Question

EverKleen Pool Services provides weekly swimming pool maintenance in Atlanta. Dozens of firms provide this service. The service is standardized, each company cleans the pool and maintains the proper levels of chemicals in the water. The service is typically sold as a four-month summer contracts. The market price for the four-month service is $125. EverKleen Pool Services has a fixed cost of $3,500. The manager of EverKleen has estimated the following cost function:

SMC = 125-0.42Q+0.0021Q²

Where SMC is measured in dollars and Q is the number of pools serviced each summer.
a. At what output level does AVC reach its minimum value? What is the value of AVC at its minimum point?
b. Should the firm continue to operate or shut down? Explain why? (15 pts)
c. What is the optimal output level? Make necessary calculations and clearly highlight the optimum level
d. How much profit/loss can the manager of the firm expect to earn?
e. The fixed costs rise to $4,000. What is the effect of such an increase? Explain.

Answer 1

EverKleen Pool Services provides weekly swimming pool maintenance in Atlanta. Dozens of firms provide this service. The service is standardized, each company cleans the pool and maintains the proper levels of chemicals in the water. The service is typically sold as a four-month summer contracts. The market price for the four-month service is

P = $125

The short run Marginal Cost of production is

And, the fixed cost is F = $3500

The total cost is TC(Q) and the Fixed Cost is when Q=0

TC(0) = F = $3500

Now, we will integrate the SMC function to get the TC or Total Cost function.

Hence,

Where F is integral Constant or Fixed Cost.

or,

Now, when Q=0,

TC(0) = 0 + F

or, F = 3500

Hence,

Now, let us answer the following questions with the help of these informations.

(a) The VC or Variable Cost is

And, AVC or Average Variable Cost is

AVC = VC/Q

or,

Now, the first order derivative of AVC is zero at that value of output for which AVC reaches its minimum.

Hence, let us put

d(AVC)/dQ = 0

or, 0 - 0.21 + 0.0014.Q = 0

or, 0.0014.Q = 0.21

or, Q = 150

Hence, AVC reaches its minimum when output is 150 units.

The value of AVC at the mimimum point is

or, AVC(150) = $109.25

Hence, the value of AVC at the minimum point is $109.25.

(b) The Shut Down point of a firm is where it should stop production. The shut down point is the Minimum Point of AVC or Average Variable Cost. When the Price level (P) reaches the minimum point of AVC, then the firm should shut down. But the firm should continue production upto that price level where,

P > AVCmin

Now, in the above question, the Market Price is

P = $125

And, the minimum point of AVC is where

AVCmin = $109.25

Hence,

P > AVCmin (As 125>109.25)

Hence, the firm should not shut down. It should continue production.

Hence, the firm should continue to operate.

Reason: The price level is above the minimum point of AVC. Hence, it should continue operating.

(c) In a competitive market, the optimal profit maximizing output level is achieved where Short run Marginal Cost (SMC) equals the Price level (P).

Hence, at equilibrum

SMC = P

or,

or,

or, 0.0021.Q = 0.42

or, Q* = 200

The optimal output level is 200 units.

(d) The Market Price level is

P = $125

The Optimal Output level is

Q* = 200 units

Hence, the Total Revenue at Q* is

TR = P.Q* = $125×200

or, TR = $25000

And, we got thr total cost function previously i.e.

At, Q*=200, thr Total Cost is

or, TC = $25700

Hence, the profit of the firm is

π = TR - TC

or, π = $25000 - $25700

or, π = -$700

Hence, the manager can expect to earn loss of $700.

(e) Now if the fixed costs rise to $4000.

Hence, the total cost of the firm will be,

But the SMC or Short run Marginal Cost curve will remain same.

Hence, the Optimal Output level will remain same at Q*=200 units.

Hence, the Total Revenue will be

TR = P.Q* = $125×200 = $25000

And, the Total Cost is

TC = $26200

Hence, the profit of the firm is

π = TR - TC

or, π = $25000 - $26200

or, π = -$1200

Due to rise in the fixed cost, the firm will now suffer more loss. The manager can expect to loss $1200.

But the firm will still continue operating. Because the Price level is above minimum point of AVC.

Hope the solutions are clear to you my friend.