Importance of Mental Toughness in Sport

Importance of Mental Toughness in Sport Mental Toughness: What is the Real Difference Between Winning and Losing? When it comes to being successful, you can never find a substitute for hard work. In the corporate world, you need to learn the ropes of the business and spend countless hours on reading books and attending seminars on self-improvement. In sports, you have to train consistently in order to reach your full potential. But what if during an important business meeting, you suddenly had a mental block and anxiety followed? As a result, you forgot to mention an important point and because of it, the business deal fell apart. Same thing can happen in a Jiu-jitsu practice session where a sudden lapse in judgment could lead to a reversal, and in a matter of seconds, you were the one tapping out. How come we can be so prepared in doing something and then lose just like that? UFC 117: Anderson Silva Vs. Chael Sonnen Sonnen dominated the first four rounds of this championship match by using his wrestling skill. In round 5, Sonnen went for another takedown. Almost two minutes is left, and the audience is in awe. A new champion will be crowned, the long-time reigning champion will be defeated, the one considered as the GOAT (Greatest of All Time) will lose— the unthinkable will happen! Sonnen is on the guard again and continued with the ground and pound to add more damage to his opponent. Silva threw a few punches from the bottom and Sonnen lifted his left arm to block. In the process, Silva quickly positioned his leg behind Sonnen’s neck and a leg triangle was in place. Silva secured the hold and one of the greatest comeback fights in sports happened†¦.Sonnen tapped out! He was only 1 minute and 50 seconds from being a champion. A lot of professional fights end up this way; one lapse in judgment can make or break a career. Some fighters learned from it but some of them never recovered. Concentration makes you apply everything that you have learned in training. Success starts in the mind but unfortunately, not everyone can sustain that concentration during a fight. When two combatants face off, you can take advantage of your skills and concentrate on executing it. However, if both fighters have a similar skill set and conditioning, the only deciding factor here is how mentally tough they are. Who will break first? Who will make the first mistake? That’s when you need to be mentally tough more than physically tough. Relaxation and Concentration Mental toughness encompasses not just fighting sport but all competitive sports and other areas in our everyday life, including our job. Several competitors can be so good in practice but underachieves in the actual competition when the pressure of the game starts to mount up. Lebron James was called â€Å"Lechoke† before, and it only stopped when he won an NBA championship not just once but twice. Now, people in the media and fans think that he can surpass Michael Jordan and be the GOAT in basketball. Concentration is the ability to have that laser-like focus on what’s important and to let go of all distractions. However, in order to concentrate, you should be relaxed. Relaxation is defined as the state of being calm despite pressure. Take note that relaxation and concentration are interacting mental skills—they work hand in hand. You cannot be mentally tough if you don’t have both. Float like a butterfly (be relaxed) and sting like a bee (concentrate and hit the target). That’s how Muhammad Ali became the GOAT in boxing! Keep in mind that being relaxed means that you are excited in a good way. If you are totally relaxed, you will fall flat during the competition. You need to be pumped up and ready for action. Like physical conditioning and learning skills, being relaxed also takes practice—the more time you invest in it, the more it will develop. During boxing, if you are relaxed, you can see the incoming punches and you will be able to dodge them easier. If you are nervous and under pressure, you are more likely to close your eyes as the opponent’s gloves hit towards your face. If you are too anxious, your body will tighten up and you won’t be able to execute all of the skills that you have learned during practice. That is the reason why fighters have their entrance music as they walk their way towards the ring or the octagon—it makes them feel relaxed. Another way to relax is to embrace and accept nervousness. It’s very normal and once you accept the reality, then you will be calmer. Pre-competition rituals such as mumbling repeated words and doing rhythmic movements that could take the fighter’s focus away from distractions are also of great help. When it comes to defense, breathing plays a very important role. You can be the best striker but without oxygen, you will be like a race car without a fuel. Thus, deep, slow breathing is needed before and during the competition. Throwing combination punches and kicks or going for a reversal during a jiu-jitsu match requires a lot of air in your oxygen tank. If you don’t breathe properly, you won’t be able to recover and soon, your legs will feel heavy, your arms will start falling, and it’s your turn to get punched in the face. Laser-Like Concentration A laser pen doesn’t dance all over the place when you focus it on an object; it will stay there as long as you allow it to. Concentration is all about focusing on the present situation. Recalling failures in the past or worrying about the future will not help. Do what you need to do at that very moment. Before you drink water, you don’t try to determine where it came from or think whether you will sweat it out or urinate afterwards. Be honest. You just drink it, right? Likewise, when you are competing, you can only control the present. So, focus on what needs to be done right now and focus on it like a laser! Don’t Drift Away Losing your focus can happen during a match but it’s up to you if you want to drift away or return to focus. A bad call from the referee, trash talk, boos from the crowd, intimidation by your opponent and a lot of other stuff like these can break your concentration. However, remember to compete the way you have been trained. A secured submission can slip by if one of the basic jiu-jitsu techniques is not well-executed. A sure win can turn to a loss, so the focus should be within YOU – you may drift away but return as quickly as you can. Conclusion All your hard work will become futile if you are not mentally tough. Remember that a sharp mind can cut deeper! So, the pain, the struggle, the exhaustion, the thought of giving up— all of these will take your game or career to the next level!

Inquiry Based Science Instruction Using Science Fair Projects Essay

In order to fully develop critical thinking skills necessary to function in the world of science, and the world in general, science teachers must incorporate more robust authentic inquiry activities, as part of their overall science curriculum. However, in my experience as a science teacher (18 years) and a science program director (3 years) in inner-city high schools in Boston, students do not have the opportunity to engage in robust authentic inquiry activities. Using my years’ experience in the public education system, it is clear to me that factors, such as teachers’ inadequate content knowledge, lack of motivation to implement inquiry, pressure to teach to high stake standardized tests, lack of experience conducting scientific research, and belief systems, hamper their ability and willingness to implement robust inquiry activities as part of their teaching. As a result, students suffer by not only missing the opportunity to fully engage in authentic inquiry, but also become unmotivated to engage in scientific research and or pursue scientific careers. The most effective means by which to overcome this authentic inquiry deficit is through the implementation of science fair projects, which go beyond traditional â€Å"cookbook lab† teaching methodologies. And while I am fully cognizant of the fact that my experience in this matter is biased based upon the workings of the Boston inner-city school system only, academics, such as Thienhuong Hoang, agree that science fair allow students to conduct scientific experiments thereby engaging them in inquiry process, rather than only allowing them to learn concepts. 2010) Inquiry and the Science Fair Project For decades, inquiry has been the focus of science teaching. The National Research Council (NRC), for example, refers to inquiry as the central strategy for teaching science‚ (1996), and defines inquiry as â€Å"the diverse way in which scientists study the natural world and propose explanations based on evidence derived from their work. † (1996) In addition, NRC notes that inquiry seeks to create opportunities for learners to engage in science and to build an in-depth understanding based on their previous ideas and experiences. Further, NRC recommends that teachers engage students in inquiry asking scientific questions, using evidence in responding to questions, formulating explanations from evidence, connecting explanations to scientific knowledge, and communicating and justifying findings. (2000) I fully concur with the position of NRC regarding inquiry, and have, throughout both my prior teaching career and current position as a science program director, included and required teachers to include inquiry – and particularly science fair – as the foundation of their teaching. As a result, students have achieved deeper content knowledge and broader understanding of scientific processes. Inquiry through science fair projects, rather than traditional â€Å"cookbook lab† methods of science teaching and learning, in which students follow a predetermined protocol and the results of the experiment are known in advance, allow students to more fully engage in critical thinking and learning. Further, only inquiry – and specifically science fair – helps students better develop and understand the need to engage in probing questions to scientific problems. This, in turn, affords students increased opportunities to propose and develop their own methodologies, and increases their scientific literacy skills. Thus, science fair projects are a key link between science education and the work of scientists in real life. An aspect of scientists’ work that I believe is crucial for students to acquire is the significance of ethics and conclusions based on data collected through scientific research, due to the potential implication of the scientist’s work on society. For example, the racial classification in the early 18th century of the superiority and inferiority of races, which became part of the Western racial ideology, was the result of scientific research involving different human characteristics, which was later determined to be falsified. (Carlton, 2008) Similarly, western gender stereotypes have biased historical research of gender studies in the south (Fennell and Arnot, 2008); thereby rendering such research limited in its historical significance and material use. Thus, exposing students to authentic scientific inquiry via science fair projects is not only necessary to the student’s academic success, it is, in my opinion, a vital component of the development of critical thinking skills necessary to separate authentic scientific data and research results from potentially questionable results; like the superiority/inferiority of different races and other biases. Teacher Beliefs and Experience  Teachers’ actions in the classroom tend to reflect their own belief systems. This holds true both in my own experience working in the field of education, and other academics. For as Wallace and Kang note in their article â€Å"An Investigation of Experienced Secondary Science Teachers’ Beliefs About Inquiry: An Examination of Competing Beliefs Sets†, â€Å"what a teacher actually does in the classroom is representative of her beliefs. (2004) Additionally, in conducting research for their article â€Å"White Male Teachers on Differences: Narratives of Contact and Tensions†, Jupp and Slattery note the comments of one of the participants in the research project as stating â€Å"†¦teachers are certainly biased against certain children, especially if they don’t know them, and often are surprised at the amount of intelligence and cultural awareness of minority children who have been raised in quote-unquote the right way, you know, the middle class family type of situation†¦Ã¢â‚¬  (2010) A teacher holding this ideology, for example, is likely to engage students in an inferior level of inquiry because the teacher’s assumption is that such students are incapable of critical thinking. Jupp and Slattery also noted in the interview of participants (Ibid), â€Å"†¦you can see that people are going out of their way to be sure that students don’t feel divorced from the curriculum†¦Ã¢â‚¬  This translates to teachers’ beliefs that if the emphasis is placed on inquiry – and specifically science fair projects, which requires a great deal of time – it will be at the expense of teaching for high stake standardized tests. Finally, Wallace and Kang, (2004) in their article â€Å"An Investigation of Experienced Secondary Science Teachers’ Beliefs About Inquiry: An Examination of Competing Beliefs Sets†, note that the teacher’s knowledge, or lack thereof, dictates his or her actions in the classroom. 2004) Therefore, it is clear to me – both in my own experiences in the field of education and based upon the literature – the possession of strong content knowledge, as well as research and science teaching e xperience, is essential to the effective teaching of inquiry, since a teacher that possesses this necessary skill set is better equipped to help students develop their ideas through probing at a deeper level. The use of science fair projects, therefore, is a key bridge in this knowledge gap, since science fair allows students the ability to conduct their own scientific research and develop their own scientific hypotheses. Barriers to Inquiry Studies show that there exist many barriers affecting the implementation of inquiry. For example, Trautmann, MaKinster, and Avery, in their article â€Å"What Makes Inquiry so Hard (And Why is it Worth it? )†, cite that the main reason teachers are reluctant to implement inquiry stems from the need to be efficient. 2004) Teachers feel they have to choose between teaching facts which students later regurgitate on standardized tests, and teaching with a focus on in-depth learning, which is more effectively attained through inquiry – and specifically through science fair projects. Trautmann, MaKinster and Avery also noted that the pressure placed upon teachers and schools to prepare students for high stake standardized tests impedes the implementation of inquiry in science classes. (Ibid) As a result, science teachers feel the need to maintain control of their classroom, as a method of controlling the curriculum required for standardized tests, rather than allow students to work independently on science fair projects. Another obstacle hampering the implementation of authentic scientific inquiry teaching – and specifically the implementation of science fair projects – is that teachers have a tendency to employ the same teaching strategies as the ones they experienced as students. (Davis, 2003) As such, they resist change and miss the opportunity to be innovative in their teaching by incorporating inquiry into their curriculum – especially with respect to science fair projects. Further, teacher training, in general, does not offer an effective model for the practice of science instruction based on inquiry. As a consequence, teachers are limited in their ability to carry out authentic inquiry, in general, in their classrooms. This inability to teach via authentic inquiry prohibits them from adopting science fair projects as part of their curriculum.

Monopoly and marginal cost Essay

Practice Questions and Answers from Lesson III-3: Monopoly Practice Questions and Answers from Lesson III-3: Monopoly The following questions practice these skills: ? Explain the sources of market power. ? Apply the quantity and price affects on revenue of any movement along a demand curve. ? Find the profit maximizing quantity and price of a single-price monopolist. ? Compute deadweight loss from a single-price monopolist. ? Compute marginal revenue. ? Define the efficiency of P = MC. ? Find the profit-maximizing quantity and price of a perfect-price-discriminating monopolist. ? Find the profit-maximizing quantity and price of an imperfect-price-discriminating monopolist. Question: Each of the following firms possesses market power. Explain its source. a. Merck, the producer of the patented cholesterol-lowering drug Zetia b. Chiquita, a supplier of bananas and owner of most banana plantations c. The Walt Disney Company, the creators of Mickey Mouse Answer to Question: a. Merck has a patent for Zetia. This is an example of a government-created barrier to entry, which gives Merck market power. b. Chiquita controls most banana plantations. Control over a scarce resource gives Chiquita market power. c. The Walt Disney Company has the copyright over animations featuring Mickey Mouse. This Is another example of a government-created barrier to entry that gives the Walt Disney Company market power. Question: Skyscraper City has a subway system, for which a one-way fare is $1. 50. There is pressure on the mayor to reduce the fee by one-third, to $1. 00. The mayor is dismayed, thinking that this will mean Skyscraper City is losing one-third of its revenue from sales of subway tickets. The mayor’s economic adviser reminds her that she is focusing only on the price effect and ignoring the quantity effect. Explain why the mayor’s estimate of a one-third loss of revenue is likely to be an overestimate. Illustrate with a diagram. Answer to Question: A reduction in fares from $1. 50 to $1. 00 will reduce the revenue on each ticket that is currently sold by one-third; this Is the price effect. But a reduction in price will lead to more tickets being sold at the lower price of $1. 00, which creates additional revenue; this is the quantity effect. The price effect is the loss of revenue on all the currently sold tickets. The quantity effect is the increase in revenue from increased sales as a result of the lower price. Question: Consider an industry with the demand curve (D) and marginal cost curve (MC) shown in the accompanying diagram. There is no fixed cost. If the industry is a single-price monopoly, the monopolist’s marginal revenue curve would be MR. Answer the following questions by naming the appropriate points or areas. Practice Questions and Answers from Lesson III-3: Monopoly a. If the industry is perfectly competitive, what will be the total quantity produced? At what price? b. Which area reflects consumer surplus under perfect competition? c. If the industry is a single-price monopoly, what quantity will the monopolist produce? Which price will it charge? d. Which area reflects the single-price monopolist’s profit? e. Which area reflects consumer surplus under single-price monopoly? f. Which area reflects the deadweight loss to society from single-price monopoly? g. If the monopolist can price-discriminate perfectly, what quantity will the perfectly price-discriminating monopolist produce? Answer to Question: a. In a perfectly competitive industry, each firm maximizes profit by producing the quantity at which price equals marginal cost. That is, all firms together produce a quantity S, corresponding to point R, where the marginal cost curve crosses the demand curve. Price will be equal to marginal cost, E. b. Consumer surplus is the area under the demand curve and above price. In part a, we saw that the perfectly competitive price is E. Consumer surplus in perfect competition is therefore the triangle ARE. c. A single-price monopolist produces the quantity at which marginal cost equals marginal revenue, that is, quantity I. Accordingly, the monopolist charges price B, the highest price it can charge if it wants to sell quantity I. d. The single-price monopolist’s profit per unit is the difference between price and the average total cost. Since there is no fixed cost and the marginal cost is constant (each unit costs the same to produce), the marginal cost is the same as the average total cost. That is, profit per unit is the distance BE. Since the monopolist sells I units, its profit is BE times I, or the rectangle BEHF. e. Consumer surplus is the area under the demand curve and above the price. In part c, we saw that the monopoly price is B. Consumer surplus in monopoly is therefore the triangle AFB. f. Deadweight loss is the surplus that would have been available (either to consumers or producers) under perfect competition but that is lost when there is a single-price monopolist. It is the triangle FRH. g. If a monopolist can price-discriminate perfectly, it will sell the first unit at price A, the second unit at a slightly lower price, and so forth. That is, it will extract from each consumer just that consumer’s willingness to pay, as indicated by the demand curve. It will sell S units, because for the last unit, it can just make a consumer pay a price of E (equal to its marginal cost), and that just covers its marginal cost of producing that last unit. For any further units, it could not make any consumer pay more than its marginal cost, and it therefore stops selling units at quantity S. Practice Questions and Answers from Lesson III-3: Monopoly Question: Bob, Bill, Ben, and Brad Baxter have just made a documentary movie about their basketball team. They are thinking about making the movie available for download on the Internet, and they can act as a single-price monopolist if they choose to. Each time the movie is downloaded, their Internet service provider charges them a fee of $4. The Baxter brothers are arguing about which price to charge customers per download. The accompanying table shows the demand schedule for their film. Price of download Quantity of downloads demanded $10 0 $8 1 $6 3 $4 6 $2 10 $0 15 a. Calculate the total revenue and the marginal revenue per download. b. Bob is proud of the film and wants as many people as possible to download it. Which price would he choose? How many downloads would be sold? c. Bill wants as much total revenue as possible. Which price would he choose? How many downloads would be sold? d. Ben wants to maximize profit. Which price would he choose? How many downloads would be sold? e. Brad wants to charge the efficient price. Which price would he choose? How many downloads would be sold? Answer to Question: a. The accompanying table calculates total revenue (TR) and marginal revenue (MR). Recall that marginal revenue is the additional revenue per unit of output Price of download Quantity of downloads TR MR demanded $10 0 $0 $8 1 $8 $8 $6 3 $18 $5 $4 6 $24 $2 $2 10 $20 $-1 $0 15 $0 $-4 b. Bob would charge $0. At that price, there would be 15 downloads, the largest quantity they can sell. c. Bill would charge $4. At that price, total revenue is greatest ($24). At that price, there would be 6 downloads. d. Ben would charge $6. At that price, there would be 3 downloads. For any more downloads, marginal revenue would be below marginal cost, and so further downloads would lose the Baxters’ money.e. Brad would charge $4. A price equal to marginal cost is efficient. At that price, there would be 6 downloads. Practice Questions and Answers from Lesson III-3: Monopoly Question: Suppose that De Beers is a single-price monopolist in the market for diamonds. De Beers has five potential customers: Raquel, Jackie, Joan, Mia, and Sophia. Each of these customers will buy at most one diamond—and only if the price is just equal to, or lower than, her willingness to pay. Raquel’s willingness to pay is $400; Jackie’s, $300; Joan’s, $200; Mia’s, $100; and Sophia’s, $0. De Beers’s marginal cost per diamond is $100. This leads to the demand schedule for diamonds shown in the accompanying table. Price of Diamond Quantity of Diamonds Demanded $500 0 $400 1 $300 2 $200 3 $100 4 $0 5 a. Calculate De Beers’s total revenue and its marginal revenue. From your calculation, draw the demand curve and the marginal revenue curve. b. Explain why De Beers faces a downward-sloping demand curve. c. Explain why the marginal revenue from an additional diamond sale is less than the price of the diamond. d. Suppose De Beers currently charges $200 for its diamonds. If it lowers the price to $100, how large is the price effect? How large is the quantity effect? e. Add the marginal cost curve to your diagram from part a and determine which quantity maximizes De Beers’s profit and which price De Beers will charge. Answer to Question: a. Total revenue (TR) and marginal revenue (MR) are given in the accompanying table. Price of Diamond Quantity of Diamonds TR Demanded $500 0 $0 $400 1 $400 $300 2 $600 $200 3 $600 $100 4 $400 $0 5 $0 MR $400 $200 $0 -$200 -$400 The accompanying diagram illustrates De Beers’s demand curve and marginal revenue (MR) curve. b. De Beers is the only producer of diamonds, so its demand curve is the market demand curve. And the market demand curve slopes downward: the lower the price, the more customers will buy diamonds. c. If De Beers lowers the price sufficiently to sell one more diamond, it earns extra revenue equal to the Practice Questions and Answers from Lesson III-3: Monopoly price of that one extra diamond. This is the quantity effect of lowering the price. But there is also a price effect: lowering the price means that De Beers also has to lower the price on all other diamonds, and that lowers its revenue. So the marginal revenue of selling an additional diamond is less than the price at which the additional diamond can be sold. d. If the price is $200, then De Beers sells to Raquel, Jackie, and Joan. If it lowers the price to $100, it will also sell a diamond to Mia. The price effect is that De Beers loses $100 (the amount by which it lowered the price) each from selling to Raquel, Jackie, and Joan. So the price effect lowers De Beers’s revenue by 3 ? $100 = $300. The quantity effect is that De Beers sells one more diamond (to Mia), at $100. So the quantity effect is to raise De Beers’s revenue by $100. e. The marginal cost (MC) curve is constant at $100, as shown in the diagram. Marginal revenue equals marginal cost at a quantity of 2 diamonds. So De Beers will sell 2 diamonds at a price of $300 each. Question: Use the demand schedule for diamonds given in the previous question. The marginal cost of producing diamonds is constant at $100. There is no fixed cost. a. If De Beers charges the monopoly price, how large is the individual consumer surplus that each buyer experiences? Calculate total consumer surplus by summing the individual consumer surpluses. How large is producer surplus? Suppose that upstart Russian and Asian producers enter the market and the market becomes perfectly competitive. b. What is the perfectly competitive price? What quantity will be sold in this perfectly competitive market? c. At the competitive price and quantity, how large is the consumer surplus that each buyer experiences? How large is total consumer surplus? How large is producer surplus? d. Compare your answer to part c to your answer to part a. How large is the deadweight loss associated with monopoly in this case? Answer to Question: a. The monopoly price is $300. At that price Raquel and Jackie buy diamonds. Raquel’s consumer surplus is $400 ? $300 = $100; Jackie’s is $300 ? $300 = $0. So total consumer surplus is $100 + $0 = $100. Producer surplus is $300 ? $100 = $200 for each diamond sold; 2 ? $200 = $400. b. In a perfectly competitive market, P = MC. That is, the perfectly competitive price is $100, and at that price 4 diamonds will be sold—to Raquel, Jackie, Joan, and Mia. c. At the competitive price, Raquel’s consumer surplus is $400 ? $100 = $300; Jackie’s, $300 ? $100 = $200; Joan’s, $200 ? $100 = $100; and Mia’s, $100 ? $100 = $0. So total consumer surplus is $300 + $200 + $100 + $0 = $600. Since the price is equal to marginal cost, there is no producer surplus. d. Under perfect competition, the sum of consumer and producer surplus is $600 + $0 = $600. Under monopoly, the sum of consumer and producer surplus is $100 + $400 = $500. So the loss of surplus to society from monopoly—the deadweight loss—is $600 ? $500 = $100. Question: Use the demand schedule for diamonds given in the previous questions. De Beers is a monopolist, but it can now price-discriminate perfectly among all five of its potential customers. De Beers’s marginal cost is constant at $100. There is no fixed cost. a. If De Beers can price-discriminate perfectly, to which customers will it sell diamonds and at what prices? b. How large is each individual consumer surplus? How large is total consumer surplus? Calculate producer surplus by summing the producer surplus generated by each sale. Practice Questions and Answers from Lesson III-3: Monopoly Answer to Question: a. If De Beers can price-discriminate perfectly, it will charge each customer that customer’s willingness to pay. That is, it will charge Raquel $400, Jackie $300, Joan $200, and Mia $100. De Beers does not want to sell to Sophia since she will only buy at a price of $0, and that would be below De Beers’s marginal cost. b. Since each consumer is charged exactly her willingness to pay, there is no consumer surplus. De Beers’s producer surplus is $400 ? $100 = $300 from selling to Raquel; $300 ? $100 = $200 from selling to Jackie; $200 ? $100 = $100 from selling to Joan; $100 ? $100 = $0 from selling to Mia. So producer surplus is $300 + $200 + $100 + $0 = $600. Question: Download Records decides to release an album by the group Mary and the Little Lamb. It produces the album with no fixed cost, but the total cost of downloading an album to a CD and paying Mary her royalty is $6 per album. Download Records can act as a single-price monopolist. Its marketing division finds that the demand schedule for the album is as shown in the accompanying table. Price of album Quantity of albums demanded $22 0 $20 1,000 $18 2,000 $16 3,000 $14 4,000 $12 5,000 $10 6,000 $8 7,000 a. Calculate the total revenue and the marginal revenue per album. b. The marginal cost of producing each album is constant at $6. To maximize profit, what level of output should Download Records choose, and which price should it charge for each album? c. Mary renegotiates her contract and now needs to be paid a higher royalty per album. So the marginal cost rises to be constant at $14. To maximize profit, what level of output should Download Records now choose, and which price should it charge for each album? Answer to Question: a. Total revenue (TR) and marginal revenue per album (MR) is shown in the following table: Price of album Quantity of albums TR MR demanded $22 0 $0 $20 1,000 $20,000 $20 $18 2,000 $36,000 $16 $16 3,000 $48,000 $12 $14 4,000 $56,000 $8 $12 5,000 $60,000 $4 $10 6,000 $60,000 $0 $8 7,000 $56,000 -$4 b. If the marginal cost of each album is $6, Download Records will maximize profit by producing 4,000 albums, since for each album up to 4,000, marginal revenue is greater than marginal cost. For any further albums, marginal cost would exceed marginal revenue. Producing 4,000 albums, Download Records will charge $14 for each album. c. If the marginal cost of each album is $14, Download Records will maximize profit by producing 2,000 albums, and it will charge $18 per album. Practice Questions and Answers from Lesson III-3: Monopoly Question: The movie theater in Collegetown serves two kinds of customers: students and professors. There are 900 students and 100 professors in Collegetown. Each student’s willingness to pay for a movie ticket is $5. Each professor’s willingness to pay for a movie ticket is $10. Each will buy at most one ticket. The movie theater’s marginal cost per ticket is constant at $3, and there is no fixed cost. a. Suppose the movie theater cannot price-discriminate and needs to charge both students and professors the same price per ticket. If the movie theater charges $5, who will buy tickets and what will the movie theater’s profit be? How large is consumer surplus? b. If the movie theater charges $10, who will buy movie tickets and what will the movie theater’s profit be? How large is consumer surplus? c. Now suppose that, if it chooses to, the movie theater can price-discriminate between students and professors by requiring students to show their student ID. If the movie theater charges students $5 and professors $10, how much profit will the movie theater make? How large is consumer surplus? Answer to Question: a. If the movie theater charges $5 per ticket, both students and professors will buy tickets. The movie theater will sell to 1,000 customers (students and professors), at a price of $5 each. Since the movie theater’s cost per ticket is $3, its profit is $2 per ticket for a total profit of 1,000 ? $2 = $2,000. Students will experience no consumer surplus, but each of the 100 professors will experience consumer surplus of $10 ? $5 = $5 for a total consumer surplus of 100 ? $5 = $500. b. If the movie theater charges $10 per ticket, only professors will buy tickets. The movie theater will sell to 100 customers (professors) at a price of $10 each. Since the movie theater’s cost per ticket is $3, its profit is $7 per ticket for a total profit of 100 ? $7 = $700. Students experience no consumer surplus since they do not buy any tickets. Each of the 100 professors experiences no consumer surplus since the price is equal to their willingness to pay. So consumer surplus is $0. c. If the movie theater charges students a price of $5, it sells 900 tickets at a profit of $5 ? $3 =$2 each for a profit from selling to students of 900 ? $2 =$1,800. Charging professors $10, it sells 100 tickets at a profit of $10 ? $3 =$7 each for a profit from selling to professors of 100 ? $7 =$700. So the theater’s total profit is $1,800 + $700 =$2,500. Since each customer is charged exactly his or her willingness to pay, there is no consumer surplus. Question: A monopolist knows that in order to expand the quantity of output it produces from 8 to 9 units that it must lower the price of its output from $2 to $1. Calculate the quantity effect and the price effect. Use these results to calculate the monopolist’s marginal revenue of producing the 9th unit. The marginal cost of producing the 9th unit is positive. Is it a good idea for the monopolist to produce the 9th unit? Answer to Question: The quantity effect is $1 (the increase in total revenue from selling the 9th unit at $1). The price effect is 8 ? (? $1) =? $8 (the decrease in total revenue from having to lower the price of 8 units by $1 each). So the marginal revenue of producing the 9th unit is $1 ? $8 =? $7. Since marginal revenue is negative, producing the 9th unit is definitely not a good idea: it lowers revenue (since marginal revenue is negative), and it increases the total cost (since marginal cost is positive). So it will definitely lower profit. Instead, the monopolist should produce less output.

Conditions for Using a Binomial Distribution

Binomial probability distributions are useful in a number of settings. It is important to know when this type of distribution should be used. We will examine all of the conditions that are necessary in order to use a binomial distribution. The basic features that we must have are for a total of n independent trials are conducted and we want to find out the probability of r successes, where each success has probability p of occurring. There are several things stated and implied in this brief description. The definition boils down to these four conditions: Fixed number of trialsIndependent trialsTwo different classificationsThe probability of success stays the same for all trials All of these must be present in the process under investigation in order to use the binomial probability formula or tables. A brief description of each of these follows. Fixed Trials The process being investigated must have a clearly defined number of trials that do not vary. We cannot alter this number midway through our analysis. Each trial must be performed the same way as all of the others, although the outcomes may vary. The number of trials is indicated by an n in the formula. An example of having fixed trials for a process would involve studying the outcomes from rolling a die ten times.  Here each roll of the die is a trial. The total number of times that each trial is conducted is defined from the outset. Independent Trials Each of the trials has to be independent. Each trial should have absolutely no effect on any of the others. The classical examples of rolling two dice or flipping several coins illustrate independent events. Since the events are independent we are able to use the multiplication rule to multiply the probabilities together. In practice, especially due to some sampling techniques, there can be times when trials are not technically independent. A binomial distribution can sometimes be used in these situations as long as the population is larger relative to the sample. Two Classifications Each of the trials is grouped into two classifications: successes and failures. Although we typically think of success as a positive thing, we should not read too much into this term. We are indicating that the trial is a success in that it lines up with what we have determined to call a success. As an extreme case to illustrate this, suppose we are testing the failure rate of light bulbs. If we want to know how many in a batch will not work, we could define success for our trial to be when we have a light bulb that fails to work. A failure of the trial is when the light bulb works. This may sound a bit backward, but there may be some good reasons for defining the successes and failures of our trial as we have done. It may be preferable, for marking purposes,  to stress that there is a low probability of a light bulb not working rather than a high probability of a light bulb working. Same Probabilities The probabilities of successful trials must remain the same throughout the process we are studying. Flipping coins is one example of this. No matter how many coins are tossed, the probability of flipping a head is 1/2 each time. This is another place where theory and practice are slightly different. Sampling without replacement can cause the probabilities from each trial to fluctuate slightly from each other. Suppose there are 20 beagles out of 1000 dogs. The probability of choosing a beagle at random is 20/1000 0.020. Now choose again from the remaining dogs. There are 19 beagles out of 999 dogs. The probability of selecting another beagle is 19/999 0.019. The value 0.2 is an appropriate estimate for both of these trials. As long as the population is large enough, this sort of estimation does not pose a problem with using the binomial distribution.