Download The Production Function

Econ Dept, UMR
Presents
The Supply Side of the Market
in
Three Parts:
I. An Introduction to Supply and
Producer Surplus
II. The Production Function
III. Cost Functions
Part II: The Production
Function
Starring
u Supply
v Production
v Cost
u Producer
Surplus
Featuring
uThe
Law of Diminishing Marginal Product
uThe MP/P Rule
uEconomic Cost vs. Accounting Cost
uEconomic Profit vs. Accounting Profit
uThe Unimportance of Sunk Cost
Behind the Supply Curve
Necessary compensation for effort is
based on cost
u And, Cost is based on the production
function and input prices
u
v The production function relates inputs to
output and is governed by technology
v The input mix required for any output
times the input prices gives output cost
v What we want is obtained efficiently only
if it is produced at minimum output cost
Production - Cost - Supply
u
u
u
u
u
u
Supply, Cost, and the Production Function are
interdependent
We assume input prices are fixed
As is Technology
Production technology relates inputs to outputs
The optimal method of production, for a profitmaximizing firm, is the one that minimizes costs
Two periods are important for decision making
v
v
The Short Run
The Long Run
Short Run vs. Long Run
The short run is a period of time
such that there is a fixed factor of
production or constraint-it is the
period we are in
u The long run is a period of time
such that there are no fixed factors
of production or constraint-it is the
period we are planning
u
Now we will look at the
production process and three
ways to measure productivity
of inputs
Total Product
u Average Product
u Marginal Product
u
Then we will see how the production
relationships link to costs
Total Product (TP)
u A mathematical or numerical
expression of a relationship
between inputs and outputs:
v q = f(K,L) is the function relating the
production of q to just two inputs: capital,
K; and labor, L
u Graphically shows units of total
product as a function of units of a
variable input with other inputs
fixed
Average Product (AP)
u The average amount of output
produced by each unit of a
variable factor of production, or
input
u Output per unit of an input, e.g.,
APL = q/L is the average
product of labor
Marginal Product (MP)
u The additional output that can be
produced by adding one more unit
of a specific input, ceteris paribus
u If Labor is the variable input:
v MPL = Îq/ÎL (over a range)
(where Î refers to change in)
v MPL
= dq/dL (using calculus,
the 1st derivative of the
production function wrt L)
Calculus?--Don’t Worry
u
u
Often to show the mathematical
relationships, we will use formulas
derived from calculus, e.g., Calculus 8
Calculus is not a prerequisite so any
formulas will be provided
Consider a lawn service with a fixed
capital base, e.g, 2 mowers, 3 trimmers, etc.
Labor
Units
0
1
2
3
4
5
6
Total
Product
0
2.67
9.30
18.00
26.67
33.33
36.00
TP = q = 3L2 - L3/3 ;
Marginal
Product
--5
8
9
8
5
0
Average
Product
--2.67
4.65
6.00
6.67
6.67
6.00
MPL = dq/dL = 6L - L2 ; APL = q/L
This data can be plotted as
follows:
Average Product (L)
Marginal Product (L)
Total Product
10
9
8
7
6
5
4
3
2
1
0
40
35
30
25
20
15
10
5
0
0
1
2
3
4
5
6
Number of employees/t
MP
AP
0
1
2
3
4
4.5
5
6
Number of employees/t
The Law of Diminishing
Returns
u After a certain point, when
additional units of a variable input
are added to fixed inputs, the
marginal product of the variable
input declines
u At this point, output starts
increasing at a decreasing rate
In the lawn service, as more
employees are added to the fixed
inputs, eventually MP falls.
Average Product (L)
Marginal Product (L)
10
9
8
7
6
5
4
3
2
1
0
MP
AP
0
1
2
3
4
4.5
5
6
Number of employees/t
Diminishing returns
sets in after the
third worker is
hired
Adding More Inputs to the
Variable Input Makes the Variable
Input More Productive
More, or better tools makes Labor more
productive
u An increase in capital stock increases:
u
v the
total product of labor
v the average product of labor
v the marginal product of labor
Returning to our lawn service, suppose
the owners can invest in four mowers
rather than two
Units of
Labor
0
1
2
3
4
5
6
Two Mowers
TP
MP
0
--2.67
5
9.30
8
18.00
9
26.67
8
33.33
5
36.00
0
Four Mowers
TP
MP
0
--3.67
7
13.33
12
27.00
15
42.67
16
58.33
15
72.00
12
With 4 mowers, q = 4L2 - L3/3 ; MPL = 8L - L2
Review: The Production
Function
u
Simplifying Assumptions we use
v Short Run, therefore at least one input is
fixed
v Output, q, depends on only two inputs
Labor, L, the variable input
u Capital, K, the fixed input
u
u
u
As the variable input is added to the fixed
input, q increases first at an increasing rate, but
ultimately at a decreasing rate due to the law of
diminishing marginal returns
More of the fixed inputs make the variable
inputs more productive
q/t
q1
q0
L0
APL,
MPL
L1
L/t
L2
Graphic View of
Typical Short
Run Production
Function
q = f (K,L) K=K
The “bar” means the
variable is fixed
MPL
APL
L0
L1
L2
L/t
q with K = K2 > K1
q/t
q with K = K1
q1
q0
L0
L1
More capital makes labor
more productive
L/t
L2
First, notice the
total product
curve. Output as
a function of
labor depends on
a given fixed
capital input.
With more K,
labor is more
productive
q/t
C
B
q1
q0
Second, find MP
by taking the
slope of TP
A
L0
L1
MPL
MPL
L0
L1
At “A” , the inflection
L/t
point, the slope is
L2
maximized; The law of
diminishing returns set
in
L2
at “B”, the slope also
equals the average
product;
L/tat “C”, the slope is zero
q/t
q1
q0
L0
APL,
MPL
L1
L/t
L2
MPL
Last find
Average Product
by drawing a ray
from the origin
to different
points on TP.
The slope of
these rays is AP
Note AP = MP at max AP
APL
L0
L1
L2
L/t
q/t
q1
q0
L0
APL,
MPL
L1
L/t
L2
Everything
Together: Typical
Short Run
Production
Function
q = f (K,L) K=K
The “bar” means the
variable is fixed
MPL
APL
L0
L1
L2
L/t
The Equal MP/P Rule
u
u
u
u
A necessary condition for minimizing cost of
any given level of an activity is to mix the
variable inputs such that their Marginal
Product/Price ratios (MPi/Pi) are equal
MP1/P1 = MP2/P2 = . . . = MPn/Pn for all n
variable inputs
If MP1/P1 > MP2/P2 you are getting more
value per dollar from input 1 than from input
2 and to produce the same level of output at
lower cost you should hire more 1 and less 2
As you hire more of input 1, MP1 falls, and as
you hire less of input 2, MP2 increases
From the Short Run to the
Long Run
In the short run at least one input is
fixed
u In the long run all inputs may be
changed
u An important property of the
production function is its “Internal
Returns to Scale”
u
Types of Internal Returns to Scale
Economies of Scale: a proportional
change in all inputs leads to a larger
proportional change in output
u Constant Returns to Scale: a
proportional change in all inputs leads
to an equal proportional change in
output
u Diseconomies of Scale: a proportional
change in all inputs leads to a smaller
proportional change in output
u
Returns to Scale
u
u
u
u
u
Changing all inputs in the same proportion is
a “scale” change, e.g., increase all by 10%,
decrease all by 5%
Increasing Returns to Scale: %Îq>%ÎR
Constant Returns to Scale: % Îq=%ÎR
Decreasing Returns to Scale: %Îq<%ÎR
(where R is all resources)
As we will see, cost curves in the long run are
based on the underlying production
technology, i.e., returns to scale
Returns to Scale, examples
IRTS: doubling all inputs leads to an
increase of 125% in q
u DRTS: an increase in all inputs by 5%
leads to a 3% increase in q
u CRTS: a decrease in all inputs by 10%
lead to a 10% fall in q
u If all inputs are decreased by 5% and
output falls by 7%, %Îq>%ÎR, therefore
IRTS
u
What if all inputs change but
not in the same proportion?
If the %Îq>%ÎCosts we use the term
Economies of Scale
u If the %Îq<%ÎCosts we use the term
Diseconomies of Scale
u IRTS implies Economies of Scale but
Economies of scale do not imply IRTS
u The same is true for the relationship
between Diseconomies of Scale and DRTS
u
Economies of Scale
u
u
Reasons for economies of scale
Greater specialization of resources
v
v
v
Divide work get benefits of specialization (lower costs).
This was the point emphasized by Adam Smith
Efficient Utilization of specialized technologies may not
be possible at small scale, e.g., Airline hubs, irrigation
systems
Arithmetic relationships, e.g.,
u the circumference of a pipe, which approximates
cost equals pi*2*radius, but the carrying capacity
depends on the area which equals pi*radius squared
Diseconomies of Scale
Reasons for diseconomies of scale
u Coordination and control problems as firm
gets large--The Principal/Agent Problem
u
v The
Principal is the person in charge and the
Agent is the person charged with carrying out
the wishes of the Principal
v Two conditions need to be present for the P/A
problem to exist
Agents and Principals must have different
objectives
u It must be costly for the Principal to monitor and
enforce
u
Now, let’s link production to
costs
u
Each production relationship has a cost
counterpart
v
v
v
v
v
u
TP:variable input
AP:variable input
MP:variable input
Fixed inputs
MP/P rule
-- Variable Cost
-- Average Variable Cost
-- Marginal Cost
-- Fixed (or Sunk) Cost
-- Equal MC rule
The production function and the MP/P rule
tells us the minimum cost of producing any
level of output, q: Cost = input price *inputs
required
The End
Go Ahead to Part III, on Costs