Econ 302: Solutions for Practice
Questions 4
Question 1:

Note that u(2,8)=(2+2)(8+1)=36.
Thus, we must solve (X+2)(Y+1)=36 for Y. We get XY+2Y+X+2=36.
Hence Y(X+2)=34X which implies Y=(34X)/(X+2). Alternatively,
XY+2Y+X+2=36. Hence Y(X+2)=36(X+2). Thus, Y=36/(X+2)1. This curve
is indicated below.

As can be seen in the graph, the indifference curve contains
consumption bundles that are in the budget set. In fact, strictly
higher utility level can also be obtained.

The partial derivatives with respect to X and Y are given by Y+1 and
X+2, respectively. Thus, the MRS(X,Y)=(Y+1)/(X+2).

If the MRS is equal to the slope of the budget line then we get
(Y+1)/(X+2)=1. This simplifies to X+2=Y+1, i.e.,

The equation of the budget line is X+Y=11.

Substituting Y=X+1 in X+Y=11 yields 2X+1=11. Thus, X=5 and Y=6.
Question 2:
 As indicated (25,0) and (16,4) give a utility of 20.
The other points are (9,8), (4,12), (1,16), (0,20).
 If the drawing is sufficiently accurate, the solution should be
16. Note that the indifference curves are red and the budget line is
blue in the graph.
 Some points that result in a utility level of 25 are
(0,25), (1,21), (4,17), (9,13), (16,9), (25,5).
 The new optimal choice is (16,9) (the new budget line is gray).
Note that the level of consumption of nuts does not change. The
reason is that the MRS is given by
MRS=2/x_{1}^{1/2}. The price ratio is
1/2. Thus, MRS=1/2 implies
2/x_{1}^{1/2}=1/2 and hence
x_{1}=16, independent of the consumer's income.

The slope of the indifference curve at (9,0) is 2/3 (see the graph below
or use the above MRS).
 The slope of the budget line is 1/2.
 At this point the indifference curve is steeper.
 No, as the graph below indicates, (9,0) is the optimal choice.
Question 3:
Recall that at an optimal choice the
MRS must be equal to the slope of the budget line if the optimal
consumption is interior (no boundary solution) and if preferences have
convex weakly preferred sets and are monotonic.
Since (2,4) is not a boundary solution we can therefore conclude that
the MRS(2,4)=p_{1}/p_{2}=3/2.
In the second case we have a boundary optimum. Hence, we cannot
conclude that the MRS is equal to the slope of the budget line.
Question 4:
 The preferences are convex and monotone. Thus,
If the price ratio is equal to the MRS then we can have an
interior optimum. The MRS is 1/4. Thus, the price ratio must be 1/4.
If you make the graph, then you will see that the budget line at this
price ratio coincides with an indifference curve.
 If the p_{1}/p_{2} is not equal to 1/4 then
there will be boundary optima. In particular, one can check by means
of a graph that
p_{1}/p_{2}<1/4 implies that the consumer
chooses x_{1}=0. Otherwise, if
p_{1}/p_{2}>1/4 we will have
x_{2}=0 for the optimal consumption choice.
Question 5:
Recall that the indifference curves are Lshaped (blue in the graph)
and have a kink at
the 45° axis, as indicated in the graph. The optimal choice is
therefore the intersection point between the 45° axis and the
budget line. The optimal consumption choice is (8.5,8.5).
The indifference curves are again Lshaped (blue in the graph). But
now the kink is at
the points for which 50x_{1}=x_{2}. The intersection
between this line and the budget line is the optimal consumption
choice. It is given by (20,1000).
Question 6:
MRS=p_{1}/p_{2} implies
x_{1}+x_{2}=1.
 The budget line is
2x_{1}+x_{2}=m=1.5. Thus,
(0.5,0.5) is optimal.
 Now the budget line is
2x_{1}+x_{2}=2. Thus,
(1,0) is optimal. Note that the MRS is equal to the slope of the
budget line although we have a boundary optimum. Specifically, there
exist examples such as this one, where at a boundary optimum the MRS
is equal to the slope of the budget line. However, there are also
cases of boundary optima (as in the last part of this question)
where the two are not the same.
 The budget line is
2x_{1}+x_{2}=3. Thus,
(2,1) solves the two equations. This, however, cannot be an optimal
consumption choice. Thus, we must have a boundary optimum at which
the MRS is not equal to p_{1}/p_{2}. The two
candidates are (1.5,0) and (0,3). The utilities are
log(1.5)+1.5 and log(3), respectively. One can easily check that the
first utility is higher. Thus, (1.5,0) is optimal.
 The consumption
of good 2 decreased from 0.5 to 0 when income was increased from 1.5
to 2). Goods with this property will be defined later in this class
as inferior goods.
Question 7:
You will find the main ideas for solving this question in your notes
from the lecture of Wed., February 7.
The black line in the graph is the budget line without any program.
The red line in the graph is the budget line with the existing
program. In order to find the budget line for the alternative
program, note that it must look like the budget line in Figure 2.6 of
the textbook. There must be a part where the line is horizontal that
corresponds to the amount of electricity which is provided for free.
After that, the budget line must have the same slope as the black
budget line. Thus, we must determine the length of the flat part of
the budget line.
Recall that the government should spend the same amount of money in
both cases. Thus, intuitively the budget line for the alternative program
must go through the optimal consumption bundle on the red budget
line (indicated by a black dot). This insight allows us to construct
the green budget line.
Formally, this can shown as follows
(c.f., Section 5.6 of the textbook).
Denote the optimal consumption bundle on
the red line by (x_{1}*,x_{2}*). Then the government
pays S=sx_{1}* as a subsidy. The budget line of the
alternative program must therefore be of the form
p_{1}x_{1}+x_{2}=m+S. In addition
p_{1}x_{1}>=S must hold as the
subsidy S must be used on electricity only (cf., the
discussion on food stamps in class). Now recall that
(x_{1}*,x_{2}*) is on the red line. Thus,
(p_{1}s)x_{1}*+x_{2}*=m. This is
equivalent to
p_{1}x_{1}*+x_{2}*=m+sx_{1}*
which is the same as
p_{1}x_{1}*+x_{2}*=m+S. In addition,
p_{1}x_{1}*>S as one can see in the graph.
However, this means that
(x_{1}*,x_{2}*) is also on the green budget line.
It can be seen in the graph that the a part of the green budget line
lies in the interior of the weakly preferred set
for
(x_{1}*,x_{2}*). Thus, the consumer is better off
under the alternative program. Secondly, it can be seen from the
graph that the consumption of good 1 is lower under the alternative
program. In particular, for all points on the part of
the green budget line which is inside
the weakly preferred set, the consumption of good 1 is lower than
x_{1}*.
