If Y=K+C, why the value of a marginal unit of capital is the same of the value of a marginal unit of output?
This is a single-good model, that is, output is of one good which can be either consumed or added to capital.  The dynamic optimisation problem is to maximise the discounted present value of utility, utility being a function of consumption meeting the standard conditions $U_C &gt; 0$ and $U_{CC} &lt; 0$. 

Perman et al identify the second of two state variables as capital $K$.  Now a state variable features in the statement of a dynamic optimisation problem in a limited number of ways:


An equation of state, in this case $dK_t/dt = Q(K_t,R_t) – C_t$.
An initial condition, in this case $K_{t=0} = K_0$.
A terminal condition, in this case by implication that (assuming an infinite
planning horizon) $K_t → 0$ as $t → \infty$.


Suppose that instead we were to specify the second state variable as the stock of the single good, say $G_t$.  Would this result in any essential changes to 1-3 above, which would change the solution of the problem?  It would of course if we allow for stocks of the good destined for consumption but not yet consumed, eg being packaged, transported or on shelves awaiting purchase.  But in a simple model such as this we will probably want to ignore such complications.  Suppose therefore we also assume that, for the part of output that is consumed, consumption is immediate.    Then the stock of the good would consist entirely of capital, and we could re-write 1-3 substituting $K$ by $G$ throughout.

This substitution would not affect the substance of the problem, so the same optimal solution would be obtained whether we make the substitution or not.  In particular, the time path of $\omega_t$ consistent with the optimal solution would be the same, whether $\omega_t$ is regarded as the shadow price of capital or the shadow price of the good.  Hence along an optimal path $\omega_t$ is also the value of a unit of output of the good.