# Writing constraint

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.

• This seems really straightforward. Could you explain where exactly your difficulty is? Dec 1 '19 at 16:11
• 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
– 1190
Dec 1 '19 at 16:16
• "the firm operates with the ... Ak production function": is it producing knowledge, or some other output? Dec 1 '19 at 19:59
• @AdamBailey I guess only knowledge. All question is this. I don’t know more or less. What is your opinion for constraint?
– 1190
Dec 1 '19 at 20:08
• 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. Dec 2 '19 at 8:54