In Tuftsville, everyone lives along Main Street, which is 10 miles long. There a

Question

In Tuftsville, everyone lives along Main Street, which is 10 miles long. There are 1,000 people uniformly spread up and down Main Street, and every day they each buy a fruit smoothie from one of the two stores located at either end of Main Street. Cus- tomers ride their motor scooters to and from the store using $0.50 worth of gas per mile. Customers buy their smoothies from the store offering the lowest price, which is the store’s price plus the customer’s travel expenses getting to and from the store. Ben owns the store at the west end of Main Street and Will owns the store at the east end of Main Street.

a) If both Ben and Will charge $1 per smoothie, how many will each of them sell in a day? If Ben charges $1 per smoothie and Will charges $1.40, how many smoothies will each sell in a day?

b) If Ben charges $3 per smoothie, what price would enable Will to sell 250 smoothies per day? 500 smoothies per day? 750 smoothies per day? 1,000 smoothies per day?

c) If Ben charges p1 and Will charges p2, what is the location of the customer who is indifferent between going to Ben’s or Will’s shop? How many customers go to Will’s store and how many go to Ben’s store? What are the demand functions that Ben and Will face?

d) Rewrite Ben’s demand function with p1 on the left-hand side. What is Ben’s marginal revenue function?

e) Assume that the marginal cost of a smoothie is constant and equal to $1 for both Ben and Will. In addition, each of them pays Tuftsville $250 per day for the right to sell smoothies. Find the equilibrium prices, quantities sold, and profits net of the $250 license fee.

Explanation / Answer

A1. When Ben and Will charge $1 per smoothie, then the customers will be indifferent towards each store. Hence, for a population of 1000 people, both Ben and Will sell 500 smoothies each per day. This means, 500+500= 1000 smoothies (for total population).

A2. Total Population of Tuftsville=1000

Total distance of Tuftsville = 10 miles

Let Ben’s store be at a distance= d miles

Thus, Will’s store will be at a distance of= (10-d) miles

(Since Ben’s store is at west end and Will’s store is at East end of 10 Miles long Main Street)

Total travel expenses per customer:

(to) for one side= $ 0.5/mile

(from) for return side= $ 0.5/mile

Thus, Total = $2(0.5)/mile = $1 per mile

So, travel expenses for Ben’s store = $1(d)……………………(i)

And, travel expenses for Will’s store = $1(10-d)…………………(ii)

Now, Ben charges $1 per smoothie. So customer’s total cost for a smoothie at Ben’s store

= $1 + travel expense

=$1 + $1(d)……….(from i)

And, Will charges $1.4 per smoothie. So customer’s total cost for a smoothie at Will’s store

= $1.4 + travel expense

=$1.4 + $1(10-d)……….(from ii)

Suppose, customers are indifferent towards buying from Ben’s store and Will’s store. Then,

$1 + $1(d) = $1.4 + $1(10-d)……………eq(iii)

Or, 1+d = 1.4+10-d

Or, 2d= 10.4

Thus, d= 5.2

And 10-d= 4.8

So, Ben’s store is charging less hence it will sell 520 smoothies and Will sells 480 smoothies (this makes total smoothies = 520+480=1000)

2.. Since Will has to sell 250 smoothies, then Ben must be selling, 1000-250=750 smoothies.

Again suppose customers are indifferent towards buying from Ben’s store and Will’s store. Then, as per the explanation in equation 3,

We have d= 7.5 (so as to enable Ben to sell 750 smoothies)

Let Will’s price per smoothie be x, then

$3+ $1(7.5) = $x + $1(10-7.5)

Or, 3+7.5= x+ 2.5

0r, x= $8

Hence, Will should charge $8 per smoothie to be able to sell 250 smoothies.

Now Will has to sell 500 smoothies, then Ben must be selling, 1000-500=500 smoothies.

Again suppose customers are indifferent towards buying from Ben’s store and Will’s store. Then, as per the explanation in equation 3,

We have d= 5 (so as to enable Ben to sell 500 smoothies)

Let Will’s price per smoothie be x, then

$3+ $1(5) = $x + $1(10-5)

Or, 3+5= x+ 5

0r, x= $3

Hence, Will should charge $3 per smoothie to be able to sell 500 smoothies.

Now Will has to sell 750 smoothies, then Ben must be selling, 1000-750=250 smoothies.

Again suppose customers are indifferent towards buying from Ben’s store and Will’s store. Then, as per the explanation in equation 3,

We have d= 2.5 (so as to enable Ben to sell 250 smoothies)

Let Will’s price per smoothie be x, then

$3+ $1(2.5) = $x + $1(10-2.5)

Or, 3+2.5= x+ 7.5

Here, x will have a negative value.

Hence, Will becomes unable to sell 750 smoothies at any price. Similarly, he won’t be able to sell 1000 smoothies either.

3.

Ben Charges- P1

Will Charges P2

Again as per the explanation given in equation 3,

P1 +d= P2 + (10-d)

Or, d= [(P2 –P1)/2] +5

Thus, Demand function of Ben = {[(P2 –P1)/2] +5}*100

= 500 +50(P2 –P1)

And Demand function of Will= 1000-demand function of Ben

= 500 - 50(P2 –P1)

P1 = P2+10-2d

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