# How do you find marginal revenue if you don't have a production function?

I was given a word problem. No formulas. So I set up the following equations:

Demand Function \begin{equation} D(p) = a - p \end{equation} Cost Function: \begin{equation} c(q) = 9 + 10q \end{equation} Marginal Cost \begin{equation} MC = 10 \end{equation}

I am supposed to find how much the man in the problem would produce for any value of $a$. But, without a production function, I am unsure how to determine how much he will produce.

Specifically, suppose I write

\begin{equation} R = pD(p) \end{equation}

I can't then differentiate R by q to find marginal revenue.

So how do I know how much he would produce?

Here are some hints. Either solve the profit function in terms of $p$, substituting out all references to $q$ (recall that $D(p) = q$), or solve the profit function in terms of $q$, substituting out all references to $p$. Then you'll find that the profit function is concave and the differentiated with respect to $p$ in the first case or $q$ in the second case will give you the profit maximizing solution.
EDIT: Yeah, $a = 2q + 10$ looks correct. Just solve the equation for $q$ and the interpretation is clear. That is, the quantity produced is $q = (a - 10)/2$. If you want to know the price, then just substitute $q$ in for $D(p)$ like so: $a - p = D(p) = q = (a - 10)/2$.
• I calculated the value of $a$: $a = 2q + 10$. But how am I supposed to interpret this? In this case, $a$ is just the $b$ in a standard 7th grade $y = mx + b$ formula, so it corresponds to the y intercept in Cartesian coordinates. But $q$ can't take negative values and presumably must be integers. Therefore, for $a < 30$ he wouldn't produce, but for $a \geq 30$ he would. Does that make any sense? Economics confuses me sometimes to the point where I forget how to add and subtract because I can't tell what I should be performing the operations on. – Stan Shunpike Jan 19 '15 at 23:26