• Which bundles are preferred over others?

• We will allow three possible answers:

• We will allow three possible answers:

• We will allow three possible answers:

• We will allow three possible answers:

• We will allow three possible answers:

• Preferences are a list of all such comparisons between all bundles

See appendix in today's class page for more.

• For each bundle, we now have 3 pieces of information:

  • amount of x$x$
  • amount of y$y$
  • preference compared to other bundles

• How to represent this information graphically?

• Cartographers have the answer for us

• On a map, contour lines link areas of equal height

• We will use “indifference curves” to link bundles of equal preference

• Indifferent between all apartments on the same curve

• Indifferent between all apartments on the same curve

• Apts above curve are preferred over apts on curve

  • D≻A∼B∼C$D\succ A\sim B\sim C$
  • On a higher curve

• Indifferent between all apartments on the same curve

• Apts above curve are preferred over apts on curve

  • D≻A∼B∼C$D\succ A\sim B\sim C$
  • On a higher curve
• Apts below curve are less preferred than apts on curve
  • E≺A∼B∼C$E\prec A\sim B\sim C$
  • On a lower curve

• Indifference curves can never cross: preferences are transitive
  • If I prefer A≻B$A\succ B$, and B≻C$B\succ C$, I must prefer A≻C$A\succ C$

• Indifference curves can never cross: preferences are transitive

  • If I prefer A≻B$A\succ B$, and B≻C$B\succ C$, I must prefer A≻C$A\succ C$

• Suppose two curves crossed:

  • A∼B$A\sim B$
  • B∼C$B\sim C$
  • But C$C$ ≻$\succ$ B$B$!
  • Doesn't make sense (not transitive)!

• If I find another apt with 1 fewer friend nearby, how many more ft2$f{t}^2$ would you need to keep you satisfied?

• If I find another apt with 1 fewer friend nearby, how many more ft2$f{t}^2$ would you need to keep you satisfied?

• Marginal Rate of Substitution (MRS): rate at which you trade away one good for more of the other and remain indifferent

• Think of this as the relative value you place on good x$x$:

“I am willing to give up (MRS)$\left(MRS\right)$ units of y$y$ to consume 1 more unit of x$x$ and stay satisfied.”

• MRS =$=$ slope of the indifference curve

MRSx,y=−ΔyΔx=riserun$$MR{S}_{x,y}=-\frac{\Delta y}{\Delta x}=\frac{rise}{run}$$

• Amount of y$y$ given up for 1 more x$x$

• Note: slope (MRS) changes along the curve!

• Budget constraint (slope) from before measured the market’s tradeoff between x$x$ and y$y$ based on market prices

• MRS here measures your personal evaluation of x$x$ vs. y$y$ based on your preferences

• Foreshadowing: what if these two rates are different? Are you truly optimizing?

• Long ago (1890s), utility considered a real, measurable, cardinal scale^†

• Utility thought to be lurking in people's brains

  • Could be understood from first principles: calories, water, warmth, etc

• Obvious problems

• More plausibly infer people's preferences from their actions!

  • “Actions speak louder than words”

• Principle of Revealed Preference: if a person chooses x$x$ over y$y$, and both are affordable, then they must prefer x⪰y$x⪰y$

• Flawless? Of course not. But extremely useful approximation!

  • People tend not to leave money on the table

• A utility function u(⋅)$u(\cdot )$^† represents preference relations (≻,≺,∼)$(\succ ,\prec ,\sim )$

• Assign utility numbers to bundles, such that, for any bundles a$a$ and b$b$: a≻b⟺u(a)>u(b)$$a\succ b⟺u\left(a\right)>u\left(b\right)$$

• Now consider a 2^nd function v(⋅)$v(\cdot )$:

• Utility numbers have an ordinal meaning only, not cardinal

• Both are valid utility functions:^†

  • u(c)>u(b)>u(a)$u\left(c\right)>u\left(b\right)>u\left(a\right)$ ✅
  • v(c)>v(b)>v(a)$v\left(c\right)>v\left(b\right)>v\left(a\right)$ ✅
  • because c≻b≻a$c\succ b\succ a$

• Only the ranking of utility numbers matters!

• Two tools to represent preferences: indifference curves and utility functions

• Indifference curve: all equally preferred bundles ⟺$⟺$ same utility level

• Each indifference curve represents one level (or contour) of utility surface (function)

• Recall: marginal rate of substitution MRSx,y$MR{S}_{x,y}$ is slope of the indifference curve

  • Amount of y$y$ given up for 1 more x$x$

• How to calculate MRS?

  • Recall it changes (not a straight line)!
  • We can calculate it using something from the utility function

• Marginal utility: change in utility from a marginal increase in consumption

• Marginal utility: change in utility from a marginal increase in consumption

• Marginal utility: change in utility from a marginal increase in consumption

• Marginal utility: change in utility from a marginal increase in consumption

• I will always derive marginal utility functions for you

• Relationship between MU$MU$ and MRS$MRS$:

ΔyΔx⏟MRS=−MUxMUy

• See proof in today's class notes

“I am willing to give up MUxMUy units of y to consume 1 more unit of x and stay satisfied.”

“I am willing to give up MUxMUy units of y to consume 1 more unit of x and stay satisfied.”

• We can't measure MU's, but we can measure MRSx,y and infer the ratio of MU's!
  • Example: if MRSx,y=5, a unit of good x gives 5 times the marginal utility of good y at the margin

• Value is subjective

  • Each of us has our own preferences that determine our ends or objectives
  • Choice is forward looking: a comparison of your expectations about opportunities

• Preferences are not comparable across individuals

  • Only individuals know what they give up at the moment of choice

• Value inherently comes from the fact that we must make tradeoffs
  • Making one choice means having to give up pursuing others!
  • The choice we pursue at the moment must be worth the sacrifice of others! (i.e. highest marginal utility)

The Law of Diminishing Marginal Utility: each marginal unit of a good consumed tends to provide less marginal utility than the previous unit, all else equal

• As you consume more x:
  • ↓MUx
  • ↓MRSx,y: willing to give up fewer units of y for x

• Always willing to substitute between Two 1-L bottles for One 2-L bottle

• Perfect substitutes: goods that can be substituted at same fixed rate and yield same utility

• MRS1L,2L=−0.5 (a constant!)

• Always consume together in fixed proportions (in this case, 1 for 1)

• Perfect complements: goods that can be consumed together in same fixed proportion and yield same utility

• MRSH,B= ?

• A very common functional form in economics is Cobb-Douglas

u(x,y)=xayb

• Extremely useful, you will see it often!
  • Lots of nice, useful properties (we'll see later)
  • See the appendix in today's class page