A cloth producing firm in a perfectly competitive market has the following short-run total cost function: TC
= 6000 + 400Q – 20Q2 + Q3. If the prevailing market price is birr 250 per unit of cloth,
A. Should the firm produce at this price in the short-run?
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To determine whether the firm should produce at the given price in the short run, we need to compare the market price (birr 250) with the firm's marginal cost.
In a perfectly competitive market, firms maximize their profits by producing at the quantity where marginal cost (MC) equals market price (P). So, we need to find the firm's marginal cost function and compare it with the market price.
To find the marginal cost, we take the derivative of the total cost function with respect to quantity (Q):
TC = 6000 + 400Q - 20Q^2 + Q^3
First, let's find the derivative of TC with respect to Q:
MC = d(TC)/dQ = 400 - 40Q + 3Q^2
Now, we can substitute the market price (P) into the marginal cost function to see if the firm should produce:
MC = P
400 - 40Q + 3Q^2 = 250
To find the quantity (Q) at which marginal cost equals the market price, we need to solve this equation. Let's rearrange the equation:
3Q^2 - 40Q + 150 = 0
To solve this quadratic equation, we can use the quadratic formula:
Q = (-b ± sqrt(b^2 - 4ac))/(2a)
Here, a = 3, b = -40, and c = 150. Plugging in these values, we can calculate Q.
Q = (-(-40) ± sqrt((-40)^2 - 4 * 3 * 150))/(2 * 3)
Q = (40 ± sqrt(1600 - 1800))/6
Q = (40 ± sqrt(-200))/6
Since the discriminant (b^2 - 4ac) is negative, the square root of a negative number is not defined, which means there are no real solutions to this equation. In other words, there is no quantity at which marginal cost equals the market price.
Therefore, the firm should not produce at this price in the short run.