How did MR/Qd= 400−0.5p become 800-4Q

The given demand function is:

$$QD = 400 - 0.5P$$

To find the inverse demand function, we need to express Price (P) in terms of Quantity Demanded (QD). Let's use $Q$ for $QD$.

$$Q = 400 - 0.5P$$

Rearrange to isolate $P$:

$$0.5P = 400 - Q$$

Multiply both sides by 2 to solve for $P$:

$$P = 2(400 - Q)$$ $$P = 800 - 2Q$$

This is the inverse demand function.

Total Revenue (TR) is calculated as Price (P) multiplied by Quantity (Q):

$$TR = P \times Q$$

Substitute the inverse demand function for $P$:

$$TR = (800 - 2Q)Q$$ $$TR = 800Q - 2Q^2$$

Marginal Revenue (MR) is the derivative of the Total Revenue (TR) function with respect to Quantity (Q):

$$MR = \frac{dTR}{dQ}$$ $$MR = \frac{d}{dQ}(800Q - 2Q^2)$$

Differentiate each term:

$$\frac{d}{dQ}(800Q) = 800$$ $$\frac{d}{dQ}(-2Q^2) = -2 \times 2Q^{2-1} = -4Q$$

So, the Marginal Revenue function is:

$$MR = 800 - 4Q$$

This shows how $QD = 400 - 0.5P$ becomes $MR = 800 - 4Q$.