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One common production function used in economics is called the Cobb-Douglas production function. The Cobb-Douglas function is:

$$q(L,K) = L^{\alpha}K^{1 - \alpha}$$

...where 0 < $\alpha$ < 1.

What is the marginal product of labor for this production function?
Select Option $L^{\alpha - 1}K^{1-\alpha}$$K^{1-\alpha}$$L^{\alpha - 1}$$\alpha (\frac{K}{L})^{1-\alpha}$$\alpha (\frac{1}{L})^{\alpha}$

What is the marginal product of capital for this production function?
Select Option $L^{\alpha }K^{1-\alpha}$$L^{\alpha}$$K^{-\alpha}$$(1-\alpha) (\frac{K}{L})^{-\alpha}$$(1-\alpha) (\frac{1}{K})^{1-\alpha}$

What is the marginal rate of technical substitution for this production function?
Select Option $-\frac{L}{K}$$-\frac{\alpha}{1-\alpha}$$-\frac{L^\alpha}{K^{1-\alpha}}$$-\frac{\alpha}{1-\alpha}(\frac{K}{L})^{1-2\alpha}$$-\frac{\alpha}{1-\alpha}(\frac{K}{L})$
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