L08_Cost_Minimization_2015

Report
Cost Minimization
Accountants vs. Economists
• Economists think of costs differently from accountants.
– Economics Cost = Explicit Costs + Implicit Costs
– Accounting Cost = Explicit Costs
– Opportunity Cost is synonymous with Cost
– Measuring implicit opportunity costs is difficult
• generally, these are resources “owned” by the firm
• Includes the value of the entrepreneur’s time
• Includes accounting profit at next best alternative use of
firm’s resources – or, accounting profit from selling all
owned resources and investing the proceeds in the best
possible alternative investment.
Fixed Cost and Sunk Cost
• Fixed vs. Sunk cost
– “fixed” means its cost does not vary with output
– “sunk” means its cost cannot be avoided
• The security guard outside the parking lot is a fixed cost,
but if they can be fired at a moment’s notice, the cost is
not sunk
• A long term lease for renting a building is fixed and sunk
– Varian calls a fixed cost that is not sunk a “quasifixed cost”
– Sunk, but not fixed? No, because if it varies, it can
be avoided.
Short Run – Long Run
• Key distinction in terms of optimizing
behavior if everything can be controlled vs.
not everything can be controlled.
• Short run, the quantities of some inputs used
in production are fixed and some are
variable.
– Short run, firms can shut down (q=0), but cannot
exit the industry
• Long run, the quantity of all inputs used are
variable.
– Long run, firms can enter or exit an industry.
To Distinguish Short Run from Long Run
• We need two inputs, one always variable, and one that
can only be adjusted periodically, but is then re-fixed.
• To accomplish that, we assume two inputs
• Homogeneous labor (L), measured in labor per time
• Homogeneous capital (K), measured in machine per
time
• Entrepreneurial costs assumed to be zero (or
included in fixed costs)
• Inputs are hired in perfectly competitive markets, so
price of L, w, and price of K, v, are not a function of L
and K
To be clear
• L is assumed always variable
• K is always fixed when the firm is in
production, but can periodically be adjusted
to a new fixed amount (so in the long run it
can be varied to a new fixed amount).
• That is, production always takes place in a
short run situation.
Notation
• K, capital; v, rental rate of capital
• L, Labor; w, wage rate
5340 Nomenclature
(Typical Intermediate Micro Abbreviations in Parentheses)
Short Run
Long Run
SC (TC)
C (LRTC)
SAC (ATC)
AC (LRAC)
Variable Cost
VC
-
Average Variable Cost
AVC
-
Fixed Cost
FC
-
Average Fixed Cost
AFC
-
SMC (MC)
MC (LRMC)
Total Cost
Average Total Cost
Marginal Cost
Cost
• Clearly, C can be stated as:
C = vK + wL
• But this is meaningless, we need cost as a
function of output, not K and L.
• To get that, back to the production function
Graphically
q
q3
q2
K
q1
q = f(K, L)
q0
An entire range
of input
combinations can
be used to
produce every
level of output.
q0
q2 q3
q1
L
Short Run, K constant
q = f(K=K3, L)
q
q = f(K=K2, L)
q = f(K=K1, L)
q3
q2
q1
Possible to produce the
same level of output at
different combinations
of K and L.
Three different slices through the production
function at different levels of K.
L
Short Run, K constant
q
Let’s look at
just one slice.
q2
q   K 2 ,L 
At some fixed level of K, for
every q, we know how much L
it will take.
L2
L
Production to Cost
Flip the axis
qL
q   K 2 ,L 
L  L  q, K 2 
Lq
Production to Cost
L
L = L(q,K)
q
Production to Cost
$/time
Multiply L by w to
get labor cost
SC = w·L(q)+vK
Add in FC,
SC=SVC+FC
VC = w· L(q)
FC
q
Production to Cost
$/time
And we can figure a
few things
SC = w· L(q)+rK
VC = w· L(q)
SM C 
SA C 
d SC
dq
SC
q

dVC
dq
, AVC 
SC  V C  FC
SA C  A V C  A FC
VC
q
FC
q
Production to Cost
$
SMC
SAC
AVC
SM C 
SA C 
d SC
dq
SC
q

dVC
dq
, AVC 
VC
q
SC  V C  FC
SA C  A V C  A FC
AFC
q
Short Run, K constant
q = f(K=K3, L)
q
q = f(K=K2, L)
Inflection
point, q3
Inflection
point, q2
q = f(K=K1, L)
With more K, you can
produce more with a
given L and the
inflection point moves
Inflection towards more L and q.
point, q1
L
What does that mean for cost curves?
Production to Cost
$
SC = w· L2(q)+rK2
SC = w· L3(q)+rK3
Inflection Points
FC
SC = w· L1(q)+rK1
q
Fixed cost is higher with more K, but the inflection point is further
to the right, with a slower build-up of crowding.
C
$
C
SC = w· L2(q)+rK2
SC = w· L3(q)+rK3
SC = w· L1(q)+rK1
q
C is the lowest point on any of the SC curves for any q
SAC
$
AC 
C
Min value at higher q with more K.
SAC = C1/q
SAC = C2/q
SAC = C3/q
q
LAC
q
AC is the lowest point on any of the SR ATC curves for any q
MC
C
$
MC is the slope of the C curve
at any q
MC 
C
q
q
Short Run – Long Run
• Firms ALWAYS produce in a short run situation
with at least one fixed and one variable input.
• Being in the long run simply means the firm
has adjusted to an optimal level of capital to
minimize the cost of producing any chosen
(profit maximizing) level of output.
Different Shapes for the long run
C Curve
C
$
There are four different
potential shapes
q
C -- CRS
C
$
CRS, as K and L are scaled upwards,
C rises at the same rate.
That is, MC is constant.
q
C -- CRS
$
Min value at higher q with more K.
ATC = TC1/q
ATC = TC2/q
ATC = TC3/q
AC=MC
q
CRS, as K and L are scaled upwards, LAC rises at the same rate.
That is, LAC is constant.
C -- IRS
C
$
IRS, as K and L are scaled
upwards, C rises more slowly
That is, MC is decreasing.
q
C -- IRS
$
AC
MC
q
CRS, as K and L are scaled upwards, AC rises at a slower rate. AC is
decreasing.
C -- DRS
$
DRS, as K and L are scaled upwards,
TC rises faster
That is, MC is increasing.
C
q
LTC -- DRS
$
Note, it is not the low point of each ATC
curve, but the simply the lowest point
on any ATC.
MC
AC
q
CRS, as K and L are scaled upwards, AC rises at a faster rate. AC is
increasing.
C – IRS, CRS, DRS
$
C
IRS, then CRS, then DRS
q
C – IRS, CRS, DRS
$
So we have a U-shaped AC curve for a
completely different reason than the
SAC curve is U-shaped.
MC
AC
q
Firm Decisions
• So which short run curve should we be on?
– That is, how much K (and then L) do we want to hire
(with K fixed in the SR)?
• Two choices
– Profit Maximization: Firms maximize profit by
choosing q*, K and L to minimize cost all at once.
– Cost minimization: Firms choose a q*, then choose K
and L to minimize the cost of producing q*.
Profit Maximization
• Economic profits () are equal to
 = total revenue - total cost
• Total costs for the firm are given by
C = wL + vK
• Total revenue for the firm is given by
total revenue = R = p·q = p·f(K,L)
• Economic profits () are equal to
 = p·f(K,L) - wL - vK
Profit Maximization
• Solving to maximize profit means jointly
choosing q = q* along with K* and L*.
– Yields profit maximizing factor (input) demand
functions which provide L* and K* that minimizes
the cost of producing q*.
– When r, w, and p change, L*, K*, and q* all change.
– L*=L(w, v, p); K*=K(w, v, p); q*=q(w, v, p)
– These functions allow for a change in q* (the
isoquant) when prices change.
Cost Minimization
• That firms minimize cost is a weaker hypothesis
of firm behavior than profit maximization.
– Yields quantity constant (quantity contingent)
factor (input) demand functions which provide L*
and K* that minimize the cost of producing q0.
– When r and w change, L* and K* do change, but q
does not, stay on one isoquant.
• Why bother? It is how we get the cost
functions and curves (i.e. what we use in
principles and intermediate micro)
New Direction in Graphing
• In all the graphs above, we have illustrated the
long run as a series of short run curves and
traced out the envelope.
• Good for intuition, but not terribly tied to the
math of the optimization
• Let’s switch to the isoquant graph.
Intuitively
K
Isoquant, all combinations of factors
that yield the same output.
Slope is -dK/dL
Isocost (total cost), all
combinations of factors
that yield the same
total cost of
production.
Slope is -w/v
L
Cost-Minimizing Input Choices
• When K and L change a small amount
dq  fK  dK  fL  dL
• Along an Isoquant,

dK
dL
 R TS 
M PL
M PK
Cost-Minimizing Input Choices
• TRS is the change in K needed to replace one L
while maintaining output.
• Minimum cost occurs where the TRS is equal to
w/v
– the rate at which K can be traded for L in the
production process = the rate at which they
can be traded in the marketplace
M PL
M PK

w
v
Intuitively
K
w=20, v=10
w/v=2
TRS=.25
• Firing one L has MB of $20 and MC of $2.5
• Lowers cost by $17.50
L
Intuitively
K
w=20, v=10
w/v=2
TRS=1
• Firing one L has MB of $20 and MC of $10
• Lowers cost by $10
L
Intuitively
K
w=20, v=10
w/v=2
TRS=1.5
• Firing one L has MB of $20 and MC of $15
• Lowers cost by $5
L
Intuitively
K
w=20, v=10
w/v=2
TRS=2
L
• Firing one L has MB of $20 and MC of $20
• Lowers cost by $0
• Further reductions in L require an increase in cost as TRS >w/r.
Intuitively
K
w=20, v=10
w/v=2
TRS=2.5
• Firing one L has MB of $20 and MC of $25
• Raises cost by $5
L
Plan
1. Figure out the quantity of K and L that
minimize total cost, holding q constant.
2. Use the resulting factor demand curves to
derive the cost functions.
Cost-Minimizing Input Choices
• We seek to minimize total costs given q =q0
q = f(K,L) = q0
• Setting up the Lagrangian:
L  w L  vK    q 0  f  K ,L  
FOCs are
L
L
L
K
L

 w    fL  0
 v    fK  0
 q 0  f  K ,L   0
Cost-Minimizing Input Choices
• Dividing the first two conditions we get
w
v

fL
fK
 TRS LK (L for K)
• The cost-minimizing firm should equate the
RTS for the two inputs to the ratio of their
prices
• But also
f
f
L
w

K
v
• Which tells us that for the last unit of all inputs
hired should provide the same bang-for-thebuck.
Cost-Minimizing Input Choices
• The inverse of this equation is also of interest
w
fL

v
fK

• The value of the Lagrange multiplier is the extra costs that
would be incurred by increasing the output constraint slightly
by hiring enough L or K to increase output by 1.
• That is λ = MC of increasing production by one unit.
Cost-Minimizing Input Choices
• SOC to ensure costs are RISING, along the
isoquant, away from the tangency:
• Bordered Hessian
 L L
 L K
H 3   LL 
  LLL
  LLK   fL
  fLL
  fLK  0
 LK 
  LKL
  LKK
  fKL
  KK
0
0
 fK

 fL
 fK

2
2
H 3   (  fL fK fLK )  (  fK fL fKL )   (  fL fKK )  (  fK fLL )   0


H 3    2 fL fK fLK  fL fKK  fK fLL  0
H2 
0
 fL
 fL
  fLL
2
  (  LL )  0
2
2
The part in brackets is the same
condition required for strict quasi
concavity of the production
function
Intuitively
w=20, r=10
w/r = 2
K
Total Cost
RTS=2
SOC satisfied as
moving along the
isoquant means
increasing TC
K*
L*
L
• Cost minimized for q0 when L=L* and K=K *
SOC fail, Intuitively
w=20, r=10
w/r = 2
K
Total Cost
RTS=2
SOC not satisfied as
moving along the
isoquant means
decreasing TC
K*
L*
L
• Cost maximized for q0 when L=L* and K=K *
Conditional Factor Demand
(aka Constant Output)
• Solve the FOC to derive
– K * =Kq (w, v, q0)
– L * =Lq (w, v, q0)
• Earlier, we considered an individual’s expenditureminimization problem
– to develop the compensated demand for a good
• In the present case, cost minimization leads to a
demand for capital and labor to produce a constant
quantity of output.
The Firm’s Expansion Path
The expansion path is the locus of cost-minimizing
tangencies
K
K=KE(w, v, L)
The curve shows how
inputs increase as
output increases
K*
K*
K*
L*
L* L*
L
The Firm’s Expansion Path
• The cost-minimizing combinations of K and L for
every level of output (according to the quantity
constant factor demand functions)
• If input prices remain constant for all amounts of
K and L, locus of cost-minimizing choices is the
expansion path
• Solve the minimization condition for K to get the
expansion path
w
v

fL
fK
K  K (w , v ,L)
E
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
Cost Functions
• Long Run Cost
C*  C(v , w , q)  v  K (v , w , q)  w  L (v , w , q)
q
q
v  K (v , w , q)  w  L (v , w , q)
q
A C  A C(v , w , q) 
q
q
d  v  K (v , w , q)  w  L (v , w , q) 
q
M C  M C(v , w , q) 
q
dq
• Short Run Cost
– The factor demand curves we have just derived are
not appropriate as they each assume L and K are
variable.
Short Run Cost Functions
• With only two inputs, once K is fixed, the production
function dictates how much L is needed for each q.
I am just using the “s”
superscript here to
denote Short Run
 
q  f K ,L

th e n , L  L K , q
*
s



VC  w L K,q , AVC 
s



w L K,q
s
q
SC  w  L K , q  w  K , A TC 
s
SM C 
d TC
dq

dVC
dq



w L K,q  w K
s
q
Short Run & Long Run
• Pick a level of output, q1 and hold w and v constant
A llo w K 1  K(q 1 ) su ch th a t K 1 is th e co st m in i m izin g le ve l o f ca p ita l to p ro d u ce q 1



SC  w  L q;K 1  w  K 1 o r SC  SC q;K 1
s

If w e a llo w K  K* = K(q ), th e n w e ge t th e lo n g ru n C * fu n ctio n
C*  SC  q,K  K(q)  a n d sin ce e ve r yth in g is n o w a fu n ctio n o f q , C * = C(q)


W e th e n kn o w SC q,K   C(q), fo r a ll q

C

SC= C q,K 1


a s SC q,K   C *  q  a t a n y q  q 1

C*  C  q,K(q) 

a n d SC q,K   C *  q  a t q  q 1
A lso ,
d SM C
dq

dM C
dq
w h e n e va lu a te d a t q 1
q1
q
Short Long Relationship
$
MC
SMC
ATC
AC
AVC
q
SMC vs. LMC
$
When q falls in
SR, cost falls
by less than in
the LR;
Means from
q1 to q2 the
SMC is lower
as C starts
higher at q1.
When q rises in SR, cost rises
by more than in the LR
K  K1
q2
q1
q3
q
Short Long Relationship
(Easier to see if assume CRS)
$
SMC actually lower from q1 to q2 in the short run as
total cost is higher to start
SMC (K=K1)
ATC (K=K1)
MC=AC
AVC (K=K1)
q1 q2 q3
q
Fixed Proportions Factor Demand and
Cost Functions
• Suppose we have a fixed proportions technology
such that
q = f(K,L) = min(aK,bL)
• To minimize cost, production will occur at the vertex
of the L-shaped isoquants where q = aK = bL (any
extra K or L only drives up cost)
b
K

• Expansion path:
 L
a
• Factor Demand Curves Kq=q/a and Lq=q/b
• Cost function: C(w,v,q) = vK + wL = v(q/a) + w(q/b)
v w
v w
C(w , v ,q)  q    , M C  AC    
a b 
a b 
Cobb-Douglas Cost Minimization
• Suppose that the production function is CobbDouglas: q = K L
• The Lagrangian expression for cost minimization of
producing q0 is
ℒ = vK + wL + (q0 - K  L )
• The FOCs for a minimum are
ℒK = v - K -1 L = 0
ℒL = w - K  L -1 = 0
ℒ = q0 - K L = 0
Cobb-Douglas Cost Minimization
• Dividing the first equation by the second gives us
input price ratio=
w
v
  1

K L
K
 1 
L

M PL
M PK
 K
input price ratio=
   TRS
v  L
w
• This production function is homothetic
– the TRS depends only on the ratio of the two inputs
Cobb-Douglas Cost Minimization
• Expansion path equation:
– the expansion path is a straight line
K
wL
v
– The K/L ratio is a function of w and v.
K
L

w
v
Cobb-Douglas Input Demand
• Using the remaining FOC (production function), solve for
the input demand equations
K* = Kq (v,w,q)
L* = Lq (v,w,q)
1
L q
q
 
1
K q
q
 

  v   


 w 

  w   



v


Cobb-Douglas Cost
• Now we can derive total costs as
C(v , w ,q)  vK  w L  q
– Where
– MC
 1 


   
Bv
  


   
(   )
B

  


   

M C(v , w ,q) 
  


   
q
 1   


   
Bv
  


   
w
  


   

– AC
A C(v , w ,q)  q
 1   


   
Bv
  


   
w
  


   
w
  


   
CES Cost Minimization
• Suppose that the production function is CES:
q = (K +L)/
• The Lagrangian expression for cost minimization of
producing q0 is
ℒ = vK + wL + [q0 - (K  +L )/]
• The FOCs for a minimum are
ℒ L = w - (/)(K +L)(-)/()L-1 = 0
ℒ K = v - (/)(K +L)(-)/()K-1 = 0
ℒ  = q0 - (K + L)/ = 0
CES Cost Minimization
• Dividing the first FOC equation by the
second gives us
L
 
v K
w
 1
L
 
K
 (    1)
K
 
L
1
 TRS
• This production function is also homothetic
CES Expansion Path
• Expansion path

1
w
K    L , =
1
 v 
• Capital Labor ratio

1
w
   , =
L  v 
1
K
CES Input Demand
• Using the remaining FOC (production function), solve for
the input demand equations
K* = Kq (v,w,q)
L* = Lq (v,w,q)
1
L 
q
q

1

 
1
1
w
v


, =
 
 w


1
1
1
K 
q
q

1

 
1
1
w
v


 
 v


, =
1
1
CES Cost Functions
• To derive the total cost, we would use the inputs
demand functions and get
C(v , w , q)  vK  w L  q
C(v , w , q)  q
w h ere  
1
 

v
1
1
 

w
  
   

 
  1 
  1 
v

w




1

 1 


 1 
1
1
M C(v , w , q) 
q
A C(v , w , q)  q
 1  


  
v
1
w
1

 1 


 1 

 1  


  
v
1
w
1

 1 


 1 
  1 


  
Law of Demand?
• With Cobb-Douglass, CES etc. we can take the
partial derivative w.r.t. price of the inputs and
see how quantity demanded responds.
• But if we don’t know the functional form, how
do we know how demand for L and K will
respond to changes in w and v?
• Comparative Statics
Comparative Statics
• Setting up the Lagrangian:
ℒ = wL + vK + [q0 - f(K,L)]
• FOCs are
ℒL = w - ·fL = 0
ℒK = v -  ·fK = 0
ℒλ = q0 - f(K,L) = 0
• Solve for
L* = Lq(w,v,q)
K* = Kq (w,v,q)
λ* = λq (w,v,q)
Comparative Statics
• Plug solutions into FOC
w - λ(w,v,q)·fL(Lq(w,v,q), Kq (w,v,q)) ≡ 0
v - λ(w,v,q)·fK(Lq (w,v,q), Kq (w,v,q)) ≡ 0
q0 - f(Kq (w,v,q), Lq (w,v,q)) ≡ 0
• These are identities because the solutions (FOC)
are substituted into the equations from which
they were solved.
• Whatever prices and output may be, the firm will
instantly adjust K and L to minimize cost of that
level of output.
Comparative Statics
• Differentiate these w.r.t. w
w   (w , v , q)·fL (L (w , v , q),K (w , v , q))  0
q
q
v   (w , v , q)·fK (L (w , v , q),K (w , v , q))  0
q
q
q 0  f(K (w , v , q),L (w , v , q))  0
q
q
To get:
1
0
*
q
2
L
*
LL w
*
q
L
2
*
0
L w
*

2
K
-
*
q K
*
q
2
-
K
K w
*
 0
L w
*
KK w
 0
 q 
*
LK w
*
KL w
q L
-
q
-
 q 
*
K w
 0
Notationally,
replace, Lq(w,v,q)
with L*, etc.
Matrix Notation and Cramer’s Rule
   * fLL

*


fKL

 f
L

  fLK
*
  fKK
*
 fK


 fL  

 fK  
0  



*
L 

w 
*
K 

w

*
 
 w 
 1 


0 ,


 0 


  fLL
  fLK
 fL
  fKL
  fKK
 fK  0
 fL
 fK
*
*
C ram er's R u le
L
*
w

1
  fLK
 fL
0
  fKK
*
 fK
0
 fK
0
*
H
 (  1(  fK ) )
2


0
*
*
0
Matrix Notation and Cramer’s Rule
   * fLL

*


fKL

 f
L



 fL  

 fK  
0  



  fLK
*
  fKK
*
 fK
*
L 

w 
*
K 

w

*
 
 w 
 1 


0 ,


 0 


  fLL
  fLK
 fL
  fKL
  fKK
 fK  0
 fL
 fK
*
*
C ram er's R u le
  fLL
1
 fL
  fKL
0
 fK
 fL
0
0
*
*
K
*
w

H

 1(  fL )(  fK )

0
*
*
0
Matrix Notation and Cramer’s Rule
C ra m e r's R u le
  fLL
  fLK
1
  fKL
  fKK
*
0
 fL
 fK
0
*
*

*
w

H
  [(fKL )(fK )]  [(fKK )(fL )]
*

*

[(  1)(   fKL )(  fK )]  [(  1)(   fKK )(  fL )]
*

*

*
  [(fKL )(fK )  (fKK )(fL )] 

0
• With a higher wage, the amount of L used will definitely fall.
• It is possible that K becomes more productive with less L being used (if fKL< 0).
• If this effect is large enough, the higher productivity of capital can more than
compensate for the higher cost of labor and MC can fall.
• Unlikely in the real world.
Comparative Statics
• Differentiate these w.r.t. q
w   (w , v , q)·fL (L (w , v , q),K (w , v , q))  0
q
q
v   (w , v , q)·fK (L (w , v , q),K (w , v , q))  0
q
q
q 0  f(K (w , v , q),L (w , v , q))  0
q
q
To get:
0
0
*
q
L
2
*
LL q
*
q
L
2
*
1
L q
*

2
K
*
-
LK q
*
-
*
q K
*
KL q
q L
-
q
q
2
K
*
KK q
K q
 0
-
 q 
*
 0
L q
 q 
*
K q
 0
Matrix Notation and Cramer’s Rule
   * fLL

*


fKL

 f
L

  fLK
*
  fKK
*
 fK


 fL  

 fK  
0  



*
L 

q 
 0 
*

K


   0 ,
q  


1


*
 

q 
  fLL
  fLK
 fL
  fKL
  fKK
 fK  0
 fL
 fK
*
*
*
*
0
C ra m e r's R u le
L
*
q

0
  fLK
 fL
0
  fKK
*
 fK
1
 fK
0
*
H
[(  1)(   fLK ) (  fK )]  [(  1)(   fKK )(  fL )]
*

*

  [fLK  fK  fKK  fL ]  0 if fLK  0 b u t co u ld b e < 0 if fLK  0
*
If labor is an inferior input, we COULD minimize cost with more K and
less L. I.e., a “backward bending” expansion path.
Matrix Notation and Cramer’s Rule
   * fLL

*


fKL

 f
L



 fL  

 fK  
0  



  fLK
*
  fKK
*
 fK
*
L 

q 
 0 
*
K  


0
,
 

q  


1


*
 

 q 
  fLL
  fLK
 fL
  fKL
  fKK
 fK  0
 fL
 fK
*
*
*
*
0
C ra m er's R u le
  fLL
  fLK
0
  fKL
  fKK
*
0
 fL
 fK
1
*
*

*
q

*
H
*
2
  [(fLL )(fKK )]  [(fKL ) ] 

[(  1) (   fLL )(  fKK )]  [(  1)(   fKL )(   fLK )]
*

*
*

0
If there is IRS, although total cost is still rising, MC may well be falling
along the expansion path.
Shephard’s Lemma (Again)
• Remember how we used Shephard’s Lemma to
derive the compensated demand curves from
the expenditure function?
• We can do it again from the cost function
L *  vK  w L  
*
 L * (v , w , q)
w
 L * (v , w , q)
w
*


*

q 0  f(K ,L )
 C(v , w , q)
w
 C(v , w , q)
v
*
*

 L  L (v , w , q 0 )
*
q
 K  K (v , w , q 0 )
*
q
Marginal Cost Function
And also:
 L * (v , w ,q)
q

 C(v , w ,q)
q
    (v , w ,q)
*
That is, λ*= λ(v,w,q) tells us the MC of increasing production
along the expansion path
Fixed Proportions Shephard’s Lemma
Results
• Suppose we have a fixed proportions technology
• The cost function is
v w
C(w , v ,q)  q   
a b 
• For this cost function, output constant demand functions
are quite simple:
K (v , w ,q) 
q
L (v , w ,q) 
q
 C(v , w ,q)
v
 C(v , w ,q)


q
a
q
w
b
 C(v , w ,q) v w
MC 
 
q
a b
Cobb-Douglas Shephard’s Lemma
Result
• Suppose we have a Cobb-Douglas technology
• The cost function is
C(v , w ,q)  vK  w L  q
• where
 1 


   
Bv
  


   
w
  


   
(   )
B

  


   

  


   
• then the output constant factor demand functions are:
L (v , w ,q) 
q

C *
w





q
 1 


   
q
 1 


   
 v 
B 
w
 v 
B 
w
  


   
  


   
K (v , w , q) 
q

C *
v





q
 1 


   
q
 1 


   
w
B 
 v 
Bv
  


   
  


   
w
  


   
CES Shephard’s Lemma Result
• Suppose we have a CES technology
• The cost function is
1
C(v , w ,q)  q

v
1
w
1

  


 1 
1
, w here  
1
• The quantity constant demand functions for
capital and labor are
1
L (v , w ,q) 
q
C
w
1

1
1
C
w

q


v
1
1
q

v
w
w

1
1

w
1


(1   )
(1   )
w

K (v , w , q) 
q
  


 1 
C
v
1

q


v
1
1
1

w
q

v
1
w
1
v
1
v


  


 1 

(1   )

  


 1 
(1   )
Properties of Cost Functions
• Homogeneity
– cost functions are all homogeneous of degree one in
the input prices
• a doubling of all input prices will double cost.
• As derivatives of HD1 functions are HD0, the
contingent demand functions must be HD0, that is,
a doubling of w and v will not affect the cost
minimizing input mix.
• inflation will shift the cost curves up and will not
change Kc, Lc
Properties of AC and MC functions
• Both AC and MC are HD1 meaning a doubling
of input prices means a doubling of MC and
AC
– True enough, C=C(v, w, q) is HD1 and
MC 
C
q
– However, C is HD1 in input prices, and MC is the
derivative w.r.t. q. So the derivative of an HD1
function being HD0 does not apply here.
Properties of Cost Functions
• The total cost function is non-decreasing in q, v,
and w.
• Fixed factor production has a linear cost function
w.r.t. w and v
• To the extent that one factor can be substituted
for another, the function will be concave to input
prices.
Concavity of Cost Function
At w1, the firm’s costs are C(v,w1,q1)
If the firm continues to buy
the same input mix as w
changes, its cost function
would be Cpseudo
C=wL1+v1K1
C
C*=C(v,w,q1)
C(v,w1,q1)
Since the firm’s input mix
will likely change, actual
costs will be less than C
such as C*=C(v,w,q1)
w1
w
Measuring 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
• In the Production lecture:
 K   fL 
K
d     dln  
 L   fK 
L



K
 fL 
 fL 
d 
dln  
L
 fK 
 fK 
• But more usefully, because RTS = w/v at cost
minimum
K w 
K
d
dln
   
 
L
v
   
L
s


K
w
w
d 
dln  
L
 v 
 v 
Input Substitution
• This 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
– s = 0 for fixed factor production
Size of Shifts in Costs Curves
• The increase in cost caused by a change in the price of
an input 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 – easy substitution
means little change in costs.
Appendix, Envelope Derivation
• The marginal cost function (MC) is found by
computing the change in total costs for a change
in output produced
M C(v , w ,q) 
 C(v , w ,q)
q
Marginal Cost Function and 
• Back to the cost function
C  w  L (w , v , q)  v  K (w , v , q)
*
q
 C(v , w , q)
q
 C(v , w , q)
q
 C(v , w , q)
q
q
L
*
w
q
v
K
*
q
* q L
* q K




fro m th e FO C
L q
K q
*
*
*



q

L

q

K
*
 


 L q K q 
*
Marginal Cost Function
 C(v , w , q)
q
*
*



q

L

q

K
*
 


 L q K q 
Take last FO C (i.e. th e p ro d u ctio n fu n ctio n )
q  f(L ,K)
q  f(L (w , v , q),K (w , v , q))  0
q
q
d ifferen tiate w .r.t. q
q L
*
1
L q
q L

*
1
L q

q K
*
K q
q K
*
K q
0
So as long as cost are
being minimized, this is
true
Marginal Cost Function and 
• And combine:
 C(v , w , q)
q
q L
*
*



q

L

q

K
*
 



L

q

K

q


*
1
L q
 C(v , w , q)
q
MC  
*

q K
*
K q

*
1 

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