<p>This is a deterministic Optimal Control problem in continuous time. Jorgenson uses nominal terms, so prices are explicitly present. To copy the original
<img src="https://i.sstatic.net/BjVKu.png" alt="enter image description here">
<img src="https://i.sstatic.net/23r9o.png" alt="enter image description here"></p>

<p>and we want to arrive at  </p>

<p><img src="https://i.sstatic.net/nF40S.png" alt="enter image description here">  </p>

<p>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, $\dot K = I-\delta K$.  </p>

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

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

<p>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</p>

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

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

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

<p>Also</p>

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

<p>from which we also obtain (to be needed next)</p>

<p>$$\dot \lambda = \dot q \tag{3}$$</p>

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

<p>$$\frac {\partial H}{\partial K} = r\lambda - \dot \lambda$$</p>

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

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

<p>The rest are algebraic manipulations (are they correct?)</p>

<p>A good educational presentation on the workings of Optimal Control in Economics is to be found <a href="http://down.cenet.org.cn/upfile/8/200751171644184.pdf" rel="nofollow noreferrer">here</a>, 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, <a href="http://www.jstor.org/stable/1810679" rel="nofollow noreferrer">Dorfman, R. (1969). An economic interpretation of optimal control theory. The American Economic Review, 817-831.</a></p>

CC BY-SA 3.0
2015-01-27T11:02:59.253Z
2015-01-27T13:38:22.007Z
2015-01-27T13:38:22.007Z
Alecos Papadopoulos
<p>I'm reading Jorgenson, Dale W. (1963), "Capital Theory and Investment Behavior", American Economic Review. Vol 53, No. 2, pp. 247-259. <a href="http://www.jstor.org/stable/pdfplus/1823868.pdf" rel="nofollow">http://www.jstor.org/stable/pdfplus/1823868.pdf</a>  </p>

<p>I would like to know how to derive the marginal productivity conditions given in pg 4.</p>

<theory><investment>
Derive marginal productivity conditions in DW Jorgenson paper
JP83
# Derive marginal productivity conditions in DW Jorgenson paper

(Asked by: username_0 on 2015-01-27)

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.

**Tags:** theory, investment


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## 1 answers

### Answer by username_1 on 2015-01-27. Score: 4

This is a deterministic Optimal Control problem in continuous time. Jorgenson uses nominal terms, so prices are explicitly present. To copy the original
![enter image description here](https://i.sstatic.net/BjVKu.png)
![enter image description here](https://i.sstatic.net/23r9o.png)

and we want to arrive at

![enter image description here](https://i.sstatic.net/nF40S.png)

A standard neoclassical production function, is a constant-returns to scale one over Labor $L$ and capital $K$, $Q = F(K,L), F\_L, F\_K >0, F\_{LL}, F\_{KK} <0$, 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](http://down.cenet.org.cn/upfile/8/200751171644184.pdf), 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.](http://www.jstor.org/stable/1810679)


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