Over the lecture we derived that the income paid to capital...

Over the lecture, we derived that the income paid to capital is MPK·K, while the income paid to labor is MPL · L. Assuming a Cobb-Douglas production function in the form of Y = AK^α L^(1−α), show that α can be interpreted as the share of total national income paid to capital.

The total production $P$ of a certain product depends on the amount $L$ of labor used and the amount $K$ of capital investment. In Sections 14.1 and 14.3 we discussed how the Cobb- Douglas model $P=b L^{\alpha} K^{1-\alpha}$ follows from certain economic assumptions, where $b$ and $\alpha$ are positive constants and $\alpha<1 .$ If the cost of a unit of labor is $m$ and the cost of a unit of capital is $n,$ and the company can spend only $p$ dollars as its total budget, then maximizing the production $P$ is subject to the constraint $m L+n K=p .$ Show that the maximum production occurs when $$L=\frac{\alpha p}{m} \quad$ and $\quad K=\frac{(1-\alpha) p}{n}$$

Partial Derivatives

Lagrange Multipliers

Question

Over the lecture, we derived that the income paid to capital is MPK·K, while the income paid to labor is MPL · L. Assuming a Cobb-Douglas production function in the form of Y = AK^α L^(1−α), show that α can be interpreted as the share of total national income paid to capital.

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