In-Depth Notes on Production, Profit Maximization, and Cost Minimization

Production Sets

• Cobb-Douglas Production Function:
  • The function is expressed as:
    $f(2z<em>1, 2z</em>2) = 2^a + B z<em>a z</em>b \text{ where }; a + b = 1$
  • Returns to Scale:
  • If $a + b = 1$: Constant returns to scale.
  • If a + b < 1: Decreasing returns to scale.
  • If a + b > 1: Increasing returns to scale.

Additivity

• Additivity Condition: If $y, y' \in Y$, then $y + y' \in Y$.
  • Implies free entry into production.
  • Notably, allows for scaling production by a positive integer k, making $ky \in Y$.

Convexity

• Convexity Assumption: The production set $Y$ must be convex.
  • This implies if $y, y' \in Y$ and $\alpha \in [0, 1]$, then $\alpha y + (1 - \alpha)y' \in Y$.
  • Highlights that input combinations that are unbalanced are not more productive than balanced ones.

Convex Cone

• Convex Cone: If for any production vectors $y, y' \in Y$ and constants \alpha > 0, \beta \geq 0, then $\alpha y + \beta y' \in Y$.
  • Proposition 5.B.1 states that if $Y$ is a convex cone, it satisfies both additivity and nonincreasing returns.

Profit Maximization and Cost Minimization

• Profit Function: Given a price vector p > 0 for $L$ goods, the profit function defined as:
  $\pi(p) = \max {p \cdot y : y \in Y}$.
• Supply Correspondence: Denoted by $y(p)$, defines the set of profit-maximizing vectors.
  • The graphical representation indicates that maximization occurs at the highest profit point tangent to the production set's boundary.

First-order Conditions

• If $y^* = y(p)$, the following holds: $p = \lambda \nabla F(y^*)$, where $\lambda$ is a Lagrange multiplier.
  • Interpretation: The price vector and the gradient of the function are proportional.

Cost Function

• Cost Minimization Problem (CMP):
  • Stated as: $\min W \cdot z$
  • Subject to: $f(z) \geq q$
• Cost Function: Denoted by $c(w, q)$ represents the minimal cost for producing output q at input prices w.

Efficient Production

• Definition: A production vector $y = Y$ is efficient if there is no $y' \in Y$ such that y' > y.
  • Efficiency captures situations where no other feasible production yields greater output using equal or fewer inputs.

Expected Utility Theory

• Lotteries: Introduces lotteries as risky alternatives represented by outcomes with known probabilities.
• Expected Utility Theorem:
  • States that if preferences over lotteries satisfy continuity and independence, they can be represented by a utility function with the expected utility form:
    $U(L) = \sum<em>{n=1}^{N} u</em>n P_n$