Advanced Microeconomics
Prot maximization and cost minimization
Jan Hagemejer
November 28, 2011
Jan Hagemejer
Advanced Microeconomics
The plan
Prot maximization and the prot function
Cost minimization and the cost function
Examples
Jan Hagemejer
Advanced Microeconomics
Introduction
We have introduced the production sets, production function, inputs
and outputs.
Now we add the market: providing the rm with prices and factor
wages.
We will be making an assumption of price taking.
What will the rm do?
Maximize prots given prices and factor wages (choose optimal input
combination AND level of output to maximize prots) - PMP
Minimize costs of production given prices and factor wages AND
desired production level (choose optimal input combination GIVEN
output level) - CMP
We will analyze the problems separately
Jan Hagemejer
Advanced Microeconomics
Prot maximization problem
The formal denition:
(with production set
vector
y ∈ RL :
the prot is
Y ) given a price vectorm p 0 and a production
π(p ) = p · y =
cost)
PL
l =1 pl yl .
(total revenue minus total
(1) the prot maximization problem (PMP):
Max
y
(with transformation function
p · y,
s.t.
y ∈ Y.
Y ):
(2) the prot maximization problem (PMP):
Max
y
p · y,
Jan Hagemejer
s.t.
F (y ) ≤ 0.
Advanced Microeconomics
Prot maximization
z1 , . . . , zL−1 .
the production function q = f (z ) = f (z1 , . . . , zL−1 )
PL−1
the prot is π(p , w1 , . . . , wM ) = pq −
l =1 wl zl . CAUTION: now
w will stand for wage per unit of employed input!
(3) (the most common way): with many inputs
the prot maximization problem (PMP):
Max
q,z
L−1
X
pq −
or easier by substitution of
l =1
s.t.
q = f (z ).
q = f (z ):
Max
z
wl zl ,
pf (z ) −
Jan Hagemejer
L−1
X
l =1
wl zl ,
Advanced Microeconomics
Prot maximization
Example with 1 input
with prots
y1 and one output y2 .
π = p1 y1 + p2 y2 ,
π ) is:
The prot is
so the isoprot line (connecting all points
y2 = π/p2 −
PMP problem is to nd a highest
π
Jan Hagemejer
p1
y
p2 1
that is feasible
Advanced Microeconomics
Prot maximization
In the easy form (one output, many inputs), the problem is:
Max
z
π(z ) = pf (z ) −
L −1
X
l =1
wl zl .
The rst order conditions:
∂π(z )
=0 :
∂ zl
Interpretation:
p
∂ f (z )
= wl ,
∂ zl
for all
l = 1, . . . , L − 1
pMPl = wl , or, in terms of real
wl
= MPl
p
Jan Hagemejer
wages:
Advanced Microeconomics
The solution to PMP
The solution to PMP is:
zl (p, w )
the supply function q (p , w ) = f (zl (p , w ))
PL−1
and the prot function π(p , w ) = pq (p , w ) −
l =1 wl zl (p , w ).
the vector of optimal factor demands
Jan Hagemejer
Advanced Microeconomics
The solution to PMP
Note that taking any
l
and
k
so that
l , k ∈ 1, . . . , L − 1 and dividing the
corresponding FOCs, we get:
∂ f (z )
∂ zl
∂ f (z )
∂ zk
=
MPl
MPk
= MRTSlk =
wl
wk
We will come to that later....
Jan Hagemejer
Advanced Microeconomics
The general case
Max
y
p · y,
F (y ) ≤ 0.
s.t.
We have to set up the Lagrange function:
L=
L
X
l =1
pl yl − λF (y )
And the FOC's are:
∂L
∂ F (z )
=0 : λ
= pl ,
∂ yl
∂ yl
for all
l = 1, . . . , L − 1
Doing the same procedure as before, we have that:
∂ F (y )
∂ yl
∂ F (y )
∂ yk
= MRT lk =
Jan Hagemejer
pl
pk
Advanced Microeconomics
The general case
The solution to the problem are the:
prot function
π(p ) = max{p · y : y ∈ Y }
net supply correspondence
y (p) = {y ∈ Y : p · y = π(p)}.
Jan Hagemejer
Advanced Microeconomics
The Hotelling lemma
If we have the prot function
function
π(p , w )
π(p )
in the general case or the prot
in the one output case we can:
p)
yl (p) = ∂π(
∂ pl
∂π(p ,w )
get the supply function: q (p , w ) =
∂p
∂π(p )
get the factor demand function zl (p , w ) = −
∂ wl
get the net supply function:
Jan Hagemejer
Advanced Microeconomics
Example
q = f (z ) = z α , price p and factor wage w .
Note: α > 1 → IRS , α < 1 → DRS , α = 1 → CRS
Prots (assume initially that 0
< α < 1):
π(p , w ) = pq − wz = pz α − wz
FOC:
∂π(·)/∂ z = αpz α−1 − w = 0
Solution:
factor demand
supply
z (p, w ) = (α wp )1/(1−α)
q (p, w ) = (α wp )α/(1−α)
prots
p 1/(1−α)
π(p , w ) = p (α wp )α/(1−α) − w (α wp )1/(1−α) = w ( 1−α
α )(α w )
Jan Hagemejer
Advanced Microeconomics
Extra (simplication)
π(p , w ) = p (α wp )α/(1−α) − w (α wp )1/(1−α) =
p(α wp )−1 (α wp )1/(1−α) − w (α wp )1/(1−α) =
= (p (α wp )−1 − w )(α wp )1/(1−α) = ( wα − w )(α wp )1/(1−α) =
p 1/(1−α)
= w ( 1−α
α )(α w )
Jan Hagemejer
Advanced Microeconomics
Second order conditions and returns to scale
∂π 2 (·)/∂ z 2 = α(α − 1)pz α−2 < 0
Only if: 0
So, if
< α < 1.
α>1
(IRS) it is actually a local minimizer and no prot
maximizing output exists (it is innite!).
What if
α=1
(CRS)?
π(p , w ) = pz − wz
The FOC is:
p=w
and the supply is:
p<w
q = z if p = w


∞
if p > w


0
Jan Hagemejer
if
Advanced Microeconomics
General conclusions
In the prot maximization problem, the optimal input choices are
such that:
pMPl = wl
The prot maximization problem with price taking works if:
DRS: we can determine supply and inputs level
CRS: we cannot determine supply but only inputs combinations
IRS: the prot maximizing solution does not exist or yields negative
prots (example)
We can back out factor demands and supply from the prot function
using the Hotelling lemma.
Jan Hagemejer
Advanced Microeconomics
The cost minimization problem (CMP)
We may redene our problem:
Given the desired output
the q at minimum cost.
q - nd the input combination that gives
Useful to derive cost function - relationship between output level
and the total cost of inputs.
Useful to nd inputs combinations when prot maximization does
not yield a determinate prodution level.
Jan Hagemejer
Advanced Microeconomics
The cost minimization problem (CMP)
Concentrate on one output case:
The total cost of production is:
level is:
q = f (z ).
C (z ) = w · z =
P
l wl zl .
The production
The problem is:
Min
z
w · z subject to q = f (z )
The Lagrange function:
L=
X
l
wl zl − λ(f (z ) − q )
The FOC's are :
wl = λ
∂ f (z )
for
∂ zl
all
∈ 1, . . . , L − 1 and f (z ) = q
Jan Hagemejer
Advanced Microeconomics
The cost minimization problem (CMP)
The solution to the problem gives
conditional factor demands zl (q , w ) for all l
P
the cost function C (q , w ) =
l wl zl (q , w )
Taking FOC's for any l and k and dividing with one another gives:
wl
wk
∂ f (z )
∂f
= ∂ f (lz ) =
∂ fk
MPl
MPk
= MRTSlk
The solution to the two problems (PMP and CMP) coincides at the
prot maximizing
q.
Jan Hagemejer
Advanced Microeconomics
The cost minimization problem (CMP)
We nd the lowest
q.
isocost line:
c
c=
is a constant
isocost line tangent to the isoquant corresponding to
P
l wl zl
in the two input case:
c = w1 z1 + w2 z2 where
Homothetic production function: the factor demands lie on rays from
the origin (factor ratios remain constant).
Jan Hagemejer
Advanced Microeconomics
Marginal cost pricing
When we have the cost function,
Max
q
c (q , w ), we can restate the PMP:
p · q − C (q , w )
The FOC is:
p=
or in other words:
∂ C (q , w )
∂q
p = MC
Jan Hagemejer
Advanced Microeconomics
Sheppard's lemma
If we have the cost function, we can recover the conditional factor
demand:
zl (q , w ) =
∂ C (q , w )
∂ wl
Analogy to the:
Hotelling lemma
Duality result of the consumer optimization
Jan Hagemejer
Advanced Microeconomics
Geometry of costs
C (q , w ) dene the:
∂ C (q ,w )
marginal cost: MC (q , w ) =
∂q
C
(q ,w )
average cost: AC (q , w ) =
q
Given the cost function
In the short run we will have xed levels of some inputs. The rm will
take their level as given (no FOC's with respect to those inputs).
wf zf where f 's are those
l ∈ 1, . . . , L − 1 for which the inputs are xed. In that case:
Emergence of xed costs (FC) -
FC =
P
f
C (q , p, w ) = FC (w , z̄ ) + VC (q , w )
Where
VC (q , w ) is the total cost of all the variable inputs (variable cost).
Jan Hagemejer
Advanced Microeconomics
Supply function
prots are
≥0
if
p > AC
p = MC
rm produces q > 0 if MC > AC
prot maximization implies:
the supply function is the segment of
MC
that is above the
curve.
Jan Hagemejer
Advanced Microeconomics
AC
Cost functions and returns to scale
CRS or f (z ) is homogeneous of degree one
f (λz ) = λf (z ))) then z (q , w ) and C (q , w ) are homogeneous of
If we have
(
degree one in output.
To increase production by λ%, we need to increase inputs by λ%,
therefore costs increase by λ%
MC = AC
f (z ) is concave (or the production set is convex - so we have
f (λz ) ≤ λf (z ), λ > 1), then
C (q , w ) is convex
If
non-increasing returns to scale,
To increase production by λ%, we need to increase inputs by more
than λ%, therefore costs increase by more than λ%
MC is non-decreasing in q (second derivative of a convex function is
≥ 0).
C (λq , w ) ≥ λC (q , w ). Therefore AC (λq ) ≥ λAC (q , w ), for λ > 1
we have non-decreasing AC (q , w )
MC ≥ AC : Proof in class....
Jan Hagemejer
Advanced Microeconomics
Cost functions and returns to scale
AC , AC < MC
CRS - cost function linear in q , AC = MC = const
IRS - concave cost function, decreasing AC , AC > MC
DRS - convex cost function, increasing
Jan Hagemejer
Advanced Microeconomics
Geometry of costs - strictly convex technology
Example: if
w = p = 1 then the cost function is the production function
ipped 90 degrees.
Example:
q = f (z ) = z 0.5 → z (q ) = q 2 . Cost: C (w , q ) = wz (q ) = wq 2 ,
AC = wq ,
MC = 2wq , MC > AC .
p, w > 0
If
p = MC , then p > AC
Jan Hagemejer
and
π > 0,
Advanced Microeconomics
at any
Geometry of costs - CRS
q = f (z ) = z → z (q ) = q . Cost: C (w , q ) = wz (q ) = wq ,
MC = w = AC . When p = MC , π = 0!!! (general result for CRS)
Example:
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Advanced Microeconomics
Geometry of costs - non-convex technology
q = f (z ) = (z − z̄ )0.5 → z (q ) = q 2 + z̄ . Cost:
C (w , q ) = wz (q ) = (q 2 + z̄ )w
MC = 2qw ,
AC = wqz̄ + wq ,
VC (q ) = wq 2 , FC (q ) = w z¯.
At a prot maximizing point p = MC . Therefore for π > 0 we need
p = MC > AC .
Example:
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Advanced Microeconomics
Cobb-Douglas technology again
q = f (z1 , z2 ) = z1α z2β
C (w1 , w2 , q ) = q
where
θ=
β
α+β
α
β
+
−α
α+β
α
β
and
1
α+β
θφ(w1 , w2 ),
φ(w1 , w2 ) = w1
α/(α+β)
Our results apply:
α + β < 1, DRS, cost function convex in q
α + β = 1, CRS, cost function linear in q
α + β > 1, IRS, cost function concave in q
Jan Hagemejer
Advanced Microeconomics
w2β/(α+β) .