A firm accumulates useful knowledge $k$ by investing in R&D activities. Specifically, if the firm invests $r > 0$ dollars into R&D, the stock of useful knowledge grows by about $2\sqrt{r}$ units, minus the forgotten knowledge $\delta k$ where $\delta \in (0, 1)$. Suppose that the firm operates with the so-called Ak production function, with A > 0 being the fixed productivity, and let the price of output be normalized to 1.

The objective function is $$\int_0^{\infty}(Ak-r)e^{-it}dt$$

$k_0>0$ and $i>0$

I just want you to help me writing CONSTRAINT (LAW OF MOTION).

I don’t want you to solve this question. Just help me how to write its constraint according to above info in order to solve for maximization problem with hamiltonian function.

  • $\begingroup$ This seems really straightforward. Could you explain where exactly your difficulty is? $\endgroup$
    – Giskard
    Dec 1 '19 at 16:11
  • $\begingroup$ Indeed, I write $$ \dot{k}=Ak-2\sqrt{r}-\delta k$$ but I am not sure. Is it true? And my another small question is which is control variable and which is state variable here?@Giskard $\endgroup$
    – 1190
    Dec 1 '19 at 16:16
  • $\begingroup$ "the firm operates with the ... Ak production function": is it producing knowledge, or some other output? $\endgroup$ Dec 1 '19 at 19:59
  • $\begingroup$ @AdamBailey I guess only knowledge. All question is this. I don’t know more or less. What is your opinion for constraint? $\endgroup$
    – 1190
    Dec 1 '19 at 20:08
  • 2
    $\begingroup$ It is impossible to give a definite answer if "All question is this". My guess is $$\dot{k} = Ak+2\sqrt{r}−\delta k$$ (You had a sign error.) My other guess would be $$\dot{k} = 2\sqrt{r}−\delta k,$$ this is if the output of the production function is measured in dollars, not knowledge. $\endgroup$
    – Giskard
    Dec 2 '19 at 8:54

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