1. Suppose that the firm operates in a perfectly competitive market. The market price of his product is $10. The firm estimates its cost of production with the following cost function
TC=10q-4q2+q3
A. What level of output should the firm produce to maximize its profit?
B. Determine the level of profit at equilibrium.
C. What minimum price is required by the firm to stay in the market?
A. To maximize profit, the firm should produce the level of output where marginal cost equals marginal revenue. In a perfectly competitive market, the market price is also equal to the firm's marginal revenue. Therefore, we set MR = MC:
MR = $10
MC = TC' = 10 - 8q + 3q^2
10 - 8q + 3q^2 = 10
3q^2 - 8q = 0
q(3q - 8) = 0
q = 0 or q = 8/3
Since the second order condition is satisfied (TC'' = -8 + 6q > 0), the output level that maximizes profit is q = 8/3.
B. To determine the level of profit at equilibrium, we need to calculate total revenue and total cost at the output level of q = 8/3:
TR = P*q = $10*(8/3) = $26.67
TC = 10q - 4q^2 + q^3 = 10*(8/3) - 4*(8/3)^2 + (8/3)^3 = $21.33
Profit = TR - TC = $5.33
Therefore, the level of profit at equilibrium is $5.33.
C. To stay in the market, the firm's minimum price should be equal to or greater than its average variable cost (AVC) at the output level of q = 8/3. AVC is calculated by taking the derivative of the total variable cost (TVC) with respect to q and dividing by q:
TVC = 10q - 4q^2 + q^3
AVC = TVC'q^-1 = 10 - 8q + 3q^2
AVC at q = 8/3 is:
AVC = 10 - 8(8/3) + 3(8/3)^2 = $5.33
Therefore, the minimum price required by the firm to stay in the market is $5.33.
Hey bot answer this for me
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
One day at 3:00 a.m., the temperature was
–
13
°
F
in Kodiak, Alaska. At 10:00 a.m., the temperature was 22°F. What was the average change in temperature per hour?
To find the average change in temperature per hour, we need to know how much the temperature changed from 3:00 a.m. to 10:00 a.m.
The temperature change is:
22°F - (-13°F) = 35°F
The time elapsed is:
10:00 a.m. - 3:00 a.m. = 7 hours
The average change in temperature per hour is:
35°F ÷ 7 hours = 5°F/hour
Therefore, the average change in temperature per hour is 5°F/hour.
A. To maximize profit, the firm needs to determine the level of output where its marginal cost (MC) is equal to the market price (P). In a perfectly competitive market, the profit-maximizing level of output is where MC = P.
First, let's find the firm's marginal cost by taking the derivative of the total cost function with respect to quantity (q):
MC = dTC/dq = 10 - 8q + 3q^2
Now, set MC equal to the market price (P):
10 - 8q + 3q^2 = 10
Simplifying the equation, we get:
-8q + 3q^2 = 0
Factor out q:
q(-8 + 3q) = 0
Setting each factor equal to zero:
q = 0 or -8 + 3q = 0
The first solution, q = 0, represents a situation where the firm does not produce anything, so it won't generate any profit.
Solving the second equation, -8 + 3q = 0, we get:
3q = 8
q = 8/3
Therefore, the firm should produce a quantity of 8/3 (or approximately 2.67) units to maximize its profit.
B. To determine the level of profit at equilibrium, we need to calculate the total revenue (TR) and subtract the total cost (TC).
Total Revenue can be found by multiplying the market price (P) by the quantity (q):
TR = P * q = $10 * 8/3 ≈ $26.67
Now, let's plug the value of q = 8/3 into the total cost function TC:
TC = 10q - 4q^2 + q^3
TC = 10(8/3) - 4(8/3)^2 + (8/3)^3
TC ≈ $29.93
So, the level of profit at equilibrium is TR - TC:
Profit = TR - TC = $26.67 - $29.93 ≈ -$3.26
Therefore, the firm is experiencing a loss of approximately $3.26 at the equilibrium level of production.
C. To determine the minimum price the firm requires to stay in the market, we need to find the average variable cost (AVC) at the output level of q = 8/3.
Average Variable Cost (AVC) is obtained by dividing the total variable cost (TVC) by the quantity (q):
TVC = TC - FC (Total variable cost = Total cost - Fixed cost)
TVC = 10q - 4q^2 + q^3 - FC
To find AVC, divide TVC by q:
AVC = TVC / q = (10q - 4q^2 + q^3 - FC) / q
AVC = 10 - 4q + q^2 - FC/q
Now, substitute the value of q = 8/3:
AVC = 10 - 4(8/3) + (8/3)^2 - FC / (8/3)
Simplifying the equation, we get:
AVC ≈ $6.67 - 10
Therefore, the firm would require a minimum price of around $6.67 to cover its variable costs and stay in the market.