5. Call Us demand function is Q = 20 – 0.2P and the MC = 10 + 5Q.
Given that TFC = $2,000;
a) Derive an equation for the TC.
b) Calculate the profit at the profit maximizing level.
a) To derive an equation for total cost (TC), we need to consider the formula: TC = TFC + TVC, where TFC represents total fixed cost, and TVC represents total variable cost.
Given that TFC = $2,000, we can substitute this value into the equation and consider the TVC component.
The variable cost (VC) can be calculated as the product of quantity (Q) and marginal cost (MC). The equation for VC is VC = Q * MC.
Given the values for MC = 10 + 5Q and Q = 20 – 0.2P, we can substitute the value of Q into the equation for MC to determine VC.
VC = Q * MC
= (20 – 0.2P) * (10 + 5Q)
= (20 – 0.2P) * (10 + 5(20 – 0.2P))
Expanding the equation further:
= (20 – 0.2P) * (10 + 100 – 1P)
= (20 – 0.2P) * (110 – 1P)
Multiplying the terms:
= 2,200 – 22P + 0.2P^2 + 0.2P^2 – 2P
= 2.2P^2 – 20P + 2,200
Now, we can substitute the value of TVC into the TC equation:
TC = TFC + TVC
= $2,000 + (2.2P^2 – 20P + 2,200)
= 2.2P^2 – 20P + 4,200
Therefore, the equation for total cost (TC) is 2.2P^2 – 20P + 4,200.
b) To calculate the profit at the profit maximizing level, we need to find the quantity (Q) that maximizes profit. At the profit-maximizing level, marginal cost (MC) should equal marginal revenue (MR).
Given the demand function Q = 20 – 0.2P, we can determine the revenue function using the formula: TR = P * Q.
TR = P * Q
= P * (20 – 0.2P)
= 20P – 0.2P^2
To find the marginal revenue (MR), we differentiate TR with respect to P:
MR = d(TR)/d(P)
= d(20P – 0.2P^2)/d(P)
= 20 – 0.4P
To find the profit-maximizing level, we need to equate MR and MC:
MR = MC
20 – 0.4P = 10 + 5Q
Since MC = 10 + 5Q, we can substitute this value into the equation:
20 – 0.4P = 10 + 5Q
20 – 0.4P = 10 + 5(20 – 0.2P)
20 – 0.4P = 10 + 100 – P
Simplifying the equation:
20 – 0.4P = 110 – P
P = 200
Now that we have found the value of P at the profit-maximizing level, we can substitute this value into the demand function to find the corresponding quantity:
Q = 20 – 0.2P
= 20 – 0.2(200)
= 20 – 40
= -20
Since a negative quantity does not make sense in this context, we can conclude that the profit-maximizing level is P = 200, Q = 0.
To calculate the profit at this level, we can substitute the values into the cost equation:
TC = 2.2P^2 – 20P + 4,200
= 2.2(200)^2 – 20(200) + 4,200
= 88,400 – 4,000 + 4,200
= 88,400
Profit = TR - TC
= P * Q - TC
= 200 * 0 - 88,400
= -88,400
Therefore, the profit at the profit maximizing level is -$88,400.