1. We begin with Charlie of the apples and bananas. Recall that Charlie's utility function is
U
(
x
A
,x
B
) =
x
A
x
B
. Suppose that the price of apples is 1, the price of bananas is 2, and Charlie's income is 40.
(a) Use blue ink to draw Charlie's budget line. (Use a ruler and try to make this line accurate.)
Plota few points on the indi erence curve that gives Charlie a utility of 150 and sketch this curve
with red ink. Now plot a few points on the indi erence curve that gives Charlie a utility of 300
and sketch this curve with black ink or pencil.
(b) Can Charlie a ord any bundles that give him a utility of 150?
(c) Can Charlie a ord any bundles that give him a utility of 300?
(d) On your graph, mark a point that Charlie can a ord and that gives him a higher utility than
150. Label that point A.
(e) Neither of the indi erence curves that you drew is tangent to Charlie's budget line. Let's try to
nd one that is. At any point,
(
x
A
,x
B
)
, Charlie's marginal rate of substitution is a function of
x
A
and
x
B
. In fact, if you calculate the ratio of marginal utilities for Charlie's utility function, you
will nd that Charlie's marginal rate of substitution is
MRS
(
x
A
,x
B
) =
ANSWER
. This is the
slope of his indi erence curve at
(
x
A
,x
B
)
. The slope of Charlie's budget line is
NUMERICAL
ANSWER
(f) Write an equation that implies that the budget line is tangent to an indi erence curve at
(
x
A
,x
B
)
:
ANSWER
. There are many solutions to this equation. Each of these solutions corresponds to
a point on a di erent indi erence curve. Use pencil to draw a line that passes through all of
these points.
(g) The best bundle that Charlie can a ord must lie somewhere on the line you just penciled in. It
must also lie on his budget line. If the point is outside of his budget line, he can't a ord it. If the
point lies inside of his budget line, he can a ord to do better by buying more of both goods. On
your graph, label this best a ordable bundle with an
E
. This happens where
x
A
=
ANSWER
and
x
B
=
ANSWER
. Verify your answer by solving the two simultaneous equations given by
his budget equation and the tangency condition.
(h) What is Charlie's utility if he consumes the bundle obained in (g).
(i) On the graph above, use red ink to draw his indi erence curve through the bundle obained in
(g). Does this indi erence curve cross Charlie's budget line, just touch it, or never touch it?
2. Ambrose, the nut and berry consumer, has a utility function
U
(
x
1
,x
2
) = 4
√
x
1
+
x
2
, where
x
1
is his
consumption of nuts and
x
2
is his consumption of berries.
(a) The commodity bundle (25, 0) gives Ambrose a utility of 20. Other points that give him the same
utility are (16, 4), (9,
ANSWER
), (4,
ANSWER
), (1,
ANSWER
), and (0,
ANSWER
).
Plot these points and draw a red indi erence curve through them.