# Implicit Differentiation & Profit Function

Here is the question:

A firm produces one output commodity. The cost of producing and selling $x$ units and spending $y$ dollars on advertising is $C = cx + y + d$. The resulting quantity demanded is given by $x = \gamma ap + b + R(y)$ where $p$ is the price per unit. We assume that $R(0) = 0, R′(y) > 0$ and $R′′(y) < 0$. The constants $a, b, c, d$ are all positive.

1. Determine the profit function from selling $x$ units and spending $y$ dollars on advertising; call it $\pi(x, y)$.

2. Write and solve the following system of equations: $$\frac{\partial \pi(x^*,y^*)}{\partial x} = 0$$ $$\frac{\partial \pi(x^*,y^*)}{\partial y} = 0$$

where $x^* > 0$ and $y^* > 0$ represent the number of units sold and amount spent on advertisement that maximize profits. In this case, solving the system means finding one equation that only depends on $y$, the function $R(\cdot)$ (and its derivative), and the parameters of the problem $(a, b,...)$ - in other words an equation where $x^*$ has disappeared.

1. The equation found in the previous question defines $y^*$ implicitly as function of $a, b,$ and $c$. Find $\frac{\partial y}{\partial b}$ by implicit differentiation.

Embarrassingly, my problem seems to be with the profit function. If we get a revenue function by $p \cdot x$ we are left only with an $x$ in the cost function, I cannot figure out how to get a profit function which would result in $y^*$ as a function of $b$. Anyone know what I'm doing wrong?

• I can't read one of the symbols in your demand equation, right before the $ap$. I assume it's "gamma"? It would be helpful in the future if you used this site's MathJAX feature to write your equations. I'll edit it to make it clearer. Nov 15, 2016 at 5:19

I will only set up the first part of your question. See if you can move on from there.

If you try and set revenue as $p \cdot x$ immediately, you will get

$$p \cdot x = p \cdot (\gamma ap + b + R(y))$$

which as you note means $x$ will not be in this term. Profit could technically still be expressed in terms of $x, y$ in this case, but...

$$\pi(x, y) = p \cdot (\gamma ap + b + R(y)) - (cx + y + d)$$

is awkward since $x$ can be expressed in terms of $p$ and vice versa, so why would we have both in our equation?

So instead, solve for the inverse demand function:

$$x = \gamma ap + b + R(y) \implies \boxed{p = \frac{x - b - R(y)}{\gamma a}}$$

So you can solve for $p \cdot x$, but instead of substituting for $x$, you substitute for $p$. Once you solve for your first order conditions, you should be able to solve for $y$ as a function of variables including $b$.

Hope this helps.