Lecture 8: Marginal Analysis, Utility, and Optimal Consumption

Objective: Introduces utility, marginal utility, and the optimal consumption rule, which explain individual consumer decision-making. They also illustrate how economic principles like marginal analysis and tax structures influence behavior, while providing essential information for an upcoming exam.

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., $U(x<em>1) = ax</em>1 + bx_1^2$).

• 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 ($x<em>1, x</em>2$) with given prices ($P<em>1, P</em>2$).

• 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 ($P<em>1/P</em>2$).

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

  • $MU<em>1/MU</em>2 = P<em>1/P</em>2$

  • This can also be interpreted as **equalizing