Course Retrospective

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 25

Plan for the Week

  • Wednesday lecture: review
  • Wednesday Q section: review
  • No other sections
  • SEA study hall: tonight 6:30-8:30, Lathrop 282
  • Office hours (rotating cast of TA's, me): 10am-6pm Thursday, Lathrop 292

Ways to Think about Learning

Strategic:

"knowing when" - given an unstructured problem, which model/framework is most relevant?

Schematic:

Procedural:

Declarative:

"knowing why" - how do different concepts/models relate to one other? 

"knowing how" - given a well-defined problem, ability to follow correct procedure to solve it

"knowing that" - facts and figures, vocabulary

Goals of the Course

Strategic:

Develop rigorous approach to economic modeling; understand how assumptions map into conclusions.

Schematic:

Procedural:

Declarative:

Understand the relationships between
words, math, and graphs.

Learn the techniques of
optimization, equilibrium, and comparative statics.

Know the definitions of key economic terms.

Two Kinds of Optimization

Tradeoffs between two goods

Optimal quantity of one good

🍎

(not feasible)

(feasible)

🍌

Optimal choice

🙂

😀

😁

😢

🙁

🍎

benefit and cost per unit

Marginal Cost

Marginal Benefit

Optimal choice

Two Kinds of Optimization

Tradeoffs between two goods

Optimal quantity of one good

  1. Modeling consumer preferences
    with multivariate calculus
  2. Utility maximization subject to a budget constraint
  3. Consumer demand

Midterm 1: April 25

  1. Cost minimization
  2. Firm profit maximization and supply
  3. Competitive equilibrium

Midterm 2: May 16

Model: Consumer Choice

Models: Firms, Markets

Remaining 2-3 weeks: real-world applications of these models

Today's Agenda

  • Constrained optimization with
    more than one choice variable
  • Unconstrained optimization with
    one choice variable
  • Comparative statics

Modules 1-4, Module 7
Analyzing Tradeoffs

Choices in general

Choices of commodity bundles

Choosing bundles of two goods

Good 1 \((x_1)\)

Good 2 \((x_2)\)

Completeness axiom:
any two bundles can be compared.

Implication: given any bundle \(A\),
the choice space may be divided
into three regions:

preferred to A

dispreferred to A 

indifferent to A 

Indifference curves cannot cross!

A

The indifference curve through A connects all the bundles indifferent to A.

Indifference curve
through A

Special Case: Good 1 - Good 2 Space

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 =
A
B
4
8
12
16
20
4
8
12
16
20
\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

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

TW: HORRIBLE STROBE EFFECT!

8 \text{ units of good 2}
4 \text{ units of good 1}
2
\displaystyle{\frac{\text{units of good 2}}{\text{units of good 1}}}
\Delta x_2
\Delta x_1

Marginal Rate of Substitution

X = (10,24)

🍏🍏

🍏🍏

🍏🍏

🍏🍏

🍏🍏

🍌🍌🍌🍌

🍌🍌🍌🍌

🍌🍌🍌🍌

🍌🍌🍌🍌

🍌🍌🍌🍌

🍌🍌🍌🍌

Y=(12,20)

🍏🍏

🍏🍏

🍏🍏

🍏🍏

🍏🍏

🍏🍏

🍌🍌🍌🍌

🍌🍌🍌🍌

🍌🍌🍌🍌

🍌🍌🍌🍌

🍌🍌🍌🍌

Suppose you were indifferent between the following two bundles:

Starting at bundle X,
you would be willing
to give up 4 bananas
to get 2 apples

Let apples be good 1, and bananas be good 2.

Starting at bundle Y,
you would be willing
to give up 2 apples
to get 4 bananas

MRS = {4 \text{ bananas} \over 2 \text{ apples}}
= 2\ {\text{bananas} \over \text{apple}}

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

Calculating 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}

UTILS

UNITS OF GOOD 1

UTILS

UNITS OF GOOD 2

Intertemporal choice: \(c_1\), \(c_2\) represent
consumption in different time periods.

Risk aversion: \(c_1\), \(c_2\) represent
consumption in different states of the world.

u(c_1,c_2) = v(c_1)+\beta v(c_2)
u(c_1,c_2) = \pi_1 v(c_1)+ \pi_2 v(c_2)

Applications from Module 7

If \(v(c)\) exhibits diminishing marginal utility:
MRS is higher if you have less money today
and/or more money tomorrow

MRS is lower if you are more patient (\(\beta\) is high)

MRS = {v^\prime (c_1) \over \beta v^\prime (c_2)}
MRS = {\pi v^\prime (c_1) \over (1-\pi) v^\prime (c_2)}

MRS is higher if state 1 is more likely to occur
(\(\pi\) is higher)

Let \(v(c)\) be the value function describing how much utility you get from money/consumption

Modeling preferences with functional forms

= \begin{cases}\infty & \text{ if } x_1 < x_2 \\ 0 & \text{ if } x_1 > x_2 \end{cases}
-\infty
1
0
MRS = \left(x_2 \over x_1\right)^\infty
MRS = {x_2 \over x_1}
MRS = 1
r
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}

[50Q only]

  • 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

Risk aversion

v(c) = c^r \Rightarrow \mathbb{E}[v(c)] = \pi c_1^r + (1 - \pi) c_2^r

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

One type of feasible set: the budget set

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
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

(1+r)c_1 + c_2 = (1+r)m_1 + m_2

How much can you consume in the future if you save all your present income \(m_1\)?

How much can you consume in the present if you borrow the maximum amount against your future income?

FV = c_2 = (1+r)m_1 + m_2
\displaystyle{PV = c_1 = m_1 + {m_2 \over 1 + r}}

Selling Labor at a Constant Wage

Leisure (R)

Consumption (C)

24
24 - L
M
M + wL

You sell \(L\) hours of labor at wage rate \(w\).

You start with 24 hours of leisure and \(M\) dollars.

You earn \(\Delta C = wL\) dollars in addition to the \(M\) you had.

L
wL
M + 24w
C = M + wL
C = M + w(24 - R)

...and you consume \(R = 24 - L\) hours of leisure.

wR + C = 24w + M

Budget Line Equation

Leisure (R)

Consumption (C)

24
M
M + 24w
wR + C = 24w + M
p_1x_1 + p_2x_2 = m

How does this compare to a normal budget line?

You sell \(L\) hours of labor at wage rate \(w\).

You start with 24 hours of leisure and \(M\) dollars.

You earn \(\Delta C = wL\) dollars in addition to the \(M\) you had.

C = M + wL
C = M + w(24 - R)

...and you consume
\(R = 24 - L\) hours of leisure.

wR + C = 24w + M

Combining preferences and constraints

If we superimpose the budget line on the utility "hill" the nature of the problem becomes clear:

Question: mathematically, how does the utility change as you spend more money on good 1?

What does it mean if you get more "bang for your buck" from good 1 than good 2?

\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.

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 (\(x_1 \rightarrow 0\))

u(x_1,x_2)
\max

The Lagrange Method

x_1,x_2
\text{s.t.}
p_1x_1 + p_2x_2
\le m

Cost of Bundle X

Income

Utility

\max

The Lagrange Method

x_1,x_2
\text{s.t.}
m - p_1x_1 - p_2x_2

Income left over

u(x_1,x_2)
u(x_1,x_2)

Utility

\ge 0

The Lagrange Method

m - p_1x_1 - p_2x_2

Income left over

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

Utility

(utils)

(dollars)

utils/dollar

\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 Lagrange Method

m - p_1x_1 - p_2x_2
\mathcal{L}(x_1,x_2,\lambda)=
\lambda
u(x_1,x_2)
+
(
)

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

\text{Objective function: } x_1x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m

Cost Minimization

\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }x_1x_2 = U

Utility Maximization

Cost Minimization

\text{Objective function: } u(x_1,x_2) = x_1x_2
\text{Objective function: } c(x_1,x_2) = p_1x_1 + p_2x_2

Solution functions:
"Ordinary" Demand functions

x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^*(p_1,p_2,m) = {m \over 2p_2}

Solution functions:
"Compensated" Demand functions

x_1^c(p_1,p_2,U) = \sqrt{p_2U \over p_1}
x_2^c(p_1,p_2,U) = \sqrt{p_1U \over p_2}

What is the optimized value of the objective function?

V(p_1,p_2,m)=u(x_1^*(p_1,p_2,m),x_2^*(p_1,p_2,m))
E(p_1,p_2,U)=c(x_1^c(p_1,p_2,U),x_2^c(p_1,p_2,U))

INDIRECT UTILITY FUNCTION

EXPENDITURE FUNCTION

Utility from utility-maximizing choice,
given prices and income

Cost of cost-minimizing choice,
given prices and a target utility

Utility Maximization

Cost Minimization

\text{Objective function: } u(x_1,x_2) = x_1x_2
\text{Objective function: } c(x_1,x_2) = p_1x_1 + p_2x_2
x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^*(p_1,p_2,m) = {m \over 2p_2}
x_1^c(p_1,p_2,U) = \sqrt{p_2U \over p_1}
x_2^c(p_1,p_2,U) = \sqrt{p_1U \over p_2}

What is the optimized value of the objective function?

V(p_1,p_2,m)=u(x_1^*(p_1,p_2,m),x_2^*(p_1,p_2,m))
E(p_1,p_2,U)=c(x_1^c(p_1,p_2,U),x_2^c(p_1,p_2,U))
= x_1^*(p_1,p_2,m) \times x_2^*(p_1,p_2,m)
= \displaystyle{{m \over 2p_1} \times {m \over 2p_2}}
= \displaystyle{{m^2 \over 4p_1p_2}}
=\displaystyle{p_1\sqrt{p_2U \over p_1} + p_2\sqrt{p_1U \over p_2}}
=p_1x_1^c(p_1,p_2,U) + p_2x_2^c(p_1,p_2,U)
=2\sqrt{p_1p_2U}

INDIRECT UTILITY FUNCTION

EXPENDITURE FUNCTION

The Lagrange Method: Utility Maximization

m - p_1x_1 - p_2x_2

Income left over

\mathcal{L}(x_1,x_2,\lambda)=
\lambda
x_1x_2
+
(
)

Utility

(utils)

(dollars)

utils/dollar

The Lagrange Method: Consumer Cost Minimization

p_1x_1 + p_2x_2
\mathcal{L}(x_1,x_2,\lambda)=
\lambda
+
(
)
U - x_1x_2

Expenditure

Utility

(utils)

(dollars)

dollars/util

The Lagrange Method: Consumer Cost Minimization

p_1x_1 + p_2x_2
\mathcal{L}(x_1,x_2,\lambda)=
\lambda
+
(
)
U - x_1x_2

Expenditure

Utility

(utils)

(dollars)

dollars/util

The Lagrange Method: Firm Cost Minimization

wL + rK
\mathcal{L}(L,K,\lambda)=
\lambda
+
(
)
q - f(L,K)

Cost

Output

(units)

(dollars)

dollars/unit

Firm Cost Minimization

\mathcal{L}(L,K,\lambda)=
wL + rK +
(q - f(L,K))
\lambda
\frac{\partial \mathcal{L}}{\partial L} = w - \lambda MP_L

First Order Conditions

\frac{\partial \mathcal{L}}{\partial K} = 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}

What happens when the Lagrange method fails?

  • Corner solutions
  • Solutions at kinks

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: \(MRS \ge {p_1 \over p_2}\) at optimum

If only consume good 2: \(MRS \le {p_1 \over p_2}\) at optimum

Corner Solutions 

\(MRS < {p_1 \over p_2}\) along
the entire budget line!

{MU_1 \over p_1} =
{MU_2 \over p_2} =
{3 \over 4}
{6 \over p_1}
{2 \over 3}
{MU_3 \over p_3} =

Key insight: you will only buy the good with the highest "boom for the buck" (i.e., \(MU/p\)).

What does this tell us about goods 2 and 3?

You will never buy good 2!

{MU_1 \over p_1} =
{3 \over 4}
{6 \over p_1}
{MU_3 \over p_3} =

Key insight: you will only buy the good with the highest "boom for the buck" (i.e., \(MU/p\)).

When will you only buy good 1?

{3 \over 4}
{6 \over p_1}
>

Key insight: you will only buy the good with the highest "boom for the buck" (i.e., \(MU/p\)).

When will you only buy good 1?

p_1 < 8
x_1^*(p_1)=\begin{cases}0 & \text{ if }p_1 > 8\\ [0,3] & \text{ if }p_1 = 8\\ {24 \over p_1} & \text{ if }p_1 < 8\end{cases}

Key insight: you will only buy the good with the highest "boom for the buck" (i.e., \(MU/p\)).

What happens if you try to equate the MU/p across goods (or equivalently, set the MRS's equal to the price ratios)?

{MU_1 \over p_1} =
{MU_2 \over p_2} =
{3 \over 4}
{6 \over p_1}
{2 \over 3}
{MU_3 \over p_3} =

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 left 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.

Summary: Constrained Optimization

  • Analysis of tradeoffs
  • Lagrange multiplier represents the "exchange rate" between the units of the objective function and the units of the constraint.
  • Lagrange method works by equating "bang for your buck" (or "buck for your bang" / marginal cost) across competing goods.
  • However, some solutions occur at corners or kinks where bang for your buck is not equated; in that case, have to apply logic.

Today's Agenda

  • Constrained optimization with
    more than one choice variable
  • Unconstrained optimization with
    one choice variable
  • Comparative statics

✅

How much of a good to produce?

  • Firm: maximize profit by setting MR(q) = MC(q)
  • Competitive firm: maximize profit by setting P = MC(q)
  • Market: maximize sum of consumer and producer surplus by setting D(p) = S(p)
  • Market with externalities, common resources: maximize social welfare by setting MSB = MSC
  • Public goods: maximize social welfare by setting  the sum of marginal benefits equal to marginal cost

Theory of the Firm

Labor

Firm

🏭

Capital

⏳

⛏

Customers

🤓

p
w
r
q
L
K

Firms buy inputs
and produce some good,
which they sell to a customer.

PRICE

QUANTITY

Labor

w
L

Capital

r
K

Output

p
q

Firm

🏭

Costs

wL + rK

Revenue

pq

Theory of the Firm

Firms buy inputs
and produce some good,
which they sell to a customer.

PRICE

QUANTITY

Labor

w
L

Capital

r
K

Output

p
q

Costs

wL + rK

Revenue

pq

Profit

\pi = pq - (wL + rK)

The difference between their revenue and their cost is what we call profits, denoted by the Greek letter \(\pi\).

Theory of the Firm

Costs

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

Revenue

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

Profit

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

Theory of the Firm

Our approach will be to write costs, revenues, and profits all as functions of the ouput \(q\).

Pigovian tax:

 

Internalize the externality so that private marginal cost equals social marginal cost.

Competitive equilibrium:
consumers set \(P = MB\),
producers set \(P = PMC \Rightarrow MB = PMC\)

With a tax: consumers set \(P = MB\),
producers set \(P - t = PMC\)

Fees

Suppose you needed to buy a fishing permit for a fee F.

What value of F would result in the optimal L*?

Taxes

Suppose the village levied a tax of t per fish caught.

What value of t would result in the optimal L*?

Solving for the Optimal Quantity

  • Figure out the profit or welfare function from the context of the question; write this as a function of the relevant choice variable
  • Take the derivative and set it equal to zero!
  • Be able to interpret this as balancing the relevant marginal benefits and marginal costs. 

Today's Agenda

  • Constrained optimization with
    more than one choice variable
  • Unconstrained optimization with
    one choice variable
  • Comparative statics

✅

✅

Optimization: What is the optimal bundle for a given situation?

Comparative Statics: What happens to the optimal bundle when something about that situation changes?

🍏

🍌

BL1

Example: solve 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

  • 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

Analyzing Comparative Statics

  • Solve the optimization problem as a function of the variables that will shift the solution
  • Try to determine if there are key points where behavior changes
  • Know how to tell a story with a graph: most times, you don't have to graph things precisely, just show how a change percolates graphically through a model!

Summary

  • Relatively few mathematical techniques
  • Lots of different applications
  • Within each application, know your definitions well so you can apply the relevant techniques.
  • Read the questions carefully, think before you write, and don't do too much work: most questions can be answered without solving for an optimum.
  • Good luck!