Lecture 4
Supply: Production
What is the optimal level of output?
How should we decide among alternative production processes?
How would investment in production equipment affect labor costs?
Will building extra production capacity increase or decrease costs?
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Supply: Production
Quantity
(widgets)
1500
Total
Output
1200
900
600
300
3
6
9
12
15
MGMT 7730 - © 2011 Houman Younessi
Labor
Lecture 4
Output
Average Output
(Average Product)
(Q/L)
Marginal Output
(Marginal
Product)
(ΔQ/ΔL)
Marginal Output
(Marginal
Product)
(dQ/dL)
5
0
--
--
--
1
5
49
49
49
67
2
5
132
66
83
98
3
5
243
81
111
123
4
5
376
94
133
142
5
5
525
105
149
155
6
5
684
114
159
162
6.67
5
792.6
118.9
162.9
163.34
7
5
847
121
163
163
8
5
1008
126
161
158
9
5
1161
129
153
147
10
5
1300
130
139
130
11
5
1419
129
119
107
12
5
1512
126
93
78
13
5
1573
121
61
43
14
5
1596
114
23
2
15
5
1575
105
-19
-45
Labor
Capital
(# of
Machines)
0
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Quantity
(widgets)
1500
B
Total
Output
1200
900
A
600
300
3
6
9
12
15
MGMT 7730 - © 2011 Houman Younessi
Labor
Lecture 4
Average and Marginal output (product) curves
B
170
150
A
130
110
90
Average Output
(Average Product)
Q/L
70
50
30
C
10
-10
-30
-50
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3
1
4
1
5
1
6
Marginal OutPut
(Marginal Product)
dQ/dL
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Average output (product)
Q
Q
L
or in general
Q
Po 
X
Where X is a given input into production
Average output is either zero or positive
Maximum Average output Point A is precisely that: the point where
average output is at its highest.
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Marginal output (product)
Q dQ
MP  lim

X 0 X
dX
Where X is a given input into production
A measure of productivity, it measures the rate of change of output
(production) as an extra unit of input is added.
A production reaches maximum productivity with respect to a given input
(assuming all other inputs are constant) when marginal output for that
particular input is maximized (Point B) .
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Marginal output (product)
The difference between
Q
X
and
dQ
dX
Assumes continuous units (e.g. labor
in labor-hours 2.73 units is allowed)
1500
1200
900
Assumes discrete units (e.g. labor in
persons n or n+1 but not 1.5n)
600
300
3
6
9
12
15
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Marginal output (product)
As expected, output is maximized when marginal output (marginal
product) is zero (point C).
In general Output = Q(L,M,N,X,Y,Z,…)
In our simple case Output=Q(L)
As such:
dQ
MP 
dL
max Q( L ) @ MP  0
dQ
0
dL
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Marginal output (product)
Exercise:
Show that in all cases, average output is maximum when marginal output is
equal to it.
e.g. point A
Average product (average output) is
Maximum average product (average
output) is when its derivative is equal to
zero
Average product is zero when:
Q
X
dP ( X ) d ( Q / X ) X ( dQ / dX )  Q( dX / dX )


dX
dX
X2
1 dQ Q
dQ Q
 (
 )0 (
 )0
X dX X
dX X
P( X ) 
dQ Q
 0
dX X
dQ Q
  MP( X )  P ( X )
dX X
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
The Law of Diminishing Marginal Returns
If equal increments of an input are added whilst at least one other input is
held constant, there is a point beyond which the resulting product output
would start to decrease.
Note:
1. This is an empirical generalization and not a law of nature
2. It is assumed that production technology does not change
Example:
Adding staff but not workstations
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Optimal Resource Utilization
How much of a resource (input variable) should a firm use?
What do want to maximize:
A. Output
B. Revenue
C. Profit
Correct answer is:
?
C
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Optimal Resource Utilization
Marginal Revenue Product (MRP) is the amount that an additional unit of
the variable input adds to the firm’s total revenue (i.e. if MRPY is the
marginal revenue product of input Y):
MRPY 
TR
Y
However note that :
TR TR Q TR Q
MRPY 




Y
Y Q Q Y
As such:
MRPY  MR( MPY )
Therefore: The marginal revenue product of an input equals to that input’s
marginal product times the firm’s marginal revenue
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Similarly:
Optimal Resource Utilization
Marginal Product Expenditure (MPE) is the amount that an additional unit of
the variable input adds to the firm’s total costs (i.e. if MPEY is the marginal
product expenditure of input Y):
MPEY 
TC
Y
TC TC Q TC Q




Y
Y Q Q Y
However note that :
MPEY 
As such:
MPEY  MC( MPY )
Therefore: The marginal product expenditure of an input equals to that
input’s marginal product times the firm’s marginal cost
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Optimal Resource Utilization
To maximize profit (globally):
or:
Multiplying both sides by:
Rearranging:
Q
Y
MR  MC
TR TC

Q
Q
TR Q TC Q



Q Y Q Y
TR Q TC Q



Y Q Y Q
or
MRPY  MPEY
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Optimal Resource Utilization
Example:
A brewery has fixed plant and equipment. The output of this brewery is shown
to relate to its labor usage according to the following equation where L is the
number of workers hired per day and Q is the quantity of beer produced.
Q  98L  3L2
This is a thirsty country and the brewery can sell all the beer it can produce
for $20 a gallon.
This is also a developing country so the brewery can hire as many workers
as it needs for $40 a day.
How many workers should the brewery hire?
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Optimal Resource Utilization
The brewery's marginal revenue is MR= $20
The brewery's marginal product expenditure is MPEL= $40
Marginal product for labor is:
MPL 
Marginal revenue product for labor is:
To maximize, we must have:
dQ
 98  6 L
dL
MRPL  MR( MPL )
MRPL  MPEL
20( 98  6 L )  40
L  16
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Multi-variable Production Functions
A production function is very rarely a function of only one variable.
For example Q may be a function of:
Q  f ( L ,M ,T )
Where L stands for cost of Labor, M stands for cost of Machinery and T
stands for cost of Transportation
In most cases – certainly all cases covered in this course - the mathematics and
formulations are exactly the same as for the case of a single variable instance
except that we assume – in turn – that all other variables except the one of our
interest are constants. We also use the partial differential notation to indicate
multi-variability.
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Multi-variable Production Functions
Example:
Our brewery has the production function:
Q  120  10L2  25M  41 T 2
How much should they spend on transportation if it costs $200 to transport
one gallon of beer?
MR  20
Q T
MPT 

T 2
MRPT  MR( MPT )  MPET
T
 20( )  200
2
T  20
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Isoquants: Choosing Combinations of Inputs
What are the possible combinations of various inputs that are capable of
producing a certain quantity of output?
Capital
Capital/Labor Isoquants
50
40
A
30
B
20
300
C
200
10
100
1
2
3
4
5
MGMT 7730 - © 2011 Houman Younessi
6
7
Labor
Lecture 4
Marginal Rate of Substitution
The rate at which one input can be substituted for another in
such a way that the output remains constant
Change in one input
Change in another input
X 1
X 2
X 2
X 1
Rate of change of one
over the other input
MRS  
At the limit
dX 2
MRS  
dX 1
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Marginal Rate of Substitution
Exercise:
Show that the marginal rate of substitution is equal to the ratio of the
marginal output (marginal product) of the inputs concerned.
By definition:
dX2
MRS  
dX1

Therefore:
dX2 dX2 Q


dX1 dX1 Q
dX 2
(Q / X 1 )
MP1


dX1
(Q / X 2 )
MP2
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Isoquants: Choosing Combinations of Inputs
Isoquants may have regions of positive slope!
These are regions where the quantities of BOTH inputs must increase in order
to maintain a fixed level of production.
For the slope of an isoquant to be positive, it means that the marginal
product of one or the other of the inputs must be negative. This is not
efficient!
As such we restrict ourselves to operating within the range where the
slope of the isoquants are negative.
Such region is called the Economic Region of Production
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Economic Region of Production
Economic Region of Production
Capital
50
300
40
30
200
20
100
10
1
2
3
4
5
MGMT 7730 - © 2011 Houman Younessi
6
7
Labor
Lecture 4
Isocosts: Choosing Combinations of Inputs
What combinations of inputs can we obtain for the same expenditure?
In a two input situation, this is very simple, for a fixed outlay (e.g. of money),
we will have:
PX X  PYY  K
Where X and Y represent the amount of each input respectively, and PX and
PY represent the unit cost for each unit of each input.
Rewriting this we get:
K PX
Y

X
PY PY
Which is the equation for a straight line called the isocost line of X and Y
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Isocosts: Choosing Combinations of Inputs
Y
K/PY
Slope = -PX/PY
K/PX
MGMT 7730 - © 2011 Houman Younessi
X
Lecture 4
Choosing Combinations of Inputs
What point on the isocost line would you pick?
Y
K/PY
Q
K/PX
X
Pick the point that lies on both the isocost curve and the highest
isoquant
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Choosing Combinations of Inputs
Analytically:
The Q point that optimizes the input combinations is the point where the
isocost line is a tangent to the isoquant.
We know that the slope of the isocost line is:
PX

PY
We also know that the slope of the
isoquant curve (at a given point) is:
MPX

MPY
Therefore the Q point is where the
slope of the isocost curve equals the
ratio of the marginal products:
PX MPX

PY MPY
In fact in general when more than two
inputs are involved:
MPa MPb MPc
MPn


... 
Pa
Pb
Pc
Pn
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Example:
An automobile manufacturer has determined that the quantity of automobiles
they manufacture relates to the number (in thousands) of factory workers (F)
and the number (in thousands) of office workers (O):
Q  20000O  O2  12000F  0.5F 2
The monthly salary for an office worker is $4000 and the monthly wage for
a factory worker is $2000.
how many office workers and how many factory workers should the
company hire if they have allotted $2,600,000 a month to wages?
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
We know that:
We calculate the
marginal products:
Substituting:
MPO MPF

PO
PF
and that
Q
 20000  2O
O
Q
MPF 
 12000  F
F
MPO 
20000  2O 12000  F

4000
2000
F  O  2000
Therefore:
And as such:
4000O  2000F  26000000
4000O  2000( O  2000 )  26000000
O
2600000 2000
6000
 4000
F  O  2000  6000
MGMT 7730 - © 2011 Houman Younessi
PO  4000
PF  2000
Lecture 4
Example:
ABCO has a production equation as follows:
Q  10( LK )
1
2
Q is the output, L is the number of workers and K is the number of machines.
The wages of a worker is $8 per hour and the price of a machine is $2 per hour.
If ABCO produces 80 units of output per hour, how many workers and machines
should it use?
Again:
and
MPL MPK

PL
PK
1
Q
K 2
MPL 
 5( L )
L
1
Q
L 2
MPK 
 5( K )
K
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
so if:
MPL MPK

PL
PK
and therefore:
as Q=80
then:
5K
8L
K
L
1
2
L
K
5( )
5( )

8
2
 52  K  4L
80  10( L( 4L ))
L4
K  4L  16
1
2
MGMT 7730 - © 2011 Houman Younessi
1
2
Lecture 4
Optimum Lot Size
It is often more economical to produce items in lots. This is because doing so
would reduce set up time for equipment and project start-up costs.
For example if it costs S dollars to set up a machine to produce widgets and
each widget manufacture cost is W, then if we produce 100 widgets we will
have a total cost of manufacture per widget of:
(S+100W)/100
If we produce 1000,000 widgets, then the manufacturing cost per unit will be
(S+1000,000W)/1000000
So, it would pay to have as large a run as possible.
However, things are not as easy. There is a force acting against us
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Optimum Lot Size
We have to carry all those widgets in inventory until we need them, which
will cost money. Let us say the cost of carrying a widget in inventory is B.
What should be our production lot size to minimize costs?
We know that total set up cost equals SQ/L where S is the set up cost, Q
is the quantity required, and L is the number of widgets produced in a
given run.
We also know that the cost of holding an item in inventory is B, as such
the average cost of holding inventory is BL/2.
We wish to find the optimum size of L that minimizes the total cost of
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Optimum Lot Size
We need to find a formula for cost.
C  CI  CM
BL SQ

2
L
Therefore:
C
To minimize:
dC B SQ
  2 0
dL 2 L
2SQ
L
B
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Output Elasticity and Scalability
What would happen to output quantity if we doubled all inputs?
A. It would double
B. It would less than double
C. It would more than double
Answer:
IT DEPENDS
MGMT 7730 - © 2011 Houman Younessi
Lecture 4
Output Elasticity and Scalability
It depends on a measure called output elasticity.
Output elasticity is the measure of percent change in output resulting from a
one percent increase in ALL inputs.
Given a quantity Q say of X and Y, that is Q(X,Y), calculate Q’(1.01X+1.01Y)
If :
Q
1
Q'
Q
1
Q'
Q
1
Q'
Then the production has constant return to scale
Then the production has increasing return to scale
Then the production has decreasing return to scale
MGMT 7730 - © 2011 Houman Younessi