ECO 207h Honors Intermediate Micro
Homework 1
Due: Friday 2 February 2007
2.1
Suppose
U
(
x, y
)=4
x
2
+3
y
2
.
1. Calculate
∂U/∂x
,
∂U/∂y
.
2. Evaluate these partial derivatives at
x
=1
,
y
=2
.
3. Write the total di
f
erential for
U
.
4. Calculate
dy/dx
for
dU
=0
— that is, what is the implied tradeo
f
between
x
and
y
holding
U
constant?
5.
6. Show
U
=16
when
x
=1
,
y
=2
.
7. In what ratio must
x
and
y
change to hold
U
constant at
16
for move
ments away from
x
=1
,
y
=2
?
8
.Moregenera
l
ly
,whatistheshapeo
fthe
U
=16
contour line for this
function? What is the slope of that line?
2.3
Suppose that
f
(
x, y
)=
xy
.F
indthemax
imumva
luefo
r
f
if
x
and
y
are constrained to sum to
1
. Solve this problem in two ways: by substitution
and by using the Lagrangian multiplier method.
2.5
The height of a ball that is thrown straight up with a certain force is a
function of the time
(
t
)
from which it is released given by
f
(
t
)=
−
o.
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View Full Document1. Howdoesthevalueof
t
at which the height of the ball is at a maximum,
depend on the parameter
g
?
2. Use your answer to part 1 to describe how maximum height changes as
the parameter
g
changes.
3. Use the envelope theorem to answer part 2 directly.
4. On the Earth
g
=32
, but this value varies somewhat around the globe.
If two locations had gravitational constants that di
f
ered by
0
.
1
,what
would be the di
f
erence in the maximum height of a ball tossed in the
two places?
2.9
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 Spring '07
 Pavan
 Utility, CobbDouglas function

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