• 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 is slope of the indifference curve

  • Amount of y given up for 1 more 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

• Math (calculus): "marginal" means "derivative with respect to"

  • I will always derive marginal utility functions for you

• How to relate MU and MRS?

• Moving along an indifference curve

  • X and Y will change
  • MUx and MUy will change
  • Utility is constant (Δu=0)

• How to relate MU and MRS?

• Moving along an indifference curve

  • X and Y will change
  • MUx and MUy will change
  • Utility is constant (Δu=0)

MUxΔx+MUyΔy=Δu

• How to relate MU and MRS?

• Moving along an indifference curve

  • X and Y will change
  • MUx and MUy will change
  • Utility is constant (Δu=0)

MUxΔx+MUyΔy=ΔuMUxΔx+MUyΔy=0

• How to relate MU and MRS?

• Moving along an indifference curve

  • X and Y will change
  • MUx and MUy will change
  • Utility is constant (Δu=0)

MUxΔx+MUyΔy=ΔuMUxΔx+MUyΔy=0MUyΔy=−MUxΔx

• How to relate MU and MRS?

• Moving along an indifference curve

  • X and Y will change
  • MUx and MUy will change
  • Utility is constant (Δu=0)

MUxΔx+MUyΔy=ΔuMUxΔx+MUyΔy=0MUyΔy=−MUxΔxΔyΔx⏟MRS=−MUxMUy

MRS=ΔyΔx⏟slope=−MUxMUy

• Observing the choices that consumers make, given their options, give us insight into their preferences

• Represented in indifference curves and MRS

• Steepness of indifference curves tells us how consumers trade off between goods

• Let's look at extremes first

• Flatter → willing to give up few Downloads per Ticket (and vice versa)
• MRST,D is small

• More precise ways to classify objects:

• A good enters utility function positively

  • ↑ good ⟹ ↑ utility
  • Willing to pay (give up other goods) to acquire more (monotonic)

• More precise ways to classify objects:

• A good enters utility function positively

  • ↑ good ⟹ ↑ utility
  • Willing to pay (give up other goods) to acquire more (monotonic)

• A bad enters utility function negatively

  • ↑ good ⟹ ↓ utility
  • Willing to pay (give up other goods) to get rid of

• More precise ways to classify objects:

• A neutral does not enter utility function at all

  • ↑,↓ has no effect on utility

MRS=ΔyΔx⏟slope=−MUxMUy

• Curvature of indifference curves tells us how goods are related

• Relatively straight curves: goods are more substitutable for one another

• Relatively bent curves: goods are more complementary with one another

• Look at extreme cases first to see

• 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

• (MRS_{H,B}=) ?

• Straighter → more substitutable

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

u(x,y)=xayb

• Where a,b>0 (and very often a+b=1)
• Extremely useful, you will see it often!
  • Strictly convex and monotonic indifference curves
  • Other nice properties (we'll see later)
  • See the appendix in today's class page