Transcript IV. Cost of Production.

Costs of Production
In the previous section, we looked at
production.
In this section, we look at the cost of
production & determining the optimal
level of output.
Isocost Curve
The set of combinations of inputs that cost
the same amount
Equation of an isocost
Suppose you have 2 inputs, capital K & labor L.
The price of a unit of capital is PK.
The price of a unit of labor is PL.
Let a particular outlay amount be R.
Then all combinations of K & L such that
PLL+ PKK = R
lie on the isocost curve associated with that outlay.
If we rewrite the equation as
K = R/PK – (PL/PK)L ,
we see that the slope of the isocost is – (PL/PK) &
the vertical intercept is R/PK .
Graph of an isocost
For example, suppose you’re interested in the
outlay amount $10,000. Suppose also that Labor
cost $10 per unit & capital cost $100 per unit.
K
R/PK= 100
Then the slope of the isocost is
– PL/PK = – 10/100 = – 0.1 .
The vertical intercept would be 10,000/100 = 100
& the horizontal intercept is 10,000/10 = 1,000.
slope = – PL/PK = – 0.1
R/PL = 1000
L
Maximizing output for a given cost level
At points A & B, we’re spending the
outlay associated with this isocost, but
we’re not producing as much as we can.
We’re only making Q1 units of output.
isoquants
We can’t produce Q3 or Q4 with this
outlay. Those output levels would cost
more.
K
At point E, we’re producing the most for
the money, where the isocost is tangent
to an isoquant.
A
isocost
E
Q4
Q2
B
Q3
Q1
L
At the tangency of the isocost & isoquant,
the slopes of those curves are equal.
We found previously that
the slope of the isoquant is – MPL/MPK ,
& the slope of the isocost is – PL/PK .
So at the tangency, – MPL/MPK = – PL/PK
or, multiplying by -1, MPL/MPK = PL/PK .
This expression is equivalent to MPL/PL = MPK/PK .
MPL/PL = MPK/PK
This condition means that to get the most output for
your money, you should employ inputs such that
the marginal product per dollar is equal for all
inputs.
(Notice the similarity to the utility maximization
condition that the marginal utility per dollar is
equal for all goods.)
Minimizing cost for a given output level
At points A & B, we’re producing the
desired quantity, but we’re not using the
cheapest combination of inputs, so we’re
spending more than necessary.
K
We can’t produce the desired output level
at cost level C1. We need more money.
At point E, we’re producing the desired
output at the lowest cost, where the
isoquant is tangent to an isocost.
A
E
Q1
B
isoquant
L
isocosts C1
C2
C3
So whether we’re maximizing output for a
given cost level, or minimizing cost for a
given output level, the condition is the same:
MPL/PL = MPK/PK
The marginal product per dollar is equal for
all inputs.
Short Run
Costs of Production
Total Fixed Cost (TFC)
Total fixed cost is the cost associated
with the fixed input.
Since TFC is constant,
its graph is a horizontal line.
$
TFC
Quantity
Average Fixed Cost (AFC)
AFC = TFC/Q
AFC is the fixed cost per unit of output.
The AFC curve slopes downward & gets
closer & closer to the horizontal axis.
$
AFC
Quantity
Total Variable Cost (TVC)
Total variable cost is the cost associated
with the variable input.
The TVC curve is upward sloping.
$
TVC
Quantity
It is often drawn like a
flipped over S, first
getting flatter & flatter,
& then steeper & steeper.
This shape reflects the
increasing & then
decreasing marginal
returns we discussed in
the section on production.
Average Variable Cost (AVC)
AVC = TVC/Q
AVC is the variable cost per unit of output.
We can determine the shape of the AVC curve based
on the shape of the average product curve (AP).
Suppose X is the amount of variable input & PX is its
price.
Then,
AVC = TVC/Q = (PXX)/Q
= PX(X/Q)
= PX [1/(Q/X)]
= PX [1/AP].
So since AP had an inverted U-shape, AVC must have
a U-shape.
Average Variable Cost
$
AVC
Quantity
Total Cost
$
TC
TC = TFC + TVC
The TC curve
looks like the TVC
curve, but it is
shifted up, by the
amount of TFC.
TFC
Quantity
Average Total Cost
$
ATC
AVC
Quantity
Like AVC, ATC is
U-shaped, but it
reaches its minimum
after AVC reaches its
minimum.
This is because
ATC = AVC +AFC &
AFC continues to fall
& pulls down ATC.
Marginal Cost (MC)
MC is the additional cost associated with an
additional unit of output.
MC = ΔTC/ ΔQ
Alternatively, MC = dTC/dQ .
MC is the first derivative of the TC curve or
the slope of the TC curve.
We can determine the shape of the MC curve based
on the shape of the marginal product curve (MP).
Suppose the firm takes the prices of inputs as given.
Then,
MC = TC/Q
= PX X/ Q
= PX [1/(Q/X)]
= PX [1/MP].
So since MP had an inverted U-shape, MC must have
a U-shape.
While MC is U-shaped, it is often drawn so it
extends up higher on the right side.
$
MC
Quantity
Important Graphing Note:
The MC must intersect the ATC at its minimum &
the AVC curve at its minimum.
$
MC
ATC
AVC
Quantity
We have a similar graphical interpretation of
ATC to the one we had for AP.
TC
TC
Since ATC = TC/Q, the ATC
of a particular value of Q1
can be interpreted as the
slope of the line from the
origin to the corresponding
point on the curve.

TC1

0

Q1
→
Q
We also have similar graphical interpretation
of MC to the ones we had for MP.
TC
TC
The continuous MC is the slope
of the total cost curve at a
particular point.
The discrete MC is the slope of
the line segment connecting 2
points on the total cost curve.
Q
Breaking Even
Recall that TR = PQ. If the price of
output is fixed for the firm (as for a
perfectly competitive firm), then TR
is a straight line with slope P.
TC
TR
When the TR curve is above the
TC curve, the firm will have positive
economic profits.
TC
When the TC curve is above the
TR curve, the firm will have
economic losses.
The firm will break even (have zero
economic profits) where TR=TC.
Q
Maximizing Profit
The firm will have the maximum
profits where the vertical distance
between TR & TC is the largest (&
TR is above TC).
TC
TR
This is also where MR = MC (which
you should recall from Micro
Principles is the profit maximizing
condition).
TC
That means that the slope of the
TR line equals the slope of the TC
curve.
Profit-maximizing
output level
Q
So the TR line will be parallel to a
tangent to the TC line at the point
where profits are maximized.
Minimum Profit
TC
Notice that the TR line is also
parallel to a tangent to the TC line
here.
TR
TR – TC reaches a minimum here,
not a maximum.
TC
Profit-minimizing
output level
Q
We’re going to digress a little to review from Calculus how to use
first and second derivatives to determine minima and maxima.
y
Consider a function y = f(x) as shown.
Notice that it has a minimum value at x1.
Notice also that the slope of the function
(which is the same as the slope of the
line tangent to the curve at that point) is
zero. That is, f (x1) = 0.
x1
x
y
f  (x) > 0
f  (x) < 0
x1
x
Just to the left of x1, the curve slopes
downward; it has a negative slope.
To the right of x1, the curve slopes
upward; it has a positive slope.
y
f  (x) < 0
f  (x) > 0
f  (x1) = 0
x1
x
So as we move from left to right in the
vicinity of x1, the slope is going from
negative to zero to positive. It is
increasing.
Recall that if a function is increasing,
its derivative is positive.
In this case, the function itself is the
slope or first derivative.
So its derivative is the second
derivative.
Then, because the first derivative is
increasing, the second derivative must
be positive: f (x1) > 0.
To put all this together:
At a minimum x1 ,
the first derivative f (x1) = 0
and the second derivative f (x1) > 0 .
Consider instead this function.
At x1, we have a maximum.
The derivative f (x1) = 0.
f  (x1) = 0
y
f  (x) > 0
f  (x) < 0
x1
x
Here, as we move from left to right in
the vicinity of x1, the slope is going
from positive to zero to negative. The
slope is decreasing.
If a function is decreasing, its
derivative is negative.
Again here, the function is the slope or
first derivative.
So its derivative is the second
derivative.
Then, because the first derivative is
decreasing, the second derivative must
be negative: f (x1) < 0.
To put all this together:
At a maximum x1 ,
the first derivative f (x1) = 0
and the second derivative f (x1) < 0 .
To summarize our conclusions on first and
second derivatives and maxima and minima:
At a minimum x1,
the first derivative f (x1) = 0
and the second derivative f (x1) > 0 .
At a maximum x1,
the first derivative f (x1) = 0
and the second derivative f (x1) < 0 .
Memory Aid
If the second derivative is positive,
we have two happy twinkly eyes and a
smiling mouth which has a minimum.
If the second derivative is negative,
we have two sad eyes and a sad
mouth which has a maximum.
Let’s return to maximizing profit and see how we use our Calculus in this context.
Example:
Suppose the price of a product is $10.
The cost of production is
TC = Q3 – 21Q2 + 49Q+100.
What is the profit maximizing output level?
We need to determine the profit function , take its 1st
derivative, set that equal to zero, & solve for Q.
 = TR –TC
= PQ – TC
= 10Q – (Q3 – 21Q2 + 49Q+100)
= 10Q – Q3 + 21Q2 – 49Q – 100
= – Q3 + 21Q2 – 39Q – 100
d/dQ = – 3Q2 + 42Q – 39.
Setting the 1st derivative equal to zero we have
– 3Q2 + 42Q – 39 = 0
This equation can be solved either by the
quadratic formula or factoring.
1. Quadratic formula:
 b  b 2  4ac  42  422  4(3)( 39)
Q

2a
2(3)
 42  1764  468  42  1296


6
6
 42  36

6
 7  (6)  1 or 13
– 3Q2 + 42Q – 39 = 0
2. Factoring:
– 3 (Q2 – 14Q +13) = 0
– 3 (Q – 1)(Q – 13) = 0
So either
&
Q -1 = 0 or Q -13 = 0 ,
Q=1
or
Q = 13,
which is what we found by the quadratic formula.
Are these both relative maxima, minima, or one of
each?
We need to look at the 2nd derivative of our profit
function.
We had
 = – Q3 + 21Q2 – 39Q – 100
d/dQ = – 3Q2 + 42Q – 39
The 2nd derivative is – 6Q + 42
To determine whether profit is maximized or
minimized at our values of 1 and 13, we need
to know if the second derivative is positive or
negative at each of those values.
When Q = 1,
– 6Q + 42 = 36 > 0
which means that  is a minimum when Q =1 .
When Q = 13,
– 6Q + 42 = – 36 < 0
which means that  is a maximum when Q =13 .
What are our maximum & minimum
profit values?
 = – Q3 + 21Q2 – 39Q – 100
Our maximum , which is when Q = 13, is:
 = – (13)3 + 21(13)2 – 39(13) – 100 = 745
Our minimum , which is when Q = 1, is:
 = – (1)3 + 21(1)2 – 39(1) – 100 = – 119
Graph of the Profit Function  = – Q3 + 21Q2 – 39Q – 100
Q
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Profit
745
456
233
70
-39
-100
-119
-102
-55
16
105
206
313
420
521
610
681
728
745
726
665
556
393
170
19
-119
20
-480
1000
Profit
800
600
400
200
0
-10
-5
0
5
10
15
20
25
-200
-400
-600
Quantity (Q)
Notice: minimum profit (or greatest loss) of -119 occurs when Q = 1,
and the maximum profit of 745 occurs when Q = 13.
As we approach Q = 1 from the left, the slope of the profit curve goes from negative to zero (at Q = 1) to positive to the right of Q =1.
The slope is increasing and the curve is convex.
As we approach Q = 13 from the left, the slope of the profit curve goes from positive to zero (at Q = 13) to negative to the right of Q =13.
The slope is decreasing and the curve is concave.
Recall that the first derivative f' tells us how the function f changes as our independent variable (X or Q) increases.
If f' > 0 , f is increasing; if f' < 0 , f is decreasing.
Similarly, the second derivative f" tells us how f' is changing as our variable (X or Q) increases.
If f" > 0 , f' is increasing (and f is convex); if f" < 0 , f' is decreasing (and f is concave).
Long Run
Costs of Production
The Long Run ATC Curve
(or the planning curve)
shows the least per unit cost at which any
output can be produced after the firm
has had time to make all appropriate
adjustments in its plant size.
Cost
SRATC1
At a relatively low output level, in the short run, the firm might
have SRATC1 curve as its short run average cost curve.
Quantity of output
Cost
SRATC2
At a slightly higher output level, in the short run, the firm might
have SRATC2 curve as its short run average cost curve.
Quantity of output
Cost
SRATC3
At a still higher output level, in the short run, the firm might have
SRATC3 curve as its short run average cost curve.
Quantity of output
Cost
LRATC
SRATC1
SRATC5
SRATC2
SRATC3
SRATC4
In the long run, the firm can pick any appropriate plant size. At
each output level, the firm picks the plant that has the SRATC
curve with the lowest value.
Quantity of output
Cost
LRATC
SRATC1
SRATC5
SRATC2
SRATC3
SRATC4
So, the LRATC curve is made up of segments of the SRATC curves.
Quantity of output
In many industries, the number of
possible plant sizes is virtually
unlimited.
Then the long-run ATC curve is made up
of points of tangency of the theoretically
unlimited number of short-run ATC
curves.
Then the long run ATC curve is smooth.
Cost
LRATC
SRATC1
SRATC5
SRATC2
SRATC4
SRATC3
Quantity of output
The downward-sloping section of the
Long Run ATC curve reflects
Economies of Scale.
Economies of Scale:
As plant size increases, there are factors which
lead to lower average costs of production.
Labor Specialization: Jobs can be subdivided and
workers performing very specialized tasks can
become very efficient at their jobs.
Managerial Specialization: Management can also
specialize in a larger firm (in areas such as
marketing, personnel, or finance).
Equipment that is technologically efficient but only
effectively utilized with a large volume of
production can be used.
The upward-sloping section of the
Long Run ATC curve reflects
Diseconomies of Scale.
Diseconomies of Scale:
As plant size increases, there are factors which
lead to higher average costs of production.
Expansion of the management hierarchy leads to
problems of communication, coordination, and
bureaucratic red tape, and the possibility that
decisions will fail to mesh. (“The left hand
doesn’t seem to know what the right hand is
doing.”) The result is reduced efficiency.
In large facilities, workers may feel alienated and
may shirk (not work as much as they should).
Then additional supervision may be required
and that adds to costs.
Sometimes there is a segment of the
LR ATC curve which is horizontal.
In that section,
the LR ATC is constant, & there are
Constant Returns to Scale.
Once we have the LR ATC, we can
determine the LR total cost TC.
Remember that ATC = TC/Q.
So TC = (ATC) Q.
From the LR TC curve, we get the LR MC,
either from
MC = ΔTC/ΔQ
or MC = dTC/dQ
As in the case of the short run MC & ATC,
it is also true for the long run curves that
MC < ATC when ATC is decreasing,
MC > ATC when ATC is increasing, &
MC = ATC when ATC is at its minimum.
Furthermore, when the firm has built the
optimal scale of plant for producing a given
level of output,
long run MC & short run marginal cost will
be equal at that output.
That is, the LR MC & SR MC will intersect at
that output.
We can also determine the LR TC curve from the
expansion path.
The expansion path shows how the
quantities of inputs change as output
increases, but the prices of inputs
remain fixed.
K
K3
K2
E2
K1
O
E3
E1
L1
L2
L3
L
In particular, suppose that the price of labor is $10.
The TC of producing output 50 at E1 is the same
as the cost of any of the point on that isocost line.
In particular, at point H1, where only labor is
used, the cost is the price of labor times the
amount of labor or (10)(25) = 250.
K
So (50, 250) will be one point on the LR TC
curve.
40
30
20
E1
150
100
H1 50
O
25
37
45
L
Similarly, the LR TC of output 100 is (10)(37) = 370.
So, (100, 370) is another point on the LR TC curve.
The LR TC of output 150 is (10)(45) = 450
So, (150, 450) is a third point on our LR TC curve.
K
40
30
E3
20
E2
E1
150
100
50
O
25
37
45
L
So our LR TC curve might look like this:
LR TC
LR TC
450
370
250
O
50 100
150
Q
Q
LR TC
50
250
100
370
150
450
We’ve discussed economies & diseconomies
of scale.
When a firm produces more than one
product, it may also experience
economies or diseconomies of scope.
Economies of scope exist when a single firm
producing multiple products jointly can
produce them more cheaply than if each
product was produced by a separate firm.
Economies of scope may occur because
1. Production of different products use
common facilities or inputs.
Example: Automobile & truck production
may use the same factory assembly line
and raw materials.
2. Production of one product produces byproducts that the producer can sell.
Example: A cattle producer raises cattle
to sell for beef, but can also sell the
hides.
A measure of economies of scope is
TC(Q 1 )  TC(Q 2 ) – TC(Q 1  Q 2 )
TC(Q 1  Q 2 )
where TC(Q1) is the total cost of producing Q1
units of product 1 only, TC(Q2) is the total cost
of producing Q2 units of product 2 only, &
TC(Q1+Q2) is the total cost of producing them
jointly.
This measure indicates the savings of joint
production compared to separate production,
as a percentage of joint production.
Example 1: The total cost of producing Q1 units of
product 1 only is 50,000. The total cost of
producing Q2 units of product 2 only is 90,000. The
total cost of producing them jointly is 120,000.
Determine if there are economies or diseconomies
of scope, and measure them.
There are economies of scope, since joint production is
less costly than the sum of the separate productions.
TC(Q 1 )  TC(Q 2 ) – TC(Q 1  Q 2 )
TC(Q 1  Q 2 )
20,000
50,000  90,000  120,000


120,000
120,000
 0.167
Example 2: The total cost of producing Q1 units of
product 1 only is 50,000. The total cost of
producing Q2 units of product 2 only is 90,000. The
total cost of producing them jointly is 150,000.
Determine if there are economies or diseconomies
of scope, and measure them.
There are diseconomies of scope, since joint production
is more costly than the sum of the separate productions.
TC(Q 1 )  TC(Q 2 ) – TC(Q 1  Q 2 )
TC(Q 1  Q 2 )
 10,000
50,000  90,000  150,000


150,000
150,000
 0.067
Example 3: The total cost of producing Q1 units of
product 1 only is 50,000. The total cost of
producing Q2 units of product 2 only is 90,000. The
total cost of producing them jointly is 140,000.
Determine if there are economies or diseconomies
of scope, and measure them.
There are neither economies nor diseconomies of scope,
since joint production costs the same amount as the sum
of the separate productions.
TC(Q 1 )  TC(Q 2 ) – TC(Q 1  Q 2 )
TC(Q 1  Q 2 )
0
50,000  90,000  140,000


140,000
140,000
0
How do you determine the profit-maximizing
output levels for a multi-product firm?
Set MR equal to MC for each product.
Two-Product Firm Example
A firm produces Q1 units of item 1 & Q2 units of item 2.
TC = 30 Q1 + 30 Q2 – 4 Q1 Q2
MC1 = dTC/dQ1 = 30 – 4Q2
MC2 = dTC/dQ2 = 30 – 4Q1
Demand for product 1: P1 = 26 – 2Q1
MR1 = dTR1/dQ1 = d(P1Q1)/dQ1
= d(26Q1 – 2Q12)/dQ1
= 26 – 4Q1
Demand for product 2: P2 = 42 – 4Q2
MR2 = dTR2/dQ2 = d(P2Q2)/dQ2
= d(42Q2 – 4Q22)/dQ2
= 42 – 8Q2
Equate
MR1 = 26 – 4Q1 to MC1 = 30 – 4Q2
& MR2 = 42 – 8Q2 to MC2 = 30 – 4Q1 .
26 – 4Q1 = 30 – 4Q2
4Q2 – 4Q1 = 4
Q2 – Q1 = 1
Q2 – 1 = Q1
42 – 8Q2 = 30 – 4Q1
12 = 8Q2 – 4Q1
3 = 2Q2 – Q1
Q1 = 2Q2 – 3
Setting the Q1 expressions equal to each other,
Q2 – 1 = 2Q2 – 3
2 = Q2
Q1 = Q2 – 1 = 2 – 1 = 1
So the profit-maximizing output levels are
Q1 = 1 and Q2 = 2
From the demand functions, the prices are
P1 = 26 – 2Q1 = 26 – 2(1) = 24, and
P2 = 42 – 4Q2 = 42 – 4(2) = 34
TR = TR1 + TR2 = P1Q1 + P2Q2
= 24(1) + 34(2) = 24 + 68 = 92
TC = 30 Q1 + 30 Q2 – 4 Q1 Q2
= 30(1) + 30(2) – 4(1)(2) = 82
 = TR – TC = 92 – 82 = 10
You probably recall from Microeconomic
Principles that accounting profit and
economic profit differ.
The difference results from the fact that the
accountant only includes explicit costs in
TC, while the economist includes both
explicit & implicit costs.
Implicit costs do not leave a paper trail.
They are opportunity costs such as the
foregone earnings of the owner, and
foregone interest on money invested in the
firm.
Because of the differences in the cost
definitions, zero accounting profit & zero
economic profit mean different things.
Zero accounting profit means that revenue is
just sufficient to cover explicit costs.
Zero economic profit means that a business
is doing no better or worse than the typical
business.
It is making a normal accounting profit.
Firms may have objectives in addition to
profit-maximization. These may include
• maintaining or increasing market share,
• achieving better social conditions in the
community,
• protecting the ecological environment, &
• establishing an image as a good employer
and a valuable part of the community.
Often these additional goals contribute
to long term profit maximization.
For example, a better image makes it
possible to attract more productive
employees and more customers.