Download total cost function

Chapter 10
Cost Functions
Explicit Costs and Implicit Costs
Explicit Costs – Costs that involve a direct
monetary outlay.
Implicit Costs – Costs that do not involve outlays
of cash.
2
Opportunity Cost
The relevant concept of cost is opportunity cost:
the value of a resource in its best alternative use.
 The only alternative we consider is the best
alternative
3
Economic Costs and Accounting Costs
Economic Costs – Sum of a firm’s explicit costs
and implicit Costs.
Accounting Costs – Total of a firm’s explicit costs.
4
Economic Cost
• The economic cost of any input is the payment required to keep
that input in its present employment
– the remuneration the input would receive in its best alternative
employment
• Sunk Cost
– Expenditure that has been made and cannot be recovered
– Because a sunk cost cannot be recovered, it should not influence
the firm’s decisions.
– E.g: Consider the purchase of specialized equipment for a plant.
– Suppose the equipment can be used to do only what it was
originally designed for and cannot be converted for alternative
use.
– The expenditure on this equipment is a sunk cost.
– Because it has no alternative use, its opportunity cost is
zero.
Two Simplifying Assumptions
• Inputs are hired in perfectly competitive markets
– firms are price takers in input markets
• There are only two inputs
– homogeneous labor (l), measured in laborhours
– homogeneous capital (k), measured in
machine-hours
Economic Profits
• Total costs for the firm are given by
Total costs = C = wl + vk
• Total revenue for the firm is given by
Total revenue = pq = pf(k,l)
• Economic profits () are equal to
 = Total revenue - Total cost
 = pq - wl - vk
 = pf(k,l) - wl - vk
Economic Profits
• Economic profits are a function of the amount
of k and l employed
– We could examine how a firm would choose k and l
to maximize profit
– We will assume that the firm has already chosen its
output level (q0) and wants to minimize its costs
Cost-Minimizing Input Choices
• We seek to minimize total costs given q =
f(k,l) = q0
• Setting up the Lagrangian:
ℒ = wl + vk + [q0 - f(k,l)]
• FOCs are
ℒ /l = w - (f/l) = 0
ℒ /k = v - (f/k) = 0
ℒ / = q0 - f(k,l) = 0
Cost-Minimizing Input Choices
• Dividing the first two conditions we get
w f / l

 MRTS (l for k )
v f / k
• The cost-minimizing firm should equate the
MRTS for the two inputs to the ratio of their
prices
Solution to cost minimization:
• Slope of isoquant = slope of isocost line
 MRTS L , K
w
  (or)
r
MPL w

MPK r
• Ratio of marginal products = ratio of input prices
MPL MPK

w
r
11
• At point E
MPL w

MPK v
MPL MPK
(or )

w
v
• This implies the firm could
spend an additional dollar
on labor and save more
than a dollar by reducing its
employment of capital and
keep output constant
12
• At point F
MPL w

MPK v
MPL MPK
(or )

w
v
• This implies the firm could
spend an additional dollar
on capital and save more
than a dollar by reducing its
employment of labor and
keep output constant
13
Cost-Minimizing Input Choices
• The inverse of this equation is also of
interest
w
v


MPl MPk
• The Lagrangian multiplier shows how the
extra costs that would be incurred by
increasing the output constraint slightly
Cost-Minimizing Input Choices
Given output q0, we wish to find the least costly point
on the isoquant
k per period
C1
C3
Costs are represented by
parallel lines with a slope of w/v
C2
C1 < C2 < C3
q0
l per period
Cost-Minimizing Input Choices
The minimum cost of producing q0 is C2
k per period
This occurs at the tangency
between the isoquant and
the total cost curve
C1
C3
C2
k*
q0
l*
The optimal choice
is l*, k*
l per period
Interior Solution
Suppose: Q = 50L1/2K1/2
Suppose that the wage rate of the laborer is
given as $5, the rental price is given as $20 and
the firm produces an output of 1000.
What is the cost minimizing level of output?
17
Corner Solution
The cost-minimizing input
combination for producing Q0
units of output occurs at point
A where the firms uses no
capital.
At this corner point the
isocost line is flatter than the
isoquant.
MPL
w
(
)  ( )
MPK
r

MPL MPK

w
r
18
Contingent Demand for Inputs
• In Chapter 4, we considered an individual’s
expenditure-minimization problem
– to develop the compensated demand for a
good
• Can we develop a firm’s demand for an
input in the same way?
Cost-Minimizing Input Choices
The minimum cost of producing q0 is C2
k per period
This occurs at the tangency
between the isoquant and
the total cost curve
C1
C3
C2
k*
q0
l*
The optimal choice
is l*, k*
l per period
Contingent Demand for Inputs
• In the present case, cost minimization leads
to a demand for capital and labor that is
contingent on the level of output being
produced
• The demand for an input is a derived demand
– it is based on the level of the firm’s output
Some Key Definitions
An increase in Q0 moves the isoquant Northeast.
• Expansion Path: A line that connects the cost-minimizing input
combinations as the quantity of output, Q, varies, holding input
prices constant.
• Normal Inputs: An input whose cost-minimizing quantity
increases as the firm produces more output.
• Inferior Input: An input whose cost-minimizing quantity decreases
as the firm produces more output.
22
The Firm’s Expansion Path
• The firm can determine the cost-minimizing
combinations of k and l for every level of output
• If input costs remain constant for all amounts of
k and l, we can trace the locus of costminimizing choices
– called the firm’s expansion path
An Expansion Path
As output increases, the cost minimization path
moves from point A to B to C when inputs are
normal
24
The Firm’s Expansion Path
The expansion path is the locus of cost-minimizing
tangencies
k per period
The curve shows how
inputs increase as
output increases
E
q1
q0
q00
l per period
The Firm’s Expansion Path
• The expansion path does not have to be a
straight line
– the use of some inputs may increase faster
than others as output expands
• depends on the shape of the isoquants
• The expansion path does not have to be upward
sloping
– if the use of an input falls as output expands,
that input is an inferior input
An Expansion Path
As output increases, the cost minimization path
moves from point A to B to C when labor is an
inferior input
27
Cost Minimization
• Suppose that the production function is
Cobb-Douglas:
q = kl
• The Lagrangian expression for cost
minimization of producing q0 is
ℒ = vk + wl + (q0 - k  l )
Cost Minimization
• The FOCs for a minimum are
ℒ /k = v - k -1l = 0
ℒ /l = w - k l -1 = 0
ℒ/ = q0 - k  l  = 0
Cost Minimization
• Dividing the first equation by the second
gives us
w k  l  1  k
  1   
v k l
 l
• The MRTS depends only on the ratio of the two
inputs
• The expansion path is a straight line
Total Cost Function
• The total cost function shows that for any set
of input costs and for any output level, the
minimum cost incurred by the firm is
C = C(v,w,q)
• As output (q) increases, total costs increases
Average Cost Function
• The average cost function (AC) is found by
computing total costs per unit of output
C(v ,w , q )
average cost  AC (v ,w , q ) 
q
Marginal Cost Function
• The marginal cost function (MC) is found by
computing the change in total costs for a
change in output produced
C(v ,w , q )
marginal cost  MC(v ,w , q ) 
q
Graphical Analysis of Total Costs
Total
costs
With constant returns to scale, total costs
are proportional to output
AC = MC
C
Both AC and
MC will be
constant
Output
Graphical Analysis of Total Costs
• Suppose that k1 units of capital and l1 units of
labor input are required to produce one unit
of output
C(q=1) = vk1 + wl1
• To produce m units of output (assuming
constant returns to scale)
C(q=m) = vmk1 + wml1 = m(vk1 + wl1)
C(q=m) = m  C(q=1)
Graphical Analysis of Total Costs
• Suppose that total costs start out as
concave and then becomes convex as
output increases
– one possible explanation for this is that there is
a third factor of production that is fixed as
capital and labor usage expands
– total costs begin rising rapidly after diminishing
returns set in
Graphical Analysis of Total Costs
Total
costs
C
Total costs rise
dramatically as
output increases
after diminishing
returns set in
Output
Graphical Analysis of Total Costs
Average
and
marginal
costs
MC is the slope of the C curve
MC
AC
min AC
If AC > MC, AC
must be
falling
If AC < MC, AC
must be
rising
Output
Shifts in Cost Curves
• Cost curves are drawn under the assumption
that input prices and the level of technology
are held constant
– any change in these factors will cause the cost
curves to shift
Some Illustrative Cost Functions
• 1) Suppose we have a fixed proportions
technology such that
q = f(k,l) = min(ak,bl)
• Production will occur at the vertex of the Lshaped isoquants (q = ak = bl)
C(w,v,q) = vk + wl = v(q/a) + w(q/b)
v w 
C(w ,v , q )  a  
a b 
Some Illustrative Cost Functions
• 2) Suppose we have a Cobb-Douglas
technology such that
q = f(k,l) = k l 
• Cost minimization requires that
w  k
 
v  l
 w
k   l
 v
Some Illustrative Cost Functions
• Suppose we have a CES technology such
that
q = f(k,l) = (k  + l )/
• To derive the total cost, we would use the
same method and eventually get
C(v ,w , q )  vk  wl  q 1/  (v  / 1  w  / 1 )( 1) / 
C(v ,w , q )  q 1/  (v 1  w 1 )1/ 1
Properties of Cost functions
• 1) Homogeneity
• 2) Total cost functions are nondecreasing in q,
v, and w.
• 3) Total cost functions are concave in input
prices.
Properties of Cost Functions
• Homogeneity
– cost functions are all homogeneous of degree
one in the input prices
• a doubling of all input prices will not change the
levels of inputs purchased
Properties of Cost Functions
• Nondecreasing in q, v, and w
– cost functions are derived from a costminimization process
• any decline in costs from an increase in one of the
function’s arguments would lead to a contradiction
Properties of Cost Functions
• Concave in input prices
– costs will be lower when a firm faces input
prices that fluctuate around a given level than
when they remain constant at that level
• the firm can adapt its input mix to take advantage
of such fluctuations
Concavity of Cost Function
At w1, the firm’s costs are C(v,w1,q1)
Costs
If the firm continues to buy
the same input mix as w
changes, its cost function
would be Cpseudo
Cpseudo
C(v,w,q1)
C(v,w1,q1)
Since the firm’s input mix
will likely change, actual
costs will be less than Cpseudo
such as C(v,w,q1)
w1
w
Properties of Cost Functions
• Some of these properties carry over to
average and marginal costs
– homogeneity
– effects of v, w, and q are ambiguous
Input Substitution
• A change in the price of an input will cause
the firm to alter its input mix
• The change in k/l in response to a change in
w/v, while holding q constant is
k 
 
l
w 
 
v 
Input Substitution
• Putting this in proportional terms as
( k / l ) w / v
 ln(k / l )
s


(w / v ) k / l  ln(w / v )
gives an alternative definition of the elasticity
of substitution
– in the two-input case, s must be nonnegative
– large values of s indicate that firms change their
input mix significantly if input prices change
Partial Elasticity of Substitution
• The partial elasticity of substitution between
two inputs (xi and xj) with prices wi and wj is
given by
( x i / x j ) w j / w i
 ln( x i / x j )
sij 


(w j / w i ) x i / x j
 ln(w j / w i )
• Sij is a more flexible concept than 
– it allows the firm to alter the usage of inputs
other than xi and xj when input prices change
Size of Shifts in Costs Curves
• The increase in costs will be largely influenced
by
– the relative significance of the input in the
production process
– the ability of firms to substitute another input for
the one that has risen in price
Technical Progress
• Improvements in technology also lower
cost curves
• Suppose that total costs (with constant
returns to scale) are
C0 = C0(q,v,w) = qC0(v,w,1)
Technical Progress
• Because the same inputs that produced one
unit of output in period zero will produce A(t)
units in period t
Ct(v,w,A(t)) = A(t)Ct(v,w,1)= C0(v,w,1)
• Total costs are given by
Ct(v,w,q) = qCt(v,w,1) = qC0(v,w,1)/A(t)
= C0(v,w,q)/A(t)
Shifting the Cobb-Douglas Cost Function
• The Cobb-Douglas cost function is
C(v ,w , q )  vk  wl  q 1/  Bv  /  w  /  
where
B  (  )   /    /  
• If we assume  =  = 0.5, the total cost
curve is greatly simplified:
C(v ,w , q )  vk  wl  2qv 0.5w 0.5
Shifting the Cobb-Douglas Cost Function
• If v = 3 and w = 12, the relationship is
C(3,12, q )  2q 36  12q
– C = 480 to produce q =40
– AC = C/q = 12
– MC = C/q = 12
Shifting the Cobb-Douglas Cost Function
• If v = 3 and w = 27, the relationship is
C(3,27, q )  2q 81  18q
– C = 720 to produce q =40
– AC = C/q = 18
– MC = C/q = 18
Shifting the Cobb-Douglas Cost Function
• Suppose the production function is
q  A( t )k
0 .5 0 .5
l
e
0.03t
k
0 .5 0 .5
l
– we are assuming that technical change takes
an exponential form and the rate of technical
change is 3 percent per year
Shifting the Cobb-Douglas Cost Function
• The cost function is then
0.5 0.5 0.03t
C 0( v ,w ,q )
C(t v ,w ,q ) 
 2qv w e
A( t )
– if input prices remain the same, costs fall at the
rate of technical improvement
Contingent demand for inputs
60
Contingent Demand for Inputs
• Contingent demand functions for all of the firms
inputs can be derived from the cost function
–Shephard’s lemma
• the contingent demand function for any
input is given by the partial derivative of the
total-cost function with respect to that
input’s price
Contingent Demand for Inputs
• Shepherd’s lemma is one result of the
envelope theorem
– the change in the optimal value in a
constrained optimization problem with respect
to one of the parameters can be found by
differentiating the Lagrangian with respect to
the changing parameter
Contingent Demand for Inputs
• Suppose we have a fixed proportions
technology
• The cost function is
v w 
C(w ,v , q )  a  
a b 
Contingent Demand for Inputs
• For this cost function, contingent demand
functions are quite simple:
C(v ,w , q ) q
k (v ,w , q ) 

v
a
c
C(v ,w , q ) q
l (v ,w , q ) 

w
b
c
Contingent Demand for Inputs
• Suppose we have a Cobb-Douglas technology
• The cost function is
C(v,w, q )  vk  wl  q
1 /  
Bv
 /  
w
 /  
Contingent Demand for Inputs
• For this cost function, the derivation is
messier:
C

1/  
 /     /   
k (v ,w , q ) 

q
Bv
w
v   
c

1/    w 

q
B 

v 
 /  
Contingent Demand for Inputs
C

l (v ,w , q ) 

 q 1/  Bv  /  w  /  
w   
c

1/    w 

q
B 

v 
  /  
• The contingent demands for inputs depend
on both inputs’ prices
Contingent Demand for Inputs
• Suppose we have a CES technology
• The cost function is
C( v ,w ,q )  q
1/ 
v
1 
w

1   /( 1  )
Contingent Demand for Inputs
• The contingent demand function for capital
is

1/ 
1 
1 
C
1
k ( v ,w ,q ) 

q v w
v 1  
1/ 
1 
1   /( 1  )  
 q v w
v
c



 /( 1  )
( 1   )v

Contingent Demand for Inputs
• The contingent demand function for labor
is


1/ 
1 
1 
C
1
l ( v ,w ,q ) 

q v w
w 1  
1/ 
1 
1   /( 1  )

 q v w
w
c


 /( 1  )
( 1   )w

Short-Run, Long-Run Distinction
• In the short run, economic actors have only
limited flexibility in their actions
• Assume that the capital input is held
constant at k1 and the firm is free to vary
only its labor input
• The production function becomes
q = f(k1,l)
Short-Run Total Costs
• Short-run total cost for the firm is
SC = vk1 + wl
• There are two types of short-run costs:
– short-run fixed costs are costs associated with
fixed inputs (vk1)
– short-run variable costs are costs associated
with variable inputs (wl)
Short-Run Total Costs
• Short-run costs are not minimal costs for
producing the various output levels
– the firm does not have the flexibility of input
choice
– to vary its output in the short run, the firm
must use nonoptimal input combinations
– the MRTS will not be equal to the ratio of input
prices
Nonoptimality of Short-Run Total
Costs
k per period
Because capital is fixed at k1,
the firm cannot equate MRTS
with the ratio of input prices
k1
q2
q1
q0
l per period
l1
l2
l3
• Long run-all variables
are variable and the
expansion path is from A
–B–C
• Short run-some
variables are fixed
(capital)-the expansion
path is from D –B –E
75
• Short run: One input is fixed, capital K .
Firm can vary the other input, labor. SO
demand for labor will be independent of
price of K
• Short run demand for labor will also depend
on quantity produced.
• As quantity increased, labor used increases
holding capital fixed.
76
Short-Run Marginal and Average Costs
• The short-run average total cost (SAC)
function is
SAC = total costs/total output = SC/q
• The short-run marginal cost (SMC) function
is
SMC = change in SC/change in output = SC/q
Long and Short Run Total Cost Functions
Total Cost ($/yr)
STC(Q,K0)
TC(Q)
The long-run C
curve can be
derived by
varying the level
of k
K0 is the LR cost-minimising
quantity of K for Q0
0
Q0
Q1
Q (units/yr)
78
Long and Short Run Total Cost Functions
STC(Q,K0)
Total Cost ($/yr)
TC(Q)
TC0
The long-run C
curve can be
derived by
varying the level
of k
A
•
K0 is the LR cost-minimising
quantity of K for Q0
0
Q0
Q1
Q (units/yr)
79
Long and Short Run Total Cost Functions
STC(Q,K0)
Total Cost ($/yr)
TC(Q)
•C
TC1
TC0
A
•
The long-run C
curve can be
derived by
varying the level
of k
K0 is the LR cost-minimising
quantity of K for Q0
0
Q0
Q1
Q (units/yr)
80
Long and Short Run Total Cost Functions
STC(Q,K0)
Total Cost ($/yr)
•
•C
TC2
B
TC1
TC0
A
•
TC(Q)
The long-run C
curve can be
derived by
varying the level
of k
K0 is the LR cost-minimising
quantity of K for Q0
0
Q0
Q1
Q (units/yr)
81
Short-Run and Long-Run Costs
Costs
SMC (k0)
SAC (k0)
MC
AC
SMC (k1)
q0
q1
SAC (k1)
The geometric
relationship
between shortrun and long-run
AC and MC can
also be shown
Output
Short-Run and Long-Run Costs
• At the minimum point of the AC curve:
– the MC curve crosses the AC curve
• MC = AC at this point
– the SAC curve is tangent to the LAC curve
• SAC (for this level of k) is minimized at the same level of
output as AC
• SMC intersects SAC also at this point
AC = MC = SAC = SMC
Returns to Scale & Economies of Scale
• 1) When the production function exhibits increasing
returns to scale, the long run average cost function
exhibits economies of scale so that AC(Q) decreases
with Q, all else equal.
•2) When the production function exhibits decreasing
returns to scale, the long run average cost function exhibits
diseconomies of scale so that AC(Q) increases with Q, all else
equal.
• When the production function exhibits constant returns to
scale, the long run average cost function is flat: it neither
increases nor decreases with output.
84
LONG-RUN COST WITH ECONOMIES AND
DISECONOMIES OF SCALE
The long-run average
cost curve LAC is the
envelope of the shortrun average cost
curves SAC1, SAC2,
and SAC3.
With economies and
diseconomies of scale,
the minimum points of
the short-run average
cost curves do not lie
on the long-run average
cost curve.
Economies of Scope
If a firm produces multiple goods, the cost of one
good may depend on the output level of the
other.
• Outputs are linked if a single input is used to
produce both.
•There are economies of scope if it is cheaper to
produce goods jointly than separately.
Economies of Scope – a production characteristic in
which the total cost of producing given quantities of
two goods in the same firm is less than the total
cost of producing those quantities in two singleproduct firms.
86
Economies of Experience
Economies of Experience – cost advantages that result from
accumulated experience, or, learning-by-doing.
Experience Curve (or Learning Curve)– a relationship
between average variable cost and cumulative production
volume
– typical relationship is AVC(N) = ANB,
where N – cumulative production volume,
A > 0 – constant representing AVC of first unit
produced,
-1 < B < 0 – experience elasticity (% change in AVC for
every 1% increase in cumulative volume
– slope of the experience curve tells us how much
AVC goes down (as a % of initial level), when
87
cumulative output doubles
Graphing the Learning Curve
A firm’s production
cost may fall over time
as managers and
workers become more
experienced and
more effective at
using the available
plant and equipment.
The learning curve
shows the extent to
which hours of labor
needed per unit of
output fall as the
cumulative output
increases.
ECONOMIES OF SCALE VERSUS LEARNING
Economies of Scale:
A firm’s average cost of
production can decline
over time because of
growth of sales when
increasing returns are
present (a move from A
to B on curve AC1),
Learning Curve:
A firm’s average cost of
production can decline
because there is a
learning curve (a move
from A on curve AC1 to
C on curve AC2).