You’re running a small firm, and you have an estimate of both your cost function and your demand curve. Your cost function is TC=892-13q+4q^2, while your inverse demand curve is P=863-0.4q, where P is the price of one unit of your output and q is the quantity of units produced and sold.
If you wanted to minimize costs, what quantity would you produce? Please round your answer to the nearest whole number (i.e., no decimal places).
To minimize costs in my small firm, I would need to find the quantity of units to produce that minimizes the total cost function while considering the given inverse demand curve. In this scenario, the cost function is represented as TC = 892 – 13q + 4q^2, and the inverse demand curve is P = 863 – 0.4q, where P represents the price of one unit of my output, and q represents the quantity of units produced and sold.
To find the quantity that minimizes costs, I would start by setting up the profit function, which can be calculated as follows:
Profit (π) = Total Revenue (TR) – Total Cost (TC)
Total Revenue (TR) is the product of the price (P) and the quantity (q):
TR = P * q = (863 – 0.4q) * q = 863q – 0.4q^2
Now, we can calculate the profit function:
π = TR – TC = (863q – 0.4q^2) – (892 – 13q + 4q^2)
Next, we can simplify the profit function:
π = 863q – 0.4q^2 – 892 + 13q – 4q^2
π = -4.4q^2 + 876q – 892
To find the quantity that minimizes costs, we need to find the derivative of the profit function with respect to q and set it equal to zero:
dπ/dq = -8.8q + 876
Now, set dπ/dq equal to zero and solve for q:
-8.8q + 876 = 0
-8.8q = -876
q = 876 / 8.8
q ≈ 99.55
Rounding the answer to the nearest whole number, I would produce approximately 100 units to minimize costs. Producing 100 units would result in the lowest cost for my small firm while taking into account the cost function and demand curve provided.
In summary, to minimize costs, I would produce approximately 100 units, as this quantity would maximize the profit by balancing the cost and demand aspects of the business.
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