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Advanced Microeconomics
Pro�t maximization and cost minimization
Jan Hagemejer
November 28, 2011
Jan Hagemejer Advanced Microeconomics
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The plan
Pro�t maximization and the pro�t function
Cost minimization and the cost function
Examples
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Introduction
We have introduced the production sets, production function, inputsand outputs.
Now we add the market: providing the �rm with prices and factorwages.
We will be making an assumption of price taking.
What will the �rm do?
Maximize pro�ts given prices and factor wages (choose optimal inputcombination AND level of output to maximize pro�ts) - PMPMinimize costs of production given prices and factor wages ANDdesired production level (choose optimal input combination GIVENoutput level) - CMP
We will analyze the problems separately
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Pro�t maximization problem
The formal de�nition:
(with production set Y ) given a price vectorm p � 0 and a productionvector y ∈ RL :
the pro�t is π(p) = p · y =∑L
l=1plyl . (total revenue minus total
cost)
(1) the pro�t maximization problem (PMP):
Maxy
p · y , s.t. y ∈ Y .
(with transformation function Y ):
(2) the pro�t maximization problem (PMP):
Maxy
p · y , s.t. F (y) ≤ 0.
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Pro�t maximization
(3) (the most common way): with many inputs z1, . . . , zL−1.
the production function q = f (z) = f (z1, . . . , zL−1)
the pro�t is π(p,w1, . . . ,wM) = pq −∑L−1
l=1wlzl . CAUTION: now
w will stand for wage per unit of employed input!
the pro�t maximization problem (PMP):
Maxq,z
pq −L−1∑l=1
wlzl , s.t. q = f (z).
or easier by substitution of q = f (z):
Maxz
pf (z)−L−1∑l=1
wlzl ,
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Pro�t maximization
Example with 1 input y1 and one output y2.
The pro�t is π = p1y1 + p2y2, so the isopro�t line (connecting all pointswith pro�ts π) is:
y2 = π/p2 −p1
p2y1
PMP problem is to �nd a highest π that is feasibleJan Hagemejer Advanced Microeconomics
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Pro�t maximization
In the easy form (one output, many inputs), the problem is:
Maxzπ(z) = pf (z)−
L−1∑l=1
wlzl .
The �rst order conditions:[∂π(z)
∂zl= 0
]: p
∂f (z)
∂zl= wl , for all l = 1, . . . , L− 1
Interpretation: pMPl = wl , or, in terms of �real� wages:
wl
p= MPl
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The solution to PMP
The solution to PMP is:
the vector of optimal factor demands zl(p,w)
the supply function q(p,w) = f (zl(p,w))
and the pro�t function π(p,w) = pq(p,w)−∑L−1
l=1wlzl(p,w).
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The solution to PMP
Note that taking any l and k so that l , k ∈ 1, . . . , L− 1 and dividing thecorresponding FOCs, we get:
∂f (z)∂zl∂f (z)∂zk
=MPl
MPk
= MRTSlk =wl
wk
We will come to that later....
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The general case
Maxy
p · y , s.t. F (y) ≤ 0.
We have to set up the Lagrange function:
L =L∑l=1
plyl − λF (y)
And the FOC's are:[∂L
∂yl= 0
]: λ
∂F (z)
∂yl= pl , for all l = 1, . . . , L− 1
Doing the same procedure as before, we have that:
∂F (y)∂yl∂F (y)∂yk
= MRT lk =pl
pk
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The general case
The solution to the problem are the:
pro�t function π(p) = max{p · y : y ∈ Y }net supply correspondence y(p) = {y ∈ Y : p · y = π(p)}.
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The Hotelling lemma
If we have the pro�t function π(p) in the general case or the pro�tfunction π(p,w) in the one output case we can:
get the net supply function: yl(p) = ∂π(p)∂pl
get the supply function: q(p,w) = ∂π(p,w)∂p
get the factor demand function zl(p,w) = −∂π(p)∂wl
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Example
q = f (z) = zα, price p and factor wage w .
Note: α > 1→ IRS , α < 1→ DRS , α = 1→ CRS
Pro�ts (assume initially that 0 < α < 1):
π(p,w) = pq − wz = pzα − wz
FOC:
∂π(·)/∂z = αpzα−1 − w = 0
Solution:
factor demand z(p,w) = (α pw
)1/(1−α)
supply q(p,w) = (α pw
)α/(1−α)
pro�tsπ(p,w) = p(α p
w)α/(1−α) − w(α p
w)1/(1−α) = w( 1−αα )(α p
w)1/(1−α)
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Extra (simpli�cation)
π(p,w) = p(α pw
)α/(1−α) − w(α pw
)1/(1−α) =
p(α pw
)−1(α pw
)1/(1−α) − w(α pw
)1/(1−α) =
= (p(α pw
)−1 − w)(α pw
)1/(1−α) = (wα − w)(α pw
)1/(1−α) =
= w( 1−αα )(α pw
)1/(1−α)
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Second order conditions and returns to scale
∂π2(·)/∂z2 = α(α− 1)pzα−2 < 0
Only if: 0 < α < 1.
So, if α > 1 (IRS) it is actually a local minimizer and no pro�tmaximizing output exists (it is in�nite!).
What if α = 1 (CRS)?
π(p,w) = pz − wz
The FOC is: p = w and the supply is:0 if p < w
q = z if p = w
∞ if p > w
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General conclusions
In the pro�t maximization problem, the optimal input choices aresuch that: pMPl = wl
The pro�t maximization problem with price taking works if:
DRS: we can determine supply and inputs levelCRS: we cannot determine supply but only inputs combinationsIRS: the pro�t maximizing solution does not exist or yields negativepro�ts (example)
We can back out factor demands and supply from the pro�t functionusing the Hotelling lemma.
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The cost minimization problem (CMP)
We may rede�ne our problem:
Given the desired output q - �nd the input combination that givesthe q at minimum cost.
Useful to derive cost function - relationship between output leveland the total cost of inputs.
Useful to �nd inputs combinations when pro�t maximization doesnot yield a determinate prodution level.
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The cost minimization problem (CMP)
Concentrate on one output case:
The total cost of production is: C (z) = w · z =∑
l wlzl . The productionlevel is: q = f (z).
The problem is:
Minz
w · z subject to q = f (z)
The Lagrange function:
L =∑l
wlzl − λ(f (z)− q)
The FOC's are :
wl = λ∂f (z)
∂zlfor all ∈ 1, . . . , L− 1 and f (z) = q
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The cost minimization problem (CMP)
The solution to the problem gives
conditional factor demands zl(q,w) for all l
the cost function C (q,w) =∑
l wlzl(q,w)
Taking FOC's for any l and k and dividing with one another gives:
wl
wk
=
∂f (z)∂fl∂f (z)∂fk
=MPl
MPk
= MRTSlk
The solution to the two problems (PMP and CMP) coincides at thepro�t maximizing q.
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The cost minimization problem (CMP)
We �nd the lowest isocost line tangent to the isoquant corresponding toq.
isocost line: c =∑
l wlzl in the two input case: c = w1z1 + w2z2 wherec is a constant
Homothetic production function: the factor demands lie on rays fromthe origin (factor ratios remain constant).
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Marginal cost pricing
When we have the cost function, c(q,w), we can restate the PMP:
Maxq
p · q − C (q,w)
The FOC is:
p =∂C (q,w)
∂q
or in other words:p = MC
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Sheppard's lemma
If we have the cost function, we can recover the conditional factordemand:
zl(q,w) =∂C (q,w)
∂wl
Analogy to the:
Hotelling lemma
Duality result of the consumer optimization
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Geometry of costs
Given the cost function C (q,w) de�ne the:
marginal cost: MC (q,w) = ∂C(q,w)∂q
average cost: AC (q,w) = C(q,w)q
In the short run we will have �xed levels of some inputs. The �rm willtake their level as given (no FOC's with respect to those inputs).Emergence of �xed costs (FC) - FC =
∑f wf zf where f 's are those
l ∈ 1, . . . , L− 1 for which the inputs are �xed. In that case:
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).
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Supply function
pro�ts are ≥ 0 if p > AC
pro�t maximization implies: p = MC
�rm produces q > 0 if MC > AC
the supply function is the segment of MC that is above the ACcurve.
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Cost functions and returns to scale
If we have CRS or f (z) is homogeneous of degree one(f (λz) = λf (z))) then z(q,w) and C (q,w) are homogeneous ofdegree one in output.
To increase production by λ%, we need to increase inputs by λ%,therefore costs increase by λ%MC = AC
If f (z) is concave (or the production set is convex - so we havenon-increasing returns to scale, f (λz) ≤ λf (z), λ > 1), thenC (q,w) is convex
To increase production by λ%, we need to increase inputs by morethan 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 > 1we have non-decreasing AC(q,w)MC ≥ AC : Proof in class....
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Cost functions and returns to scale
DRS - convex cost function, increasing AC , AC < MC
CRS - cost function linear in q, AC = MC = const
IRS - concave cost function, decreasing AC , AC > MC
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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) = z0.5 → z(q) = q2. Cost: C (w , q) = wz(q) = wq2,
AC = wq,
MC = 2wq, MC > AC . If p = MC , then p > AC and π > 0, at anyp,w > 0
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Geometry of costs - CRS
Example: 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)
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Geometry of costs - non-convex technology
Example: q = f (z) = (z − z̄)0.5 → z(q) = q2 + z̄ . Cost:C (w , q) = wz(q) = (q2 + z̄)w
MC = 2qw ,
AC = wz̄q
+ wq,
VC (q) = wq2, FC (q) = wz̄ .
At a pro�t maximizing point p = MC . Therefore for π > 0 we needp = MC > AC .
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Cobb-Douglas technology again
q = f (z1, z2) = zα1zβ2
C (w1,w2, q) = q1
α+β θφ(w1,w2),
where θ =(αβ
) βα+β
+(αβ
) −αα+β
and φ(w1,w2) = wα/(α+β)1
wβ/(α+β)2
.
Our results apply:
α+ β < 1, DRS, cost function convex in qα+ β = 1, CRS, cost function linear in qα+ β > 1, IRS, cost function concave in q
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