# Utility Maximization with Budget Constraints

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 7

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## This Week: Maximize utility subject to a (parameterized) budget line

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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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"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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### 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)

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# “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}

"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!

# 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.

By Chris Makler

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