Utility Maximization and Concavity Quiz

Concavity of the Objective Function

• A function f : Rn → R is strictly concave if for all k ∈ (0, 1) f (kx + (1 − k)y) > kf (x) + (1 − k)f (y). The value of the average is greater than the average of the values.
• f : R2 → R differentiable is strictly concave if:
  ∂ 2 f (x) ∂ 2 f (x) ∂ 2 f (x) ∂ 2 f (x)
  ∂x2i ∂x21 ∂x22 ∂x1 ∂x2

Convexity of the Decision Set

• The set D ⊂ Rm is convex if for all x, y ∈ D and all k ∈ (0, 1) z = kx + (1 − k)y ∈ D. Every average of two elements in D is itself an element of the set D.
• Convexity excludes holes.

Application: Utility Maximization

• The Cobb-Douglas utility maximization problem:
  max xa11 xa22.x∈D={x∈R2+ |g(x)=p1 x1 +p2 x2 −m=0}
• Checking strict concavity: ∂2 a = ai (ai − 1)xai i −2 xj j < 0 ⇔ ai ∈ (0, 1) ∂x2i 2 ∂2 ∂2 ∂2 2 2 − = a1 a2 (1 − a1 − a2 )x2a 1 x2 > 0? ∂x1 ∂x2 ∂x1 ∂x2
• The maximum theorem implies that for ai ∈ (0, 1), a1 + a2 < 1 the Cobb-Douglas demand functions xi (p, m) are well-defined and the indirect Cobb-Douglas utility function f (p, m) is strictly concave (⇒ risk aversion).

Parametric Dependence of the Maximum Value

• How does the maximized value f ∗ (θ) = maxx∈[x,x̄] f (x, θ) vary with the parameter θ?
• Consider the total differential: df ∗ (θ) df (x∗ (θ), θ) ∂f (x∗ (θ), θ) ∂f (x∗ (θ), θ) dx∗ (θ) = = +. dθ dθ ∂θ ∂x dθ ∂f (x∗ (θ),θ)
• The first-order condition requires that ∂x = 0.
• Envelope Theorem: df ∗ (θ) ∂f (x∗ (θ), θ) =. dθ ∂θ The total derivative of the envelope f ∗ (θ) is equal to the partial derivative.
• The name of the theorem comes from the fact that f ∗ (θ) represents the envelope of the family of curves {f (x, θ), x ∈ [x, x̄]}.

Summary

• Continuous objective functions take on a maximum on compact decision sets.
• Unique maxima exist for concave objective functions on convex decision sets.
• The parallelism of the gradients of the objective function and the constraint is a necessary (and often sufficient) condition for determining the maxima of constrained optimization problems.
• With a concave objective function, the optimal decision increases (decreases) with a parameter if it increases (decreases) the marginal effect of the decision variable.

Determination

Application: Utility Maximization

• Consider the utility maximization problem: max f (x1 , x2 ).x∈D={x∈R2+ |g(x)=p1 x1 +p2 x2 −m=0}
• Lagrange: If there exists a unique solution x∗1 > 0, x∗2 > 0, λ∗ ∈ R of the system ∂f (x∗1 , x∗2 ) ∂g(x∗1 , x∗2 ) = λ∗ = λp1 ∂x1 ∂x1 ∂f (x∗1 , x∗2 ) ∂g(x∗1 , x∗2 ) = λ∗ = λp2 ∂x2 ∂x2 g(x∗1 , x∗2 ) = 0 then (x∗1 , x∗2 ) is the solution to the utility maximization problem.
• Reminder: At x∗, the budget line and the indifference curve have the same slope: ∂f (x∗1 ,x∗2 ) ∂x1 p1 p1 − ∂f (x ∗ ,x∗ ) =− ⇔ GRS =. ∂x2 p2 p2

A Look at the Exercises

• The first exercise sheet contains a recipe for Lagrange problems.
• There are also a few practice examples for setting up the Lagrange problem itself and determining the solution.

Uniqueness

Maximum Theorem

• Under what conditions is the solution to a parameterized optimization problem max f (x, θ).x∈D(θ) unique, so that we can understand not only f ∗ (θ) but also D∗ (θ) = {x∗ (θ)} as a function of the parameter θ?
• Maximum Theorem Assuming
  • f (., θ) is continuous and strictly concave in x for all θ.
  • D(.) is continuous and for all θ D(θ) is compact and convex. Then f ∗ (θ) and D∗ (θ) = {x∗ (θ)} are continuous functions and f ∗ (θ) is strictly concave. If f (., θ) is only concave, then D∗ (θ) does not necessarily consist of a single element, but f ∗ (θ) is still continuous and concave.

Test your understanding of the concepts of concavity and convexity as they apply to objective functions and decision sets. This quiz covers definitions, properties, and applications, including the Cobb-Douglas utility maximization problem.