Topic 4 – Production and the Choice of Inputs
Outline:
I) Motivation
II) Relevant Concepts
III) Optimal Combination of Inputs
A) In the short-run
B) In the long-run
IV) Applications
I)
Motivation / Introduction
Two Goals:
-
Understand optimal use of inputs in production
process
-
Provide a foundation for understanding pricing
and output decisions of managers / firms
II) Relevant Concepts (most of this should be review)
-
Need a mathematical description of the available
production technology
Production Function – An engineering relation that
defines the maximum output that can be produced
with a given set of inputs.
Simplifying Assumption – Two inputs: Labor and
Capital
Example:
Q  F (K , L)
The maximum amount of output Q that can be
produced with K units of capital and L units of labor.
-
Manager’s job is to use the production
technology efficiently, i.e. to determine how
much of each input to use to produce a given
level of output
Two Relevant Time Horizons
Short Run: Defined as the time period during which
some inputs are fixed (e.g. assembly line at Ford)
Long Run: All inputs are variable (i.e. can be
adjusted)
Usually, we assume that capital is fixed in the shortrun, while labor is variable
Three Measures of Productivity
A) Total Product – The maximum level of output that
can be produced with a given amount of inputs.
Example: Cobb-Douglas Production Function
Q  F (K , L)  K .5L.5
Suppose K is fixed at 16 units.
Short-Run production function is:
Q  (16).5 L.5  4L.5
Q: What is the total product when 100 units of labor
are used?
Q  4(100).5  40 units
B) Average Product – A measure of the output
produced per unit of input (average productivity
of the input).
Q
Q
AP 
, AP 
L L
K K
C) Marginal Product – The change in total output
arising from a 1-unit change in one of the inputs,
holding the quantity of the other input constant.
Q
MP 
,
L L
Q
MP 
K K
Stages of Production
-
Most inputs exhibit three stages of production,
characterized by the value of their marginal
product (holding constant the quantities of all
other inputs).
a) Increasing marginal returns – Initially, as the
usage of an input increases, marginal product rises.
b) Decreasing / Diminishing marginal returns – As
more of the input is used, the marginal product
typically begins to fall.
c) Negative marginal returns – As greater quantities
of the input is used, eventually the marginal
product will become negative.
Example: Assembly line workers
Stages of Production
Q
Increasing
Marginal
Returns
Diminishing
Marginal
Returns
Negative
Marginal
Returns
Q=F(K,L)
=TP
AP
MP
L
Relationship between marginal and average product
Notice that:
When MP > AP, AP is rising
When MP < AP, AP is falling
When MP = AP, AP is at its maximum
Example: GPAs and test scores
Law of Diminishing (Marginal) Returns – Beyond
some point, output grows at a diminishing rate with
increases in the variable input.
It has been said that, “if it weren’t for the law of
diminishing returns, the whole world could be fed out
of a flowerpot.”
Returns to Scale
Q: In the long-run (when all inputs are variable),
what happens to output when all inputs are increased
by exactly the same proportion?
Increasing Returns to Scale – A proportional increase
in all inputs yields a more than proportional increase
in output (aka “Economies of Scale”).
Constant Returns to Scale – A proportional increase
in all inputs yields an equal proportional increase in
output.
Decreasing Returns to Scale – A proportional
increase in all inputs yields a less than proportional
increase in output (aka “Diseconomies of Scale”).
Note: A production function can exhibit different
degree of returns to scale over different levels of
output.
Common pattern:
- Increasing returns to scale at low levels of output
- Constant returns to scale at medium levels of output
- Decreasing returns to scale at high levels of output
Caveat: Decreasing returns to scale and the law of
diminishing returns are completely unrelated!
-
“Decreasing returns to scale” is a long-run
concept based on simultaneously varying all
inputs.
-
“Diminishing returns” is short-run concept based
on varying one input while all others are held
fixed.
III) Optimal Combination of Inputs
A) In the Short-Run (capital fixed)
With capital fixed, this reduces to asking how much
labor should be hired?
First, define a new concept…
Value Marginal Product (VMP) – The value of the
output produced by the last unit of an input.
VMP  P * MP
L
L
VMP  P * MP
K
K
where P is the price the firm gets for it’s product.
Note: Also sometimes referred to as the “marginal
revenue product.”
Profit-Maximizing Input Usage Rule: When the cost
of each additional unit of labor is w, continue to hire
labor up until the point where
VMP  w
L
(in the range of diminishing marginal product).
This is just a form of the marginal benefit = marginal
cost rule we saw earlier in the semester.
Where VMP is the marginal benefit of hiring an
L
additional unit of labor and w is the marginal cost of
hiring an additional unit of labor.
Example 1:
Quantity
Of
Labor
0
1
2
3
4
5
6
7
8
9
10
11
Price
Of
Output
$3
3
3
3
3
3
3
3
3
3
3
3
MPL
VMPL
76
172
244
292
316
316
292
244
172
76
-44
$228
516
732
876
948
948
876
732
516
228
-132
Wage
Rate
(weekly)
$400
400
400
400
400
400
400
400
400
400
400
Initially, w > VMPL , but this is in the region where
MPL is increasing
Then, VMPL > w up through the 9th worker hired.
Example 2:
Price of output = $10
Q  F (K , L)  K1/ 2L1/ 2
Q: If capital is fixed at 1 unit in the short run, how
much labor should the firm hire if the wage rate is $2
and the marginal product of labor is as shown below?
K 1/ 2
MP 
L
2L1/ 2
Rule is:
VMP  w
L
in region where MPL is declining.
Notice that MPL is everywhere declining in L.
With K = 1,
MP 
L
1
2L1/ 2

5
VMP 
L 1/ 2
L
Then
VMP  w
L

5
2
L1/ 2

L
25
 6.25
4
Note: The downward-sloping portion of the VMPL is
the firm’s labor demand curve.
The fact that it is downward sloping follows from the
property of diminishing marginal product.
Thus, in the firm’s short-run labor demand curve will
always be downward-sloping (in contrast to
consumer demand curves).
B) In the Long-Run (both inputs variable)
Motivation
1) To maximize profits, the firm must produce its
output in the least cost manner.
2) Cost-minimization is more fundamental than
profit-maximization; applies to behavior of nonprofit
organizations as well.
We will study the manager’s cost minimization
problem.
Goal of the manager is to find the cost-minimizing
combination of inputs to produce a given level of
output.
Manager’s cost-minimization problem is analogous
to the consumer’s utility maximization problem that
we saw earlier.
Two familiar concepts:
i) Isoquants (similar to indifference curves)
ii) Isocost Line (similar to the budget line)
“Iso” = “same”
“Isoquant” = “same quantity”
“Isocost” = “same cost”
Isoquants
-
Defines the quantities of inputs (K,L) that yield
the same level of output (Q).
-
Isoquants farther from the origin are associated
wit higher levels of output.
Marginal Rate of Technical Substitution (MRTS) –
The rate at which a producer can substitute between
two inputs and maintain the same level of output.
-
Given by the slope of the isoquant.
Equal to the ratio of the marginal products.
MP
L
MRTS

KL
MP
K
Law of Diminishing Marginal Rate of Technical
Substitution – As less of one input is used, increasing
amounts of another input must be used to produce the
same level of output.
-
The more of one input you use, the less
productive it is relative to the other input.
-
Yields typical “bowed-in” isoquants.
Examples of Isoquants
- Next page
Linear Isoquants
Capital and Labor are
perfect substitutes
K
Increasing
Output
Q1
Q2
Q3
L
Leontieff Isoquants
Capital and labor
are perfect
complements, i.e.
capital and labor
are used in fixed
proportions
Q3
K
Q2
Q1
Increasing
Output
L
Benchmark Case
Inputs are not perfectly K
substitutable
Q3
Q2
Diminishing marginal rate
of technical substitution
Increasing
Output
Q1
Most production processes
have isoquants with this
shape
L
Isocost Line
-
A line that represents the combinations of inputs
(K,L) that cost the producer the same amount of
money.
-
For given input prices, isocosts farther from the
origin are associated with higher costs.
-
Equation similar to budget line:
wL  rK  C
-
Slope of the isocost line is the ratio of the input
prices.
-
To see this, rewrite equation as:
K
-
C  w 
  L
r  r 
Changes in input prices rotate the isocost line
(see diagram)
Cost Minimization
For typically-shaped isoquants, marginal product per
dollar spent should be equal for all inputs
MP
MP
L 
K
w
r
Expressed differently
MP
L 
MRTS 
MP
K
w
r
Graphically, the cost-minimizing input combination
occurs at a point of tangency between the isocost line
and an isoquant.
K
Point of Cost
Minimization
Slope of Isocost
=
Slope of Isoquant
Q
L
Optimal Input Substitution
To minimize the cost the producing a given level of
output, the firm should use less of an input whose
relative price has risen (see diagram).
Nothing akin to an income effect here; just a
substitution effect.
 Long-run input demand curves are downward
sloping.
IV) Applications
A) Fringe Benefit Costs and Labor Substitution.
-
Federal tax laws prevents firms from offering
different benefit packages to high and low
income workers.
-
Goal is to ensure that low-income workers
receive adequate benefits.
-
Unintended consequences?
-
Consider a firm that employs
programmers and secretaries.
computer
-
Suppose programmers earn $60,000 per year
while secretaries earn $20,000 per year.
-
Suppose that prior to the law, both types of
workers have a health plan that costs 10% of
their salary
 The programmers’ plan costs $6000 and the
secretaries’ plan costs $2000.
 Relative price of a secretary to a programmer is
22,000/66,000 = .333
-
Now suppose the tax law requires that both types
of workers receive the same health plan.
-
Suppose the firm chooses a plan costing $4000.
Q: What is the new relative price?
A: Now, the relative price of a secretary is
24,000/64,000 = .375
-
To minimize costs, firms will substitute away
from secretaries (by hiring temps, contracting
work out, or making secretaries part-time
workers, etc.)
Q: Does this really happen?
A: See article by Scott, Berger, and Black,
“Effects of the Tax Treatment of Fringe Benefits
on Labor Market Segmentation.”
B) Union support for minimum wage legislation.
Q: Why do unions lobby for increases in the
minimum wage when virtually all of their members
already earn above the minimum wage?