1
$\begingroup$

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)?

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ Have you actually tried solving the FOC for $d_3$? $\endgroup$
    – Michael Greinecker
    Mar 19, 2022 at 20:50
  • 1
    $\begingroup$ 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 $d_2-d_3$ ... $\endgroup$
    – Jesper Hybel
    Mar 20, 2022 at 12:21
  • 2
    $\begingroup$ @JesperHybel That wasn't OP. $\endgroup$
    – Michael Greinecker
    Mar 20, 2022 at 13:49
  • $\begingroup$ Haha ok my bad .... $\endgroup$
    – Jesper Hybel
    Mar 20, 2022 at 17:04

1 Answer 1

Reset to default
3
$\begingroup$

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 $d_3$.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.