Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 9
Demand curves:
plot demand for one good
in price-quantity space
Offer curves ("consumption loci"):
plot bundles chosen
in good 1-good 2 space
What is the effect of a 1% change
in the price of good 1 \((p_1)\) on the quantity demanded of good 1 \((x_1^*)\)?
no change
perfectly inelastic
less than 1%
inelastic
exactly 1%
unit elastic
more than 1%
elastic
What is the effect of an increase
in the price of good 2 \((p_2)\) on the quantity demanded of good 1 \((x_1^*)\)?
no change
independent
decrease
complements
increase
substitutes
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.
What is the effect of an increase
in income \((m)\) on the quantity demanded of good 1 \((x_1^*)\)?
decrease
good 1 is inferior
increase
good 1 is normal
When your income goes up,
demand for the good increases.
When your income goes up,
demand for the good decreases.
Plot relationship
between \(p_1\) and \(x_1^*\),
holding \(p_2\) and \(m\) constant
(ceteris paribus)
Change in \(p_1\): movement along
the demand curve for good 1
Change in \(p_2\) or \(m\): shift of
the demand curve for good 1
Think about how the behavior
described by the demand function translates into the overall shape of the demand curve:
Choose prices strategically and plot points.
Quantity of Good 1 \((x_1)\)
Price of Good 1 \((p_1)\)
All demand curves must be in this region
Quantity bought at each price if you spent all your money on good 1
Leisure (R)
Consumption (C)
You trade \(L\) hours of labor for some amount of consumption, \(\Delta C\).
You start with 24 hours of leisure and \(M\) dollars.
You end up consuming \(R = 24 - L\) hours of leisure,
and \(C = M + \Delta C\) dollars worth of consumption.
Better to produce
more good 1
and less good 2.
Better to produce
more good 2
and less good 1.
These forces are always true.
In certain circumstances, optimality occurs where MRS = MRT.
Think about maximizing each of these functions subject to the constraint \(0 \le x \le 10\).
Plot the graph on that interval; then find and plot the derivative \(f'(x)\) on that same interval.
Which functions have a maximum at the point where \(f'(x) = 0\)? Why?
Sufficient conditions for an interior optimum characterized by \(f'(x)=0\) with constraint \(x \in [0,10]\)
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
Discontinuities in the MRS
(e.g. Perfect Complements utility function)
Discontinuities in the MRT
(e.g. homework question with two factories)
If preferences are nonmonotonic,
you might be satisfied consuming something within the interior of your feasible set.
FISH
COCONUTS
PPF
If preferences are nonconvex,
the tangency condition might find a minimum rather than a maximum.
FISH
COCONUTS
PPF
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)
Lectures 2 & 3: Derive feasible set (PPF)
from production functions and resource constraints.
Lectures 5 & 6: Solve constrained optimization problem:
find most preferred choice in the feasible set.
Lecture 4: Describe preferences using utility functions.
Lecture 7: Derive feasible set (budget set)
as a function of prices and income.
Lectures 9-12: Analyze the comparative statics of how a consumer responds to changes in prices and income.
Lecture 8: Derive the demand function: the optimal bundle as a function of prices and income.