Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 7
🍏
🍌
BL
🐟
🥥
PPF
🍏
🍌
BL
All the math we did last week holds,
but the slope of the constraint is the price ratio,
not the MRT. (Still opportunity cost!)
🍏
🍌
BL
"Gravitational pull" argument:
Indifference curve is
steeper than the budget line
Indifference curve is
flatter than the budget line
Moving to the right
along the budget line
would increase utility.
Moving to the left
along the budget line
would increase utility.
🍏
🍌
BL
Can sometimes use the tangency condition
\(MRS = p_1/p_2\), sometimes you have to use logic.
🍏
🍌
BL1
We will be solving for the optimal bundle
as a function of income and prices:
The solutions to this problem will be called the demand functions. We have to think about how the optimal bundle will change when \(p_1,p_2,m\) change.
BL2
The consumer receives more utility per additional unit of good 1 than the price reflects, relative to good 2.
The consumer receives more
"bang for the buck"
(utils per dollar)
from good 1 than good 2.
Regardless of how you look at it, the consumer would be
better off moving to the right along the budget line --
i.e., consuming more of good 1 and less of good 2.
The consumer is more willing to give up good 2
to get good 1
than the market requires.
MRS and the Price Ratio: Cobb-Douglas
The budget line and indifference curves describe different things.
Indifference curves describe the "shape of the utility hill."
They do not change when prices or income change.
They do change when preferences change, but we usually assume preferences are fixed.
The budget line describes the boundary of affordable bundles;
we can think of it as a fence over the utility hill.
IF...
THEN...
The consumer's preferences are "well behaved"
-- smooth, strictly convex, and strictly monotonic
\(MRS=0\) along the horizontal axis (\(x_2 = 0\))
The budget line is a simple straight line
The optimal consumption bundle will be characterized by two equations:
More generally: the optimal bundle may be found using the Lagrange method
\(MRS \rightarrow \infty\) along the vertical axis (\(x_1 \rightarrow 0\))
First Order Conditions
"Bang for your buck" condition: marginal utility from last dollar spent on every good must be the same!
What happens when the price of a good increases or decreases?
What happens when income decreases?
Next class, we'll derive the demand functions: that is, the optimal choice as a function of prices and income.
Remember what you learned about demand curves in Econ 1:
"The demand curve shows the quantity demanded of a good at different prices"
DEMAND CURVE FOR GOOD 1
"Good 1 - Good 2 Space"
"Quantity-Price Space for Good 1"
"The demand curve represents the marginal benefit of an additional unit,
or alternatively the marginal willingness to pay for another unit"
Let's look at the FOC with respect to good 1:
Solve for \(p_1\):
Wednesday's class:
derive the demand functions for four "canonical" utility functions:
Cobb-Douglas, perfect complements, perfect substitutes, and quasilinear.
Friday's "skill section":
draw the demand curves for those same functions
The demand functions describe the optimal bundle
as a function of prices and income.
Demand curves illustrate how the quantity demanded of a good changes as its price changes, holding income and the prices of other goods constant.