Assignment 1, Part B

The instructor of an elementary-school class tries to teach his students some simple principles of economics by having them play a trading game.He starts out by giving each student small quantities of up to five different goods:

M&Ms, in small "Fun-Size" packages containing 17 M&Ms each: abbreviation MM,

pencil erasers (the hand-held rectangular kind): abbreviation PE,

simple mechanical calculators: abbreviation C,

fidget spinners: abbreviation FS,

small bottles of juice: abbreviation JB.

After he hands out the goods, on a Monday, he lets the students trade them freely for 15 minutes each school day, through Friday of the following week.The students quickly open up some of the M&M packages so they can eat or trade individual M&Ms.The instructor provides them with small food-storage bags that they can use to store "loose" M&Ms.(These bags are not a trade item).Each day he hands out a few additional M&M packages and juice bottles, to replace the ones that are consumed.

After a week or so of trading, stable "market" trade rates for the goods, when traded one for another, have emerged.These include:

three juice bottles trade for one fidget spinner,

one package of M&Ms, plus 13 loose M&Ms, trade for two juice bottles,

fifteen pencil erasers trade for two calculators,

two calculators trade for three fidget spinners.

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Question 1

[1a-1]Type out the formula, presented in one of my lectures and in my lecture notes, that can be used to calculate the total number of distinctrelative pricesin an economy.(Hint:See page 13 of my 1-AB lecture notes; as in those notes, usento represent the total number of goods traded in the economy.)

Ans- n(n-1)/2

Question 2

[1a-2]Use this formula to calculate the total number of distinct relative prices in the classroom economy.(Hint:How many different goods are traded?)

Question 3

[1-b]Identify the distinct pairs of goods that can be traded for each other.Verify that the number of distinct pairs is equal to your answer to Part a of this question.

(Hint:For distinct pairs, the order of the goods listed doesn't matter.If we think of juice and fidget spinners as a pair of goods then fidget spinners and juice are not another distinct pair.But juice and M&Ms could be another distinct pair.)

Report the distinct pairs like this:JB and FS, MM and JB, etc.For full credit, list all the distinct pairs associated with the first good, alphabetically according to its abbreviation (C), then all the remaining distinct pairs associated with the second good, alphabetically (FS), etc.

Question 4

[2]Use the price information you were given to calculateallthe distinct relative prices in the classroom economy.(Notethat there should be as many such prices as your answer to Question 1a-2 indicates.)

Remember that the relative prices we are interested in areunit prices.A unit price is the price of a single unit of one good in units of another good.For example, you are given the unit price of fidget spinners in juice bottles, but not the unit price of calculators in pencil erasers (although it is very easy to calculate from the information you were given).

In the classroom economy, prices involving M&Ms are quoted in terms of individual M&Ms,not packages.(That is, individual M&Ms are the relevant M&M units.)For example, the students think of two juice bottles as costing 30 M&Ms (why?).

Notethat each distinct pair of goods has only onedistinctrelative price.For example, if we count the price of calculators in pencil erasers as one relative price, then we don't count the price of pencil erasers in calculators as a second distinct relative price, because the two prices have a fixed relationship (see below).

For each distinct pair of goods, use the relative price that produces the highest unit price.In the case ofcalculators and pencil erasers, for example, if the price of a calculator in pencil erasers is higher than the price of a pencil eraser in calculators then report the first price but not the second one.

In calculating unit prices, don't worry about issues of divisibility.Assume that any good can be divided in any way, so that fractions or decimals can be components of unit prices.(This assumption will be unrealistic in many cases: see below.)If a unit price is not a whole number then write it as a fraction and also as a decimal.

Any time you report a fraction as all or part of your answer, make sure the fraction is a ratio of natural numbers (no decimals), and make sure it's in lowest terms.If the decimal value you report is not exactly equal to the fraction, round the decimal value to the third decimal place.

Report the relative prices like this: 3 JB per FS, 7 PE per C, etc.For full credit, report them from the highest value to the lowest.(Hint:All the values you report should be greater than 1:why?)

In many cases, you will have to use to of the relative prices your were given to calculate a relative price you were not given.Examples:[i] Since three juice bottles trade for a fidget spinner, and 1 fidget spinners trade for a calculator (why?), we know that3112=4123112=412(4.5) juice bottles trade for a calculator: 4 JB per C.[ii] Since 7 pencil erasers trade for a calculator (why?) and 1 fidget spinners trade for a calculator (see above), we know that 7/1 = 5 pencil erasers trade for a fidget spinner: 5 PE per FS.

Question 5

[3-a]For each relative price you just calculated, calculate the relative price if the order of the two items is switched.(In the scenario above, for example, you would now calculate the price of a single pencil eraser in calculators.)

Again, if the unit price is not a whole number then write it as a fraction and also as a decimal.

Example:Since the unit price of a fidget spinner in juice bottles is 3 (3 bottles per spinner), the unit price of a juice bottle in fidget spinners is 1/30.333 (1/3 spinners per bottle), reported as 1/30.333 FS per JB.

Question 6

[3-b]What very simple formula did you use to perform these calculations?Explain briefly.

Question 7

[4]Now suppose that, after a week or so of trading, the students in the class start using M&Ms asmoney.This situation does not change the stable relative prices that have emerged.

[4a-1]Type out the formula, presented in one of my lectures and in my lecture notes, that can be used to calculate the total number ofabsolutepricesin an economy.(Hint:Again, see page 13 of my 1-AB lecture notes.)

Question 8

[4a-2]Use this formula to calculate the number of absolute prices in the classroom economy.

Question 9

[4-b]Report these absolute prices, in descending numerical order (highest value to lowest).

(Hint:The absolute price of a good is the number of units of money needed to buy a unit of the good.You shouldn't have to calculate the absolute prices, because you've already done it as part of the answer to a previous question.)

Question 10

[5]In practice, many of these goods can't easily be divided for trading.Calculators and fidget spinners are obvious examples.Juice bottles can't easily be divided, either, because the juice goes bad quickly after a bottle is opened.Pencil erasers are still useful if they're divided in half, and half an eraser can be traded easily.But smaller pieces are not very useful.Individual M&Ms are too hard to divide to make it worth the effort.

[5a-1a]In this classroom economy, a student who wants a single calculator won't buy it using money.Why not?

Question 11

[5a-1b]What will s/he trade for it instead?Why?

Question 12

[5a-2]If a student has a bunch of M&Ms and wants to obtain a single calculator, what sequence of trades is s/he likely to use to get it?How much "change" is she likely to get, in what form?

Question 13

[5b-1]On the other hand, a student who has two juice bottles and wants pencil erasers won't try to make that trade directly.Why not?

Question 14

[5b-2] What will s/he do instead?(Hint:Indirect exchange using money.)How many erasers will s/he buy, and how much change will s/he get after paying?

Question 19

[7]Now suppose that, in the last couple of days of trading, after the students have started using money for most trades (see Question 4 above), the instructor plays the role of a merchant, in order to make a point.She announces that she is willing to trade pencil erasers for calculators (or vice-versa) directly, at a rate of eight pencil erasers per calculator.

In this question, we'll work through a simple procedure a student could use to take advantage of the instructor-merchant's willingness to trade these goods, at this rate, to make a profit ofone eraser.The student's trading strategy should involve (1) using money to buy one of the goods in the market, (2) trading that good to the merchant in exchange for the other good, and (3) selling enough of the other good in the market, for money, to cover the cost of the initial purchase.Identify the items traded and the quantities, and verify the amount of the profit.

(Hint:Given the way the instructor-merchant values pencil erasers and calculators, compared to their market valuation, which good will the student want to trade to him in exchange for the other good?The student will want to start by buying the other good with money.And s/he can buy more than one unit of that good.)

[7a-1]Which good should the student purchase, using money?How many units of the good should s/he buy, and how many units of money will she pay?

Question 202pts

[7a-2] Which good should the student obtain in trade from the instructor-merchant, in exchange for the good s/he purchased in the market, using money?How many units of the good will s/he be able to obtain in trade?Why?

Question 21

[7a-3] How can the student now obtain enough money to cover the original purchase (from Part 7a-1)?And explain why her/his profit from this sequence of transactions is one eraser (or its market value in money).

Question 22

[7-b]Finally, explain what the student would need to do in order to make a larger profit: twice as large, for example.

Question 23

[8-a]Based on the prices that prevailed at the end of the week, which of the goods that were distributed by the instructor was the most valuable, per unit?The least valuable?(Note:For this purpose, think of apackageof M&Ms as one unit.)How do you know?

Question 24

[8-b]The instructor numbered the students in order to help him keep records of the outcome of the trading game.On the first day, he gave student #1 one unit of each of the five goods.He gave student #2 two calculators and three pencil erasers, and he gave student #3 two fidget spinners, two juice bottles and a package of M&Ms.

Based on the prices that prevailed at the end of the week, which student had the most initial wealth?The least?How do you know?

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