Transcript Cost - MSUMainEcon160

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Economists consider both explicit costs and
implicit costs.
Explicit costs are a firm’s direct, out-ofpocket payments for inputs to its production
process during a given time period such as a
year.
These costs include production worker’s
wages, manager’s salaries, and payments for
materials.
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However, firms use inputs that may not have
an explicit price.
These implicit costs include the value of the
working time of the firm’s owner and the
value of other resources used but not
purchased in a given period.
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The economic costs or opportunity cost is the
value of the best alternative use of a
resource.
The economic or opportunity cost includes
both explicit and implicit costs.
If a firm purchases and uses an input
immediately, that input’s opportunity cost is
the amount the firm pays for it.
If the firm uses an input from its inventory,
the firm’s opportunity cost is not necessarily
the price it paid for the input years ago.
Rather, the opportunity cost is what the firm
could buy or sell that input for today.
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The classic example of an implicit
opportunity cost is captured in the phrase
“There’s no such thing as a free lunch.”
Suppose that your friend offer to take you to
lunch tomorrow.
You know that they’ll pay for the meal, but
you also know that this lunch will not really
be free for you.
Your opportunity cost for the lunch is the
best alternative use of your time.
This might be studying, working at a job or
watching TV.
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If you start your own firm, you should be very
concerned about opportunity costs.
Suppose that your explicit cost is P40,000,
including the rent for you work space, the
cost of materials, and the wage payments to
your employees.
Because you do not pay yourself a salary –
instead, you keep any profit at the year’s end
– the explicit cost does not include the value
of your time.
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Your firm’s full economic cost is the sum of
of the explicit cost plus the opportunity value
of your time.
If the highest wage you could have earned
working for some other firm is P25,000, your
full economic cost is P65,000.
If your annual revenue is P60,000 after you
pay your explicit cost of P40,000, you keep
P20,000 at the end of the year.
The opportunity cost of your time P25,000,
exceeds P20,000, so you can earn more
working for someone else.
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Determining the cost of capital, such as land
or equipment, requires special
considerations.
Capital is a durable good: a good that is
usable for years.
Two problems arise in measuring the cost of
capital.
1. Allocating Capital Costs over Time
2. Actual and Historical Costs
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Capital may be rented or purchased.
For example, a firm may rent a truck for P200
a month or buy it outright for P18,000.
If the firm rents the truck, the rental payment
is the relevant opportunity cost.
If the firm buys the truck, the firm may
expense the cost by recording the full
P18,000 when the purchase is made, or may
amortize the cost by spreading the P18,000
over the life of the truck.
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An economists amortizes the cost of the
truck on the basis of its opportunity cost at
each moment of time, which is the amount
that the firm could charge others to rent the
truck.
Regardless of whether the firm buys or rents
the truck, an economist views the opportunity
cost of this capital good as a rent per time
period: the amount the firm will receive if it
rents its truck to others at the going rental
rate.
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A piece of capital may be worth much more
or much less today than when it was
purchased.
To maximize profit, a firm must properly
measure the cost of a piece of capital – its
current opportunity cost of the capital good –
and not what the firm paid for it – its
historical cost.
Suppose that a firm paid P30,000 for a piece
of land that it can resell for only P20,000.
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Also suppose that is uses the land itself and
that the current value of the land to the firm
is only P19,000.
Should the firm use the land or sell it?
The firm should ignore how much it paid for
the land in making its decision.
Because the value of the land to the firm,
P19,000, is less than the opportunity cost of
the land, P20,000, the firm can make more
money by selling the land.
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The firm’s current opportunity cost of capital
may be less than what it paid if the firm
cannot resell the capital.
A firm that bought a specialized piece of
equipment that has no alternative use cannot
resell the equipment.
Because the equipment has no alternative
use, the historical cost of buying that capital
is a sunk cost: an expenditure that cannot be
recovered.
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Because this equipment has no alternative
use, the current or opportunity cost of the
capital is zero.
In short, when determining the rental value of
capital, economists use the opportunity value
and ignore the historical price.
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A firm’s cost rises as the firm increases its
output.
A firm cannot vary some of its input, such as
capital, in the short run.
As a result, it is usually more costly for a firm
to increase output in the short run than in the
long run when all inputs can be varied.
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To produce a given level of output in the
short run, a firm incurs costs for both its
fixed and variable inputs.
A firm’s fixed cost (F) is its production
expense that does not vary with output.
The fixed cost includes the cost of inputs that
the firm cannot practically adjust in the short
run, such as land, a plant, large machines,
and other capital good.
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A firm’s variable cost (VC) is the production
expense that changes with the quantity of
output produced.
The variable cost is the cost of the variable
inputs – the inputs the firm can adjust to alter
its output level, such as labor and materials.
A firm’s cost (or total cost, C) is the sum of a
firm’s variable cost and fixed cost:
C VC  F
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Because variable cost changes with the level
of output, total cost also varies with the level
of output.
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A firm’s marginal cost (MC) is the amount by
which a firm’s cost changes if the firm
produces one more unit of output. The
marginal cost is
dC (q )
MC 
dq
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Because only the variable cost changes with
output, we can also define marginal cost as
the change in variable cost from a small
increase in output.
dVC (q )
MC 
dq
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The average fixed cost (AFC) is the fixed cost
divided by the units of output produced
F
AFC 
q
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The average fixed cost falls as output rises
because the fixed cost is spread over more
units:
dAFC
F
 2 0
dq
q
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It approaches zero as the output level grows
very large.
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The average variable cost (AVC) is the
variable cost divided by the units of output
produced
VC
AVC 
q
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Because the variable cost increases with
output, the average variable cost may either
increase or decrease as output rises.
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The average cost (AC) – or average total cost
– is the total cost divided by the units of
output produced:
C
AC 
q
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Because total cost equals variable cost plus
fixed cost, C = VC + F, when we divide both
sides of the equation by q, we learn that
C VC F
AC  
  AVC  AFC
q
q q
Illustration:
A manufacturing plant has a short-run cost
function of
C q   100q  4q 2  0.2q 3  450
What is the firm’s short-run fixed cost and
variable cost function?
Derive the formulas for its marginal cost,
average fixed cost, average variable cost, and
average cost.
The fixed cost is F = 450, the only part that
does not vary with q.
The variable cost function
VC q   100q  4q 2  0.2q 3
is the part of the cost function that varies with
q.
Differentiating the short-run cost function or
variable cost function, we find that
dC q 
 MC 
dq
d 100q  4q 2  0.2q 3  450
dq
MC  100  8q  0.6q 2
F 450
AFC  
q
q
V q  100q  4q 2  0.2q 3
AVC 

q
q
AVC  100  4q  0.2q 2
C q  100q  4q 2  0.2q 3  450
AC 

q
q
AC  100  4q  0.2q 
2
AC  AVC  AFC
450
q
cost
C
A
1,725
800
B
1
cost per unit
F
10
80
15
a
125
115
q
MC
AC
AVC
b
45
0
VC
80
450
0
1
115
AFC
10
15
q
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The production function determines the
shape of a firm’s cost curves.
The production function shows the amount of
inputs needed to produce a given level of
output.
The firm calculates its cost by multiplying the
quantity of each input by its price and
summing.
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If a firm produces output using capital and
labor and its capital is fixed in the short run,
the firm’s variable cost is its cost of labor.
Its labor cost is the wage per hour, w, times
the number of hours of labor, L, employed by
the firm: VC = wL
If input prices are constant, the production
function determines the shape of the variable
cost curve.
Because capital does not vary, we can write
the production function as
q  f L,K   g L 
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By inverting, we know that the amount of
labor we need to produce any given amount
of output is L = g-1(q).
If the wage of labor is w, the variable cost
function is VC(q) = wL = wg-1(q)
Similarly, the cost function is
C q  VC q   F  wg 1 q   F
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In the short run, when the firm’s capital is
fixed, the only way the firm can increase its
output is to use more labor.
If the firm increases its labor enough, it
reaches the point of diminishing marginal
returns to labor, where each extra worker
increases output by a smaller amount.
If the production function exhibits
diminishing marginal returns, then the
variable cost rises more than in proportion as
output increases.
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The marginal cost is the change in variable
cost as output increases by one unit:
dVC q 
MC 
dq
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In the short run, capital is fixed, so the only
way a firm can produce more output is to use
extra labor.
The extra labor required to produce one more
unit of output is
dL
1

dq
MPL
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The extra labor costs the firm w per unit, so
the firm’s cost rises by
 dL 
w

 dq 
As a result, the firm’s marginal cost is
dVC q 
 dL 
MC 
w 

dq
dq
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The marginal cost equals the wage times the
extra labor necessary to produce one more
unit of output.
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Since the marginal product of labor, the
amount of extra output produced by another
unit of labor, holding other input fixed is
dq
MPL 
dL
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Thus, the extra labor needed to produce one
more unit of output is
dL
1

dq

MPL
So the firm’s marginal cost is
w
MC 
MPL
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The marginal cost equals the wage divided by
the marginal product of labor.
If it takes four extra hours of labor services to
produce one more unit of output, the
marginal product of an hour is ¼.
Given a wage of P5 an hour, the marginal cost
of one more unit of output is P5 divided by ¼,
or P20.
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For the firm whose only variable input is
labor, variable cost is wL, so average variable
cost is
VC wL
AVC 

q
q
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Because the average product of labor is
q
APL 
L
Average variable cost is the wage divided by
the average product of labor
w
AVC 
APL
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In the long run, a firm adjusts all its inputs so
that its cost of production is as low as
possible.
The firm can change its plant size, design
and build new machines, and otherwise
adjusts inputs that were fixed in the short
run.
The rent of F per month that a restaurant
pays is a fixed cost because it does not vary
with the number of meals served.
In the short run, this fixed cost is sunk.
The firm must pay F even if the restaurant
does not operate.
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In the long run, this fixed cost is avoidable.
The firm does not have to pay this rent if it
shuts down.
The long run is determined by the length of
the rental contract, during which time the
firm is obligated to pay rent.
All inputs can be varied in the long run, so
there are no long-run fixed costs (F = 0).
As a result, the long-run total cost equals the
long-run variable cost: C = VC.
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In the long run, the firm chooses how much
labor and capital to use, whereas in the short
run, when capital is fixed, it chooses only
how much labor to use.
As a consequence, the firm’s long-run costs
is lower than its short-run cost of production
if it has to use the ‘wrong’ level of capital in
the short run.
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A firm can produce a given level of output
using many different technologically efficient
combinations of inputs, as summarized by an
isoquant.
From among the technologically efficient
combinations of inputs, a firm wants to
choose the particular bundle with the lowest
cost of production, which is the economically
efficient combination of inputs.
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To do so, the firm combines the information
about technology from the isoquant with
information about the cost of labor and
capital
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The cost of producing a given level of output
depends on the price of labor and capital.
The firm hires L hours of labor services at a
wage of w per hour, so its labor cost is wL.
The firm rents K hours of machine services at
a rental rate of r per hour, so its capital cost
is rK.
The firm’s total cost is the sum of its labor
and capital costs:
C  wL  rK
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The firm can hire as much labor and capital
as it wants at these constant input prices.
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The firm can use many combinations of labor
and capital that cost the same amount.
These combinations of labor and capital are
plotted on an isocost line, which indicates all
the combinations of inputs that require the
same (iso) total expenditure (cost).
Along an isocost line, cost is fixed at a
particular level.
We can write the equation for isocost line
with cost fixed at C
C  wL  rK
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Which we can rewrite as
C w
K   L
r r
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By differentiating with respect to L, we find
that the slope of any isocost line is
dK
w

dL
r
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Thus, the slope of the isocost line depends
on the relative prices of the inputs.
K
w = 24, r = 8
375
C = 3,000
250
C = 2,000
125
C = 1,000
0
41.67
83.33
125
L
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By combining the information about costs
that is contained in the isocost lines with
information about efficient production that is
summarized by an isoquant, a firm chooses
the lowest-cost way to produce a given level
of output.
K
375
w = 24, r = 8
q = 100
C = 3,000
y
303
250
C = 2,000
125
C = 1,000
x
100
z
28
0
24
41.67
50
83.33
116
125
L
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The firm is minimizing cost subject to the
information in the production function
contained in the isoquant expression:
q  f L,K 
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The corresponding Lagrangian problem is
min L  wL  rK   q  f L,K 
L, K , 
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The first-order conditions are
L
f
w  
0
L
L
L
f
r 
0
K
K
L
 q  f L,K   0

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By rearranging terms, we obtain
f
MPL
w

L


f
r
MPK
K
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We find that cost is minimized where the
factor price ratio equals the ratio of the
marginal products.
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We could equivalently examine the dual
problem of maximizing output for a given
level of cost,

max L  f L,K    C  wL  rK
L, K , 
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The first-order conditions are
L
f

 w  0
L
L
L
f

 r  0
K
K
L
 C  wL  rK  0


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By rearranging terms, we obtain
f
MPL
w

L


f
r
MPK
K

The same condition as when we minimized
cost by holding output constant.
K
q = 175
q = 75
w = 24, r = 8
q = 100
250
100
0
x
50
83.33
L
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Suppose that the wage falls but that the
rental rate of capital stays constant, the firm
now minimizes its new cost by substituting
away from the now relatively more expensive
input, capital, toward the now relatively less
expensive input, labor.
Because of the wage decrease, the new
isocost lines have a flatter slope.
K
100
q = 100
x
v
52
0
50
77
L
w = 24, r = 8
K
Expansion path
z
200
150
100
y
q = 200
x
q = 150
q = 100
0
50
75
100
L
Illustration:
What is the expansion path for a constantreturns-to-scale Cobb-Douglas production
function
q  ALaK 1a
Show the special case for A = 1.52 and a = 0.6
given that w = 24 and r = 8.
Use the tangency condition between the
isocost and isoquant that determines the factor
ratio when the firm is minimizing cost to derive
the expansion path.
q
q
and MPK  1 a 
Since MPL  a
L
K
The tangency condition is
q
a
L
w MPL


q
r MPK
1 a 
K
w
a K

r 1 a L
Rearranging, we obtain the expansion path
function
1 a  w

K 
L
a
r
For the case of a = 0.6, since w = 24 and
r = 8, we have
 0.4   24 
K 
L



 0.6   8 
K  2L
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As the expansion path plot shows, to produce
q units of output takes K = q units of capital
and L = q/2 units of labor.
Thus the long-run cost of producing q units
of output is
C q   wL  rK  w
q
 rq
2
w

 24

C q     r  q  
 8 q
2

 2

C q   20q
C
C q   20q
Long-run cost curve
4,000
Z
3,000
2,000
0
Y
X
100
150
200
q
Illustration:
Derive the long-run cost as a function of only
output and factor prices for a Cobb-Douglas
production function
q  AL K
a
1a
Show the special case for A = 1.52 and a = 0.6
given that w = 24 and r = 8.
From the expansion path, we know that
rK
1 a 


wL
a
Substituting for rK in the cost identity gives
C
1 a 

 wL 
wL
a
Simplifying shows that
C
L a
w
Repeating this process to solve for K, we find
that
C
K  1 a 
r
Substituting L and K into the production
function, we have;
a
1a
C
 C  
q  A  a  1 a  
r 
 w 
We can rewrite as
C  q
where
w ar 1a

1a
a
Aa 1 a 
For the case of A = 1.52 and a = 0.6 given that
w = 24 and r = 8, we have
24 8 


0.6
0.4
1.520.6 0.4
0.6

Thus,
  20
C  20q
0.4
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In the long run, returns to scale play a major
role in determining the shape of the average
cost curve and the other cost curves.
If a production function exhibits increasing
returns to scale at low levels of output,
constant returns to scale at intermediate
levels of output, and decreasing returns to
scale at high levels of output, then the longrun average cost curve must be U-shaped.
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A cost function is said to exhibit economies
of scale if the average cost of production falls
as output expands, as we would expect in the
range where the production function had
increasing returns to scale.
In the range where the production function
has constant returns to scale, the average
cost remains constant, so the cost function
has no economies of scale.
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Finally, in the range where the production
function has decreasing returns to scale,
average cost increases.
A firm suffers from diseconomies of scale if
average cost rises when output increases.