Question

Suppose that you won an exclusive bid to sell Christmas trees from National Park Service (NPS). However, NPS requires that you plant one and a half multiple of any number of trees you cut. For example, if you cut 4 trees, you have to plant 8 trees. If you cut 9, you have to plant 27 trees. The NPS may argue that the number of trees that survive is proportional to the number of tree that you grow. Or, they might just do so to prevent you from cutting all trees. Assume further that the cost of cutting and transporting a tree is $2. The cost of growing a tree is $1.  
Questions:
A.) Write an equation that describes total cost of cutting any number of trees.
B.) Graph total cost function
C.) Derive and graph the average cost equation
D.) Derive and graph the marginal cost equation

Now, assume that as a monopolist, you can sell Christmas trees directly to customers and charge them a higher retail price. Or, on the other hand, you can sell Christmas tree to retail stores and charge them lower wholesale price. It is logical to assume that ordinary customer demand tends to be less elastic than the demand of retail stores. Assume that you estimate the demand of retail stores and customers and find:

P= 3000 – 0.5Q  (Wholesale demand)
P= 2000-2Q   (Retail Demand)

Questions
E) Write the equations that describe total revenue for each market
F.) Graph your total revenue equations
G.) Derive the marginal revenue equations for each market
H.) Graph your answer
E
I.) Find the profit maximizing price and quantity for each market
J.) Write the equation that describe total revenue for the two market combined
K.) Derive the marginal revenue equations for the two markets combined
L.) Graph your answer
M.) Find the profit maximizing price and quantity for both markets combined.
N.) What is better, to combine both market and charge a single price or segregate the two markets and charge different prices? Explain your answer

Answer 1

Solution:

A)Suppose Q is the number of trees that is supposed to be cut. The number of trees supposed to be planted = Q^1.5

Cost of cutting and transporting a tree = $2

Cost of planting a tree = 1

Total cost = (Cost of cutting and transporting * number of trees supposed to be cut) + (Cost of planting * number of trees supposed to be plant)

Total cost = 2*Q + 1*Q^1.5

= 2Q + Q^1.5

C)Average total cost = Total cost / quantity =( 2Q + Q^1.5)/ Q = Q + Q^1.5/Q

D)Marginal cost = dTC(q)/ dq = 2 + 1.5Q

E) to I)Given, P= 3000 – 0.5Q (wholesale demand)

Total revenue in wholesale market= PQ = (3000 – 0.5Q) * Q = 3000Q – 0.5Q²

Marginal revenue in wholesale market= dTC(q)/ dq = 3000 – Q

Profit will be maximise where MC will be equals to MR

MC =MR

2 + 1.5Q = 3000 – Q

2Q = 2998

Q = 1499

P= 3000- 0.5Q = 3000- 749.5 = 2250.5

P= 2000 – 2Q (Retail demand)

Total revenue in retail market= PQ = (2000 – 2Q) * Q = 2000Q – 2Q²       

Marginal revenue in retail market = dTC(q)/ dq = 2000 - 4Q

Profit will be maximise where MC will be equals to MR

MC =MR

2 + 1.5Q = 2000 – 4Q

5.5Q = 1998

Q = 363.27

P= 2000- 2Q = 2000- 726.54 = 1273.45

J) Total revenue for both market combined = revenue for wholesale market + revenue for retail market

= 3000Q – 0.5Q² + 2000Q – 2Q²

= 5000Q – 2.5Q²

K)Marginal revenue in retail market = dTC(q)/ dq = 5000 - 5Q

M)Profit will be maximise where MC will be equals to MR

MC =MR

2 + 1.5Q = 5000 – 5Q

6.5Q = 4998

Q = 768.9

P= 5000- 2.5Q = 5000- 1537.8 = 3462.15