Analyzing a
Price Change

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 11

Consumer Theory

Utility maximization
subject to a budget constraint

Derive demand functions

Cost minimization

Analyze the effects
of a price change

Plot demand, offer curves

Decomposition

Breaking down the total effect of a price change into its component parts
(income effect and
substitution effect)

Welfare Analysis

How does a change in price affect people's well-being?

Use the same decomposition to develop a dollar value for how much a price change harms a consumer.

Elasticity

How responsive is a consumer to a change in a price or income?

Own-price elasticity

Cross-price elasticity

Income elasticity

Today's Agenda

  • Decomposition of a price change into income
    and substitution effects
  • Welfare analysis:
    compensating and equivalent variation
  • Elasticity

Verbal Analysis: MRS, MRT, and the “Gravitational Pull" towards Optimality 

Model 1: Fish vs. Coconuts

  • Can spend your time catching fish (good 1)
    or collecting coconuts (good 2)
  • What is your optimal division of labor
    between the two?
  • Intuitively: if you're optimizing, you
    couldn't reallocate your time in a way
    that would make you better off.
  • The last hour devoted to fish must
    bring you the same amount of utility
    as the last hour devoted to coconuts

Marginal Rate of Transformation (MRT)

  • The  number of coconuts you need to give up in order to get another fish
  • Opportunity cost of fish in terms of coconuts

Marginal Rate of Substitution (MRS)

  • The number of coconuts you are willing to give up in order to get another fish
  • Willingness to "pay" for fish in terms of coconuts

Both of these are measured in
coconuts per fish
(units of good 2/units of good 1)

Marginal Rate of Transformation (MRT)

  • The  number of coconuts you need to give up in order to get another fish
  • Opportunity cost of fish in terms of coconuts

Marginal Rate of Substitution (MRS)

  • The number of coconuts you are willing to give up in order to get another fish
  • Willingness to "pay" for fish in terms of coconuts

Opportunity cost of marginal fish produced is less than the number of coconuts
you'd be willing to "pay" for a fish.

Opportunity cost of marginal fish produced is more than the number of coconuts
you'd be willing to "pay" for a fish.

Better to spend less time fishing
and more time making coconuts.

Better to spend more time fishing
and less time collecting coconuts.

MRS
>
MRT
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.

\frac{MU_1}{MU_2}
\frac{MP_{L2}}{MP_{L1}}
=
MU_1
MP_{L1}
=
\times
MU_2
MP_{L2}
\times
MRS
=
MRT

Suppose we only have one input (labor)
in the production of each good,
so \(MRT = MP_{L2}/MP_{L1}\)

\frac{\text{utils}}{\text{units of 1}}
\frac{\text{good 1}}{\text{hour of }L}
\frac{\text{utils}}{\text{units of 2}}
\frac{\text{good 2}}{\text{hour of }L}

Utility from last hour spent producing good 1

Utility from last hour spent producing good 2

Model 2: Labor vs. Coconuts

  • Choose how much of your time (good 1)
    to spend collecting coconuts (good 2)
  • You dislike working but like coconuts.
  • If you're optimizing, the marginal cost 
    (disutility from the last hour worked)
    must exactly offset the marginal benefit
    (utility from the coconuts produced in that hour).

Marginal Product of Labor (\(MP_L\))

  • The  number of coconuts you can produce if you work for another hour

Marginal Rate of Substitution (MRS)

  • The minimum number of coconuts you would be willing to work for another hour to get

Both of these are measured in
coconuts per hour
(units of good 2/units of good 1)

Marginal Product of Labor (\(MP_L\))

  • The  number of coconuts you can produce if you work for another hour

Marginal Rate of Substitution (MRS)

  • The minimum number of coconuts you would be willing to work for another hour to get
\frac{MU_1}{MU_2}
=
MU_1
=
MU_2
MP_{L}
\times
MRS
=
MP_L
\frac{\text{utils}}{\text{hour}}
\frac{\text{utils}}{\text{coconut}}
\frac{\text{coconuts}}{\text{hour}}

Disutility from last hour worked
(opp. cost of leisure)

Utility from coconuts produced in the last hour

MP_L

Graphical Analysis:
PPFs and Indifference Curves

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

Mathematical Analysis:
Lagrange Multipliers 

 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

Let's see where this comes from in the math.

Next Time

Examine cases where the optimal bundle is not characterized by a tangency condition.

New concepts:
corner solutions and kinks.