What is the last dollar rule for cost-minimization? Provide a ...

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What is the last dollar rule for cost-minimization? Provide a brief explanation (in words) as well as the corresponding mathematical equality.
 
  If the firm is producing at a point where the isocost line is steeper than the isoquant, what does the last dollar rule imply (i.e., where is the last dollar most productive, L or K) and how should the firm alter its capital and labor in the long run?

Question 2

The above figure shows the demand and supply curves in the market for milk. If the government imposes a quota at 500 gallons, calculate the deadweight loss.
 
  What will be an ideal response?

[fromAlphatoOmega22]

Answer to Question 1

The last dollar a firm spends on capital should have the same impact on output as the last dollar a firm spends on labor in order to be minimizing costs:
MPL/w = MPK/r
If the isocost is steeper than the isoquant, then
MRTS < w/r
This implies MPL/w < MPK/r, in which case the last dollar is more productive when employing capital. The firm should increase the amount of capital and decrease the amount of labor in order to minimize its costs.

Answer to Question 2

DWL = .5(4  2 )(1000  500 ) = 500