CobbDouglas: Constant marginal costs and constant 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 &lt; 1, there will be decreasing returns to scale (but still homotheticity) and you will get increasing marginal cost.