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Attendance trivia question:
The name of what company is formed from a portmanteau of Danish words that translate to English as "play well"?
Utility Maximization with Budget Constraints
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 10
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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Part I:
The Market Economy
Resource
Owners
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⛏
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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 Four 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)
p_1 = \text{price of good 1}
p_2 = \text{price of good 2}
x_1 = \text{quantity of good 1}
x_2 = \text{quantity of good 2}
p_1x_1 = \text{amount spent on good 1}
p_2x_2 = \text{amount spent on good 2}
p_1x_1 + p_2x_2 = \text{cost of buying bundle }X
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
p_1x_1 + p_2x_2 \le m
The feasible set, or budget set, is the set of all affordable bundles.
Example: suppose you have \(m = \$240\) to spend on two goods.
Good 1 costs \(p_1 = \$3\) per unit.
Good 2 costs \(p_2 = \$4\) per unit.
Is the bundle (10,40) affordable (in your budget set)? What about the bundle (40,40)?
Draw your budget set.
How would it change if the price of good 1 rose to \(p_1' = \$6\) per unit?
How would it change if your income dropped to \(m' = \$120\)?
m = \text{money income}
p_1 = \text{price of good 1}
p_2 = \text{price of good 2}
\text{horizontal intercept} = \frac{m}{p_1}
\text{vertical intercept} = \frac{m}{p_2}
\text{slope of budget line} = -\frac{p_1}{p_2}
Budget Line
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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 theprice 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.
-\frac{p_1}{p_2} = -2 \text{ bananas per apple}
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.
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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?
Kinked Budget Constraints
- Nonlinear electricity rates
- Gift cards
- Quantity discounts
- "Buying" lower prices
(more on these in the interactive lecture notes)
Utility Maximization
This Week:
Maximize utility subject to a (parameterized) budget line
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🍌
BL
Last Week:
Maximize utility subject to a (fixed) PPF
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PPF
This Week:
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!)
This Week:
Maximize utility subject to a (parameterized) budget line
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🍌
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
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🍌
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
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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.
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?
pollev.com/chrismakler
If your utility function is
\(u(x_1,x_2) = 4x_1 + 2x_2\),
when will you buy only good 1?
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 | Spring 23 | Lecture 10
By Chris Makler
Econ 50 | Spring 23 | Lecture 10
The Consumer's Problem: Utility Maximization subject to a Budget Constraint
