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Ch06-1 - summary and sample questions
Microeconomics 1 (The University of Western Ontario)
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Chapter 6
Inputs and Production Functions
Solutions to Review Questions
1. We said that the production function tells us the maximum output that a firm can
produce with its quantities of inputs. Why do we include the word maximum in this
definition?
The production function tells us the maximum volume of output that may be produced
given a combination of inputs. It is possible that the firm might produce less than this
amount of output due to inefficient management of resources. While it is possible to
produce many levels of output with the same level of inputs, some of which are less
technically efficient than others, the production function gives us the upper bound on (the
maximum of) the level of output.
2. Suppose a total product function has the “traditional shape” shown in Figure 6.2. Sketch
the shape of the corresponding labor requirements function (with quantity of output on the
horizontal axis and quantity of labor on the vertical axis).
The labor requirements function, which is the inverse of the production function, tells us
the minimum amount of labor that is required to produce a given amount of output.
30
25
L
20
15
10
5
0
0
50
100
150
200
250
Q
3. What is the difference between average product and marginal product? Can you sketch
a total product function such that the average and marginal product functions coincide
with each other?
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The average product of labor is the average amount of output per unit of labor.
APL 
Total Product
Q

Quantity of Labor L
The marginal product of labor is the rate at which total output changes as the firm
changes its quantity of labor.
MPL 
Change in Total Product
Q

Change in Quantity of Labor L
The total product function in the graph below Q 3L , which is linear, would have the
average and marginal products coincide. In particular, for all values of Q we would have
APL MPL 3 .
250
Q
200
150
100
50
0
0
10
20
30
40
50
60
70
L
4. What is the difference between diminishing total returns to an input and diminishing
marginal returns to an input? Can a total product function exhibit diminishing marginal
returns but not diminishing total returns?
With diminishing total returns to an input, increasing the level of the input will decrease
the level of total output holding the other inputs fixed. Diminishing marginal returns to
an input means that as the use of that input increases holding the quantities of the other
inputs fixed, the marginal product of that input will become less and less. Essentially,
diminishing total returns implies that output is decreasing while with diminishing
marginal returns we could have output increasing, but at a decreasing rate as the amount
of the input increases.
It is entirely plausible to have a total product function exhibit diminishing marginal
returns but not diminishing total returns. This would occur when each additional unit of
an input increased the total level of output, but increased the level of output less than the
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previous unit of the input did. Essentially, this occurs when output is increasing at a
decreasing rate as the level of the input increases.
5. Why must an isoquant be downward sloping when both labor and capital have positive
marginal products?
If the marginal product of labor is positive, then when we increase the level of labor
holding everything else constant this will increase total output. To keep the level of
output at the original level, we need to stay on the same isoquant. To do so, since the
marginal product of capital is positive we would then need to reduce the amount of
capital being used. So, to keep output constant, when the level of one input increases the
level of the other input must decrease. This negative relationship between the inputs
implies the isoquant will have a negative slope, i.e., be downward sloping.
6. Could the isoquants corresponding to two different levels of output ever cross?
No, as with indifference curves, isoquants can never cross. For example, suppose we
draw isoquants for two levels of output Q1 and Q2 with Q2  Q1 . In addition, suppose
that these isoquants crossed at some point A as in the following diagram.
Capital
C
B
A
Q2
Q1
Labor
Because A and B are on Q2, both achieve the same level of output. Since A and C are on
Q1, both achieve the same level of output. This would imply that B and C achieve the
same level of output. However, this is not possible since point C contains more of both
inputs which would achieve a higher level of output. Therefore, isoquants cannot cross.
7. Why would a firm that seeks to minimize its expenditures on inputs not want to operate
on the uneconomic portion of an isoquant?
By operating on the uneconomic portion of an isoquant, the firm would be using a
combination of inputs in which one of the inputs has a negative marginal product, i.e.,
increasing the input decreases the level of total output. At a point such as this, the firm
could increase output by decreasing the level of the input. By decreasing the level of the
input, the firm could decrease total cost. Thus, if a firm were operating on the
uneconomic region of an isoquant it could simultaneously increase output and decrease
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total cost. Thus, a cost-minimizing firm would never operate on this portion of an
isoquant because it would always take advantage of this opportunity.
8. What is the elasticity of substitution? What does it tell us?
The elasticity of substitution measures how the marginal rate of technical substitution of
labor for capital changes as we move along an isoquant. Essentially this value tells us the
level of substitutability between capital and labor, i.e., how easily the firm can substitute
capital for labor to maintain the same level of total output.
9. Suppose the production of electricity requires just two inputs, capital and labor, and
that the production function is Cobb–Douglas. Now consider the isoquants corresponding
to three different levels of output: Q = 100,000 kilowatt-hours, Q = 200,000 kilowatt-hours,
and Q = 400,000 kilowatt-hours. Sketch these isoquants under three different assumptions
about returns to scale: constant returns to scale, increasing returns to scale, and decreasing
returns to scale.
Decreasing Returns to Scale
Capital
1,000
750
Q=400,000
500
250
Q=200,000
Q=100,000
-
250
500
750
1,000
Labor
With decreasing returns to scale the firm needs to more than double inputs to double
output. Equivalently, doubling inputs less than doubles output.
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Constant Returns to Scale
Capital
1,000
750
500
Q=400,000
250
Q=200,000
Q=100,000
-
250
500
750
1,000
Labor
With constant returns to scale the firms needs to double inputs to double output.
Increasing Returns to Scale
Q=100,000
Q=200,000
Q=400,000
Capital
1,000
750
500
250
-
250
500
750
1,000
Labor
With increasing returns to scale the firm needs to less than double the inputs to double the
output. Equivalently, doubling inputs more than doubles output.
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Solutions to Problems
6.1 A firm uses the inputs of fertilizer, labor, and hothouses to produce roses. Suppose that
when the quantity of labor and hothouses is fixed, the relationship between the quantity of
fertilizer and the number of roses produced is given by the following table:
a) What is the average product of fertilizer when 4 tons are used?
b) What is the marginal product of the sixth ton of fertilizer?
c) Does this total product function exhibit diminishing marginal returns? If so, over what
quantities of fertilizer do they occur?
d) Does this total product function exhibit diminishing total returns? If so, over what
quantities of fertilizer do they occur?
Q 2200

550 .
F
4
a)
APF 
b)
MPF 
c)
Diminishing marginal returns set in when MPF for some unit is lower than MPF
for the previous unit. This occurs for F  3 .
d)
Diminishing total returns set in at the point where total output begins to fall. This
occurs for F  6 .
Q 2600  2500

100 .
F
6 5
6.2. A firm is required to produce 100 units of output using quantities of labor and capital
(L, K) = (7, 6). For each of the following production functions, state whether it is possible to
produce the required output with the given input combination. If it is possible, state
whether the input combination is technically efficient or inefficient.
a) Q = 7L + 8K
b) Q = 20√KL
c) Q = min(16L, 20K)
d) Q = 2(KL + L + 1)
a)
The input combination gives Q = 97 so it is infeasible.
b)
Q = 129.6 which is greater than 100, so feasible, but inefficient.
c)
Q = 112 so again feasible but inefficient.
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d)
Q = 100 therefore the required output is feasible and the input combination is
efficient.
6.3. For the production function Q = 6L2 − L3, fill in the following table and state how much
the firm should produce so that:
a) average product is maximized
b) marginal product is maximized
c) total product is maximized
d) average product is zero
The completed table is shown below:
L
Q
0
0
1
5
2
16
3
27
4
32
5
25
6
0
a)
You can calculate the average product at each point by just dividing total output
by L. The values obtained are 0,5,8,9,8,5,0. Therefore Average Product is
maximized when L = 3.
b)
The marginal product at values 1 through 6 are respectively: 5,11,11,5,–7, –25.
Therefore both the second and the third unit of L give the greatest marginal
increase in output [if you use calculus techniques it can be seen that marginal
product is maximized when L = 2].
c)
From the Table it is clear that total product is maximized when L = 4.
d)
Average Product will be zero only when Total Product is zero. This happens when
L = 6.
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6.4. Suppose that the production function for DVDs is given by Q = KL2 − L3, where Q is the
number of disks produced per year, K is machine-hours of capital, and L is man-hours of
labor.
a) Suppose K = 600. Find the total product function and graph it over the range L = 0 to L =
500. Then sketch the graphs of the average and marginal product functions. At what level
of labor L does the average product curve appear to reach its maximum? At what level does
the marginal product curve appear to reach its maximum?
b) Replicate the analysis in (a) for the case in which K = 1200.
c) When either K = 600 or K = 1200, does the total product function have a region of
increasing marginal returns?
a)
Total Product
Total Product with K=600
40,000,000
30,000,000
20,000,000
10,000,000
0
100
200
300
400
500
600
700
Labor
AP and MP with K=600
200000
AP/MP
AP
0
0
100
200
300
400
500
600
700
-200000
-400000
MP
Labor
Based on the figure above it appears that the average product reaches its
maximum at L = 300. The marginal product curve appears to reach its maximum
at L = 200.
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b)
Total Product
Total Product with K=1200
300,000,000
200,000,000
100,000,000
0
500
1000
1500
Labor
AP/MP
AP and MP with K=1200
1000000
500000
0
-500000 0
-1000000
-1500000
-2000000
AP
200
400
600
800
1000
1200
1400
MP
Labor
Based on the figure above it appears that the average product curve reaches its
maximum at L = 600. The marginal product curve appears to reach its maximum
at L = 400.
c)
In both instances, for low values of L the total product curve increases at an
increasing rate. For K = 600, this happens for L < 200. For K = 1200, it happens
for L < 400.
6.5. Are the following statements correct or incorrect?
a) If average product is increasing, marginal product must be less than average product.
b) If marginal product is negative, average product must be negative.
c) If average product is positive, total product must be rising.
d) If total product is increasing, marginal product must also be increasing.
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a)
Incorrect. When MP  AP we know that AP is increasing. When MP  AP we
know that AP is decreasing.
b)
Incorrect. If MP is negative, MP < 0. But AP = Q / L can never be negative
since total product Q and the level of input L can never be negative. Thus, MP <
0 < AP, which only implies that AP is falling.
c)
Incorrect. Average product is always positive, so this tells us nothing about the
change in total product. Whether or not total product is rising depends on whether
or not marginal product is positive.
d)
Incorrect. If total product is increasing we know that MP  0 . If diminishing
marginal returns have set in, however, marginal product will be positive but
decreasing, but that does not preclude MP > 0.
6.6. Economists sometimes “prove” the law of diminishing marginal returns with the
following exercise: Suppose that production of steel requires two inputs, labor and capital,
and suppose that the production function is characterized by constant returns to scale.
Then, if there were increasing marginal returns to labor, you or I could produce all the steel
in the world in a backyard blast furnace. Using numerical arguments based on the
production function shown in the following table, show that this (logically absurd)
conclusion is correct. The fact that it is correct shows that marginal returns to labor cannot
be everywhere increasing when the production function exhibits constant returns to scale.
To develop the answer, suppose that we were initially producing 64 units of steel.
According to the table, we could do this with 8 units of labor and 100 units of capital.
Now, since we have constant returns to scale, if we double the amount of labor and
capital, i.e., L = 16 and K = 200, we can double output, i.e., produce Q = 128 units of
steel.
But notice from the table that the input combination L = 16 and K = 100 results in an
even greater output of Q = 256 units of steel. Thus, by reducing the amount of capital it
uses (from K = 200 to K = 100), holding the quantity of labor fixed, the firm can produce
more output! That is, the marginal product of capital is negative over this range.
We can see the same thing if we start with any other input combination. For example,
suppose the firm is initially producing 4 units of steel using 2 units of labor and 100 units
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of capital. Because of constant returns to scale, if we double the amount of labor and
capital, i.e., L = 4 and K = 200, we can double output, i.e., produce Q = 8 units of steel.
But notice from the table that the input combination L = 4 and K = 100 results in an even
greater output of Q = 16 units of steel. Again, by reducing the amount of capital it uses
(holding the quantity of labor fixed), the firm can produce more output. Again, we see
that the marginal product of capital is negative.
The above calculations illustrate that a two-input production function with (a) constant
returns to scale and (b) increasing marginal returns to labor must necessarily imply that
the marginal product of capital is negative. And, of course, if the marginal product of
capital is negative, the firm can expand output by reducing the amount of capital it uses.
It could, theoretically, produce an enormous amount of steel in a backyard blast furnace.
Because this conclusion is absurd, the point of the illustration is that with constant returns
to scale, marginal returns to labor cannot be everywhere increasing. Eventually the law
of diminishing marginal returns must set in.
6.7. The following table shows selected input quantities, total products, average products,
and marginal products. Fill in as much of the table as you can:
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The correct answers are shown in bold face red type.
Labor, L
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Total product, Q
0
19
72
153
256
375
504
637
768
891
1000
1089
1152
1183
1176
1125
APL
0
19
36
51
64
75
84
91
96
99
100
98
96
91
84
75
MPL
---19
53
81
103
119
129
133
131
123
109
89
63
31
-7
-51
6.8. Widgets are produced using two inputs, labor, L, and capital, K. The following table
provides information on how many widgets can be produced from those inputs:
a) Use data from the table to plot sets of input pairs that produce the same number of
widgets. Then, carefully, sketch several of the isoquants associated with this production
function.
b) Find marginal products of K and L for each pair of inputs in the table.
c) Does the production function in the table exhibit decreasing, constant, or increasing
returns to scale?
a)
Three sample isoquants: red for production of 4 widgets (Q = 4), green for
production of 8 widgets (Q = 8), and blue for production of 12 widgets (Q = 12).
The dots represent particular combinations of inputs.
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K
4
3
(4,2)
2
(1,1)
1
Q=8
Q=8
Q=4
0
1
2
3
4
L
b)
Recall that marginal product of an input, say labor, is given by Q/L. If we
compute the marginal product of labor and capital at any point in the table, we
find that it always equals 2. For example, in moving from input combination (2,2)
to (3,2), we increase output from 8 to 10. Hence, MPL = (10 – 8)/(3 – 2) = 2.
c)
From the table, we see that as we increase the quantity of each input by a given
proportion, the quantity produced increases by the same proportion. Hence, in
moving from input combination (1,1) to (3,3), we are tripling the quantity of labor
and capital used. As a result, the quantity of output produced triples as well.
where Q is the quantity of
6.9 Suppose the production function for automobiles is
automobiles produced per year, L is the quantity of labor (man-hours) and K is the
quantity of capital (machine-hours).
a. Sketch the isoquant corresponding to a quantity of Q = 100?
b. What is the general equation for the isoquant corresponding to any level of output Q?
c. Does the isoquant exhibit diminishing marginal rate of technical substitution?
a. The Q = 100 isoquant looks like this:
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b. We find the general equation of an isoquant for this production function by starting with
the production function and solving for K in terms of L. Thus, since Q = LK, the general
equation of an isoquant for this production function is
The isoquants for this production function exhibit diminishing marginal rate of technical
substitution. We can see this from the graph above which shows that the isoquant is convex to
the origin.
Capital
6.10. Suppose the production function is given by the equation Q = L√K. Graph the
isoquants corresponding to Q = 10, Q = 20, and Q = 50. Do these isoquants exhibit
diminishing marginal rate of technical substitution?
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
Q=50
Q=10 Q=20
0
20
40
60
80
Labor
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Because these isoquants are convex to the origin they do exhibit diminishing marginal
rate of technical substitution.
6.11. Consider again the production function for DVDs: Q = KL2 − L3.
a) Sketch a graph of the isoquants for this production function.
b) Does this production function have an uneconomic region? Why or why not?
a)
50.0
Q=150
Capital
40.0
30.0
Q=100
20.0
Q=50
10.0
0.0
0
10
20
30
40
Labor
b)
Because each of these isoquants has an upward-sloping portion beyond some
level of labor, each one does indeed have an uneconomic region.
6.12. Suppose the production function is given by the following equation (where a and b
are positive constants): Q = aL + bK. What is the marginal rate of technical substitution of
labor for capital (MRTSL,K) at any point along an isoquant?
For this production function MPL a and MPK b . The MRTS L , K is therefore
MRTS L , K 
MPL a

MPK b
6.13. You might think that when a production function has a diminishing marginal rate of
technical substitution of labor for capital, it cannot have increasing marginal products of
capital and labor. Show that this is not true, using the production function Q = K2L2, with
the corresponding marginal products MPK = 2KL2 and MPL = 2K2L.
First, note that MRTSL,K = L/K, which diminishes as L increases and K falls as we move
along an isoquant. So MRTSL,K is diminishing. However, the marginal product of capital
MPK is increasing (not diminishing) as K increases (remember, the amount of labor is
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held fixed when we measure MPK.) Similarly, the marginal product of labor is increasing
as L grows (again, because the amount of capital is held fixed when we measure MPL).
This exercise demonstrates that it is possible to have a diminishing marginal rate of
technical substitution even though both of the marginal products are increasing.
6.14 Consider the following production functions and their associated marginal products.
For each production function, determine the marginal rate of technical substitution of
labor for capital, and indicate whether the isoquants for the production function exhibit
diminishing marginal rate of substitution.
Production
function
MPL
MPK
MRTSL,K
Diminishing
marginal
product of
labor?
Diminishing
marginal
product of
capital?
Diminishing
marginal
rate of
technical
substitution
NO
YES
NO
YES
NO
YES
YES
YES
YES
NO
NO
YES
NO
NO
NO
6.15. Suppose that a firm’s production function is given by Q = KL + K, with MPK = L + 1
and MPL = K . At point A, the firm uses K = 3 units of capital and L = 5 units of labor. At
point B, along the same isoquant, the firm would only use 1 unit of capital.
a) Calculate how much labor is required at point B.
b) Calculate the elasticity of substitution between A and B. Does this production function
exhibit a higher or lower elasticity of substitution than a Cobb–Douglas function over this
range of inputs?
a)
At point A, the firm produces 18 units of output. Therefore, since B is on the same
isoquant, it must be that L = 17 at B.
b)
The capital-to-labor ratio at A is 3/5 and MRTSL,K = ½. At B, the capital-to-labor
ratio is 1/17, and MRTSL,K = 1/18.
Therefore the elasticity of substitution is
(1 / 17  3 / 5) /(3 / 5) 69
 .
(1 / 18  1 / 2) /(1 / 2) 68
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A Cobb-Douglas production function has an elasticity of substitution of 1.
Therefore this production function has a slightly higher elasticity of substitution,
indicating a slightly greater ease of substitutability of inputs.
6.16. Two points, A and B, are on an isoquant drawn with labor on the horizontal axis and
capital on the vertical axis. The capital–labor ratio at B is twice that at A, and the elasticity
of substitution as we move from A to B is 2. What is the ratio of the MRTSL,K at A versus
that at B?
Since the capital-labor ratio at B is twice that at A, it implies that the percent change in
this ratio as we move from A to B is 100%. If we denote the percent change in the MRTS
over these two points as x then using the definition of elasticity of substitution,
100
2, which means that %MRTS L , K 50.
%MRTS L , K
Equivalently,
MRTS B  MRTS A
MRTS A 2
100 50. Solving,
 .
MRTS A
MRTS B 3
6.17. Let B be the number of bicycles produced from F bicycle frames and T tires. Every
bicycle needs exactly two tires and one frame.
a) Draw the isoquants for bicycle production.
b) Write a mathematical expression for the production function for bicycles.
a)
This isoquants for this situation will be L-shaped as in the following diagram
Tires
6
Q=3
4
Q=2
2
Q=1
1
2
3
Frames
These L-shaped isoquants imply that once you have the correct combination of
inputs, say 2 frames and 4 tires, additional units of one resource without more
units of the other resource will not result in any additional output.
b)
Mathematically this production function can be written
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1
Q min( F , T )
2
where F and T represent the number of frames and tires.
6.18. To produce cake, you need eggs E and premixed ingredients I. Every cake needs
exactly one egg and one package of ingredients. When you add two eggs to one package of
ingredients, you produce only one cake. Similarly, when you have only one egg, you can’t
produce two cakes even though you have two packages of ingredients.
a) Draw several isoquants of the cake production function.
b)Write a mathematical expression for this production function. What can you say about
returns to scale of this function?
a)
mix
1
b)
2
3
eggs
The formula is # of cakes = Min{# of eggs, # of ingredients’ packages}. This
production function has constant returns to scale. To see why, let x and y denoted
the quantities of eggs and mix, respectively, and let Q denote the number of cakes
produced. The equation of our production function is: Q = Min(x,y). If we
increase each input by a factor of a, we have the following quantity of cake:
min{ax, ay} = a min{x, y} = aQ. Hence, increasing the quantities of inputs by a
given proportion results in the same proportionate increase in output, and the
production function thus exhibits constant returns to scale.
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6.19. What can you say about the returns to scale of the linear production function Q = aK
+ bL, where a and b are positive constants?
If we were to scale up all inputs by a factor  (that is, replace K by K, and L by L), the
resulting output would equal Q. Therefore a linear production function has constant
returns to scale.
6.20. What can you say about the returns to scale of the Leontief production function Q =
min(aK, bL), where a and b are positive constants?
A general fixed proportions production function is of the form Q min(aK , bL) . If we
were to scale up all inputs by a factor  (that is, replace K by K, and L by L), the
resulting output would be min(aK , bL)  min(aK , bL) = Q. Therefore the
production function has constant returns to scale.
6.21. A firm produces a quantity Q of breakfast cereal using labor L and material M with
the production function Q = 50√ML + M+ L. The marginal product functions for this
production function are
a) Are the returns to scale increasing, constant, or decreasing for this production function?
b) Is the marginal product of labor ever diminishing for this production function? If so,
when? Is it ever negative, and if so, when?
a)
To determine the nature of returns to scale, increase all inputs by some factor 
and determine if output goes up by a factor more than, less than, or the same as 
.
Q 50  M  L   M   L
Q 50 ML   M   L
Q   50 ML  M  L 
Q Q
By increasing the inputs by a factor of  output goes up by a factor of  . Since
output goes up by the same factor as the inputs, this production function exhibits
constant returns to scale.
b)
The marginal product of labor is
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M
1
L
Suppose M  0 . Holding M fixed, increasing L will have the effect of
decreasing MPL . The marginal product of labor is decreasing for all levels of L .
The MPL , however, will never be negative since both components of the equation
above will always be greater than or equal to zero. In fact, for this production
function, MPL 1 .
MPL 25
6.22 Consider a production function whose equation is given by the formula
which has corresponding marginal products,
and
Show that the
elasticity of substitution for this production function is exactly equal to 1, no matter what
the values of K and L are.
ANSWER
First note that
In this case that implies,
which simplifies to
Now recall that the definition of the elasticity of substitution is
Since
it follows that
will be exactly equal to
(You can
verify this
by plugging in particular values for
, an increase of 66.66 percent. Since
. Suppose, for example,
, and then changes to
it follows that MRTSL,K changes from 1 to
1.5, (also an increase of 66.66 percent.) In other words, since the marginal rate of substitution of
labor for capital is proportional to the capital-labor ratio, the percentage change in the marginal
rate of substitution of labor for capital must equal the percentage change in the capital-labor
ratio. Since
then using the definition of the elasticity of substitution, it
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follows that,
6.23. A firm’s production function is Q = 5L2/3K1/3 with MPK = (5/3)L2/3K−2/3 and MPL =
(10/3)L−1/3K1/3.
a) Does this production function exhibit constant, increasing, or decreasing returns to
scale?
b) What is the marginal rate of technical substitution of L for K for this production
function?
c) What is the elasticity of substitution for this production function?
a)
Notice that (aK)1/3(aL)2/3 = a1/3a2/3 K1/3 L2/3 = a K1/3 L2/3 = aQ. This production
function exhibits constant returns to scale.
b)
MRTSL,K = MPL / MPK = 2 K/L .
c)
Because this is a Cobb-Douglas production function, its elasticity of substitution
equals 1.
6.24. Consider a CES production function given by Q = (K0.5 + L0.5)2.
a) What is the elasticity of substitution for this production function?
b) Does this production function exhibit increasing, decreasing, or constant returns to
scale?
c) Suppose that the production function took the form Q = (100 + K0.5 + L0.5)2. Does this
production function exhibit increasing, decreasing, or constant returns to scale?
a)
For a CES production function of the form

 1  1
  1

Q  aL  bK  


the elasticity of substitution is  . In this example we have a CES production
function of the form
2
Q  K 0.5  L0.5  .
To determine the elasticity of substitution, either set (  1) /  0.5 or
 /(  1) 2 and solve for  .
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1
0.5

  1 0.5
0.5 1
 2.
In either case, the elasticity of substitution is 2.
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b)
Q  ( K ) 0.5  ( L)0.5 
2
Q  ( 0.5 )( K 0.5  L0.5 ) 
2
2
Q   K 0.5  L0.5 
Q Q.
Since output goes up by the same factor as the inputs, this production function
exhibits constant returns to scale.
c)
Q  100  ( K )0.5  ( L)0.5 
Q  100   0.5 ( K 0.5  L0.5 ) 
2
2
2
 100

Q   0.5  K 0.5  L0.5    Q.


When the inputs are increased by a factor of  , where   1 output goes up by a
factor less than  implying decreasing returns to scale.
Intuitively, in this production function, while you can increase the K and L
inputs, you cannot increase the constant portion. So output cannot go up by as
much as the inputs.
6.25 Consider the following production functions and their associated marginal products.
For each production function, indicate whether (a) the marginal product of each input is
diminishing, constant, or increasing in the quantity of that input; (b) the production
function exhibits decreasing, constant, or increasing returns to scale.
Production
function
MPL
MPK
Marginal
product of
labor?
CONSTANT in
L
DIMINISHING
in L
Marginal
product of
capital?
CONSTANT in
K
DIMINISHING
in K
Returns to
scale?
CONSTANT
CONSTANT
DIMINISHING DIMINISHING DECREASING
in L
in K
INCREASING
in L
CONSTANT in
L
INCREASING
in K
CONSTANT in
K
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INCREASING
INCREASING
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6.26. The following table presents information on how many cookies can be produced from
eggs and a mixture of other ingredients (measured in ounces):
Recently, you found a new way to mix ingredients with eggs. The same amount of
ingredients and eggs produces different numbers of cookies, as shown in the following
table:
a) Verify that the change to the new production function represents technological progress.
b) For each production fuction find the marginal products of eggs when mixed ingredients
is held fixed at 8. Verify that when mixed ingredients is held fixed at 8, the technological
progress increases the marginal product of eggs.
a)
For each pair of inputs, except those where there are no eggs or no other
ingredients, new recipe produces more cookies. Hence, the new recipe represents
technological progress.
b)
MPE =Q/E, where E denotes the quantity of eggs. With mixed ingredients held
fixed at 8, we have:
MPE = (8 – 0)/(1 – 0) = 8, when E goes from 0 to 1.
MPE = (16 – 8)/(2 – 1) = 8, when E goes from 1 to 2.
MPE = (16 – 16)/(3 – 2) = 0, when E goes from 2 to 3.
MPE is zero for all subsequent changes in E.
After the technological progress we have:
MPE = (10 – 0)/(1 – 0) = 10, when E goes from 0 to 1.
MPE = (19 – 10)/(2 – 1) = 9, when E goes from 1 to 2.
MPE = (22 – 19)/(3 – 2) = 3, when E goes from 2 to 3.
MPE = (23 – 22)/(4 – 3) = 1, when E goes from 3 to 4.
Comparing the marginal products, we see that MPE (when mix equals 8 is
higher after the technological progress.
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6.27. Suppose a firm’s production function initially took the form Q = 500(L + 3K).
However, as a result of a manufacturing innovation, its production function is now Q =
1000(0.5L + 10K).
a) Show that the innovation has resulted in technological progress in the sense defined in
the text.
b) Is the technological progress neutral, labor saving, or capital saving?
a)
It is possible to write the two production functions as
Q1 500 L  1,500 K
Q2 500 L  10, 000 K
Since Q2  Q1 for given quantities of K and L , the firm can achieve more
output for a given combination of inputs. This innovation has therefore resulted
in technological progress as defined in the text.
b)
Initially MPK 1,500 and MPL 500 implying the MRTS L , K 0.33 . After the
innovation the MPK 10, 000 and MPL 500 implying the MRTS L , K 0.05 .
Since the marginal rate of technical substitution of labor for capital has decreased
after the innovation this is labor-saving technological progress.
6.28. A firm’s production function is initially Q = KL, with MPK = 0.5(√L/√K) and MPL =
0.5(√K/√L). Over time, the production function changes to Q = KL, with MPK = L and MPL =
K.
a) Verify that this change represents technological progress.
b) Is this change labor saving, capital saving, or neutral?
a)
With any positive amounts of K and L, KL  KL so more Q can be produced
with the final production function. So there is indeed technological progress.
b)
With the initial production function
MRTS L , K 
MPL K
 .
MPK
L
With the final production function
MRTS L , K 
MPL K
 .
MPK
L
For any ratio of capital to labor, MRTSL,K is the same for the two production
functions. Thus, the technological progress is neutral.
6.29. A firm’s production function is initially Q = KL, with MPK = 0.5(√L/√K) and MPL =
0.5(√K/√L). Over time, the production function changes to Q = K√L, with MPK = √L and
MPL = 0.5(K/√L).
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a) Verify that this change represents technological progress.
b) Is this change labor saving, capital saving, or neutral?
a)
With any positive amounts of K and L, KL  K L so more Q can be produced
with the final production function. So there is indeed technological progress.
b)
With the initial production function
MRTS L , K 
MPL K
 .
MPK
L
With the final production function
MRTS L , K 
MPL 0.5K

.
MPK
L
For any ratio of capital to labor, MRTSL,K is lower with the second production
function. Thus, the technological progress is labor-saving.
6.30 Suppose that in the 21st century the production of semiconductors requires two
inputs: capital (denoted by K) and labor (denoted by L). The production function takes the
form
However, in the 23rd century, suppose the production function for
semiconductors will take the form
In other words, in the 23rd century it will be
possible to produce semiconductors entirely with capital (perhaps because of robots).
a. Does this change in the production function change the returns to scale?
b. Is this change in the production function an illustration of technological progress?
a. No. In both the 21st and 23rd centuries, the production function for this good exhibited
constant returns to scale. In both cases, increasing inputs by a given proportion
increases output by the same proportion.
b. The change in the production function is not an example of technological progress.
This is because we do not get more output from a given combination of inputs. For
example, if L = 100, K=100, in the 21st century
. In the 21st
century, with the same input combination, we would get the same output, Q = 100.
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