Chapter 19 NAME
Technology
Introduction. In this chapter you work with production functions, re-
lating output of a firm to the inputs it uses. This theory will look familiar
to you, because it closely parallels the theory of utility functions. In utility
theory, an indifference curve is a locus of commodity bundles, all of which
give a consumer the same utility. In production theory, an isoquant is a lo-
cus of input combinations, all of which give the same output. In consumer
theory, you found that the slope of an indifference curve at the bundle
(x1,x
2) is the ratio of marginal utilities, MU1(x1,x
2)/M U2(x1,x
2). In
production theory, the slope of an isoquant at the input combination
(x1,x
2) is the ratio of the marginal products, MP1(x1,x
2)/M P2(x1,x
2).
Most of the functions that we gave as examples of utility functions can
also be used as examples of production functions.
There is one important difference between production functions and
utility functions. Remember that utility functions were only “unique up to
monotonic transformations.” In contrast, two different production func-
tions that are monotonic transformations of each other describe different
technologies.
Example: If the utility function U(x1,x
2)=x1+x2represents a person’s
preferences, then so would the utility function U∗(x1,x
2)=(x1+x2)2.
A person who had the utility function U∗(x1,x
2) would have the same
indifference curves as a person with the utility function U(x1,x
2)and
would make the same choices from every budget. But suppose that one
firm has the production function f(x1,x
2)=x1+x2, and another has the
production function f∗(x1,x
2)=(x1+x2)2. It is true that the two firms
will have the same isoquants, but they certainly do not have the same
technology. If both firms have the input combination (x1,x
2)=(1,1),
then the first firm will have an output of 2 and the second firm will have
an output of 4.
Now we investigate “returns to scale.” Here we are concerned with
the change in output if the amount of every input is multiplied by a
Example: Consider the production function f(x1,x
2)=x1/2
1x3/4
2.Ifwe
multiply the amount of each input by t, then output will be f(tx1,tx
2)=
(tx1)1/2(tx2)3/4. To compare f(tx1,tx
2)tof(x1,x
2), factor out the
240 TECHNOLOGY (Ch. 19)
Example: Let the production function be f(x1,x
2)=min{x1,x
2}.Then
Therefore when all inputs are multiplied by t, output is also multiplied by
t. It follows that this production function has constant returns to scale.
You will also be asked to determine whether the marginal product
of each single factor of production increases or decreases as you increase
the amount of that factor without changing the amount of other factors.
Those of you who know calculus will recognize that the marginal product
of a factor is the first derivative of output with respect to the amount
of that factor. Therefore the marginal product of a factor will decrease,
increase, or stay constant as the amount of the factor increases depending
on whether the second derivative of the production function with respect
to the amount of that factor is negative, positive, or zero.
Example: Consider the production function f(x1,x
2)=x1/2
1x3/4
2.The
marginal product of factor 1 is 1
2x−1/2
1x3/4
2. This is a decreasing function
19.0 Warm Up Exercise. The first part of this exercise is to cal-
culate marginal products and technical rates of substitution for several
frequently encountered production functions. As an example, consider
the production function f(x1,x
2)=2x1+√x2. The marginal product of
x1is the derivative of f(x1,x
2) with respect to x1, holding x2fixed. This
is just 2. The marginal product of x2is the derivative of f(x1,x
2)with
respect to x2, holding x1fixed, which in this case is 1
2√x2.TheTRS is
−MP1/M P2=−4√x2. Those of you who do not know calculus should
fill in this table from the answers in the back. The table will be a useful
reference for later problems.
NAME 241
Marginal Products and Technical Rates of Substitution
f(x1,x
2)MP1(x1,x
2)MP2(x1,x
2)TRS(x1,x
2)
242 TECHNOLOGY (Ch. 19)
Returns to Scale and Changes in Marginal Products
For each production function in the table below, put an I,C,orDin
the first column if the production function has increasing, constant, or
decreasing returns to scale. Put an I,C,orDin the second (third)
column, depending on whether the marginal product of factor 1 (factor
2) is increasing, constant, or decreasing, as the amount of that factor
alone is varied.
f(x1,x
2)Scale MP1MP2
19.1 (0) Prunella raises peaches. Where Lis the number of units of
labor she uses and Tis the number of units of land she uses, her output
is f(L, T )=L1
2T1
2bushels of peaches.
(a) On the graph below, plot some input combinations that give her an
output of 4 bushels. Sketch a production isoquant that runs through these
NAME 243
04812
16
2
4
6
L
T
8
(b) This production function exhibits (constant, increasing, decreasing)
(c) In the short run, Prunella cannot vary the amount of land she uses.
On the graph below, use blue ink to draw a curve showing Prunella’s
output as a function of labor input if she has 1 unit of land. Locate the
points on your graph at which the amount of labor is 0, 1, 4, 9, and
16 and label them. The slope of this curve is known as the marginal
04812
16
2
4
6
Labour
Output
8
Blue line
Red line
Red MPL line
244 TECHNOLOGY (Ch. 19)
(d) Assuming she has 1 unit of land, how much extra output does she
get from adding an extra unit of labor when she previously used 1 unit of
(e) In the long run, Prunella can change her input of land as well as
of labor. Suppose that she increases the size of her orchard to 4 units
of land. Use red ink to draw a new curve on the graph above showing
output as a function of labor input. Also use red ink to draw a curve
showing marginal product of labor as a function of labor input when the
amount of land is fixed at 4.
19.2 (0) Suppose x1and x2are used in fixed proportions and f(x1,x
2)=
min{x1,x
2}.
(a) Suppose that x1<x
2. The marginal product for x1is 1
and (increases, remains constant, decreases) remains constant
(b) Suppose that f(x1,x
2)=min{x1,x
2}and x1=x2= 20. What is
the marginal product of a small increase in x1?0. What is the
Calculus 19.3 (0) Suppose the production function is Cobb-Douglas and
f(x1,x
2)=x1/2
1x3/2
2.
(a) Write an expression for the marginal product of x1at the point
(x1,x
2). 1
2x−1/2
1x3/2
2.