cost function - relates cost to output level for future prediction  

• VC = bQ
  • linear function, implies constant marginal cost
• VC = bQ + gQ2
  • quadratic function, implies linear marginal cost
• VC = bQ + gQ2 + dQ3
  • cubic function, implies quadratic (U-shaped) marginal cost

Cobb-Douglas cost/production function - when production in form Q = AKaLb  

• C(Q) = wL(Q) + rK(Q)
  • use production function and MRTS to find L and K functions in terms of Q
• MRTS = w/r = MPL / MPK
  • w/r = (AbKaLb-1) / (AaKa-1Lb) = (bK) / (aL)
  • waL = rbK
  • L = (rb)K / (wa)
  • K = (wa)L / (rb)
  • substitute these back into the production function to find both L and K in terms of Q
  • no need to use lagrangian method
• substitute L and K functions back into the initial cost function to get final outcome
• note that in short-run, either K or L will be fixed
  • leaves the production function in terms of just K or L and makes it easy to solve
  • in finding total cost, don't forget to calculate the fixed cost as well