• Step 1: Axioms of consumer preferences (today)
• Step 2: Translate axioms into a utility function (today)
• Step 3: Introduce budget constraint → derive demand (next lecture)

Note on Non-satiation:

• Does NOT mean the 10th unit makes you as happy as the 9th
• Just means the 10th unit is better than not having it
• Applies to "goods" — things you want (not "bads" like getting poked in the eye)

• Choice A: 2 slices of pizza, 1 cookie
• Choice B: 1 slice of pizza, 2 cookies
• Choice C: 2 slices of pizza, 2 cookies

If indifferent between A and B, but prefer C to both → A and B are on the same indifference curve, C is on a higher one.

Property 1: Higher ICs are preferred

Reason: Non-satiation (more is better)

IC₂ is farther from the origin than IC₁ → more pizza AND more cookies → IC₂ is preferred.

Property 2: ICs are downward sloping

Reason: Non-satiation (can't be indifferent between less and more)

Therefore, to stay indifferent when gaining cookies, you must give up pizza → IC slopes downward.

Property 3: ICs never cross

Reason: Transitivity + Non-satiation

Therefore, indifference curves can never cross.

Property 4: Only one IC through any point

Reason: Completeness (must know how you feel)

Therefore, exactly one IC passes through each point.

A graduate student choosing between jobs based on two dimensions:

• X-axis: Weather quality
• Y-axis: School/job quality

Options: Princeton (great quality, mediocre weather) vs. Santa Cruz (good quality, great weather)

Decision: Chose IMF (DC) — better weather than Princeton, better job than Santa Cruz

Note: There's no "right" answer — preferences are personal.

U = √(S × C)

where S = slices of pizza, C = cookies

Student question: "U = S × C also gives the same ranking. Why use √?"

Answer: Different forms have different properties:

All three give the same ranking, but √ reflects real-world behavior: the 10th slice isn't as satisfying as the 1st.

Utility is meaningful only for ranking, not for measuring absolute happiness.

Key point: U = S×C, U = √(S×C), U = ln(S×C) all give the same ranking → they're all valid utility functions for the same preferences.

Compare bundles (2,3) vs (3,2):

Numbers differ, but ranking is identical!

Total Utility vs Marginal Utility:

• U(3) = Total satisfaction from 3 units
• MU(3) = Additional satisfaction from the 3rd unit = U(3) - U(2)

MU_C = ∂U/∂C = S / (2√(S×C))

MU_S = ∂U/∂S = C / (2√(S×C))

Partial derivative = "Hold other variables constant, how does U change with C?"

• Regular derivative (d/dx): only 1 variable
• Partial derivative (∂/∂x): multiple variables, hold others constant

The symbol ∂ (instead of d) means: treat S as a constant, then differentiate with respect to C.

MU_C = "If I hold pizza constant and eat one more cookie, how much happier am I?"

U = (S × C)^1/2

Treat S as constant, differentiate with respect to C:

∂U/∂C = (1/2) × S^1/2 × C^-1/2
      = (1/2) × √S / √C
      = (1/2) × √(S/C)
      = S / (2√(S×C))

Intuitive Example — Hunger:

You haven't eaten all day. Pizza arrives!

1st slice: "Amazing!!!"     → MU = 😍😍😍😍😍 (5)
2nd slice: "Still great"    → MU = 😍😍😍😍 (4)
3rd slice: "Pretty good"    → MU = 😍😍😍 (3)
4th slice: "Getting full"   → MU = 😍😍 (2)
5th slice: "Can't eat more" → MU = 😍 (1)

Key: Total utility keeps increasing (5 slices > 4 slices), but marginal utility decreases.

MRS = ΔS/ΔC = −MU_C/MU_S

= Slope of the indifference curve = Ratio of marginal utilities

Key idea: Moving along an IC means utility doesn't change (ΔU = 0)

Step 1: Total change in utility from changing both goods:

ΔU = (ΔS × MU_S) + (ΔC × MU_C)

= (change in pizza × utility per pizza) + (change in cookies × utility per cookie)

Step 2: On an IC, utility doesn't change:

ΔU = 0

∴ ΔS × MU_S + ΔC × MU_C = 0

Step 3: Solve for ΔS/ΔC:

ΔS × MU_S = −ΔC × MU_C

ΔS/ΔC = −MU_C/MU_S

Result:

MRS = −MU_C/MU_S

MRS = −MU_C/MU_S = −[S/(2√SC)] / [C/(2√SC)] = −S/C

As you move from A → B → C:

• Pizza decreases (4 → 2 → 1) → Pizza becomes more valuable
• Cookies increase (1 → 2 → 4) → Cookies become less valuable
• |MRS| decreases (4 → 1 → 0.25) → Less willing to give up pizza for cookies

This is why indifference curves are convex to the origin!

Problem:

• At (8 pizza, 1 cookie): MRS = −1/8 → barely willing to trade pizza for cookies
• At (4 pizza, 7 cookies): MRS = −7/4 → willing to give up almost 2 pizzas for 1 cookie

This doesn't make sense! Why would you give up MORE pizza when you have LESS of it?

Why?

• The first 16 oz satisfies most of your thirst
• The additional 16 oz is nice, but not as valuable
• Companies know consumers won't pay proportionally more for larger sizes
• Marginal cost to McDonald's: ~$0.03 (small) vs ~$0.04 (large)

Business implication: Companies price based on diminishing MRS — proof that they believe preferences are convex!