Lecture 3: Budget Constraints and Constrained Choice

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MIT 14.01 Principles of Microeconomics | Fall 2023 | Prof. Jonathan Gruber

Core Message

"Last time we let you have as much money as you want. Today is the bad news. Today we impose a budget constraint."

Consumer optimization occurs where the indifference curve is tangent to the budget constraint. This is the point where MRS = MRT.

Where We Are in Consumer Theory

Step 1 (Lec 2) Preferences → Utility Function → Indifference Curves ✓ Done
Step 2 (Lec 3) Budget Constraint → Constrained Choice ← Today
Step 3 (Lec 4) Derive Demand Curves Next

1. Budget Constraint

Key Assumption: No Saving or Borrowing

Budget = Income

You spend all your income on consumption goods. No saving, no borrowing.

"Sadly, this is not a terrible approximation for Americans. The average American has less than $1,000 in available savings to access."

Budget Constraint Equation

Y = PC × C + PS × S

Y Your income (given)
PC × C Total spending on cookies = (price per cookie) × (number of cookies)
PS × S Total spending on pizza = (price per slice) × (number of slices)

Numerical Example

Parameter Value
Income (Y) $24
Price of cookies (PC) $2
Price of pizza slice (PS) $4
X-intercept (all cookies) Y/PC = 24/2 = 12 cookies
Y-intercept (all pizza) Y/PS = 24/4 = 6 slices
Slope −PC/PS = −2/4 = −1/2

The Opportunity Set

Definition: The area under the budget constraint represents all affordable combinations.

Q: "If we assume all income is spent, why look at the area below the line?"

A: You will always be on the line. But the opportunity set matters because when it shrinks, you have fewer choices available → you're worse off.

2. Marginal Rate of Transformation (MRT)

Definition

MRT = −PC/PS = Slope of the budget constraint

The rate at which you can transform one good into another through the market.

"This is not a course about alchemy. We're not going to teach you how to transform cookies into pizza."

But given prices and income, opportunity cost means:

  • Every cookie you buy = 1/2 slice of pizza you can't buy
  • Every pizza slice you buy = 2 cookies you can't buy

MRS vs MRT: The Critical Distinction

Concept Formula What It Represents Determined By
MRS −MUC/MUS Rate you're willing to trade Your preferences (internal)
MRT −PC/PS Rate you can trade Market prices (external)

3. Real-World Application: Weight Watchers

Why diets don't work: Telling people "eat this, not that" doesn't work.

Why Weight Watchers works: It establishes a budget constraint and lets you optimize!

How It Works

  • Points (like prices): Each food item gets assigned points based on weight-gain potential
  • Budget (like income): You get a daily point budget based on your weight goal
  • Choice (optimization): You decide how to spend your points

McDonald's Lunch Example (30-point budget)

Option A: Classic Combo Option B: Lighter Choice
Big Mac 14 pts 10-piece Nuggets 12 pts
Fries 10 pts Apple slices 1 pt
Coke 6 pts Diet Coke 0 pts
Total 30 pts (done for the day!) Total 13 pts (17 left!)

"It empowers the consumer to not follow arbitrary rules but to follow their hearts within the constraint. That's constrained optimization."

4. Changes in the Budget Constraint

Two Ways the Budget Constraint Can Change

Change Effect Slope
Price change Budget line pivots Changes
Income change Budget line shifts parallel Unchanged

Case 1: Price Change (Pizza $4 → $6)

  • X-intercept (cookies): Unchanged at 12
  • Y-intercept (pizza): Decreased from 6 to 4
  • New slope: −2/6 = −1/3 (was −1/2)
  • Opportunity set shrinks → You are worse off

"Your income hasn't changed. You've still got 24 bucks. But you're worse off because your income is just a representation of what you can buy."

Case 2: Income Change ($24 → $20)

  • Slope: Unchanged at −1/2 (prices didn't change)
  • Both intercepts decrease
  • Opportunity set shrinks → You are worse off

What About Inflation?

  • Uniform inflation (all prices rise by same %): Looks like income decrease (parallel shift)
  • Differential inflation (prices rise by different %): Looks like price change (pivot)

5. Constrained Choice: Finding the Optimum

The Core Problem

  • Lecture 2: More is better (reach highest IC possible)
  • Lecture 3: You can't consume more than your budget allows

Solution: Choose the highest indifference curve that still touches the budget constraint.

Mathematical Condition: MRS = MRT

At tangency, the slopes are equal:

MRS = MRT

−MUC/MUS = −PC/PS

Or equivalently:

MUC/MUS = PC/PS

The "Bang for the Buck" Interpretation

Rearranging the optimization condition:

MUC/PC = MUS/PS

  • Bang = MU (happiness from next unit)
  • Buck = P (cost of next unit)
  • Bang per Buck = MU/P (happiness per dollar)

Rule: Consume until the happiness-per-dollar is equal for all goods.

Graphical Comparison of Points

Point Location Utility (U = √(S×C)) Why Not Optimal?
A On budget line √10 ≈ 3.16 Lower IC than D
D Tangent to budget line √18 ≈ 4.24 Optimal!
E Beyond budget line √32 ≈ 5.66 Can't afford it

6. Why Point A Is Not Optimal: Mathematical Proof

Setup

  • U = √(S × C), Point A: S = 5, C = 2
  • PC = $2, PS = $4

Step 1: Calculate MRS at Point A

MUC = 0.5 × 5 / √10 = 2.5 / √10
MUS = 0.5 × 2 / √10 = 1 / √10

MRS = −MUC/MUS = −2.5/1 = −2.5

Step 2: Calculate MRT

MRT = −PC/PS = −2/4 = −0.5

Step 3: Interpret the Disequilibrium

MRS = −2.5 You're willing to give up 2.5 pizzas for 1 cookie
MRT = −0.5 Market only requires 0.5 pizza for 1 cookie

Conclusion: You value cookies highly but the market says they're cheap! Great deal → Buy more cookies!

General Rule

Condition Action
|MRS| > |MRT| Buy more of x-axis good (cookies)
|MRS| < |MRT| Buy more of y-axis good (pizza)

7. Policy Application: SNAP vs Cash Transfers

What is SNAP?

SNAP = Supplemental Nutrition Assistance Program (formerly "food stamps")

  • Gives poor people a debit card that can only be used on food
  • Eligibility: Below or near poverty line (~$14,000/year)

Question: Why not just give people cash?

Two Types of People

Person Preferences Original Choice ($5,000 income)
Person X Loves shelter, doesn't eat much $4,800 shelter, $200 food
Person Y Loves food, minimal shelter needs $100 shelter, $4,900 food

Effect of $500 SNAP (Food Only)

Person Y: No change from cash! Already spending $4,900 on food.

"Money is fungible. I'll just relabel $500 of my food spending as SNAP."

Person X: Forced to change behavior!

  • Wanted (with cash): $5,200 shelter, $300 food
  • Gets (with SNAP): $5,000 shelter, $500 food

Lower indifference curve → Worse off!

Why Do Policymakers Use SNAP?

"Cross out the word 'shelter' and put in the word 'cocaine'."

Policymakers worry: "What if poor people spend cash on drugs instead of food?"

Empirical Evidence (J-PAL Experiments)

  • Positive: SNAP does change behavior — $1 in SNAP → $0.15 more food than cash
  • Normative: Cash recipients invest in education, businesses, health — rarely on "bad" things
  • Uganda: $150 cash → doubled earnings in 18 months

Key Takeaways

# Concept Key Point
1 Budget Constraint Y = PC×C + PS×S; Slope = −PC/PS
2 MRT Rate the market allows you to trade; slope of budget line
3 Optimization MRS = MRT (tangency of IC and budget line)
4 Bang for Buck MUC/PC = MUS/PS
5 Price Change Budget line pivots; slope changes
6 In-Kind vs Cash In-kind constrains choice; may reduce welfare

Key Terms

Term Definition
Budget Constraint The limit on consumption bundles imposed by income and prices
Opportunity Set All affordable consumption bundles (area under/on budget line)
MRT Marginal Rate of Transformation; rate market allows trading goods
Constrained Optimization Maximizing utility subject to budget constraint
Tangency Condition MRS = MRT; optimal point where IC touches budget line
Fungibility Property that money can be relabeled; $1 is $1 regardless of source
In-Kind Benefits Benefits given as specific goods/services rather than cash (e.g., SNAP)

Last updated: 2025-01-05

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