Download Chapter 8 Costs Functions

Chapter 8
Costs Functions
The economic cost of an input is the minimum payment
required to keep the input in its present employment. It is the
payment the input would receive in its best alternative
employment. This cost concept is closely related to the
opportunity cost concept (not talking about accounting costs).
We usually assume that inputs are hired in perfectly competitive
markets. The firm can get all the input it wants without
affecting prices. The supply curve for an input is horizontal at
the prevailing price.
w

Supply of the input = price of the input.
Firm’s demand for the input
L
Total Cost, Revenue, and Profit
• Total cost = C = wL + vK (with only 2
inputs, capital and labor)
• TR = pq (with only 1 output)
• Then, economic profit is:
  pq  wL vK  pf (K, L)  wL vK
• Thus, economic profit is simply a function
of K and L, given that all prices (p, w, and
v) and technology are fixed.
Cost Minimizing Input Choices (for
given q)
• Assume for now that output has been
determined to be q0 and the firm wishes to
minimize its cost. That is, the firm must
choose a specific point on the q0 isoquant.
K


q0
L
• Cost will be minimized by choosing the point
where RTSLK (-slope of isoquant) equals the ratio
of input prices (w/v). This happens when the rate
of substitution in production equals the rate of
substitution in the market.
Min: C = wL + vK
 is the increase in C
st: q0 = f(K, L) or q0 – f(K, L) = 0 when q0 increases by
one unit.  is marginal
 = wL + vK + (q0 –f(K, L))
cost, MC.
FOC

f
 w 
0
L
L

f
 v
0
K
K

 q 0  f (K , L)  0

The SOC require diminishing
RTSLK or q=f(K, L) strictly quasiconcave or K = f(L,q0) strictly
convex.
Then
dK
w f L MPL

 RTS L , K  

dL
v f K MPK
So, RTSLK should equal w/v for the minimum
cost combination of inputs, or the slope of the
isoquant (dK dL) equals the slope of the isocost
line ( w v).
MPL MPK 1


w
v
  the marginal product per dollar
spent is equal for all inputs.
Also,  = marginal cost is the inverse of the above,
w
v

   MC.
MPL MPK
K
Graphically
C3 > C2 > C1
C1
v
K*
0

L*
C1
C1
w
C2
Minimum cost is C1, which
is the minimum cost to
achieve q0.
The isocost line shows
combinations of K and L
q0
that can be purchased with
C
L fixed total cost.
3
C w
K   L  isocost line.
v v
The solution can be a corner point, but not usually unless the inputs
are close substitutes (close to linear isoquants). Assuming perfect
substitutes and RTSLK > w/v, which input would not be used? (K!)
Dual: Output maximization subject to a cost
constraint:
Max: q=f(K, L)
s.t.: C1 = wL + vK or C1 – wL – vK = 0
= f(K, L) + D(C1 – wL – vK)
D represents the marginal product of one additional
dollar of expenditure on inputs. It equals 1/.
Solving the FOC yields K* and L* as did the primal.
C1
K
K*
0

L*
Output Maximum
qo
q-1
L
Can we derive a demand curve for L
by changing price (w) and looking at
the resulting change in L*?
• The answer is “yes”, but the logic is somewhat different from
the consumer’s demand for a good. As w changes and L*
changes, the output level changes, which will change the market
for q, which will change p (price of q). We cannot investigate
the demand for an input without also considering the interaction
of supply and demand for the output. The demand for the
input is derived from the output market. Along the demand
curve for L, v and p are held constant.
• Therefore, the analogy between consumer and firm optimization
is not exact. (Isoquants are not directly interpretable as revenue
whereas indifference curves represent utility.)
py ΔU for consumer
w  q for firm  p in market for q.
Demand for L is a derived demand
from the market for q.
• Expansion Path – As the firm expands q, the
cost minimization points trace out the
expansion path. The Expansion Path shows how optimal
K
0
C4
C3
C2
C1



input usage changes as output expands with
v, w, and technology constant (isocost lines
are parallel because v and w are constant).
q2
q1
q3
q4
L
If the production function is homothetic, the
Expansion Path will be linear. The shape of
the isoquants determines the shape of the
Expansion Path. The Expansion Path shows
points of equal RTSLK on the isoquants
because the isocost lines are parallel.
An Expansion Path that slopes toward an axis indicates an
K
inferior input on the other axis; ie., use of the input actually
declines as q increases. For example, as you produce larger and
larger acreages of vegetables, your use of hand-held
0
implements would decrease; hoes and unskilled labor are
inferior inputs.
L
Cost Functions come directly from
the production function and prices.
• Total cost: C = C(v, w, q)
Minimum Total Cost is a function of input prices and
output quantity. Thus, the C function represents the
minimum cost necessary to produce output q with fixed
input prices. C represents the minimum isocost line for
any level of q. It reflects the cost minimizing
combination of inputs (K*, L*) for any given q.
A total cost function is analogous to an expenditure
function in consumer theory. Someone define an
expenditure function.
• Average Cost
AC
C

q
AC is the cost per unit of output;
AC = AC(v, w, q).
• Marginal Cost
C
MC 
q
MC is the change in C as output
changes. It is the added cost for
producing an additional unit of
output. MC = MC(v, w, q).
Initially, we will hold v and w constant and look at how
cost varies as q changes. This will give us standard
two-dimensional graphs.
With a production function that shows constant returns
to scale (homogeneous of degree 1, or linear
homogeneous), C will be linear with fixed input prices.
MC will be constant and equal to AC.
$
C
$
AC = MC
q
q
However, typical (in theory) cost curves are
C'  0
sloped as follows:
C is cubic
$
C ''  0 and then  0
C
C is concave; C convex
Inflection
point
AC = the slope of a cord from
the origin to any point on C.
Based on C, can draw
q
$
MC
AC
Slope of the cord
= slope of C.
q
MC   C  q .
Thus, when C is concave, MC is
declining; when C is convex, MC is
increasing; and MC is at a minimum at
the C inflection point.
AC = MC for first unit of q. AC reflects
lower MC of first units of q. Once MC
crosses AC, AC turns up reflecting higher
MC of later units of q. MC crosses AC at
minimum AC.
Changes in Input Prices
•
•
•
•
When input prices change relative to each other, the expansion path
changes and the cost curves shift.
But C is homogeneous of degree 1 in input prices, so doubling all input
prices doubles C. This doubling of input prices would not affect q, L*, K*
or the Expansion Path,.
Because C is homogeneous of degree 1, MC and AC will be also.
“Pure inflation” will not affect input combinations or q, but it will affect
C, AC, and MC.
Mathematically and Graphically
If C1 = vK1 + wL1 and v and w are multiplied by m,
C2 = mvK1 + mwL1 = mC1.
If v, w, and C double, optional point
remains at A.
K
C2/2v=C1/v

A
If w and v double, this isocost line changes from C1 to
2C1 = C2, but L*, K* and q* do not change. 2w  w  RTS
LK
2v v
C1/w = 2C1/2w =C2/2w is the intercept.
L
Changes in a Single Input Price
(relative prices of K and L change).
• The slope of the isocost line will change, resulting in changes in
the optimal input combination and changes in the expansion path.
Input Substitution (q constant)– Deals with how the optimal combination of
K and L (or K/L) changes as w/v changes with q constant.
Total Effect and its Direction (q changes)– An increase in v or w will
increase C. AC will also rise. MC will rise if the input is not inferior.
From Footnote 9 on page 228 and Footnote 7 on page 226:
MC  ( q)  2 
 2   ( v) k




 ;
v
v
qv vq
q
q
MC k
MC
 , so if k is inferior,
 0,
v
q
v
where from the Envelope Theorem  q  λ  MC,  v  k, and  is the optimal
Lagrangian function from the cost minimization problem subject to an output constraint.
K
  w %Δ K
L
L.
An alternative elasticity of substitution is
s   v 
 w  K %Δ w
with q, v, and w constant (Partial Elasticity of Substitution).  v  L
v
s is positive in a two-input world, but can be negative if three or more inputs.
This elasticity is similar to the elasticity of substitution () developed earlier
from the production function if we remember that at the optimal combination
of K and L w
dK
 K/L  RTS
 RTSLK   .
 

 RTS  K/L
dL
v
LK
LK
In practical application, which would be easier to estimate,  or s?
A large s means the optimal input combination changes a lot as the price ratio
changes, suggesting close to linear isoquants (close substitutes).
K

w
v
Large s implies a
 large change in K/L

for a change in w/v close substitutes.
q0
L
K

w

v

q0
L
Small s implies a
small change in
K/L for a change
in w/v – not close
substitutes.
For many inputs
 xi  w
j

 
 w
x
j
 
i
s ij  
 w j  xi

 
 wi  xj
This formula gives a partial
approach for use where many
inputs are involved.
Other input usages (other than inputs i and j) are not
held constant in sij but they are in  ij. sij does not
have to be non-negative. Other inputs’ usages may
change to give a net negative sign on sij; q is
constant.
Contingent Demand for Inputs and
Sheppard’s Lemma
Quantity of output is under the firm’s control and actual input demand changes as
output changes. However, cost minimization, subject to an output constraint,
creates an implicit demand for inputs with quantity of output held constant.
Contingent demand for an input (output-constant input demand) holds output
constant similar to compensated demand for a good.
Sheppard’s Lemma is a result of the Envelop Theorem for constrained
optimization. Sheppard’s Lemma is that the partial derivative of C with respect to
an input price gives the contingent demand function for that input (q constant).
The envelop theorem and Sheppard’s Lemma says that when the Lagrangian
expression is at its optimum,
C/v  */v and C/w  */w.
Use the Lagrangian method to find K*
If *  vK*  wL*  λ*[q  f(K* , L* )],
C(v,w,q) * (v,w,q, λ)

 Kc (v,w,q)
v
v
C(v,w,q) * (v,w,q, λ) c
 L (v,w,q)

w
w
and L*. Or if you are given C, just take
the partial derivative of C with respect
to w and v to get the contingent demand
functions for L and K, respectively.
Size of Shifts in Cost curves – two factors:
• Cost Share – The more important the input, the
larger the cost curve response to a change in the
input price. If the input makes up a large portion
of total cost, an increase in its price will raise
total cost substantially.
• Input substitutability – High substitutability with
other inputs will reduce the effect on the cost
curves of a change in the price of single input,
ie., large partial elasticity of substitution (s) will
reduce the effect on C of a change in v or w.
Short-Run versus Long-Run Costs
• Short run is a time period in which some inputs are
variable, but at least one is fixed. This brings about
the “Law of Diminishing Returns” to the variable
inputs.
• Long run is a time period in which all inputs are
variable. So far, evaluation of Economies of Scale
and cost minimization have assumed the long run.
• In short run, we have some fixed cost, while in the
long run all costs are variable.
• Not the same time period for all firms and industries.
• Not a specific time unit.
• The text considers K as the fixed input in the short run.
• SC = vK1 + wL where K is fixed at K1.
• Then vK1 is short-run fixed cost, because it will not change
in the short run. wL represents short-run variable cost
because L can be changed in the short run to affect q.
SFC  vK 1 
 So SC = SFC + SVC
SVC  wL 
• SFC has to be paid regardless of level of q (0 to ). SVC
changes as L changes to change q.
• If q = 0, SVC = 0 because L = 0.
• If q = 0, SFC = vK1.
Assuming any level of q
requires some L.
$
This SC represents minimum
short-run cost for each level of
SC output, but not minimum C for
SVC each level of output.
SFC same over all q.
vK1
SVC = 0 if q = 0.
SFC
vK1
0
q
SC = SFC + SVC.
SFC has no effect on
the shape of SC.
Only one of the output levels in this graph shows the minimum
total cost (C not SC) for K1. Because K is fixed at K1 and
there is only one level of L that is optimal for K1, there will be
only one point on SC where total cost (C) is minimum for K1.
KSC
1
= C1
SC2
C2
Expansion Path
C0
K1
SC0

  
A
L0
L1
SC0 > C0
q3
q0
q1
q2
L
L2
SC1 = C1
SC2 > C2
With K fixed at K1, only
at output q1 is total cost
(C) minimized because
only at point A is w/v =
RTS. At any other
quantity along the line K
= K1, SC is minimized
but C is not minimized.
For C minimization, we
would need to use less K
and more L for points on
K = K1 to the left of L1
and more K and less L
for points on K = K1 to
the right of L1.
Per Unit Costs
SC
SAC 
q
SC SVC

SMC 
q
q
The concepts and
relationships between
SC and SAC and
SMC are the same as
described earlier for
C, AC, and MC.
Only variable costs
affect MC.
If SFC = α and SVC = q2, then SC = α + q2,
= α/q + q, and
SC
SVC
 2β q 
SMC 
q
q
SAC
SAC can be separated into
fixed and variable parts.
SMC
SAC
SAVC
$
SAFC (Same
vertical distance)
0
SAFC
q
SVC
SAVC 
 βq
q
SFC α
SAFC 

q
q
SC α
SAC 
  βq
q
q
SAC  SAFC  SAVC
•SAFC is a rectangular hyperbola that is asymptotic to 0.
Thus, as q increases, SAVC approaches SAC, ie., SAVC is
asymptotic to SAC.
•SMC passes through minimum SAVC and SAC.
•The cost curves (total and variable) reflect the shapes of
corresponding physical product curves (production function,
etc.).
Switching to the Long Run
By changing the fixed factor (K) we can draw a large number
SC1
of SC curves.
SC0 is drawn for K = K0
SC2
SC0
$



C
SC1 is drawn for K = K1
SC2 is drawn for K = K2
Optimal q and
C for K = K1
Optimal q and C for K = K0
q0
q1
q2
For a constant returns to scale
production function, C would
be a ray from the origin.
q
•At the q that corresponds to optimal L for the fixed level of K, the SC curve will
be tangent to C curve. At all other points, the particular SC is above C, reflecting
the fact that SC represents costs higher than would be necessary if K were variable.
•The long run C is the “envelope” of the SC curves when K is allowed to vary.
•Every point on C is taken from a SC at its optimal combination of K and L.
Average and marginal curves - For every level of K
(fixed) there exists different SAC and SMC.
SAC0
SMC0
$
SMC2
SAC0
SMC1
SAC0'
SAC1

A

q0 = q0(L0,K0) q0' = q0'(L0',K0)
q0' = q0'(L0',K0')
q1 = q1(L1,K1)

MC
SAC2
AC
Level of K indicates
the size of “plant”.
K2 > K1 > K0' >K0
q2 = q2(L2,K2)
q
•If the firm has K0 and is producing q0, it could reduce cost per unit of output by
increasing labor to L0' at minimum SAC0; but q0' could be produced at a lower cost
with K0' reflected by SAC0'. Further reduce average cost by increasing L to
minimum SAC0′, which q could be produced with more K, etc.
•Each SAC is tangent to AC at the optimal amount of L for that level of fixed K,
but that point is not the minimum point for that SAC except at point A where SAC
is tangent to the AC curve at minimum AC and at minimum SAC1.
• All SMC curves cross their respective SAC curves at minimum
SAC.
• SMC curves cross MC at the C minimizing level of L for the fixed
level of K.
• MC is made up of many optimal points from the SMC curves. At
q1, minimum SAC1 equals minimum AC, and SMC1 equals MC.
• The most efficient scale is K1 capital with q1 produced.
• The slopes of the SMC curves are greater than the slope of the MC
curve because of diminishing returns with fixed K. The slopes of
the SAC curves reflect diminishing returns also.
• Only at q1 will the firm be “satisfied” in the long run because, if the
firm is at a point other than the minimum on any SAC curve, it can
reduce costs for that level of fixed K by changing L and q to
minimum SAC. But for all SAC curves other than SAC1, the q
produced at the minimum point on the SAC curve could be
produced with another level of K at a lower cost.