In standard microeconomics textbooks they usually assume that the cost curve consists of just two variables which are Capital and Labor ( I'm talking about this equation: TC = rK + wL)
So when we derive the marginal cost curve from the equation above and eventually get MC = w/MPL , the textbook I'm reading stated that when the price of labor (w) rises, the MC curve and AVC curve will shift upwards. I have two questions regarding this:
When the price of labor w rises, does the MC curve shift upwards without any change to its shape at all? Because when we assume that Labor (L) is a cubic function of Quantity (Q) (which means that the Total Variable Cost curve "TVC = wL" will also be a cubic function), with the MC and AVC curve being a quadratic function, when w rises in "wL" the shape of MC will definitely change because the constant number "w" attached as the leading coefficient to the MC curve equation will rise. So I want to ask if my logic is right and if the the shape of the MC curve will change while shifting upwards (To be more exact, I want to know if when "w" rises, the U-shaped MC curve will be folded more towards the middle mathematically, as I expressed above. For instance, f y=x^2 becomes y=2x^2, the U-shape will be folded more inwards. That's what I'm asking about above).
What happens if other Variable Costs besides the price of Labor shifts? I know that only "w" is included in the TVC model for simplicity, so I'm curious how we apply the changes to other Variable Costs (such as the price of raw materials) to the MC curve. If the price of raw materials rises, will the MC curve shift upwards? If so, does it shift upwards while changing the shape of the MC curve, like the first question above? I wish to know how this is mathematically applied to the original MC curve equation which only has "w" as the Variable Cost variable and nothing else.