A golf-course operator must decide what greens fees (prices) to set on rounds of golf. Daily demand during the week is Pd = 36 –Qd/10 where Qd is the number of 18 hole rounds and Pd is the price per round. Daily demand on the weekend is Pw = 50 – Qw/12. As a practical matter, the capacity of the course is 240 rounds per day. Wear and tear on the golf course is negligible.
a) Can the operator profit by charging different prices during the week and on the weekend? What greens fee should the operator set on weekdays and how many rounds will be played on the weekend?
b) When weekend prices skyrocket, some weekend golfers choose to play during the week instead. The greater the difference between weekday and weekend prices; the greater are the number of these defectors. How might this factor affect the operator’s pricing policy
As weekend prices skyrocket more people will play golf during the week and eventually the weekend prices will drop and weekday prices.
To determine whether the golf-course operator can profit by charging different prices during the week and on the weekend, we need to consider the demand and capacity constraints.
a) Firstly, let's determine the weekday greens fee and the number of rounds played on the weekend. We can use the demand equations provided:
Weekday demand: Pd = 36 – Qd/10
Weekend demand: Pw = 50 – Qw/12
Given that the capacity of the course is 240 rounds per day, we can set up an equation to find the number of rounds played on the weekend:
Weekday rounds + Weekend rounds = Total capacity
Qd + Qw = 240
To maximize profit, the operator will set prices that maximize revenue. Revenue is calculated by multiplying the price per round by the number of rounds played:
Weekday revenue: Rd = Pd * Qd
Weekend revenue: Rw = Pw * Qw
Now, we need to differentiate between the weekday and weekend rounds to find the optimal pricing strategy.
Since profit is equal to revenue minus costs, we need to consider any additional costs associated with playing on weekdays or weekends.
If we assume that the costs are the same regardless of whether it is a weekday or weekend round (i.e., negligible wear and tear), then we can simply focus on maximizing revenue.
To find the optimal pricing strategy, we need to maximize the revenue equations. This can be done by finding the derivative of the revenue equations with respect to the number of rounds and setting them equal to zero.
d(Rd)/dQd = 36 - Qd/5 = 0
d(Rw)/dQw = 50 - Qw/6 = 0
Solving these equations, we find:
Qd = 180 rounds
Qw = 120 rounds
Substituting these values into the demand equations, we can find the corresponding prices:
Pd = 36 - 180/10 = 36 - 18 = $18
Pw = 50 - 120/12 = 50 - 10 = $40
Therefore, the operator should set the greens fee to $18 on weekdays and $40 on weekends. The number of rounds played on the weekend will be 120.
b) When weekend prices skyrocket and the price difference between weekdays and weekends increases, more people are likely to choose to play during the week instead. This is because the higher weekend prices make playing on weekdays comparatively cheaper.
As more golfers choose to play during the week, the demand for weekday rounds increases (Qd increases) and the demand for weekend rounds decreases (Qw decreases).
As a result, the operator may need to adjust the pricing policy to find a balance between maximizing revenue on weekends and weekdays. In this case, to attract more golfers back to the weekends, the operator may consider reducing the weekend prices or narrowing the price difference between weekdays and weekends.
Ultimately, the pricing policy will depend on the operator's objectives, market conditions, and the balance between revenue generation and customer satisfaction.
The problem doesn't state what Pw is, so assuming that is the price per round on the weekend, then:
weekend profit = Pw x Qw
daily profit = Pd x Qd
Take the 1st derivative of each of these, set equal to zero, and solve for the maximum value of the profit.