Transcript sam and max

Production. Costs
Marginal cost
While dealing with discrete data,
If cost is given as a function of Q, then
For example:
TC = 10,000 + 200 Q + 1.5 Q2
MC = ?
TC VC
MC 

Q
Q
d (TC )
MC 
dQ
Problem 2 on p.194.
“Diminishing 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 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
Amount of input used
Calculating the marginal product (of capital)
for the data in Problem 2:
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
Calculating the marginal product (of capital)
for the data in Problem 2:
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
Calculating the marginal product (of capital)
for the data in Problem 2:
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
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 =
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 …
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 =
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)
Profit is believed to be the ultimate goal of any firm.
In Problem 2, what number of units of capital
maximizes the firm’s profit?
PK = $75
PL = $15
POUTPUT = $2
The aggregate approach:
K
0
1
2
3
4
5
6
L
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
VC
FC
Profit
Profit is believed to be the ultimate goal of any firm.
In Problem 2, what number of units of capital
maximizes the firm’s profit?
PK = $75
PL = $15
POUTPUT = $2
The aggregate approach:
K
0
1
2
3
4
5
6
L
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
VC
0
75
150
225
300
375
450
FC
Profit
Profit is believed to be the ultimate goal of any firm.
In Problem 2, what number of units of capital
maximizes the firm’s profit?
PK = $75
PL = $15
POUTPUT = $2
The aggregate approach:
K
0
1
2
3
4
5
6
L
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
VC
0
75
150
225
300
375
450
FC
300
300
300
300
300
300
300
Profit
Profit is believed to be the ultimate goal of any firm.
In Problem 2, what number of units of capital
maximizes the firm’s profit?
PK = $75
PL = $15
POUTPUT = $2
The aggregate approach:
K
0
1
2
3
4
5
6
L
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
VC
0
75
150
225
300
375
450
FC
300
300
300
300
300
300
300
Profit
- 300
- 225
- 150
75
200
225
200
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
and stopping right before the unit for which
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 MR > MC
and stopping right before the unit for which MR < MC
In our case, 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
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 MR > MC
and stopping right before the unit for which MR < MC
In our case, 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
The marginal approach:
To find the profit maximizing amount of input (part d), we need
to compare the marginal benefit from a change to the
marginal cost of than change.
More specifically, we compare VMPK, the value of marginal
product of capital, to the price 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
r
Back to problem 2, p.194.
To find the profit maximizing amount of input (part d), we will
once again use the marginal approach, which compares the
marginal benefit from a change to the marginal cost of than
change.
More specifically, we compare VMPK, the value of marginal
product of capital, to the price 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
Back to problem 2, p.194.
To find the profit maximizing amount of input (part d), we will
once again use the marginal approach, which compares the
marginal benefit from a change to the marginal cost of than
change.
More specifically, we compare VMPK, the value of marginal
product of capital, to the price 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
Back to problem 2, p.194.
To find the profit maximizing amount of input (part d), we will
once again use the marginal approach, which compares the
marginal benefit from a change to the marginal cost of than
change.
More specifically, we compare VMPK, the value of marginal
product of capital, to the price 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
What if the cost of labor (the fixed cost in this case)
changes?
How does the profit maximization point change?
K
0
1
2
3
4
5
6
L
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
VC
0
75
150
225
300
375
450
FC
300
300
300
300
300
300
300
Profit
- 300
- 225
- 150
75
200
225
200
What if the cost of labor (the fixed cost in this case)
changes?
How does the profit maximization point change?
K
0
1
2
3
4
5
6
L
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
VC
0
75
150
225
300
375
450
300
300
300
300
300
300
300
FC
600
600
600
600
600
600
600
Profit
- 300
- 225
- 150
75
200
225
200
What if the cost of labor (the fixed cost in this case)
changes?
How does the profit maximization point change?
K
0
1
2
3
4
5
6
L
20
20
20
20
20
20
20
Q
0
50
150
300
400
450
475
VC
0
75
150
225
300
375
450
300
300
300
300
300
300
300
FC
600
600
600
600
600
600
600
- 300
- 225
- 150
75
200
225
200
Profit
- 600
- 525
- 450
- 225
- 100
- 75
- 100
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.)
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.
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!
Surprised?
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
(K, L, Q) 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:
Sam 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:
Sam 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
Cost of ‘input’
Think of the following analogy:
Sam 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’
Bottled Frappucino, 9.5oz
80
$2.00
Coca-Cola, 12oz
40
$0.80
Mountain Dew, 12oz
60
$1.00
Think of the following analogy:
Sam 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’
Bottled Frappucino, 9.5oz
80
$2.00
Coca-Cola, 12oz
40
$0.80
Mountain Dew, 12oz
60
$1.00
Mg/$
Think of the following analogy:
Sam 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
$1.00
60
Think of the following analogy:
Sam needs his 240 mg of caffeine a day or he will fall asleep
while driving, and something bad will happen.
He can get his caffeine fix from several options listed below:
Option
Caffeine, mg
Cost of ‘input’
Mg/$
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 Sam’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 will make the firm
better off regardless of the demand it is facing!
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.
Profit maximization in
different market structures
In the Elasticity topic, we discussed a firm facing
downward-sloping demand – if a company wants to
attract more customers, it has to lower its price.
Tonight, we considered examples where the price the
firm could get for each unit of out put did not depend
on the number of units produced.
Which way is right?
Which way is more realistic therefore more relevant?
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
P
nothing about the price.
Q
This is the story of any small-size firm that cannot
differentiate itself from the others.
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 =
1. Aggregate:
Profit = TR – TC = 160 Q – (100 + 40 Q + 5 Q2)
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
What if the market is NOT perfectly competitive?
(This happens if some or all of the attributes of perfect
competition are not present. For example:
- The firm in question is large (takes up a large portion of
the market);
- The firm produces a good that consumers perceive as
different from the others;
- Searching for the best deal is costly for consumers;
Etc.)
For now, we will consider
Monopolistic competition
and
Monopoly
Many small firms
-
One LARGE firm
Fairly easy entry and exit
-
Entry is very costly
or impossible
In both markets, a firm can vary its price to some extent
As a firm in either of these markets raises its price, the
quantity it is able to sell drops –
Demand curve is downward sloping!
Demand curve of an individual firm:
P
Q
Total Revenue:
TR
Q
Marginal Revenue as a function of quantity:
MR
-
Starts out positive;
Gradually decreases;
Hits zero when TR = MAX;
Goes on into the negative
range.
Q
In fact, when demand is linear, so is MR
MR and demand curves share
the same vertical intercept
but MR decreases twice as fast.
P
Q
MR
The profit maximization story told graphically:
In aggregate terms:
TC
TR
Q
Profit
In marginal terms:
Profit-maximizing
price
Note that for a profit-maxing
firm with market power
price is always higher than
the MC of the last item
produced
MC
P
Profit-maximizing
quantity
MR
The more market
power a firm has, the
greater that difference
(“profit margin”,
Q
“markup”)
Recall also that profit is maximized where MR=MC…
…whereas revenue is maximized where MR=0
Profit-maximizing
price
MC
Revenuemaximizing price
P
Q
Profit-max quantity
MR
Revenue-max quantity
Recall also that profit is maximized where MR=MC…
…whereas revenue is maximized where MR=0
For an imperfectly competitive firm,
profit is always maximized at a smaller
MC
output quantity
Revenue(therefore
at
a
higher
price)
maximizing price
than revenue is maximized.
P
Profit-maximizing
price
Q
Profit-max quantity
MR
Revenue-max quantity
Analytically:
TC = 100 + 40 Q + 5 Q2 ,
And the demand facing the firm is given by
QD = 25 – 0.1 P
What is the profit maximizing quantity AND price?