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A company has a production function: $$y=x_1^{\alpha}x_2^{1-\alpha}$$

where $0<\alpha<1$. Factor input 1 costs $w_1> 0$ and factor input 2 costs $w_2> 0$. The company wants to minimize its production costs when it produces y> 0 units of the output.

I have in a problem found that MC: $$MC=\frac{\partial c}{\partial y}=w_1((\frac{w_2\alpha}{w_1(1-\alpha)})^{1-\alpha})+w_2((\frac{w_1(1-\alpha)}{w_2 \alpha})^\alpha)$$

We see that marginal costs only depending on $w_1$ and $w_2$ and thus not on y. So we have that the marginal costs are constant. But why is that the case? What is the economic interpretation? I think maybe I should use returns to scale?

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Since the exponents add to one the production function has constant returns to scale, which means that, given factor prices, total cost is linear, which means that it's derivative (= marginal cost) is contant. If you change the exponent 1-alpha to beta where alpha+beta < 1, there will be decreasing returns to scale (but still homotheticity) and you will get increasing marginal cost.

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  • $\begingroup$ Another important result of constant returns to scale (and hence, constant marginal cost) is that it implies there are zero profits in a perfectly competitive economy $\endgroup$
    – Brennan
    Jan 3, 2021 at 5:57

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