Production. Costs
Problem 6 on p.194.
Output
FC
0
10,000
VC
TC
AFC
AVC
ATC
---
100
200
200
125
300
133.3
400
150
500
200
600
250
MC
Production. Costs
Problem 6 on p.194.
MC =
Output
FC
VC
TC
AFC
AVC
0
10,000
0
10,000
---
∆TC ∆VC
=
∆Q
∆Q
ATC
MC
---
---
---
100
10,000 10,000
20,000
100
100
200
100
200
10,000 15,000
25,000
50
75
125
50
300
10,000 30,000
40,000
33.3
100
133.3
150
400
10,000 50,000
60,000
25
125
150
200
500
10,000 90,000 100,000
20
180
200
400
600
10,000 140,000 150,000
16.7
233.3
250
500
If cost is given as a function of Q, then
For example:
TC = 10,000 + 200 Q + 150 Q2
MC = 0 + 200 + 300 Q
MC =
d (TC )
dQ
Profit is believed to be the ultimate goal of any firm.
If the production unit described in the problem
above can sell as many units as it wants for P=$360,
what is the best quantity to produce (and sell)?
Output
FC
VC
TC
AFC
AVC
ATC
MC
0
10,000
0
10,000
---
---
---
---
100
10,000 10,000
20,000
100
100
200
100
200
10,000 15,000
25,000
50
75
125
50
300
10,000 30,000
40,000
33.3
100
133.3
150
400
10,000 50,000
60,000
25
125
150
200
500
10,000 90,000 100,000
20
180
200
400
600
10,000 140,000 150,000
16.7
233.3
250
500
Doing it the “aggregate” way,
by actually calculating the profit:
Output
FC
VC
TC
0
10,000
0
10,000
100
10,000
10,000
20,000
200
10,000
15,000
25,000
300
10,000
30,000
40,000
400
10,000
50,000
60,000
500
10,000
90,000
100,000
600
10,000
140,000
150,000
Doing it the “aggregate” way,
by actually calculating the profit:
P=$360
Output
FC
VC
TC
TR
Profit
0
10,000
0
10,000
0
–10,000
100
10,000
10,000
20,000
36,000
16,000
200
10,000
15,000
25,000
72,000
47,000
300
10,000
30,000
40,000
108,000
68,000
400
10,000
50,000
60,000
144,000
84,000
500
10,000
90,000
100,000 180,000
80,000
600
10,000
140,000
150,000 216,000
66,000
Doing it the “aggregate” way,
by actually calculating the profit:
P=$360
Output
FC
VC
TC
TR
Profit
0
10,000
0
10,000
0
–10,000
100
10,000
10,000
20,000
36,000
16,000
200
10,000
15,000
25,000
72,000
47,000
300
10,000
30,000
40,000
108,000
68,000
400
10,000
50,000
60,000
144,000
84,000
500
10,000
90,000
100,000 180,000
80,000
600
10,000
140,000
150,000 216,000
66,000
Alternative: The Marginal Approach
The firm should produce only units that are worth
producing, that is, those for which the selling price
exceeds the cost of making them.
Output
FC
VC
TC
AFC
AVC
ATC
MC
0
10,000
0
10,000
---
---
---
---
100
10,000 10,000
20,000
100
100
200
100
200
10,000 15,000
25,000
50
75
125
50
300
10,000 30,000
40,000
33.3
100
133.3
150
400
10,000 50,000
60,000
25
125
150
200
500
10,000 90,000 100,000
20
180
200
400
600
10,000 140,000 150,000
16.7
233.3
250
500
The marginal approach to decision making
Least profit
Most profit
Costs (–)
Revenue (+)
The cost-benefit approach to decision making
(an “artistic” view)
Most profit
Least profit
Costs (–)
Revenue (+)
Alternative: The Marginal Approach
The firm should produce only units that are worth
producing, that is, those for which the selling price
exceeds the cost of making them.
Output
FC
VC
TC
AFC
AVC
ATC
MC
0
10,000
0
10,000
---
---
---
---
100
10,000 10,000
20,000
100
100
200
100 < 360
200
10,000 15,000
25,000
50
75
125
50
300
10,000 30,000
40,000
33.3
100
133.3
150
400
10,000 50,000
60,000
25
125
150
200
500
10,000 90,000 100,000
20
180
200
400 > 360
600
10,000 140,000 150,000
16.7
233.3
250
500
Principle (Marginal approach to profit maximization):
If data is provided in discrete (tabular) form, then
profit is maximized by producing all the units for
which
The revenue they generate exceeds the costs of
producing them, MR>MC
and stopping right before the unit for which MR<MC
Note that in this problem, price of output stays
constant throughout therefore MR = P
(an extra unit increases TR by the amount it sells for)
If costs are continuous functions of QOUTPUT, then profit
is maximized where MR=MC
What if FC is $100,000 instead of $10,000? How does the
profit maximization point change?
Output
FC
VC
TC
TR
Profit
0
10,000
0
10,000
0
–10,000
100
10,000
10,000
20,000
36,000
16,000
200
10,000
15,000
25,000
72,000
47,000
300
10,000
30,000
40,000
108,000
68,000
400
10,000
50,000
60,000
144,000
84,000
500
10,000
90,000
100,000 180,000
80,000
600
10,000
140,000
150,000 216,000
66,000
What if FC is $100,000 instead of $10,000? How does the
profit maximization point change?
Output
FC
VC
TC
TR
Profit
0
100,000
0
10,000
0
–10,000
100
100,000
10,000
20,000
36,000
16,000
200
100,000
15,000
25,000
72,000
47,000
300
100,000
30,000
40,000
108,000
68,000
400
100,000
50,000
60,000
144,000
84,000
500
100,000
90,000
100,000 180,000
80,000
600
100,000 140,000
150,000 216,000
66,000
What if FC is $100,000 instead of $10,000? How does the
profit maximization point change?
Output
FC
VC
TC
TR
Profit
0
100,000
0
100,000
0
–10,000
100
100,000
10,000
110,000
36,000
16,000
200
100,000
15,000
115,000
72,000
47,000
300
100,000
30,000
130,000 108,000
68,000
400
100,000
50,000
150,000 144,000
84,000
500
100,000
90,000
190,000 180,000
80,000
600
100,000 140,000
240,000 216,000
66,000
What if FC is $100,000 instead of $10,000? How does the
profit maximization point change?
Output
FC
VC
TC
TR
Profit
0
100,000
0
100,000
0
–100,000
100
100,000
10,000
110,000
36,000
–74,000
200
100,000
15,000
115,000
72,000
–43,000
300
100,000
30,000
130,000 108,000 –22,000
400
100,000
50,000
150,000 144,000
500
100,000
90,000
190,000 180,000 –10,000
600
100,000 140,000
240,000 216,000 –24,000
–6,000
What if FC is $100,000 instead of $10,000? How does the
profit maximization point change?
Output
FC
VC
TC
TR
Profit
0
100,000
0
100,000
0
–100,000
100
100,000
10,000
110,000
36,000
–74,000
200
100,000
15,000
115,000
72,000
–43,000
300
100,000
30,000
130,000 108,000 –22,000
400
100,000
50,000
150,000 144,000
500
100,000
90,000
190,000 180,000 –10,000
600
100,000 140,000
240,000 216,000 –24,000
–6,000
Principle:
Fixed cost does not affect the firm’s optimal shortterm output decision and can be ignored while
deciding how much to produce today.
Consistently low profits may induce the firm to close down
eventually (in the long run) but not any sooner than your fixed
inputs become variable
( your building lease expires,
your equipment wears out and new equipment needs to be purchased,
you are facing the decision of whether or not to take out a new loan,
etc.)
Sometimes, it is more convenient to formulate a problem not
through costs as a function of output but through output
(product) as a function of inputs used.
Problem 2 on p.194.
“Diminishing marginal returns” – what are they?
In the short run, every company has some inputs fixed
and some variable. As the variable input is added,
every extra unit of that input increases the total output
by a certain amount; this additional amount is called
“marginal product”.
The term, diminishing marginal returns, refers to the
situation when the marginal product of the variable
input starts to decrease (even though the total output
may still keep going up!)
Total output, or Total Product, TP
Amount of input used
Marginal product, MP
Range of diminishing
returns
The growth in total
product slows down
Marginal product of
input starts
decreasing
Amount of input used
In other words, we know we are in the range of
diminishing returns when the marginal product of the
variable input starts falling, or, the rate of increase in
total output slows down.
(Ex: An extra worker is not as useful as the one before him)
Implications for the marginal cost relationship:
Worker #10 costs $8/hr, makes 10 units. MCunit = $0.80
Worker #11 costs $8/hr, makes 8 units. MCunit = $1
In the range of diminishing returns, MP of input is falling
and MC of output is increasing
Marginal cost, MC
Amount of output
Marginal product, MP
This amount of output
corresponds to
this amount of input
Amount of input used
When MP of input is decreasing, MC of output is
increasing and vice versa.
Therefore the range of diminishing returns can be
identified by looking at either of the two graphs.
(Diminishing marginal returns set in at the max of
the MP graph, or at the min of the MC graph)
Back to problem 2, p.194.
Aggregate approach:
K
0
1
2
3
4
5
6
7
L
20
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
475
VC
0
75
150
225
300
375
450
525
FC
300
300
300
300
300
300
300
300
TC
300
375
450
525
600
675
750
825
TR
0
100
300
600
800
900
950
950
Profit
- 300
- 275
- 150
75
200
225
200
125
The marginal approach compares the benefit from a change
(in this case, a change in the amount of capital) to the cost
of than change.
First, calculate the additional output (marginal product of
capital, MPK):
K
0
1
2
3
4
5
6
L
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
MPK
--50
100
150
100
50
25
Next, convert MP, the benefit expressed in terms of units of
output, into VMP, or “Value of MP”, expressed in dollars...
… and compare it to the price the firm is paying for each unit
of capital, or the “rental rate”, r.
K
0
1
2
3
4
5
6
L
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
MPK
--50
100
150
100
50
25
VMPK
--100
200
300
200
100
50
r
>
>
>
>
>
<
75
75
75
75
75
75
STOP
Why would we ever want to be in the range of
diminishing returns?
Consider the simplest case when the price of output
doesn’t depend on how much we produce.
Until we get to the DMR range, every next worker is
more valuable than the previous one, therefore we
should keep hiring them one after another.
Only after we get to the DMR range and the MP starts
falling, we should consider stopping.
Therefore, the profit maximizing point is always in the
diminishing marginal returns range!
(Surprise?)
Note also that the starting point of the DMR range is
NOT the profit maximizing point!!!
DMR deal exclusively with the input, or the cost, side.
Discussion of profit involves both costs and revenue.
Here is how this works:
-The firm is increasing output by gradually increasing
the use of variable inputs;
- At first, the marginal product of a unit of input may be
increasing – GOOD, keep going!
- Then you enter the range of DMR (MP starts to
decrease) – you should keep going a bit further;
- Finally, MP drops to the level below the cost of that
unit of input. This is an indication that you have
reached the profit-max point.
STOP.
In the problems we did above, the price the firm
could get for each unit of output did not depend on
the number of units produced.
This is typical of a market structure called “perfect
competition”.
Traditionally, economics textbooks distinguish four
types of markets, or of market structures.
They differ in the degree of market power an
individual firm has:
• Perfect competition
the least market power
• Monopolistic competition
• Oligopoly
• Monopoly
the most market power
“Market power” also known as “pricing power” is defined
in the managerial literature as the ability of an individual
firm to vary its price while still remaining profitable or as
the firm’s ability to charge the price above its MC.
Perfect competition
The features of a perfectly competitive market are:
•Large number of competing firms;
•Firms are small relative to the entire market;
•Products different firms make are identical;
•Information on prices is readily available.
As a result, the price is set by the interaction of supply
and demand forces, and an individual firm can do
nothing about the price.
P
P
$1
Q, mln lb
Entire market
Q, thousand lb
Individual firm
This is the story of any small-size firm that cannot
differentiate itself from the others.
(Individual firm’s demand is
perfectly elastic
).
What does a Total Revenue (TR) graph look
like for such a firm, if plotted against quantity
produced/sold?
TR
Every unit sells at the
same price so…
Slope equals price
Q
How about the Marginal Revenue graph?
MR
Every unit sells at the
same price so…
MR = P
Q
The profit maximization story told graphically:
In aggregate terms:
TC
TR
Profit
FC
Q
max capacity
max profit
In marginal terms:
MC
MR
Q
max profit
Doing the same thing mathematically:
TC = 100 + 40 Q + 5 Q2 ,
And the market price is $160,
What is the profit maximizing quantity (remember, price is
determined by the market therefore it is given)?
Just like in the case with tabular data, there are two
approaches.
1. Aggregate:
Profit = TR – TC = 160 Q – (100 + 40 Q + 5 Q2) =
=120 Q – 100 – 5 Q2
A function is maximized when its derivative is zero;
Specifically, when it changes its sign from ( + ) to ( – )
d(Profit)/dQ = 0
120 – 10 Q = 0
Q = 12
2. Marginal (looking for the MR = MC point)
MR = Price = $160
MC = d(TC)/dQ ;
TC = 100 + 40 Q + 5 Q2
MC = 40 + 10 Q
MR = MC
160 = 40 + 10 Q
120 = 10 Q
Q = 12
Cost minimization
(Another important aspect of being efficient.)
Suppose that, contrary to the statement of the last
problem, we ARE ABLE to change not just the amount of
capital but the amount of labor as well.
(Recall the distinction between the long run and the short run.)
Given that extra degree of freedom, can we do better?
(In other words, is there a better way to allocate our
budget to achieve our production goals?)
In order not to get lost in the multiple possible
capital-labor-output combinations, it is useful to have
some of them fixed and focus on the question of
interest.
In our case, we can either:
• Fix the total budget spent on inputs and see if
we can increase the total output;
or,
• Fix the target output and see if we can reduce
the total cost by spending our money differently.
Think of the following analogy:
Gary needs his 240 mg of caffeine a day or he will fall asleep
while driving, and something bad will happen.
He can get his caffeine fix from several options listed below:
Option
Caffeine, mg
Bottled Frappucino, 9.5oz
80
Coca-Cola, 12oz
40
Mountain Dew, 12oz
60
Which one should he choose?
Think of the following analogy:
Gary needs his 240 mg of caffeine a day or he will fall asleep
while driving, and something bad will happen.
He can get his caffeine fix from several options listed below:
Option
Caffeine, mg
Cost of ‘input’
Mg/$
Bottled Frappucino, 9.5oz
80
$2.00
40
Coca-Cola, 12oz
40
$0.80
50
Mountain Dew, 12oz
60
$1.00
60
Next, think of caffeine as Gary’s ‘target output’ (what
he is trying to achieve)
and drinks as his inputs, which can to a certain
extent be substituted for each other.
The same principle holds for any production unit
that is trying to allocate its resources wisely:
In order to achieve the most at the lowest
cost possible, a firm should go with the
option with the highest MPinput/Pinput ratio.
Note that following this principle while producing the
desired target output will always make the firm better off!
The textbook refers to “wage” as the shortcut name for the cost
of each unit of labor;
“rent” – the cost of each unit of capital
If MPL > MPK
wage
rent
then - reduce the amount of capital;
- increase the amount of labor.
If
As you do that,
- MPL will decrease;
- MPC will increase;
- the LHS will get smaller,
- the RHS bigger
MPL MPK
<
wage rent
then - reduce the amount of labor;
- increase the amount of capital.
If
MPL
MPK
=
wage rent
then inputs are used in the right proportion.
No need to change anything.