CobbDouglas: Constant marginal costs and constant returns to scale

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?

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.

• 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 Jan 3 at 5:57