3. Suppose that the total monthly demand for golf services is given by
Q = 20 – P
The marginal cost to the firm of each golfer is $1. If this demand function is based on the individual demands of 10 golfers, what is the optimal two part pricing strategy for this golf club? How much profit will the firm earn? When should this firm adopt this particular pricing strategy?
The optimal per-unit charge = Marginal cost when P=20-q and MC=1
equating P=MC
20-q=1
q=19
thus 19rounds of golf will be played each month.
The consumer surplus received by 10 golfers=1/2*(20-1)*19=$180.5
consumer surplus received by each golfer=180.5/10=$18.05
Thus for optimal two part strategy, each golfer should be charged a price of $18.05 plus a fee of $1.
The total profit of the firm=$180.5-fixed cost of the firm.
This strategy is used to extract maximum profits.
Well, it seems like this golf club is in quite a putt-ickle situation! Let's break it down and find the optimal strategy.
The demand function is given as Q = 20 - P, which means that for every golfer, the quantity demanded decreases by 1 for every $1 increase in price. We also know that the marginal cost per golfer is $1.
To find the optimal two-part pricing strategy, we need to find the price at which the club should charge for the golf services. To do that, we can set marginal cost equal to marginal revenue, which is the derivative of the total revenue function.
Since we know that the total monthly demand is based on the individual demands of 10 golfers, we can rewrite the demand function as Q = 10(20 - P), considering the number of golfers.
Let's find the marginal revenue (MR) function:
MR = dTR/dQ
To find the total revenue (TR) function, we need to multiply the quantity (Q) by the price (P):
TR = PQ
Taking the derivative of the total revenue function with respect to Q, we get:
MR = (d/dQ)(PQ)
MR = P + Q(dP/dQ)
Since the demand function is Q = 10(20 - P), we can substitute this into the MR function:
MR = P + 10(20 - P)(dP/dQ)
Setting MR equal to the marginal cost per golfer ($1), we have:
P + 10(20 - P)(dP/dQ) = 1
Now, solving this equation for P will give us the optimal price.
As for the profit, we can find it by subtracting the total cost from the total revenue. Since the marginal cost per golfer is $1 and each golfer brings in a revenue of P, the total cost for 10 golfers is 10.
Profit = Total revenue - Total cost
Profit = (P)(10) - 10
Now, the firm should adopt this pricing strategy when the profit is maximized. To find this optimal point, we can take the derivative of the profit function with respect to P and set it equal to zero.
To summarize, the optimal two-part pricing strategy and the amount of profit earned by the firm will depend on solving these equations.
To find the optimal two part pricing strategy for the golf club, we need to consider the demand function and the marginal cost.
The demand function is given by:
Q = 20 - P
Where Q is the quantity of golf services demanded and P is the price per golfer.
First, let's find the Price Elasticity of Demand (PED) to determine the optimal price. The formula for PED is:
PED = (% change in quantity demanded) / (% change in price)
Since the demand function is based on the individual demands of 10 golfers, we can assume an infinitely small change in price and quantity. Therefore, the PED is given by the derivative of the demand function:
PED = dQ/dP
Taking the derivative of the demand function with respect to P:
dQ/dP = -1
Since the marginal cost is given as $1, the optimal price would be set at the point where marginal cost equals marginal revenue. In this case, the marginal revenue is equal to the marginal cost, which is $1.
Therefore, the optimal price is P = $1.
To find the quantity demanded at this price, we substitute P = $1 into the demand function:
Q = 20 - 1
Q = 19
So, the quantity demanded at the optimal price is 19.
Next, we calculate the fixed fee (T) that the golf club should charge. The fixed fee is calculated by subtracting the variable cost (VC) from the total cost (TC), and then dividing by the quantity demanded at the optimal price:
TC = T + VC
VC = (variable cost per golfer) * (quantity demanded at the optimal price)
In this case, the variable cost per golfer is $1, and the quantity demanded at the optimal price is 19:
VC = $1 * 19 = $19
TC = T + $19
To maximize profit, the fixed fee should be set as high as possible. Hence, the fixed fee is equal to the total cost at the optimal price:
T = TC - VC
T = $19 - $19
T = $0
Therefore, the optimal two part pricing strategy for this golf club is to charge a fixed fee of $0 and a price of $1 per golfer.
To calculate the profit earned by the firm, we need to find the total revenue and subtract the total cost:
Total Revenue = (fixed fee) * (quantity demanded at the optimal price) + (price per golfer) * (quantity demanded at the optimal price)
Total Revenue = $0 * 19 + $1 * 19
Total Revenue = $19
Profit = Total Revenue - Total Cost
Profit = $19 - $19
Profit = $0
The firm will earn zero profit with this pricing strategy.
The firm should adopt this particular pricing strategy when it wants to maximize its revenue and fill as many spots as possible, as it charges a fixed fee of $0 and a price of $1 per golfer, which is the optimal price for maximizing revenue.
To determine the optimal two-part pricing strategy for the golf club and calculate the firm's profit, you can follow these steps:
Step 1: Find the individual consumer surplus using the demand function.
Consumer Surplus = 0.5 * Quantity * (Price - Marginal Cost)
Substituting the given values:
Consumer Surplus = 0.5 * [20 - P] * (P - 1)
Step 2: Calculate the social welfare or total surplus by multiplying the individual consumer surplus with the number of consumers.
Social Welfare = Number of Consumers * Consumer Surplus
Substituting the given value of the number of consumers (10):
Social Welfare = 10 * [0.5 * (20 - P) * (P - 1)]
Step 3: Determine the optimal price (P) that maximizes the total surplus (Social Welfare).
To do this, you can take the first derivative of the social welfare function with respect to P and set it equal to zero to find the critical points. Then, check which critical point maximizes the social welfare.
Step 4: Calculate the quantity (Q) corresponding to the optimal price (P) using the demand function.
Substitute the optimal price (P) into the demand function:
Q = 20 - P
Step 5: Determine the membership or entry fee (F) needed to maximize profit.
Membership Fee (F) = Marginal Cost * Quantity
Substitute the marginal cost (given as $1) and the optimal quantity (Q) obtained from Step 4:
Membership Fee (F) = $1 * Quantity
Step 6: Calculate the total profit (Profit) of the firm.
Profit = Social Welfare - Total Cost
Total Cost = Membership Fee (F) * Number of Consumers
Substitute the values obtained from the previous steps into the profit equation.
Step 7: Decide when to adopt this pricing strategy.
To determine when to adopt this particular pricing strategy, you should consider factors such as market conditions, competition, and customer demand. If the profit calculated in Step 6 is acceptable or higher than alternative pricing strategies, you could adopt this two-part pricing strategy.
By following these steps and performing the necessary calculations, you can find the optimal two-part pricing strategy and the firm's profit.