Derive marginal productivity conditions in DW Jorgenson paper I'm reading Jorgenson, Dale W. (1963), "Capital Theory and Investment Behavior", American Economic Review. Vol 53, No. 2, pp. 247-259. http://www.jstor.org/stable/pdfplus/1823868.pdf  

I would like to know how to derive the marginal productivity conditions given in pg 4.
This is a deterministic Optimal Control problem in continuous time. Jorgenson uses nominal terms, so prices are explicitly present. To copy the original



and we want to arrive at  

  

A standard neoclassical production function, is a constant-returns to scale one over Labor $L$ and capital $K$, $Q = F(K,L), F_L, F_K &gt;0, F_{LL}, F_{KK} &lt;0$, satisfying also the Inada limit conditions, while the law of motion of capital is described verbally and it is the standard one, $\dot K = I-\delta K$.  

To solve this we can set up the current value Hamiltonian (analogous to the Lagrangean)

$$H = R(t) - D(t) + \lambda \cdot [I-\delta K]$$

where $\lambda$ is a multiplier, and we maximize it over the decision variables $L$ and $I$, and the state variable $K$.
The first order conditions are

$$\frac {\partial H}{\partial L} = 0 \Rightarrow \frac {\partial R(t)}{\partial L}   -\frac {\partial D(t)}{\partial L}=0$$

$$\Rightarrow \frac {\partial }{\partial L} \left(pQ-sL\right)  -u\frac {\partial }{\partial L}\left(pQ-sL\right)=0 $$

$$\implies (1-u)\left(p\frac {\partial Q}{\partial L} - s\right) = 0 \implies \frac {\partial Q}{\partial L} = s/p  \tag{1}$$ 

Also

$$\frac {\partial H}{\partial I} = 0 \implies -q +\lambda =0 \implies q = \lambda \tag{2}$$

from which we also obtain (to be needed next)

$$\dot \lambda = \dot q \tag{3}$$

The first-order condition related to the state variable $K$, when we use the current value Hamiltonian is

$$\frac {\partial H}{\partial K} = r\lambda - \dot \lambda$$

$$\implies \frac {\partial R(t)}{\partial K}   -\frac {\partial D(t)}{\partial K} = r\lambda - \dot \lambda$$

$$\implies p\frac {\partial Q }{\partial K} -up\frac {\partial Q }{\partial K} + (v\delta q + wrq - x\dot q) -\lambda \delta = r\lambda - \dot \lambda  \tag {4}$$

The rest are algebraic manipulations (are they correct?)

A good educational presentation on the workings of Optimal Control in Economics is to be found here, mathematical appendix A.3, p. 604. Dorfman's 1969 paper remains a classic in mapping the mathematical structure of Optimal Control to economic concepts and intuition, Dorfman, R. (1969). An economic interpretation of optimal control theory. The American Economic Review, 817-831.