nhi 111lecture_2

Utility

• Lecture given by Dr. Leon Vinokur in ECN 111 - Microeconomics 1, Lecture 2.

Objectives

• Focus on representing preferences through a utility function.

Utility Functions and Indifference Curves

• Utility Function:

  • Defined as: U(x_1,x_2) = x_1x_2

  • Example Calculations:

  • U(2,3) = 6

  • Other combinations yielding the same utility:

    • (3,2), (1,6), (6,1), (0.5,12)…

• Continuous goods yield an infinite combination that results in the same utility level (e.g., K = 6).

Utility Functions Explained

• Alternative Utility Function Representation:

  • Monotonic Transformation: Define V = U^2

  • New function: V(x_1,x_2) = x_1^2 x_2^2

  • Example Calculation:

  • V(2,3) = 36, which is greater than V(4,1) = V(2,2) = 16

  • Preference order remains: (2,3) ≻ (4,1) ≻ (2,2).

• Another Transformation:

  • Define W = 2U + 10 (monotonic transformation)

  • New function: W(x_1,x_2) = 2x_1x_2 + 10

  • Comparison: W(2,3) = 22 > W(4,1) = W(2,2) = 18

  • Preservation of preference order is verified.

Other Utility Functions and Indifference Curves

• Perfect Substitution:

  • Scenario: Equal happiness from green and blue M&Ms; utility based on total quantity.

  • Utility Function: V(x_1,x_2) = x_1 + x_2

  • Indifference Curves:

  • Shaped as linear and parallel. Example curves:

    • x_1 + x_2 = 5

    • x_1 + x_2 = 9

    • x_1 + x_2 = 13.

• Different Valuation of M&Ms:

  • Hypothetical: 2 green M&Ms replace 1 blue M&M.

  • Utility Function: V(x_G,x_B) = x_G + 2x_B

  • General Representation: V(x_1,x_2) = ax_1 + bx_2

• Perfect Complementarity:

  • Utility Function: W(x_1,x_2) = ext{min}
    olimitsigreak{x_1,x_2ig}.

  • Indicates utility from complete pairs only.

• Indifference Curves for Perfect Complements:

  • Curved as right-angled with a vertex on a ray from the origin.

  • Representation of the function:

  • Example curves at ext{min}(x_1,x_2) = 8, ext{min}(x_1,x_2) = 5, ext{min}(x_1,x_2) = 3.

• Generalization for Perfect Complements:

  • Utility Function: u(x_1,x_2) = ext{min}(ax_1, bx_2) with coefficients indicating necessary proportions.

  • Example: For a complete pair of 2 units of x_1 with 1 unit of x_2.

  • Utility function form: u(x_1,x_2) = ext{min}(0.5x_1, x_2)

  • Alternative Monotonic Transformation: u(x_1,x_2) = ext{min}(x_1, 2x_2).

Quasi-linear Utility Functions

• Form: U(x_1,x_2) = f(x_1) + x_2

  • Example: U(x_1,x_2) = 2x_1^{1/2} + x_2.

• Indifference Curves:

  • Each curve acts as a vertically shifted copy of the others.

Cobb-Douglas Utility Functions

• Form: U(x_1,x_2) = x_1^a x_2^b with a > 0 and b > 0.

  • Example:

  • U(x_1,x_2) = x_1^{1/2} x_2^{1/2}

  • Or V(x_1,x_2) = x_1 x_2^3

• Indifference Curves:

  • Shape: Hyperbolic, asymptoting towards, but never touching, any axis.

  • Convexity indicates well-behaved indifference curves offering Cobb-Douglas preferences adaptive for various economic scenarios.

Marginal Utility

• Concept: Marginal Utility indicates the utility change when an additional unit of a good is consumed.

  • Marginal utility concerning good 1 is defined as the change in utility (∆U) from a small change in good 1 (∆x_1), holding good 2 constant.

  • Expressed mathematically: ∆U = MU_1∆x_1

• Specific utility function dependency affects the magnitude of marginal utility; values do not convey utility directly.

  • General formulation:

  • rac{ ext{MU}_x}{x} = rac{U(x_1, x_2)}{U(x_1 + ∆x_1, x_2)}.

Typical Assumptions About Utility Functions

1. Utility Increases with Goods:

   • Increase in either good leads to increased utility.

2. Positive Marginal Utility:

   • Positive marginal utility indicates that as one good increases, utility also increases.

   • For a utility function U(x,y):

   • rac{ ext{MU}_x}{ ext{MU}_y} > 0.

Extra Assumptions About Utility Functions

1. Diminishing Marginal Utility:

   • Utility increases but at a decreasing rate; the first hamburger may provide high utility, the second less so, and the third offers just okay satisfaction.

2. Continuity and Smoothness:

   • Utility functions are assumed to be continuous and smooth without abrupt changes.

Utility Curve Characteristics

• Positive Slope:

  • Utility curves rise due to positive marginal utility.

• Bowed Shape:

  • Indicates diminishing marginal utility; as consumption increases, the addition to utility becomes less significant.

  • Example points where marginal utility differs (e.g., MU at A is greater than MU at B).

Marginal Rate of Substitution (MRS) Relationship to MU

• MRS defined as:

  • MRS_{1,2} = - rac{MU_2}{MU_1}

  • Change in utility when adjusting consumption of the two goods keeping total utility consistent (ΔU=0), expressed through MRS.

Differential Calculus and MRS

• For any indifference curve represented by U(x_1,x_2)
  ightarrow k (a constant), deriving allows for partial differentiations that relate changes in each variable to the total utility.

• Total differentiation yields insights into utility's underlying structure.

Examples and Calculus in Utility Functions

• Example Application:

  • For U(x_1,x_2) = x_1x_2

  • Derivatives yield MRS through transformations reflecting the relationships among variables under modification.

Monotonic Transformations and MRS

• Monotonic transformations create new utility functions while preserving the original preference relations.

• It affects marginal rates of substitution only in their magnitude, while their ratio remains unchanged despite the constant transformation.

Key Takeaways

1. Understand Ordinal Utility and its implications.

2. Recognize the connection between utility functions and indifference curves.

3. Familiarize with monotonic transformations.

4. Differentiate types of preferences, including perfect substitutes, complements, quasi-linear, and Cobb-Douglas.

5. Comprehend Marginal Utility and its foundational roles in economics.

6. Explore the relationship between Marginal Utility and Marginal Rate of Substitution (MRS) in various contexts.