Maximization of the Utility Function against the Budget Constraints.

Perfect Substitutes

With the form:

⇒ Go to the corner that gives the highest utility (=buy only one good!)

Perfect Compliments

  1. Get the formula that is the set of all kinked points in the utility function ⇒
  2. Get the Constraint formula ⇒
  3. Solve for the two equations (=get the intersection)

Quasilinear Optimization

Use Lagrangian optimization, but beware that the blue line might happen (=maximum point lies outside ): ⇒ In this case, go to the red (*) corner solution.

Kinked Budget Constraint

  1. Case 1: Inner kink
    1. Lagrangian for blue section
    2. Lagrangian for red section
    3. Choose the better one
  2. Case 2: Outer Kink
    1. Lagrangian for blue and red section
    2. If both solutions are unaffordable (= in the graph i.e. in graph,), go to the kink.

More than Two Goods

  1. Case 1: ← Pure three-var cobb-douglas
    • ⇒ Use 3-var lagrangian
  2. ==Case 2: ==
    1. Check that each term is Homogenous Degree of 1.
    2. Try the two-var lagrangian on the first term (assumping isn’t consumed)
    3. Try to maximize (assuming isn’t used)
    4. Choose the higher of the two utilities
      • If there is no concrete number, the cases differ on the conditions

Solution for Case 2:

Graphing a budget line with an indifference map, we can see that the bundle is where the consumer can achieve the most possible utility; where what is affordable = most possible utility

To find the bundle(=point) of maximum utility that is affordable, you can rephrase the problem as…

Worked Example: Constrained Optimization Problem

Using the Lagrange Method for ,

To simplify further: and thus let:

to construct a set of equations where:

and solve the three equations. Note that the Lagrange method doesn’t work when:

  • One or more goods are non-essential, meaning that the budget line crosses the axes → It’s a corner solution; i.e. the maximum point is at one of the intercepts, or at points where quantities of goods are negative
  • Tastes are non-convex, where there will be multiple solutions
  • Utility Function are kinked or otherwise non-differentiable