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Question Determine the output level that maximizes profit given the demand curve q=90−2Pq=90-2P and total cost curve TC=2q2+25q+60TC=2q^2+25q+60.

Studdy Solution

STEP 1

Assumptions
1. The demand curve is given by q=90−2P q = 90 - 2P , where q q is the quantity demanded and P P is the price.
2. The total cost (TC) curve is given by TC=2q2+25q+60 TC = 2q^2 + 25q + 60 .
3. Profit (Π \Pi ) is defined as total revenue (TR) minus total cost (TC), where total revenue is P×q P \times q .
4. To maximize profit, we need to find the output level (q q ) that results in the highest profit.
5. The output level should be one of the given options: a) 4, b) 5, c) 20, d) 4.4.

STEP 2

First, we need to express the total revenue (TR) in terms of q q . To do this, we need to find the price P P from the demand curve and then multiply it by q q .
TR=P×q TR = P \times q

STEP 3

To find P P in terms of q q , we rearrange the demand curve equation:
q=90−2P q = 90 - 2P
2P=90−q 2P = 90 - q
P=90−q2 P = \frac{90 - q}{2}

STEP 4

Now, substitute P P in the total revenue equation:
TR=q×90−q2 TR = q \times \frac{90 - q}{2}
TR=90q−q22 TR = \frac{90q - q^2}{2}

STEP 5

Next, we need to express profit (Π \Pi ) in terms of q q by subtracting total cost (TC) from total revenue (TR):
Π=TR−TC \Pi = TR - TC
Π=90q−q22−(2q2+25q+60) \Pi = \frac{90q - q^2}{2} - (2q^2 + 25q + 60)

STEP 6

Now, simplify the profit equation by combining like terms:
Π=90q−q22−2q2−25q−60 \Pi = \frac{90q - q^2}{2} - 2q^2 - 25q - 60
Π=90q−q2−4q2−50q−1202 \Pi = \frac{90q - q^2 - 4q^2 - 50q - 120}{2}
Π=−5q2+40q−1202 \Pi = \frac{-5q^2 + 40q - 120}{2}
Π=−52q2+20q−60 \Pi = -\frac{5}{2}q^2 + 20q - 60

STEP 7

To maximize profit, we need to take the derivative of the profit function with respect to q q and set it equal to zero. This will give us the critical points where the profit could be maximized.
dΠdq=−5q+20 \frac{d\Pi}{dq} = -5q + 20

STEP 8

Set the derivative equal to zero to find the critical points:
−5q+20=0 -5q + 20 = 0

STEP 9

Solve for q q :
−5q=−20 -5q = -20
q=−20−5 q = \frac{-20}{-5}
q=4 q = 4

STEP 10

We found a critical point at q=4 q = 4 . To ensure that this point is a maximum, we need to check the second derivative of the profit function or examine the profit at this quantity level compared to the other options given.

STEP 11

Since we have discrete options for q q given in the problem, we can calculate the profit for each option and compare them to determine which one gives the highest profit.

STEP 12

Calculate the profit for q=4 q = 4 :
Π(4)=−52(4)2+20(4)−60 \Pi(4) = -\frac{5}{2}(4)^2 + 20(4) - 60
Π(4)=−52(16)+80−60 \Pi(4) = -\frac{5}{2}(16) + 80 - 60
Π(4)=−40+80−60 \Pi(4) = -40 + 80 - 60
Π(4)=80−100 \Pi(4) = 80 - 100
Π(4)=−20 \Pi(4) = -20

STEP 13

Calculate the profit for q=5 q = 5 :
Π(5)=−52(5)2+20(5)−60 \Pi(5) = -\frac{5}{2}(5)^2 + 20(5) - 60
Π(5)=−52(25)+100−60 \Pi(5) = -\frac{5}{2}(25) + 100 - 60
Π(5)=−62.5+100−60 \Pi(5) = -62.5 + 100 - 60
Π(5)=100−122.5 \Pi(5) = 100 - 122.5
Π(5)=−22.5 \Pi(5) = -22.5

STEP 14

Calculate the profit for q=20 q = 20 :
Π(20)=−52(20)2+20(20)−60 \Pi(20) = -\frac{5}{2}(20)^2 + 20(20) - 60
Π(20)=−52(400)+400−60 \Pi(20) = -\frac{5}{2}(400) + 400 - 60
Π(20)=−1000+400−60 \Pi(20) = -1000 + 400 - 60
Π(20)=400−1060 \Pi(20) = 400 - 1060
Π(20)=−660 \Pi(20) = -660

STEP 15

Since q=4.4 q = 4.4 is not an integer, it does not make sense in the context of this problem because a firm cannot produce a fraction of a unit. Therefore, we do not need to calculate the profit for q=4.4 q = 4.4 .

STEP 16

Comparing the calculated profits, we see that the profit for q=4 q = 4 is −20-20, for q=5 q = 5 is −22.5-22.5, and for q=20 q = 20 is −660-660. The highest profit among these is −20-20, which corresponds to q=4 q = 4 .

STEP 17

Therefore, the firm should produce 4 units to maximize profit.
The output level the firm should produce to maximize profit is: a. 4

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