# In the real world, do companies try to maximize return via maximizing profits or maximizing profits/costs (or neither)?

Suppose I buy $$s$$ shares of a stock at price $$p_0$$ and then later sell at price $$p_t$$. The initial and final values of my investment are then $$I_0=sp_0$$ and $$I_t=sp_t$$ respectively. The return is then
$$r=\frac{I_t-I_0}{I_0} =\frac{p_t-p_0}{p_0}$$ If we make the unrealistic assumption that stock price $$p_t$$ and earnings $$\pi_t$$ were perfectly linearly correlated using just a scalar, then $$p_t=\lambda\pi_t$$ Then we have $$\frac{p_t-p_0}{p_0} =\frac{\pi_t-\pi_0}{\pi_0}$$ Under this linear assumption then, the return would be maximized by maximizing $$\pi_t$$. In this circumstance then, profit maximization is identical to return maximization. Thus, we would say here that the goal of a company is to maximize the return for investors by maximizing profit in the future at time t. We would write this as $$y^*=\underset{y}{\text{argmax }} ⁡r(y)=\underset{y}{\text{argmax }}⁡\frac{\pi_t (y)-\pi_0}{\pi_0}= \underset{y}{\text{argmax }}\pi_t (y)=\underset{y}{\text{argmax }}⁡R(y)-C(y)$$ where $$y$$ is the amount you choose to produce, $$\pi_0$$ is the profit from the prior period, $$\pi_t (y)$$ is the profit from the next period, $$R(y)$$ is revenue, $$C(y)$$ is costs (both fixed and variable), and $$y^*$$ is the optimum quantity to produce.

However, let’s now consider a different way to think about return. Suppose you’ve got a bunch of money and want to start a company. You don’t get any investors and instead fund the entire company yourself. Every dollar you spend then is a cost because it costs you money to fund the company. Intuitively, the return should be $$\frac{\text{the amount you made net}}{\text{the amount you invested}}$$

In this case, the return would be

$$r(y)=\frac{R(y)-C(y)}{C(y)}$$

where $$y$$ is the amount of the good produced. Thus, your optimization problem would now be

$$y^*=\underset{y}{\text{argmax }}r(y)=\underset{y}{\text{argmax }}\frac{R(y)-C(y)}{C(y)}$$

Notice how this is not equivalent to the prior example for maximizing profit.

My question

My question is, in the real world, which logic do companies use?

Do they view maximizing return as the same as maximizing profit? Or do they use the second type of formula where return is profit/costs? Or is neither right?

# Example

Let \begin{align*} R(y)&=y+4y^2-y^3\\ C(y)&=y\\ \end{align*} Let our two different definitions of return be \begin{align*} \pi(y)&=R(y)-C(y)\\ r(y)&=\frac{R(y)-C(y)}{C(y)}\\ \end{align*}

Optimum for return type 1

$$\frac{\partial \pi}{\partial y} = 8y-3y^2\equiv 0$$ Then $$y^*=\frac{8}{3}$$

Optimum for return type 2

$$\frac{\partial r}{\partial y}=4-2y\equiv 0$$ Then $$y^*=2$$.

Plotting we can see these two optimums do not match in the horizontal dimension.

Therefore, the solutions are not identical because they do not predict the same optimum output.