Cost Function and Scheme Lecture Notes

Introduction

• Discussion on cost function and its relation to cost minimization.
• Recap of cost minimization problem.

Cost Minimization Problem

• Objective: Minimize (W1X1 + W2X2)
  • (X1): Amount of input 1
  • (W1): Cost of one unit of input 1
  • (X2): Amount of input 2
  • (W2): Cost of one unit of input 2
  • Constraint: Produce at least (Y) amount of output

Two-Dimensional Case

• Used for easier visualization.
• N-dimensional generalization is possible with summation of (W_i X_i).
• The problem involves minimizing the total cost given these inputs and their costs.

Solved Cases

• Linear Technology
  • Isoquant/Production function: (AX1 + BX2)
  • Minimized cost: ( \min(W1/a , W2/b) , y )
  • Optimal amounts of inputs:
    • (X1^* = (W1/a) y) if (W1/a < W2/b)
    • (X1^* = 0) if (W1/a > W2/b)
    • (X2^*) similarly defined
  • General case scenarios discussed.

Conditional Input Demand Function

• Solution gives optimal input amounts, (X1^) and (X2^), as functions of (W1), (W2), and (Y).
• Named Conditional Input Demand function.
  • Function: Dependence on (W1), (W2), and (Y)
  • Demand: Reflects the producer's demand for inputs
  • Input: Refers to inputs (X1) and (X2)
  • Conditional: Conditioned by the need to produce at least (Y) units of output

Cost Versus Minimized Cost

• General cost dependence on market prices and input amounts: (C = f(X1, X2, W1, W2))
• Minimized cost: Use optimal inputs ((X1^, X2^)), making cost a function of (W1), (W2), and (Y).
• Cost Function: (C^*(W1, W2, Y))
  • Defines minimized cost given input prices and desired output
  • A notion of optimization inherent in the definition

Example Cost Functions

• Leontief Function: (Y = \min(aX1, bX2))
  • Cost function: (C^*(W1, W2, Y) = (W1/a + W2/b)Y)
• Cobb-Douglas Function: (Y = X1^\alpha X2^\beta)
  • Cost function: Similar derived form

Average Cost

• Average cost derived by dividing total cost by (Y).
• Observations in linear and Leontief technologies: Average cost independent of (Y).
• Important caution: Independence from (Y) is not universal.

Conclusion

• Understanding scale helps contextualize the behavior of cost functions.
• Key takeaway: Differentiation between general costs and optimized (minimized) costs.