Consumer Theory: Preferences, Utility and the Consumer Problem

Overview

• Consumer Theory focuses on the preferences, utility, and decision-making processes of consumers based on available resources.

Part 1: Optimization

• Unconstrained Optimization: Finding argument $x$ to maximize or minimize a function over a domain $D$.
  • Local Maximum: Function achieves a local maximum at $x^<em>$ if $f(x^</em>) \geq f(x)$ for all $x$ close to $x^*$.
  • Global Maximum: Achieved at $x^<em>$ if $f(x^</em>) \geq f(x)$ for all $x$ in $D$.
  • Local Minimum: Function achieves a local minimum at $\tilde{x}$ if $f(\tilde{x}) \leq f(x)$ for all $x$ close to $\tilde{x}$.
  • Global Minimum: Achieved at $\tilde{x}$ if $f(\tilde{x}) \leq f(x)$ for all $x$ in $D$.

Uniqueness of Solutions

• A function has:
  • Unique Local Maximum at $x^<em>$ if f(x^) > f(x) for all $x \neq x^*$;
  • Global Maximum if $f(x^<em>) > f(x)$ for all $x \neq x^</em>$ in the domain.
  • Equivalent definitions for unique local and global minima.

Necessary Conditions for Local Optima

• For a twice continuously differentiable function $f(x)$:
  • Local maximum at $x^*$:
  • $f' (x^<em>) = 0$ (First Order Condition), $f'' (x^</em>) \leq 0$ (Second Order Condition)
  • Local minimum at $\tilde{x}$:
  • $f' (\tilde{x}) = 0$, $f'' (\tilde{x}) \geq 0$.

Part 2: The Consumer Problem

• Marginal Utility (MU): Change in utility from consuming an additional unit of a good. (MU = \frac{\Delta U}{\Delta Q} )

Consumer Problem as a Model

• Represents consumer behavior in economic choices; conclusions can be approximate or imperfect.

Utility Function with Two Goods

• For utility function $U(x,y)$:
  • Marginal utility for each good:
  • $MU<em>x = \frac{\partial U}{\partial x}$, $MU</em>y = \frac{\partial U}{\partial y}$.
  • Example: $U(x, y) = x^{1/2} y^{1/2}$ gives $MU<em>x = \frac{1}{2} x^{-1/2} y^{1/2}$ and $MU</em>y = \frac{1}{2} x^{1/2} y^{-1/2}$.

Constraints in Consumer Problems

• Constraints reflect allowable combinations of goods under economic limits (e.g., $ax<em>1 + bx</em>2 \leq M$).

Lagrangian Method

• For optimization:
  • Lagrangian: $L = f(x<em>1, x</em>2) + \lambda c(x<em>1, x</em>2)$.
  • First order conditions:
  • $\frac{\partial L}{\partial x<em>1} = 0$, $\frac{\partial L}{\partial x</em>2} = 0$, $\frac{\partial L}{\partial \lambda} = 0$.

Economic Interpretation

• First-order condition reflects the Marginal Rate of Substitution (MRS) equating with price ratios.
• Solution graphically found at tangency of the budget line and the highest indifference curve.

Consumer Problem: General Form

• Consumption space defined by commodity prices and income, maximizing utility subject to budget constraints:
  • $\text{max} \ u(x)\ \ ext{s.t. } p \cdot x \leq y$ (where $x \geq 0$).

Properties of Indirect Utility Function

• Continuous and Homogeneous of degree zero:
  • Strictly increasing in income.
  • Decreasing in prices.
  • Satisfies Roy's Identity: (x(p, y) = \frac{\partial v(p, y)}{\partial p_i})

Primal and Dual Problems

• Utility Maximization (Primal Problem): Maximize utility with budget constraints.
• Expenditure Minimization (Dual Problem): Minimize cost to achieve a desired utility level.

Hicksian Demand Function

• Minimum expenditure for a given utility level:
  • $H(p,u) = \text{arg min } {p \cdot x : | : u(x) \geq u}$.

Conclusion

• Consumer Theory explores how individuals select consumption bundles to maximize utility with budget limitations, applying calculus methods to derive and understand these choices.
• Key concepts include Marshallian demand, indirect utility functions, dual problem interpretations, and analytical tools for deriving individual demand functions.