The Solow growth model

Question

I wonder why at the beginning $sf(k(t))$ is steeper than $\delta k(t)$, but at some point, it starts to become flatter than $\delta k(t)$?

Answer 1

The answer: it is an assumption
Solow-models usually assume that $f$ fulfills the Inada conditions, points 3. and 4. of which state $$ \begin{equation*} \lim_{k \to 0} f'(k) = \infty \\ \lim_{k \to \infty} f'(k) = 0. \end{equation*} $$

A frequent special case
If the production function $F(K,L)$ is of Cobb-Douglas type with constant returns to scale and has positive parameters, then the derived $f(k) = F(K,L)/L$ function will satisfy the Inada conditions.

Caveat
Based on the graphs they see, students of economics sometimes mistakenly assume that all strictly concave functions have these properties, but this is not the case. $f(k) = \sqrt{k} + k$ does not have the second one, while $f(k) = \sqrt{1+k}$ does not have the first one. Thus properties like $f$ being concave or non-linear are not sufficient, one has to assume the conditions.

Answer 2

Because $\delta k$ is linear function, so the slope of the function is always constant, whereas $sf(k)$ is typically nonlinear function exponential function like $f(k)=k^\alpha$ with $0<\alpha<1$, so mathematically such function starts with steep slope, but this slope always gets smaller (this can be seen from $d/dk= \alpha k^{\alpha -1}$.

However, note depending on value of parameter $\delta$, $\delta k$ could have equal or steeper slope but in such case capital accumulation is impossible because capital depreciates faster than it can be accumulated given $s$.