Microeconomics
Claudia Vogel
EUV
Winter Term 2009/2010
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
1 / 36
The Cost of Production
Lecture Outline
Part II Producers, Consumers, and Competitive Markets
7
The Cost of Production
Measuring Cost: Which Costs Matter?
Cost in the Short Run
Cost in the Long Run
Long-Run versus Short-Run Cost Curves
Production with Two Outputs - Economies of Scope
Mathematical Appendix
Summary
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
2 / 36
The Cost of Production
Measuring Cost: Which Costs Matter?
Measuring Cost: Which Costs Matter?
accounting cost: Actual expenses plus depreciation charges for capital
equipment.
economic cost: Cost to a rm utilizing economic resources in production,
including opportunity cost.
opportunity cost: Cost associated with opportunities that are forgone when
a rm's resources are not put to their best alternative use.
sunk cost: Expenditure that has been made and cannot be recovered.
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
3 / 36
Measuring Cost: Which Costs Matter?
Fixed Costs and Variable Costs
total costs (TC or C): Total economic cost of production, consisting of
xed and variable costs.
xed cost (FC): Cost that does not vary with the level of output and that
can be eliminated only by shutting down.
variable cost (VC): Cost that varies as output varies.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
4 / 36
The Cost of Production
Measuring Cost: Which Costs Matter?
Marginal and Average Cost
marginal cost (MC): Increase in cost resulting from the production of one
extra unit of output.
Because xed cost does not change as the rm's level of output changes, marginal
cost is equal to the increase in variable cost or the increase in total cost that
results from an extra unit of output.
We can therefore write marginal cost as
MC
=
4VC
4TC
=
4q
4q
average total costs (ATC): Firm's total cost divided by its level of output.
average xed cost (AFC): Fixed cost divided by the level of output.
average variable cost (AVC): Variable cost divided by the level of output.
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
5 / 36
Measuring Cost: Which Costs Matter?
Example: A Firm's Cost
Rate of
Fixed
Variable
Total
Marginal
Average
Average
Average
Output
Cost
Cost
Cost
Cost
Fixed Cost
Variable Cost
Total Cost
(U. per Y.)
($ per Y.)
($ per Y.)
($ per Y.)
($ per Y.)
($ per U.)
($ per U.)
($ per U.)
(FC)
(VC)
(TC)
(MC)
(AFC)
(AVC)
(ATC)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
0
50
0
50
-
-
-
-
1
50
50
100
50
50
50
100
2
50
78
128
28
25
39
64
3
50
98
148
20
16.7
32.7
49.3
4
50
112
162
14
12.5
28
40.5
5
50
130
180
18
10
26
36
6
50
150
200
20
8.3
25
33.3
7
50
175
225
25
7.1
25
32.1
8
50
204
254
29
6.3
25.5
31.8
9
50
242
292
38
5.6
26.9
32.4
10
50
300
350
58
5
30
35
11
50
385
435
85
4.5
35
39.5
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
6 / 36
The Cost of Production
Cost in the Short Run
The Determinants of Short-Run Cost
The change in variable cost is the per-unit cost of the extra labor w times the
amount of extra labor needed to produce the extra output 4L. Because
4VC = w 4L, it follows that
MC
=
4VC
w 4L
=
4q
4q
4L
1
The extra labor needed to obtain an extra unit of output is 4
q = MPL . As a
result,
MC
=
w
MPL
Diminishing Marginal Returns and Marginal Cost
Diminishing marginal returns means that the marginal product of labor declines as
the quantity of labor employed increases.
As a result, when there are diminishing marginal returns, marginal cost will
increase as output increases.
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
7 / 36
Winter Term 2009/2010
8 / 36
Cost in the Short Run
The Shapes of the Cost Curves
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Cost in the Short Run
Example: The Short-Run Cost of Aluminium Smelting 1/2
Production Costs for Aluminium Smelting ($/ton)
(based on an output of 600 tons/day
Per-ton costs that are constant
for all output levels
Electricity
Alumina
Other Raw Materials
Plant Power and fuel
Subtotal
Per-ton costs that increase when
output exceeds 600 tons/day
Labor
Maintenance
Freight
Subtotal
Total per-ton production costs
Claudia Vogel (EUV)
Output ≤600
tons/day
$316
369
125
10
$820
Output > 600
tons/day
$316
369
125
10
$820
$150
120
50
$320
$1140
$225
180
75
$480
$1300
Microeconomics
The Cost of Production
Winter Term 2009/2010
9 / 36
Cost in the Short Run
Example: The Short-Run Cost of Aluminium Smelting 2/2
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
10 / 36
The Cost of Production
Cost in the Long Run
User Cost of Capital
user cost of capital: Annual cost of owning and using capital asset, equal to
economic depreciation plus forgone interest.
User Cost of Capital = Economic Depreciation + (Interest Rate)(Value of Capital)
We can also express the user cost of capital (i.e. the price of capital) a a rate per
dollar of capital:
r = Depreciation rate + Interest rate
rental rate: Cost per year of renting one unit of capital.
If the capital market is competitive, the rental rate should be equal to the user
cost r. Why? Firms that own capital expect to earn a competitive return when
they rent it. This competitive return is the user cost of capital.
Capital that is purchased can be treated as though it were rented at a rental rate
equal to the user cost of capital.
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
11 / 36
Cost in the Long Run
The Isocost Line
isocost line: Graph showing all possible combinations of labor and capital
that can be purchased for a given total cost.
Total cost C of producing any particular
output:
C
= wL + rK
Rewrite the total cost equation:
K
=
C
r
−
w
r
L
K
w
The isocost line has a slope of 4
4L = − r ,
which is the ratio of the wage rate to the
rental capital.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
12 / 36
The Cost of Production
Cost in the Long Run
Choosing Inputs
MRTS
=−
4K
=
4L
MPL
MPK
When a rm minimizes the cost of producing a particular output, the following
condition holds:
MPL
MPK
=
w
r
We can rewrite this condition as follows:
MPL
w
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
=
MPK
r
Winter Term 2009/2010
13 / 36
Cost in the Long Run
Example: The Eect of Euent Fees on Input Choices
When a rm is not charged for dumping
its wastewater in a river, it chooses to produce a given output using 10000 gallons
of wastewater and 2000 machine-hours of
capital at A.
However, an euent fee raises the cost of
wastewater, shifts the isocost curve from
FC to DE, and causes the rm to produce
at B - a process that results in much less
euent.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
14 / 36
The Cost of Production
Cost in the Long Run
Cost Minimization with Varying Output Levels
expansion path: Curve passing through points of tangency between a rm's
isocost lines and its isoquants.
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
15 / 36
Long-Run versus Short-Run Cost Curves
The Inexibility of Short-Run Production
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
16 / 36
The Cost of Production
Long-Run versus Short-Run Cost Curves
Long-Run Average Cost
long-run average cost curve
(LAC): Curve relating average cost
of production to output when all
inputs, including capital, are
variable.
short-run average cost curve
(SAC): Curve relating average cost
of production to output when level
of capital is xed.
long-run marginal cost curve
(LMC): Curve showing the change
in long-run total cost as output is
increased incrementally by 1 unit.
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
17 / 36
Long-Run versus Short-Run Cost Curves
Economies and Diseconomies of Scale 1/3
As output increases, the rm's average cost of producing that output is likely to
decline, at least to a point.
This can happen for the following reasons:
1 If the rm operates on a larger scale, workers can specialize in the activities
at which they are most productive.
2 Scale can provide exibility. By varying the combination of inputs utilized to
produce the rm's output, managers can organize the production process
more eectively.
3 The rm may be able to acquire some production inputs at lower cost
because it is buying them in large quantities and can therefore negotiate
better prices. The mix of inputs might change with the scale of the rm's
operation if managers take advantage of lower-cost inputs.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
18 / 36
The Cost of Production
Long-Run versus Short-Run Cost Curves
Economies and Diseconomies of Scale 2/3
At some point, however, it is likely that the average cost of production will begin
to increase with output.
There are three reasons for this shift:
1 At least in the short run, factory space and machinery may make it more
dicult for workers to do their jobs eectively.
2 Managing larger rms may become more complex and inecient as the
number of tasks increases.
3 The advantages of buying in bulk may have disappeared once certain
quantities are reached. At some point, available supplies of key inputs may
be limited, pushing their costs up.
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
19 / 36
Long-Run versus Short-Run Cost Curves
Economies and Diseconomies of Scale 3/3
economies of scale: Situation in which output can be doubled for less than
a doubling of cost.
diseconomies of scale: Situation in which a doubling of output requires
more than a doubling of cost.
: Output more than doubles when
the quantities of all inputs are doubled.
Increasing Returns to Scale
Economies of Scale
doubling of cost.
Claudia Vogel (EUV)
: A doubling of output requires less than a
Microeconomics
Winter Term 2009/2010
20 / 36
The Cost of Production
Long-Run versus Short-Run Cost Curves
Cost-Output Elasticity
Economies of scale are often measured in terms of a cost-output elasticity, EC .
is the percentage change in cost of production resulting from a 1-percent
increase in output:
EC
EC
=
4C /C
4q / q
To see how EC relates to our traditional measures of cost, rewrite the equation as
follows:
EC
Claudia Vogel (EUV)
=
4C /4q
=
C /q
Microeconomics
The Cost of Production
MC
AC
Winter Term 2009/2010
21 / 36
Long-Run versus Short-Run Cost Curves
The Relationship between Short-Run and Long-Run Cost
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
22 / 36
The Cost of Production
Production with Two Outputs - Economies of Scope
Product Transformation Curve
product transformation curve: Curve showing the various combinations of
two dierent outputs (products) that can be produced with a given set of
inputs.
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
23 / 36
Production with Two Outputs - Economies of Scope
Economies and Diseconomies of Scope
economies of scope: Situation in which joint output of a single rm is
greater than output that could be achieved by two dierent rms when each
produces a single product.
diseconomies of scope: Situation in which joint output of a single rm is
less than could be achieved by separate rms when each produces a single
product.
To measure the degree to which there are economies of scope, we should ask what
percentage of the cost of production is saved when two (or more) products are
produced jointly rather than individually.
SC
=
C (q1 ) + C (q2 ) − C (q1 , q2 )
C (q1 , q2 )
degree of economies of scope (SC): Percentage of cost savings resulting
when two or more products are produced jointly rather than individually.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
24 / 36
The Cost of Production
Mathematical Appendix
Cost Minimization
Firm's Minimization problem: C = wL + rK → min
Firm's constraint: F (K , L) = q0
Solving by Lagrange Multiplier Method:
The Lagrangian:
φ (L, K , λ) = wL + rK − λ (F (K , L) − q0 )
Derivatives of the Lagrangian:
∂φ(L,K ,λ)
= r − λ ∂ F∂(KK,L) = r − λMPK (K , L) = 0
∂K
∂φ(L,K ,λ)
= w − λ ∂ F ∂(KL,L) = w − λMPL (K , L) = 0
∂L
∂φ(L,K ,λ)
= F (K , L) − q = 0
0
∂λ
Solving the Resulting Equations:
MPK (K ,L) = MPL (K ,L)
r
w
r = λMPK (K , L) = 0
→
λ = MPKr(K ,L)
r = λMPL (K , L) = 0
→
λ = MPLw(K ,L)
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
25 / 36
Mathematical Appendix
Output Maximization
Firm's Maximization problem: F (K , L) → max
Firm's constraint: wL + rK = C0
Solving by Lagrange Multiplier Method:
The Lagrangian:
φ (K , L, λ) = F (K , L) − µ (wL + rK − C0 )
Derivatives of the Lagrangian:
∂φ(L,K ,λ)
= MPK (K , L) − µr = 0
∂K
∂φ(L,K ,λ)
= MPL (K , L) − µw = 0
∂L
∂φ(L,K ,λ)
= wL + rK − C = 0
∂λ
→
→
µ = MPKr(K ,L)
µ = MPLw(K ,L)
0
Solving the Resulting Equations:
MPK (K ,L) = MPL (K ,L)
r
w
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
26 / 36
The Cost of Production
Mathematical Appendix
The Cobb-Douglas-Function 1/2
Cobb-Douglas-Function: F (K , L) = AK α Lβ
Firm's Minimization problem: C = wL + rK → min
Firm's constraint: F (K , L) = AK α Lβ = q0
Solving by Lagrange Multiplier Method:
A, α, β
> 0;
α, β < 1
The Lagrangian:
φ (L, K , λ) = wL + rK − λ
AK α Lβ − q0
Derivatives of the Lagrangian:
∂φ(L,K ,λ)
α β−1
=
r
−
λ
β
AK
L
=0
∂K
∂φ(L,K ,λ)
= r − λ αAK α−1 Lβ = 0
∂L
∂φ(L,K ,λ)
= AK α Lβ − q0 = 0
∂λ
→
λ = β AK αwLβ−1
Solving the Resulting Equations:
I in II: L = wr βα K
→
expansion path
β
1
in III: K = qA0 α+β wr βα α+β
1
α
in L: L = qA0 α+β wr βα α+β
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
27 / 36
Mathematical Appendix
The Cobb-Douglas-Function 2/2
Cost-Function:
C
C
C
= wL + rK
β
α
α+β
α+β
1 1 q α+β
q α+β
rβ
rβ
0
0
=w
+r
A
wα
A
wα
β
α
= w α+β + r α+β

β
α 
α+β
− α+β
1
q α+β
α
α
0


+
β
β
A
constant returns to scale: α + β = 1
C
Claudia Vogel (EUV)
= wβ
" −α # β
α
α
q0
+ rα
+
β
β
A
Microeconomics
Winter Term 2009/2010
28 / 36
The Cost of Production
Summary
Summary 1/3
Managers, investors, and economists must take into account the opportunity
cost associated with the use of a rm's resources: the cost associated with
the opportunities forgone when the rm uses its resources in its next best
alternative.
A sunk cost is an expenditure that has been made and cannot be recovered.
After it has been incurred, it should be ignored when making future economic
decisions.
In the short run, one or more of the rm's inputs are xed. Total cost can be
divided into xed cost and variable cost. A rm's marginal cost is the
additional variable cost associated with each additional unit of output. The
average variable cost is the total variable cost divided by the number of units
of output.
Claudia Vogel (EUV)
Microeconomics
The Cost of Production
Winter Term 2009/2010
29 / 36
Summary
Summary 2/3
In the short run, when not all inputs are variable, the presence of diminishing
returns determines the shape of the cost curves. In particular, there is an
inverse relationship between the marginal product of a single variable input
and the marginal cost of production. The average variable cost and average
total cost curves are U-shaped. The short-run marginal cost curve increases
beyond a certain point, and cuts both average cost curves from below at
their minimum points.
In the long-run, all inputs to the production process are variable. As a result,
the choice of inputs depends both on the relative costs of the factors of
production and on the extent to which the rm can substitute among inputs
in its production process. The cost-minimizing input choice is made by
nding the point of tangency between the isoquant representing the level of
desired output and an isocost line.
The rm's expansion path shows how its cost-minimizing input choices vary
as the scale or output of its operation increases. As a result, the expnasion
path provides useful information relevant for long-run planning decisions.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
30 / 36
The Cost of Production
Summary
Summary 3/3
A rm enjoys economies of scale when it can double its output at less than
twice the cost. Correspondingly, there are diseconomies of scale when a
doubling of output requires more than twice the cost. Scale economies and
diseconomies apply even when input proportions are variable; returns to scale
applies only when input proportions are xed.
When a rm produces two (or more) outputs, it is important to note whether
there are economies of scope in production. Economies of scope arise when
the rm can produce any combination of the two outputs more cheaply than
could two independent rms that each produced a single product. The degree
of economies of scope is measured by the percentage reduction in cost when
one rm produces two products relative to the cost of producing them
individually.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
31 / 36
Exerxises 6
Problem 1
Please explain whether the following statements are true or false.
1
If the owner of a business pays himself no salary, then the accounting cost is
zero, but the economic cost is positive.
2
A rm that has positive accounting prot does not necessarily have positive
economic prot.
3
If a rm hires a currently unemployed worker, the opportunity cost of
utilitizing the worker's services is zero.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
32 / 36
Exerxises 6
Problem 2
1
Assume that the marginal cost of production is increasing. Can you determine
whether the average variable cost is increasing or decreasing? Explain.
2
Assume that the marginal cost of production is greater than the average
variable cost. Can you determine whether the average variable cost is
increasing or decreasing? Explain.
3
Distinguish between economies of scale and economies of scope. Why can
one be present without the other?
4
What is the dierence between economies of scale and returns to scale?
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
33 / 36
Exerxises 6
Problem 3
1
Is the rm's expansion path always a straight line?
2
Suppose the economy takes a downturn, and that labor costs fall by 50
percent and are expected to stay at that level for a long time. Show
graphically how this change in the relative price of labor and capital aects
the rm's expansion path.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
34 / 36
Exerxises 6
Problem 4
1
1
Suppose that a rm's production function is q = 10L 2 K 2 . The cost of a unit of
labor is $20 and the cost of a unit of capital is $80.
1 The rm is currently producing 100 units of output and has determined that
the cost-minimizing optimal quantities of labor and capital are 20 and 5,
respectively. Graphically illustrate this using isoquants and isocost lines.
2
The rm now wants to increase output to 140 units. If capital is xed in the
short run, how much labor will the rm need? Illustrate this point graphically
and nd the rm's new total cost.
3
Graphically identify the optimal cost-minimizing level of capital and labor in
the long run if the rm wants to produce 140 units.
4
If the marginal rate of technical substitution is KL , nd the optimal level of
capital and labor required to produce the 140 units of output.
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
35 / 36
Exerxises 6
Problem 5
Suppose the process of producing lightweight parkas by Polly's Parkas is described
by the function q = 10K 0.8 (L − 40)0.2 where q is the number of parkas produced,
K the number of computerized stitching-machine hours, and L the number of
person-hours of labor. In addition to capital and labor, $10 worth of raw material
is used in the production of each parka.
1 By minimizing cost subject to the production function, derive the
cost-minimizing demands for K and L as a function of output(q), wage rates
(w), and rental rates on machines (r). Use these results to derive the total
cost function: that is, costs as a function of q, r, w, and the constant $10 per
unit material cost.
2 This process requires skilled workers, who earn $32 per hour. The rental rate
on the machines used in the process is $64 per hour. At these factor prices,
what are total costs as a function of q? Does this technology exhibit
decreasing, constant, or increasing returns to scale?
3 Polly's Parkas plans to produce 2000 parkas per week. At the factor prices
given above, how many workers should the rm hire (at 40 hours per week)
and how many machines should it rent (at 40 machine-hours per week)?
What are the marginal and average costs at this level of production?
Claudia Vogel (EUV)
Microeconomics
Winter Term 2009/2010
36 / 36