Utility Maximization with Budget Constraints
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 12
pollev.com/chrismakler
How did the test go, relative to how you thought it would go?
Resources
Technology
Stuff
Happiness
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Part I:
The Real Economy
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⛏
Resources
Firms
Stuff
Consumers
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🏭
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Part I:
The Market Economy
Resource
Owners
👷🏽♀️
⛏
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Resources
Firms
Stuff
Consumers
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🏭
⌚️
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Part I:
The Market Economy
Resource
Owners
👷🏽♀️
⛏
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Firms pay wages for labor
Firms pay rent on capital
Consumers pay prices for goods
Demand
Supply
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Next Five Weeks:
How do consumers and firms respond to prices?
The Consumer's Problem
Prices and Expenditure
Suppose each good has a constant price
(so every unit of the good costs the same)
Affordability
Suppose you have a given income \(m\)
to spend on goods 1 and 2.
Then bundle \(X = (x_1,x_2)\) is affordable if
The feasible set, or budget set, is the set of all affordable bundles.
Example: suppose you have \(m = \$24\) to spend on two goods.
Good 1 costs \(p_1 = \$4\) per unit.
Good 2 costs \(p_2 = \$2\) per unit.
Is the bundle (2,4) affordable (in your budget set)? What about the bundle (4,6)?
Draw your budget set.
How would it change if the price of good 2 rose to \(p_2' = \$6\) per unit?
How would it change if your income rose to \(m' = \$32\)?
Budget Line
pollev.com/chrismakler
Holding income and the price of good 2 constant, an increase in the price of good 1 will cause the budget line to become:
steeper
flatter
it depends on thelevel of income
it depends on the price of good 2
Interpreting the Slope of the Budget Line
Example:
Apples cost 50 cents each
Bananas cost 25 cents each
Slope of the budget line represents the opportunity cost of consuming good 1, as dictated by market prices.
In other words: it is the amount of good 2 the market requires you to give up in order to get another unit of good 1.
amount of labor required to produce another unit of good 1
amount of labor required to produce another unit of good 2
amount of money required to
buy another unit of good 1
amount of money required to
buy another unit of good 2
PPF
Budget Line
pollev.com/chrismakler
If apples (good 1) cost $0.80 each,
and bananas (good 2) cost $0.20 each, what is the magnitude (absolute value) of the slope of the budget line?
Composite Goods
You have $100 in your pocket.
You see a cart selling apples (good 1) for $2 per pound.
- Plot your budget line.
- What is "good 2"?
- What does the bundle (10,80) signify?
- What is the slope of the budget line, and what are its units?
Utility Maximization
Next two weeks:
Maximize utility subject to a (parameterized) budget line
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BL
Last two lectures:
Maximize utility subject to a (fixed) PPF
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PPF
Next two weeks:
Maximize utility subject to a (parameterized) budget line
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🍌
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!)
Next two weeks:
Maximize utility subject to a (parameterized) budget line
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🍌
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.
Next two weeks:
Maximize utility subject to a (parameterized) budget line
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🍌
BL
Can sometimes use the tangency condition
\(MRS = p_1/p_2\), sometimes you have to use logic.
Next two weeks:
Maximize utility subject to a (parameterized) budget line
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🍌
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.
BL2
“Gravitational Pull" with a Budget Line
What does it mean if the MRS is
greater than the price ratio?
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
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:
More generally: the optimal bundle may be found using the Lagrange method
\(MRS \rightarrow \infty\) along the vertical axis (\(x_1 \rightarrow 0\))
How do you tell if a preferences are "well behaved"?
Strictly monotonic
Strictly convex
Smooth
\(MU_1 > 0\) and \(MU_2 > 0\) for any \(x_1,x_2\)
\(\frac{\partial MRS}{\partial x_1} \le 0\) and \(\frac{\partial MRS}{ \partial x_2} \ge 0\), with at least one strict
MRS has no "jumps" (not defined piecewise)
Continuous
Utility function has no "jumps" (not defined piecewise)
(i.e., indifference curves get flatter as you move down and to the right)
The Lagrange Method
Cost of Bundle X
Income
Utility
The Lagrange Method
Income left over
Utility
The Lagrange Method
Income left over
Utility
(utils)
(dollars)
utils/dollar
First Order Conditions
"Bang for your buck" condition: marginal utility from last dollar spent on every good must be the same!
The Lagrange Method
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?
pollev.com/chrismakler
If your utility function is
\(u(x_1,x_2) = 4x_1 + 2x_2\),
when will you buy only good 1?
Cobb-Douglas
Quasilinear
Perfect Substitutes
TANGENCY CONDITION
Ray from origin, will always intersect budget line =>
Lagrange always works
Vertical line, may or may not intersect BL in first quadrant =>
Lagrange sometimes works
Tries to equate two constants, which you just can't do =>
Lagrange never works
Kinked Budget Constraints
- Nonlinear electricity rates
- Gift cards
- Quantity discounts
- "Buying" lower prices
(more on these in the interactive lecture notes)
How to Solve a Kinked Constraint Problem
- Evaluate the MRS at the kink
- Compare it to the price ratio on either side of the kink
- If \(MRS > {p_1 \over p_2}\) to the right of the kink, the solution is to the right of the kink; use the equation for that line and maximize.
- If \(MRS < {p_1 \over p_2}\) to the left of the kink, the solution is to the right of the kink; use the equation for that line and maximize.
- If the MRS is between the price ratios, then the solution is at the kink.
- Note: if the price ratio to the left of the kink is greater than the price ratio to the left, it's more complicated...you could have two potential solutions! Maximize subject to each of the constraints and compare utility at the respective maxima.
Next class, we'll derive the demand functions: that is, the optimal choice as a function of prices and income.
Notice that your optimal choice depends on the prices of goods and your income.
Econ 50 | Lecture 12
By Chris Makler
Econ 50 | Lecture 12
The Consumer's Problem: Utility Maximization subject to a Budget Constraint
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