advanced microeconomics -...

Advanced Microeconomics

Pro�t maximization and cost minimization

Jan Hagemejer

November 28, 2011

Jan Hagemejer Advanced Microeconomics

The plan

Pro�t maximization and the pro�t function

Cost minimization and the cost function

Examples


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


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.


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 ,



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


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


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).


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....


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


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)}.


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


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)


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)


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


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.


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.



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



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.



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).


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


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


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).


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.


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....


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


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


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)


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 .


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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