MICROECONOMIC PROBLEMS
SOLUTIONS FOR HOMEWORK #2
Problem 2 (Problem 5 not solved in class)
Utility functions in the above problems represent some ‘model’ preferences. Of course, we
can be dealing with other types of preferences, so the utility functions can take other forms.
Below you can find some examples, that you probably did not see during the lecture. Draw
the utility map, find marginal utilities and MRS. Are marginal utilities positive? Does law of
the diminishing rate of marginal substitution work? Are the axioms or rational choice
satisfied? What type of preferences these functions represent? Can they illustrate our real
preferences? Under what conditions?
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U = 3*X12 + 5*X2
U = 20 - (X1-10)2 - (X2-10)
U = (X1 - X2)2
SOLUTION 1:
U=3X12+5X22
MU1=δU/δX1=6X1
MU2= δU/δX2=10X2
MRS= -MU1/ MU2=-3X1/5X2
Marginal utilities are positive.
Completeness – yes.
Reflexivity – yes.
Transitivity – yes.
Monotonicity – yes (a negative
slope).
Convexity – no (an example of
concave preferences – the shape
of the curve!)
The law of diminishing rate of marginal substitution doesn’t work (the more X1, the steeper
the curve).
That’s an example of concave preferences.
This can illustrate our real preferences – one can like both milk and cherries while consuming
them separately. However, the person doesn’t want to consume them together.
U=20-(X1-10)2-(X2-10)
MU1=δU/δX1=-2X1+20
MU2= δU/δX2=-1
MRS= -MU1/ MU2=-2X1+20
Marginal utilities are not positive.
Completeness – yes.
Reflexivity – no (cannot be derived
from transitivity).
Transitivity – no (for certain X2
there are two values of X1).
Monotonicity – no (changes in the
slope’s sign, sometimes the slope is
positive).
Convexity – no (the shape of the
curve).
The law of diminishing rate of marginal substitution doesn’t work (irregularities, finally the
more X1 we have, the steeper the curve).
That’s an example of concave preferences.
The consumer’s preferences are inconsistent. The consumer is irrational.
On the other hand, the curves remind me of an example of satiation, when there is some
overall best bundle for the consumer and the closer he is to that best bundle, the better off he
is. The satiation point would have an X1 coordinate of the peaks of the curves and X2
coordinate lower than the X2 coordinate of the peak of the lowest curve.
I cannot judge whether it’s an example of consumer’s inconsistency or a satiation case…
JT: This is consumer inconsistency – he should not be equally well off with two different
consumption bundles. Other than that, a perfect solution.
U=(X1-X2)2
MU1=δU/δX1=2X1-2X2
MU2= δU/δX2=2X2-2X1
MRS= -MU1/ MU2=-(X1-X2)/(X2-X1)
Marginal utilities aren’t positive.
Completeness – yes.
Reflexivity – yes.
Transitivity – yes.
Monotonicity – no (positive slope).
Convexity – no (X2 are ‘bads’ which we
don’t want to consume while the
convexity axiom implies that we would
like to consume two goods at the same
time).
The law of diminishing rate of marginal substitution doesn’t work (adding X1 doesn’t change
the slope of the curve).
X2 are bads while X1 are goods. We want to consume as much as possible of X1 (which we
like) and as little as possible of X2 (which we dislike).
Author: Wojtek Deja
SOLUTION 2:
U = 20 - (X1-10)2 - (X2-10) is a very interesting function. It can be drawn as demonstrated
below (x is X1, while y is X2 thereafter).
It would really be difficult to interpret it. It was probably meant to be: U = 20 - (X1-10)2 (X2-10)2 i.e. a straight quadratic function like this.
U = (X1 - X2)2 can be represented by:
C’est a dire, X2 is not totally an economic ‘bad’. It is better visible in the third dimension:
X2 might be thought of as an economic ‘bad’ in a slightly different setting, e.g.:
U = X1 – (X2)2 .
It would have the following graphical representation.
Author: mgr Anna Kukla-Gryz