Say a Cobb Douglas like:

$$\max_{X,Y\: s.t. X \cdot P_x+ Y \cdot P_y=I} U=X^\alpha Y^\beta$$

When it comes to maximization I would do the following way (for the fastest result):

x: $\alpha/(\alpha + \beta) = r; r*income = r_2$; $r_2$/price good = x$

Is there a similar (or really fast) method to solve for minimization problems?

  • 2
    $\begingroup$ What are you minimizing? $\endgroup$ – Herr K. Oct 25 '19 at 18:33
  • $\begingroup$ Some restriction or cost, say for exaplme PriceX + PriceY >= budget restriction. A standard Cobb Douglas minimization. $\endgroup$ – Omar Charming Khodr Oct 25 '19 at 20:13
  • $\begingroup$ I coudn't follow your math so I tried to clean it up with latex. Please look and confirm that the updated version is correct. $\endgroup$ – BKay Oct 25 '19 at 21:02
  • $\begingroup$ I believe you meant maximize utility subject to the constraint. Or did you imply minimize cost subject to utility? $\endgroup$ – Brennan Oct 26 '19 at 23:16

A shortcut to solve $$\min_{x,y} x^{\alpha} y^{\beta}$$ s.t. $$xp_x + yp_x \leq m$$ and $$0 \leq x,y$$ is to set $x = 0$ and/or $y = 0$, as this results in $U(x,y) = 0$, and the function does not map to negative values.


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