Chapter 21: The Discounted Utility Model


 Erica Bailey
 4 years ago
 Views:
Transcription
1 Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal preferences. In fact, it is not only empirically relevant but it also has the important normative property that individuals with such preferences are never dynamically inconsistent. What is meant by this is that the relative evaluation of consumption in period t and in period s remains constant through time for instance, if today an individual regards 1000 in 2015 as equivalent to him or her as 1500 in 2017, they should think the same way tomorrow and in 2003, and so on. This means that it represents the kind of intertemporal preferences that people ought to have if they want to be consistent in their intertemporal decisionmaking. These normative properties are not relevant in the twoperiod world that we have been analysing so far, but become relevant in a morethantwo period world. We shall consider such a world after we have described the model and explored its implications in a twoperiod world. 21.2: The Discounted Utility Model in a TwoPeriod World This is a model of preferences over bundles of consumption (c 1, c 2 ) where c 1 denotes consumption in period 1 and c 2 denotes consumption in period 2. It is simply given as follows: U(c 1, c 2 ) = u(c 1 ) + u(c 2 )/(1+ρ) (21.1) Do note carefully that there are two utility functions here: one is U, which is defined over the consumption bundle (c 1, c 2 ) and which represents the utility of that bundle; the other is u, which is defined over a single period s consumption and which represents the utility gained from consuming a particular amount in a particular period. The Discounted Utility Model states that the utility of a consumption bundle (c 1, c 2 ) is given by the utility gained from consuming the amount c 1 (in period 1) plus the utility gained from consuming the amount c 2 (in period 2) divided by (1+ρ). In this model, ρ is a parameter of the model and usually varies from individual to individual. For most individuals the parameter ρ is positive, which means that (1+ ρ) is greater than 1, which means that u(c 2 )/(1+ρ) is smaller than u(c 2 ) and hence that greater weight is attached to a particular level of consumption if it consumed in period 1 rather than in period 2. Why is it called the Discounted Utility Model? Because it discounts the utility gained from consumption in period 2 at the rate ρ. This parameter for a particular individual is called the individual s discount rate. It is called this for the following reason. Consider the present value of the stream of income m 1 in period 1 and m 2 in period 2 when the market interest rate is r. The present value is (see section 20.5) is m 1 + m 2 /(1+r) As we have already noted in chapter 20, as viewed from period 1 the future income m 2 is discounted by the market at the rate r because it will not be received for a period. Now notice the similarity between this expression and that for the Discounted Utility Model in (21.1). In the latter the individual discounts the future utility because it will not be received for a period. He or she discounts it at the rate ρ. We should emphasise that this discount factor is individual specific some individuals have a high value for ρ, some a low value. What it depends upon is how the
2 individual regards the utility of consumption in period 2 relative to the utility of consumption in period 1. It may be the case that the individual regards them equally in which case the parameter ρ takes the value 0 the individual in this case does not discount the future. However for most of us it seems to be the case that we weight the present more highly than the future in which case the value of the parameter ρ is positive. Furthermore, the more the individual regards the present relative to the future the higher is ρ  alternatively we can say the higher the individual regards the present relative to the future, the more the individual discounts the future. We should also point out that, as a parameter representing preferences, the value of ρ is independent of the value of the market discount rate r. The parameter ρ captures the individual s preferences regarding the relative weighting of present and future consumption. To complete the specification of the Discounted Utility Model we also need to specify the function u. This will also be individual specific. Normally we expect it to be a concave function as consumption rises then so does the utility gained from that consumption, but it rises less than proportionately. Or, in more casual terms, for every increase of 1 in consumption there is an increase in utility but these increases get smaller as the amount consumed increases. In what follows we will assume that the function u is the square root function (which is concave for positive consumption 1 ). An alternative is that the u function is the logarithmic function (which is also concave for positive consumption). With this function we get slightly different demand functions but the important property concerning the relationship of r with ρ (which we shall prove shortly) is true for all concave u functions, including the logarithmic function. Of course, in practice, the form of the function depends upon the individual s preferences. 21.4: Indifference Curves with the Discounted Utility Model We have now specified the Discounted Utility Model. In the next section we explore its implications. But first, as we are going to use the framework for analysis developed in chapter 20, we need to look at the properties of the indifference curves in (c 1, c 2 ) space implied by this model. There follows a little bit of mathematics; if you do not like maths, look away until we discuss the implications of the results. An indifference curve in (c 1, c 2 ) space is given, as ever, by U(c 1, c 2 ) = constant If we substitute in the specification of the Discounted Utility Model from (21.1) we get the following equation for an indifference curve in (c 1, c 2 ) space. u(c 1 ) + u(c 2 )/(1+ρ) = constant From this we can find the slope of the indifference curve (a proof is provided in the Mathematical Appendix to this chapter). The slope is given by: slope of indifference curve =  (1+ρ) [du(c 1 )/dc 1 ]/[du(c 2 )/dc 2 ] (21.2) where du(c)/dc denotes the derivative of u(c) with respect to c that is the rate at which utility is increasing with consumption. 1 It is difficult to conceive of negative consumption.
3 The slope is negative so the indifference curves are downward sloping. Moreover if u is concave then, as we move down and rightwards along an indifference curve, c 1 is rising while c 2 is falling, and hence du(c 1 )/dc 1 is falling while du(c 2 )/dc 2 is rising, and so the magnitude of the slope is falling. From this it follows that if u is concave then the indifference curves are convex. If however u is linear then both du(c 1 )/dc 1 and du(c 2 )/dc 2 are constant and so the slope of the indifference curves are constant that is they are linear. If we continue this line of argument with a convex function u, then we get the following result: If u is concave, linear or convex then the indifference curves in (c 1, c 2 ) space are convex, linear or concave. As we have already argued the concave u function is the empirically more realistic as people s consumption increases, their utility rises, but at a decreasing rate. One final result is of particular importance. If, in (21.2) we put c 1 = c 2 we get that the slope of an indifference curve is (1+ρ). Calling the line c 1 = c 2 the equal consumption line we get the important result that: Along the equal consumption line the slope of every indifference curve of an individual with Discounted Utility Model preferences is equal to (1+ρ). You should remember this. We illustrate in figure (21.1). In this figure I have used the square root utility function for u, and I have put ρ = 0.2. (that is, the individual discounts the future at 20%). In figure 21.1 I draw some of his or her indifference curves in (c 1, c 2 ) space. I have also inserted the equal consumption line. It can be seen that the slope of every indifference curve along this line is 1.2. For an individual with a higher discount rate than 20%, his or her indifference curves are everywhere steeper along the equal consumption line. 21.5: The Implications in a TwoPeriod World You may be able to guess some of the implications. We know that along the equal income line the slope of the indifference curves are all (1+ρ). It follows therefore that to the right of the equal income line the magnitude of the slope is everywhere less than (1+ρ) and to the left of the equal income line the magnitude of the slope is everywhere more than (1+ρ). Why is this important? Because we know that the magnitude of the slope of the budget line is (1+r) and we know that at the optimum point the magnitude of the slope of the budget line is equal to the magnitude of the slope of the indifference curve at the optimum point. (Because the budget line must be tangential to
4 the indifference curve at the optimal point.) So at the optimal point the slope of the indifference curve must be (1+r). There are three cases to consider. The simplest is when r = ρ. In this case the optimum point must lie somewhere along the equal consumption line for we know that the slopes of the indifference curves there are (1+ρ), and if this is equal to (1+r) (because r = ρ) it follows that the optimum point (where the slopes of the indifference curve and the budget line are equal) must be along the equal consumption line. In this case the individual consumes the same in both periods the reason being that the market discounts the future at exactly the same rate as the individual. When r > ρ we can argue that the optimal point must lie to the left of the equal consumption line so that the individual consumes more in the second period than the first. And when r < ρ we can argue that the optimal point must lie to the right of the equal consumption line so that the individual consumes more in the first period than the second. What is crucial is whether the market discounts the future more heavily than the individual. These properties are confirmed in the following example, in which we keep fixed the initial incomes of 40 (in period 1) and 40 (in period 2) and vary the rate of interest from zero (as in figure 21.3) upwards. We assume a value of ρ equal to 0.2 (20%) that is, we use the preferences of (21.1) above. You might like to verify that when the market rate of interest reaches 20% the optimal point is on the equal consumption line. The implied borrowing and lending are pictured in figure (Recall that the rate of return is one plus the rate of interest. The downward sloping line is the net demand for consumption in period 1 and the upward sloping line the net demand for consumption in period 2. Note that they cross the axis at a rate of interest 20%.) If we repeat this exercise for an individual with ρ = 0.4 we get figure Note the similarities and the differences between this and figure 21.4.
5 21.6: The Discounted Utility Model in a ManyPeriod World Although a little outside the scope of this book, it is interesting to consider the extension of the Discounted Utility Model to a manyperiod world. The extension is the natural one and follows the extension of the discounting formula discussed in section Suppose we are in a world which lasts T periods, where T may be finite, infinite or random. Suppose the individual has consumption c 1 in period 1, c 2 in period 2,, c t in period t,, and c T in period T. We consider his or her preferences over consumption bundles over these T periods bundles which we denote by (c 1, c 2,, c t,, c T ). The Discounted Utility Model states that preferences over these bundles are given by U(c 1, c 2,, c t,, c T ) = u(c 1 ) + u(c 2 )/(1+ρ) + + u(c t )/(1+ρ) t1 + +u(c T )/(1+ρ) T1 (21.3) Once again the model is specified by a discount rate ρ and a utility function u. The model is the natural extension of the twoperiod model. The utility of the bundle is the sum of the utilities of consumption in each period discounted at the rate ρ. If ρ is positive the weight attached to future consumption declines with the time from the present: period 1 s consumption is given a weight of 1; period 2 s consumption is weighted by 1/(1+ρ); ; period t s consumption by 1/(1+ρ) t1 ; ; and period T s by 1/(1+ρ) T1. You may find it constructive to compare (21.3) with the formula in section 20.5 giving the present value of an income stream there future incomes are discounted by the market at the rate r because they are received in the future. In the Discounted Utility Model future utilities are discounted by the individual at the rate ρ because they are experienced in the future. Whether this is a good description of actual intertemporal preferences is obviously an empirical issue but it may be instructive to give the normative reason why such preferences might be appropriate. In essence the idea is that an individual with such preferences is dynamically consistent in the sense that they carry out plans that they formulate. In a certain world there is no reason why one should ever want to change plans once formulated, so this property is an appealing one. Let us discuss it in more detail. Suppose we start in period 1 and we know that the income stream is going to be m 1 in period 1, m 2 in period 2,, m t in period t,, m T in period T. Then the individual in period 1 formulates a plan for consumption (c 1, c 2,, c t,, c T ) through time on the basis of maximising the utility (21.3) subject to the income stream and the rate of interest (which we are assuming is fixed and known). Accordingly in period 1 the individual consumes the planned c 1. In period 2 what happens? Does the individual implement the planned consumption c 2? It depends. Suppose the individual replans at this stage. At this stage he or she has a different objective
6 function because period 1 has been and gone. Now only periods 2 through T remain. If we update (21.3) to take account of this the objective function is now U(c 2,, c t,, c T ) = u(c 2 ) + u(c 3 )/(1+ρ) + + u(c t )/(1+ρ) t2 + +u(c T )/(1+ρ) T2 (21.4) Does the maximisation of this, given the remaining stream of income, lead to the same optimal values c 2,, c t,, c T that were the optimal values in the original plan c 1, c 2,, c t,, c T formulated at time period 1? The answer to this question is yes though the reason for this answer may not be obvious. A proof is provided in the Mathematical Appendix, which shows that we can express the utility from period 1 onwards in the following form: U(c 1, c 2,, c t,, c T ) = u(c 1 ) + U(c 2,, c t,, c T )/(1+ρ) (21.5) So the choice of the best values of c 2,, c t,, c T given the optimal value of c 1 from the maximisation of utility as viewed from the first period (with respect to the choice of c 1, c 2,, c T ) leads to the same choice of c 2,, c t,, c T as in the original choice of c 1, c 2,, c t,, c T in the maximisation of utility as viewed from the first period. The individual would not want to change the plans originally made. In essence and this should be clear from (21.5)  the reason is that if we compare any two periods s and t the relative weight attached to the utility of consumption in period s is always the same value (1+ρ) st irrespective of which period we are viewing it from. This means that the individual never wishes to revise any plans that he or she has formulated. (Of course, in a certain world, there are no reasons why one should ever wish to revise any plans.) If, instead, the individual used a varying discount rate through time, then it is possible that the individual might want to revise plans made earlier. In a sense this is because the individual, with a varying discount rate, is not just one individual but several and they have conflicting views what is best to do. This is exactly the kind of person who resolves every morning not to drink in the evening and then ends up doing so. 21.7: Summary Most of this chapter has been concerned with the twoperiod Discounted Utility Model though the final section contained an extension to the manyperiod model in which we provided a normative justification for it. In the twoperiod case we showed the following. The Discounted Utility model states that the intertemporal utility function is given by U(c 1, c 2 ) = u(c 1 ) + u(c 2 )/(1+ρ) where ρ is the individual's discount rate. With the Discounted Utility Model the slope of all indifference curves along the equalconsumption line are (1+ρ) Now we recalled that the perfect capital market budget constraint has a slope of (1+r) where r is the market interest rate. Combining these two results we showed that: An individual with Discounted Utility preferences would consume the same in both periods if his or her ρ equalled the market r... and that he or she will consume more in period 1 than in period 2 if r < ρ.. and that he or she will consume more in period 2 than in period 1 if r > ρ
7 We then briefly extended the model to a manyperiod world and showed that one attractive normative property of the model is that an agent with such preferences is never dynamically inconsistent. 21.8: At what rate do you discount the future? The Discounted Utility Model incorporates the notion that individuals do not value consumption in all periods equally. In general, it would appear to be the case that individuals care more about the present than the future, though the extent to which they do this differs from individual to individual. Here you are invited to try and discover how the extent to which you do this. The model in a Tperiod world takes the form of equation (21.3) and in a 2period world, the form of equation (21.1). This latter says that a twoperiod bundle of consumption (c 1,c 2 ) is valued by the function U(c 1, c 2 ) = u(c 1 ) + u(c 2 )/(1+ρ) where u(c) is the utility gained from consuming c in some period and ρ is the individual s discount rate. Both of these are specific to the individual. Here we will concentrate on discovering your value of ρ, assuming, of course, that your preferences are in accordance with the Discounted Utility Model 2. The method used is simple we exploit the implications of the equation above, particularly those results that we have already derived concerning the slope of indifference curves. If we can find two points on the same indifference curve, then we can use these results to try to infer the value of ρ. Specifically, let us take equation (21.2): slope of indifference curve =  (1+ρ) [du(c 1 )/dc 1 ]/[du(c 2 )/dc 2 ] If we do not know the individual s utility function, then we start at a position where c 1 and c 2 are equal. This enables us to derive the result stated in the text: Along the equal consumption line the slope of every indifference curve of an individual with Discounted Utility Model preferences is equal to (1+ρ). It follows that if we start at a point where c 1 = c 2 = c and we find a nearby point about which the individual feels the same (that is, is on the same indifference curve) then we can infer the value of ρ. To fix ideas suppose that the point (c  a, c + b) is such a point that is, the individual feels indifferent between the twoperiod consumption bundle (c, c) and the twoperiod consumption bundle (c  a, c + b). Then the slope of the indifference curve is approximately equal to b/a. This is an estimate of (1+ ρ). Hence we have that an estimate of ρ is b/a 1 = (b a)/a. If the individual puts more weight on present than on future consumption, then b will be bigger than a (because to compensate the individual for consuming a less today he or she will require more than a next period), and hence the estimate of ρ is positive. Obviously the value of ρ depends upon the length of the period we are considering. Let us assume here that the period is of length one year. You can now try and implement the above ideas and 2 If they are not, then there is not a ρ to discover.
8 hence find your yearly discount rate. You have to do the following introspection. Suppose you start from a position where you are consuming the same both this year and next year: for example, you are consuming 5000 each year. Now suppose someone suggests that your consumption this year will fall by say 100 (to 4900) but that you will be given some extra consumption next year to compensate. You should ask yourself: what is the minimum compensation I would require? This is quite a difficult introspection, but you should attempt it. Try and narrow it down. Would 1 compensation be enough? (Probably not.) Would 1000 compensation be enough? (Probably more than enough.) Would 50 be enough? And so on. The minimum compensation that you require gives an estimate of your discount rate ρ. Suppose this minimum compensation is 120. Then we have that a = 100 and b = 120, so that your ρ is 0.2 ( = ( )/100). It should be clear that we can derive the following table of examples. Minimum Compensation required for a decrease in period 1 consumption of 100 Implied value of the discount rate ρ Note carefully that we are talking about changes in consumption and not about changes in money income. If it were the latter and there was a perfect capital market in which you could freely borrow and lend at the constant rate of interest r, then the answer to the question would have to be 100(1+r). We would not learn anything about your discount rate only about the rate of interest in the perfect capital market! Note the assumptions carefully: (1) we start along the equal consumption line (so that we do not have to worry about your utility function, which is difficult to infer); (2) we consider a small reduction in period 1 consumption (otherwise we are moving around the indifference curve and its slope may change); (3) you tell us honestly the minimum amount of compensation in terms of period 2 consumption you require. If you do all this, you can find your discount rate. If you are interested, you could explore the implications for a manyperiod world. In the extension to T periods, as given in equation (21.3), you will see that the Discounted Utility Model assumes that the same discount rate is used throughout. Obviously this is a strong assumption, but one that can be tested. You can try one such test yourself. Suppose you have found that the minimum compensation you require (under the assumptions listed above) is 120. Then, starting from an equal consumption point, you regard having 100 less today as being compensatable with an extra 120 in one year s time. Now do the same exercise, but now ask what is the minimum compensation in two years time for having 100 less today. Denote again by b this minimum compensation, but do remember that this will be consumed in two years time. Repeating the argument that we used above it follows that b/a is an estimate of (1+ ρ) 2. Thus to be consistent with the Discounted Utility Model and with your previously derived estimate of ρ (which is 0.2), it must be the case that the minimum compensation you require in two years time is 144. (So that b/a = (1+ ρ) 2.) At first glance you may this odd or, at least inconsistent with your introspection. Let us discover why the Discounted Utility Model makes this prediction. We begin with your first introspection you needed 120 in one year s time to compensate you for having 100 less today, that is, for each
9 1 less today you needed 1.20 in compensation in one year s time. If this story applies not only to consumption deferred for one year from today, it should also apply to consumption deferred for one year from next year. So, if you are to be compensated for having 100 less today, but will receive the compensation in two years time, you can argue as follows: I would need 120 more in one year s time, and to defer each of these 120 for a further year, I will need a compensation of 1.20 for each of those 120 that is a compensation of 1.2 times 120 = 144 in two year s time. Was your introspection consistent with this? You may be interested to know that there have been many experimental tests of the Discounted Model, and particularly its central assumption that the discount rate is constant 3. The great strength of experimental economics is its central tenet that participants should be given appropriate incentives to behave in such a way that their behaviour reveals their preferences. In many areas it is easy to give appropriate incentives as we will see in the chapter on Game Theory but in the area of intertemporal choice it is more difficult. We have already noted some strong assumptions that underlie the inferences we have made. These are difficult to enforce in the laboratory. The greatest problem, however, is that correct incentives in intertemporal choice experiments necessary involve the passage of time. It may be difficult to ensure that participants and experimenters are still around after that passage of time Mathematical Appendix We first derive the proposition concerning the slope of the indifference curves implied by the Discounted Utility Model. As stated in the text, an indifference curve in (c 1, c 2 ) space is given by U(c 1, c 2 ) = constant If we substitute in the specification of the Discounted Utility Model from (21.1) we get the following equation for an indifference curve in (c 1, c 2 ) space. u(c 1 ) + u(c 2 )/(1+ρ) = constant To find the slope of the indifference curve we differentiate this totally, thus getting u'(c 1 ) dc 1 + u'(c 2 ) dc 2 /(1+ρ) = 0 where u'(c) denotes the derivative of u(c) with respect to c. From this we get the slope of an indifference curve This is equation (21.2) of the text. We now derive equation (21.5). Suppose c 1 *, c 2 *,, c t *,, c T * maximise dc 2 /dc 1 =  (1+ρ) u'(c 1 )/u'(c 2 ) 3 Many of these studies suggest that the discount rate is not constant.
10 U(c 1, c 2,, c t,, c T ) = u(c 1 ) + u(c 2 )/(1+ρ) + + u(c t )/(1+ρ) t1 + +u(c T )/(1+ρ) T1 given an income stream m 1, m 2,, m t,, m T, and hence subject to the intertemporal budget constraint m 1 + m 2 /(1+r) + + m T1 /(1+r) T2 + m T /(1+r) T1 = c 1 + c 2 /(1+r) + + c T1 /(1+r) T2 + c T /(1+r) T1 then it must be the case that c * 2,, c * t,, c * T maximise U(c 2,, c t,, c T ) = u(c 2 ) + u(c 3 )/(1+ρ) + + u(c t )/(1+ρ) t2 + +u(c T )/(1+ρ) T2 subject to the constraint m 1 + m 2 /(1+r) + + m T1 /(1+r) T2 + m T /(1+r) T1 = c * 1 + c 2 /(1+r) + + c T1 /(1+r) T2 + c T /(1+r) T1 because U(c 1, c 2,, c t,, c T ) = u(c 1 ) + U(c 2,, c t,, c T )/(1+ρ).
11
The fundamental question in economics is 2. Consumer Preferences
A Theory of Consumer Behavior Preliminaries 1. Introduction The fundamental question in economics is 2. Consumer Preferences Given limited resources, how are goods and service allocated? 1 3. Indifference
More information. In this case the leakage effect of tax increases is mitigated because some of the reduction in disposable income would have otherwise been saved.
Chapter 4 Review Questions. Explain how an increase in government spending and an equal increase in lump sum taxes can generate an increase in equilibrium output. Under what conditions will a balanced
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationLecture notes for Choice Under Uncertainty
Lecture notes for Choice Under Uncertainty 1. Introduction In this lecture we examine the theory of decisionmaking under uncertainty and its application to the demand for insurance. The undergraduate
More informationWeek 7  Game Theory and Industrial Organisation
Week 7  Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number
More informationSecond degree price discrimination
Bergals School of Economics Fall 1997/8 Tel Aviv University Second degree price discrimination Yossi Spiegel 1. Introduction Second degree price discrimination refers to cases where a firm does not have
More information1 Uncertainty and Preferences
In this chapter, we present the theory of consumer preferences on risky outcomes. The theory is then applied to study the demand for insurance. Consider the following story. John wants to mail a package
More informationEC247 FINANCIAL INSTRUMENTS AND CAPITAL MARKETS TERM PAPER
EC247 FINANCIAL INSTRUMENTS AND CAPITAL MARKETS TERM PAPER NAME: IOANNA KOULLOUROU REG. NUMBER: 1004216 1 Term Paper Title: Explain what is meant by the term structure of interest rates. Critically evaluate
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationInsurance. Michael Peters. December 27, 2013
Insurance Michael Peters December 27, 2013 1 Introduction In this chapter, we study a very simple model of insurance using the ideas and concepts developed in the chapter on risk aversion. You may recall
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationIn recent years, Federal Reserve (Fed) policymakers have come to rely
LongTerm Interest Rates and Inflation: A Fisherian Approach Peter N. Ireland In recent years, Federal Reserve (Fed) policymakers have come to rely on longterm bond yields to measure the public s longterm
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More information1. Briefly explain what an indifference curve is and how it can be graphically derived.
Chapter 2: Consumer Choice Short Answer Questions 1. Briefly explain what an indifference curve is and how it can be graphically derived. Answer: An indifference curve shows the set of consumption bundles
More informationThe CobbDouglas Production Function
171 10 The CobbDouglas Production Function This chapter describes in detail the most famous of all production functions used to represent production processes both in and out of agriculture. First used
More informationCHAPTER 4 Consumer Choice
CHAPTER 4 Consumer Choice CHAPTER OUTLINE 4.1 Preferences Properties of Consumer Preferences Preference Maps 4.2 Utility Utility Function Ordinal Preference Utility and Indifference Curves Utility and
More informationChapter 25: Exchange in Insurance Markets
Chapter 25: Exchange in Insurance Markets 25.1: Introduction In this chapter we use the techniques that we have been developing in the previous 2 chapters to discuss the trade of risk. Insurance markets
More informationChapter 4 Inflation and Interest Rates in the ConsumptionSavings Framework
Chapter 4 Inflation and Interest Rates in the ConsumptionSavings Framework The lifetime budget constraint (LBC) from the twoperiod consumptionsavings model is a useful vehicle for introducing and analyzing
More information11.3 BREAKEVEN ANALYSIS. Fixed and Variable Costs
385 356 PART FOUR Capital Budgeting a large number of NPV estimates that we summarize by calculating the average value and some measure of how spread out the different possibilities are. For example, it
More informationECON 312: Oligopolisitic Competition 1. Industrial Organization Oligopolistic Competition
ECON 312: Oligopolisitic Competition 1 Industrial Organization Oligopolistic Competition Both the monopoly and the perfectly competitive market structure has in common is that neither has to concern itself
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationInsurance and Public Pensions : (b) Adverse Selection
Insurance and Public Pensions : (b) Adverse Selection Adverse selection is said to occur if potential buyers of insurance know their own probabilities of loss better than do insurance companies. So suppose
More information$2 4 40 + ( $1) = 40
THE EXPECTED VALUE FOR THE SUM OF THE DRAWS In the game of Keno there are 80 balls, numbered 1 through 80. On each play, the casino chooses 20 balls at random without replacement. Suppose you bet on the
More informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More informationAnswers to Text Questions and Problems. Chapter 22. Answers to Review Questions
Answers to Text Questions and Problems Chapter 22 Answers to Review Questions 3. In general, producers of durable goods are affected most by recessions while producers of nondurables (like food) and services
More informationc 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.
Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions
More informationCONSUMER PREFERENCES THE THEORY OF THE CONSUMER
CONSUMER PREFERENCES The underlying foundation of demand, therefore, is a model of how consumers behave. The individual consumer has a set of preferences and values whose determination are outside the
More informationElasticity. I. What is Elasticity?
Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in
More informationUniversidad de Montevideo Macroeconomia II. The RamseyCassKoopmans Model
Universidad de Montevideo Macroeconomia II Danilo R. Trupkin Class Notes (very preliminar) The RamseyCassKoopmans Model 1 Introduction One shortcoming of the Solow model is that the saving rate is exogenous
More informationMidterm Exam:Answer Sheet
Econ 497 Barry W. Ickes Spring 2007 Midterm Exam:Answer Sheet 1. (25%) Consider a portfolio, c, comprised of a riskfree and risky asset, with returns given by r f and E(r p ), respectively. Let y be the
More informationEconomics 2020a / HBS 4010 / HKS API111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4
Economics 00a / HBS 4010 / HKS API111 FALL 010 Solutions to Practice Problems for Lectures 1 to 4 1.1. Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with
More informationWHY THE LONG TERM REDUCES THE RISK OF INVESTING IN SHARES. A D Wilkie, United Kingdom. Summary and Conclusions
WHY THE LONG TERM REDUCES THE RISK OF INVESTING IN SHARES A D Wilkie, United Kingdom Summary and Conclusions The question of whether a risk averse investor might be the more willing to hold shares rather
More informationKEELE UNIVERSITY MIDTERM TEST, 2007 BA BUSINESS ECONOMICS BA FINANCE AND ECONOMICS BA MANAGEMENT SCIENCE ECO 20015 MANAGERIAL ECONOMICS II
KEELE UNIVERSITY MIDTERM TEST, 2007 Thursday 22nd NOVEMBER, 12.0512.55 BA BUSINESS ECONOMICS BA FINANCE AND ECONOMICS BA MANAGEMENT SCIENCE ECO 20015 MANAGERIAL ECONOMICS II Candidates should attempt
More informationECO364  International Trade
ECO364  International Trade Chapter 2  Ricardo Christian Dippel University of Toronto Summer 2009 Christian Dippel (University of Toronto) ECO364  International Trade Summer 2009 1 / 73 : The Ricardian
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More informationTwoState Options. John Norstad. jnorstad@northwestern.edu http://www.norstad.org. January 12, 1999 Updated: November 3, 2011.
TwoState Options John Norstad jnorstad@northwestern.edu http://www.norstad.org January 12, 1999 Updated: November 3, 2011 Abstract How options are priced when the underlying asset has only two possible
More informationChapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The PreTax Position
Chapter 27: Taxation 27.1: Introduction We consider the effect of taxation on some good on the market for that good. We ask the questions: who pays the tax? what effect does it have on the equilibrium
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationECON 305 Tutorial 7 (Week 9)
H. K. Chen (SFU) ECON 305 Tutorial 7 (Week 9) July 2,3, 2014 1 / 24 ECON 305 Tutorial 7 (Week 9) Questions for today: Ch.9 Problems 15, 7, 11, 12 MC113 Tutorial slides will be posted Thursday after 10:30am,
More informationMultivariable Calculus and Optimization
Multivariable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multivariable Calculus and Optimization 1 / 51 EC2040 Topic 3  Multivariable Calculus
More informationFACTORING QUADRATIC EQUATIONS
FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods
More informationThe term marginal cost refers to the additional costs incurred in providing a unit of
Chapter 4 Solutions Question 4.1 A) Explain the following The term marginal cost refers to the additional costs incurred in providing a unit of product or service. The term contribution refers to the amount
More informationWhy is Insurance Good? An Example Jon Bakija, Williams College (Revised October 2013)
Why is Insurance Good? An Example Jon Bakija, Williams College (Revised October 2013) Introduction The United States government is, to a rough approximation, an insurance company with an army. 1 That is
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of EquationsGraphically and Algebraically Solving Systems  Substitution Method Solving Systems  Elimination Method Using Dimensional Graphs to Approximate
More informationLecture 1: The intertemporal approach to the current account
Lecture 1: The intertemporal approach to the current account Open economy macroeconomics, Fall 2006 Ida Wolden Bache August 22, 2006 Intertemporal trade and the current account What determines when countries
More information.4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.
Chapter 16. Risk and Uncertainty Part A 2009, Kwan Choi Expected Value X i = outcome i, p i = probability of X i EV = pix For instance, suppose a person has an idle fund, $100, for one month, and is considering
More informationThe Time Value of Money
The Time Value of Money This handout is an overview of the basic tools and concepts needed for this corporate nance course. Proofs and explanations are given in order to facilitate your understanding and
More informationOligopoly: How do firms behave when there are only a few competitors? These firms produce all or most of their industry s output.
Topic 8 Chapter 13 Oligopoly and Monopolistic Competition Econ 203 Topic 8 page 1 Oligopoly: How do firms behave when there are only a few competitors? These firms produce all or most of their industry
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationRolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
More informationInfinitely Repeated Games with Discounting Ù
Infinitely Repeated Games with Discounting Page 1 Infinitely Repeated Games with Discounting Ù Introduction 1 Discounting the future 2 Interpreting the discount factor 3 The average discounted payoff 4
More informationPrice Discrimination: Part 2. Sotiris Georganas
Price Discrimination: Part 2 Sotiris Georganas 1 More pricing techniques We will look at some further pricing techniques... 1. Nonlinear pricing (2nd degree price discrimination) 2. Bundling 2 Nonlinear
More informationWHAT IS CAPITAL BUDGETING?
WHAT IS CAPITAL BUDGETING? Capital budgeting is a required managerial tool. One duty of a financial manager is to choose investments with satisfactory cash flows and rates of return. Therefore, a financial
More information4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4
4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Nonlinear functional forms Regression
More informationPolynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005
Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationSlutsky Equation. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Slutsky Equation 1 / 15
Slutsky Equation M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Slutsky Equation 1 / 15 Effects of a Price Change: What happens when the price of a commodity decreases? 1 The
More informationThe Relation between Two Present Value Formulae
James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 6174. The Relation between Two Present Value Formulae James E. Ciecka,
More informationTable 1: Field Experiment Dependent Variable Probability of Donation (0 to 100)
Appendix Table 1: Field Experiment Dependent Variable Probability of Donation (0 to 100) In Text Logit Warm1 Warm2 Warm3 Warm5 Warm4 1 2 3 4 5 6 Match (M) 0.731 0.446 0.965 1.117 0.822 0.591 (0.829) (0.486)
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More information13 EXPENDITURE MULTIPLIERS: THE KEYNESIAN MODEL* Chapter. Key Concepts
Chapter 3 EXPENDITURE MULTIPLIERS: THE KEYNESIAN MODEL* Key Concepts Fixed Prices and Expenditure Plans In the very short run, firms do not change their prices and they sell the amount that is demanded.
More informationUsing clients rejection to build trust
Using clients rejection to build trust Yukfai Fong Department of Economics HKUST Ting Liu Department of Economics SUNY at Stony Brook University March, 2015 Preliminary draft. Please do not circulate.
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationChapter 6 CostVolumeProfit Relationships
Chapter 6 CostVolumeProfit Relationships Solutions to Questions 61 The contribution margin (CM) ratio is the ratio of the total contribution margin to total sales revenue. It can be used in a variety
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationFactoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
More informationUGBA 103 (Parlour, Spring 2015), Section 1. Raymond C. W. Leung
UGBA 103 (Parlour, Spring 2015), Section 1 Present Value, Compounding and Discounting Raymond C. W. Leung University of California, Berkeley Haas School of Business, Department of Finance Email: r_leung@haas.berkeley.edu
More informationAP Physics 1 and 2 Lab Investigations
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
More informationPascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question.
Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive
More informationNo Claim Bonus? by John D. Hey*
The Geneva Papers on Risk and Insurance, 10 (No 36, July 1985), 209228 No Claim Bonus? by John D. Hey* 1. Introduction No claim bonus schemes are a prominent feature of some insurance contracts, most
More informationChapter 3: Commodity Forwards and Futures
Chapter 3: Commodity Forwards and Futures In the previous chapter we study financial forward and futures contracts and we concluded that are all alike. Each commodity forward, however, has some unique
More informationLecture 6: Price discrimination II (Nonlinear Pricing)
Lecture 6: Price discrimination II (Nonlinear Pricing) EC 105. Industrial Organization. Fall 2011 Matt Shum HSS, California Institute of Technology November 14, 2012 EC 105. Industrial Organization. Fall
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationChapter 2 An Introduction to Forwards and Options
Chapter 2 An Introduction to Forwards and Options Question 2.1. The payoff diagram of the stock is just a graph of the stock price as a function of the stock price: In order to obtain the profit diagram
More informationManagerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture  13 Consumer Behaviour (Contd )
(Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture  13 Consumer Behaviour (Contd ) We will continue our discussion
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationLecture 12/13 Bond Pricing and the Term Structure of Interest Rates
1 Lecture 1/13 Bond Pricing and the Term Structure of Interest Rates Alexander K. Koch Department of Economics, Royal Holloway, University of London January 14 and 1, 008 In addition to learning the material
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitelyrepeated prisoner s dilemma
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationThe relationship between exchange rates, interest rates. In this lecture we will learn how exchange rates accommodate equilibrium in
The relationship between exchange rates, interest rates In this lecture we will learn how exchange rates accommodate equilibrium in financial markets. For this purpose we examine the relationship between
More information3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial
More informationConsumer Theory. The consumer s problem
Consumer Theory The consumer s problem 1 The Marginal Rate of Substitution (MRS) We define the MRS(x,y) as the absolute value of the slope of the line tangent to the indifference curve at point point (x,y).
More informationC(t) (1 + y) 4. t=1. For the 4 year bond considered above, assume that the price today is 900$. The yield to maturity will then be the y that solves
Economics 7344, Spring 2013 Bent E. Sørensen INTEREST RATE THEORY We will cover fixed income securities. The major categories of longterm fixed income securities are federal government bonds, corporate
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More informationIndifference Curves: An Example (pp. 6579) 2005 Pearson Education, Inc.
Indifference Curves: An Example (pp. 6579) Market Basket A B D E G H Units of Food 20 10 40 30 10 10 Units of Clothing 30 50 20 40 20 40 Chapter 3 1 Indifference Curves: An Example (pp. 6579) Graph the
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More informationNotes from Week 1: Algorithms for sequential prediction
CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 2226 Jan 2007 1 Introduction In this course we will be looking
More informationScreening by the Company You Keep: Joint Liability Lending and the Peer Selection Maitreesh Ghatak presented by Chi Wan
Screening by the Company You Keep: Joint Liability Lending and the Peer Selection Maitreesh Ghatak presented by Chi Wan 1. Introduction The paper looks at an economic environment where borrowers have some
More informationIntroduction (I) Present Value Concepts. Introduction (II) Introduction (III)
Introduction (I) Present Value Concepts Philip A. Viton February 19, 2014 Many projects lead to impacts that occur at different times. We will refer to those impacts as constituting an (inter)temporal
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationhttp://www.jstor.org This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions
A Significance Test for Time Series Analysis Author(s): W. Allen Wallis and Geoffrey H. Moore Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 36, No. 215 (Sep., 1941), pp.
More informationMBA Finance PartTime Present Value
MBA Finance PartTime Present Value Professor Hugues Pirotte Spéder Solvay Business School Université Libre de Bruxelles Fall 2002 1 1 Present Value Objectives for this session : 1. Introduce present value
More informationLecture Note 7: Revealed Preference and Consumer Welfare
Lecture Note 7: Revealed Preference and Consumer Welfare David Autor, Massachusetts Institute of Technology 14.03/14.003 Microeconomic Theory and Public Policy, Fall 2010 1 1 Revealed Preference and Consumer
More informationMathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationIntermediate Microeconomics (22014)
Intermediate Microeconomics (22014) I. Consumer Instructor: Marc TeignierBaqué First Semester, 2011 Outline Part I. Consumer 1. umer 1.1 Budget Constraints 1.2 Preferences 1.3 Utility Function 1.4 1.5
More information