Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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?
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Supply: Production
1500
Quantity(widgets)
Labor
1200
900
600
300
3 6 9 12 15
Total Output
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
LaborCapital (# of
Machines)Output
Average Output(Average Product)
(Q/L)
Marginal Output (Marginal Product) (ΔQ/ΔL)
Marginal Output (Marginal Product) (dQ/dL)
0 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
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
1500
Quantity(widgets)
Labor
1200
900
600
300
3 6 9 12 15
Total Output
A
B
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Average and Marginal output (product) curves
1 2 3 4 5 6 7 8 9 10
11
12
13
14
15
16
-10
-30
-50
10
30
50
70
90
110
130
150
170
A
B
Marginal OutPut (Marginal Product)
dQ/dL
Average Output (Average Product)
Q/L
C
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Average output (product)
L
QQ or in general
X
QPo
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.
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Marginal output (product)
dX
dQ
X
QMP
X
0lim
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) .
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Marginal output (product)
X
Q
dX
dQThe difference between and
1500
1200
900
600
300
3 6 9 12 15
Assumes continuous units (e.g. labor in labor-hours 2.73 units is allowed)
Assumes discrete units (e.g. labor in persons n or n+1 but not 1.5n)
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Marginal output (product)
As expected, output is maximized when marginal output (marginal product) is zero (point C).
0
0)(max @
dL
dQ
MPLQ
dL
dQMP
In general Output = Q(L,M,N,X,Y,Z,…)
In our simple case Output=Q(L)
As such:
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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:
X
QXP )(
0)(0)(1
)/()/()/()(2
X
Q
dX
dQ
X
Q
dX
dQ
X
X
dXdXQdXdQX
dX
XQd
dX
XPd
)()(
0
XPXMPX
Q
dX
dQX
Q
dX
dQ
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Optimal Resource UtilizationHow 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
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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):
Y
TRMRPY
However note that :Y
Q
Q
TR
Q
Q
Y
TR
Y
TRMRPY
As such: )( YY MPMRMRP
Therefore: The marginal revenue product of an input equals to that input’s marginal product times the firm’s marginal revenue
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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):
Y
TCMPEY
However note that :Y
Q
Q
TC
Q
Q
Y
TC
Y
TCMPEY
As such: )( YY MPMCMPE
Therefore: The marginal product expenditure of an input equals to that input’s marginal product times the firm’s marginal cost
Similarly:Optimal Resource Utilization
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Optimal Resource UtilizationTo maximize profit (globally): MCMR
or:
Q
TC
Q
TR
Multiplying both sides by:Y
Q
Y
Q
Q
TC
Y
Q
Q
TR
Rearranging:
YY MPEMRP
orQ
Q
Y
TC
Q
Q
Y
TR
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Optimal Resource UtilizationExample:
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.
2398 LLQ
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?
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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: LdL
dQMPL 698
Marginal revenue product for labor is: )( LL MPMRMRP
To maximize, we must have:
16
40)698(20
L
L
MPEMRP LL
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Multi-variable Production Functions
A production function is very rarely a function of only one variable.
For example Q may be a function of: ),,( TMLfQ
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.
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Multi-variable Production FunctionsExample:
Our brewery has the production function:
How much should they spend on transportation if it costs $200 to transport one gallon of beer?
20
200)2(20
)(2
20
T
T
MPEMPMRMRP
T
T
QMP
MR
TTT
T
2412 2510120 TMLQ
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Isoquants: Choosing Combinations of InputsWhat are the possible combinations of various inputs that are capable of
producing a certain quantity of output?
50
Capital
Labor6
40
30
20
10
1 2 3 4 5
100
200
A
B
Capital/Labor Isoquants
C300
7
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Marginal Rate of Substitution
The rate at which one input can be substituted for another in such a way that the output remains constant
2
1
X
X
Change in one input
Change in another input
1
2
1
2
dX
dXMRS
X
XMRS
Rate of change of one
over the other input
At the limit
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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.
€
MRS =−dX2
dX1
−dX2
dX1
=dX2
dX1
×∂Q∂Q
dX2
dX 1
= −(∂Q / ∂X 1)
(∂Q / ∂X 2)= −
MP1
MP2
By definition:
Therefore:
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Isoquants: Choosing Combinations of InputsIsoquants 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
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
50
Capital
6
40
30
20
10
1 2 3 4 5 7
200
300
100
Labor
Economic Region of Production
Economic Region of Production
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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:
KYPXP YX
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: XP
P
P
KY
Y
X
Y
Which is the equation for a straight line called the isocost line of X and Y
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Isocosts: Choosing Combinations of Inputs
Y
X
K/PY
K/PX
Slope = -PX/PY
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Choosing Combinations of InputsWhat point on the isocost line would you pick?
Pick the point that lies on both the isocost curve and the highest isoquant
Y
X
K/PY
K/PX
Q
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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:
We also know that the slope of the isoquant curve (at a given point) is:
Therefore the Q point is where the slope of the isocost curve equals the ratio of the marginal products:
In fact in general when more than two inputs are involved:
Y
X
P
P
Y
X
MP
MP
Y
X
Y
X
MP
MP
P
P
n
n
c
c
b
b
a
a
P
MP
P
MP
P
MP
P
MP ...
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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):
22 5.01200020000 FFOOQ
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?
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
F
F
O
O
P
MP
P
MPWe know that:
FF
QMP
OO
QMP
F
O
12000
220000We calculate the marginal products:
2000
4000
F
O
P
Pand that
2000
2000
12000
4000
220000
OF
FO
26000000)2000(20004000
2600000020004000
OO
FO
Substituting:
Therefore:
And as such:60002000
4000600020002600000
OF
O
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Example:
ABCO has a production equation as follows:
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:
K
K
L
L
P
MP
P
MP
21
)(10 LKQ
and
€
MPL =∂Q∂L
=5(KL)
12
MPK =∂Q
∂K= 5( L
K)
12
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
so if:K
K
L
L
P
MP
P
MP then:
2
)(5
8
)(5 21
21
KL
LK
and therefore: LKLK 42
585
as Q=80
164
4
))4((1080 21
LK
L
LL
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Optimum Lot SizeIt 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
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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 wish to find the optimum size of L that minimizes the total cost of
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.
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
Optimum Lot Size
We need to find a formula for cost.
MI CCC
Therefore:L
SQBLC
2
To minimize:
B
SQL
L
SQB
dL
dC
2
02 2
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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
Lecture 4
MGMT 7730 - © 2011 Houman Younessi
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 :
1'
1'
1'
Q
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