Costs of Production

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Costs of Production
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In the previous section, we looked at production.

In this section, we look at the cost of
production determining the optimal level of
output. Well start with an interesting cost
example, then focus on determining the optimal
level of output in general.
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Optimal Level of Law Enforcement Crime
Prevention.
Consider the total cost of crime (CT), including
both the cost of the criminal act itself (CA)
the cost of law enforcement crime prevention
(CP). Wed like to know the optimal level of law
enforcement crime prevention (L) that will
minimize the total cost CT CP CA .
Remember from calculus that to minimize a
function, we take the first derivative and set it
equal to zero. So we will have dCT/dL dCP/dL
dCA/dL 0 or dCP/dL dCA/dL . This means
that the optimal enforcement level is where the
cost of preventing an additional crime is equal
to the cost of an additional criminal act. To
finish solving for the optimal level (L), wed
need to know the specific form of the cost
functions involved.
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Isocost Curve

• The set of combinations of inputs that cost the
  same amount

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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 .

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Graph of an isocost
For example, suppose youre interested in the
outlay amount 10,000. Suppose also that Labor
cost 10 per unit capital cost 100 per
unit. 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.
K
R/PK 100
slope PL/PK 0.1
L
R/PL 1000
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Maximizing output for a given cost level
At points A B, were spending the outlay
associated with this isocost, but were not
producing as much as we can. Were only making
Q1 units of output. We cant produce Q3 or Q4
with this outlay. Those output levels would cost
more. At point E, were producing the most for
the money, where the isocost is tangent to an
isoquant.
isoquants
K
A
isocost
E
Q4
Q3
Q2
B
Q1
L
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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 .
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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.)
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Minimizing cost for a given output level
At points A B, were producing the desired
quantity, but were not using the cheapest
combination of inputs, so were spending more
than necessary. We cant produce the desired
output level at cost level C1. We need more
money. At point E, were producing the desired
output at the lowest cost, where the isoquant is
tangent to an isocost.
K
A
E
isoquant
Q1
B
L
isocosts C1 C2 C3
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So whether were 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.

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Short Run Costs of Production
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Total Fixed Cost (TFC)
Total fixed cost is the cost associated with the
fixed input.
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Since TFC is constant, its graph is a horizontal
line.
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Average Fixed Cost (AFC)
AFC TFC/Q AFC is the fixed cost per unit of
output.
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The AFC curve slopes downward gets closer
closer to the horizontal axis.

AFC
Quantity
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Total Variable Cost (TVC)
Total variable cost is the cost associated with
the variable input.
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The TVC curve is upward sloping.
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.
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Average Variable Cost (AVC)
AVC TVC/Q AVC is the variable cost per unit of
output.
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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.
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Average Variable Cost

AVC
Quantity
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Total Cost
TC TFC TVC The TC curve looks like the TVC
curve, but it is shifted up, by the amount of TFC.
Quantity
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Average Total Cost
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.

ATC
AVC
Quantity
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Marginal Cost (MC)
MC is the additional cost associated with an
additional unit of output. MC ?TC/ ?Q
Alternatively, MC dTC/dQ . MC is the first
derivative of the TC curve or the slope of the TC
curve.
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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.
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While MC is U-shaped, it is often drawn so it
extends up higher on the right side.
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Important Graphing Note The MC must intersect
the ATC at its minimum the AVC curve at its
minimum.
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We have a similar graphical interpretation of
ATC to the one we had for AP.
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.
TC
TC
? TC1 ?
0
? Q1 ?
Q
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We also have similar graphical interpretation of
MC to the ones we had for MP.
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.
TC
TC
Q
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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. When the TR curve is above the TC
curve, the firm will have positive economic
profits. 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
TRTC.
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Maximizing Profit
The firm will have the maximum profits where the
vertical distance between TR TC is the largest
( TR is above TC). This is also where MR MC
(which you should recall from Micro Principles is
the profit maximizing condition). That means that
the slope of the TR line equals the slope of the
TC curve. So the TR line will be parallel to a
tangent to the TC line at the point where profits
are maximized.
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Minimum Profit
Notice that the TR line is also parallel to a
tangent to the TC line here. TR TC reaches a
minimum here, not a maximum.
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Example Suppose the price of a product is 10.
The cost of production is TC Q3 21Q2
49Q100.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 49Q100)
• 10Q Q3 21Q2 49Q 100
• Q3 21Q2 39Q 100
• d?/dQ 3Q2 42Q 39.

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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
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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.
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We had? Q3 21Q2 39Q 100 d?/dQ
3Q2 42Q 39
The 2nd derivative is 6Q 42 When Q 1,
6Q 42 36 gt 0 which means that Q 1 is
a minimum. When Q 13, 6Q 42 36 lt
0 which means that Q 13 is a maximum.
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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

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Long Run Costs of Production
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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.

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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
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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
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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
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Cost
LRATC
SRATC1
SRATC5
SRATC2
SRATC4
SRATC3
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
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Cost
LRATC
SRATC1
SRATC5
SRATC2
SRATC4
SRATC3
Quantity of output
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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.
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Cost
LRATC
SRATC1
SRATC5
SRATC2
SRATC4
SRATC3
Quantity of output
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The downward-sloping section of the Long Run ATC
curve reflects Economies of Scale.
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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.

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The upward-sloping section of the Long Run ATC
curve reflects Diseconomies of Scale.
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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
  doesnt 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.

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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.
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Once we have the LR ATC, we can determine the LR
total cost TC.

• Remember that ATC TC/Q.
• So TC (ATC) Q.

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From the LR TC curve, we get the LR MC, either
from

• MC ?TC/?Q
• or MC dTC/dQ

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As in the case of the short run MC ATC, it is
also true for the long run curves that
MC lt ATC when ATC is decreasing, MC gt ATC
when ATC is increasing, MC ATC when ATC is
at its minimum.
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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.
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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 K1
E3
E2
E1
O L1 L2 L3
L
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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. So (50, 250)
will be one point on the LR TC curve.
K
40 30 20
E1
150
100
50
H1
O 25
37 45
L
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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 20
E3
E2
E1
150
100
50
O 25
37 45
L
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So our LR TC curve might look like this
LR TC
LR TC
450 370 250
O 50 100 150
Q
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Weve 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.
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Economies of scope may occur because

• Production of different products use common
  facilities or inputs.
• Example Automobile truck production may
  use the same factory assembly line and raw
  materials.
• Production of one product produces by-products
  that the producer can sell.
• Example A cattle producer raises cattle to
  sell for beef, but can also sell the hides.

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A measure of economies of scope is
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(Q1Q2) 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.
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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 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.
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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 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.
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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 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.
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How do you determine the profit-maximizing output
levels for a multi-product firm?

• Set MR equal to MC for each product.

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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

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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
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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

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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.

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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.
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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.

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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.