Notes on Budget Lines, Demand Decomposition, and Consumer Surplus

Budget Line Analysis and Price Changes

• Initial Budget Line Division: When a "wrinkle bundle" is placed on the budget line, vertical and horizontal lines split the new budget line into three distinct areas:

  • Area 1: More b, less a.
  • Area 2: Less b, less a.
  • Area 3: More b, less a.

• Determining Optimal Area based on Price Change of a:

  • Ordinary Good a: If good a is ordinary, consumers buy less when it's more expensive. This restricts the optimal bundle to either of the two segments with less a (Area 1 or Area 3 in the initial three-way split, assuming a is on the horizontal axis).
  • Giffen Good a: If good a is Giffen, consumers buy more when it's more expensive, leading to the area with more a (Area 2).

• Considering Good b as a Substitute: To differentiate between the remaining ordinary good a areas, one considers good b as a substitute. If b is a substitute for a, and the price of a increased, more b should be consumed, implying being above a certain dashed line.

• Intermediate Budget Line Segmentation: The "vertical horizontal line thing" further splits the intermediate budget line into three segments:

  • Lower Segment: Less b, more a.
  • Middle Segment: More b, more a.
  • Upper Segment: Less a, but still more b.

• Income Shift and Good Classification: This segmentation relates to a shift in income, determining if goods a and b are normal or inferior. In a given example, both goods are assumed to be normal goods.

Demand Functions and Decomposition

• Step 1: Deriving Demand Functions: A crucial first step in analyzing consumer behavior, often requiring practice to go from a given utility function to derived demand functions.

• Finding Bob's New Optimal Bundle: Once demand functions are established, the new optimal bundle is found by plugging in the new price(s) into these functions.

• Decomposition (Income and Substitution Effects): A key analytical task is to decompose the total change in a consumer's demand for a good (e.g., x) into its income and substitution components. This is due to the income and substitution effects of a price change.

• Mathematical Clarification (Cross-out Terms): In algebraic manipulation, terms like $p_y$ and $p_y^2$ can be canceled out if they are factors of independent terms; however, $p_x x$ and $p_x^2$ cannot be canceled if they are part of separate terms connected by an addition operator (e.g., $p_x x + p_x^2$ versus $p_y / p_y^2$).

Calculating Bundles and Effects

• Bundle A (Original Bundle): This represents the initial consumption values of x and y before any price changes.

• Final Bundle ($X_{final}, Y_{final}$): This is the bundle chosen after the price change. Its cost is calculated as the amount needed to buy the original quantity of x ($x_A$) at the new price of x ($p_x'$), plus the amount needed to buy $y_A$ ($p_y y_A$).

• Intermediate Bundle Calculation Example: For a specific y value, the calculation yields $Y = 90 / (1+1^2 * 2) = 90 / (1+2) = 90 / 3 = 60$.

• Comparing Bundles to Determine Effects:

  • Substitution Effect (SE): Measures the change in quantity demanded due to a change in relative prices, holding utility constant. It is calculated by comparing the intermediate bundle ($x_{intermediate}$) to the original bundle ($x_{original}$).

    • Formula: $SE = x_{intermediate} - x_{original}$
    • Example: If the intermediate bundle $x$ is $10$ and the original $x$ was $15$, then $SE = 10 - 15 = -5$.

  • Total Change in x: This can be calculated in two ways:

    1. Summing the income effect ($IE$) and the substitution effect ($SE$): $Total Change = IE + SE$ (e.g., $IE + SE = -20$).
    2. Directly comparing the final consumption $x$ to the original consumption $x$.

• Graphical Representation: The substitution effect is visually represented by finding an indifference curve that is tangent to the 'dashed' (hypothetical) budget line at the intermediate bundle. This tangency means the indifference curve is parallel to the new budget line but just touches the old indifference curve, rather than passing through the original bundle.

Consumer Surplus Introduction

• Definition: Consumer surplus measures the welfare or how well-off consumers are from consuming a good or service.

• Derivation from a Special Utility Function: To quantitatively measure consumer surplus, we often start with a specific type of utility function:

  • $U(x, m) = g(x) + m$
  • Here, $x$ is the good of interest, and $m$ represents money spent on all other goods. This form implies that the benefit from an extra dollar spent on other goods is exactly $1$ dollar, essentially constraining the utility function to measure happiness in monetary terms.

Application to Market Equilibrium (Williamsburg Apartments)

• Losing Surplus: Surplus can be lost for distinct reasons in different market areas.

• Downward Sloping Demand in Apartment Market: Even for essential goods like apartments in Williamsburg, demand is assumed to be downward sloping. This means as apartment prices increase, the quantity demanded decreases. Reasons include: tenants choosing not to sublet extra rooms, effectively reducing the number of available 'apartments' or units of housing.

• Equilibrium and Consumer Behavior: At an equilibrium rent $r^*$, no one wishes to change their consumption or supply. Students not renting are on the higher portion of the demand curve, indicating they are unwilling to pay the equilibrium price. Students who secure apartments at $r_{bar}$, especially if they were willing to pay more, experience a consumer surplus (represented by a triangular area above $r_{bar}$ and below their demand curve).

• Total Surplus: The sum of consumer surplus and producer surplus is defined as total surplus ($Total Surplus = Consumer Surplus + Producer Surplus$).