Utility Maximization with Budget Constraints
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 7
This Week:
Maximize utility subject to a (parameterized) budget line
🍏
🍌
BL
Last Week:
Maximize utility subject to a (fixed) PPF
🐟
🥥
PPF
This Week:
Maximize utility subject to a (parameterized) budget line
🍏
🍌
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!)
This Week:
Maximize utility subject to a (parameterized) budget line
🍏
🍌
BL
"Gravitational pull" argument:
MRS > \frac{p_1}{p_2}
MRS < \frac{p_1}{p_2}
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.
This Week:
Maximize utility subject to a (parameterized) budget line
🍏
🍌
BL
Can sometimes use the tangency condition
\(MRS = p_1/p_2\), sometimes you have to use logic.
This Week:
Maximize utility subject to a (parameterized) budget line
🍏
🍌
BL1
Big difference:
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.
x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)
BL2
“Gravitational Pull" with a Budget Line
What does it mean if the MRS is
greater than the price ratio?
\frac{MU_1}{MU_2} > \frac{p_1}{p_2}
The consumer receives more utility per additional unit of good 1 than the price reflects, relative to good 2.
\frac{MU_1}{p_1} > \frac{MU_2}{p_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.
MRS > \frac{p_1}{p_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
Important and Difficult Distinction
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:
MRS = \frac{p_1}{p_2}
p_1x_1 + p_2x_2 = m
More generally: the optimal bundle may be found using the Lagrange method
\(MRS \rightarrow \infty\) along the vertical axis (\(x_1 \rightarrow 0\))
The Lagrange Method
\mathcal{L}(x_1,x_2,\lambda)=
u(x_1,x_2)+
(m - p_1x_1 - p_2x_2)
\lambda
\frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial u}{\partial x_1} - \lambda p_1
First Order Conditions
\frac{\partial \mathcal{L}}{\partial x_2} = \frac{\partial u}{\partial x_2} - \lambda p_2
\frac{\partial \mathcal{L}}{\partial \lambda} = m - p_1x_1 - p_2x_2
= 0 \Rightarrow \lambda = \frac{MU_1}{p_1}
= 0 \Rightarrow \lambda = \frac{MU_2}{p_2}
"Bang for your buck" condition: marginal utility from last dollar spent on every good must be the same!
\text{Also: }\frac{\partial \mathcal{L}}{\partial m} = \lambda = \text{bang for your buck, in }\frac{\text{utils}}{\$}
\text{Set two expressions }
\text{for }\lambda \text{ equal:}
\frac{MU_1}{p_1} = \frac{MU_2}{p_2}
The Tangency Condition
What happens when the price of a good increases or decreases?
The Tangency Condition when Lagrange sometimes works
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.
Part II: Demand Curves
Remember what you learned about demand curves in Econ 1:
- The demand curve shows the quantity demanded of a good at different prices
- A change in the price of a good results in a movement along its demand curve
- A change in income or the price of other goods results in a shift of the demand curve
- The demand curve represents the marginal benefit of an additional unit,
or alternatively the marginal willingness to pay for another unit
Demand Curve for Good 1
\text{Fix }p_2\text{ and }m
\text{Plot }(x_1^*(p_1),p_1)
Demand Functions
x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)
"The demand curve shows the quantity demanded of a good at different prices"
x_1
x_1
x_2
p_1
DEMAND CURVE FOR GOOD 1
BL_{p_1 = 2}
BL_{p_1 = 3}
BL_{p_1 = 4}
2
3
4
"Good 1 - Good 2 Space"
"Quantity-Price Space for Good 1"
BL
- A change in the price of a good results in a movement along its demand curve
- A change in income or the price of other goods results in a shift of the demand curve
"The demand curve represents the marginal benefit of an additional unit,
or alternatively the marginal willingness to pay for another unit"
\mathcal{L}(x_1,x_2,\lambda)=
u(x_1,x_2)+
(m - p_1x_1 - p_2x_2)
\lambda
\frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial u}{\partial x_1} - \lambda p_1
Let's look at the FOC with respect to good 1:
= 0 \Rightarrow \lambda = \frac{MU_1}{p_1}
Solve for \(p_1\):
\lambda = \text{bang for your buck, in }\frac{\text{utils}}{\$}
Summary and Next Steps
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.
Econ 50 | 7 | Utility Max with Budget Constraints
By Chris Makler
Econ 50 | 7 | Utility Max with Budget Constraints
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