Intermediate Microeconomics
Professor Yongmin Chen
Topic 5: Inputs and Production Functions
Firm Owners and Managers: A Principle-agent Problem
A firm is usually assumed to maximize profit. This is what we would expect if the firm is
run by its owners. Most modern
corporations, however, are run by managers who are not owners of the firm. Of course, there is no reason why the
managers would have the same objectives as the owners. Much of the economic research in the last
twenty years or so is concerned with how to solve this type of principle-agent
problem. There are two basic
mechanisms: the internal mechanism and the external mechanism.
Internal mechanism:
through compensation scheme and through the monitoring by the board of
directors.
External mechanism: the market for corporate control.
Discussion: the forms
and effectiveness of these mechanisms.
Production Function
Just as a consumer faces the budget constraint in consumption,
a firm faces the constraint of technology in production. Given a technology, there will be a set of
feasible combinations of inputs and outputs.
Fixed inputs and variable inputs.
Suppose the production process requires only one variable
input, say labor (L), and the output is denoted by X. When L=0, X=0. When L
increases, the maximum X that can be produced also increases. See a graphic explanation in class.
The technologically feasible set of all input and output
combinations is called a production set.
For any level of input, there is a maximum level of output
that can be produced under the existing technology. These maximum levels of output constitute the boundary of the
production set, which can be denoted by X = X(L) and is called a production function. Thus a production function specifies the
maximum levels of outputs as a function of inputs.
Example 5-1: X = L1/2
The production function passes through the origin. The shape of the production function is
determined by technology. For a given technology,
a production function is obtained.
When there are more than one input, the production function is
the maximum level of output that is attainable as a function of all
inputs. For example, if inputs are L and
K, a production function is X = X(L,K).
When we have only one input, it is easy to graph the
production function. How can we express
the production function graphically when there are two inputs?
Let L be on the horizontal axis and K be on the vertical
axis. Let's first keep L and K in fixed
proportion (L/K constant). As we increase
both L and K, a straight line is generated in the input space. Each input combination along the line
corresponds to a maximum level of output that can be produced. We call such a line an activity ray.
If we change the ratio L/K, we will have another activity
ray. Given an output level, say, 100,
we can find all the input combinations under which the maximum level of output
is 100 in the input space. The curve
which connects all such points for a given output level is called an
isoquant. That is, an isoquant is a
graph of all input bundles that result in the same level of output. The production function of different levels
of output can then be represented by an isoquant map.
Compare an isoquant and
an indifference curve.
Examples of technologies (production functions):
(1) Leontief Technology: the
inputs have to be kept in fixed proportion in production. No substitution among inputs is
possible.
Example 5-2: X = Min {K, 2L}
How can you graph it with isoquants? Can you think of a real-world situation
where such technology prevails?
(2) Perfect input
substitution: one input can be used to substitute another input at a constant
rate to maintain a certain level of output.
Example 5-3: X = L + 2K
How to graph its isoquants?
(4) Cobb-Douglas technology:
X = aLbK1-b
The isoquant map of Cobb-Douglas function.
Example 5-4: X=10L0.5K0.5
In his pioneering study of production in the U.S. economy
between 1899 and 1922, P.H.Douglas found the value of b is about .75 for the
U.S. economy.
Production in the Short-run
Economists often distinguish short-run and long-run. Short-run refers to such a time period that
technology is constant and at least one input is fixed. In the long-run, all inputs can be changed.
A total product (TP) curve is a graph of maximum output as a
function of a variable input when other inputs remain constant.
Labor is often considered as a variable input. From a map of isoquants, we can derive the
TP curve. See derivation in class.
Average product (AP): total product divided by the quantity of
the input required.
The average product of labor: APL = X/L
Marginal product (MP): the change in output divided by the
incremental change in the input as that input changes alone.
The marginal product of labor can be measured by the slope of
the TP curve:
MPL = DX/DL
Example 5-5: Suppose the production function is given by X =
10L0.7K0.3 and K=1.
What is the TP function of labor? The average product of labor? The
marginal product of labor? What will be
the answers to these questions if K=2?
Example 5-6: Suppose your company has 1000 employees and you
have two plants which produce the same products. You want to allocate your employees so that the total output of
your company is maximized. How can you
do that?
Example 5-7: Suppose your company has recently installed some
new equipment. How will this affect
your company's TP curve?
The law of diminishing
marginal productivity (diminishing marginal returns): If you increase the use
of one input while keeping all other inputs fixed, the marginal product of that
variable input will eventually decrease.
Production in the Long-run
1. Input substitution
In the long run all inputs are variable. There are often many input combinations that
can produce certain amount of output, as is represented by an isoquant. In other words, it is often possible to
substitute one input for another while keeping the output at a constant
level. Such substitution between inputs
are movements along an isoquant.
Marginal rate of technical substitution of labor for capital
(MRTSLK): the amount by which capital input can be reduced when one more
unit of labor input is used while holding quantity produced constant. It is the negative of the slope of an
isoquant.
MRTSLK = - DK/DL (for output constant)
Example 5-8: If you are given two points on an isoquant, can
you tell at which point MRTSLK is higher? If the isoquant is convex, will MRTSLK be increasing
or diminishing as more L is used?
There is a relation between MRTSLK and MPL
and MPK:
MRTSLK = MPL/MPK
Example 5-9: If the production function is Cobb-Douglas: X =
aLbK1-b, then
MRTSLK = [b/(1-b)](K/L)
Example 5-10: If the production function is given by X = L +
aK, where a (> 0) is a constant, then
MRTSLK = 1/a
2. Returns to scale
When all inputs change in the same proportion, we say the
scale of production changes. So any
movement along an activity ray represents a change in production scale. The relationship between output and
production scale is called returns to scale.
Suppose the production function is X = X(L,K). There are three possible types of returns to scale:
(1) Constant returns to
scale: Output changes by the same proportion as all inputs.
For any constant s > 0, constant returns to scale imply
X(sL,sK) = sX(L,K)
Examples 5-11:
production process with constant returns to scale:
X = aLbK1-b
X = Min {L, 2K}
X = L + 3K
(2) Decreasing returns to
scale: Output changes by a smaller proportion than all inputs.
For any constant s > 1, decreasing returns to scale imply
X(sL,sK) < sX(L,K)
Example 5-12: X = 2L.6K.3
(3) Increasing returns to
scale: Output changes by a larger proportion than all inputs.
For any constant s > 1, increasing returns to scale imply
X(sL,sK) > sX(L,K)
Example 5-13: X = 4LK2
Caution: Not to confuse
marginal product and average product with returns to scale. When we talk about marginal product and
average product, we change only one input at a time while holding all other
inputs fixed. When we talk about
returns to scale, all inputs are changed by the same proportion at the same
time. For instance, a Cobb-Douglas
production function has constant returns to scale, but has diminishing marginal
and average products.