Understanding Marginal Revenue and Marginal Cost

This problem asks us to analyze the relationship between marginal revenue (MR) and marginal cost (MC) at a specific output level (Q = 100 units), given the total revenue (TR) and total cost (TC) functions.

To find marginal revenue (MR), we need to find the derivative of the total revenue (TR) function with respect to the quantity of output (Q). Marginal revenue is the change in total revenue from selling one additional unit of output.

The given total revenue function is: $TR = 1400Q - 6Q^2$

The marginal revenue (MR) function is the derivative of TR with respect to Q:

$MR = \frac{d(TR)}{dQ} = \frac{d(1400Q - 6Q^2)}{dQ}$

$MR = 1400 - 12Q$

Now, we need to calculate the marginal revenue at Q = 100 units:

$MR(Q=100) = 1400 - 12(100) = 1400 - 1200 = 200$

So, the marginal revenue at Q = 100 units is 200.

Next, to find marginal cost (MC), we need to find the derivative of the total cost (TC) function with respect to the quantity of output (Q). Marginal cost is the change in total cost from producing one additional unit of output.

The given total cost function is: $TC = 1500 + 80Q$

The marginal cost (MC) function is the derivative of TC with respect to Q:

$MC = \frac{d(TC)}{dQ} = \frac{d(1500 + 80Q)}{dQ}$

$MC = 0 + 80 = 80$

So, the marginal cost is constant at 80 for any level of output Q, including Q = 100 units.

Now we compare the calculated marginal revenue and marginal cost at Q = 100 units:

$MR = 200$

$MC = 80$

Comparing these values, we see that $MR (200) > MC (80)$ at Q = 100.

Analyzing the Statements

Let's evaluate the truthfulness of the statements given in the options based on our calculations at Q = 100:

• Statement A: MR > MC
  At Q = 100, MR = 200 and MC = 80. Since $200 > 80$, this statement is true.
• Statement B: MC = 80
  Our calculation of the marginal cost function showed $MC = 80$. This statement is true.
• Statement C: MR < MC
  At Q = 100, MR = 200 and MC = 80. Since $200$ is not less than $80$, this statement is false.
• Statement D: MR = MC
  At Q = 100, MR = 200 and MC = 80. Since $200$ is not equal to $80$, this statement is false.

Based on our analysis, statements A and B are true, while statements C and D are false at Q = 100 units of output.

Therefore, the correct option is the one stating that A and B are true only.

Revision Table: Marginal Revenue and Cost Analysis

Additional Information: Profit Maximization and MR/MC

In economics, firms aim to maximize profit. Profit ($\Pi$) is calculated as Total Revenue (TR) minus Total Cost (TC): $\Pi = TR - TC$.

Profit is maximized at the output level where marginal revenue equals marginal cost ($MR = MC$).

If $MR > MC$, as is the case at Q = 100 in this problem, it means producing one more unit adds more to revenue than it adds to cost. This implies that increasing output will increase profit (or reduce loss). Therefore, a firm currently producing where MR > MC should increase its output towards the level where MR = MC.

If $MR < MC$, producing one more unit adds less to revenue than it adds to cost. This implies that increasing output will decrease profit (or increase loss). A firm producing where MR < MC should decrease its output towards the level where MR = MC.

The condition $MR = MC$ is the necessary condition for profit maximization. The second-order condition is that the MC curve must be rising faster than the MR curve at the point of intersection.