Econ 4808 - Spring 2008 Review Set 1 - Answers ORY ELEMENTS OF LOGIC. NECESSARY AND SUFFICIENT. SET THE- 1. De ne a thing or action in words. Refer to this thing or action as A. Then de ne a condition that is necessary but not su cient for A to occur; explain why. Then, de ne a condition that is su cient but not necessary for A to occur; explain why. Now de ne a condition that is both necessary and su cient for A to occur; explain why. Some general comments: You needed to begin by "de ning the action or thing you would treat as A. De ne means you need to give it a name and then associate a de nition with that name. E.g. "A is a duck and I am de ning a duck as..." Just saying A is a duck is not enough. I can t tell whether your following answers are true until I know your de nition of a duck. Further note that the set of conditions that are necessary and su cient for something to be A is the de nition of A. I basically asked you to give me the de nition of A twice. Let me stress again, whether something is necessary but not su cient for A, or su cient but not necessary for A, completely depends on how A is de ned. For example, let A be a duck and I de ne a duck as a bird that can oat. I de ne B as a bird and I de ne C as a yellow bird that can oat. 1) B is necessary but not su cient for A: A ) B. Being a bird is necessary for being a duck (every duck is a bird), but being a bird is not su cient for being a duck (not every bird is a duck). 2) C is a su cient but not necessary for A: C ) A. Being a yellow bird that can oat is su cient for being a duck (every yellow bird that can oat is a duck), but being a yellow bird that can oat is not necessary for being a duck (a duck can be a di erent color). 3) Being a bird that can oat is necesary and su cient for being a duck: a bird that can oat, A 2. Consider a table: Identify a condition that is necessary but not su cient for something to be a table. Why is it necessary but not su cient. Being a piece of furniture with a at top is a necesary condition for something to be a table: table=>furniture with a at top. 1
However, not every furniture with a at top is a table and so it is not a su cient condition for something to be a table. 3. Explain why Donald Duck is an example of a duck using the terminology of necessary and su cient. Being Donald Duck is su cient to make one a duck, but it is not necessary (lots of ducks are not named Donald). Donald Duck is a subset of all ducks, so is an example of a duck. 4. Consider the following implications and decide: (i) if the implication is true, and (ii) if the converse implication is true: x > y 2 ) x > 0. x > y 2 and y 2 > 0, so by transitivity x > y 2 ) x > 0; that is, the rst implication is true. Now consider the converse x > 0 ) x > y 2, which is not always true. E.g. if x = 1 and y = 3. 5. Consider the di erent ways of expressing equivalent, necessary and su cient: (A, B), (A and B are equivalent), (A iff B), (B iff A) (A ) B), (A is su cient for B), (B if A) (B ) A), (A is necessary for B), (B only if A) if. Note that i stands for "if and only if". Also note the di erence between i, if, or only Now, ll in the the following blanks with either i, if, or only if. x 2 > 0::::::::::x > 0 x 2 < 9::::::::::x < 3 x(x 2 + 1) = 0::::::::::x = 0 x(x + 3) < 0::::::::::x > 3 Explain, in words, each of your four choices. 2
1) The correct statement is x 2 > 0 if x > 0, which is equivalent to x > 0 =) x 2 > 0 and also equivalent to: x > 0 is su cient but not necessary for x 2 > 0. It is not necessary because, for example, even if x = 4, it is still true that x 2 > 0. 2) The correct statement is x 2 < 9 only if x < 3, which is equivalent to x 2 < 9 ) x < 3 and also equivalent to: x < 3 is necessary, but not su cient for x 2 < 9. (Consider x = 3:1 and x = 5). Only if means that the condition after the only if is necessary for the condition before the only if. 3) Assuming that x is a real nunber, the correct statement is x(x 2 + 1) = 0 i x = 0, which is equivalent to x(x 2 + 1) = 0, x = 0, and aslo equivalent to: x(x 2 + 1) = 0 is necessary and su cie 4) The correct statement is fx(x + 3) < 0 only if x > 3g, which is equivalent to fx(x + 3) < 0 ) x > 3g and also equivalent to: fx > 3 is necessary but not su cient for x(x + 3) < 0g. (Consider x = 1 and x = 1). 6. Try to use three di erent methods to prove that x 2 + 4 > 0 ) jxj < 2 Direct proof : x 2 + 4 > 0, x 2 < 4, jxj < 2. Note that two things have been proved. The initial assertion x 2 + 4 > 0 ) jxj < 2, and jxj < 2 ) x 2 + 4 > 0. Note that x 2 < 4 ) x < 2 but that x < 2 ; x 2 < 4. Indirect proof: Let A x 2 + 4 > 0, and B jxj < 2. We have to show that notb ) nota. Assume notb, that is assume that jxj 2. Squaring notb gives jxj 2 4, x 2 4, x 2 + 4 0 nota. Proof by contradiction: Let A x 2 + 4 > 0, and B jxj < 2. Assume that A ; B, that is x 2 + 4 > 0 ; jxj < 2. Then there must exist an x such that jxj 2, which satis es A. Squaring both sides of jxj 2 gives jxj 2 4, x 2 4, x 2 + 4 0. But that contradicts A. Threfore it must be the case that A ) B. 7. State two methods of proving the statement A is a necessary condition for B. Show that nota ) notb, which is the same as B only if A, or show that B ) A, which is the same as B is su cient for A. 8. What is an indirect proof of A ) B. Provide an example of an indirect proof. This method of proof follows from 3
fa ) Bg, fnotb ) notag That is, one way to show that A implies B is to show that the absence of B implies the absence of A. Example: Prove that A 5x > 0 implies B x > 0. This is a very trivial example, in that it is obvious that A implies B. But let s work through the steps. Direct proof: want to directly manipulate A into B divide 5x > 0 by 5 to get x > 0 So, A ) B Indirect proof: show notb implies nota assume notb, that is, assume x 0 if x 0 then 5x > 0 is false So, notb ) nota Proof by contradiction: Start by assuming A ; B ( or A ) notb), that is, assume 5x > 0 ; x > 0 but x 0 ) 5x 0 which contradicts A 5x > 0. 9. Prove that x 2 > 2 ) x 6= 1 using an indirect proof. In this case A x 2 > 2 and B x 6= 1. x 2 = 1 2. So notb ) nota. So notb is x = 1. If x = 1, then 10. Suppose that A denotes x 2 +5x 4 > 0 and B denotes x > 0. Prove that A ) B using proof by contradiction. One proceeds by assuming that A ; B, and then showing that this contradicts A. So, if x 2 +5x 4 > 0 ; x > 0 there must be an x 0 for which x 2 +5x 4 > 0. However, the solution to (4 x) (x 1) > 0 is x 2 (1; 4) and so there can t be an x 0 for which x 2 + 5x 4 > 0. This contradicts our assumption. Therefore, A ) B. 11. Consider the statement: One is a good economist only if one hates math. With the di erent characterizations of the concepts of necessary and su cient in mind, write down a number of equivalent statements, rst in terms of G and H, then in terms of words, where G denotes a good economist and H denotes those who hate math. 4
(G only if H) () (G ) H) () (noth ) notg) () (H is necessary for G) () (G is su cient for H) All of these statements are equivalent. Given my de nition of G and H, (One is a good economist only if one hates math),( G only if H ). Therefore the statement is equivalent to all of the following statements "Being a good economist implies that one hates math, (G ) H)," "If you don t hate math, you are not a good economist, (noth ) notg)," "Hating math is a necessary condition to be a good economist, (H is necessary for G)," "Being a good economist is su cient to guarantee that one hates math, (G is su cient for H)," Whether the original statement is, or is not true, is immaterial to the question. The following is an attempt at a Venn diagram of the statement "one is a good economist only if one hates math". Good economists are a subset of economists (G E). Good 5
economists are also a subset of things that hate math (G H). By de nition, economists are a subset of the universal set (E ); everything is a subset of the universal set. Note some statements that do not follow from the statement that "one is a good economist only if one hates math". Hating math does not imply that one is a good economist or even an economist. Not being a good economist does not imply that you don t hate math. One can be something other than a good economist and hate math or not hate math (H ; E). That is, there are lots of math haters who are not economists. Not hating math does not imply that one is an economist (noth ; E). That is, there are lots of things out there that do not hate math who are not economists. Not being a good economist does not imply that you are a bad economist; you might be an OK economist (notg ; bade) unless, of course, one assumes that all economists are either good or bad; that is, mediocre economists are impossible. 12. De ne four sets: S 1 = fx 1 ; x 2 : p 1 x 1 + p 2 x 2 yg, S 2 = fx 1 ; x 2 : p 1 x 1 + p 2 x 2 < yg, S 3 = fx 1 ; x 2 : p 1 x 1 + p 2 x 2 = yg, and S 4 = fx 1 ; x 2 : p 1 x 1 + p 2 x 2 > yg. What if anything, is the relationship (in terms of subsets, unions, intersections, etc.) between these four sets? Do any of these sets de ne a function? What do we call these sets if x i is the quantity purchased of good i, i = 1; 2, p i is the price of good i, and y is the individual s income. Assuming that individuals always prefer more to less, what is the most restrictive thing that you can say about the bundle, x = (x 1 ; 2 ), that the individual will choose. To narrow it down further we would either have to place more restrictions on the individual s budget set, such as his mother makes him consume at least 3 units of x, or know something about her preferences. (S 1 = S 2 [ S 3 ) () (S 1 ns 2 = S 3 ) S 1 + S 4 = S 1 \ S 4 =Ø Set S 3 is the only set that de nes a function. Set S 1 is the feasible (or a ordable) budget set. Set S 3 is the budget line. Set S 4 is the set of bundles that the consumer cannot a ord. Assuming that individuals always prefer more to less, the most restrictive thing that you can say about the chosen bundle, x = (x 1 ; 2 ), is that it is on the budget line, set S 3. 13. A) Are the greatest economist among the mathematicians and the greatest mathematician among the economists one and the same person? 6
B) Is the oldest economist among the mathematicians and the oldest mathematician among the economists one and the same person? Answer in terms of set theory and demonstrate your answers with Venn diagrams. A) No. Let E denote the set of all economists. Let M denote the set of all mathematicians. The set of individuals that are both mathematicians and economists is EM = E \ M; that is, the intersection of the two sets. The greatest economist among the mathematicians, e, is a member of EM; that is e 2 EM. The greatest mathematician among the economists, m, is a member of EM; that is m 2 EM. But e and m are not necessarily the same person. For example, Sue might be the greatest economist in EM and Peggy might be the the greatest mathematician in EM. B) Yes. Everyone in EM is both an economist and a mathematician, so the oldest person in EM is both the oldest economist and the oldest mathematician. In terms of a Venn diagram EM is the intersection of the set of economists and the set of mathematicians, so consists of all of those individuals that are both economists and mathematicians. 14. Assume that neither set A nor set B are the universal set. Further assume that A and B have the following relationship, nota ) B. Does this imply that notb ) A? Why or why not? Yes. (nota ) B) () (notb =) not (nota)) () (notb =) A) WHAT IS A THEORY (MODEL)? CONSUMER THEORY. 15. De ne, in a few sentences, the concept of an economic theory. 16. A) The relationship between a temperature measured in degrees Celsius (or Centigrade) (C) and Fahrenheit (F ) is given by C = 5 9 (F 32). Find C when F is 32. Find F when C = 100. Find a general expression for F in terms of C. If it is 40 o F in Boulder and 80 o F in Miami is it correct to say that it is twice as hot in Miami? Write down the above problem and its solution using the structure of an economic model (de nitions, assumptions, predictions). 7
B) If one bundle of goods provides 20 utils and another bundle 10 utils, can one say that the individual likes the rst bundle twice as much as the second bundle? Why or why not? A) De nitions: C denotes temperature measured in degrees Celsius (or Centigrade) F denotes temperature measured in degrees Fahrenheit Assumptions: C = 5 9 (F 32) Predictions: Find C when F is 32. Plug 32 into C = 5 9 (F all know, is 0. 32) and solve for C. The answer, as we To answer the second question, rst solve C = 5 9 (F 32) for F. The answer is F = 32 + 9 5C. Plug 100 for C, then F = 212, the boiling point on the Fahrenheit scale. 40 and 80 on the Celsius scale are 4:4 and 26:7 on the Fahrenheit scale. Note that 4:4 2 6= 26:7. It is not correct to say that Miami is twice as hot. Twice as hot is a meaningless statement. Temperature is not a cardinal concept; that is we can look at temperature at two di erent points in time or two di erent places and determine which is hotter (an ordinal concept), but not how much hotter. B) If we assume the individual has only ordinal preferences, all we can say is that the bundle that provides 20 utils is preferred to the one that provides 10 utils. Alternatively, one could assume preferences have cardinal properties, but we typically don t do this because it adds nothing to the predictive power of consumer theory. 17. What is a utility function, what is its purpose, and how does it ful ll this purpose? A utility function assigns a number to every possible bundle of goods. Its goal is to represent an individual s preferences (ranking over the bundles of goods) in a compact way. If ful lls this goal if it assigns the same number to two bundles if the individual is indi erent between those two bundles, and a larger number to the bundle the individual prefers if the individual prefers one bundle over the other. 8
18. Consider George the consumer. George lives on a xed income, m, and only has so much time, t. De ne B m as all those bundles of goods that George can a ord to buy with his income. De ne B t as all those bundles of goods George has enough time to consume. Assume that B m \ B t is not empty. Further assume George has preferences such that he can compare any two bundles of goods and say which he prefers. Also assume George chooses the bundle he most prefers, b, from those he has the time and money to consume. Part 1: Is the most-preferred bundle from those he can a ord to buy, b m, necessarily the same bundle as the most-preferred bundle from among those he has time to consume, b t? Part 2: Is the chosen bundle, b, always either b m or b t? Part 3: Will George necessarily spend all of his time and money? Part 4: Would your answer to Part 2 be di erent if one adds the assumption that George has enough time to consume b m? Answer each of the rst three parts of the question, 1-3, either Yes or No. Explain each of those answers in words and using a diagram. You can probably use the same diagram for all three questions. Now answer Part 4 in words and in terms of your diagram. Your diagram is a Venn diagram in the sense that it will show the intersection of two sets. However, in this case the shape and dimensions of these two sets have meaning which you should incorporate. Think a world of two goods and what budget sets look like. Part1: Draw the two sets with an non-empty intersection. Note that no one assumed that George had enough time to consume b m or enough money to purchase b t, so neither one has to be in the intersection of the two sets. Place them in the diagram such that neither is in the intersection. This proves that b m and b t are not always the same bundle. So, the answer to part 1 is no. Part 2: The bundle that George consumes, b, must be in the intersection of the two sets. Since, in my example neither b m nor b t are in the intersection, then in my example neither b m nor b t is b. So, one of them does not have to be the consumed bundle and the answer to part 2 is no. Part 3: There is no reason his most preferred bundle among those he can a ord has to use up all of his time and money. It could be, but this is pretty unlikely. Part 4: Yes, it would change the answer because now we have added an assumption that puts b m in the intersection of the two sets (in the bundles he has time and money to consume). So, now b m = b. Note that it could be the case that b t 6= b. The rst step in answering this question is to understand the notation (language). If you don t understand the concepts and notation, all is lost. B is a set of bundles of goods and b is a particular bundle. For example, B m is the set of all bundles of goods that George can a ord, b m is a bundle in B m, and b m is a speci c bundle of goods; it is the one he likes the best from those he can a ord to buy. Do you understand the distinction between a 9
bundle of goods and a set of bundles of goods? b t is the bundle George would choose if money was not a constraint. E.g., a weekend in Paris at the Ritz with lots of good food, wine and shopping. b m is the bundle George would choose if time was not a constraint. E.g. an around-the-world backpacking trip, eating lots of rice. There is no reason they are the same. Consider the following example. I have 10 hours of time and $100. There are two goods, x 1 and x 2. Assume good 1 is very expensive in terms of money, but takes little time to consume. Assume the opposite for good 2. Ignoring, the money constraint, I would choose a bundle with more good 1 than good 2. Ignoring the time constraint, I would choose a bundle with more good 2 than 1. Neither of these bundles is necessarily my most preferred bundle from the set of bundles I can a ord in terms of both time and money. 19. Consider Wilber. Wilber has preferences over bundles of goods. Let X 0 denote the set of bundles that Wilber can currently a ord. Further assume that Wilber always chooses the bundle that he likes the best from those he can a ord. Call the bundle that he chooses given the constraint X 0 ; x 0. Now things change such that the set of bundles he can a ord is X 1 where X 0 X 1. Convince the reader, in words, that this change in the constraint, cannot make Wilbur worse o. Since the original set of a ordable bundles is a subset of the new set of a ordable bundles, after the change the individual can still a ord his best bundle given X 0. 10
That is, he can duplicate how well he did in the original case. do at least as well as in the old case. So, in the new case he will WHAT IS A FUNCTION? 20. De ne the term mathematical function in one or two sentences. Provide a speci c example of a mathematical function. Is a circle in two-dimensional space a function? Why or why not? 21. Describe, in words, what the following expression means: f(x; y) : x + y = 0g : Graph this relationship in (x; y) space with x on the horizontal axis and y on the vertical axis. Numerically identify at least one point on your graph. Does this relationship identify a function? Do the same for : f(x; y) : y xg: f(x; y) : x + y = 0g identi es a straight line through the origin with a slope of negative one. Yes, this set does identify a function. That is, ff(x; y) : x + y = 0g () y = x In contrast, f(x; y) : y xg does not identify y as a function of x. In terms of a graph of the second set, it is all the points on and below a straight line through the origin with a slope of positive one. 22. Consider the function y = f(x), where x is a scalar. What is a function and what does this function do? The function f(x) associates with each value of the variable x that is in the domain of the function a unique value of the variable y. Note that y = f(x) does not necessarily associate a unique x with each y. 11