Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 5
pollev.com/chrismakler
Insight #1:
If preferences are monotonic, the optimal bundle must lie along the budget line.
If preferences are nonmonotonic,
you might be satisfied consuming something within the interior of your feasible set...
Apples
Bananas
BL
BL'
...or it might not!
Better to buy
more good 1
and less good 2.
Better to buy
more good 2
and less good 1.
These forces are always true.
In certain circumstances, optimality occurs where MRS = p1/p2.
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.
More willing to give up good 2
than the market requires.
Less willing to give up good 2
than the market requires.
IF...
THEN...
The consumer's preferences are "well behaved"
The indifference curves do not cross the axes
The budget line is a simple straight line
The optimal consumption bundle will be characterized by two equations:
More generally: the optimal bundle may be found using the Lagrange method
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 \(x \in [0,10]\)
How does the "Gravitational Pull" argument apply in each of these cases?
Is the solution characterized by \(MRS = p_1/p_2\)?
UTILITY FUNCTION
BUDGET CONSTRAINT
What is the MRS if you spend half your money on good 1?
What is the MRS if you spend all your money on good 1?
UTILITY FUNCTION
BUDGET CONSTRAINT
What would you get if you set the MRS equal to the price ratio?
You: Lagrange, I'd like you to find me a maximum please.
Lagrange: Here you go.
You: but that has a negative quantity of good 1! That's impossible!
Lagrange:
Lagrange will only find you
the point along the mathematical description of the constraint where the slope of the constraint is equal to the slope of the level set of the objective function passing through that point.
It doesn't care about your petty insistence on positivity.
UTILITY FUNCTION
BUDGET CONSTRAINT
What would you get if you tried to set the MRS equal to the price ratio?
What is the relationship between the MRS and the price ratio along the budget constraint?
Concave Utility
Budget Line:
12
6
12
6
Plot some indifference curves.
What is the "gravitational pull" showing you?
Concave Utility
1. Plot utility along the budget line
2. Plot MRS vs. price ratio along the same budget line
(12,0)
(6,6)
utility
(12,0)
(6,6)
units of 2
units of 1
(0, 12)
(0, 12)
How does the "Gravitational Pull" argument apply in each of these cases?
Is the solution characterized by \(MRS = p_1/p_2\)?
UTILITY FUNCTION
BUDGET CONSTRAINT
If \(3x_1 < 2x_2\), you would get more utility from additional good 1, and not give up any utility by having less good 2; you should move to the right along your budget constraint.
If \(3x_1 > 2x_2\), you got no utility from the last unit of good 1, but you would get utility from additional good 2; you should move to the left along your budget constraint.
avoids a satiation point within the constraint
At the left corner of the constraint, \(MRS > p_1/p_2\)
avoids a corner solution when \(x_1 = 0\)
Monotonicity (more is better)
At the right corner of the constraint, \(MRS < p_1/p_2\)
avoids a corner solution when \(x_2 = 0\)
MRS and price ratio 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)
Bringing preferences and budget sets together, we find the most preferred bundle in a budget set.
Under certain important conditions
the optimal consumption bundle will be the point along the budget line
where the consumer's MRS is equal to the price ratio.
However, if those conditions are not met, we need to apply logic:
always compare MRS and price ratio!!!
Along a budget line, if the MRS is greater than the price ratio,
the consumer gets more "bang for their buck" from good 1 than good 2;
so they can be made better off by choosing more of good 1 and less of good 2; and vice versa.
Next time: examine what happens when prices and income change