EC487 Advanced Microeconomics, Part I: Lecture 2econ.lse.ac.uk/staff/lfelli/teach/EC487 Slides...

EC487 Advanced Microeconomics, Part I:Lecture 2

Leonardo Felli

32L.LG.04

6 October, 2017

Properties of the Profit Function

Recall the following property of the profit function

π(p,w) = maxx

p f (x)− w x = p f (x(p,w))− w x(p,w)

I Hotelling’s Lemma:

∂p= y(p,w) ≥ 0 and

∂wi= −xi (p,w) ≤ 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 2 / 49

Hessian of the Profit Function

Consider now the Hessian matrix of the profit function π(p,w):

∂2π∂p2

∂2π∂p∂w1

· · · ∂2π∂p∂wh

∂2π∂w1∂p

∂2π∂w2

1· · · ∂2π

∂w1∂wh

......

. . ....

∂2π∂wh∂p

∂2π∂wh∂w1

· · · ∂2π∂w2

By Hotelling’s Lemma the matrix H is:

∂y∂p

∂y∂w1

· · · ∂y∂wh

−∂x1∂p − ∂x1

∂w1· · · − ∂x1

∂wh...

.... . .

−∂xh∂p − ∂xh

∂w1· · · − ∂xh

Properties of the Hessian of the Profit Function

Result

Let F : A→ R where A ⊂ Rn is a convex open set. Let F (·) betwice differentiable. Then the function F (·) is convex if and only ifthe Hessian matrix of F (·) is positive semi-definite on A.

I Recall that the profit function π(p,w) is convex in (p,w).

I This together with Young theorem implies that H issymmetric and has a non negative main diagonal.

I Assume now that π(p,w) is defined on an open and convexset S of prices (p,w) then convexity of π(p,w) is equivalentto the Hessian Matrix H being positive semi-definite.

Profit Maximization

Profit maximization can be achieved in two sequential steps:

1. Given y , find the choice of inputs that allows the producer toobtain y at the minimum cost;

this generates conditional factor demands and the costfunction;

2. Given the cost function, find the profit maximizing outputlevel.

Comments

I Notice: step 1 is common to firms that behave competitivelyin the input market but not necessarily in the output market.

I Only in step 2 we impose the competitive assumption on theoutput market.

Cost Minimization

We shall start from step 1:

s.t. f (x) ≥ y

The necessary first order conditions are:

y = f (x∗),

w ≥ λ∇f (x∗)

and[w − λ∇f (x∗)] x∗ = 0

Cost Minimization (cont’d)

or for every input ` = 1, . . . , h:

w` ≥ λ∂f (x∗)

with equality if x∗` > 0.

The first order conditions are also sufficient if f (x) isquasi-concave (the input requirement set is convex).

Alternatively, a set of sufficient conditions for a local minimum arethat f (x) is quasi-concave in a neighborhood of x∗.

This can be checked (sufficient condition) by means of thebordered hessian matrix and its minors.

In the case of only two inputs f (x1, x2) we have:

w` ≥ λ∂f (x∗)

∂x`, ∀` = 1, 2

with equality if x∗` > 0

and SOC: ∣∣∣∣∣∣0 f1(x∗) f2(x∗)

f1(x∗) f11(x∗) f12(x∗)f2(x∗) f21(x∗) f22(x∗)

∣∣∣∣∣∣ > 0

In the case the two first order conditions are satisfied with equality(no corner solutions) we can rewrite the necessary conditions as:

MRTS =

∣∣∣∣dx2dx1

∣∣∣∣ =∂f (x∗)/∂x1∂f (x∗)/∂x2

andy = f (x∗)

Notice a close formal analogy with consumption theory(expenditure minimization).

Conditional Factor Demands and Cost Function

This leads to define:

I the solution to the cost minimization problem:

x∗ = z(w , y) =

z1(w , y)...

zh(w , y)

the conditional (to y) factor demands (correspondences).

I the minimand function of the cost minimization problem:

c(w , y) = w z(w , y)

the cost function.

Properties of the Cost Function

1. c(w , y) is non-decreasing in y .

Proof: Suppose not. Then there exist y ′ < y ′′ such that(denote x ′ and x ′′ the corresponding solution to the costminimization problem)

w x ′ ≥ w x ′′ > 0

If the latter inequality is strict we have an immediatecontradiction of x ′ solving the cost minimization problem.

Properties of the Cost Function (cont’d)

If on the other hand

w x ′ = w x ′′ > 0

then by continuity and monotonicity of f (·) there existsα ∈ (0, 1) close enough to 1 such that

f (α x ′′) > y ′

andw x ′ > w αx ′′

which contradicts x ′ solving the cost minimizationproblem.

2. c(w , y) is non-decreasing with respect to w` for every` = 1, . . . , h.

Proof: Consider w ′ and w ′′ such that w ′′` ≥ w ′` but w ′′k = w ′kfor every k 6= `.

Let x ′′ and x ′ be the solutions to the cost minimizationproblem with w ′′ and w ′ respectively.

Then by definition of c(w , y)

c(w ′′, y) = w ′′ x ′′ ≥ w ′ x ′′ ≥ w ′ x ′ = c(w ′, y).

3. c(w , y) is homogeneous of degree 1 in w .

Proof: The feasible set of the cost minimization problem

f (x) ≥ y

does not change when w is multiplied by the factor t > 0.

Hence ∀t > 0, minimizing (t w) x on this set leads to thesame answer as minimizing w x .

Let x∗ be the solution, then:

c(t w , y) = (t w) x∗ = t c(w , y).

4. c(w , y) is a concave function in w .

Proof: Let w = t w + (1− t) w ′ for t ∈ [0, 1].

Let x be the solution to the cost minimization problem for w .Then

c(w , y) = w x = t w x + (1− t) w ′ x

≥ t c(w , y) + (1− t) c(w ′, y)

by definition of c(w , y) and c(w ′, y) and f (x) ≥ y .

Properties of the Conditional Factor Demands

5. Shephard’s Lemma:

If z(w , y) is single valued with respect to w then c(w , y) isdifferentiable with respect to w and

∂c(w , y)

∂w`= z`(w , y)

Proof: By constrained Envelope Theorem and the definitionof cost function

c(w , y) = w z(w , y) = w z(w , y)− λ(w , y) [f (z(w , y)− y ]

Properties of the Conditional Factor Demands (cont’d)

6. z(w , y) is homogeneous of degree 0 in w .

Proof: By Shepard’s Lemma and the following result.

Result

If a function G (x) is homogeneous of degree r in x then (∂G/∂x`)is homogeneous of degree (r − 1) in x for every ` = 1, . . . , L.

Proof: Differentiate with respect to x` the identity that defineshomogeneity of degree r :

G (k x) ≡ k r G (x) ∀k > 0

We obtain:

k∂G (k x)

∂x`= k r

∂G (x)

∂x`∀k > 0

or∂G (k x)

∂x`= k(r−1)

∂G (x)

∂x`∀k > 0

This is the definition of homogeneity of degree (r − 1) of the

function∂G (k x)

∂x`.

Notice that

6. The Lagrange multiplier of the cost minimization problem isthe marginal cost of output:

∂c(w , y)

∂y= λ(w , y)

Proof: By constrained Envelope Theorem applied to the costfunction:

c(w , y) = w z(w , y)− λ(w , y) [f (z(w , y)− y ]

Law of Supply

7. Let the set W of input prices w be open and convex, ifz(w , y) is differentiable in w then:

∂2c∂w2

1· · · ∂2c

∂w1 ∂wh

.... . .

...∂2c

∂wh ∂w1· · · ∂2c

∂w2h

∂z1∂w1

· · · ∂z1∂wh

.... . .

...∂zh∂w1

· · · ∂zh∂wh

is a symmetric and negative semi-definite matrix.

Law of Supply

Proof: Symmetry follows from Shephard’s lemma and YoungTheorem:

∂z`∂wi

(∂c(w , y)

∂zi∂w`

While negative semi-definiteness follows from the concavity ofc(w , y), the fact that W is open and convex and thefollowing result.

Result

Let F : A→ R where A ⊂ Rn is a convex open set. Let F (·) betwice differentiable. Then the function F (·) is concave if and onlyif the Hessian matrix of F (·) is negative semi-definite on A.

Euler Theorem

Result (Euler Theorem)

If a function G (x) is homogeneous of degree r in x then:

r G (x) = ∇G (x) x

Proof: Differentiating with respect to k the identity:

G (k x) ≡ k r G (x) ∀k > 0

we obtain:

∇G (kx) x = rk(r−1) G (x) ∀k > 0

for k = 1 we obtain:

∇G (x) x = r G (x).

Returns to Scale

We now introduce a set of new properties closely related to theones of the expenditure function in consumer theory.

8. If f (x) is homogeneous of degree one (i.e. exhibits constantreturns to scale), then c(w , y) and z(w , y) are homogeneousof degree one in y .

Proof: Let k > 0 and consider:

c(w , k y) = minx

s.t. f (x) ≥ k y(1)

By definition of c(w , y) if x∗ is the solution to the costminimization problem we have y = f (x∗).

Returns to Scale (cont’d)

Hence by homogeneity of degree 1 of f (x) we obtain:

k y = k f (x∗) = f (k x∗)

which implies that k x∗ is feasible in Problem (1).

Therefore:

k c(w , y) = k [w x∗] = w (k x∗) ≥ c(w , k y).

Let now x be the solution to Problem (1). Necessarily:f (x) = k y

or, by homogeneity of degree 1:

f [(1/k) x ] = (1/k) f (x) = y

which implies that [(1/k) x ] is feasible in the problem thatdefines c(w , y).

Therefore we get:

c(w , k y) = w x = k w [(1/k) x ] ≥ k c(w , y)

which concludes the proof.

8′. In other words, a technology that exhibits CRS has a costfunction that is linear in y : c(w , y) = c(w)y .

9. A technology that exhibits CRS has equal and constantmarginal (∂c(w , y)/∂y) and average cost functions

(∂c(w , y)/∂y) = (c(w , y)/y).

Proof: Homogeneity of degree 1 and Euler theorem imply

cy (w)y = c(w , y) or cy (w) =c(w , y)

-,,,,,,,,,,,,,

cy (w) = c(w ,y)y

c(w , y) = cy (w) y

10. If f (x) is convex (IRS technology), then the cost functionc(w , y) is concave in y .

11. A technology that exhibits IRS has a decreasing marginal costfunction (∂c(w , y)/∂y) and average cost function

(∂c(w , y)/∂y) ≤ (c(w , y)/y)

c(w ,y)y

c(w , y)

cy (w , y)

12. If f (x) is concave (DRS technology), then the cost functionc(w , y) is convex in y .

13. A technology that exhibits DRS has an increasing marginalcost function (∂c(w , y)/∂y) and average cost function

(∂c(w , y)/∂y) ≥ (c(w , y)/y)

cy (w , y)c(w ,y)

yc(w , y)

Profit Maximization with Single Output

I Assume that the output market is competitive.

I The profit maximization problem is then:

p y − c(w , y)

The necessary FOC are:

p − ∂c(w , y∗)

∂y≤ 0

with equality if y∗ > 0.

The sufficient SOC for a local maximum:

∂2c(w , y∗)

∂y2> 0

Profit Maximization with Single Output (cont’d)

I Clearly the SOC imply at least local DRS in a neighborhoodof y∗.

I Notice that if y∗ > 0 the optimal choice of the firm is:

p =∂c(w , y∗)

∂y= MC(y∗)

I In words, price equal to marginal cost.

I This condition defines the solution to the profit maximizationproblem: the supply function: y∗(w , p)

Profit Maximization with Single Output (cont’d)

The two profit maximization problems we have considered so farproduce the same outcome for equal (w , p).

In fact:maxy

py − c(w , y)

wherec(w , y) = min

s.t. f (x) ≥ y

yieldsmaxx

p y − w x

s.t. f (x) = y

which is the very first problem we considered.

Short Run

Short run cost minimization arises when one or more inputs maybe fixed, xh = xh, while the remaining inputs may be varied at will.

The short run variable cost function:

cS(w , y , xh) = wh xh + minx1,...,xh−1

h−1∑`=1

s.t. f (x1, . . . , xh−1, xh) ≥ y

Alternatively:

cS(w , y , xh) = minx

s.t. f (x) ≥ y

xh = xh

Short Run (cont’d)

If z(w , y) denotes the long run conditional factor demands, thatsolve:

c(w , y) = minx

s.t. f (x) ≥ y

Let x = (x1, . . . , xh) be the input vector that achieves theminimum long run cost of producing y with prices w :

x = (x1, . . . , xh) = z(w , y)

Short Run vs. Long Run

We can then characterize the relationship between short and longrun total costs, or alternatively, short run and long run variablecosts (more familiar).

Notice thatc(w , y) ≡ cS(w , y , zh(w , y)) (2)

orc(w , y)

y≡ cS(w , y , zh(w , y))

Short Run vs. Long Run (cont’d)

moreover by Envelope Theorem

∂c(w , y)

∂y≡ ∂cS(w , y , zh(w , y))

∂y(3)

We shall now focus on a neighborhood of (w , y) and setxh = zh(w , y). From (2) above by Envelope Theorem we get:

∂c(w , y)

∂y=∂cS(w , y , xh)

Recall that Envelope Theorem implies that only the first ordereffect is zero.

Since (3) is an identity in (w , y) we can differentiate both sideswith respect to y :

∂2cS(w , y , xh)

∂y2+∂2cS(w , y , xh)

∂y ∂xh

∂zh(w , y)

∂y=∂2c(w , y)

and with respect to wh:

∂2cS(w , y , xh)

∂y ∂wh+∂2cS(w , y , xh)

∂y ∂xh

∂zh(w , y)

∂wh=∂2c(w , y)

∂y ∂wh

Now∂2cS(w , y , xh)

∂y ∂wh= 0

since∂cS(w , y , xh)

∂wh= xh

is independent of y .

Hence by Shephard’s Lemma:

∂2cS(w , y , xh)

∂y ∂xh=

∂zh(w , y)/∂y

∂zh(w , y)/∂wh

which implies by substitution:

∂2cS(w , y , xh)

∂y2+

(∂zh(w , y)/∂y)2

∂zh(w , y)/∂wh=∂2c(w , y)

which delivers:

∂2cS(w , y , xh)

∂y2≥ ∂2c(w , y)

since(∂zh(w , y)/∂y)2

∂zh(w , y)/∂wh≤ 0

This allows us to conclude that the loss function:

l(w , y) = c(w , y)− cS(w , y , xh) ≤ 0

reaches a local maximum at x .

Proof: We have shown above that the FOC are satisfied:

∂cS(w , y , xh)

∂y=∂c(w , y)

and we just proved that the SOC hold:

∂2c(w , y)

∂y2≤ ∂2cS(w , y , xh)

Le Chatelier Principle

In other words:c(w , y) ≤ cS(w , y , xh)

A similar approach proves that for every ` ∈ {1, . . . , h}:

0 ≥∂zS`∂w`

≥ ∂z`∂w`

Moving to profit maximization:

0 ≥∂xS`∂w`

≥ ∂x`∂w`

0 ≤ ∂yS

∂p≤ ∂y

All these results are known as Le Chatelier Principle.Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 44 / 49

Aggregation

I The question we address is when can we speak of anaggregate supply function in similar terms as the aggregatedemand you have seen before?

I The question above is closely related to the question: underwhich conditions can we speak of a representative producer?

I Recall that the key problem when constructing an aggregatedemand was the presence of income effects in consumertheory.

Aggregate Supply

I The absence of a budget constraint implies that individualfirms’ supply are not subject to income effects.

I Hence aggregation of production theory is simpler andrequires less restrictive conditions.

I Consider J production technologies: (Z 1, . . . ,Z J)

I Let z j(p,w) =

(−x j(p,w)y j(p,w)

)be firm j ’s production plan.

Law of Supply

I We have seen that the matrix of cross and own price effects ofproduction plan z j(p.w):

Dz j(p,w)

is symmetric and positive semi-definite.

I This is also known as law of supply.

Aggregate Law of Supply

I Define now the following aggregate optimal production plan:

z(p,w) =J∑

z j(p,w) =

(−∑

j xj(p,w)∑

j yj(p,w)

I Does an aggregate law of supply hold?

I Since both symmetry and positive semi-definiteness arepreserved under sum then

Dz(p,w)

is also symmetric and positive semi-definite.

Representative Producer

Result

In a purely competitive environment the maximum profit obtainedby every firm maximizing profits separately is the same as theprofit obtained if all J firms where they coordinate their choices ina joint profit maximization:

π(p,w) =J∑

πj(p,w)

In other words, there exists a representative producer.

EC487 Advanced Microeconomics, Part I: Lecture 2econ.lse.ac.uk/staff/lfelli/teach/EC487 Slides...

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