5.5 Summary

Things you should know

• Definitions: Optimal choice
  • Choice space: the set of all possible options, regardless of whether or not they’re feasible/affordable
  • Feasible set: the set of all options that are feasible/affordable to an agent
  • Preferences: a ranking by an economic agent of all options in the choice space
  • Optimal choice: the option from among the feasible set that the agent prefers the most
• Maximizing utility subject to a PPF
  • If $MRS > MRT$ at some point along a PPF constraint, an agent could improve their utility by moving to the right along the constraint, if possible
  • If $MRS < MRT$ at some point along a PPF constraint, an agent could improve their utility by moving to the left along the constraint, if possible
  • Under certain conditions, an agent’s optimal choice is the point along their PPF where their MRS is equal to the MRT. In other words, their optimum is characterized by two conditions: the tangency condition and the constraint.
• Lagrange multiplier method
  • The Lagrange multiplier method is method of solving a constrained optimization problem by converting it into an unconstrained optimization problem
  • The Lagrangian is formed by adding the objective function to an expression which is equal to zero along the constraint
  • The Lagrange multiplier measures the amount by which the objective function would increase if the constraint were relaxed; it is measured in the units of the objective function per unit of the constraint

Things you should be able to do

• Given any utility function — even functional forms you haven’t seen — and the equation of a constraint, solve for the utility-maximizing bundle along the constraint in cases where that bundle is characterized by a tangency condition.
• More generally, given any objective function and any constraint, set up the Lagrangian and solve for the optimal choice.