Suppose a chemical company is in a perfectly competitive industry and has a short run total cost curve of TC = q3 + 5q2 + 10q + 10 and a short run marginal cost of SMC = q2 + 10q + 10. At the price of 49, how many will be produced?
What Is The Answer
yes
To determine the quantity that will be produced in a perfectly competitive industry, we need to identify the level of output where the short-run marginal cost (SMC) equals the price (P). In this scenario, the price is given as $49.
The SMC is represented by the equation SMC = q^2 + 10q + 10, where q represents the quantity produced.
To find the quantity produced, we need to equate the SMC to the price and solve for q:
q^2 + 10q + 10 = 49
To solve this quadratic equation, we can rearrange it to the standard quadratic form:
q^2 + 10q + 10 - 49 = 0
q^2 + 10q - 39 = 0
Now, we can solve this equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
q = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 1, b = 10, and c = -39:
q = (-(10) ± √((10)^2 - 4(1)(-39))) / (2(1))
q = (-10 ± √(100 + 156)) / 2
q = (-10 ± √256) / 2
q = (-10 ± 16) / 2
Now, we have two solutions for q:
q1 = (-10 + 16) / 2 = 6 / 2 = 3
q2 = (-10 - 16) / 2 = -26 / 2 = -13
Since we're dealing with a quantity of output, negative values do not make sense in this context. Therefore, we disregard the negative solution.
Hence, the chemical company will produce 3 units of output when the price is $49 in a perfectly competitive industry.