Lecture 8: Marginal Analysis, Utility, and Optimal Consumption

Historical Tax Example

• Window Tax: Taxes levied on homes based on the number of windows (e.g., in the UK from 1747-1830).

  • Tax rate increased dramatically for 8 or more windows, then again at 12 and 20 windows.

  • Observed distribution shows a huge spike at 7 windows, indicating a behavioral response to avoid the tax.

• Hearth Tax: A Similar property tax based on the number of fireplaces (Byzantine Empire).

Exam Information

• Duration: 1 hour 10 minutes in class.

• Points: 60 points; budget 1 minute per point.

• Coverage: Everything up to the end of Wednesday's lecture.

• Preparation: Problem sets are crucial; don't just look at answers, work through them and relate to lectures. Supplementary video worked examples are available.

• Accommodation: Students with SAS accommodations must send their letter to Anya.

• Logistics: Write your name and UPI carefully on the exam for Gradescope scanning.

Marginal Analysis

• Core Principle: Determine the optimal level of an action by setting Marginal Benefit (MB) equal to Marginal Cost (MC) to maximize net benefit.

  • If MB > MC, increase the action. If MB < MC, decrease the action.

Utility and Marginal Utility

• Utility Function: A measure of happiness or enjoyment from consuming a good or service (e.g., U(x1) = ax1 + bx_1^2).

• Assumption: More X is good (always thinking of X as something positive or the absence of something negative).

• Types of Marginal Utility:

  • Diminishing: Each additional unit gives less happiness than the previous one (most pervasive).

  • Constant: Each additional unit gives the same happiness.

  • Increasing: Each additional unit gives more happiness.

• Marginal Utility (MU): The change in utility as consumption of a good (x) increases a little bit.

  • Mathematically, it's the slope of the utility function (MU_x = dU/dx).

• Marginal Benefit vs. Marginal Utility: Marginal utility is a specific application of marginal benefit when the benefit is measured in terms of individual happiness.

Optimal Consumption Rule (Two-Good Case)

• Problem: How to divide a fixed budget (y) between two goods (x1, x2) with given prices (P1, P2).

• Marginal Cost of a Good (in terms of another good): The number of units of the other good you must give up. This is the price ratio (P1/P2).

• Optimal Consumption Rule: Consumers choose allocations such that the ratio of marginal utilities equals the ratio of prices.

  • MU1/MU2 = P1/P2

  • This can also be interpreted as **equalizing