Econ 604.
Suggested Repsonses
Spring, 2006
Problem Set #3. Chapter 3, Problems 3.2, 3.4, 3.5, 3.7
3.2. Suppose the utility function for two goods, X and Y, has the Cobb-Douglas form
utility =
U(X,Y)
=
(XY)1/2
a. Graph the U=10 indifference curve associated with this utility function
When U=10, we have
100 = XY
Or
Y = 100/X
Thus
- 100/X2
dY/dX =
20
15
10
U=10
5
0
0
5
10
15
20
b. If X = 5, what must Y equal to be on the U=10 indifference curve? What is the MRS
at this point?
X
5
10
15
20
Y
20
10
6.666667
5
U
10
10
10
10
MRS
4
1
0.44
¼
With X=5, MRS = 100/25 = 4. For reference, I also list other values.
Reading down in the above table observe that as we move from the consumption of
relatively few X’s to relatively more, the MRS falls. That is, you must give up
progressively fewer units of Y to gain constant increments of X – a diminishing MRS that
indicates a preference for a mix of X and Y.
c. In general, develop an expression for the MRS for this utility function. Show how this
can be interpreted as the ratio of the marginal utilities for X and Y.
U(X,Y) =
dU
=
(XY)1/2. Taking the total differential
UXdX+UYdY =
.5(Y/X).5dX +.5(X/Y).5dY
=
0
Solving the middle equality
dY/dX =
-UX/UY
Solving the rightmost expression
dY/dX =
-Y/X so the MRS = Y/X (the opposite of dY/dX)
.
d. Consider a logarithmic transformation of this utility function
U’
=
logU
Where log is the logarithmic function to base 10. Show that for this transformation the
U’=1 indifference curve has the same properties as the U=10 curve calculated in parts (a)
and (b). What is the general expression for the MRS of this transformed utility function?
U’
=
log (XY)1/2
=
.5logX + .5logY
Plotting ordered pairs when U’=1 yields
X
5
10
15
20
Y
20
10
6.666667
5
U
100
100
100
100
logX
0.69897
1
1.176091
1.30103
logY
1.30103
1
0.823909
0.69897
U'
1
1
1
1
MRS
4
1
.44
.25
Obviously indifference curves are the same for each utility function.
One can totally differentiate U’ to obtain the same general expression for the MRS as
before:
dU’
Solving
=
dY/dX =
(.5/X)dX
-Y/X
+ (.5/Y)dY
so the MRS = Y/X
=
0
3.4 For each of the following expressions, state the formal assumption that is being made
about the individual’s utility function.
a.
It (margarine) is just as good as the high-price spread (butter).
MRSmb =
b.
1, where m = margarine and b = butter.
Peanut butter and jelly go together like a horse and carriage
Peanut butter and jelly are perfect complements. That is
U(peanut butter, jelly) =
min{peanut butter, jelly}
Where the terms “peanut butter” and “jelly” refer to servings of each product.
c.
Things go better with Coke.
Coca Cola is a complement for all goods. That is, for any good x
Ux, coca cola>0
d.
Popcorn is addictive – the more you eat, the more you want.
Popcorn consumption exhibits increasing marginal utility, e.g.,
Upopcorn >0.
e.
Mosquitoes ruin a nice day at the beach.
Mosquito avoidance and a (mosquito free) day at the beach are perfect
substitutes. Let the incremental utility of the day at the beach be U(beach)>0.
Then the incremental utility of a day at the beach with mosquitoes is U(beach,
mosquitoes) < 0. In other words, the marginal utility of a day at the beach is less
than or equal to the marginal disutility of mosquitoes.
f.
A day without wine is like a day without sunshine. The marginal (more precisely
the incremental) utility of a “wine” just equals the marginal (incremental) utility
of sunshine in a day.
g.
It takes two to tango. “tango” dancing and a partner are perfect complements in
consumption. U(tango, partner)
=
min(tango, partner)
3.5
Graph a typical indifference curve for the following utility functions and
determine whether they have convex indifference curves (that is, whether they
obey the assumption of a diminishing MRS)
a.
U = 3X + Y
Here the MRS = -dY/dX = 3. The MRS is a
constant, and does not exhibit
diminishing MRS.
20
15
10
5
U=35
0
0
5
10
15
20
b. U = (XY).5
X
5
10
15
20
Y
20
10
6.6666667
5
U
10
10
10
10
MRS
4
1
0.444444
0.25
Here MRS is –dY/dX = X/Y. As seen in the rightmost column of the above table, this does
exhibit diminishing MRS
20
15
10
5
U=10
0
0
5
10
15
20
c.
U=
(X2 + Y2).5
Suppose we confine attention to constant increments of X and a utility level of 28.28.
X
20
15
10
5
Y
20
23.976
26.455
27.836
U
28.284271
28.281594
28.28192
28.281494
MRS
1
0.625626
0.378
0.179624
Here the utility function is obviously concave,
implying an increasing MRS. More formally,
dU = X(X2 + Y2)-.5dX+ Y(X2 + Y2)-.5dY =0
implies
30
25
dY/dX =
- X/Y.
20
U=28.82
15
10
0
5
10
15
Values are shown in the rightmost column of
the above table. Notice that the MRS moves
directly with X (Constant increments of X
require giving up increasing increments of Y)
20
Notice: What does this imply about the
preferred consumption bundle? It implies that for a given budget constraint,
utility will be maximized at one corner or the other. For example, an agent may
want to select either a collection of modern black and silver furniture for a room,
or 18th century Jaocbian antiques, depending on the relative price. But the
consumer may be worse of with a combination of the two.
d.
U= (X2 - Y2).5
Plotting some points
X
20
15
12
11
10
Y
17.315
11.175
6.63
4.55
0
U
10.009534
10.005967
10.002155
10.014864
10
MRS
-1.15507
-1.34228
-1.80995
-2.41758
#DIV/0!
Graphically
Here, notice the Y is a “bad.” Thus, the
slope of the MRS is negative. More
formally,
20
dU = X(X2-Y2)-.5dX –Y(X2-Y2)-.5dY = 0
implies
dY/dX = X/Y
15
10
U=10
5
0
10
12
14
16
18
20
Given a budget constraint, with positive prices for both, the consumer would maximize
utility by purchasing only X. On the other hand, if we framed the problem in
terms of Y removal, then U = (X2+Y2).5 as we established in the part c, the MRS
for such a problem–X/Y exhibits an increasing MRS. So even were Y presented as
a “good” the consumer would consume either X or ‘not’ Y.
e. U = X2/3Y1/3
X
20
15
10
5
Y
2.5
4.45
10
40
U
10
10.00417
10
10
MRS
0.25
0.593333
2
16
This is another variant of a Cobb-Douglas
function. The function does exhibit
diminishing MRS. Formally,
20
dU = (2/3)X-1/3Y1/3)dX
+ (1/3) X2/3Y-1/3)dY =0
15
10
implies
U=10
5
dY/dX =
-2Y/X.
Values are shown in the rightmost column
of the above table.
0
0
5
10
15
20
f. U = log X + log Y. We analyzed this function in problem 3.2(d). Looking the table
shown below, it is obvious that the MRS for this function is the same as for 3.5(b).
X
20
15
10
5
Y
5
6.6666667
10
20
Formally,
dU
SolvingdY/dX =
U
10
10
10
10
=
-Y/X
MRS
0.5
0.888889
2
8
dX/X
+
logX
1.30103
1.1760913
1
0.69897
dY/Y
logY
0.69897
0.8239087
1
1.30103
=
0
U'
1
1
1
1
3.7. Consider the following utility functions. Show that each of these has a diminishing
MRS, but that they exhibit constant, increasing and decreasing marginal utility,
respectively. What can you conclude?
a.
U(X,Y) =
XY
MRS: dU
=
Implies that dY/dX
YdX
=
+
XdY =
0
-Y/X. This is diminishing MRS.
Not consider marginal utility.
U1
=
Y, U2 =
X. These are both positive for any positive
combination of Y and X. The second order conditions U11 = 0, U22 = 0
suggest that they have constant marginal utility for each good.
b.
U(X,Y)
=
X2Y2
MRS: dU
=
2XY2dX
+
2YX2dY
=
Implies that dY/dX
=
-Y/X. This is the same as above,
diminishing.
0
Utility. In the above function U11 =2Y2, U22 = 2X2. These are both positive,
so the function exhibits increasing marginal utility. (More generally, with
U11 >0, U22 >0 and U12 = 4XY, we have U11 U22 - U122 = 4X2Y2 2 2
16X Y <0. Thus, this function as a whole is not convex. It does,
however, satisfy quasi convexity, since the indifference curve is convex.
(More mechanically,
U11U22 - 2 U12 U1U2+ U22U12 =
2Y2(4X4Y2) -2(4XY)(2XY2)(2YX2) + 2X2(4Y4X2)
8X4Y4 - 32 X4Y4 <0 )
c.
U(X,Y)
=
lnX + lnY
MRS: dU
=
Implies that dY/dX
MRS.
dX/X
=
+
dY/Y =
0
-Y/X. This is the same as above, diminishing
Utility. In the above function U11 =-dX/X2 0, U22 = -dY/Y2 and U12 = 0.
Thus
U11 <0,. This implies diminishing marginal utility (and indeed, the
function is concave since U11 U22 - U122 = 1/X2Y2 >0.)
Result: Convexity in X1 X2 space does not imply concavity in U(X1 X2 ) space.