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

Jan Hagemejer Advanced Microeconomics

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

Jan Hagemejer Advanced Microeconomics

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.

Jan Hagemejer Advanced Microeconomics

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 ,

Jan Hagemejer Advanced Microeconomics

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

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

Jan Hagemejer Advanced Microeconomics

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).

Jan Hagemejer Advanced Microeconomics

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....

Jan Hagemejer Advanced Microeconomics

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

Jan Hagemejer Advanced Microeconomics

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)}.

Jan Hagemejer Advanced Microeconomics

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

Jan Hagemejer Advanced Microeconomics

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)

Jan Hagemejer Advanced Microeconomics

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)

Jan Hagemejer Advanced Microeconomics

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

Jan Hagemejer Advanced Microeconomics

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.

Jan Hagemejer Advanced Microeconomics

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.

Jan Hagemejer Advanced Microeconomics

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

Jan Hagemejer Advanced Microeconomics

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.

Jan Hagemejer Advanced Microeconomics

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).

Jan Hagemejer Advanced Microeconomics

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

Jan Hagemejer Advanced Microeconomics

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

Jan Hagemejer Advanced Microeconomics

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).

Jan Hagemejer Advanced Microeconomics

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.

Jan Hagemejer Advanced Microeconomics

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....

Jan Hagemejer Advanced Microeconomics

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

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) = 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

Jan Hagemejer Advanced Microeconomics

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)

Jan Hagemejer Advanced Microeconomics

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 .

Jan Hagemejer Advanced Microeconomics

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

Jan Hagemejer Advanced Microeconomics