Suppose that a consumer has $100 to spend during the week and likes to use some of this money to play tennis. She can buy time at the local court for $1 per hour, but the local court will only let her play for a maximum of 20 hours. If she wants to play for more than 20 hours, she has to pay a fee of $20 at the private tennis club. Then she can buy additional time for $2 per hour. Suppose that her preferences are represented by a Cobb-Douglas utility function \begin_inset Formula $u(x,y)=x^{\alpha}y^{1-\alpha}$ \end_inset where \begin_inset Formula $x$ \end_inset is the number of hours of tennis she plays and \begin_inset Formula $y$ \end_inset is the money she has left over for other stuff. Draw a picture to describe the budget line she faces. Label each of the important points in the diagram. Give a formula to show how much time she will spend playing tennis at the local court and how much time she will spend at the private court as a function of the number \begin_inset Formula $\alpha$ \end_inset . \end_layout \begin_layout Standard 4. Our consumer has $ \begin_inset Formula $20$ \end_inset per month to spend on her cell phone and comic books. Comic books cost $ 1 each. The phone company has offered her two different deals - for $ 6 per month she can have 100 'free' minutes on the phone. Each minute thereafter will cost her 10 cents. Or, if she wants, he she can take the deluxe plan, $ 10 per month which entitles her to 150 free minutes. As a bonus, extra minutes are then charged at the lower rate of 5 cents. Draw the consumer's budget set in this case. Suppose her preferences are given by \begin_inset Formula $u(x,y)=x^{\alpha}y^{1-\alpha}$ \end_inset . Explain how her choice of plan and telephone use should depend on \begin_inset Formula $\alpha$ \end_inset . \end_layout \begin_layout Section Best Reply Behavior \end_layout \begin_layout Enumerate Alice and Bob both produce Caesar salad dressing in the same kitchen. Each of them has an endowment of one unit of labor that can be used as effort in producing the dressing, or can be used to play video games. Both Alice and Bob have identical preferences given by: \begin_inset Formula \begin{equation} U(x,y)=\ln(x)+y,\end{equation} \end_inset where \begin_inset Formula $x$ \end_inset is the hours of TV watched and \begin_inset Formula $y$ \end_inset is the output of Caesar salad dressing. \end_layout \begin_deeper \begin_layout Standard If an amount of effort, \begin_inset Formula $z_{i}$ \end_inset , is used in Caesar salad dressing production by person \begin_inset Formula $i$ \end_inset , then the amount of dressing produced is given by \begin_inset Formula \begin{equation} y_{i}=f_{i}(z_{i})=S_{i}z_{i},\end{equation} \end_inset for \begin_inset Formula $i\in\left\{ \text{alice},\text{bob}\right\} $ \end_inset . Note that \begin_inset Formula $S_{i}$ \end_inset is the marginal product of production effort for person \begin_inset Formula $i$ \end_inset . An hour of video game playing is produced one-for-one from the labor endowment. That is (by substituting in the resource constraint), \begin_inset Formula $x_{i}=1-z_{i}$ \end_inset . \end_layout \begin_layout Standard The thing is that making the dressing is hard when the other person is not in the kitchen very much (because it is time-consuming to look for the spoons etc, and it gets very boring and lonely). In particular, it turns out that \begin_inset Formula \begin{equation} S_{a}=cz_{b},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\, S_{b}=cz_{a},\end{equation} \end_inset where \begin_inset Formula $c>2$ \end_inset . \end_layout \begin_layout Standard Suppose that Alice guesses that Bob will spend \begin_inset Formula $z_{b}^{*}$ \end_inset hours in the kitchen. Write out Alice's optimization problem. \end_layout \end_deeper \begin_layout Enumerate What is her best response number of hours in the kitchen, \begin_inset Formula $z_{a}(z_{b}^{*})$ \end_inset ? Graph this best response in \begin_inset Formula $(z_{a},z_{b})$ \end_inset space. \end_layout \begin_deeper \begin_layout Enumerate Do the same exercise for Bob, but place his best response on the same graph that you have just drawn. \emph on Hint: \emph default The two are identical, so the problem (and graph) should be very similar. \end_layout \begin_deeper \begin_layout Enumerate Graphically identify the Nash equilibrium levels of production effort, \begin_inset Formula $(z_{a}^{*},z_{b}^{*})$ \end_inset . Explicitly solve for the solution when \begin_inset Formula $c=3$ \end_inset . \emph on Hint \emph default : You will need to use the quadratic formula. \end_layout \end_deeper \end_deeper \begin_layout Enumerate Consider the following (extreme form of) consumption externality. Alice is going to Bob's house for dinner, and has agreed to bring something to drink. After arriving at the store, she realizes that she has no idea what Bob is planning to cook. Neither she or Bob cares much what they eat. If Bob is cooking pizza, she (and Bob) would prefer that she bring beer. If Bob is cooking pasta, then she (and Bob) would prefer that she bring wine. Bob has a similar problem - what he wants to cook will depend on what he thinks Alice will bring. Which outcomes (choices of a food and a drink) seem reasonable? Does it seem possible to predict exactly what kind of drink Alice will buy? \end_layout \begin_layout Section Expected Utility \end_layout \begin_layout Enumerate What are the two objects that define a \emph on lottery \emph default ? Define the lottery associated with rolling a fair die. \end_layout \begin_layout Enumerate How is a \emph on simple lottery \emph default different from a \emph on compound lottery \emph default ? What is a \emph on reduced lottery \emph default ? \end_layout \begin_layout Enumerate Write the reduced lottery associated with the following (compound) lotteries. \end_layout \begin_deeper \begin_layout Enumerate A coin is tossed. If it comes up heads, you roll a fair die and get an amount of money equal to the number that turns up (e.g. if you roll a four, you get $4). If it comes up tails, you toss the coin six more times and get an amount of money equal to the number of tails that comes up in the six tosses (e.g. if four of the six were tails, you get $4). \end_layout \begin_layout Enumerate A coin is tossed. If it comes up heads, you roll a fair die and get an amount of money equal to the number that turns up (e.g. if you roll a four, you get $4). If it comes up tails, you start the process over again. \end_layout \begin_layout Enumerate A coin is tossed. If it comes up heads, you toss a second coin. If this second coin comes up heads, you get $10 today. If the second coin comes up tails, you get $10 in one years time. If the first coin instead comes up tails, you get $10 in two years time. \end_layout \end_deeper \begin_layout Enumerate Consider the following compound lottery, with probability \begin_inset Formula $\frac{1}{4}$ \end_inset you will play the lottery \begin_inset Formula $\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\}$ \end_inset , with probability \begin_inset Formula $\frac{3}{8}$ \end_inset you play the lottery \begin_inset Formula $\{0,1,0\}$ \end_inset , while with probability \begin_inset Formula $\frac{3}{8}$ \end_inset you play the lottery \begin_inset Formula $\left\{ 0,0,1\right\} $ \end_inset where each element in the list is the probability with which you receive payoffs \begin_inset Formula $\{1000,500,0\}$ \end_inset respectively. What is the reduced lottery associated with this compound lottery. \end_layout \begin_layout Enumerate You tell your friend the Monty Hall problem (game show host revealing doors etc.). They are convinced that it is irrelevant whether or not you switch doors based on the following argument. \begin_inset Quotes eld \end_inset No matter what door I choose, I will end up in a situation in which I am choosing between two doors, one of which will have the prize behind it. Since the prize is equally likely to be behind all doors, the chance that it is behind the door I chose is 1/2, which is the same as the chance of it being behind the other door. Therefore, it doesn't make a difference if I change doors \begin_inset Quotes erd \end_inset . Explain where your friend is making an error in their reasoning. \end_layout \begin_layout Enumerate Consider again the Monty Hall problem. At first glance, what did you think the probability of winning the prize was? What is the actual probability of winning the prize (assuming you switch doors as suggested)? \end_layout \begin_layout Enumerate We know that it is always better to switch doors in the 3-door version of the Monty Hall problem. Will this remain true for \begin_inset Formula $n$ \end_inset doors? If no, at what value of \begin_inset Formula $n$ \end_inset does it stop being worthwhile to switch doors? If yes, does the benefit to switching increase or decrease as \begin_inset Formula $n$ \end_inset increases? \end_layout \begin_layout Enumerate Change the Monty Hall problem so that the prize is initially placed behind door A with probability \begin_inset Formula $\frac{1}{2}$ \end_inset (instead of probability \begin_inset Formula $\frac{1}{3}$ \end_inset as in the problem we discussed in class). The prize is placed behind doors B and C with equal probability \begin_inset Formula $\frac{1}{4}$ \end_inset . Suppose you choose door A. Monty then opens one of the other doors and shows you there is nothing behind it. You are offered the chance to switch doors. What is the probability that you will find the prize if you keep door A? What is the probability if you choose to switch to the other door? Do you think you could increase your chances of finding the prize by choosing a door other than door A to start with? \end_layout \begin_layout Enumerate Consider the St. Petersberg Paradox. When considering how much to pay for this lottery, it was argued that you could not lose by paying $2 or less since you are guaranteed to win at least this amount. On the other hand, there is no limit to how much you could win. However, if you win after \begin_inset Formula $t$ \end_inset rounds the amount that you win will be finite. That is, if you win in round \begin_inset Formula $t$ \end_inset , you get \begin_inset Formula $1/2^{t}$ \end_inset which is a finite number if \begin_inset Formula $t$ \end_inset is finite. That is, the most you can win is strictly less than infinity. However, it was shown that the expected value of the lottery (and therefore how much a risk-neutral individual would be willing to pay) is infinite. How could it be that someone is willing to pay an amount for a lottery that is greater than the highest possible amount they could win? \end_layout \begin_layout Enumerate Suggest two reasons why no individual would be willing to pay an infinite amount for the lottery described in the St. Petersberg Paradox (besides things like no-one has that much money). \end_layout \begin_layout Enumerate A simplifying assumption used throughout the chapter was that all the lotteries in \begin_inset Formula ${\mathcal{L}}$ \end_inset had the same set of possible outcomes, \begin_inset Formula ${\mathcal{X}}$ \end_inset . In what way does this assumption simplify the analysis? Is the assumption restrictive? Why or why not? \end_layout \begin_layout Enumerate State the independence axiom both formally and intuitively. Can you think of a scenario where this axiom is likely to fail? \end_layout \begin_layout Enumerate The independence axiom is concerned with `mixing' or `combining' lotteries with a third lottery. Care must be taken in understanding exactly what it means to `mix' two lotteries. The question is designed to highlight a common error. \end_layout \begin_deeper \begin_layout Standard Suppose there are two possible and equally-likely states of the world: I toss a coin, and it either comes up heads or comes up tails. Lottery 1 gives you $2 if it is heads, and $0 if it is tails. Lottery 2 is the reverse - you get $2 if it is tails, and $0 if it is heads. \end_layout \end_deeper \begin_layout Enumerate These two lotteries are clearly different (by name if by nothing else). Are the differences meaningful in terms of the treatment of lotteries given in the text? The following questions help with this issue. \end_layout \begin_deeper \begin_layout Enumerate What is the difference between the \begin_inset Formula ${\mathcal{X}}$ \end_inset associated with these two lotteries? \end_layout \begin_layout Enumerate What is the difference between the \begin_inset Formula $p$ \end_inset associated with these two lotteries? \end_layout \begin_layout Enumerate Given your answers to the above two questions, what is the difference (if any) between the lotteries? \end_layout \end_deeper \begin_layout Enumerate what are the three assumptions on the preference relation over \begin_inset Formula ${\mathcal{L}}$ \end_inset ? Explain what each of these means. for each assumption think of a realistic situation in which the assumption would be inappropriate. \end_layout \begin_layout Enumerate In the text, the set of feasible lotteries over three possible outcomes is drawn as a triangle in two dimensions. What would the set of feasible lotteries look like if there were only two outcomes? Can you imagine how to draw the set of feasible outcomes when there are 4 outcomes? How many dimensions are needed in order to draw the set of feasible lotteries when there are \begin_inset Formula $k$ \end_inset outcomes? Why? \end_layout \begin_layout Enumerate Consider the following lotteries \begin_inset Formula $q=\left\{ \frac{1}{3},\frac{1}{3},\frac{1}{3}\right\} $ \end_inset , \begin_inset Formula $q^{\prime}=\left\{ \frac{1}{10},\frac{3}{10},\frac{3}{5}\right\} $ \end_inset and \begin_inset Formula $q^{\prime\prime}=\left\{ \frac{1}{4},\frac{1}{4},\frac{1}{2}\right\} $ \end_inset . Create a compound lottery over these three lotteries in which each of them is run with equal probability. What is the reduced lottery associated with this compound lottery? \end_layout \begin_layout Enumerate A coin is flipped at most three times. If it comes up heads on the first try, the game ends and you win $2. If it comes up tails, it is flipped again. If it comes up heads on the second try, you win $2. If it comes up tails, it is flipped again. If it comes up heads on the third try, you lose $2. If it comes up tails, the game is over an no one wins anything. Draw a tree depicting the compound lottery over compound lotteries that is involved in this process. Compute the reduced lottery associated with it. What is the expected value of this lottery? \end_layout \begin_layout Enumerate In the Monty Hall problem, use Bayes rule to compute the posterior probability that the prize is behind door \begin_inset Formula $A$ \end_inset once you have chosen \begin_inset Formula $A$ \end_inset and the show host opens door \begin_inset Formula $C$ \end_inset . (Bayes rule - \begin_inset Formula $\Pr\left\{ A|C\right\} =\Pr\left\{ A\cap C\right\} /\Pr\left\{ C\right\} $ \end_inset . In words, the probability of event A conditional on event C is the probability that both A and C occur, divided by the probability that event C occurs. Draw yourself a picture to see if you can understand why this should be so). \end_layout \begin_layout Enumerate Suppose that the set of lotteries is equal to all lotteries over three outcomes, \begin_inset Formula $x_{1}$ \end_inset , \begin_inset Formula $x_{2}$ \end_inset and \begin_inset Formula $x_{3}$ \end_inset as above, with \begin_inset Formula $x_{1}$ \end_inset being the best outcome, and \begin_inset Formula $x_{3}$ \end_inset the worst. What are the best and worst lotteries \begin_inset Formula $b$ \end_inset and \begin_inset Formula $w$ \end_inset used the in the expected utility theorem in this case? What are the utility values assigned to these lotteries by the construction in the theorem? \end_layout \begin_layout Enumerate Show that the utility function constructed in the expected utility theorem above actually represents preferences in the sense that if \begin_inset Formula $u\left(p\right)>u\left(p^{\prime}\right)$ \end_inset then \begin_inset Formula $p\succ p^{\prime}$ \end_inset and conversely. \end_layout \begin_layout Enumerate Draw the lotteries that Allais suggested in the last section above in a diagram and show that anyone who prefers \begin_inset Formula $q$ \end_inset to \begin_inset Formula $q^{\prime}$ \end_inset must also prefer \begin_inset Formula $p$ \end_inset to \begin_inset Formula $p^{\prime}$ \end_inset if they have expected utility preferences. \end_layout \begin_layout Section Risk Aversion \end_layout \begin_layout Enumerate What is a probability distribution function? \end_layout \begin_layout Enumerate Suppose that the lottery \begin_inset Formula $(p,{\mathcal{X}})$ \end_inset can represented by a probability distribution function, \begin_inset Formula $F$ \end_inset . \end_layout \begin_deeper \begin_layout Enumerate Does this impose any restriction on the set of objects in the set \begin_inset Formula ${\mathcal{X}}$ \end_inset ? For example, could \begin_inset Formula ${\mathcal{X}}=\left\{ \mbox{red ball},\mbox{green ball},\mbox{blue ball}\right\} $ \end_inset ? \end_layout \begin_layout Enumerate Suppose in addition to representing \begin_inset Formula $(p,{\mathcal{X}})$ \end_inset , \begin_inset Formula $F$ \end_inset happens to be such that it has a \emph on density \emph default , in other words, it has a derivative. What additional properties does \begin_inset Formula ${\mathcal{X}}$ \end_inset have to possess for this to be true? For example, could \begin_inset Formula ${\mathcal{X}}=\left\{ -2,-1,0,1,2\right\} $ \end_inset ? \end_layout \end_deeper \begin_layout Enumerate If \begin_inset Formula $F$ \end_inset does have a density (derivative) and that \begin_inset Formula $\mathcal{X}=[0,1]$ \end_inset . What does \begin_inset Formula $\int_{0}^{x}F'(x')dx'$ \end_inset equal? \end_layout \begin_layout Enumerate Again, using the assumption that \begin_inset Formula $\mathcal{X}=[0,1]$ \end_inset , consider a degenerate lottery that gives $ \begin_inset Formula $z$ \end_inset for sure. Write the \begin_inset Formula $F$ \end_inset that characterizes this lottery. \end_layout \begin_layout Enumerate Write the \begin_inset Formula $F$ \end_inset that characterizes the following lottery. A coin is tossed, and you get $1 if it comes up heads and -$1 if it comes up tails. \end_layout \begin_layout Enumerate Intuitively explain what a `risk premium' is. Explain how to calculate it. \end_layout \begin_layout Enumerate What is a `fair bet'? A person is said to be \emph on risk averse \emph default if they have a positive risk premium when faced with a fair bet. Suppose that I tell you that a person associates a risk premium of -$5 with some lottery \begin_inset Formula ${\mathcal{L}}$ \end_inset (not necessarily a fair lottery). What can we say about the person's attitude toward risk? \end_layout \begin_layout Enumerate Is it possible that an individual has a different risk premium associated with different lotteries? Explain. \end_layout \begin_layout Enumerate Is it possible that two individuals with the exact same preferences over all lotteries have a different risk premium when faced with identical lotteries ? If not, explain why not. If so, what must be different across individuals? \end_layout \begin_layout Enumerate Daron is unsure how much he will get through a student loan this year. There is a 50% chance that he will get $2000 and 50% chance that he will get $3000. He already has $1000 saved. \end_layout \begin_deeper \begin_layout Enumerate Express the lottery that Daron faces, \begin_inset Formula ${\mathcal{L}}$ \end_inset , in terms of \begin_inset Formula $(p,{\mathcal{X}})$ \end_inset . Now write it in terms of \begin_inset Formula $(W,F)$ \end_inset . \end_layout \begin_layout Enumerate What allows us to express his relative preference for the lottery \begin_inset Formula ${\mathcal{L}}$ \end_inset in the form: \begin_inset Formula \begin{equation} U({\mathcal{L}})=p_{1}u(x_{1})+p_{2}u(x_{2})\,\,\,\,?\end{equation} \end_inset \end_layout \begin_layout Enumerate Suggest an interpretation of the function \begin_inset Formula $u(x)$ \end_inset . Using your answer to the first part, fill in what \begin_inset Formula $(p_{1},p_{2},x_{1},x_{2})$ \end_inset are in this case. \end_layout \begin_layout Enumerate Write an expression for the \emph on risk premium \emph default associated with \begin_inset Formula ${\mathcal{L}}$ \end_inset . Is this positive or negative? Does this make sense in terms of your intuitive understanding of what a risk premium represents? \end_layout \begin_layout Enumerate Calculate the risk premium if we assumed that \begin_inset Formula $u(x)=\sqrt{x}$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Explain carefully how the expected utility theorem is useful in the process of calculating a risk premium. \end_layout \begin_layout Enumerate What is the Arrow-Pratt measure of risk aversion? What is the relationship between the risk premium and the Arrow-Pratt measure of risk aversion? What simplifications were made in deriving this relationship? Under what cases would the simplifications be inappropriate? \end_layout \begin_layout Enumerate What is the `Portfolio Problem'? In what way does each choice of the \begin_inset Formula $(i_{s},i_{r})$ \end_inset pair generate a lottery? Write this lottery in terms of a probability distribut ion function, \begin_inset Formula $F$ \end_inset . \end_layout \begin_layout Enumerate What does the \emph on Diversification Theorem \emph default say? Does it only apply to those investors that are not too risk averse? What was the intuitive argument based on? Describe an aspect of the real world that might cause individuals to behave in a manner that is seemingly inconsistent with the theorem. \end_layout \begin_layout Enumerate The \emph on Diversification Theorem \emph default was derived without actually ever solving for the optimal portfolio. This question verifies the Theorem by solving for the optimal portfolio. \end_layout \begin_deeper \begin_layout Standard We are going to (implicitly) find the optimal investment in the risky asset, \begin_inset Formula $i_{r}$ \end_inset , under the assumption of a specific form of \begin_inset Formula $u(x)$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Argue that the wealth constraint \begin_inset Formula $i_{r}+i_{s}\leq W$ \end_inset can safely be assumed to hold with equality (given the existence of a risk-free asset). \end_layout \begin_deeper \begin_layout Enumerate Use this to write the objective function in the form: \begin_inset Formula \begin{equation} \int u(W+i_{r}s)F'(s)ds\end{equation} \end_inset \end_layout \begin_layout Enumerate Show that if the agent is risk-neutral (i.e. \begin_inset Formula $u(x)=x$ \end_inset ), the objective function can be written in the form: \begin_inset Formula \begin{equation} \alpha_{1}+\alpha_{2}\cdot i_{r},\end{equation} \end_inset where \begin_inset Formula $\alpha_{1}$ \end_inset and \begin_inset Formula $\alpha_{2}$ \end_inset are constants that are specific to \begin_inset Formula $W$ \end_inset and \begin_inset Formula $F$ \end_inset . \end_layout \begin_layout Enumerate Remaining in the risk-neutral case, what condition on \begin_inset Formula $F$ \end_inset will ensure that \begin_inset Formula $i_{r}$ \end_inset is positive? Does the level of \begin_inset Formula $W$ \end_inset matter (apart from \begin_inset Formula $W>0)$ \end_inset ? Is this general, or a feature of risk-neutrality? \end_layout \begin_layout Enumerate Now suppose that there is some risk-aversion. In particular, suppose that \begin_inset Formula $u(x)=x^{\alpha}$ \end_inset , where \begin_inset Formula $\alpha\in(0,1)$ \end_inset . Suppose that the optimal portfolio had the property that \begin_inset Formula $i_{r}=0$ \end_inset . Use the FOC to derive the property of the lottery that must have induced this. Does this make sense? \end_layout \end_deeper \begin_layout Enumerate Comparative static techniques were used to show that if an investor's preference s are such that the Arrow-Pratt measure of (absolute) risk-aversion is decreasin g in wealth, then the investor will invest more in the risky asset as their wealth increases. \end_layout \begin_deeper \begin_layout Enumerate The fact that an investor invests more as wealth increases might have nothing to do with risk aversion, but rather simply that they have more available wealth. Does the amount invested in the risky asset increase with wealth for investors that have an Arrow-Pratt measure of risk aversion that is \emph on constant \emph default over wealth levels? \end_layout \begin_layout Enumerate What happens to the \emph on proportion \emph default of wealth allocated to the risky asset as wealth increases (for an investor with a decreasing Arrow-Pratt measure of risk aversion)? That is, calculate the comparative static \begin_inset Formula $\partial(i_{r}(w)/w)/\partial w$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Suppose it was observed that wealthier individuals tended to invest more in risky assets. Mike claims that this is evidence of investors having preferences that are adequately described by a utility function that exhibits a decreasing level of risk aversion (as measured by the Arrow-Pratt measure). Eric disagrees. He claims that wealthier people invest more because they are less affected by transactions costs. \end_layout \begin_deeper \begin_layout Enumerate For Eric's claim to make sense, what kind of transactions costs must he be thinking of? For example, would his argument apply if one had to pay $t for every dollar invested? What about if it cost $t per transaction (regardless of size)? \end_layout \begin_layout Enumerate Describe how you go about determining who (Mike or Eric) had the better interpretation of the observation that wealthier individuals invest more in the risky asset. \end_layout \end_deeper \begin_layout Enumerate Compute the Arrow Pratt Measure of Absolute Risk Aversion for the following utility functions \end_layout \begin_deeper \begin_layout Itemize \begin_inset Formula $u\left(w\right)=\ln\left(w\right)$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $u\left(w\right)=a+bw-cw^{2}$ \end_inset (For what values of \begin_inset Formula $b$ \end_inset and \begin_inset Formula $c$ \end_inset is this concave? What happens to the Arrow Pratt Measure outside this region?) \end_layout \begin_layout Itemize \begin_inset Formula $u\left(w\right)=-w^{-\beta};\beta>0$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $u\left(w\right)=-e^{-w}$ \end_inset \end_layout \end_deeper \begin_layout Enumerate Evaluate \begin_inset Formula \[ \int u^{\prime}\left(w+i_{r}s\right)sF^{\prime}\left(s\right)ds\] \end_inset and \begin_inset Formula \[ \int u^{\prime\prime}\left(w+i_{r}s\right)sF^{\prime}\left(s\right)ds\] \end_inset for each of the utility functions above assume that \begin_inset Formula $s$ \end_inset is distributed uniformly on the interval \begin_inset Formula $\left[-1,3\right]$ \end_inset (which means that \begin_inset Formula $F\left(s\right)=\frac{s+1}{4}.$ \end_inset \end_layout \begin_layout Enumerate A more interesting question than the one in the reading is to ask whether an individual invests a higher \emph on proportion \emph default of his or her wealth in the risky asset as their wealth rises. Use the method in the reading to prove that this is true provided the Arrow Pratt measure of \emph on relative risk aversion \emph default is decreasing in wealth where the Arrow Pratt measure of relative risk aversion is given by \begin_inset Formula \[ -\frac{u^{\prime\prime}\left(w\right)}{u^{\prime}\left(w\right)}w\] \end_inset \end_layout \begin_layout Section Insurance Theory \end_layout \begin_layout Enumerate Write an expression for the profits of the insurance company in terms of \begin_inset Formula $p$ \end_inset , \begin_inset Formula $q$ \end_inset and \begin_inset Formula $b$ \end_inset , and derive the \emph on actuarially fair \emph default premium, \begin_inset Formula $q^{*}$ \end_inset , as a function of \begin_inset Formula $p$ \end_inset and \begin_inset Formula $b$ \end_inset . \end_layout \begin_deeper \begin_layout Enumerate Draw the `budget set' for the consumer when the firm charges \begin_inset Formula $q^{*}$ \end_inset per unit of net payout, \begin_inset Formula $b$ \end_inset . \end_layout \begin_layout Enumerate Show how this set would change when the firm charges a premium higher than \begin_inset Formula $q^{*}$ \end_inset . \end_layout \begin_layout Enumerate How does this increase in the premium affect the consumer's optimal insurance policy choice? \end_layout \end_deeper \begin_layout Enumerate Martha is considering holding an outdoor cooking show for which she will get ticket sales of \begin_inset Formula $\$y$ \end_inset (Martha has no other wealth - due to recent kind donations to the Expensive Lawyers Foundation). However, there is some chance that it will rain, in which case she has to partially refund the tickets to the show. In total, she will have to refund \begin_inset Formula $\$d$ \end_inset . Martha is risk averse and has preferences represented by \begin_inset Formula $u(\cdot)$ \end_inset . \end_layout \begin_deeper \begin_layout Standard She is considering buying insurance against the possibility of rain. She discovers that all insurance firms agree that the chance of rain on the day of her show is \begin_inset Formula $p$ \end_inset , and as such are willing to offer any contract that will pay a net amount of \begin_inset Formula $b$ \end_inset in the event of rain, in exchange for a premium of \begin_inset Formula $q$ \end_inset such that their profits are zero. \end_layout \begin_layout Standard On a recent holiday to Jailstown, Martha met a very clever meteorologist that specialized in rain predictions. Martha phones this guy to get his inside \begin_inset Foot status collapsed \begin_layout Plain Layout The opinion is `inside' in the sense that the insurance firms do not know it. \end_layout \end_inset opinion on what the chance of rain is. He says that his extensive and top-secret analysis reveals that the chance of rain is \begin_inset Formula $p'$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Based on the insurance firms' belief that the probability of rain is \begin_inset Formula $p$ \end_inset , write out the zero-profit relationship between \begin_inset Formula $q$ \end_inset , \begin_inset Formula $b$ \end_inset , and \begin_inset Formula $p$ \end_inset . \end_layout \begin_deeper \begin_layout Enumerate Martha trusts the meteorologist, and believes that the probability of rain is not \begin_inset Formula $p$ \end_inset , but in fact \begin_inset Formula $p'$ \end_inset . Does Martha still buy full insurance? How does this depend on the advice, \begin_inset Formula $p'$ \end_inset ? \end_layout \end_deeper \begin_layout Enumerate Let the utility for wealth function be given by: \begin_inset Formula \begin{equation} u(x)=\left(\alpha+x\right)^{\beta},\end{equation} \end_inset where \begin_inset Formula $\alpha,\beta\geq0$ \end_inset are parameters. \end_layout \begin_deeper \begin_layout Enumerate For what values of \begin_inset Formula $\beta$ \end_inset is \begin_inset Formula $u$ \end_inset a concave function? Does this depend on \begin_inset Formula $\alpha$ \end_inset ? \end_layout \begin_layout Enumerate Suppose that \begin_inset Formula $\beta<1$ \end_inset . For what values of \begin_inset Formula $\alpha$ \end_inset is \begin_inset Formula $u'(x)_{|x=0}=\infty$ \end_inset ? \end_layout \begin_layout Enumerate Using the standard notation from the text, suppose that \begin_inset Formula $d=y$ \end_inset (so that in the accident state the consumer has zero wealth). Use the first two parts to determine the parameter values such that a consumer with preferences that are given by \begin_inset Formula $u$ \end_inset will buy some insurance \emph on regardless of \begin_inset Formula $y$ \end_inset and \begin_inset Formula $q/B$ \end_inset \emph default (as long as they are both positive). If you are not able to provide an intuitive answer, use Lagrangian techniques. \end_layout \end_deeper \begin_layout Enumerate Ben loves hockey. He really enjoys it when his favorite team, the Calgary Flames, are playing well. He has an income of \begin_inset Formula $y$ \end_inset , and gets a utility of \begin_inset Formula $u(y)$ \end_inset when the Flames win. However, when the team loses, he gets upset and says that he would give up \begin_inset Formula $\$d$ \end_inset dollars in order for the Flames to have played better and won (so that his utility when the team loses is \begin_inset Formula $u(y-d)$ \end_inset ). Ben is \series bold risk averse \series default , so that \begin_inset Formula $u''(\cdot)<0$ \end_inset (and \begin_inset Formula $u'(\cdot)>0$ \end_inset ). \end_layout \begin_deeper \begin_layout Standard A betting agency offers the following deal: you can pay \begin_inset Formula $\$q$ \end_inset , and if the Flames lose you get a net payout of \begin_inset Formula $\$b$ \end_inset (i.e. \begin_inset Formula $b$ \end_inset is in addition to getting your \begin_inset Formula $\$q$ \end_inset bet back). There is competition among betting agencies which implies that their profits are zero. Everyone (Ben, the betting agencies, and everyone else) evaluates the probabili ty that the Flames will win at \begin_inset Formula $1-p$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Write out the betting agency's profit function, and write the zero-profit relationship between \begin_inset Formula $q$ \end_inset and \begin_inset Formula $b$ \end_inset . How does \begin_inset Formula $q/b$ \end_inset change as the Flames become more likely to win (i.e. as \begin_inset Formula $p$ \end_inset decreases)? \end_layout \begin_deeper \begin_layout Enumerate Does Ben find it worthwhile to take the bet? That is, solve Ben's optimal choice of \begin_inset Formula $b$ \end_inset and \begin_inset Formula $q$ \end_inset (subject to the zero-profit condition) and determine whether \begin_inset Formula $b^{*}>0$ \end_inset . \end_layout \begin_layout Enumerate Does the answer to the last part surprise you? In particular, how would you reconcile the fact that Ben is betting on his favorite team losing? What about the fact that Ben is risk-averse? \end_layout \begin_layout Enumerate What would \begin_inset Formula $p$ \end_inset have to be in order for Ben to not make any bets ( \begin_inset Formula $b^{*}=0$ \end_inset )? \end_layout \end_deeper \begin_layout Enumerate In what ways is a standard health insurance policy really insurance in the sense introduced in the text? In what ways is it not? \end_layout \begin_layout Enumerate Suppose that instead of there being just two states (accident or no accident), there are three states. There is a no-accident state ( \begin_inset Formula $S_{1}$ \end_inset ), an insurable-accident state \begin_inset Formula $S_{2}$ \end_inset , and a non-insurable accident state, \begin_inset Formula $S_{3}$ \end_inset . These states occur with probability \begin_inset Formula $p_{1}$ \end_inset , \begin_inset Formula $p_{2}$ \end_inset , and \begin_inset Formula $1-p_{1}-p_{2}$ \end_inset respectively. The consumer loses \begin_inset Formula $d_{2}$ \end_inset in state 2, and loses \begin_inset Formula $d_{3}$ \end_inset in state \begin_inset Formula $3$ \end_inset . \end_layout \begin_deeper \begin_layout Standard An insurance firm offers a policy which pays net benefits of \begin_inset Formula $b$ \end_inset in the event that the state is \begin_inset Formula $S_{2}$ \end_inset , and collects a premium of \begin_inset Formula $q$ \end_inset in both of the other two states. Using Lagrangian methods, describe how the consumer's demand for insurance is affected by the introduction of the uninsurable-accident state. \end_layout \end_deeper \begin_layout Section First Welfare Theorem \end_layout \begin_layout Enumerate The function that describes how a firm transforms \begin_inset Formula $x$ \end_inset into \begin_inset Formula $y$ \end_inset (and vice versa) was left quite general in the text. Name some sensible properties of \begin_inset Formula $f$ \end_inset . For example, (in terms of the first figure) which quadrant(s) should the function never pass through? Why not? \end_layout \begin_layout Enumerate Draw the \begin_inset Formula $f$ \end_inset function associated with the following scenario. The two goods are bread ( \begin_inset Formula $x$ \end_inset ) and toast ( \begin_inset Formula $y$ \end_inset ). By exposing slices of bread to a fire, a firm is able to transform one slice of bread into one piece of toast. However, once turned to toast, the firm is not able to `un-expose' the piece of toast so that it turns back into a slice of bread. \end_layout \begin_layout Enumerate If there are \begin_inset Formula $\omega_{x}$ \end_inset units of \begin_inset Formula $x$ \end_inset before production, and \begin_inset Formula $z_{x}\leq\omega_{x}$ \end_inset units after production, how many units of \begin_inset Formula $x$ \end_inset were used as inputs in the firm's production process? How many units of \begin_inset Formula $y$ \end_inset are produced using this level of input? Therefore, how many units of \begin_inset Formula $y$ \end_inset are there after production (that is, what must \begin_inset Formula $z_{y}$ \end_inset be)? \end_layout \begin_layout Enumerate The previous question demonstrated that we can determine \begin_inset Formula $z_{y}$ \end_inset if we know \begin_inset Formula $z_{x}$ \end_inset (and, of course, we must know the endowment, \begin_inset Formula $(\omega_{y},\omega_{x})$ \end_inset , and production function, \begin_inset Formula $f$ \end_inset ). To be sure, explicitly write \begin_inset Formula $z_{y}$ \end_inset as a function of \begin_inset Formula $z_{x}$ \end_inset . What is this function commonly known as? \end_layout \begin_layout Enumerate Must the endowment \begin_inset Formula $(\omega_{x},\omega_{y})$ \end_inset lie on the production possibilities frontier? If not, what `unusual' property must \begin_inset Formula $f$ \end_inset possess at either \begin_inset Formula $z_{x}=\omega_{x}$ \end_inset or at \begin_inset Formula $z_{x}:f(z_{x}-\omega_{x})=0$ \end_inset ? \end_layout \begin_layout Enumerate In reality, there are two main benefits to being a shareholder in a firm. First, you get a share of the firm's profits. Second, you get to vote on how the firm operates. \end_layout \begin_deeper \begin_layout Standard Under these maintained assumptions, we claimed that the second benefit is illusory because all shareholders would like the firm to maximize profits. \end_layout \end_deeper \begin_layout Enumerate Explain the logic behind why all investors would vote to maximize profits. \end_layout \begin_deeper \begin_layout Enumerate What are the precise assumptions (there are at least three) that are creating this divergence between reality and the model? \end_layout \begin_deeper \begin_layout Standard \emph on Hint: \emph default Think of real-world motivations for individuals to hold voting rights in a firm, and then think about the assumptions used that would remove such motivations. \end_layout \end_deeper \end_deeper \begin_layout Enumerate Carefully define a Walrasian equilibrium in a production economy. In particular, describe the objects (e.g. a consumption choice for each consumer) and the properties of these objects (e.g. the consumption plan is affordable). What is the main difference between a production economy and an exchange economy? \end_layout \begin_layout Enumerate Briefly outline the argument explaining why equilibrium in an exchange economy is Pareto efficient (the first welfare theorem). \end_layout \begin_deeper \begin_layout Standard Since a production economy is basically an exchange economy in which these things called `firms' effectively determine the `endowments' that are to be exchanged, what must be true if the first welfare theorem still holds in a production economy? \end_layout \end_deeper \begin_layout Enumerate Prove that the first welfare theorem (Walrasian equilibrium is Pareto efficient) is still true in a production economy. \end_layout \begin_deeper \begin_layout Standard \emph on Hint \emph default : Try a proof by contradiction: derive a contradiction that follows from the supposition that there exists alternative consumption plans such that these new plans make at least one consumer better off without making any other worse off. \end_layout \end_deeper \begin_layout Enumerate Can a consumer in a production economy be made worse off than she would have been if she had refused to trade and instead consumed her endowment? Can a consumer be made worse off in a production economy relative to the welfare she experiences in the corresponding exchange economy? \end_layout \begin_layout Enumerate Consider some equilibrium in an exchange economy. Now, suppose that a firm arises and has the technology to transform the goods into each other according to a production function \begin_inset Formula $f$ \end_inset . \end_layout \begin_deeper \begin_layout Enumerate Is it possible that \emph on all \emph default consumers have a lower utility after the firm arises? \end_layout \begin_layout Enumerate Is it possible for \emph on some \emph default consumer to have a lower utility after the firm arises? Does this depend on whether the consumer owns shares in the firm? \end_layout \begin_layout Enumerate Suppose that some consumer is made worse off by the introduction of the firm, and that this consumer owns some shares in the firm. Surely, the consumer that is made worse off would not vote for the firm to pursue profit maximization (since this outcome makes him worse off by construction). In particular, he would surely rather try to convince the firm to produce something close to the exchange economy allocations (since he was better off in that equilibrium). What assumption makes this reasoning invalid? \end_layout \begin_layout Enumerate What do you find appealing about a Pareto efficient allocation? Why might a Pareto efficient allocation be undesirable? \end_layout \end_deeper \begin_layout Section Public Goods \end_layout \begin_layout Enumerate Define a \emph on public good \emph default , and give three examples. Is noise pollution a public good? \end_layout \begin_layout Enumerate Do public goods tend to be over- or under-provided? Explain, and describe one possible solution. \end_layout \begin_layout Enumerate Which assumption underlying the first welfare theorem is violated when studying an economy with a public good? \end_layout \begin_layout Enumerate Consider the voluntary contribution game. Describe how agent 2's consumption of the private good affects agent 1's utility (i.e. write \begin_inset Formula $u_{1}(x_{1},y)$ \end_inset in the form \begin_inset Formula $u_{1}(x_{1},x_{2},\cdot)$ \end_inset ). Is agent 1's utility increasing or decreasing in \begin_inset Formula $x_{2}$ \end_inset ? Explain. \end_layout \begin_layout Enumerate Carefully define a \emph on Nash Equilibrium \emph default . How is it different from a Walrasian equilibrium? \end_layout \begin_layout Enumerate There are two agents with identical utility functions given by \begin_inset Formula \begin{equation} u_{i}(x_{i},y)=x_{i}^{\beta}+y,\end{equation} \end_inset where \begin_inset Formula $x_{i}$ \end_inset is the level of consumption of the private good for agent \begin_inset Formula $i$ \end_inset , \begin_inset Formula $y$ \end_inset is the consumption of the public good, and \begin_inset Formula $\beta\in(0,1)$ \end_inset . The public good is produced using the production function: \begin_inset Formula \begin{equation} y=f\left(\underbrace{\overbrace{\omega_{1}-x_{1}}^{\text{1's contribution}}+\overbrace{\omega_{2}-x_{2}}^{\text{2's contribution}}}_{\text{Total Contribution}}\right)=\left(\omega_{1}-x_{1}+\omega_{2}-x_{2}\right)^{\beta},\end{equation} \end_inset where \begin_inset Formula $\omega_{i}$ \end_inset is agent \begin_inset Formula $i$ \end_inset 's endowment of the private good, and \begin_inset Formula $\beta\in(0,1)$ \end_inset . \end_layout \begin_deeper \begin_layout Enumerate Write agent 1's utility in terms of \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $x_{2}$ \end_inset (and \begin_inset Formula $\omega_{1}$ \end_inset and \begin_inset Formula $\omega_{2}$ \end_inset ). In \begin_inset Formula $(x_{1},x_{2})$ \end_inset space, draw (roughly) a family of indifference curves for agent 1. What general shape do they have? \end_layout \begin_layout Enumerate Suppose that agent 1 thinks that agent 2 is going to consume \begin_inset Formula $\bar{x}_{2}$ \end_inset units of the private good. What is the optimal choice of \begin_inset Formula $x_{1}$ \end_inset (in terms of \begin_inset Formula $\bar{x}_{2}$ \end_inset )? \end_layout \begin_deeper \begin_layout Enumerate How is this optimal choice affected by \begin_inset Formula $\bar{x}_{2}$ \end_inset ? Intuitively, what is going on? \end_layout \begin_layout Enumerate How is this optimal choice affected by \begin_inset Formula $\omega_{1}$ \end_inset ? \begin_inset Formula $\omega_{2}$ \end_inset ? \end_layout \begin_layout Enumerate Draw a graph in \begin_inset Formula $x_{1},x_{2}$ \end_inset space that shows agent 1's optimal choice of \begin_inset Formula $x_{1}$ \end_inset for any given level of \begin_inset Formula $\bar{x}_{2}$ \end_inset . \end_layout \end_deeper \begin_layout Enumerate Suppose that \begin_inset Formula $\omega_{1}=\omega_{2}$ \end_inset so that there is no quantitative difference between the two agents. Given this symmetry, what would you expect the relationship between the optimal \begin_inset Formula $x_{1}$ \end_inset and the optimal \begin_inset Formula $x_{2}$ \end_inset to be? Use this to calculate the Nash equilibrium private consumption pair ( \begin_inset Formula $x_{1}$ \end_inset , \begin_inset Formula $x_{2}$ \end_inset ). \end_layout \begin_layout Enumerate Suppose that a planner suspected that the Nash equilibrium was inefficient. The planner would like to set the private consumption levels so that it maximizes the sum of the two agents' utilities. Set up the planner's problem, and solve for the socially optimal private consumption levels \begin_inset Formula $(x_{1}^{**},x_{2}^{**})$ \end_inset . \end_layout \begin_layout Enumerate Compare the socially optimal private consumption levels to the Nash equilibrium levels: \end_layout \begin_deeper \begin_layout Enumerate How does \begin_inset Formula $\beta$ \end_inset affect the Nash equilibrium private consumption levels? How does it affect the socially optimal levels? \end_layout \begin_layout Enumerate Is the Nash equilibrium outcome inefficient (i.e. not Pareto efficient)? Is there too much or too little private consumption? Intuitively explain this result. \end_layout \begin_layout Enumerate How is the inefficiency affected by \begin_inset Formula $\beta$ \end_inset ? That is, do the Nash equilibrium consumption levels get closer to the optimal levels when \begin_inset Formula $\beta$ \end_inset approaches zero, or when \begin_inset Formula $\beta$ \end_inset approaches one? Any intuition for why this is so? \end_layout \end_deeper \begin_layout Enumerate Suppose that instead of assuming that each agent has a personal endowment, \begin_inset Formula $\omega_{i}$ \end_inset , we instead assumed that the pair were given an endowment of \begin_inset Formula $\omega$ \end_inset to `share'. That is, each agent is free to consume from \begin_inset Formula $\omega$ \end_inset but must take into account that whatever was left over (after both had consumed) was to be used in the production of the public good. Would this assumption change any of the above results? Why or why not? \end_layout \begin_layout Enumerate Suppose that some outside body (like a government) wanted to help these agents out by donating \begin_inset Formula $z$ \end_inset units of the private good (as an endowment that can be allocated to the production of the public good as desired). Due to some kind of favoritism, the total of \begin_inset Formula $z$ \end_inset is split as follows: agent 1 is to get \begin_inset Formula $\lambda z$ \end_inset and agent 2 is to get \begin_inset Formula $(1-\lambda)z$ \end_inset , where \begin_inset Formula $\lambda\in[0,1]$ \end_inset . \end_layout \begin_deeper \begin_layout Enumerate Suppose that there was no public good at all. Would agent 1 care about what \begin_inset Formula $\lambda$ \end_inset was? Which would he most prefer? \end_layout \begin_layout Enumerate Now introduce the public good in question. Now does agent 1 care? Why is this so? \end_layout \begin_layout Enumerate Suggest a slight change to the model that would result in agent 1 caring about \begin_inset Formula $\lambda$ \end_inset in the presence of a public good. \end_layout \end_deeper \end_deeper \begin_layout Enumerate What happens to the Nash equilibrium of the voluntary contribution game as the number of agents increases? In particular, does efficiency get improved or worsened? \end_layout \begin_layout Enumerate What is a \emph on Lindahl price \emph default ? Describe how such prices are used to resolve the public goods problem. \end_layout \begin_layout Enumerate \begin_inset CommandInset label LatexCommand label name "long" \end_inset How do you think agent 1's Lindahl price is affected by her endowment (keeping agent 2's endowment fixed)? What is your reasoning? \end_layout \begin_deeper \begin_layout Standard To derive this relationship in a particular economy, consider the following scenario. There are two agents, each with Cobb-Douglas preferences: \begin_inset Formula \begin{equation} U(x_{i},y_{i})=x_{i}^{\alpha}y_{i}^{1-\alpha},\end{equation} \end_inset for \begin_inset Formula $i=1,2$ \end_inset . Each agent is endowed with \begin_inset Formula $\omega_{i}$ \end_inset units of the private good which is turn given to the firm in exchange for shares which gives the agent the rights to a proportion, \begin_inset Formula $\omega_{i}/(\omega_{1}+\omega_{2})$ \end_inset of the firm's profits. \end_layout \begin_layout Standard The firm transforms the private good into the public good with the simple linear technology (recall that \begin_inset Formula $y_{1}=y_{2}=y$ \end_inset ): \begin_inset Formula \begin{equation} y=\omega_{1}+\omega_{2}-x_{1}-x_{2}.\end{equation} \end_inset Taking the price of the public good as the numeraire, the firms profits are given by: \begin_inset Formula \begin{equation} \pi=x_{1}+x_{2}+(p_{1}+p_{2})y.\end{equation} \end_inset \end_layout \end_deeper \begin_layout Enumerate Write the firm's constrained maximization problem. \end_layout \begin_deeper \begin_layout Enumerate \begin_inset CommandInset label LatexCommand label name "b" \end_inset Rather than using a Lagrangian approach, try substituting the constraint in directly. That is, write the firm's profit function in terms of endowments, prices, and \begin_inset Formula $(x_{1},x_{2})$ \end_inset . \end_layout \begin_layout Enumerate If an Walrasian equilibrium were to exist in this economy, one requirement is that the firm is choosing \begin_inset Formula $(x_{1},x_{2})$ \end_inset to maximize profits. Another is that \begin_inset Formula $(x_{1},x_{2})$ \end_inset must be feasible (i.e. \begin_inset Formula $x_{1}+x_{2}\in[0,\omega_{1}+\omega_{2}]$ \end_inset ). The latter implies the sensible restriction that the optimal production of the private goods can not be infinity or negative infinity. \end_layout \begin_deeper \begin_layout Standard Using this information, and the profit function derived in ( \begin_inset CommandInset ref LatexCommand ref reference "b" \end_inset ), argue that the Lindahl prices must satisfy the following: \begin_inset Formula \begin{equation} p_{1}+p_{2}=1.\end{equation} \end_inset What does this imply that equilibrium profits will be? \end_layout \end_deeper \begin_layout Enumerate Leaving the firm's production problem for now, let's look at the agents' consumption problem. Let \begin_inset Formula $M_{i}$ \end_inset denote the income of agent \begin_inset Formula $i$ \end_inset . Set up agent \begin_inset Formula $i$ \end_inset 's constrained maximization problem, and derive the demand functions (for \begin_inset Formula $x_{i}$ \end_inset and \begin_inset Formula $y_{i}$ \end_inset ). \end_layout \begin_layout Enumerate Now that we have the demands for \begin_inset Formula $y_{1}$ \end_inset and \begin_inset Formula $y_{2}$ \end_inset in terms of incomes and their (Lindahl) prices, we are able to derive another property of the Lindahl prices. What is the equilibrium relationship between \begin_inset Formula $y_{1}$ \end_inset and \begin_inset Formula $y_{2}$ \end_inset ? What does this relationship imply about how \begin_inset Formula $p_{1}$ \end_inset and \begin_inset Formula $p_{2}$ \end_inset are related to \begin_inset Formula $M_{1}$ \end_inset and \begin_inset Formula $M_{2}$ \end_inset ? Use the fact that \begin_inset Formula $M_{i}$ \end_inset equals the proportion of firms profits that agent \begin_inset Formula $i$ \end_inset has rights over to show that \begin_inset Formula \begin{equation} \frac{p_{1}}{p_{2}}=\frac{\omega_{1}}{\omega_{2}}.\end{equation} \end_inset \end_layout \begin_layout Enumerate Use the two above restrictions on \begin_inset Formula $p_{1}$ \end_inset and \begin_inset Formula $p_{2}$ \end_inset to derive \begin_inset Formula $p_{1}$ \end_inset as a function of \begin_inset Formula $\omega_{1}$ \end_inset and \begin_inset Formula $\omega_{2}$ \end_inset . How is \begin_inset Formula $p_{1}$ \end_inset affected by an increase in \begin_inset Formula $\omega_{1}$ \end_inset ? How does this compare with your initial reasoning? \end_layout \end_deeper \begin_layout Enumerate How do you think the Lindahl price facing agent 1 is affected by agent 2's relative desire for the public good (relative to her desire for the private good)? Use the following modified structure provided in Question \begin_inset CommandInset ref LatexCommand ref reference "long" \end_inset to derive this. \end_layout \begin_deeper \begin_layout Enumerate Suppose that instead of \begin_inset Formula $\alpha$ \end_inset being constant in the utility functions, that agent \begin_inset Formula $i$ \end_inset has a preference parameter of \begin_inset Formula $\alpha_{i}$ \end_inset . Re-derive the demand functions under this change. \end_layout \begin_layout Enumerate Exploit the equilibrium relationship between \begin_inset Formula $y_{1}$ \end_inset and \begin_inset Formula $y_{2}$ \end_inset to derive a relationship between \begin_inset Formula $p_{1}/p_{2}$ \end_inset , and \begin_inset Formula $(\alpha_{1},\alpha_{2},M_{1},M_{2})$ \end_inset . \end_layout \begin_layout Enumerate Use the fact that \begin_inset Formula $M_{i}$ \end_inset equals the proportion of firms profits that agent \begin_inset Formula $i$ \end_inset has rights over to show that \begin_inset Formula \begin{eqnarray} \frac{p_{1}}{p_{2}}=\frac{(1-\alpha_{1})\omega_{1}}{(1-\alpha_{2})\omega_{2}}.\end{eqnarray} \end_inset \end_layout \begin_layout Enumerate Has anything changed on the production side? Re-iterate the arguments that can be used to show that \begin_inset Formula \begin{eqnarray} p_{1}+p_{2}=1.\end{eqnarray} \end_inset \end_layout \begin_layout Enumerate Use the above two conditions to derive \begin_inset Formula $p_{1}$ \end_inset as a function of \begin_inset Formula $(\omega_{1},\omega_{2},\alpha_{1},\alpha_{2})$ \end_inset . How is \begin_inset Formula $p_{1}$ \end_inset affected by \begin_inset Formula $\alpha_{2}$ \end_inset ? How does this compare to your initial reasoning? \end_layout \end_deeper \begin_layout Enumerate We can verify that the Lindahl prices produce optimal choices. Consider the framework introduced in Question \begin_inset CommandInset ref LatexCommand ref reference "long" \end_inset . For ease of calculation, suppose that \begin_inset Formula $\omega_{1}=\omega_{2}$ \end_inset (and retain the assumption that the preference parameter, \begin_inset Formula $\alpha$ \end_inset , is the same across agents), so that the agents are completely symmetric. \end_layout \begin_deeper \begin_layout Enumerate Set up the planner's problem (do not solve yet). What does the planner maximize? Which variables does she choose? What are the constraints she faces? \end_layout \begin_layout Enumerate Rather than setting up the Lagrangian, try substituting the public good production function in directly so that the objective function is completely in terms of \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $x_{2}$ \end_inset (and endowments of course). \end_layout \begin_layout Enumerate Since the agent's are identical in every way (e.g. in preferences and endowments), what would you expect the relationship between the socially optimal \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $x_{2}$ \end_inset ? Use this relationship to simplify the objective function. \end_layout \begin_layout Enumerate Solve for the optimal level of private consumptions, \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $x_{2}$ \end_inset . \end_layout \begin_layout Enumerate Compare this with the \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $x_{2}$ \end_inset induced by the Lindahl prices. Are they the same? Does this suggest an alternative method by which to calculate the Lindahl prices? \end_layout \begin_layout Enumerate Does the fact that the levels of private consumption are efficient imply that the level of the public good is efficient? Either make an argument for or against this, or manually verify whether it is true. \end_layout \end_deeper \begin_layout Enumerate Informally speaking, Lindahl prices are a means by which to distort agents' incentives to contribute to the public good. The most obvious way to do this is to make the private good relatively more expensive. Consider again the framework in Question \begin_inset CommandInset ref LatexCommand ref reference "long" \end_inset , and recall that \begin_inset Formula $p_{1}$ \end_inset is the (Lindahl) price of agent 1's public good when the private good has price 1. \end_layout \begin_deeper \begin_layout Enumerate If the problem is that agents do not contribute enough to the public good, then given the above argument, would you expect \begin_inset Formula $p_{1}$ \end_inset to be greater than or less than 1? What about \begin_inset Formula $p_{2}$ \end_inset ? Was this the case? \end_layout \begin_layout Standard If we let \begin_inset Formula $M_{1}$ \end_inset be agent 1's income (his share of profits), then you should have calculated his demand for the private good to be \begin_inset Formula \begin{equation} x_{1}^{*}=\alpha M_{1}.\end{equation} \end_inset This demand does not seem to be a function of \begin_inset Formula $p_{1}$ \end_inset - a feature of the Cobb-Douglas specification. \end_layout \end_deeper \begin_layout Enumerate How do you reconcile the fact that the Lindahl prices induced the efficient demand for the private good when the demand function depends only on income, and not prices directly? \end_layout \begin_layout Enumerate Name two problems in using Lindahl prices as a solution to the public goods problem. In particular, think about problems that are likely to arise when trying to implement the mechanism. Can you think of a better solution to the public goods problem? \end_layout \begin_layout Enumerate Consider an economy with two consumers. The total of the private good is \begin_inset Formula $W$ \end_inset , and each consumer has \begin_inset Formula $\frac{W}{2}$ \end_inset of this endowment. The public good can be produced from the private good using the simple linear technology \begin_inset Formula $y=x$ \end_inset , i.e., each unit of the private good can be used to produce exactly one unit of the public good. Both consumers have preferences given by \begin_inset Formula $u\left(y,x\right)=y+\ln\left(x\right)$ \end_inset . Find the equilibrium of the voluntary contribution game. Show that if \begin_inset Formula $W<2$ \end_inset , then no public good is produced in this equilibrium. Find the Lindahl equilibrium allocation. Show that if the aggregate endowment of the private good is less than \begin_inset Formula $1$ \end_inset , no public good will be produced in the Lindahl equilibrium. Interpret. \end_layout \begin_layout Enumerate In the patent problem, assume that each dollar spent on a public good produces exactly one unit of the public good. There are two consumers. The aggregate endowment of money of both consumers is \begin_inset Formula $\omega$ \end_inset , each consumer's individual endowment is \begin_inset Formula $\frac{\omega}{2}$ \end_inset . Suppose both consumers have identical utility functions given by \begin_inset Formula \[ \ln\left(x\right)+\ln\left(1+y\right).\] \end_inset Draw the production possibilities frontier for this problem. Locate consumer 1's endowment in this diagram. Compute the equilibrium of the voluntary contribution game and label it in your diagram, showing consumption of good \begin_inset Formula $x$ \end_inset by consumer 1 along with the total output of the public good. \end_layout \begin_layout Enumerate Following the information given in problem 17, suppose that consumer 1 enjoys consumption \begin_inset Formula $x_{1}$ \end_inset of the private good, while output of the public good is \begin_inset Formula $y_{1}$ \end_inset . Write down the payoff that consumer 2 must receive in this outcome. Find the slope of consumer 2's indifference curve (in the space of all \begin_inset Formula $\left(x_{1},y_{1}\right)$ \end_inset pairs by totally differentiating consumer 2's payoff as function of \begin_inset Formula $x_{1}$ \end_inset and \begin_inset Formula $y_{1}$ \end_inset . What is the slope of this indifference curve? Does it always slope the same way? Why or why not? \end_layout \begin_layout Enumerate Once consumer 2 has been awarded a patent, suppose consumer 2 sets the price of the public good at \begin_inset Formula $p$ \end_inset . Consumer 1 is now able to purchase whatever quantity of the public good that he likes. For each \begin_inset Formula $x_{1}<\frac{\omega}{2}$ \end_inset , find some quantity \begin_inset Formula $y_{1}$ \end_inset such that the slope of consumer 1's indifference curve at the point \begin_inset Formula $\left(x_{1},y_{1}\right)$ \end_inset is equal to the slope of the line running from the point \begin_inset Formula $\left(x_{1},y_{1}\right)$ \end_inset to the point \begin_inset Formula $\left(\frac{\omega}{2},0\right)$ \end_inset . Use the corresponding solution to draw a picture representing consumer 1's offer curve for this problem. Explain the sense in which this curve describes all the consumption bundles \begin_inset Formula $\left(x_{1},y_{1}\right)$ \end_inset that consumer 2 could force consumer 1 to take by setting an appropriate price \begin_inset Formula $p$ \end_inset for the public good. \end_layout \begin_layout Enumerate Find the point where this offer curve is tangent to consumer 2's indifference curve in \begin_inset Formula $\left(x_{1},y_{1}\right)$ \end_inset space, and compare this point to the point corresponding to the equilibrium of the voluntary contribution game. \end_layout \begin_layout Enumerate In the environment described in Problem 17, suppose that consumer 1 has preferences given by \begin_inset Formula \[ \ln\left(x\right)+2\ln\left(1+y\right)\] \end_inset while consumer 2 has the same preferences as before. Would it be possible to make both consumers better off by taxing money from consumer 2 and giving it consumer 1? \end_layout \begin_layout Section Music Downloading \end_layout \begin_layout Enumerate Consider an economy with 2 goods and two consumers. Good \begin_inset Formula $x$ \end_inset is a private good. Each consumer owns exactly one unit of good \begin_inset Formula $x$ \end_inset . The other good is a public good which is produced using good \begin_inset Formula $x$ \end_inset as an input. There is no endowment of the public good, however each unit of good \begin_inset Formula $x$ \end_inset can be used to produce exactly one unit of the public good. The complication is that only consumer 1 is able to produce the public good - consumer 2 can produce nothing on his own. The consumers have identical quasi linear utility functions given by \begin_inset Formula $U(x,y)=\ln(y)+x$ \end_inset . Negative consumption of good \begin_inset Formula $x$ \end_inset is okay (think of it as borrowing). \end_layout \begin_deeper \begin_layout Enumerate What is the equilibrium of the voluntary contribution game. \end_layout \begin_layout Enumerate Starting from the equilibrium of the voluntary contribution game, suppose that consumer 2 (the non-producer) proposes to match any additional contributio ns that consumer 1 makes to production of the public good one for one, provided that consumer 1 agrees to use these matching grants to produce the public good. Show that as long as consumer 2 can limit consumer 1's contributions to the public good, then this scheme can be used to make both consumers better off. \end_layout \begin_layout Enumerate What is the Lindahl equilibrium price for the public good. How much public good is produced in the Lindahl equilibrium. \end_layout \begin_layout Enumerate Interpret the Lindahl equilibrium as one in which a single price taking competitive firm is given the exclusive right to sell both public and private good. What happens if the firm instead acts as a monopolist? In particular, show that a monopoly firm (who maximizes its own profit) will produce less of the public good than the producer does in the voluntary contribution game. \end_layout \end_deeper \begin_layout Section Constrained Optimization \end_layout \begin_layout Enumerate As described in class, let the utility function for good \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset be \begin_inset Formula $y+\ln\left(x\right)$ \end_inset (natural logarithm). Write out the Lagrangian function and first order conditions. Find the demand function. Find the income and substitution effects (in derivative form) of an increase in the price of good \begin_inset Formula $x$ \end_inset , and \begin_inset Formula $y$ \end_inset . \end_layout \begin_layout Enumerate Solve for the Cobb Douglas demand functions for utility function \begin_inset Formula $u\left(x,y\right)=x^{\alpha}y^{\beta}$ \end_inset (in this case let \begin_inset Formula $\alpha+\beta>1$ \end_inset ). Also suppose that the consumer has an endowment of \begin_inset Formula $x_{0}$ \end_inset units of good \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y_{0}$ \end_inset units of good \begin_inset Formula $y$ \end_inset (so the consumer maximizes utility subject to the constraint that the value of chosen consumption is less than or equal to \begin_inset Formula $px_{0}+qy_{_{0}}$ \end_inset . Find the income and substitution effects in derivative form. \end_layout \begin_layout Enumerate For utility function \begin_inset Formula $y+bx$ \end_inset find the demand function when \begin_inset Formula $p