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:
∂π
∂p= y(p,w) ≥ 0 and
∂π
∂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):
H =
∂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
h
By Hotelling’s Lemma the matrix H is:
H =
∂y∂p
∂y∂w1
· · · ∂y∂wh
−∂x1∂p − ∂x1
∂w1· · · − ∂x1
∂wh...
.... . .
...
−∂xh∂p − ∂xh
∂w1· · · − ∂xh
∂wh
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 3 / 49
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 4 / 49
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 5 / 49
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 6 / 49
Cost Minimization
We shall start from step 1:
minx
w x
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 7 / 49
Cost Minimization (cont’d)
or for every input ` = 1, . . . , h:
w` ≥ λ∂f (x∗)
∂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∗.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 8 / 49
Cost Minimization (cont’d)
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 9 / 49
Cost Minimization (cont’d)
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
=w1
w2
andy = f (x∗)
Notice a close formal analogy with consumption theory(expenditure minimization).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 10 / 49
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 11 / 49
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 12 / 49
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 13 / 49
Properties of the Cost Function (cont’d)
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 14 / 49
Properties of the Cost Function (cont’d)
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 15 / 49
Properties of the Cost Function (cont’d)
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 .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 16 / 49
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 ]
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 17 / 49
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 18 / 49
Properties of the Conditional Factor Demands (cont’d)
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`.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 19 / 49
Properties of the Conditional Factor Demands (cont’d)
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 ]
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 20 / 49
Law of Supply
7. Let the set W of input prices w be open and convex, ifz(w , y) is differentiable in w then:
H =
∂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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 21 / 49
Law of Supply
Proof: Symmetry follows from Shephard’s lemma and YoungTheorem:
∂z`∂wi
=∂
∂wi
(∂c(w , y)
∂w`
)=
∂
∂w`
(∂c(w , y)
∂wi
)=
∂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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 22 / 49
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 23 / 49
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
w x
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∗).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 24 / 49
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 25 / 49
Returns to Scale (cont’d)
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 26 / 49
Returns to Scale (cont’d)
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).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 27 / 49
Returns to Scale (cont’d)
Proof: Homogeneity of degree 1 and Euler theorem imply
cy (w)y = c(w , y) or cy (w) =c(w , y)
y.
-
6 6
-,,,,,,,,,,,,,
y
cy (w) = c(w ,y)y
y
c(w , y) = cy (w) y
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 28 / 49
Returns to Scale (cont’d)
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)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 29 / 49
Returns to Scale (cont’d)
-
6 6
-
y
c(w ,y)y
y
c(w , y)
cy (w , y)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 30 / 49
Returns to Scale (cont’d)
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)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 31 / 49
Returns to Scale (cont’d)
-
6 6
-
y y
cy (w , y)c(w ,y)
yc(w , y)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 32 / 49
Profit Maximization with Single Output
I Assume that the output market is competitive.
I The profit maximization problem is then:
maxy
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 33 / 49
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)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 34 / 49
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
xw x
s.t. f (x) ≥ y
yieldsmaxx
p y − w x
s.t. f (x) = y
which is the very first problem we considered.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 35 / 49
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
w` x`
s.t. f (x1, . . . , xh−1, xh) ≥ y
Alternatively:
cS(w , y , xh) = minx
w x
s.t. f (x) ≥ y
xh = xh
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 36 / 49
Short Run (cont’d)
If z(w , y) denotes the long run conditional factor demands, thatsolve:
c(w , y) = minx
w x
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)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 37 / 49
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))
y
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 38 / 49
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)
∂y
Recall that Envelope Theorem implies that only the first ordereffect is zero.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 39 / 49
Short Run vs. Long Run (cont’d)
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)
∂y2
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 40 / 49
Short Run vs. Long Run (cont’d)
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
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 41 / 49
Short Run vs. Long Run (cont’d)
which implies by substitution:
∂2cS(w , y , xh)
∂y2+
(∂zh(w , y)/∂y)2
∂zh(w , y)/∂wh=∂2c(w , y)
∂y2
which delivers:
∂2cS(w , y , xh)
∂y2≥ ∂2c(w , y)
∂y2
since(∂zh(w , y)/∂y)2
∂zh(w , y)/∂wh≤ 0
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 42 / 49
Short Run vs. Long Run (cont’d)
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)
∂y
and we just proved that the SOC hold:
∂2c(w , y)
∂y2≤ ∂2cS(w , y , xh)
∂y2
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 43 / 49
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`
and
0 ≤ ∂yS
∂p≤ ∂y
∂p
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 45 / 49
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 46 / 49
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 47 / 49
Aggregate Law of Supply
I Define now the following aggregate optimal production plan:
z(p,w) =J∑
j=1
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.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 48 / 49
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∑
j=1
πj(p,w)
In other words, there exists a representative producer.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 49 / 49
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