Final answer:
The firm's total cost function is the sum of fixed costs and variable costs. The average variable cost (AVC) is constant since the production function implies a constant ratio of $1 of input to output. Therefore, the most accurate statement about the firm's cost function is that the marginal cost is constant.
Step-by-step explanation:
The firm's production function is given by y=ax, where a>0 is a constant, and x is the variable input. Because the firm incurs a fixed cost of $100 and the price of the input is $1 per unit, the total cost (TC) function for the firm can be represented as TC = $100 + $1x. The average variable cost (AVC) is calculated by dividing the variable cost by the total output (y), which in this case is just $1 per unit, because the variable cost is equal to the price of the input times the number of units of input.
The total variable cost (TVC) in this case is $1x which equals y since y=ax. Hence, the AVC is $1x / y, which simplifies to $1/a, a constant, because a is a constant in the production function, and x/y equals 1/a. Therefore, the AVC does not change with the level of output y. Since AVC is constant, and fixed costs remain constant at $100 regardless of the output level, the average total cost (ATC) decreases as output increases, due to the spreading effect of the fixed cost over a larger number of units. After reaching a minimum point, ATC will then begin to increase if there were increasing variable costs at higher levels of output, but in this case, the cost function does not indicate such behavior.
Given this information, the most accurate of the given statements about the firm's cost function is B) Marginal cost is constant. This is because the marginal cost (MC) is the change in total cost when one more unit is produced, and since variable cost is constant at $1 per unit, marginal cost is also $1 per unit.