The cost of a firm producing colour television has worked out the total cost function for the firm as TC=120Q-Q^2+0.02Q^3.A sales manager has provided the sales forecasting function as P=114-0.25Q where P is price and Q the quantity sold.Required:
i)Find the level of production that will yield minimum average cost per unit and determine whether this level of output maximizes profit for the firm.
ii)Determine the price that will maximize profit for the firm.
iii)Determine the maximum revenue for this firm.
Tc =230
What is the answer
To find the level of production that will yield minimum average cost per unit and determine whether this level of output maximizes profit for the firm, we need to calculate the average cost function and the total revenue function.
i) Average Cost per unit:
Average Cost (AC) is calculated by dividing Total Cost (TC) by the quantity produced (Q).
AC = TC/Q
Given that TC = 120Q - Q^2 + 0.02Q^3, we can substitute this equation into the average cost equation.
AC = (120Q - Q^2 + 0.02Q^3) / Q
Simplifying the equation, we get:
AC = 120 - Q + 0.02Q^2
To find the level of production that will yield minimum average cost per unit, we need to find the value of Q that minimizes the average cost function. We can do this by taking the derivative of the average cost function and setting it equal to zero.
d(AC)/dQ = -1 + 0.04Q = 0
Solving for Q, we get:
0.04Q = 1
Q = 1/0.04
Q = 25
Therefore, the level of production that will yield minimum average cost per unit is 25 units.
To determine whether this level of output maximizes profit for the firm, we need to compare the marginal cost and the marginal revenue at this level of output.
ii) Maximum Profit:
Profit is calculated by subtracting the total cost from the total revenue.
Total Revenue (TR) is calculated by multiplying the price (P) by the quantity sold (Q).
TR = P * Q
Given that P = 114 - 0.25Q, we can substitute this equation into the total revenue equation.
TR = (114 - 0.25Q) * Q
Now, we have the total revenue function.
Profit (π) = TR - TC
Profit (π) = (114 - 0.25Q) * Q - (120Q - Q^2 + 0.02Q^3)
To determine the level of production that maximizes profit, we need to find the value of Q that maximizes the profit function. We can do this by taking the derivative of the profit function and setting it equal to zero.
d(π)/dQ = 114 - 0.5Q - 120 + 2Q - 0.06Q^2 = 0
Simplifying the equation, we get:
-0.06Q^2 + 1.5Q - 6 = 0
Using the quadratic formula to solve for Q, we get:
Q = (-b ± √(b^2 - 4ac))/(2a)
Considering only the positive value for Q, we find that Q ≈ 52.63 units.
Therefore, the level of production that maximizes profit for the firm is approximately 52.63 units.
iii) Maximum Revenue:
To determine the maximum revenue for this firm, we need to find the price that will maximize the revenue function.
We already have the total revenue function:
TR = (114 - 0.25Q) * Q
To find the price that maximizes revenue, we need to take the derivative of the revenue function with respect to Q and set it equal to zero.
d(TR)/dQ = 114 - 0.5Q = 0
Solving for Q, we get:
0.5Q = 114
Q = 228
To find the price, we can substitute this value of Q back into the sales forecasting function:
P = 114 - 0.25Q
P = 114 - 0.25 * 228
P ≈ 57
Therefore, the price that will maximize profit for the firm is approximately 57.
To answer these questions, we need to determine the average cost, profit, and revenue functions first. Let's break down each step:
i) Find the level of production that will yield minimum average cost per unit and determine whether this level of output maximizes profit for the firm.
To find the level of production that yields the minimum average cost per unit, we need to calculate the average cost (AC) function. The formula for average cost is:
AC = TC / Q
Where AC is the average cost and TC is the total cost.
Given that TC = 120Q - Q^2 + 0.02Q^3, we can substitute this into the equation for AC:
AC = (120Q - Q^2 + 0.02Q^3) / Q
Simplifying:
AC = 120 - Q + 0.02Q^2
To find the level of production that minimizes the average cost, we take the derivative of the average cost with respect to Q and set it equal to zero:
d(AC)/dQ = -1 + 0.04Q = 0
Solving for Q:
0.04Q = 1
Q = 25
Therefore, to yield the minimum average cost per unit, the level of production should be 25 units.
To determine whether this level of output maximizes profit for the firm, we need to calculate the profit function. The profit (π) function is given by:
π = TR - TC
Where TR is the total revenue and TC is the total cost.
Given the sales forecasting function P = 114 - 0.25Q, we can substitute this into the equation for TR:
TR = P * Q
TR = (114 - 0.25Q) * Q
TR = 114Q - 0.25Q^2
Now we can calculate the profit:
π = TR - TC
π = (114Q - 0.25Q^2) - (120Q - Q^2 + 0.02Q^3)
π = -0.02Q^3 + 0.25Q^2 - 6Q + 0
To determine if this level of output maximizes profit, we need to find the maximum point of the profit function. We take the derivative of the profit with respect to Q and set it equal to zero:
d(π)/dQ = -0.06Q^2 + 0.5Q - 6 = 0
This is a quadratic equation, which can be solved to find the value of Q that maximizes profit.
ii) Determine the price that will maximize profit for the firm.
To determine the price that will maximize profit, we need to substitute the value of Q obtained in the previous step into the sales forecasting function P = 114 - 0.25Q. This will give us the corresponding price.
iii) Determine the maximum revenue for this firm.
To determine the maximum revenue for the firm, we need to calculate the total revenue (TR). We can substitute the value of Q obtained in the first step into the sales forecasting function P = 114 - 0.25Q. Then, we multiply this price by the quantity sold (Q) to get the total revenue (TR).