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.
50V | 11 | Analyzing a Price Change
By Chris Makler
