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I'm attempting to solve a utility maximization problem for return-on-investment (ROI) across two different products, where each product experiences a different linear ROI curve.

  • For product one, the ROI curve is y=-2x+300 where x is the amount invested, and y is the ROI, with x > $0

  • For product two, the ROI curve is y=-1.25x+200 where x > $0.

I want to find the optimal allocation of budget across these two products such that total profit is maximized, where profit=(amount invested * ROI)-amount invested. I'm having a hard time finding the right approach to this problem: the standard geometric utility maximization approach doesn't work because there are no "prices" to use, and because the utility function isn't really known a priori.

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  • $\begingroup$ Hi Lando! Not sure where utility comes in here; aren't you trying to maximize profit or returns? $\endgroup$
    – Giskard
    Sep 29 at 18:57
  • $\begingroup$ Also, do you know calculus? Specifically first order derivatives? $\endgroup$
    – Giskard
    Sep 29 at 18:57
  • $\begingroup$ Yes, I know calculus. Yes, I'm trying to maximize profit but I thought utility might be the right framing. $\endgroup$
    – EBS
    Sep 29 at 19:05

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Your maximization problem is $$ \max_{x_1,x_2} x_1 \cdot (300 - 2x_1) + x_2 \cdot (200-1.25x_2) - x_1 - x_2 $$ subject to $x_1 + x_2 \leq \overline{x}$ where $\overline{x}$ denotes the budget. The first part is called the goal function, the inequality is a constraint.

Assume the constraint is not met (not all of the budget will be used) and calculate optimal values for $x_1,x_2$. Check if their sum is higher than the budget. If not, congrats you are done!
If the sum is larger than the budget, you will have to do constrained optimization or use some trick.

Approach 1. (You can simply use derivatives.)
Here you can just write that $x_2 = \overline{x} - x_1$, i.e. we always use the exact amount given by the budget, thus the constraint is met. Substitute this equation for $x_2$ into the goal function. Calculate maximum according to $x_1$.

Approach 2. (The geometric approach.)
The constraint $x_1 + x_2 = \overline{x}$ defines the budget line; in optimum this should be tangential to a level curve of the goal function. If you take the partial derivatives of the goal function, and set their ratio equal to the slope of the budget line, you get one optimality condition. The other equation is that of the budget line itself. Two equations, two unknowns. (A remark: the budget line does define the relative prices, to be exact the rate at which $x_1$ can be "transformed" into $x_2$. This rate is 1.)

Approach 3. (The Lagrange-method.)
This is more general and involved than the above, which is why they teach the geometric method instead. For a general treatment see the Wikipedia article.

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    $\begingroup$ Appreciate it! I was hung up on using profit vs. ROI in the maximization equation. This makes sense, thanks for the assistance. $\endgroup$
    – EBS
    Sep 29 at 19:45

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