Production

1
Production

• In this section we want to explore ideas about
  production of output from using inputs. We will
  do so in both a short run context and in a long
  run context.

2
Production function
Here we will assume output is made with the
inputs capital and labor. K amount of capital
used and L amount of labor. The production
function is written in general as Q F(K,
L), where Q output,and F and the parentheses
are general symbols that mean output is a
function of capital and labor. The output, Q,
from the production function is the maximum
output that can be obtained form the inputs.
3
Time Frame In production, we have said that firms
have the ability to use both capital and
labor. When you consider the fact that capital is
basically the production facility the building,
equipment, machines and the like you can get
the feeling that it is probably less easy to
change the capital than it is to change the
amount of labor used. When you look at how long
it takes to change the amount of capital in
production, during that time when capital can not
be changed in amount the time period of
production is said to be the SHORT RUN. When all
inputs can be changed we are in the LONG RUN.
4
Example
Say production of units of output follow the
function Q 2KL. This means output is the
multiplication of 2, the units of capital used,
and the units of labor used. You can probably
envision a table of numbers that puts units of
labor across the top, units of capital down the
side and inside the table is the output
amount. For example if we went down 1unit of
capital and over to 2 units of labor and we would
have output Q 2(1)(2) 4
5
Long run
Capital
On a curve we have different combinations of L
and K that give the same amount of output.
Curves farther out in the northeast direction
have more output. Later we will say more about
what the firm uses as a guide to choice of
position in the graph. The position chosen will
have implications for the amount of labor
demanded.
Labor
6
Short Run
Capital
In the short run the firm would have a given
amount of capital, say K here. Production would
occur along the dotted line.
K
Labor
7
Short run/long run
The notion of a fixed or variable input is
related to the time frame of production. The
short run is that period of time when at least
one input is fixed in amount. The long run is
that period of time in which all inputs are
variable. As an example of this consider fast
food in Wayne. About any store in town could
remodel and increase floor space in about 3
months. So after 3 months we have the long run,
all inputs can vary - even floor space. But less
than three months is the short run because there
is only so much floor space to use.
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example
Say Q 2KL. In short run say K1. Then if L
1, Q 2 and if L 2, Q 4, and if L 3, Q
6, and so on.
Capital
K1
Q 6
Q 4
Q 2
Labor
1 2 3 and so on
9
Short run production function
Typically in the short run we use the graph here
instead of the previous one. We put the variable
input on the horizontal axis and the output
amount on the horizontal axis. Implicitly we
have the capital amount fixed at a level when we
draw the short run production function.
Units of output Q
Labor amount
10
Short run production functionexample Q 2KL
If K 1 we have the production function Q 2L.
Some points would be if L 1 Q 2, If L2, Q4
and so on. The graph is on the left here
Units of output Q
4 2
1 2
Labor amount
11
Short run Production example Q sqrt(KL) when K
4
12
General short run example

Here is a general short run production function.
Notice 1) if labor input 0 output 0, 2)
initially output grows at an increasing rate when
labor input rises, then 3) output grows at a
decreasing rate (called diminishing returns),
and 4) finally more labor may even make output
start to decline.
Units of output Q
Labor amount
13
Malthus and diminishing returns
It has been suggested that enough production
processes in the short run exhibit diminishing
returns that we should take it seriously. Malthus
argued way back in late 1700s that because of
diminishing returns we would eventually starve to
death. The fixed land (which places us in the
short run) would eventually not be able to add
food production at the rate at which the
population increased. In our model labor is the
only variable input. In the real world there are
many variable inputs. Technology has increased
so much that Malthus has not been proven right,
yet. Will he ever be proven right? Only time
will tell (but I think NOT!)
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General example continued
In the general example, the relationship between
the labor used and the total product (TP), or
output Q, is called the short run production
function. Behind the scenes we assume there is a
given amount of capital. The marginal product of
labor (MPL)is the additional output forthcoming
from the additional unit of labor. Note that as
the units of labor increases the marginal product
first increases, but then begins to diminish
after more labor is employed.
15
example continued
The marginal product curve has the pattern it
does because of the way the fixed input is used.
Remember that the variable input is used in
conjunction with only so much of the fixed
input. In the beginning, as more labor is added,
specialization of labor can occur and increasing
returns to labor can result, but eventually as
more labor is added there will be less of the
fixed input to work with and thus additions to
output have to diminish. The way output changes
as the variable input is changed, with a given
amount of a fixed input, is summarized with the
phrase diminishing marginal product.
16
Example continued
The average product of labor (APL)is for each
amount of labor the output produced divided by
the labor amount. The average product mimics,
or follows, the marginal product. It is just a
math thing. Next lets look at some graphs.
Definitions APL Q/L MPL change in Q / change
in L
17
TP and MPL, APL
APL curve
18
Notes about MPL and APL

• Note
• When the MPL is above the APL the APL rises.
• When the MPL is below the APL the APL falls.
• The APL continues to rise while the MPL is
  falling only when the MPL is above the APL.

19
APL from the graph of TP

I have reproduced the general short run
production function, also called the total
product, TP, curve. I have also put out a ray
line through the origin. Note at L1 we get Q1.
The APL Q/L so at the point shown APL1
Q1/L1. Also note as you go along the ray line
from the origin to the TP the slope of the ray is
Q1/L1.
Units of output Q
Q1
L1
Labor amount
So the slope of the ray from the origin to the TP
curve is the APL.
20
APL from the graph of TP

Note 1) as labor is first increased the ray lines
are moving from right to left and since the
slopes are getting bigger the APL is rising, 2)
at L the ray is tangent to the TP curve and we
can see the APL is at a maximum, and 3) beyond Y
the ray lines have less steep slopes and thus APL
is falling.
Units of output Q
Q1
L1
L
Labor amount
21
MPL from the graph of TP

The MPL is the change in output divided by the
change in labor. The MPL of an amount of labor
is really the slope of the TP curve at the level
of L used. A way to see the slope is to look at
the slope of the tangent line at that point
Units of output Q
Q1
L1
Labor amount
22
MPL from the graph of TP


Crazy graph, I know. Before L the tangent
lines have slopes that get bigger. But when the
curve switches from increasing at an increasing
rate to increasing rate at a decreasing rate it
is before L. So MP reaches it peak and begins
to diminish before AP has reached its peak. At
L MP AP.
Units of output Q
Units of output Q
Q1
Q1
L1
L1
L
Labor amount
Labor amount
23
Marginal analysis
It has been said economics is a science of
marginal analysis. With this in mind, we see
later MPL is the more interesting idea. As an
example of this, firms might ask, should another
economist be hired? By the way, when Q 2KL
and if K 1, for example, we are in the short
run. The APL Q/L 2L/L 2, a constant, and
MPL2 as well. You have to read pages 272-275 for
a great example of why we focus more on margins
than averages in economics.
24
Long Run - isoquants
In the long run all inputs can be varied. On the
next slide you see what we call an isoquant.
Along the curve the amount of output is the same,
but we have different combinations of K and
L. Again say Q 2KL. To get one isoquant you
pick a level of output. Say we pick Q 100.
Then we have 100 2KL. Since capital is on the
vertical axis we might re-express this function
as K 100/2L 50/L. If L 1 K 50 to get Q
100. If L 2 K 25 to get Q 100. Isoquants
have properties similar to indifference curves
for consumers.
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Marginal Rate of Technical Substitution - MRTS
Capital
Change in K Change in L On the next slide I
will refer to a change with the use of a
triangle.
slope
Labor
The MRTS absolute value of the slope.
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MRTS
The slope of the curve at a point is K/
L Now, if the marginal product of an input is
defined as the change in output divided by the
change in the input, the slope can be manipulated
to be K Q and since K 1 L
Q Q MPK
So the slope is MPL/MPK and is called the MRTS
(in absolute value) and it is a measure of the
rate at which inputs can be substituted and
output remains the same.
27
A few slides back I showed an isoquant. I also
put a tangent line at a point on the curve. The
slope of a curved line at a point is really the
slope of a tangent line at the point. You will
notice that as you move along the curved line
from left to right that the slope of a tangent
line gets smaller (in absolute value). Isoquants
for perfect substitutes in production will be
straight, downward sloping from right to left,
lines. Isoquants for perfect complements are L
shaped. The production process is often called a
fixed proportion process. An example would be
you need a computer and a computer operator in
many cases.
28
LR returns to scale
Remember in short run at least one input is fixed
in amount. In LR all inputs can vary. Returns
to scale is idea that if all inputs are increased
by a proportionate amount what happens to the
amount of output. Since all inputs are changed
it is a long run concept. Example We might be
interested what would happen to output if all
inputs were doubled. It is not a done deal that
if all inputs are increased in the same
proportion that output will grow by that same
proportion. But if it does the we say there are
constant returns to scale.
29
LR returns to scale
If inputs are all increased by a proportionate
amount and the resulting output grows by more
than this proportion, then increasing returns to
scale exists. If inputs are all increased by a
proportionate amount and the resulting output
grows by less than this proportion, then
decreasing returns to scale exists. A given
production function may exhibit each of these
returns to scale at different ranges of output,
with increasing returns happening first, then
constant and finally decreasing returns happening.
30
NOTE
Returns to scale is a long run concept when
changing all inputs in the same
proportion. Diminishing returns is a short run
concept when at least one input is fixed in
amount and another input is changed in
amount. The two ideas are really not related in
any general way.