A firm's demand function is p = 10 e^(-x/400), where p is the price per unit and x is quantity demanded. Find at what level of output is the Marginal revenue zero.

Steps to Solve

1. Express Revenue in Terms of Quantity Revenue ($R$) is calculated as the product of price ($p$) and quantity ($x$). From the demand function, we have: $$ R = p \cdot x = 10 e^{-x/400} \cdot x $$

2. Differentiate Revenue to Find Marginal Revenue To find marginal revenue, we need to take the derivative of the revenue function with respect to quantity ($x$): $$ R = 10x e^{-x/400} $$

Using the product rule, we differentiate: $$ MR = \frac{dR}{dx} = \frac{d}{dx}(10x) \cdot e^{-x/400} + 10x \cdot \frac{d}{dx}(e^{-x/400}) $$

3. Apply the Derivative Calculating the derivatives, we have:

• For $10x$: $\frac{d}{dx}(10x) = 10$
• For $e^{-x/400}$: $\frac{d}{dx}(e^{-x/400}) = -\frac{1}{400} e^{-x/400}$

So, $$ MR = 10 e^{-x/400} - \frac{10x}{400} e^{-x/400} $$

Factoring out $e^{-x/400}$, we get: $$ MR = e^{-x/400}\left(10 - \frac{10x}{400}\right) $$

4. Set Marginal Revenue to Zero Setting $MR$ equal to zero: $$ e^{-x/400}\left(10 - \frac{10x}{400}\right) = 0 $$

Since $e^{-x/400} \neq 0$ for any finite $x$, we focus on the term in parentheses: $$ 10 - \frac{10x}{400} = 0 $$

5. Solve for Quantity $x$ Rearranging the equation, we get: $$ \frac{10x}{400} = 10 $$

Multiply both sides by $400$: $$ 10x = 4000 $$

Now, divide by 10: $$ x = 400 $$