1.4 — Utility Maximization

ECON 306 • Microeconomic Analysis • Spring 2023

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microS23
microS23.classes.ryansafner.com

Constrained Optimization

Constrained Optimization I

We model most situations as a constrained optimization problem:
People optimize: make tradeoffs to achieve their objective as best as they can
Subject to constraints: limited resources (income, time, attention, etc)

Constrained Optimization II

One of the most generally useful mathematical models
Endless applications: how we model nearly every decision-maker

consumer, business firm, politician, judge, bureaucrat, voter, dictator, pirate, drug cartel, drug addict, parent, child, etc

Key economic skill: recognizing how to apply the model to a situation

Remember!

Constrained Optimization III

All constrained optimization models have three moving parts:

Constrained Optimization III

All constrained optimization models have three moving parts:

Choose: < some alternative >

Constrained Optimization III

All constrained optimization models have three moving parts:

Choose: < some alternative >
In order to maximize: < some objective >

Constrained Optimization III

All constrained optimization models have three moving parts:

Choose: < some alternative >
In order to maximize: < some objective >
Subject to: < some constraints >

Constrained Optimization: Example I

Example: A Hood student picking courses hoping to achieve the highest GPA while getting an Econ major.

Choose:
In order to maximize:
Subject to:

Constrained Optimization: Example II

Example: How should FedEx plan its delivery route?

Choose:
In order to maximize:
Subject to:

Constrained Optimization: Example III

Example: The U.S. government wants to remain economically competitive but reduce emissions by 25%.

Choose:
In order to maximize:
Subject to:

Constrained Optimization: Example IV

Example: How do elected officials make decisions in politics?

Choose:
In order to maximize:
Subject to:

The Utility Maximization Problem

The individual's utility maximization problem we've been modeling, finally, is:

Choose: < a consumption bundle >
In order to maximize: < utility >
Subject to: < income and market prices >

The Utility Maximization Problem: Tools

We now have the tools to understand individual choices:
Budget constraint: individual’s constraints of income and market prices
- How market trades off between goods
- Marginal cost (of good $x$ , in terms of $y)$
Utility function: individual’s objective to maximize, based on their preferences
- How individual trades off between goods
- Marginal benefit (of good $x$ , in terms of $y)$

The Utility Maximization Problem: Verbally

The individual's constrained optimization problem:

choose a bundle of goods to maximize utility, subject to income and market prices

The Utility Maximization Problem: Mathematically

$max_{x, y \geq 0} u (x, y)$ $s . t . p_{x} x + p_{y} y = m$

This requires calculus to solve.^† We will look at graphs instead!

^† See the mathematical appendix in today's class notes on how to solve it with calculus, and an example.

The Individual's Optimum: Graphically

Graphical solution: Highest indifference curve tangent to budget constraint
- Bundle A!

The Individual's Optimum: Graphically

Graphical solution: Highest indifference curve tangent to budget constraint
- Bundle A!
B or C spend all income, but a better combination exists

The Individual's Optimum: Graphically

Graphical solution: Highest indifference curve tangent to budget constraint
- Bundle A!
B or C spend all income, but a better combination exists
D is higher utility, but not affordable at current income & prices

The Individual's Optimum: Why Not B?

$\begin{aligned} indiff. curve slope & > budget constr. slope \end{aligned}$

The Individual's Optimum: Why Not B?

$\begin{aligned} indiff. curve slope & > budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & > \frac{p_{x}}{p_{y}} \\ 2 & > 0.5 \end{aligned}$

Consumer views MB of $x$ is 2 units of $y$
- Consumer’s “exchange rate:” 2Y:1X
Market-determined MC of $x$ is 0.5 units of $y$
- Market exchange rate is 0.5Y:1X

The Individual's Optimum: Why Not B?

$\begin{aligned} indiff. curve slope & > budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & > \frac{p_{x}}{p_{y}} \\ 2 & > 0.5 \end{aligned}$

Consumer views MB of $x$ is 2 units of $y$
- Consumer’s “exchange rate:” 2Y:1X
Market-determined MC of $x$ is 0.5 units of $y$
- Market exchange rate is 0.5Y:1X
Can spend less on y, more on x for more utility!

The Individual's Optimum: Why Not C?

$\begin{aligned} indiff. curve slope & < budget constr. slope \end{aligned}$

The Individual's Optimum: Why Not C?

$\begin{aligned} indiff. curve slope & < budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & < \frac{p_{x}}{p_{y}} \\ 0.125 & < 0.5 \end{aligned}$

Consumer views MB of $x$ is 0.125 units of $y$
- Consumer’s “exchange rate:” 0.125Y:1X
Market-determined MC of $x$ is 0.5 units of $y$
- Market exchange rate is 0.5Y:1X

The Individual's Optimum: Why Not C?

$\begin{aligned} indiff. curve slope & < budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & < \frac{p_{x}}{p_{y}} \\ 0.125 & < 0.5 \end{aligned}$

Consumer views MB of $x$ is 0.125 units of $y$
- Consumer’s “exchange rate:” 0.125Y:1X
Market-determined MC of $x$ is 0.5 units of $y$
- Market exchange rate is 0.5Y:1X
Can spend less on y, more on x for more utility!

The Individual's Optimum: Why A?

$\begin{aligned} indiff. curve slope & = budget constr. slope \end{aligned}$

The Individual's Optimum: Why A?

$\begin{aligned} indiff. curve slope & = budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & = \frac{p_{x}}{p_{y}} \\ 0.5 & = 0.5 \end{aligned}$

Marginal benefit = Marginal cost
- Consumer exchanges at same rate as market
No other combination of (x,y) exists that could increase utility!^†

^† At current income and market prices!

The Individual's Optimum: Two Equivalent Rules

Rule 1

$\frac{M U_{x}}{M U_{y}} = \frac{p_{x}}{p_{y}}$

Easier for calculation (slopes)

The Individual's Optimum: Two Equivalent Rules

Rule 1

$\frac{M U_{x}}{M U_{y}} = \frac{p_{x}}{p_{y}}$

Easier for calculation (slopes)

Rule 2

$\frac{M U_{x}}{p_{x}} = \frac{M U_{y}}{p_{y}}$

Easier for intuition (next slide)

The Individual's Optimum: The Equimarginal Rule

$\frac{M U_{x}}{p_{x}} = \frac{M U_{y}}{p_{y}} = \dots = \frac{M U_{n}}{p_{n}}$

Equimarginal Rule: consumption is optimized where the marginal utility per dollar spent is equalized across all $n$ possible goods/decisions
Always choose an option that gives higher marginal utility (e.g. if $M U_{x} < M U_{y})$ , consume more $y$ !
- But each option has a different price, so weight each option by its price, hence $\frac{M U_{x}}{p_{x}}$

An Optimum, By Definition

Any optimum in economics: no better alternatives exist under current constraints
No possible change in your consumption that would increase your utility

Practice I

Example: You can get utility from consuming bags of Almonds $(a)$ and bunches of Bananas $(b)$ , according to the utility function:

$\begin{aligned} u (a, b) & = a b \\ M U_{a} & = b \\ M U_{b} & = a \end{aligned}$

You have an income of $50, the price of Almonds is $10, and the price of Bananas is $2. Put Almonds on the horizontal axis and Bananas on the vertical axis.

What is your utility-maximizing bundle of Almonds and Bananas?
How much utility does this provide? [Does the answer to this matter?]

Practice II, Cobb-Douglas!

Example: You can get utility from consuming Burgers $(b)$ and Fries $(f)$ , according to the utility function:

$\begin{aligned} u (b, f) & = \sqrt{b f} \\ M U_{b} & = 0.5 b^{- 0.5} f^{0.5} \\ M U_{f} & = 0.5 b^{0.5} f^{- 0.5} \end{aligned}$

You have an income of $20, the price of Burgers is $5, and the price of Fries is $2. Put Burgers on the horizontal axis and Fries on the vertical axis.

What is your utility-maximizing bundle of Burgers and Fries?
How much utility does this provide?

1.4 — Utility Maximization

ECON 306 • Microeconomic Analysis • Spring 2023

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microS23
microS23.classes.ryansafner.com

Constrained Optimization

Constrained Optimization I

We model most situations as a constrained optimization problem:
People optimize: make tradeoffs to achieve their objective as best as they can
Subject to constraints: limited resources (income, time, attention, etc)

Constrained Optimization II

One of the most generally useful mathematical models
Endless applications: how we model nearly every decision-maker

consumer, business firm, politician, judge, bureaucrat, voter, dictator, pirate, drug cartel, drug addict, parent, child, etc

Key economic skill: recognizing how to apply the model to a situation

Remember!

Constrained Optimization III

All constrained optimization models have three moving parts:

Constrained Optimization III

All constrained optimization models have three moving parts:

Choose: < some alternative >

Constrained Optimization III

All constrained optimization models have three moving parts:

Choose: < some alternative >
In order to maximize: < some objective >

Constrained Optimization III

All constrained optimization models have three moving parts:

Choose: < some alternative >
In order to maximize: < some objective >
Subject to: < some constraints >

Constrained Optimization: Example I

Example: A Hood student picking courses hoping to achieve the highest GPA while getting an Econ major.

Choose:
In order to maximize:
Subject to:

Constrained Optimization: Example II

Example: How should FedEx plan its delivery route?

Choose:
In order to maximize:
Subject to:

Constrained Optimization: Example III

Example: The U.S. government wants to remain economically competitive but reduce emissions by 25%.

Choose:
In order to maximize:
Subject to:

Constrained Optimization: Example IV

Example: How do elected officials make decisions in politics?

Choose:
In order to maximize:
Subject to:

The Utility Maximization Problem

The individual's utility maximization problem we've been modeling, finally, is:

Choose: < a consumption bundle >
In order to maximize: < utility >
Subject to: < income and market prices >

The Utility Maximization Problem: Tools

We now have the tools to understand individual choices:
Budget constraint: individual’s constraints of income and market prices
- How market trades off between goods
- Marginal cost (of good $x$ , in terms of $y)$
Utility function: individual’s objective to maximize, based on their preferences
- How individual trades off between goods
- Marginal benefit (of good $x$ , in terms of $y)$

The Utility Maximization Problem: Verbally

The individual's constrained optimization problem:

choose a bundle of goods to maximize utility, subject to income and market prices

The Utility Maximization Problem: Mathematically

$max_{x, y \geq 0} u (x, y)$ $s . t . p_{x} x + p_{y} y = m$

This requires calculus to solve.^† We will look at graphs instead!

^† See the mathematical appendix in today's class notes on how to solve it with calculus, and an example.

The Individual's Optimum: Graphically

Graphical solution: Highest indifference curve tangent to budget constraint
- Bundle A!

The Individual's Optimum: Graphically

Graphical solution: Highest indifference curve tangent to budget constraint
- Bundle A!
B or C spend all income, but a better combination exists

The Individual's Optimum: Graphically

Graphical solution: Highest indifference curve tangent to budget constraint
- Bundle A!
B or C spend all income, but a better combination exists
D is higher utility, but not affordable at current income & prices

The Individual's Optimum: Why Not B?

$\begin{aligned} indiff. curve slope & > budget constr. slope \end{aligned}$

The Individual's Optimum: Why Not B?

$\begin{aligned} indiff. curve slope & > budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & > \frac{p_{x}}{p_{y}} \\ 2 & > 0.5 \end{aligned}$

Consumer views MB of $x$ is 2 units of $y$
- Consumer’s “exchange rate:” 2Y:1X
Market-determined MC of $x$ is 0.5 units of $y$
- Market exchange rate is 0.5Y:1X

The Individual's Optimum: Why Not B?

$\begin{aligned} indiff. curve slope & > budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & > \frac{p_{x}}{p_{y}} \\ 2 & > 0.5 \end{aligned}$

Consumer views MB of $x$ is 2 units of $y$
- Consumer’s “exchange rate:” 2Y:1X
Market-determined MC of $x$ is 0.5 units of $y$
- Market exchange rate is 0.5Y:1X
Can spend less on y, more on x for more utility!

The Individual's Optimum: Why Not C?

$\begin{aligned} indiff. curve slope & < budget constr. slope \end{aligned}$

The Individual's Optimum: Why Not C?

$\begin{aligned} indiff. curve slope & < budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & < \frac{p_{x}}{p_{y}} \\ 0.125 & < 0.5 \end{aligned}$

Consumer views MB of $x$ is 0.125 units of $y$
- Consumer’s “exchange rate:” 0.125Y:1X
Market-determined MC of $x$ is 0.5 units of $y$
- Market exchange rate is 0.5Y:1X

The Individual's Optimum: Why Not C?

$\begin{aligned} indiff. curve slope & < budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & < \frac{p_{x}}{p_{y}} \\ 0.125 & < 0.5 \end{aligned}$

Consumer views MB of $x$ is 0.125 units of $y$
- Consumer’s “exchange rate:” 0.125Y:1X
Market-determined MC of $x$ is 0.5 units of $y$
- Market exchange rate is 0.5Y:1X
Can spend less on y, more on x for more utility!

The Individual's Optimum: Why A?

$\begin{aligned} indiff. curve slope & = budget constr. slope \end{aligned}$

The Individual's Optimum: Why A?

$\begin{aligned} indiff. curve slope & = budget constr. slope \\ \frac{M U_{x}}{M U_{y}} & = \frac{p_{x}}{p_{y}} \\ 0.5 & = 0.5 \end{aligned}$

Marginal benefit = Marginal cost
- Consumer exchanges at same rate as market
No other combination of (x,y) exists that could increase utility!^†

^† At current income and market prices!

The Individual's Optimum: Two Equivalent Rules

Rule 1

$\frac{M U_{x}}{M U_{y}} = \frac{p_{x}}{p_{y}}$

Easier for calculation (slopes)

The Individual's Optimum: Two Equivalent Rules

Rule 1

$\frac{M U_{x}}{M U_{y}} = \frac{p_{x}}{p_{y}}$

Easier for calculation (slopes)

Rule 2

$\frac{M U_{x}}{p_{x}} = \frac{M U_{y}}{p_{y}}$

Easier for intuition (next slide)

The Individual's Optimum: The Equimarginal Rule

$\frac{M U_{x}}{p_{x}} = \frac{M U_{y}}{p_{y}} = \dots = \frac{M U_{n}}{p_{n}}$

Equimarginal Rule: consumption is optimized where the marginal utility per dollar spent is equalized across all $n$ possible goods/decisions
Always choose an option that gives higher marginal utility (e.g. if $M U_{x} < M U_{y})$ , consume more $y$ !
- But each option has a different price, so weight each option by its price, hence $\frac{M U_{x}}{p_{x}}$

An Optimum, By Definition

Any optimum in economics: no better alternatives exist under current constraints
No possible change in your consumption that would increase your utility

Practice I

Example: You can get utility from consuming bags of Almonds $(a)$ and bunches of Bananas $(b)$ , according to the utility function:

$\begin{aligned} u (a, b) & = a b \\ M U_{a} & = b \\ M U_{b} & = a \end{aligned}$

You have an income of $50, the price of Almonds is $10, and the price of Bananas is $2. Put Almonds on the horizontal axis and Bananas on the vertical axis.

What is your utility-maximizing bundle of Almonds and Bananas?
How much utility does this provide? [Does the answer to this matter?]

Practice II, Cobb-Douglas!

Example: You can get utility from consuming Burgers $(b)$ and Fries $(f)$ , according to the utility function:

$\begin{aligned} u (b, f) & = \sqrt{b f} \\ M U_{b} & = 0.5 b^{- 0.5} f^{0.5} \\ M U_{f} & = 0.5 b^{0.5} f^{- 0.5} \end{aligned}$

You have an income of $20, the price of Burgers is $5, and the price of Fries is $2. Put Burgers on the horizontal axis and Fries on the vertical axis.

What is your utility-maximizing bundle of Burgers and Fries?
How much utility does this provide?

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1.4 — Utility Maximization

ECON 306 • Microeconomic Analysis • Spring 2023

Ryan Safner Associate Professor of Economics safner@hood.edu ryansafner/microS23 microS23.classes.ryansafner.com

Constrained Optimization

Constrained Optimization I

Constrained Optimization II

Remember!

Constrained Optimization III

Constrained Optimization III

Constrained Optimization III

Constrained Optimization III

Constrained Optimization: Example I

Constrained Optimization: Example II

Constrained Optimization: Example III

Constrained Optimization: Example IV

The Utility Maximization Problem

The Utility Maximization Problem: Tools

The Utility Maximization Problem: Verbally

The Utility Maximization Problem: Mathematically

The Individual's Optimum: Graphically

The Individual's Optimum: Graphically

The Individual's Optimum: Graphically

The Individual's Optimum: Why Not B?

The Individual's Optimum: Why Not B?

The Individual's Optimum: Why Not B?

The Individual's Optimum: Why Not C?

The Individual's Optimum: Why Not C?

The Individual's Optimum: Why Not C?

The Individual's Optimum: Why A?

The Individual's Optimum: Why A?

The Individual's Optimum: Two Equivalent Rules

Rule 1

The Individual's Optimum: Two Equivalent Rules

Rule 1

Rule 2

The Individual's Optimum: The Equimarginal Rule

An Optimum, By Definition

Practice I

Practice II, Cobb-Douglas!

Constrained Optimization

Help

1.4 — Utility Maximization

1.4 — Utility Maximization

ECON 306 • Microeconomic Analysis • Spring 2023

Ryan Safner Associate Professor of Economics safner@hood.edu ryansafner/microS23 microS23.classes.ryansafner.com

Constrained Optimization

Constrained Optimization I

Constrained Optimization II

Remember!

Constrained Optimization III

Constrained Optimization III

Constrained Optimization III

Constrained Optimization III

Constrained Optimization: Example I

Constrained Optimization: Example II

Constrained Optimization: Example III

Constrained Optimization: Example IV

The Utility Maximization Problem

The Utility Maximization Problem: Tools

The Utility Maximization Problem: Verbally

The Utility Maximization Problem: Mathematically

The Individual's Optimum: Graphically

The Individual's Optimum: Graphically

The Individual's Optimum: Graphically

The Individual's Optimum: Why Not B?

The Individual's Optimum: Why Not B?

The Individual's Optimum: Why Not B?

The Individual's Optimum: Why Not C?

The Individual's Optimum: Why Not C?

The Individual's Optimum: Why Not C?

The Individual's Optimum: Why A?

The Individual's Optimum: Why A?

The Individual's Optimum: Two Equivalent Rules

Rule 1

The Individual's Optimum: Two Equivalent Rules

Rule 1

Rule 2

The Individual's Optimum: The Equimarginal Rule

An Optimum, By Definition

Practice I

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microS23
microS23.classes.ryansafner.com

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microS23
microS23.classes.ryansafner.com