Final answer:
The firm's average variable cost function is AVC(y) = y^2 - 8y + 30. The AVC decreases until y < 4 and increases for y > 4. Marginal cost equals AVC at an output of y = 6, and the firm will supply zero output if the price is below $16.
Step-by-step explanation:
The firm’s average variable cost (AVC) function can be found by subtracting the fixed cost from the total cost function and then dividing by the output (y). Since the fixed cost is 5 (the constant term in the total cost function), we subtract this from the total cost function and then divide by y. Therefore, the AVC function is AVC(y) = (y3 - 8y2 + 30y + 5 - 5) / y, which simplifies to AVC(y) = y2 - 8y + 30.
To address part (c) and (d), we look at the slope of the AVC function. Since it is a quadratic equation, the AVC curve will be U-shaped and reach its minimum where its derivative is zero. Taking the derivative of AVC and setting it to zero gives us the output level where AVC starts to increase. After calculating, we find that AVC is decreasing when y < 4 and increasing when y > 4.
For part (e), we set the marginal cost (MC) function equal to the AVC function to find the output level where MC = AVC. This occurs at the intersection point, and upon solving, we find it to be at y = 6.
For the firm to supply zero output, the price must be below the minimum of the AVC curve. Substituting y = 4 into the AVC function, we find that the price level must be below $16 (since that is the minimum AVC). This is found in part (f).
Next, in part (g), we're looking for the smallest positive amount that the firm will supply at any price. This is typically the point where MC starts to rise, which, for this cost function, occurs right from the start. Thus, the smallest positive output is just above zero (approaching zero).
Finally, for part (h), to determine the price at which the firm will supply exactly 6 units of output, we plug y = 6 into the MC function. MC(6) gives us the price which is $54.