Profit Maximization in Perfect Competition

Profit Maximization in Perfect Competition Market

Producing and selling at profit maximization output at particular price is very important to ensure our resource and capital are utilized at optimum level. Economic theory say that in perfect competition market in long run Price=demand function=Marginal Revenue=Marginal Cost. This is because the efficient market condition that allow easy entry and out.

1.0       Question No. 1  

The Green company produces chemicals in a perfectly

competitive market. The current market price is $40;

the firm’s total cost is C=100 + 4Q + Q2.

                                                         

a.      Determine the firm’s profit –maximizing output.                   

More generally, write down the equation for the firm’s supply curve in terms of price P.

b.      Complying with more stringent environmental                      

regulations increases the firm’s fixed cost from 100

to 144. Would this affect the firm’s output?

Its supply curve?

2.0       Question No. 2  

Suppose that the manager of a firm operating in a

competitive market has estimated the firm’s average

variable cost function to be

AVC = 10 – 0.03Q + 0.00005Q2

Total fixed cost is $600.                                                        

a.      What is the corresponding marginal cost function?             

b.      At what output is AVC at its minimum?                                  

c. What is the minimum value for AVC?                                        

If the forecasted price of the firm’s output is $10 per unit:

d. How much output will the firm produce in the short run?    

e. How much profit (loss) will the firm earn?                                

                   

If the forecasted price of the firm’s output is $7 per unit:

f. How much output will the firm produce in the short run?     

g. How much profit (loss) will the firm earn? If the                       forecasted price is $5 per unit:

If the forecasted price is $5 per unit:

h. How much output will the firm produce in the short run?     

i. How much profit (loss) will the firm earn?                                  

  

1.0 Question 1

Solution 1(a)

Total cost    C= 100 + 4Q + Q²

Fixed Cost (FC) = 100

Variable Cost (VC) = 4Q + Q²

Marginal Cost (MC) = 4 + 2Q

Average Cost (AC)	=	100	+	4Q	+	Q2
		Q
	=	100	+	4	+	Q
		Q

Average Variable Cost (AVC)	=	4Q	+	Q2
		Q
	=	4	+	Q

The current market price is $40. To solve the Q, the firm will set the P = MC to maximise profits.

(P)	40	=	4	+	2Q	(MC)
	2Q	=	40	-	4
	Q	=	36 / 2
	Q	=	18

AVC	=	4 + Q
	=	4 + 18
	=	$22

               

ATC	=	(100 / Q) + 4 + Q
	=	(100 / 18) + 4 + 18
	=	$27.55

Profit = TR – TC

= (PxQ) – (FC + VC)

                                                = (PxQ) – (100 + 4Q + Q²)

= (40 x 18) – (100 + 4(18) + 18²)

= 720 – 100 – 72 – 324

= $224

From here we can see at the market price $40, the company will produce 18 unit of outputs more than cover its Total Cost (FC & VC).  Thus, the Green Company makes a profit as shown in Figure 1.

Solution 1(b)

If the fixed cost increases from 100 to 144?

ATC	=	(144 / Q) + 4 + Q
	=	(100 / 18) + 4 + 18
	=	$30

Profit = TR – TC

   = (PxQ) – (FC + VC)

                                                   = (PxQ) – (144 + 4Q + Q²)

   = (40 x 18) – (144 + 4(18) + 18²)

   = 720 – 144 – 72 – 324

= $180

As the input price given of $40 and output of 18 units is fix, an increase in Fixed Cost from $100 to $144 wouldn’t have any effect on the supply curve. The company just recorded less profit as the Total Revenue earned need to cover slightly higher Total Cost from $27.5 to $30. The company still making profit at $180 which less than $224 previously.

2.0 Question 2

Solution 2(a) – What is the corresponding marginal cost function

AVC = 10-0.03Q+0.00005Q²

VC = 10Q-0.03Q²+0.00005Q³                                    (multiply by Q)

TC = 600+10Q-0.03Q²+0.00005Q³

MC = 10-0.06Q+0.00015Q²

Solution 2(b)–Minimum Output for AVC

10-0.03Q+0.00005Q² = 10-0.06Q+0.00015Q²

0.03Q+0.00010Q²= 0

Q(0.03+0.00010Q)= 0

Q = 0, or Q = 0.03/0.00010 = 300 units

Solution 2(c) - Minimum value for AVC

At Q = 300,          AVC       = 10-0.03(300)+0.00005(300²)

                                                = $5.5 (minimum)

To get AVC at its minimum can be found by setting AVC = MC. At Q = 300, AVC = $5.5 and is a minimum. So Q=300 is where the AVC at its minimum. So, in the short run firm will not produce if price are less than $5.5.

Solution 2(d) - If P = $10 per unit

P=MC

$10 = 10-0.06Q+0.00015Q²

0.06Q - 0.00015Q²= 0

Q(0.06 - 0.00015Q) = 0

Q = 0, or Q = 0.06/0.00015 =400 units

At Q = 400,          AVC       = 10-0.03(400) +0.00005(400²)

                                                = $6 (minimum)

In the short run, firm will only produce 400 outputs at the price of $6 in the short run since it’s the minimum AVC.

Solution 2(e) - Profit / Loss

 Profit = TR – TC

   = (PxQ) – (FC + VC)

   = (PxQ) – (600+10Q-0.03Q²+0.00005Q³)

   = (10 x 400) – (600+10(400)-0.03(400²)+0.00005(400³)

   = 4000 - 3000

= $1000 (Profit)

Solution 2(f) - If P = $7 per unit

P=MC

$7 = 10-0.06Q+0.00015Q²

0.06Q - 0.00015Q²+ 3= 0

6000Q-15Q²+300000 = 0                                                 (round the number)

We solve the Q using quadratic formula below:

Q = 58.57 @ 59unit ; or Q = 341.43 @ 341units

At Q = 341,          AVC       = 10-0.03(341)+0.00005(341)²

                                                = $5.58 (Minimum)

In the short run, firm will only produce 341 outputs at the price of $5.58.

Solution 2(g) - Profit / Loss

 Profit = TR – TC

   = (PxQ) – (FC + VC)

   = (PxQ) – (600+10Q-0.03Q²+0.00005Q³)

   = (7 x 341) – (600+10(341)-0.03(341²)+0.00005(341³)

   = 2387 - 2504

= - $117 (Loss)

Even though the firm loss at RM117, it still less than fixed cost of RM600 (FC > Loss), as such the firm can still continue operating in the short run.

Solution 2(h) - If P = $5 per unit

P=MC

$5 = 10-0.06Q+0.00015Q²

0.06Q - 0.00015Q²+5= 0

6000Q-15Q²+500000 = 0                                                 (round the number)

We solve the Q using quadratic formula below:

Q = 118.35 @ 118 units ; or Q = 281.65 @ 282units

At Q = 282,          AVC       = 10-0.03(282)+0.00005(282)²

                                                = $5.51 (Minimum)

In the short run, firm will only produce 282 outputs at the price of $5.51.

Solution 2(i) - Profit / Loss

Profit = TR – TC

   = (PxQ) – (FC + VC)

   = (PxQ) – (600+10Q-0.03Q²+0.00005Q³)

   = (7 x 282) – (600+10(282)-0.03(282²)+0.00005(282³)

   = 1974–2155.60

= - $181.60 (Loss)

Even though the firm loss at RM181.60, it still less than fixed cost of RM600 (FC > Loss), as such the firm can still continue operating in the short run.