I am struggling to understand the optimal choices of firms in the Ramsey-Cass-Koopmans model. Specifically, everywhere I see there is the following optimality condition for the interest rate, $r_{t}$,

$$ r_{t} = f'(k_{t}) - \delta_{k} $$

The information given to me is the following: Only one good is produced by competitive firms in the economy with the production function $F(K_{t}, \xi_{t}N_{t})$ $$ Y_{t} = K_{t}^{\alpha}(\xi_{t}N_{t})^{1 - \alpha} $$

where $\alpha$ denotes the share of capital income and $\xi_{t}$ a productivity term that grows at $g = \frac{\dot{\xi_{t}}}{\xi_{t}}$. Output can be consumed or invested $$ Y_{t} = C_{t} + I_{t} $$

Capital accumulates according to $$ \dot{K}_{t} = I_{t} - \delta_{k}K_{t} $$

where $\delta_{k}$ is the depreciation rate of physical capital.

Firms take input prices as given and choose how much labour to hire and capital rent to maximize their profit.

Then letting $f(k_{t}) = F(k_{t}, 1)$ with $k_{t} = \frac{K_{t}}{\xi_{t}N_{t}}$, assuming firms pay a price $r_{t}$ for renting a unit of capital, optimal capital requires the above mentioned condition (i.e. $r_{t} = f'(k_{t}) - \delta_{k} $).

Without making some sort of assumption about the interest rate I fail to see how to arrive to this FOC from the firm's problem. I found a text online () that stated the following:

Since we assume that households understand precisely how the economy works, they can predict the future path of wages and interest rates. That is, we assume perfect foresight. Owing to this absence of uncertainty, the consequences of a choice are known. And rates of return on alternative assets must in equilibrium be the same. So the (short-term) interest rate in the bond market must equal the rate of return on real capital in equilibrium, that is $$ r_{t} = \hat{r_{t}} - \delta_{k} $$ where $\delta$ (≥ 0) is a constant rate of capital depreciation. This no-arbitrage condition shows how the interest rate is related to the rental rate of capital.

My question then is, is this a condition that is given in the model and that my professor failed to mention? Or I am supposed to derive this from the information given?

Thanks in advance!


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