উত্তরঃ
Financial Costs vs. Economic Costs
Financial costs (also known as explicit costs) are the direct monetary payments made by a firm for the factors of production and inputs it uses. These are actual out-of-pocket expenses that can be clearly identified and recorded in a firm's accounting books. Examples include wages paid to employees, rent for premises, cost of raw materials, utility bills, interest payments on loans, and depreciation of equipment.
Economic costs, on the other hand, encompass both explicit (financial) costs and implicit costs. Implicit costs represent the opportunity cost of using resources that the firm already owns or that are provided by the owners themselves, for which no direct monetary payment is made. These are the foregone benefits from the next best alternative use of those resources. Examples include the salary the owner could have earned working elsewhere, the rent the firm could have received by leasing out its owned building, or the interest that could have been earned on capital invested in the business. Economic costs are crucial for making rational business decisions as they provide a more complete picture of the true cost of doing business.
Example:
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Financial Costs: A baker pays ₹500 for flour, ₹300 for sugar, and ₹2000 for oven electricity. Total financial cost = ₹2800.
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Economic Costs: In addition to the ₹2800 financial costs, the baker uses a kitchen she owns (could have rented it out for ₹1000) and forgoes a ₹1500 salary she could have earned working at another bakery. Total economic cost = ₹2800 (explicit) + ₹1000 (implicit rent) + ₹1500 (implicit wage) = ₹5300.
Profit Maximization Problem Solution:
Given:
Total Revenue (TR) function: \(TR = 4350Q - 13Q^2\)
Average Cost (AC) function: \(AC = Q^2 - 5.5Q + 150 + 675Q^2\)
First, simplify the Average Cost (AC) function:
\(AC = (1 + 675)Q^2 - 5.5Q + 150\)
\(AC = 676Q^2 - 5.5Q + 150\)
Step 1: Determine the Total Cost (TC) function.
\(TC = AC \times Q\)
\(TC = (676Q^2 - 5.5Q + 150) \times Q\)
\(TC = 676Q^3 - 5.5Q^2 + 150Q\)
Step 2: Determine the Profit (\(\pi\)) function.
\(\pi = TR - TC\)
\(\pi = (4350Q - 13Q^2) - (676Q^3 - 5.5Q^2 + 150Q)\)
\(\pi = 4350Q - 13Q^2 - 676Q^3 + 5.5Q^2 - 150Q\)
Combine like terms:
\(\pi = -676Q^3 + (-13 + 5.5)Q^2 + (4350 - 150)Q\)
\(\pi = -676Q^3 - 7.5Q^2 + 4200Q\)
Step 3: Find the first derivative of the Profit function with respect to Q and set it to zero (First Order Condition).
\(\frac{d\pi}{dQ} = \frac{d}{dQ}(-676Q^3 - 7.5Q^2 + 4200Q)\)
\(\frac{d\pi}{dQ} = -676 \times 3Q^{3-1} - 7.5 \times 2Q^{2-1} + 4200Q^{1-1}\)
\(\frac{d\pi}{dQ} = -2028Q^2 - 15Q + 4200\)
Set \(\frac{d\pi}{dQ} = 0\) to find the critical points:
\(-2028Q^2 - 15Q + 4200 = 0\)
Multiply by -1 to make the leading coefficient positive:
\(2028Q^2 + 15Q - 4200 = 0\)
Step 4: Solve the quadratic equation for Q using the quadratic formula \(Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Here, \(a = 2028\), \(b = 15\), \(c = -4200\).
\(Q = \frac{-15 \pm \sqrt{15^2 - 4(2028)(-4200)}}{2(2028)}\)
\(Q = \frac{-15 \pm \sqrt{225 + 34070400}}{4056}\)
\(Q = \frac{-15 \pm \sqrt{34070625}}{4056}\)
\(Q = \frac{-15 \pm 5837.005}{4056}\)
Two possible values for Q:
\(Q_1 = \frac{-15 + 5837.005}{4056} = \frac{5822.005}{4056} \approx 1.4354\)
\(Q_2 = \frac{-15 - 5837.005}{4056} = \frac{-5852.005}{4056} \approx -1.4428\)
Since quantity cannot be negative, we take the positive value: \(Q \approx 1.4354\).
Step 5: Verify if this level of output maximizes profit using the Second Order Condition.
Find the second derivative of the Profit function:
\(\frac{d^2\pi}{dQ^2} = \frac{d}{dQ}(-2028Q^2 - 15Q + 4200)\)
\(\frac{d^2\pi}{dQ^2} = -2028 \times 2Q - 15\)
\(\frac{d^2\pi}{dQ^2} = -4056Q - 15\)
Substitute \(Q \approx 1.4354\) into the second derivative:
\(\frac{d^2\pi}{dQ^2} = -4056(1.4354) - 15\)
\(\frac{d^2\pi}{dQ^2} = -5822.0944 - 15\)
\(\frac{d^2\pi}{dQ^2} = -5837.0944\)
Since \(\frac{d^2\pi}{dQ^2} < 0\), the profit is indeed maximized at \(Q \approx 1.4354\).
Step 6: Calculate the maximum profit at the profit-maximizing level of output.
Substitute \(Q \approx 1.4354\) into the profit function:
\(\pi = -676Q^3 - 7.5Q^2 + 4200Q\)
\(\pi = -676(1.4354)^3 - 7.5(1.4354)^2 + 4200(1.4354)\)
\(\pi = -676(2.9609) - 7.5(2.06097) + 6028.68\)
\(\pi = -2001.0764 - 15.457275 + 6028.68\)
\(\pi \approx 4012.1463\)
Therefore, the profit maximizing level of output is approximately \(1.4354\) units, and the maximum profit is approximately \(4012.15\).
