Notes on Constrained Optimization with Leontief Utility Function

Introduction

• Discussion on constrained optimization faced by consumers using Leontief utility function.
• Named after economists W. Paul Samuelson and others.
• Commonly used for goods that are complements.

Leontief Utility Function

• Defined as:
  [ U(x_1, x_2) = \min(a x_1, b x_2) ]
  where a and b are strictly positive real numbers.
• Utility depends on the minimum of either component, signifying complementarity in goods.

Optimal Consumption Bundle

• To find the optimal consumption bundle, set:
  [ a x_1 = b x_2 ]
• Solving for x2 gives:
  [ x_2 = \frac{a}{b} x_1 ]

Budget Constraint

• Budget constraint formula:
  [ P_1 x_1 + P_2 x_2 = M ]
• Substitute x2 into the budget constraint:
  [ P_1 x_1 + P_2 \left( \frac{a}{b} x_1 \right) = M ]
• Factor out x1:
  [ x_1 \left(P_1 + \frac{a}{b} P_2 \right) = M ]
• Solve for optimal x1:
  [ x_1^* = \frac{M}{P_1 + \frac{a}{b} P_2} ]*

Finding x2 Star

• From the optimal condition:
  [ x_1 = \frac{b}{a} x_2 ]
• Substitute into budget constraint:
  [ P_1 \left( \frac{b}{a} x_2 \right) + P_2 x_2 = M ]
• Factor out x2:
  [ x_2 \left( \frac{b}{a} P_1 + P_2 \right) = M ]
• Solve for optimal x2:
  [ x_2^* = \frac{M}{\frac{b}{a} P_1 + P_2} ]*

Summary of Optimal Consumption Bundle

• Optimal consumption bundle in terms of x1 and x2:
  • [ x^* = \left( \frac{M}{P_1 + \frac{a}{b} P_2}, \frac{M}{\frac{b}{a} P_1 + P_2} \right) ]*

Indifference Curves and Budget Constraints

• Graphical representation of indifference curves with budget constraints.
• Touchpoint indicates optimal consumption (x1*, x2*).

Special Cases

• If the price of one good is zero, the budget line shifts.
• Assumes strictly positive prices for goods under normal conditions.

Bonus - Maximizing Utility with Different Function

• Alternate utility function:
  [ U(x_1, x_2) = \max(a x_1, b x_2) ]
• Decision based on bang for the buck (marginal utility divided by price).
  • Compare ( \frac{a}{P_1} ) and ( \frac{b}{P_2} ) to determine allocation of budget.

Optimal Consumption Decisions

• If ( \frac{a}{P_1} > \frac{b}{P_2} ): Spend all on good 1:
  [ x_1^* = \frac{M}{P_1}, x_2^* = 0 ]
• If ( \frac{a}{P_1} < \frac{b}{P_2} ): Spend all on good 2:
  [ x_1^* = 0, x_2^* = \frac{M}{P_2} ]
• If equal, any combination is valid:
  • ( (M/P_1, 0) ) or ( (0, M/P_2) )

Conclusion

• Covered consumer choice problem under Leontief utility function.
• Noted differences in decision-making compared to linear utility functions.