In order to "ask my question", I have to solve a model first. I will omit some steps but still, this will unavoidably make this post very long -so this is also a test to see whether this community likes such kind of questions.
Before starting, I clarify that this may look totally like a standard neoclassical growth model in continuous time, but it is not: It is concerned with a single individual, which does not "represent" anybody else in the economy around him, an economy which is not modeled. The framework here is "application of Optimal Control to the maximization problem of a single individual". This is about the Optimal Control solution framework and method itself.
We solve the intertemporal utility maximization problem of a small businessman that owns the capital in his firm, while he purchases labor services in a perfectly competitive labor market, and he sells his product (fresh doughnuts) in a perfectly competitive goods market. We set the model in continuous time without uncertainty (socioeconomic conditions are steady), and with infinite horizon (the businessman envisions many future copies of him in a row):
$$\max_{c,\ell,k}\int_0^{\infty}e^{-\rho t}\ln c\,\text{d}t\\ \text{s.t.}\;\; \dot k = f(k,\ell) - w\ell - \delta k - c\\ \lim_{t\rightarrow \infty}e^{-\rho t}\lambda(t) k(t) = 0$$
where $c$ is the businessman's consumption, $\ln c$ is instantaneous utility from consumption, $\rho>0$ is the rate of pure time preference, $k$ is the firm's capital, $\delta$ is the capital depreciation rate, and $f(k,\ell)$ is the production function of the business. Initial level of capital is given, $k_0$. The businessman's own occupation with the business is subsumed into capital. The production function is standard neoclassical (constant returns to scale, positive marginal products, negative second partials, Inada conditions). The constraints are the law of motion of capital, and the Transversality condition using the current value multiplier.
Setting up the current value Hamiltonian
$$\hat H = \ln c +\lambda[f(k,\ell) - w\ell - \delta k - c]$$
we calculate the first-order conditions
$$\frac {\partial \hat H}{\partial c} = 0 \Rightarrow \frac 1c =\lambda \Rightarrow \frac {\dot c}{c} = -\frac {\dot \lambda}{\lambda}$$
$$\frac {\partial \hat H}{\partial \ell} = 0 \Rightarrow \lambda [f_{\ell} -w]=0 \Rightarrow f_{\ell} =w$$
$$\frac {\partial \hat H}{\partial k} = \rho\lambda - \dot \lambda \Rightarrow \lambda [f_{k} -\delta]=\rho\lambda - \dot \lambda$$
and combining them we obtain the law of evolution of consumption of our businessman ,
$$\dot c = \big(f_k-\delta -\rho \big)c \tag{1}$$
From the optimal rule for labor demand $\ell: f_{\ell} =w$ (static) and the constant returns to scale implication ($f = f_kk + f_{\ell}\ell$) we obtain $f-w\ell = f_kk$. Inserting this into the law of motion of capital we obtain
$$\dot k = f_kk - \delta k - c \tag {2}$$
Equations $(1)$ and $(2)$ form a system of differential equations. The steady-state values for the consumption and the capital of the businessman are
$$c^* = f_k^*k^* - \delta k^*,\;\;\; k^*: f^*_k = \delta +\rho \tag{3}$$
$$ \Rightarrow c^* = \rho k^* \tag{3a}$$
...which is a pretty familiar expression.
$k^*$ is sometimes called the "modified golden rule" level of capital. The Jacobian of the system evaluated at the steady state values has a negative determinant for any value of the model parameters, which is a necessary and sufficient condition for the system to exhibit saddle-path stability.
The maximum of the $\dot k =0$ locus is at the point, $\tilde k$ (sometimes called the "golden rule" level of capital)
$$\tilde k : f_{kk}(\tilde k)\tilde k + f_k(\tilde k) - \delta = 0\Rightarrow f_k(\tilde k) = \delta - f_{kk}(\tilde k)\tilde k \tag{4}$$
The $\tilde k$ value is important as a benchmark: it is the level of capital where $\dot k =0$ and $c$ is at a maximum (not optimal or steady state).
The $\dot c=0$ loci crosses the horizontal axis of the phase diagram (that measures capital) at the steady-state capital level $k^*$.
If $k^* > \tilde k$, which requires $f_k^* < f_k(\tilde k)$ due to negative second partials, we will have "over-accumulation of capital" (too many doughnuts): the businessman could enjoy more steady-state consumption with a lower level of capital. Using $(3)$ and $(4)$ we have
$$f_k^* < f_k(\tilde k) \Rightarrow \delta + \rho < \delta - f_{kk}(\tilde k)\tilde k$$
$$\Rightarrow \rho < - f_{kk}(\tilde k)\tilde k \tag {5}$$
Inequality $(5)$ is the condition for sub-optimal steady-state level of capital. And the thing is, we cannot rule it out. It simply requires that the businessman is "sufficiently patient", with a sufficiently small rate of pure time preference, but still positive.
Here starts the problem: overaccumulation of capital is effectively excluded in the representative agent model. It is possible in overlapping generation models, but as an unintended consequence at the macroeconomic level, one of the earliest examples that the macro-economy may be micro-founded and still behave differently than the micro-world.
But our model falls in neither category: it is a partial equilibrium model of a single agent in an implicitly heterogeneous environment -and general equilibrium here won't alter the results: this person represents only himself. So the problem is that if $(5)$ holds, then the Optimal Control solution will be obviously sub-optimal, because here we have a single person, a single will, a single mind: by looking at the solution our businessman will say, "hey, this method is worthless, if I follow its advice I will end up with a sub-optimally high level of capital".
And I am not satisfied to simply say "well, Optimal Control is not suitable for this problem, try another method", because I cannot see why we should consider it unsuitable. But if it is suitable, then the method should signal that something is wrong, it should at some point require that $(5)$ does not hold, in order to be able to offer a solution (if it so happens that $(5)$ does not hold, everything looks swell).
One could wonder "maybe the Transversality condition is violated if $(5)$ holds?" -but it doesn't look that it does, since $\lambda(t)k(t) = k(t)/c(t)$, which goes to a positive constant, while $e^{-\rho t}$ goes to zero, requiring only that $\rho>0$.
My questions:
1) Can somebody offer some insight here?
2) I would be grateful if somebody solved this using Dynamic Programming and reported the results.
ADDENDUM
From a mathematical point of view, the crucial difference of this model is that the optimized law of motion of capital, eq. $(2)$ includes not the whole output $f(k)$ as in the standard model, but only the returns to capital $f_kk$. And this happens because we have separated property rights over the output, which in the "individual business maximization problem" framework, is to be expected.
