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How secure are you feeling about the material from Wednesday?
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 6
Example: draw the indifference curve for \(u(x_1,x_2) = \frac{1}{2}x_1x_2^2\) passing through (4,6).
Step 1: Evaluate \(u(x_1,x_2)\) at the point
Step 2: Set \(u(x_1,x_2)\) equal to that value.
Step 4: Plug in various values of \(x_1\) and plot!
\(u(4,6) = \frac{1}{2}\times 4 \times 6^2 = 72\)
\(\frac{1}{2}x_1x_2^2 = 72\)
\(x_2^2 = \frac{144}{x}\)
\(x_2 = \frac{12}{\sqrt x_1}\)
How to Draw an Indifference Curve through a Point: Method I
Step 3: Solve for \(x_2\).
How would this have been different if the utility function were \(u(x_1,x_2) = \sqrt{x_1} \times x_2\)?
\(u(4,6) =\sqrt{4} \times 6 = 12\)
\(\sqrt{x_1} \times x_2 = 12\)
\(x_2 = \frac{12}{\sqrt x_1}\)
Example: draw the indifference curve for \(u(x_1,x_2) = \frac{1}{2}x_1x_2^2\) passing through (4,6).
Step 1: Derive \(MRS(x_1,x_2)\). Determine its characteristics: is it smoothly decreasing? Constant?
Step 2: Evaluate \(MRS(x_1,x_2)\) at the point.
Step 4: Sketch the right shape of the curve, so that it's tangent to the line at the point.
How to Draw an Indifference Curve through a Point: Method II
Step 3: Draw a line passing through the point with slope \(-MRS(x_1,x_2)\)
How would this have been different if the utility function were \(u(x_1,x_2) = \sqrt{x_1} \times x_2\)?
This is continuously strictly decreasing in \(x_1\) and continuously strictly increasing in \(x_2\),
so the function is smooth and strictly convex and has the "normal" shape.
We've asserted that all (rational) preferences are complete and transitive.
There are some additional properties which are true of some preferences:
Strict monotonicity: any increase in any good strictly increases utility (\(MU > 0\) for all goods)
Weak monotonicity: no increase in any good will strictly decrease utility (\(MU \ge 0\) for all goods)
Example: Pfizer's COVID-19 vaccine has a dose of 0.3mL, Moderna's has a dose of 0.5mL
Take any two bundles, \(A\) and \(B\), between which you are indifferent.
Would you rather have a convex combination of those two bundles,
than have either of those bundles themselves?
If you would always answer yes, your preferences are convex.
1. Convexity does not imply you always want equal numbers of things.
2. It's preferences which are convex, not the utility function.
Continuous: utility functions don't have "jumps"
Smooth: marginal utilities don't have "jumps"
Counter-example: vaccine dose example
Counter-example: Leontief/Perfect Complements utility function
If preferences are strictly monotonic, strictly convex, continuous, and smooth, then:
Indifference curves are smooth, downward-sloping, and bowed in toward the origin
The MRS is diminishing as you move down and to the right along an indifference curve
Good 1 \((x_1)\)
Good 2 \((x_2)\)
"Law of Diminishing MRS"
Suppose one liter of soda gives you 20 "utils" of utility; so each one-liter bottle (good 1) gives you 20 utils,
and each two-liter bottle (good 2) gives you 40 utils.
Which other bundles give you the same utility as
12 one-liter bottles and 6 two-liter bottles (A)?
Which other bundles give you the same utility as
12 one-liter bottles and 2 two-liter bottle (B)?
12
24
Any combination that has 24 total liters
Any combination that has 16 total liters
16
20
4
8
12
24
16
20
4
8
A
B
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Suppose one liter of soda gives you 20 "utils" of utility; so each one-liter bottle (good 1) gives you 20 utils,
and each two-liter bottle (good 2) gives you 40 utils.
Which other bundles give you the same utility as
12 one-liter bottles and 6 two-liter bottles (A)?
Which other bundles give you the same utility as
12 one-liter bottles and 2 two-liter bottle (B)?
Any combination that has 24 total liters
Any combination that has 16 total liters
What utility function represents these preferences?
Red pencils and blue pencils, if you con't care about color
One-dollar bills and five-dollar bills
One-liter bottles of soda and two-liter bottles of soda
Charlotte always has 2 biscuits with every cup of tea;
if she has a plate of biscuits and one cup of tea, she'll only eat two. (And if she has a whole pot of tea and 6 biscuits, she'll stop pouring after 3 cups of tea.)
Each (cup + 2 biscuits) gives her 10 utils of joy.
Which other bundles give her the same utility as
3 cups of tea and 4 biscuits (A)?
Which other bundles give her the same utility as
1 cup of tea and 4 biscuits (B)?
3
6
Any combination that has 4 biscuits
and 2 or more cups of tea
Any combination that has 1 cup of tea and
at 2 or more biscuits
4
5
1
2
3
6
4
5
1
2
A
B
One reason to transform a utility function is to normalize it.
This allows us to describe preferences using fewer parameters.
[ multiply by \({1 \over a + b}\) ]
[ let \(\alpha = {a \over a + b}\) ]
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What utility function represents these preferences?
Charlotte always has 2 biscuits with every cup of tea;
if she has a plate of biscuits and one cup of tea, she'll only eat two. (And if she has a whole pot of tea and 6 biscuits, she'll stop pouring after 3 cups of tea.)
Each (cup + 2 biscuits) gives her 10 utils of joy.
Which other bundles give her the same utility as
3 cups of tea and 4 biscuits (A)?
Which other bundles give her the same utility as
1 cup of tea and 4 biscuits (B)?
Any combination that has 4 biscuits
and 2 or more cups of tea
Any combination that has 1 cup of tea and
at 2 or more biscuits
Left shoes and right shoes
Sugar and tea
Used for two independent goods (neither complements nor substitutes) -- e.g., t-shirts vs hamburgers
Also called "constant shares" for reasons we'll see later.
One reason to transform a utility function is to normalize it.
This allows us to describe preferences using fewer parameters.
[ multiply by \({1 \over a + b}\) ]
[ let \(\alpha = {a \over a + b}\) ]
One reason to transform a utility function is to normalize it.
This allows us to describe preferences using fewer parameters.
[ raise to the power of \({1 \over a + b}\) ]
[ let \(\alpha = {a \over a + b}\) ]
Marginal utility of good 2 is constant
If good 2 is "dollars spent on other things," utility from good 1 is often given as if it were in dollars.
MRS is increasing in \(x_1\) and/or decreasing in \(x_2\)
Indifference curves are bowed away from the origin.
Not monotonic
Realistic, but often the satiation point is far out of reach.
Can be a combination of quasilinear and satiation point
Can generate the familiar linear demand curve
Good 1: chocolate chip cookies
Good 2: snickerdoodle cookies
Good 3: vanilla ice cream
Good 4: strawberry ice cream
Good 1: burritos
Good 2: burgers
Good 3: fries
Good 1: fries
Good 2: onion rings
Good 3: burgers
When considering two goods, there are lots of ways you might feel about them — especially how substitutable the goods are for one another, which is captured by the MRS.
Different functional forms have different MRS's; so they're good for modeling different kinds of preferences.
Take the time to understand this material well.
It's foundational for many, many economic models.