Ecn 100  Intermediate Microeconomic Theory
University of California  Davis
November 3, 2010
John Parman
Problem Set 4  Solutions
This problem set will be not be graded and does not need to be turned in. It will help
you prepare for the Chapter 18 material that will be on the second midterm. It is highly
recommended that you also use the old midterms on Smartsite to get additional practice
with the new material.
1.
Production Functions
For the ﬁrst two production functions given below, derive expressions for the marginal
product of
x
, the marginal product of
y
and the technical rate of substitution, graph
three isoquants and determine whether the technology exhibits increasing, constant or
decreasing returns to scale. For the last production function, you only need to graph
the isoquants and determine whether the technology exhibits increasing, constant or
decreasing returns to scale.
(a)
f
(
x,y
) = 5
x
3
4
y
3
4
(b)
f
(
x,y
) = 2
x
+ 3
y
1
2
(c)
f
(
x,y
) = min (
x,
3
y
)
For production function (a):
MP
x
=
df
(
x,y
)
dx
= 5
3
4
x

1
4
y
3
4
MP
y
=
df
(
x,y
)
dy
= 5
3
4
x
3
4
y

1
4
TRS
=

MP
x
MP
y
=
5
3
4
x

1
4
y
3
4
5
3
4
x
3
4
y

1
4
=

y
x
f
(
λx,λy
) = 5 (
λx
)
3
4
(
λy
)
3
4
=
λ
3
4
λ
3
4
5
x
3
4
y
3
4
f
(
λx,λy
) =
λ
6
4
f
(
x,y
)
From the above algebra, we can see that increasing inputs by a factor of
λ
will increase output by a factor greater than
λ
, so this production function
exhibits increasing returns to scale. As for the isoquants, they will be
standard convex isoquants. We know this because the marginal products
of both inputs are positive, guaranteeing downward sloping isoquants,
and the technical rate of substitution gets smaller in magnitude as we
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Problem Set 4  Solutions
move down and to the right, telling us that the isoquants get ﬂatter as
we move to the right.
For production function (b):
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 Winter '08
 PARMAN
 Microeconomics, Economics of production, Monotonic function, Convex function

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