Economics - Graphs on MC, MR, etc.

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I am having trouble understanding the theory behind “what” intersects “another” at its “max/min” like the question below. If anyone could share their tips on remembering the graphs, that would be much appreciated!
Q44) In the short run, the average product of labor:
Answer: is at the maximum where it intersects the marginal product of labor.
 
The graph of the marginal whatever always intersects the graph of the average whatever at either the maximum or minimum value on the average whatever curve.
Huh?
Substitute anything for “whatever” (but the same thing all three times):
  • Marginal cost intersects average cost (total or variable) at the minimum value on the average cost curve.
  • Marginal revenue intersects average revenue at the maximum value on the average revenue curve.
  • Marginal revenue product intersects average revenue product at the maximum value on the average revenue product curve.
Note that this works as well for things not related to finance; for example:
  • Marginal speed intersects average speed at the minimum or maximum value on the average speed curve.
  • Marginal acceleration intersects average acceleration at the minimum or maximum value on the average acceleration curve.
  • and so on
All that remains is to characterize how to determine whether the intersection point is the maximum average value, or the minimum average value. That one’s easy:
  • If the intersection occurs at the maximum value, then:
    • The marginal curve will start above the average curve
    • The marginal curve will end below the average curve
    • Therefore, the marginal values must be decreasing
  • If the intersection occurs at the minimum value, then:
    • The marginal curve will start below the average curve
    • The marginal curve will end above the average curve
    • Therefore, the marginal values must be increasing
For example:
  • Marginal revenue (as a function of quantity), generally starts out increasing (and above average revenue), but, after some point, it starts to decrease (diminishing marginal revenue); the intersection point will be the maximum average revenue.
  • Marginal cost (as a function of quantity) generally starts out decreasing (economies of scale) and below average cost, but, after some point, it starts to increase (diseconomies of scale); the intersection point will be the minimum average cost.
  • Marginal revenue product generally starts out increasing (and above average revenue product), but, after some point, it starts to decrease; the intersection point will be the maximum average revenue product.
 
For the problem “The graph of the marginal X always intersects the graph of the average X at either the maximum or minimum value on the average X curve”, I would like to bring another approach which uses derivative in calculus. We must only know the derivative of f(x)/x .
- Suppose X is Cost , we have
AVC = VC/ q (VC - Variable Cost, which is a function of q and q - products)
- So d(AVC)/ dq = d(VC/q) /dq = (dVC/dq) *1/q - VC/q^2
- But we have MC = dVC/dq (MC - Marginal cost) and AVC = VC/q
=> So, d(AVC)/dq = (MC - AVC)/q (remember that MC and AVC are always functions of q
- the AVC(q) function attains its maximum (or minimum) when d(AVC)/dq = 0 at the value q = q*
- At the value q = q*, MC(q*) - AVC(q*) = 0 => the curve AVC cuts the curve MC at the point q = q* which minimize (or maximize) the AVC.
- We obtain the same result with MC and ATC (Average total cost)
I use this method when I forget the relation between AVC, ATC and MC.
 
pivpomars wrote: For the problem “The graph of the marginal X always intersects the graph of the average X at either the maximum or minimum value on the average X curve”, I would like to bring another approach which uses derivative in calculus. We must only know the derivative of f(x)/x .
- Suppose X is Cost , we have
AVC = VC/ q (VC - Variable Cost, which is a function of q and q - products)
- So d(AVC)/ dq = d(VC/q) /dq = (dVC/dq) *1/q - VC/q^2
- But we have MC = dVC/dq (MC - Marginal cost) and AVC = VC/q
=> So, d(AVC)/dq = (MC - AVC)/q (remember that MC and AVC are always functions of q
- the AVC(q) function attains its maximum (or minimum) when d(AVC)/dq = 0 at the value q = q*
- At the value q = q*, MC(q*) - AVC(q*) = 0 => the curve AVC cuts the curve MC at the point q = q* which minimize (or maximize) the AVC.
- We obtain the same result with MC and ATC (Average total cost)
I use this method when I forget the relation between AVC, ATC and MC.
Click to expand...
For those who know calculus, this is a good approach. However, many CFA candidates never had to learn calculus (or have long since forgotten it).
 
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