The cost accountant 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 is the quantity sold. Required, 1. 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. 2. Determine the price that will maximize profit for the firm. 3.Determine the maximum revenue for this firm.
#1. avg cost is just TC/Q. So, the avg cost A is
A = 120-Q+0.02Q^2
minimum A is just the vertex of that parabola.
revenue = price * quantity = PQ
R = (114-0.25Q)Q = 114Q - 0.25Q^2
profit = revenue - cost, so the profit
y = (114Q - 0.25Q^2)-(120Q-Q^2+0.02Q^3) = -0.02Q^3 + 0.75Q^2 - 6Q
maximum revenue/profit is, of course, where
dR/dQ = 0 and dy/dQ = 0
To find the level of production that will yield minimum average cost per unit, we need to find the point where the average cost curve (AC) is at its lowest. The average cost per unit is calculated by dividing the total cost (TC) by the quantity (Q).
1. Find the average cost function:
AC = TC / Q
AC = (120Q - Q^2 + 0.02Q^3) / Q
AC = 120 - Q + 0.02Q^2
To find the minimum of this function, we can take its derivative with respect to Q and set it equal to zero:
dAC/dQ = -1 + 0.04Q = 0
0.04Q = 1
Q = 25
So, the level of production that will yield the minimum average cost per unit is 25 units.
To determine whether this level of output maximizes profit for the firm, we need to calculate the profit function. The profit (π) is calculated by subtracting the total cost (TC) from the total revenue (TR). Total revenue is equal to the price (P) times the quantity (Q).
Profit function:
π = TR - TC
π = (P * Q) - (120Q - Q^2 + 0.02Q^3)
Since we have the sales forecasting function P = 114 - 0.25Q, we can substitute it into the profit function:
π = [(114 - 0.25Q) * Q] - (120Q - Q^2 + 0.02Q^3)
To determine whether the level of production that yields the minimum average cost per unit also maximizes profit, we need to find the derivative of the profit function with respect to Q and evaluate it at the quantity Q = 25.
dπ/dQ = 114 - 0.5Q - 120 + 2Q - 0.06Q^2
dπ/dQ = -6 - 0.5Q - 0.06Q^2
Now, substitute Q = 25 into the derivative:
dπ/dQ = -6 - (0.5 * 25) - (0.06 * 25^2)
dπ/dQ = -6 - 12.5 - 37.5
dπ/dQ = -56
Since the derivative is negative (-56), this indicates that the level of production that yields the minimum average cost per unit does not maximize profit for the firm. To maximize profit, the level of output should be adjusted.
2. To determine the price that will maximize profit for the firm, we need to differentiate the profit function with respect to P and set it equal to zero:
dπ/dP = Q - 120 = 0
Substituting the value of Q = 25, we get:
25 - 120 = 0
-95 = 0 (contradiction)
This implies that the price cannot be optimized to maximize profit for the firm. The profit maximization lies in adjusting the level of production, not the price.
3. To determine the maximum revenue for the firm, we need to find the revenue function. Revenue (R) is calculated by multiplying the price (P) by the quantity (Q).
Revenue function:
R = P * Q
R = (114 - 0.25Q) * Q
R = 114Q - 0.25Q^2
To find the maximum revenue, we can differentiate the revenue function with respect to Q and set it equal to zero:
dR/dQ = 114 - 0.5Q = 0
0.5Q = 114
Q = 228
By substituting Q = 228 into the revenue function, we can find the maximum revenue:
R = 114 * 228 - 0.25(228^2)
R = $26,136
Therefore, the maximum revenue for this firm is $26,136.
To find the level of production that will yield minimum average cost per unit, we need to find the derivative of the total cost function with respect to quantity (Q) and set it equal to zero. Let's calculate it step by step:
1. The total cost function is given by: TC = 120Q - Q^2 + 0.02Q^3.
2. The average cost per unit (AC) is calculated by dividing the total cost (TC) by the quantity sold (Q): AC = TC / Q.
3. Taking the derivative of the average cost function with respect to quantity (Q), we get:
d(AC)/dQ = d(TC/Q)/dQ
= (dTC/dQ * Q - TC)/Q^2
= (dTC/dQ - TC/Q)/Q.
4. Now, let's find dTC/dQ (derivative of total cost with respect to quantity):
dTC/dQ = d(120Q - Q^2 + 0.02Q^3)/dQ
= 120 - 2Q + 0.06Q^2.
5. Substitute this value back into the derivative of the average cost function:
d(AC)/dQ = (120 - 2Q + 0.06Q^2 - (120Q - Q^2 + 0.02Q^3))/Q^2.
6. Simplify the equation further:
d(AC)/dQ = (120 - 2Q + 0.06Q^2 - 120Q + Q^2 - 0.02Q^3)/Q^2
= (120 - 119Q + 1.06Q^2 - 0.02Q^3)/Q^2
= (1.06Q^2 - 0.02Q^3 - 119Q + 120)/Q^2.
7. Set d(AC)/dQ equal to zero and solve for Q:
(1.06Q^2 - 0.02Q^3 - 119Q + 120)/Q^2 = 0.
8. Multiply both sides by Q^2 to eliminate the denominator:
1.06Q^2 - 0.02Q^3 - 119Q + 120 = 0.
9. Rearrange the equation to get it in standard form:
0.02Q^3 - 1.06Q^2 + 119Q - 120 = 0.
10. Unfortunately, solving this equation explicitly is difficult. We'll need to use numerical methods or a graphing calculator to find the values of Q at which the equation equals zero.
Once you find the value of Q that yields the minimum average cost per unit, you can substitute that value into the profit function to determine whether it maximizes profit for the firm. Similarly, you can use the profit function to find the price that maximizes profit and calculate the maximum revenue for the firm.