# 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?

In a simple model of welfare dynamic optimisation with non-renewable natural resource where:

$$\dot S_t = -R_t$$ $$\dot K_t = Q(K_t,R_t)-C_t$$

one of the first order condition is that $P_t=\omega_t Q_{R}$, where $P_t$ is the shadow price of the natural resource stock $S$, $Q_{R}$ is the marginal product of the natural resource and $\omega_t$ is the shadow price of the human-capital stock $K$.

In the Perman et al. (2011) "Natural resource and environmental economics" this equation is explained as "the marginal value (or shadow price) of the natural resource stock must be equal to the value of the marginal product of the natural resource". In particular $\omega_t$ is redefined as the value of one unit of output, with the justification that units of outputs and units of capital are in effect identical in this economy among the optimal path, as any output that is not consumed is added to capital.

It is this redefinition that I do not understand.

• Welcome to the site. By Pergman, did you mean Perman (& Ma, Common, Maddison & McGilvray)? – Adam Bailey Sep 15 '16 at 11:04
• yep, sorry.. I have corrected the typo.. – Antonello Sep 15 '16 at 11:30

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 $U_C > 0$ and $U_{CC} < 0$.
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:
1. An equation of state, in this case $dK_t/dt = Q(K_t,R_t) – C_t$.
2. An initial condition, in this case $K_{t=0} = K_0$.
3. 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.