3 Candice and Sally both have a surf lesson company Each sum

3. Candice and Sally both have a surf lesson company. Each summer season they employ a oertain mimber of suf instructors beawed on their demand for surf lessons. Lets assume it takes one surf instructor to xive one lesson. Because you need a license to give surf lessons, Candice and Sally are the only companies that give surf lessons. The demand function for surf lessons are q 3,200- 1,600p. The total ber of surf lessons demanded is q ?+g, where is the amount of surf cssons Candice will giwe each season and , is the amount of surf lessons Sally will give in a given season. The cost of giving surf lessons are the same for both women, e(qi)s 500 + 0.50 x ?, i-c, s, meaning, MC ,\" 0.50. (a) The inverse demand function will have the following functional form, p at bge + q,). Find a and b. (b) After every summer, Candice and Saly have to decidle how many instractors to employ based on market demand and olserving how many surf lessons their com petitor gave. Lets assume euch person takes the levet of business their competitor had the prior summer, and assumes the same output this summer Menning, if Candice observed that Sally gave 400 surf lessons last summer (meaning she employed 400 workens), she will usaume Sally will give 400 lessons this year. i. What type of market is Candice and Sally in? ii. What is the marginal revenue function for Candice and Sally? ili. What are the reauction fuuctions for Candice nd Sally? iv. In wpilibrium, what will the market price be for surf lesons?

Solution

It has been given that the total demand for surfing is: q=3200 – 1600p-----1 where q is the market demand consisting of Sally’s and Candice’s demands such that q=qc+qs-----2 and p is the price.

The cosr function for surf lessons for both is : c(qi) = 500+0.50qi-----3 (i=s,c) and marginal cost (MC) = 0.50---4

a) The inverse demand function is written as: p = a-bq or p=a-b(qc+qs)

From equation 1, q=3200-1600p

Or 1600p=3200-q

Or p=3200/1600 – q/1600

Or p=2-(qc+qs)/1600 (Substituting equation 2 for q)

Therefore the inverse demand function is: p=2- (1/1600)*(qc+qs) which takes the form p=a-b(qc+qs)

Where a= 2 and b= 1/1600

b) Here each person observes the other person’s output as given. Since Candice observes Sally’s output, Candice is the follower and Sally is the leader in this Stackelberg market. It is important to note that there is no question of observing anyone’s output under Cournot Duopoly.

(i) Since Candice and Sally are the only two licensed surf lessons provider, the type of market they are in is Oligopoly. This is because there are a large number of buyers or surfers, but very few sellers. Precisely, since there are 2 sellers here, Candice and Sally, the market type is Duopoly.

(ii) The total revenue (TR) function for Candice is: TRc= {a-b(qc+qs)}*qc

Or TRc = aqc – bqc² – bqcqs

Therefore Marginal Revenue (MR) for Candice is: delta(TRc)/delta qc

Or MRc = a-2bqc-bqs

Similarly, The total revenue (TR) function for Sally is: TRs= {a-b(qc+qs)}*qs

Or TRs = aqs – bqs² – bqcqs

Therefore Marginal Revenue (MR) for Sally is: delta(TRs)/delta qs

Or MRs = a-2bqs-bqc

(iii) The reaction function of Candice (qc) in terms of Sally’s output is given by:

qc = (a-c-bqs)/2b where a = 2, b = 1/1600 and c=0.50

qc = (2-0.50)/1600 – qs/2-----5

Similarly, the reaction function of Sally(qs) in terms of Candice’s output is given by:

qs = (a-c-bqc)/2b where a = 2, b = 1/1600 and c=0.50

qs = (2-0.50)/1600 – qc/2------6

(iv) Since Candice observes Sally’s output and then decides, the formula for Sally’s output is: qs* = (a-c)/2b

Or qs* = 1.5/3200

The formula for Candice’s output is:

qc* = (a-c)/4b

Or qc* = 1.5/6400

The total output is: 1.5/3200 + 1.5/6400

Or q = 4.5/6400

Therefore price is: p= 2- (1/1600)*4.5/6400

Or p = 1.99 or p~2.