Course Retrospective

Christopher Makler

Stanford University Department of Economics

Econ 50

Far out in the uncharted backwaters of the unfashionable end of the western spiral arm of the Galaxy lies a small unregarded yellow sun.

SCIEPRO/GETTY IMAGES

Orbiting this at a distance of roughly ninety-two million miles is an utterly insignificant little blue green planet whose ape-descended life forms are so amazingly primitive that they still think digital watches are a pretty neat idea.

This planet has — or rather had —
a problem, which was this:

😢

most of the people on it were unhappy for pretty much of the time.

Many solutions were proposed
for this problem...

...but most of these were largely concerned with the movements
of small green pieces of paper,

which is odd because on the whole
it wasn't the small green pieces of paper that were unhappy.

Resources

Technology

Stuff

Happiness

🌎

🏭

⌚️

🤓

Part I: The Real Economy

Demand

Supply

Equilibrium

🤩

🏪

⚖

Part II: Little Green
Pieces of Paper

Three Fundamental Tools of Analysis

Optimization

Given a fixed set of circumstances (prices, technology, preferences), how do economic agents (consumers, firms) make choices?

Comparative Statics

How do changes in circumstances (changing prices, shifting technology, preferences, etc.) translate into changes in behavior?

Equilibrium

How do economic systems converge toward certain outcomes?

Unit I Overview

Unit I: Scarcity and Choice

Economics is the study of how
we use scarce resources
to satisfy our unlimited wants

Resources

Goods

Happiness

🌎

⌚️

🤓

Week 1: Modeling Production
with Multivariable Functions

Labor

Fish

🐟

Capital

Coconuts

🥥

[GOODS]

⏳

⛏

[RESOURCES]

Unit I: Scarcity and Choice

Economics is the study of how
we use scarce resources
to satisfy our unlimited wants

Resources

Goods

Happiness

🌎

⌚️

🤓

🐟

🥥

Production Possibilities Fronier

Feasible

Week 2: Setting Up the Problem

Unit I: Scarcity and Choice

Economics is the study of how
we use scarce resources
to satisfy our unlimited wants

Resources

Goods

Happiness

🌎

⌚️

🤓

🐟

🥥

🙂

😀

😁

😢

🙁

Unit I: Scarcity and Choice

Economics is the study of how
we use scarce resources
to satisfy our unlimited wants

Resources

Goods

Happiness

🌎

⌚️

🤓

Week 2: Setting Up the Problem

🐟

🥥

Optimal choice

🙂

😀

😁

😢

🙁

Unit I: Scarcity and Choice

Economics is the study of how
we use scarce resources
to satisfy our unlimited wants

Resources

Goods

Happiness

🌎

⌚️

🤓

Week 3: Solving the Constrained Optimization problem

Derivative Along a Level Set

f(x,y)=z

Take total derivative of both sides with respect to x:

Solve for $dy/dx$:

\displaystyle{{\partial f \over \partial x}}

\displaystyle{{\partial f \over \partial y}}

\displaystyle{\times \ \ {dy \over dx}}

\displaystyle{{dy \over dx}}

= -

\displaystyle{{\partial f \over \partial x}}

\displaystyle{{\partial f \over \partial y}}

IMPLICIT FUNCTION THEOREM

Production Functions

A mathematical form describing how much output is produced as a function of inputs.

Labor $(L)$

Capital $(K)$

Production Function $f(L,K)$

Output ($q$ or $x$)

Marginal Products of Labor and Capital

Economic definition: how much more output is produced if you increase labor or capital?

Mathematical definition:
partial derivatives of the production function

\displaystyle{MP_L = {\partial f(L,K) \over \partial L}}

These are both rates: they are measured in terms of units of ouptut per unit of input.

\displaystyle{= \lim_{\Delta L \rightarrow 0} {f(L + \Delta L, K) - f(L, K) \over \Delta L}}

\displaystyle{MP_K = {\partial f(L,K) \over \partial K}}

\displaystyle{=\lim_{\Delta K \rightarrow 0} {f(L, K + \Delta K) - f(L, K) \over \Delta K}}

Isoquant: combinations of inputs that produce a given level of output

Isoquant map: a contour map showing the isoquants for various levels of output

Marginal Rate of Technical Substitution (MRTS)

Economic definition: the rate at which a producer can substitute one input for another while keeping output at the same level

Mathematical definition: slope of an isoquant

Recall: by implicit function theorem,
the slope of a level set is given by

\displaystyle{MRTS = {MP_L \over MP_K}}

\displaystyle{\left.{dy \over dx} \right|_{f(x,y) = z} = -{\partial f/\partial x \over \partial f/\partial y}}

Therefore the formula for the MRTS is

(absolute value)

Examples of Production Functions

f(L,K) = aL + bK

Linear

f(L,K) = \min\{aL, bK\}

Leontief
(Fixed Proportions)

Cobb-Douglas

f(L,K) = AL^aK^b

Constant Elasticity of Substitution (CES)

f(L,K) = (aL^\rho + bK^\rho)^{1 \over \rho}

Linear Production Function

MP_L =

MP_K =

MRTS =

\displaystyle{a\ {\text{output} \over \text{unit of L}}}

\displaystyle{b\ {\text{output} \over \text{unit of K}}}

\displaystyle{= {a \text{ units of K} \over b \text{ units of L}}}

q=f(L,K)=aL + bK

Leontief (Fixed Proportions) Production Function

f(L,K)=\begin{cases}aL \text{\ \ if }aL \le bK\\bK \text{\ \ if }aL \ge bK\end{cases}

=\min\{aL,bK\}

MP_L =

MP_K =

MRTS =

\begin{cases}a \text{\ \ if }aL < bK\\0 \text{\ \ if }aL > bK\end{cases}

\begin{cases}0 \text{\ \ if }aL < bK\\b \text{\ \ if }aL > bK\end{cases}

= \begin{cases}\infty \text{\ \ if }aL < bK\\0 \text{\ \ if }aL > bK\end{cases}

Cobb-Douglas Production Function

q=f(L,K)=AL^aK^b

MP_L = aAL^{a-1}K^b

MP_K = bAL^aK^{b-1}

MRTS =

\displaystyle{= {a \over b} \times {K \over L}}

CES Production Function

q=f(L,K)=(aL^\rho + bK^\rho)^{1 \over \rho}

\displaystyle{MRTS= {a \over b} \times \left({K \over L}\right)^{1-\rho}}

Scaling Production in the Short Run

Suppose $K$ is fixed at some $\overline K$ in the short run.
Then the production function becomes $f(L\ |\ \overline K)$

Scaling Production in the Long Run

What happens when we increase all inputs proportionally?

For example, what happens if we double both labor and capital?

Does doubling inputs -- i.e., getting $f(2L,2K)$ -- double output?

f(2L,2K) > 2f(L,K)

f(2L,2K) = 2f(L,K)

f(2L,2K) < 2f(L,K)

Decreasing Returns to Scale

Constant Returns to Scale

Increasing Returns to Scale

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 4

Resource Constraints and Production Possibilities

Good 1 - Good 2 Space

x_1

x_2

Two "Goods" (e.g. fish and coconuts)

A bundle is some quantity of each good

\text{Bundle }X = (x_1,x_2)

x_1 = \text{quantity of good 1 in bundle }X

x_2 = \text{quantity of good 2 in bundle }X

Can plot this in a graph with $x_1$ on the horizontal axis and $x_2$ on the vertical axis

A = (4,16)

B = (8,8)

Good 1 - Good 2 Space

x_1

x_2

What tradeoff is represented by moving
from bundle A to bundle B?

\text{Give up }\Delta x_2 =

\text{Gain }\Delta x_1 =

\text{Rate of exchange }=

ANY SLOPE IN
GOOD 1 - GOOD 2 SPACE
IS MEASURED IN
UNITS OF GOOD 2
PER UNIT OF GOOD 1

TW: HORRIBLE STROBE EFFECT!

Multiple Uses of Resources

Labor

Fish

🐟

Coconuts

🥥

[GOOD 1]

⏳

[GOOD 2]

L_1

L_2

\overline L

Resource Constraint

L_1

L_2

L_1 + L_2 = \overline L

\overline L

\text{Total labor available }=\overline L

A PPF with Linear Technologies

x_1 = f_1(L_1) = 3L_1

x_2 = f_2(L_2) = 2L_2

Fish production function

Coconut production function

Resource Constraint

L_1 + L_2 = 150

A PPF with Diminishing $MP_L$

x_1 = f_1(L_1) = 10\sqrt{L_1}

x_2 = f_2(L_2) = 6\sqrt{L_2}

Fish production function

Coconut production function

Resource Constraint

L_1 + L_2 = 100

Suppose we're allocating 100 units of labor to fish (good 1),
and 50 of labor to coconuts (good 2).

Now suppose we shift
one unit of labor
from coconuts to fish.

How many fish do we gain?

\Delta_2

\Delta_1

100

300

303

How many coconuts do we lose?

\Delta_2 = MP_{L_2} = 2\text{ coconuts}

\Delta_1 = MP_{L_1} = 3\text{ fish}

\left|\text{slope}\right| = \frac{MP_{L_2}}{MP_{L_1}}

Relationship between MPL's and MRT

x_1 = f_1(L_1) = 3L_1

x_2 = f_2(L_2) = 2L_2

Fish production function

Coconut production function

Resource Constraint

L_1 + L_2 = 150

PPF

Diminishing $MP_L$'s
and Increasing $MRT$

Preferences and Utility

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 5

Preferences: Ordinal Ranking of Options

Given a choice between option A and option B, an agent might have different preferences:

A \succ B

A \succeq B

A \sim B

A \preceq B

A \prec B

The agent strictly prefers A to B.

The agent strictly disprefers A to B.

The agent weakly prefers A to B.

The agent weakly disprefers A to B.

The agent is indifferent between A and B.

Visually: the MRS is the magnitude of the slope
of an indifference curve

Utility Functions

How do we model preferences mathematically?

Approach: assume consuming goods "produces" utility

Production Functions

Labor

Fish

🐟

Capital

⏳

⛏

[RESOURCES]

Utility Functions

Utility

😀

[GOODS]

Fish

🐟

Coconuts

🥥

Representing Preferences with a Utility Function

u(a_1,a_2) > u(b_1,b_2)

u(a_1,a_2) \ge u(b_1,b_2)

u(a_1,a_2) = u(b_1,b_2)

u(a_1,a_2) \le u(b_1,b_2)

u(a_1,a_2) < u(b_1,b_2)

u(x_1,x_2)

"A is strictly preferred to B"

Words

Preferences

Utility

A \succ B

A \succeq B

A \sim B

A \preceq B

A \prec B

"A is weakly preferred to B"

"A is indifferent to B"

"A is weakly dispreferred to B"

"A is strictly dispreferred to B"

Suppose the "utility function"

assigns a real number (in "utils")
to every possible consumption bundle

We get completeness because any two numbers can be compared,
and we get transitivity because that's a property of the operator ">"

Marginal Utility

MU_1(x_1,x_2) = {\partial u(x_1,x_2) \over \partial x_1}

MU_2(x_1,x_2) = {\partial u(x_1,x_2) \over \partial x_2}

Given a utility function $u(x_1,x_2)$,
we can interpret the partial derivatives
as the "marginal utility" from
another unit of either good:

Indifference Curves and the MRS

Along an indifference curve, all bundles will produce the same amount of utility

In other words, each indifference curve
is a level set of the utility function.

The slope of an indifference curve is the MRS. By the implicit function theorem,

MRS(x_1,x_2) = \frac{\partial u(x_1,x_2) \over \partial x_1}{\partial u(x_1,x_2) \over \partial x_2} = {MU_1 \over MU_2}

(Note: we'll treat this as a positive number, just like the MRTS and the MRT)

Transformations and the MRS

Applying a positive monotonic transformation to a utility function doesn't affect
its MRS at any bundle (and therefore generates the same indifference map).

u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}

\hat u(x_1,x_2) = {1 \over 2}\ln x_1 + {1 \over 2}\ln x_2

MU_1 = {1 \over 2}x_1^{-{1 \over 2}}x_2^{1 \over 2}

MU_2 = {1 \over 2}x_1^{{1 \over 2}}x_2^{-{1 \over 2}}

MU_1 = {1 \over 2x_1}

MU_2 = {1 \over 2x_2}

MRS =

\displaystyle{= {x_2 \over x_1}}

\hat{MRS} =

\displaystyle{= {x_2 \over x_1}}

Example: $\hat u(x_1,x_2) = \ln(u(x_1,x_2))$

Desirable Properties of Preferences

We've asserted that all (rational) preferences are complete and transitive.

There are some additional properties which are true of some preferences:

Monotonicity
Convexity
Continuity
Smoothness

Monotonic Preferences: “More is Better"

\text{For any bundles }A=(a_1,a_2)\text{ and }B=(b_1,b_2)\text{, }A \succeq B \text{ if } a_1 \ge b_1 \text{ and } a_2 \ge b_2

\text{In other words, all marginal utilities are positive: }MU_1 \ge 0, MU_2 \ge 0

Convex Preferences: “Variety is Better"

Take any two bundles, $A$ and $B$, between which you are indifferent.

Would you rather have a convex combination of those two bundles,
than have either of those bundles themselves?

If you would always answer yes, your preferences are convex.

Concave Preferences: “Variety is Worse"

Take any two bundles, $A$ and $B$, between which you are indifferent.

Would you rather have a convex combination of those two bundles,
than have either of those bundles themselves?

If you would always answer no, your preferences are convex.

Perfect Substitutes

Goods that can always be exchanged at a constant rate.

Red pencils and blue pencils, if you con't care about color
One-dollar bills and five-dollar bills
One-liter bottles of soda and two-liter bottles of soda

u(x_1,x_2) = ax_1 + bx_2

Preferences over Tea and Biscuits

Charlotte always has 2 biscuits with every cup of tea;
if she has a plate of biscuits and one cup of tea, she'll only eat two. (And if she has a whole pot of tea and 6 biscuits, she'll stop pouring after 3 cups of tea.)

Each (cup + 2 biscuits) gives her 10 utils of joy.

Which other bundles give her the same utility as
3 cups of tea and 4 biscuits (A)?

Which other bundles give her the same utility as
1 cup of tea and 4 biscuits (B)?

Any combination that has 4 biscuits
and 2 or more cups of tea

Any combination that has 1 cup of tea and
at 2 or more biscuits

Perfect Complements

Goods that you like to consume
in a constant ratio.

Left shoes and right shoes
Sugar and tea

u(x_1,x_2) = \min \left\{\frac{x_1}{a},\frac{x_2}{b}\right\}

Cobb-Douglas

An easy mathematical form with interesting properties.

Used for two independent goods (neither complements nor substitutes) -- e.g., t-shirts vs hamburgers
Also called "constant shares" for reasons we'll see later.

u(x_1,x_2) = a \ln x_1 + b \ln x_2

u(x_1,x_2) = x_1^ax_2^b

Quasilinear

Generally used when Good 2 is
"dollars spent on other things."

Marginal utility of good 2 is constant
If good 2 is "dollars spent on other things," utility from good 1 is often given as if it were in dollars.

\text{e.g., }u(x_1,x_2) = \sqrt x_1 + x_2 \text{ or }u(x_1,x_2) = \ln x_1 + x_2

u(x_1,x_2) = v(x_1) + x_2

Constrained Optimization

Choice space:
all possible options

Feasible set:
all options available to you

Optimal choice:
Your best choice(s) of the ones available to you

Constrained Optimization

Canonical Constrained Optimization Problem

f(x_1,x_2)

\text{s.t. }

g(x_1,x_2) = k

k - g(x_1,x_2) = 0

\mathcal{L}(x_1,x_2,\lambda)=

\displaystyle{\max_{x_1,x_2}}

f(x_1,x_2)

k - g(x_1,x_2)

+ \lambda\ (

)

Suppose $g(x_1,x_2)$ is monotonic (increasing in both $x_1$ and $x_2$).

Then $k - g(x_1,x_2)$ is negative if you're outside of the constraint,
positive if you're inside the constraint,
and zero if you're along the constraint.

OBJECTIVE

FUNCTION

CONSTRAINT

\mathcal{L}(x_1,x_2,\lambda)=

f(x_1,x_2)

k - g(x_1,x_2)

+ \lambda\ (

)

\displaystyle{\partial \mathcal{L} \over \partial x_1} =

FIRST ORDER CONDITIONS

\displaystyle{\partial \mathcal{L} \over \partial x_2} =

\displaystyle{\partial \mathcal{L} \over \partial \lambda} =

\displaystyle{\partial f \over \partial x_1}

\displaystyle{\partial f \over \partial x_2}

k - g(x_1,x_2)

- \lambda\ \times

\displaystyle{\partial g \over \partial x_1}

\displaystyle{\partial g \over \partial x_2}

- \lambda\ \times

3 equations, 3 unknowns

Solve for $x_1$, $x_2$, and $\lambda$

How does the Lagrange method work?

It finds the point along the constraint where the
level set of the objective function passing through that point
is tangent to the constraint

The story so far, in two graphs

Production Possibilities Frontier
Resources, Production Functions → Stuff

Indifference Curves
Stuff → Happiness (utility)

Both of these graphs are in the same "Good 1 - Good 2" space

Better to produce
more good 1
and less good 2.

MRS

MRT

Better to produce
less good 1
and more good 2.

MRS

MRT

Better to produce
more good 1
and less good 2.

MRS

MRT

MRS

MRT

“Gravitational Pull" Towards Optimality

Better to produce
more good 2
and less good 1.

These forces are always true.

In certain circumstances, optimality occurs where MRS = MRT.

We've just seen that, at least under certain circumstances, the optimal bundle is
"the point along the PPF where MRS = MRT."

CONDITION 1:
CONSTRAINT CONDITION

CONDITION 2:
TANGENCY
CONDITION

This is just an application of the Lagrange method!

Example: Linear PPF, Cobb-Douglas Utility

Chuck has 150 hours of labor, and can produce 3 coconuts per hour or 2 fish per hour.

His preferences may be represented by the utility function $u(x_1,x_2) = x_1^2x_2$

\text{s.t. }

\mathcal{L}(L,W,\lambda)=

\displaystyle{\max_{x_1,x_2}}

x_1^2x_2

150 - {1 \over 3}x_1 - {1 \over 2}x_2

+ \lambda\ (

)

OBJECTIVE

FUNCTION

CONSTRAINT

x_1^2x_2

150 - {1 \over 3}x_1 - {1 \over 2}x_2 = 0

{1 \over 3}x_1 + {1 \over 2}x_2 = 150

\mathcal{L}(L,W,\lambda)=

x_1^2x_2

150 - {1 \over 3}x_1 - {1 \over 2}x_2

+ \lambda\ (

)

\displaystyle{\partial \mathcal{L} \over \partial x_1} =

FIRST ORDER CONDITIONS

\displaystyle{\partial \mathcal{L} \over \partial x_2} =

\displaystyle{\partial \mathcal{L} \over \partial \lambda} =

2x_1x_2

x_1^2

150 - {1 \over 3}x_1 - {1 \over 2}x_2

- \lambda\ \times

{1 \over 3}

{1 \over 2}

- \lambda\ \times

\displaystyle{\Rightarrow \lambda\ = \ \ \ \ \ \ \ \ \ \ \ \times}

{1 \over 3}x_1 + {1 \over 2}x_2 = 150

\displaystyle{\Rightarrow \lambda\ = \ \ \ \ \ \ \ \ \ \ \ \times}

2x_1x_2

x_1^2

MU_1

MU_2

MP_{L1}

MP_{L2}

\Rightarrow

Utility from last hour spent fishing

Utility from last hour spent collecting coconuts

Equation of PPF

\displaystyle{\lambda\ = \ \ \ \ \ \ \ \ \ \ \ \times}

{1 \over 3}x_1 + {1 \over 2}x_2 = 150

\displaystyle{ = \ \ \ \ \ \ \ \ \ \ \ \times}

2x_1x_2

x_1^2

MU_1

MU_2

MP_{L1}

MP_{L2}

Utility from last hour spent fishing

Utility from last hour spent collecting coconuts

Equation of PPF

{1 \over 3}x_1 + {1 \over 2}x_2 = 150

2x_1x_2

x_1^2

MU_1

MU_2

MP_{L1}

MP_{L2}

Equation of PPF

TANGENCY
CONDITION

MRS

MRT

CONSTRAINT

MRS

MRT

MRS

MRT

Corner Solutions

Interior Solution:

Corner Solution:

Optimal bundle contains
strictly positive quantities of both goods

Optimal bundle contains zero of one good
(spend all resources on the other)

If only consume good 1: $MRS \ge MRT$ at optimum

If only consume good 2: $MRS \le MRT$ at optimum

What would Lagrange find...?

Kinked Indifference Curve:

Kinked Constraint:

Discontinuities in the MRS
(e.g. Perfect Complements utility function)

Discontinuities in the MRT
(e.g. homework question with two factories)

Monotonicity and Convexity

If preferences are nonmonotonic,
you might be satisfied consuming something within the interior of your feasible set.

FISH

COCONUTS

PPF

Conditions for Calculus to Work

avoids a satiation point within the constraint

At the left corner of the constraint, $MRS > MRT$

avoids a corner solution when $x_1 = 0$

Monotonicity (more is better)

At the right corner of the constraint, $MRS < MRT$

avoids a corner solution when $x_2 = 0$

MRS and MRT are continuous as you move along the constraint

avoids a solution at a kink

ensures FOCs find a maximum, not a minimum

Convexity (variety is better)

Utility Maximization with Budget Constraints

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 10

Resources

Technology

Stuff

Happiness

🏭

⌚️

🤓

Part I:
The Real Economy

⏳

⛏

Resources

Firms

Stuff

Consumers

⏳

🏭

⌚️

🤓

Part I:
The Market Economy

Resource
Owners

👷🏽‍♀️

⛏

🏦

Resources

Firms

Stuff

Consumers

⏳

🏭

⌚️

🤓

Part I:
The Market Economy

Resource
Owners

👷🏽‍♀️

⛏

🏦

💵

Firms pay wages for labor

Firms pay rent on capital

Consumers pay prices for goods

Demand

Supply

🤩

🏪

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

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.

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?

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

🍏

🍌

"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

🍏

🍌

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

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?

Demand Functions and Demand Curves

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 11

Last Two Classes: What is the optimal bundle for a given budget line?

Today: What happens to the optimal bundle when prices/income change?

🍏

🍌

BL1

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

Specific Prices & Income

General Prices & Income

\text{Constraint: }2 x_1 + x_2 = 12

Plug tangency condition back into constraint:

Tangency Condition: $MRS = p_1/p_2$

\text{Constraint: }p_1x_1 + p_2x_2 = m

\text{Objective function: } x_1^{1 \over 2}x_2^{1 \over 2}

MRS(x_1,x_2) = {x_2 \over x_1}

{x_2 \over x_1}

{2 \over 1}

\Rightarrow x_2 = 2x_1

{x_2 \over x_1}

{p_1 \over p_2}

\Rightarrow x_2 = {p_1 \over p_2}x_1

2x_1 + 2x_1 = 12

x_1^* = 3

p_1x_1 + p_2\left[{p_1 \over p_2}x_1\right] = m

x_1^*(p_1,p_2,m) = {m \over 2p_1}

4x_1 = 12

2p_1x_1 = m

x_2^* = 2x_1^* = 6

x_2^*(p_1,p_2,m) = {m \over 2p_2}

Specific Prices & Income

General Prices & Income

\text{Constraint: }2 x_1 + x_2 = 12

\text{Constraint: }p_1x_1 + p_2x_2 = m

\text{Objective function: } x_1^{1 \over 2}x_2^{1 \over 2}

MRS(x_1,x_2) = {x_2 \over x_1}

x_1^* = 3

x_1^*(p_1,p_2,m) = {m \over 2p_1}

x_2^* = 2x_1^* = 6

x_2^*(p_1,p_2,m) = {m \over 2p_2}

OPTIMAL BUNDLE

DEMAND FUNCTIONS

(optimization)

(comparative statics)

x_1

x_2

p_1

DEMAND CURVE FOR GOOD 1

BL_{p_1 = 2}

BL_{p_1 = 3}

BL_{p_1 = 4}

"Good 1 - Good 2 Space"

"Quantity-Price Space for Good 1"

Remember what you learned about demand and demand curves in Econ 1 / high school:

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
- If two goods are substitutes, an increase in the price of one will increase the demand for the other (shift the demand curve to the right).
- If two goods are complements, an increase in the price of one will decrease the demand for the other (shift the demand curve to the left).
- If a good is a normal good, an increase in income will increase demand for the good
- If a good is an inferior good, an increase in income will decrease demand the good

x_1^*(p_1,p_2,m)\ \

Three Relationships

...its own price changes?

Movement along the demand curve

...the price of another good changes?

Complements

Substitutes

Independent Goods

How does the quantity demanded of a good change when...

...income changes?

Normal goods

Inferior goods

Giffen goods

(possible) shift of the demand curve

x_1^*(p_1,p_2,m)\ \

Three Relationships

...its own price changes?

Movement along the demand curve

How does the quantity demanded of a good change when...

The demand curve for a good

shows the quantity demanded of that good

as a function of its own price

holding all other factors constant

(ceteris paribus)

The price offer curve shows how the optimal bundle changes in good 1-good 2 space as the price of one good changes.

x_1

x_2

p_1

DEMAND CURVE FOR GOOD 1

BL_{p_1 = 2}

BL_{p_1 = 3}

BL_{p_1 = 4}

"Good 1 - Good 2 Space"

"Quantity-Price Space for Good 1"

PRICE OFFER CURVE

x_1^*(p_1,p_2,m)\ \

Three Relationships

...the price of another good changes?

How does the quantity demanded of a good change when...

Substitutes

Complements

When the price of one good goes up, demand for the other increases.

When the price of one good goes up, demand for the other decreases.

Independent

Demand not related

x_1

x_2

x_1

x_2

Complements: $p_2 \uparrow \Rightarrow x_1^* \downarrow$

What happens to the quantity of good 1 demanded when the price of good 2 increases?

Substitutes: $p_2 \uparrow \Rightarrow x_1^* \uparrow$

COMPLEMENTS:

UPWARD-SLOPING

PRICE OFFER CURVE

SUBSTITUTES:

DOWNWARD-SLOPING

PRICE OFFER CURVE

x_1^*(p_1,p_2,m)\ \

Three Relationships

How does the quantity demanded of a good change when...

...income changes?

Normal Goods

Inferior Goods

When your income goes up,
demand for the good increases.

When your income goes up,
demand for the good decreases.

The income offer curve shows how the optimal bundle changes in good 1-good 2 space as income changes.

x_1

x_2

x_1

x_2

Good 1 normal: $m \uparrow \Rightarrow x_1^* \uparrow$

What happens to the quantity of good 1 demanded when the income increases?

Good 1 inferior: $m \uparrow \Rightarrow x_1^* \downarrow$

BOTH NORMAL GOODS:

UPWARD-SLOPING

INCOME OFFER CURVE

ONE GOOD INFERIOR:

DOWNWARD-SLOPING

PRICE OFFER CURVE

CES Utility

= \begin{cases}\infty & \text{ if } x_1 < x_2 \\ 0 & \text{ if } x_1 > x_2 \end{cases}

-\infty

MRS = \left(x_2 \over x_1\right)^\infty

MRS = {x_2 \over x_1}

MRS = 1

u(x_1,x_2) = \min\{x_1,x_2\}

u(x_1,x_2) = x_1x_2

u(x_1,x_2) = x_1 + x_2

PERFECT
SUBSTITUTES

PERFECT
COMPLEMENTS

INDEPENDENT

PERFECT
SUBSTITUTES

u(x_1,x_2) = (x_1^r+x_2^r)^{1 \over r}

Constant Elasticity of Substitution (CES) Utility

MRS = \left(x_2 \over x_1\right)^{1-r}

u(x_1,x_2) = (x_1^r+x_2^r)^{1 \over r}

Constant Elasticity of Substitution (CES) Utility

MRS = \left(x_2 \over x_1\right)^{1-r}

-\infty

COMPLEMENTS: $r < 0$

SUBSTITUTES: $r > 0$

-1

MRS = \left(x_2 \over x_1\right)^2

{1 \over 2}

MRS = \left(x_2 \over x_1\right)^{1 \over 2}

u(x_1,x_2) = (x_1^{-1}+x_2^{-1})^{-1}

u(x_1,x_2) = \left(x_1^{1 \over 2} + x_2^{1 \over 2}\right)^2

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

How to Plot a Price Offer Curve

Pick a few values of the price which is being varied along the offer curve.
Be sure to think of extreme cases (when the price approaches zero or infinity)
Find the optimal bundle for each of those prices
Connect the dots
Do not try to find an equation

How to Plot an Income Offer Curve

Think about the "rule" that you plug into the budget line: e.g. tangency condition, ridge condition, "buy only good 1," "buy only good 1 if income is below a certain threshold," etc.
That rule describes the income offer curve.

Income and Substitution Effects of a Price Change

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 13

Two Effects

Substitution Effect

Effect of change in relative prices, holding utility constant.

Effect of change in real income,
holding relative prices constant.

Income Effect

Decomposition Bundle

Suppose that, after a price change,
we compensated the consumer
just enough to afford some bundle
that would give the same utility
as they had before the price change?

The Hicks decomposition bundle
is the lowest-cost bundle
that satisfies this condition.

Approach

TOTAL EFFECT

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

INCOME EFFECT

TOTAL EFFECT

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

INCOME EFFECT

Movement along POC

Shift of IOC

Movement along IOC

Cost Minimization

Utility Maximization

Cost Minimization

\max \ u(x_1,x_2)

\min \ p_1x_1 + p_2x_2

\text{s.t. }p_1x_1 + p_2x_2 = m

\text{s.t. }u(x_1,x_2) = U

Solution functions:
"Ordinary" Demand functions

x_1^*(p_1,p_2,m)

x_2^*(p_1,p_2,m)

Solution functions:
"Compensated" Demand functions

x_1^c(p_1,p_2,U)

x_2^c(p_1,p_2,U)

Utility Maximization

Cost Minimization

\text{Objective function: } x_1x_2

\text{Objective function: } p_1x_1 + p_2x_2

\text{Constraint: }p_1x_1 + p_2x_2 = m

\text{Constraint: }x_1x_2 = U

Plug tangency condition back into constraint:

\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}

Same tangency condition, different constraints

When Lagrange Doesn't Work: Perfect Complements

\text{Objective function: } p_1x_1 + p_2x_2

\text{Constraint: }\min\left\{\frac{x_1}{2},\frac{x_2}{3}\right\} = U

Hicks Decomposition

Hicks Decomposition Bundle

Suppose the price of good 1 increases from $p_1$ to $p_1^\prime$.

The price of good 2 ($p_2$) and income ($m$) remain unchanged.

Initial Bundle (A):
Solves
utility maximization
problem

Final Bundle (C):
Solves
utility maximization
problem

\max \ u(x_1,x_2)

\min \ p_1^\prime x_1 + p_2x_2

\text{s.t. }p_1x_1 + p_2x_2 = m

\text{s.t. }u(x_1,x_2) = U(A)

Decomposition Bundle (B):
Solves
cost minimization
problem

\max \ u(x_1,x_2)

\text{s.t. }p_1^\prime x_1 + p_2x_2 = m

"Compensated Budget Line"

The "compenated budget line" shows the budget line as if the consumer was given just enough money to achieve their initial utility at the new prices.

If a consumer's preferences are well behaved, her compensated budget line

Complements and Substitutes (one last time)

Complements

Substitutes

When the price of good 1 goes up...

Net effect: buy less of both goods

Net effect: buy less good 1 and more good 2

Substitution effect: buy less of good 1 and more of good 2

Income effect (if both goods normal): buy less of both goods

Substitution effect dominates

Income effect dominates

Which of the following would be true if these goods were substitutes rather than complements?

pollev.com/chrismakler

Production and Cost

Christopher Makler

Stanford University Department of Economics

Econ 50 : Lecture 15

Unit I: The “Real Economy"

Labor

Fish

🐟

Capital

Coconuts

🥥

[GOODS]

⏳

⛏

[RESOURCES]

Utility

🤓

The story thus far...

Unit IIa: Consumers and Prices

Labor

Fish

🐟

Capital

Coconuts

🥥

[GOODS]

⏳

⛏

[RESOURCES]

🤓

p_1

p_2

Consumer

The story thus far...

Unit IIb: Theory of the Firm

Labor

Firm

🏭

Capital

⏳

⛏

Customers

🤓

This unit: analyze the firms
consumers buy things from

Unit IIb: Theory of the Firm

Firm

🏭

Costs

wL + rK

Customers

🤓

This unit: analyze the firms
consumers buy things from

Unit IIb: Theory of the Firm

Firm

🏭

Costs

wL + rK

Revenue

From the firm's perspective, they get revenue and pay costs...

Unit IIb: Theory of the Firm

Costs

wL + rK

Revenue

Profit

pq - (wL + rK)

Next week: Solve the optimization problem
finding the profit-maximizing quantity $q^*$

...which is what we call profits

Unit IIb: Theory of the Firm

Costs

c(q) = wL(q) + rK(q)

Revenue

Profit

pq - c(q)

Today: Solve the cost minimization problem
and derive the cost function $c(q)$

Unit IIb: Theory of the Firm

Costs

c(q) = wL(q) + rK(q)

Revenue

r(q) = p(q) \times q

Profit

r(q) - c(q)

Today: Solve the cost minimization problem
and derive the cost function $c(q)$

Friday: Derive the revenue function $r(q)$

Unit IIb: Theory of the Firm

Costs

c(q) = wL(q) + rK(q)

Revenue

r(q) = p(q) \times q

Profit

r(q) - c(q)

Today: Solve the cost minimization problem
and derive the cost function $c(q)$

Friday: Derive the revenue function $r(q)$

Next Monday: Solve the profit maximization problem

Unit IIb: Theory of the Firm

Output Supply

q^*(p)

Input Demands

L^\star(w)

Next Wednesday: Analyze the comparative statics
of how the optimal choice changes with prices
for the special case of a price-taking firm

K^\star(r)

Cost Minimization

Cost Minimization Subject to a Utility Constraint

Cost Minimization Subject to an Output Constraint

\min p_1x_1 + p_2x_2 \text{ s.t. } u(x_1,x_2) = U

\min wL + rK \text{ s.t. } f(L,K) = q

\text{solutions}: x_1^c(p_1,p_2,U), x_2^c(p_1,p_2,U)

\text{solutions}: L^c(w,r,q), K^c(w,r,q)

Hicksian Demand

Conditional Demand

Cost Minimization: Lagrange Method

\mathcal{L}(L,K,\lambda)=

wL + rK +

(q - f(L,K))

\lambda

\frac{\partial \mathcal{L}}{\partial x_1} = w - \lambda MP_L

First Order Conditions

\frac{\partial \mathcal{L}}{\partial x_2} = r - \lambda MP_K

\frac{\partial \mathcal{L}}{\partial \lambda} = q - f(L,K) = 0 \Rightarrow q = f(L,K)

= 0 \Rightarrow \lambda = w \times {1 \over MP_L}

= 0 \Rightarrow \lambda = r \times \frac{1}{MP_K}

MRTS (slope of isoquant) is equal to the price ratio

\text{Also: }\frac{\partial \mathcal{L}}{\partial q} = \lambda = \text{Marginal cost of producing last unit using either input}

\text{Set two expressions }

\text{for }\lambda \text{ equal:}

\frac{MP_L}{MP_K} = \frac{w}{r}

Expansion Path

A graph connecting the input combinations a firm would use as it expands production: i.e., the solution to the cost minimization problem for various levels of output

Exactly the same as the income offer curve (IOC) in consumer theory.

(And, if the optimum is found via a tangency condition, exactly the same as the tangency condition.)

Long-Run Total Cost of $q$ Units

c^{LR}(q) = wL^c(w,r,q) + rK^c(w,r,q)

Conditional demand for labor

Conditional demand for capital

"The total cost of producing $q$ units in the long run
is the cost of the cost-minimizing combination of inputs
that can produce $q$ units of output."

Exactly the same as the expenditure function in consumer theory.

Conditional Demand
in the Short Run

If there's only one variable input,
it's perfectly inelastic -- there's only one choice!

Short-Run Total Cost of $q$ Units

TC(q\ |\ \overline{K}) = wL(q,\overline{K}) + r\overline{K}

Variable cost

"The total cost of producing $q$ units in the short run is the variable cost of the required amount of the input that can be varied,
plus the fixed cost of the input that is fixed in the short run."

Fixed cost

Total, Fixed and Variable Costs

c(q,\overline{K}) =

Fixed Costs $(F)$: All economic costs
that don't vary with output.

Variable Costs $(VC(q))$: All economic costs
that vary with output

explicit costs ($r \overline K$) plus
implicit costs like opportunity costs

r\overline{K}

wL(q,\overline{K})

TC(q) =

VC(q)

e.g. cost of labor required to produce
$q$ units of output given $\overline K$ units of capital

\text{Total Cost}: TC(q) = F + VC(q)

\text{Average Cost}: ATC(q) = \frac{TC(q)}{q} = \frac{F}{q} + \frac{VC(q)}{q}

Fixed Costs

Variable Costs

Average Fixed Costs (AFC)

Average Variable Costs (AVC)

Average Costs

\text{Total Cost}: TC(q) = F + VC(q)

\text{Marginal Cost}: MC(q) = TC'(q) = 0 + VC'(q)

Fixed Costs

Variable Costs

Marginal Cost

(marginal cost is the marginal variable cost)

\text{Total Cost}: TC(q) = F + VC(q)

\text{Marginal Cost}: MC(q) = TC'(q)

Marginal Cost

TC(q) = 64 + {1 \over 4}q^2

MC(q) = {1 \over 2}q

Marginal cost tends to "pull" average cost toward it:

MC > AC \Rightarrow AC \text{ increasing}

MC = AC \Rightarrow AC \text{ constant}

MC < AC \Rightarrow AC \text{ decreasing}

Marginal grade = grade on last test, average grade = GPA

Relationship between Average and Marginal Costs

Relationship between Marginal Cost and Marginal Product of Labor

TC(q) = wL^c(q | \overline K) + r \overline K

{dTC(q) \over dq} = w \times {dL^c(q) \over dq}

= w \times {1 \over MP_L}

Elasticity and Revenue

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 17

Notation

\epsilon_{Y,X} = \frac{\% \Delta Y}{\% \Delta X}

\text{Elasticity }(\epsilon) = \frac{\% \text{ change in endogenous variable}}{\% \text{ change in exogenous variable}}

"X elasticity of Y"
or "Elasticity of Y with respect to X"

Examples:
"Price elasticity of demand"
"Income elasticity of demand"
"Cross-price elasticity of demand"
"Price elasticity of supply"

Perfectly Inelastic

Inelastic

Unit Elastic

Elastic

Perfectly Elastic

|\epsilon| = 0

|\epsilon| < 1

|\epsilon| = 1

|\epsilon| > 1

|\epsilon| = \infty

Doesn't change

Changes by less than the change in X

Changes proportionally to the change in X

Changes by more than the change in X

Changes "infinitely" (usually: to/from zero)

How does the endogenous variable Y respond to a
change in the exogenous variable X?

\text{Elasticity }(\epsilon) = \frac{\% \text{ change in endogenous variable}}{\% \text{ change in exogenous variable}}

(note: all of these refer to the ratio of the perentage change, not absolute change)

\text{General formula: }\epsilon_{Y,X} = \frac{\Delta Y}{\Delta X}\times \frac{X}{Y}

\text{If }Y = a + bX \text{, then }\frac{\Delta Y}{\Delta X} =

\epsilon_{Y,X} = b\times \frac{X}{a + bX}

Note: the slope of the relationship is $b$.

Elasticity is related to, but not the same thing as, slope.

= \frac{bX}{a + bX}

\text{General formula: }\epsilon_{y,x} = \frac{dy}{dx}\times \frac{x}{y}

\text{If }y = ax^b \text{, then }\frac{dy}{dx} =

abx^{b-1}

\epsilon_{y,x} = abx^{b-1} \times \frac{x}{ax^b}

= b

This is related to logs, in a way that you can explore in the homework.

This is a super useful trick and one that comes up on midterms all the time!

Revenue for a Firm

Profit

The profit from $q$ units of output

\pi(q) = r(q) - c(q)

PROFIT

REVENUE

COST

is the revenue from selling them

minus the cost of producing them.

Revenue

We will assume that the firm sells all units of the good for the same price, $p$. (No "price discrimination")

r(q) = p(q) \times q

The revenue from $q$ units of output

REVENUE

PRICE

QUANTITY

is the price at which each unit it sold

times the quantity (# of units sold).

The price the firm can charge may depend on the number of units it wants to sell: inverse demand $p(q)$

Usually downward-sloping: to sell more output, they need to drop their price
Special case: a price taker faces a horizontal inverse demand curve;
can sell as much output as they like at some constant price $p(q) = p$

\text{revenue} = r(q) = p(q) \times q

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p(q)

\displaystyle{\text{average revenue} = \frac{r(q)}{q} = p(q)}

If the firm wants to sell $q$ units, it sells all units at the same price $p(q)$

Since all units are sold for $p$, the average revenue per unit is just $p$.

By the product rule...
let's delve into this...

Total, Average, and Marginal Revenue

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

dr = dp \times q + dq \times p

p(q)

The total revenue is the price times quantity (area of the rectangle)

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

dr = dp \times q + dq \times p

p(q)

Note: $MR < 0$ if

dq \times p

{dq \over q} < {dp \over p}

\% \Delta q < \% \Delta p

|\epsilon| < 1

dp \times q

The total revenue is the price times quantity (area of the rectangle)

If the firm wants to sell $dq$ more units, it needs to drop its price by $dp$

Revenue loss from lower price on existing sales of $q$: $dp \times q$

Revenue gain from additional sales at $p$: $dq \times p$

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

Marginal Revenue and Elasticity

= \left[{dp \over dq} \times {q \over p}\right] \times p + p

= {p \over {dq \over dp} \times {p \over q}} + p

= - {p \over |\epsilon_{q,p}|} + p

(multiply first term by $p/p$)

(simplify)

(since $\epsilon < 0$)

Notes

Elastic demand: $MR > 0$

Inelastic demand: $MR < 0$

In general: the more elastic demand is, the less one needs to lower ones price to sell more goods, so the closer $MR$ is to $p$.

Profit Maximization

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 18

Profit Maximization with Market Power

\pi(q) = r(q) - c(q)

Optimize by taking derivative and setting equal to zero:

\pi'(q) = r'(q) - c'(q) = 0

\Rightarrow r'(q) = c'(q)

Profit is total revenue minus total costs:

"Marginal revenue equals marginal cost"

Example:

p(q) = 20 - q \Rightarrow r(q) = 20q - q^2

c(q) = 64 + {1 \over 4}q^2

What is the profit-maximizing value of $q$?

\pi(q) = r(q) - c(q)

Multiply right-hand side by $q/q$:

\pi(q) = \left[{r(q) \over q} - {c(q) \over q}\right] \times q

= (AR - AC) \times q

Profit is total revenue minus total costs:

"Profit per unit times number of units"

AVERAGE PROFIT

Elasticity and Profit Maximization

MR = p\left[1 - \frac{1}{|\epsilon_{q,p}|}\right]

Recall our elasticity representation of marginal revenue:

MR = MC

Let's combine it with this
profit maximization condition:

p\left[1 - \frac{1}{|\epsilon_{q,p}|}\right] = MC

p = \frac{MC}{1 - \frac{1}{|\epsilon_{q,p}|}}

Really useful if MC and elasticity are both constant!

Inverse elasticity pricing rule:

One more way of slicing it...

P = \frac{MC}{1 - \frac{1}{|\epsilon|}}

1 - \frac{1}{|\epsilon|} = \frac{MC}{P}

\frac{P - MC}{P} = \frac{1}{|\epsilon|}

Fraction of price that's markup over marginal cost
(Lerner Index)

What if $|\epsilon| \rightarrow \infty$?

Competitive (Price-Taking) Firms

\text{marginal revenue} = {dr \over dq} = {dp \over dq} \times q + p

Marginal Revenue for Perfectly Elastic Demand

= \left[{dp \over dq} \times {q \over p}\right] \times p + p

= {p \over {dq \over dp} \times {p \over q}} + p

= - {p \over |\epsilon_{q,p}|} + p

(multiply first term by $p/p$)

(simplify)

(since $\epsilon < 0$)

Note

Perfectly elastic demand: $MR = p$

c(q) = 64 + {1 \over 4}q^2

r(q) = pq = 12q

\text{Let's assume }p = 12:

\text{Profit function is:}

\text{Total cost}

\pi(q) = 12q - [64 + {1 \over 4}q^2]

\text{Take derivative with respect to } q \text{ and set equal to zero:}

\pi'(q) = \ \ \ \ \ \ - \ \ \ \ \ \ = 0

{1 \over 2}q

Price

$q$

$/unit

P = MR

MC = {1 \over 2}q

\Rightarrow q^* = 24

Summary

All firms maximize profits by setting MR = MC
If a firm faces a downward-sloping demand curve,
the marginal revenue is less than the price.
The more elastic a firm's demand curve,
the less it will optimally raise its price above marginal cost.
A competitive firm faces a perfectly elastic demand curve,
so its marginal revenue is equal to the price.
Next time: establish a competitive firm's output supply and labor demand as functions of $p$ and $w$

Output Supply
and Labor Demand

Christopher Makler

Stanford University Department of Economics

Econ 50 | Lecture 19

The Competitive Firm

L^c(w,r,q)

K^c(w,r,q)

Exogenous Variables

Endogenous Variables

technology, f()

level of output, q

conditional
input demands

Cost Minimization

Isoquant

Isocost

lines

factor prices (w, r)

q^*(w,r,p)

profit-maximizing output supply

Profit Maximization

output price, p

Total Revenue

Total Cost

profit-maximizing input demands

total cost

TC(w,r,q)

L^*(w,r,p)

K^*(w,r,p)

Optimization

What is an agent's optimal behavior for a fixed set of circumstances?

x_1^*

x_2^*

q^*

L^*

Utility-maximizing bundle for a consumer

Profit-maximizing quantity for a firm

Profit-maximizing input choice for a firm

Comparative Statics

How does an agent's optimal behavior change when circumstances change?

x_1^*

x_2^*

q^*

L^*

Utility-maximizing bundle for a consumer

Profit-maximizing quantity for a firm

Profit-maximizing input choice for a firm

x_1^*(p_1\ |\ p_2,m)

x_2^*(p_2\ |\ p_2,m)

q^*(p\ |\ w,r)

L^*(w\ |\ p,r)

Assumptions

We will be analyzing a
competitive (price-taking) firm

output price $p$
wage rate $w$
rental rate $r$

f(L,K) = \sqrt{LK}, \overline K = 32

64 + {wq^2 \over 32}

{wq \over 16}

\pi(q) = \ \ \ \ \ - [\ \ \ \ \ \ \ \ \ \ \ \ \ \ ]

\pi^\prime(q) = \ \ \ - \ \ \ \ \ \ \ \ = 0

Take derivative and set = 0:

Solve for $q^*$:

q^*(p\ |\ w) = {16p \over w}

SUPPLY FUNCTION

Profit as a function of quantity

Profit as a function of labor

c(q) = wL(q) + r \overline K

1. Total costs = cost of required inputs

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(q) = pq - [wL(q) + r\overline K]

\pi^\prime(q) = p - w \times {dL \over dq} = 0

r(L) = p \times f(L)

1. Total revenue = value of output produced

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

\pi(L) = p\times f(L) - [wL + r\overline K]

\pi^\prime(L) = p \times {dq \over dL} - w = 0

p = w \times {dL \over dq}

p \times {dq \over dL} = w

\text{Profit }\pi = pq - [wL + rK]

Profit as a function of quantity

Profit as a function of labor

1. Total revenue = value of output produced

2. Profit = total revenues minus total costs

3. Take derivative of profit function, set =0

"Keep producing output as long as the marginal revenue from the last unit produced is at least as great as the marginal cost of producing it."

p = MC

r(L) = p \times f(L)

\pi(L) = p\times f(L) - [wL + r\overline K]

\pi^\prime(L) = p \times {dq \over dL} - w = 0

p \times {dq \over dL} = w

Profit as a function of quantity

Profit as a function of labor

"Keep producing output as long as the marginal revenue from the last unit produced is at least as great as the marginal cost of producing it."

"Keep hiring workers as long as the marginal revenue from the output of the last worker is at least as great as the cost of hiring them."

p = MC

w = MRP_L

f(L,K) = \sqrt{LK}, \overline K = 32

wL + 64

p\sqrt{32L}

\pi(L) = \ \ \ \ \ \ \ \ \ \ \ \ \ - [\ \ \ \ \ \ \ \ \ \ \ \ \ \ ]

\pi^\prime(q) = \ \ \ \ \ \ \ \ \ \ - \ \ \ \ = 0

p\sqrt{8 \over L}

MRPL

Take derivative and set = 0:

Solve for $L^*$:

L^*(w\ |\ p) = 8 \left({p \over w}\right)^2

LABOR DEMAND FUNCTION

L^*(w\ |\ p) = 8 \left({p \over w}\right)^2

LABOR DEMAND FUNCTION

f(L,K) = \sqrt{LK}, \overline K = 32

L^*(w\ |\ p) = 8 \left({p \over w}\right)^2

LABOR DEMAND FUNCTION

q^*(p\ |\ w) = {16p \over w}

SUPPLY FUNCTION

the conditional labor demand
for the profit-maximizing supply:

L^*(w\ |\ p) = L^c(q^*(p\ |\ w))

L^c(q) = {1 \over 32} q^2

= {1 \over 32}\left(16p \over w\right)^2

= 8\left(p \over w\right)^2

The profit-maximizing labor demand is

Partial Equilibrium
and Welfare Analysis

Christopher Makler

Stanford University Department of Economics

Econ 50: Lectures 20 and 21

Responding to Prices

Weeks 4-5: Consumer Theory

Firms face prices and
choose how much to produce

Consumers face prices and
choose how much to buy

Weeks 6-7: Theory of the Firm

Individual demand curve, $d^i(p)$: quantity demanded by consumer $i$ at each possible price

Market demand sums across all consumers:

\displaystyle D(p) = N_Cd(p)

\displaystyle D(p) = \sum_{i=1}^{N_C}{d^i(p)}

If all of those consumers are identical and demand the same amount $d(p)$:

There are $N_C$ consumers, indexed with superscript $i \in \{1, 2, 3, ..., N_C\}$.

Market demand curve, $D(p)$: quantity demanded by all consumers at each possible price

Market demand sums across all consumers:

\displaystyle D(p) = \sum_{i=1}^{N_C}{d^i(p)}

Firm supply curve, $s^j(p)$: quantity supplied by firm $j$ at each possible price

Market supply sums across all firms:

\displaystyle S(p) = N_Fs(p)

\displaystyle S(p) = \sum_{j=1}^{N_F}{s^j(p)}

If all of those firms are identical and supply the same amount $s(p)$:

There are $N_F$ competitive firms, indexed with superscript $j \in \{1, 2, 3, ..., N_F\}$.

Market supply curve, $S(p)$: quantity supplied by all firms at each possible price

Calculating Partial Equilibrium

\displaystyle \sum_{j=1}^{N_F}{q_j^*}

\displaystyle \sum_{i=1}^{N_C}{x_i^*}

Price $p^*$ is an equilibrium price in a market if:

1. Consumer Optimization: each consumer $i$ is consuming a quantity $x_i^*(p^*)$ that solves their utility maximization problem.

2. Firm Optimization: each firm $j$ is producing a quantity $q_j^*(p^*)$ that solves their profit maximization problem.

3. Market Clearing: the total quantity demanded by all consumers equals the total quantity supplied by all firms.

p^* = \frac{MU_i(x_i^*)}{\lambda_i}

p^* = MC_j(q_j^*)

"Marginal benefit in dollars per unit of good 1"

\underbrace{S(p^*)}

\underbrace{D(p^*)}

\displaystyle \sum_{j=1}^{N_F}{s^j(p)}

\displaystyle \sum_{i=1}^{N_C}{d^i(p)}

Important Note: Three Kinds of “=" Signs

1. Mathematical Identity: holds by definition

2. Optimization condition: holds when an agent is optimizing

3. Equilibrium condition: holds when a system is in equilibrium

MRS = {MU_1 \over MU_2}

MRS = {p_1 \over p_2}

Is this the “right" price?

If you were an omniscient social planner, could you do "better"
than the price the market "chooses"?

Welfare Analysis:
Consumer and Producer Surplus

TOTAL WELFARE

(dollars)

Marginal welfare,
in dollars per unit:

Total benefit to consumers minus total cost to firms

Marginal benefit to consumers minus marginal cost to firms

Relationships between Markets

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 22

Unit IV: Equilibrium

This Week:
General Equilibrium

Analyze a single market, taking everything determined outside that market as given

(prices of other goods,
consumer income, wages)

Last Week:
Partial Equilibrium

Today: examine linkages between markets

Analyze all markets simultaneously

Thursday: solve for equilibrium quantities in all markets simultaneously as a function only of production functions, resource constraints, and consumer preferences.

(endogenize all prices, income, wages)

Equilibrium in One Market

Market for Good 2

Market for Good 1

Equilibrium in Two Markets with Related Demand

S_1(p_1)

D_1(p_1,p_2)

S_2(p_2)

D_2(p_1,p_2)

S(p)

D(p)

Market for Good 2

Market for Good 1

Equilibrium in Two Markets with Related Demand

S_1(p_1) = {p_1 \over a}

D_1(p_1,p_2) = {m \over p_1 + p_2}

S_2(p_2) = {p_2 \over b}

D_2(p_1,p_2) = {m \over p_1 + p_2}

Notation

$Y_1$ = total amount of good 1 produced by all firms in an economy

$Y_2$ = total amount of good 2 produced by all firms in an economy

$GDP(Y_1,Y_2) = p_1Y_1 + p_2Y_2$
= market value of all final goods and services produced in an economy

Resource Allocation

Narrow question:
How many productive resources should we devote to a single good?

Broader question:
How should we allocate productive resources across goods?

Firms will choose the quantity at which $p = MC$

Firms will choose the point along the PPF at which ${p_1 \over p_2} = MRT$

GDP maximizing point!!

Firms will choose the point along the PPF at which ${p_1 \over p_2} = MRT$

How will they do this?

In this lecture, we'll show:

MRT = {MC_1 \over MC_2}

p_1 = MC_1

p_2 = MC_2

=MRT

PROFIT MAX FOR GOOD 1

PROFIT MAX FOR GOOD 2

Input prices signal resource constraints, keep production on PPF.

Conditions for
GDP Maximization

{p_1 \over p_2} = {MC_1 \over MC_2}

TANGENCY CONDITION

CONSTRAINT CONDITION

L_1(Y_1) + L_2(Y_2) = \overline L

Firms in industry 1 set $p_1 = MC_1$

Firms in industry 2 set $p_2 = MC_2$

How does competition achieve this?

Wages adjust until the
labor market clears

The Circular Flow

Consumers

Good 1 Firms

Market for Good 1

Market for Good 2

Market for Labor

Good 2 Firms

Money flows clockwise

Goods, labor flow counter-clockwise

General Equilibrium: Everyone optimizes, all markets clear simultaneously.

1. Given prices $p_1,p_2$, firms will choose the point $(Y_1^*,Y_2^*)$ along the PPF where $MRT = \frac{p_1}{p_2}$

2. All money received by firms $(p_1Y_1^* + p_2Y_2^*)$ will become income $M$ for consumers.

3. Given prices $p_1,p_2$ and income $M$, consumers will choose the point $(X_1^*,X_2^*)$ along the budget line where $MRS = \frac{p_1}{p_2}$

Equilibrium in and Disequilibrium in the Short Run

MRS =

p_1 = MC_1

p_2 = MC_2

= MRT

If consumers and firms all face the same price, and if they choose the same quantity in response to that price, then MRS = MRT.

u(X_1,X_2) = \alpha \ln X_1 + (1-\alpha) \ln X_2

Key Takeaways

In general equilibrium, everything having to do with money has been endogenized.

We are left with the same things Chuck had on his desert island:
resources, production technologies, and preferences.

As an individual in autarky, Chuck solved his maximization problem by setting
the marginal benefit of any activity he undertook equal to its opportunity cost.

Markets solve the problem of how to resolve scarcity in the same way:
by having everyone equate their own MB or MC to a common price,
which represents the opportunity cost of using resources in some other way.

Course Retrospective

Part I: The Real Economy

Part II: Little Green Pieces of Paper

Three Fundamental Tools of Analysis

Unit I Overview

Unit I: Scarcity and Choice

Week 1: Modeling Production with Multivariable Functions

Unit I: Scarcity and Choice

Week 2: Setting Up the Problem

Unit I: Scarcity and Choice

Unit I: Scarcity and Choice

Week 2: Setting Up the Problem

Unit I: Scarcity and Choice

Week 3: Solving the Constrained Optimization problem

Derivative Along a Level Set

Production Functions

Production Functions

A mathematical form describing how much output is produced as a function of inputs.

Marginal Products of Labor and Capital

Economic definition: how much more output is produced if you increase labor or capital?

Mathematical definition: partial derivatives of the production function

These are both rates: they are measured in terms of units of ouptut per unit of input.

Marginal Rate of Technical Substitution (MRTS)

Economic definition: the rate at which a producer can substitute one input for another while keeping output at the same level

Mathematical definition: slope of an isoquant

Recall: by implicit function theorem, the slope of a level set is given by

Therefore the formula for the MRTS is

Examples of Production Functions

Linear

Leontief (Fixed Proportions)

Cobb-Douglas

Constant Elasticity of Substitution (CES)

Linear Production Function

Leontief (Fixed Proportions) Production Function

Cobb-Douglas Production Function

CES Production Function

Scaling Production in the Short Run

Suppose \(K\) is fixed at some \(\overline K\) in the short run. Then the production function becomes \(f(L\ |\ \overline K)\)

Scaling Production in the Long Run

What happens when we increase all inputs proportionally?

For example, what happens if we double both labor and capital?

Does doubling inputs -- i.e., getting \(f(2L,2K)\) -- double output?

Decreasing Returns to Scale

Constant Returns to Scale

Increasing Returns to Scale

Resource Constraints and Production Possibilities

Good 1 - Good 2 Space

Good 1 - Good 2 Space

Multiple Uses of Resources

Resource Constraint

A PPF with Linear Technologies

A PPF with Diminishing \(MP_L\)

Relationship between MPL's and MRT

Diminishing \(MP_L\)'s and Increasing \(MRT\)

Preferences and Utility

Preferences: Ordinal Ranking of Options

Utility Functions

Production Functions

Utility Functions

Representing Preferences with a Utility Function

Marginal Utility

Indifference Curves and the MRS

Transformations and the MRS

Desirable Properties of Preferences

Monotonic Preferences: “More is Better"

Convex Preferences: “Variety is Better"

Concave Preferences: “Variety is Worse"

Perfect Substitutes

Goods that can always be exchanged at a constant rate.

Preferences over Tea and Biscuits

Perfect Complements

Goods that you like to consume in a constant ratio.

Cobb-Douglas

An easy mathematical form with interesting properties.

Quasilinear

Generally used when Good 2 is "dollars spent on other things."

Constrained Optimization

Constrained Optimization

Canonical Constrained Optimization Problem

How does the Lagrange method work?

Part II: Little Green
Pieces of Paper

Week 1: Modeling Production
with Multivariable Functions

Mathematical definition:
partial derivatives of the production function

Recall: by implicit function theorem,
the slope of a level set is given by

Leontief
(Fixed Proportions)

Suppose \(K\) is fixed at some \(\overline K\) in the short run.
Then the production function becomes \(f(L\ |\ \overline K)\)

Diminishing \(MP_L\)'s
and Increasing \(MRT\)

Goods that you like to consume
in a constant ratio.

Generally used when Good 2 is
"dollars spent on other things."

It finds the point along the constraint where the
level set of the objective function passing through that point
is tangent to the constraint

Part I:
The Real Economy

Part I:
The Market Economy

Part I:
The Market Economy

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

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

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