This is the first-order condition of a dynamic programming problem where I am trying to get the Euler equation from a sequential problem.

(1) $$\frac{\partial V(d_2)}{\partial d_3} = \frac{-1}{d_2-d_3} + \frac{\beta}{d_3} = 0$$

(2)$$d_3 = \frac{\beta d_2}{1+\beta}$$

Can anyone plz show the steps of this simplification from stage (1) to stage (2)?

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    $\begingroup$ Have you actually tried solving the FOC for $d_3$? $\endgroup$ Mar 19, 2022 at 20:50
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    $\begingroup$ Since the OP took the time to rewrite the question from a picture into math-setting I assume the OP has also tried the algebra and is desperate. Hint: Try multiplying with $d_2-d_3$ ... $\endgroup$ Mar 20, 2022 at 12:21
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    $\begingroup$ @JesperHybel That wasn't OP. $\endgroup$ Mar 20, 2022 at 13:49
  • $\begingroup$ Haha ok my bad .... $\endgroup$ Mar 20, 2022 at 17:04

1 Answer 1


Rearrange $$\frac{-1}{d_2-d_3} + \frac{\beta}{d_3} = 0$$ to $$\frac{\beta}{d_3} = \frac{1}{d_2-d_3}$$ Flip it: $$\frac{d_3}{\beta} = d_2-d_3$$ The equation is now linear in $d_3$.


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