The instructor of an elementary$-$school class tries to teach her students some simple principles

of economics by having them play a trading game. She 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 she hands out the goods, on a Monday, she 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$.$ She provides

them with small food$-$storage bags they can use to store $$loose$$ M&Ms$.($These bags are

not a trade item.$)$ Each day, she hands out a few additional M&M packages and juice

bottles, to replace ones that have been consumed.

After a week or so of trading, stable $$market$$ trade rates for the goods, when traded one for

another, have emerged. These include $($among others$)$:

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.

$1.$ A$.$ Calculate the total number of distinct relative prices in this classroom economy. $($Note

that, at this stage, I$$m looking for the number of distinct prices, not which prices they

are, or their numerical values.$)$

Hint: Use a formula presented on page $19$ of the latest $($revised$)$ version of my $1-$AB

lecture notes. Remember that there are five different goods.

B$.$ In this question, we$$ll use a systematic procedure to identify the distinct pairs of goods

that can be traded for each other. We$$ll report distinct pairs like this: FS and JB$,$ JB

and MM$,$ FS and MM$,$ etc.

For distinct pairs, the order of the goods listed doesn$$t matter. For example, if we think

of juice and fidget spinners $($JB and FS$)$ as a pair of goods, then fidget spinners and

juice are not another distinct pair. But juice and M&Ms $($JB and MM$)$ are another

distinct pair, and fidget spinners and M&Ms $($FS and MM$)$ are a third distinct pair.

We$$ll use alphabetical order to determine the order in which we report the pairs, based on

the first letter of the abbreviation for the first good in a pair. If two pairs have the same

first good, we$$ll order them using the first letter in the abbreviation for the second good.

Note that I followed both rules in ordering the three example pairs: JB and FS precedes

JB and MM$,$ because F precedes M in alphabetical order; both of these pairs precede FS

and MM$,$ because J precedes F in alphabetical order.

$2$

$1.$ a$.$ How many distinct pairs include calculators $($C$)?$

b$.$ Report these pairs.

$2.$ a$.$ How many additional distinct pairs include fidget spinners $($FS$)?$ Note that you

should already have one pair that includes them.

b$.$ Report these additional pairs.

$3.$ a$.$ How many additional distinct pairs include juice bottles $($JB$)?$ Note that you

should already have two pairs that include them.

b$.$ Report these additional pairs.

$4.$ a$.$ How many additional distinct pairs include M&Ms $($MM$)?$ Note that you

should already have three pairs that include them.

b$.$ Report these additional pairs.

$5.$ Are there any additional distinct pairs that include pencil erasers $($PE$)?$

Explain briefly.

Note that if you$$ve answered these five questions correctly, the total number of distinct

pairs you$$ve identified should be equal to your answer to Question $1.$A$.$

$2.$ In this question, you$$ll use the price information you$$re given in the last four lines of the

introduction to this assignment to calculate the values of all the distinct relative prices in

this classroom economy. Note that there should be exactly as many relative prices as your

answer to Question $1.$A$.$

The relative prices we$$re interested in are unit prices: the price of one unit of a good, in

units of another good. For example, we$$re given the unit price of fidget spinners in juice

bottles. But we$$re not given the unit price of calculators in pencil erasers, although it$$s

very easy to calculate from the information given.

In the classroom economy, the students think of a single M&M$,$ not a package, as one unit of

M&Ms$.$ For example, the students think of two juice bottles as costing $30$ M&Ms $($why$?).$

Note that each distinct pair of goods produces only one distinct relative price. Consider,

for example, calculators and pencil erasers. 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 to each

other: see below.

For each distinct pair of goods, we$$ll start by calculating the version of their relative price

that produces the highest unit price. In the case of calculators and pencil erasers, for

exam