### Comparative statics

```c 2001 Michael Carter
โ
Lecture notes based on
Foundations of Mathematical Economics
Comparative statics
1 The maximum theorems
max ๐ (x, ๐ฝ)
xโ๐บ(๐ฝ)
Let
๐ฃ(๐ฝ) = max ๐ (x, ๐ฝ)
๐(๐ฝ) = arg max ๐ (x, ๐ฝ)
xโ๐บ(๐ฝ)
Objective
function
Constraint
correspondence
Value
function
Solution
correspondence
Monotone
maximum
theorem
Theorem 2.1
supermodular,
increasing
weakly
increasing
increasing
increasing
xโ๐บ(๐ฝ)
Continuous
maximum
theorem
Theorem 2.3
continuous
Convex
maximum
theorem
Theorem 3.10
concave
Smooth
maximum
theorem
Theorem 6.1
smooth
continuous,
compact-valued
continuous
convex
compact-valued
nonempty, uhc
convex-valued
smooth
regular
locally
smooth
locally
smooth
2 The envelope theorems
2.1 Envelope theorem 1
๐ฃ(๐ฝ) = max
๐ (x, ๐ฝ)
โ
๐ฅ โ๐บ
โ
= ๐ (x (๐ฝ), ๐ฝ)
so that
๐ฃ โฒ (๐ฝ) = ๐x
โxโ
+ ๐๐ฝ
โ๐ฝ
The ๏ฌrst-order conditions determining xโ are
๐x = ๐๐x
1
concave
c 2001 Michael Carter
โ
Lecture notes based on
Foundations of Mathematical Economics
Moveover, xโ (๐ฝ) satis๏ฌes the constraint as a identity
๐(xโ (๐ฝ)) = 0 =โ ๐x
โxโ
=0
โ๐ฝ
Substituting, we conclude that
๐ฃ โฒ (๐ฝ) = ๐๐ฝ
Example 1 (Chip producer) It is characteristic of microchip production technology that a proportion of output is defective. Consider a small producer for
whom the price of good chips ๐ is ๏ฌxed. Suppose that proportion 1 โ ๐ of the
๏ฌrmโs chips are defective and cannot be sold. Let ๐(๐ฆ) denote the ๏ฌrmโs total
cost function where ๐ฆ is the number of chips (including defectives) produced.
Suppose that with experience, the yield of good chips ๐ increases. How does this
a๏ฌect the ๏ฌrmโs production ๐ฆ? Does the ๏ฌrm compensate for the increased yield
by reducing production, or does it celebrate by increasing production?
The ๏ฌrmโs optimization problem is
๐ฃ(๐) = max ๐๐๐ฆ โ ๐(๐ฆ)
๐ฆ
= ๐๐๐ฆ โ โ ๐(๐ฆ โ)
โ๐ฆ โ
โ๐ฆ โ
โ ๐โฒ (๐ฆ โ)
๐ฃ โฒ (๐) = ๐๐ฆ โ + ๐๐
โ๐
โ๐
โ
โ๐ฆ
= ๐๐ฆ โ + (๐๐ โ ๐โฒ (๐ฆ โ ))
โ๐
But the ๏ฌrst-order condition de๏ฌning ๐ฆ โ (๐) is
๐๐ โ ๐โฒ (๐ฆ โ ) = 0
so that
๐ฃ โฒ (๐) = ๐๐ฆ โ > 0
Further, we can deduce that
๐ฆ โ (๐) =
so that
๐ฃ โฒ (๐)
๐
๐ฃ โฒโฒ (๐)
โ๐ฆ โ (๐)
=
โฅ0
โ๐
๐
since the pro๏ฌt function is convex.
2
c 2001 Michael Carter
โ
Lecture notes based on
Foundations of Mathematical Economics
2.2 Envelope theorem 2
๐ฃ(๐ฝ) = โmax ๐ (x, ๐ฝ)
๐ฅ โ๐บ(๐ฝ)
= ๐ (xโ (๐ฝ), ๐ฝ)
โxโ
๐ฃ โฒ (๐ฝ) = ๐x
+ ๐๐ฝ
โ๐ฝ
The ๏ฌrst-order conditions determining xโ are
๐x = ๐๐x
Moveover, xโ (๐ฝ) satis๏ฌes the constraint as a identity
๐(xโ (๐ฝ), ๐ฝ) = 0 =โ ๐x
or
โxโ
+ ๐๐ฝ = 0
โ๐ฝ
โxโ
= โ๐๐ฝ
๐x
โ๐ฝ
Substituting, we conclude that
๐ฃ โฒ (๐ฝ) = ๐๐ฝ โ ๐๐๐ฝ = ๐ฟ๐ฝ
Example 2 (Consumer problem)
๐ฃ(p, ๐) = max ๐ข(x)
(1)
subject to p x = ๐
(2)
xโ๐
๐
๐ฟ = ๐ข(x) โ ๐(p๐ x โ ๐)
โ๐ฃ
= ๐ฟ๐ = ๐
โ๐
โ๐ฃ
= ๐ฟ๐๐ = โ๐๐ฅโ๐
โ๐๐
which leads immediately to Royโs identity
xโ๐ (p, ๐)
3
=โ
โ๐ฃ
โ๐๐
โ๐ฃ
โ๐
Lecture notes based on
Foundations of Mathematical Economics
c 2001 Michael Carter
โ
2.3 Smooth envelope theorem (Corollary 6.1.1)
Assume that x0 is a strict local maximum of
max ๐ (x, ๐ฝ)
xโ๐บ(๐ฝ)
where ๐บ(๐ฝ) = { x โ ๐ : g(x, ๐ฝ) โค 0 }. By the smooth maximum theorem,
there exists a neighbourhood ฮฉ around ๐ฝ 0 and function xโ such that
๐ฃ(๐ฝ) = ๐ (xโ (๐ฝ), ๐ฝ) for every ๐ฝ โ ฮฉ
and ๐ฃ is di๏ฌerentiable. Applying the chain rule
๐ท๐ฝ ๐ฃ[๐ฝ] = ๐x xโ๐ฝ + ๐๐ฝ
โ
โ
indirect direct
What do we know of the indirect e๏ฌect?
First If xโ is optimal, it must satisfy the Kuhn-Tucker conditions
๐x = ๐๐0 gx and ๐๐0 g(x, ๐ฝ) = 0
(3)
at (x0 , ๐0 ) where ๐0 is the unique Lagrange multiplier associated with
x0 .
Second The solution xโ (๐ฝ) satis๏ฌes the constraint g(xโ (๐ฝ), ๐ฝ) = 0 for all
๐ฝ โ ฮฉ. Another application of the chain rule gives
gx xโ๐ฝ + g๐ฝ = 0 =โ ๐๐0 gx xโ๐ฝ = โ๐๐ g๐ฝ
(4)
Using (3) and (4), the indirect e๏ฌect is ๐x xโ๐ฝ = ๐๐0 ๐x xโ๐ฝ = โ๐๐ g๐ฝ and therefore
๐ท๐ฝ ๐ฃ[๐ฝ] = ๐๐ฝ โ ๐๐0 g๐ฝ = ๐ฟ๐ฝ
(5)
where ๐ฟ denotes the Lagrangean ๐ฟ(x, ๐ฝ, ๐) = ๐ (x, ๐ฝ)โ๐๐ g(x, ๐ฝ). This is the
envelope theorem, which states that the derivative of the value function
is equal to the partial derivative of the Lagrangean evaluated at the optimal
solution (x0 , ๐0 ).
In the special case in which the feasible set ๐บ is independent of the parameters, g๐ฝ = 0 and (5) becomes
๐ท๐ฝ ๐ฃ[๐ฝ] = ๐๐ฝ
The indirect e๏ฌect is zero, and the only impact on ๐ฃ of a change in ๐ฝ is the
direct e๏ฌect f๐ฝ .
4
Lecture notes based on
Foundations of Mathematical Economics
c 2001 Michael Carter
โ
2.4 General envelope theorem (Theorem 6.2)
The assumptions required for Corollary 6.1.1 are stringent. Where the feasible set is independent of the parameters, a more general result can be given.
Let xโ be the solution correspondence of the constrained optimization problem
max ๐ (x, ๐ฝ)
xโ๐บ
in which ๐ : ๐บ × ฮ โ โ is continuous and ๐บ compact. Suppose that ๐ is
continuously di๏ฌerentiable in ๐, that is ๐ท๐ฝ ๐ [x, ๐ฝ] is continuous in ๐บ × ฮ.
Then the value function
๐ฃ(๐) = sup ๐ (x, ๐ฝ)
๐ฅโ๐บ
is di๏ฌerentiable wherever xโ is single-valued with ๐ท๐ฝ ๐ฃ[๐] = ๐ท๐ฝ ๐ [x(๐ฝ), ๐ฝ].
Proof.
To simplify the proof, assume that xโ is single-valued for every
๐ฝ โ ฮ Then
๐ฃ(๐ฝ) = ๐ (xโ (๐ฝ), ๐ฝ) for every ๐ฝ โ ฮ
For any ๐ฝ โ= ๐ฝ 0 โ ฮ
)
(
)
(
๐ฃ(๐ฝ) โ ๐ฃ(๐ฝ 0 ) = ๐ xโ (๐ฝ), ๐ฝ โ ๐ xโ (๐ฝ 0 ), ๐ฝ0
)
(
)
(
โฅ ๐ xโ (๐ฝ 0 ), ๐ฝ โ ๐ xโ (๐ฝ 0 ), ๐ฝ0
= ๐ท๐ฝ ๐ [xโ (๐ฝ 0 ), ๐ฝ 0 ](๐ฝ โ ๐ฝ 0 ) + ๐(๐ฝ) โฅ๐ฝ โ ๐ฝ 0 โฅ
with ๐(๐ฝ) โ 0 as ๐ฝ โ ๐ฝ 0 . On the other hand, by the mean value theorem
¯ โ (๐ฝ, ๐ฝ 0 ) such that
(Theorem 4.1) there exist ๐ฝ
)
(
)
(
๐ฃ(๐ฝ) โ ๐ฃ(๐ฝ 0 ) = ๐ xโ (๐ฝ), ๐ฝ โ ๐ xโ (๐ฝ 0 ), ๐ฝ0
)
(
)
(
โค ๐ xโ (๐ฝ), ๐ฝ โ ๐ xโ (๐ฝ), ๐ฝ 0
¯
โ ๐ฝ0)
= ๐ท๐ฝ ๐ [xโ (๐ฝ), ๐ฝ](๐ฝ
Letting ๐ฝ โ ๐ฝ 0
๐ท๐ฝ ๐ [xโ (๐ฝ 0 ), ๐ฝ 0 ](๐ฝ โ ๐ฝ 0 )
๐ฃ(๐ฝ) โ ๐ฃ(๐ฝ 0 )
๐ท๐ฝ ๐ [xโ (๐ฝ 0 ), ๐ฝ 0 ](๐ฝ โ ๐ฝ 0 )
โค lim
โค lim
lim
๐ฝโ๐ฝ0
๐ฝโ๐ฝ0
๐ฝโ๐ฝ0
โฅ๐ฝ โ ๐ฝ 0 โฅ
โฅ๐ฝ โ ๐ฝ 0 โฅ
โฅ๐ฝ โ ๐ฝ 0 โฅ
๐ฃ is di๏ฌerentiable (Exercise 4.3) and
๐ท๐ฃ[๐] = ๐ท๐ฝ ๐ [xโ (๐ฝ), ๐ฝ]
where ๐ท๐ฝ ๐ [xโ (๐ฝ), ๐ฝ] denotes the partial derivative of ๐ with respect to ๐ฝ
โก
holding x constant at x = xโ (๐ฝ).
5
c 2001 Michael Carter
โ
Lecture notes based on
Foundations of Mathematical Economics
โณ Note that there is no requirement in Theorem 6.2 that ๐ is di๏ฌerentiable with respect to the decision variables x, only with respect to the
parameters. The practical importance of dispensing with di๏ฌerentiability with respect to x is that Theorem 6.2 applies even when the feasible
set is discrete (See Example 6.2).
โ
๐ฃ(๐)
๐ (๐ฅ1 , ๐)
๐ (๐ฅ2 , ๐)
๐ (๐ฅ3 , ๐)
๐
3 Comparative statics of optimization models
There are four di๏ฌerent approaches to comparative statics of optimization
models
โ Revealed preference approach
โ Envelope theorem approach
โ Monotone maximum theorem approach
โ Implicit function theorem approach
3.1 Revealed preference approach
A competitive ๏ฌrmโs optimization problem is to choose a feasible production
plan y โ ๐ to maximize total pro๏ฌt
max p โ y
yโ๐
Consequently, if y1 maximizes pro๏ฌt when prices are p1 , then
p1 โ y1 โฅ p โ y for every y โ ๐
Similarly, if y2 maximizes pro๏ฌt when prices are p2 , then
p2 โ y2 โฅ p โ y for every y โ ๐
6
c 2001 Michael Carter
โ
Lecture notes based on
Foundations of Mathematical Economics
In particular
p1 โ y1 โฅ p1 โ y2
and
p2 โ y2 โฅ p2 โ y1
p1 โ y1 + p2 โ y2 โฅ p1 โ y2 + p2 โ y1
Rearranging
p2 โ (y2 โ y1 ) โฅ p1 โ (y2 โ y1 )
and therefore
(p2 โ p1 ) โ (y2 โ y1 ) โฅ 0
or
๐
โ
(๐1๐ โ ๐2๐ )(๐ฆ๐2 โ ๐ฆ๐2 ) โฅ 0
(6)
๐=1
If prices change from p1 to p2 , the optimal production plan must change in
such a way as to satisfy the inequality (6). For a change in the price of a
single good ๐ (๐2๐ = ๐1๐ for every ๐ โ= ๐), (6) implies that
(๐2๐ โ ๐1๐ )(๐ฆ๐2 โ ๐ฆ๐1) โฅ 0
or
๐2๐ > ๐1๐ =โ ๐ฆ๐2 โฅ ๐ฆ๐1
3.2 The envelope theorem approach
Letting ๐ (y, p) = p โ y denote the objective function, the competitive ๏ฌrm
solves
max ๐ (y, p)
yโ๐
Note that ๐ is di๏ฌerentiable with ๐ทp ๐ [y, p] = y. Applying the envelope
theorem 6.2, the pro๏ฌt function
ฮ (p) = sup ๐ (y, p)
yโ๐
is di๏ฌerentiable wherever the supply correspondence yโ is single-valued with
๐ทp ฮ [p] = ๐ทp ๐ [yโ (p), p] = yโ (p)
or
(7)
yโ (p) = โฮ (p)
which is known as Hotellingโs lemma.
โณ The practical signi๏ฌcance of Hotellingโs lemma is that, if we know the
pro๏ฌt function, we can calculate the supply function by straightforward
di๏ฌerentiation instead of solving a constrained optimization problem.
7
c 2001 Michael Carter
โ
Lecture notes based on
Foundations of Mathematical Economics
โณ Its theoretical signi๏ฌcance is more important. Hotellingโs lemma enables us to deduce the properties of the supply function yโ from the
already established properties of the pro๏ฌt function. In particular, we
know that the pro๏ฌt function is convex (Example 3.42).
From Hotellingโs lemma (7), we deduce that the derivative of the supply
function is equal to the second derivative of the pro๏ฌt function
๐ทyโ [p] = ๐ท 2 ฮ [p]
or equivalently that the Jacobian of the supply function is equal to the Hessian of the pro๏ฌt function.
๐ฝyโ (p) = ๐ปฮ  (p)
Since ฮ  is smooth and convex, its Hessian ๐ป(p) is symmetric (Theorem 4.2)
and nonnegative de๏ฌnite (Proposition 4.1) for all p. Consequently, the Jacobian of the supply function ๐ฝyโ is also symmetric and nonnegative de๏ฌnite.
This implies for all goods ๐ and ๐
๐ท๐๐ ๐ฆ๐โ [p] โฅ 0
๐ท๐๐ ๐ฆ๐โ [p] = ๐ท๐๐ ๐ฆ๐โ[p]
Nonnegativity
Symmetry
In a similar fashion, we can deduce
โ Shephardโs lemma (Example 6.7)
โ Royโs identity (Example 6.8)
From the latter, we can easily derive the Slutsky equation (Example 6.9).
3.3 The implicit function theorem approach
The ๏ฌrst-order conditions of an equality constrained optimization problem
constitute a system of equations.
๐(x; ๐ฝ) = 0
Provided the Jacobian (๐ทx ๐[x; ๐ฝ]) of this system is non-singular, we can use
the implicit function theorem to solve for xโ in terms of ๐ฝ. We illustrate by
means of an example.
8
Lecture notes based on
Foundations of Mathematical Economics
c 2001 Michael Carter
โ
Example Recall again the chip maker, whose optimization problem is
max ๐๐๐ฆ โ ๐(๐ฆ)
๐ฆ
The ๏ฌrst-order and second-order conditions for pro๏ฌt maximization are
๐(๐ฆ, ๐, ๐) = ๐๐ โ ๐โฒ (๐ฆ) = 0 and ๐ท๐ฆ ๐[๐ฆ, ๐, ๐] = โ๐โฒโฒ (๐ฆ) < 0
The second-order condition requires increasing marginal cost. Assuming ๐ is
๐ถ 2 , the ๏ฌrst-order condition implicitly de๏ฌnes a function ๐ฆ(๐). Di๏ฌerentiating
the ๏ฌrst-order condition with respect to ๐, we deduce that
๐ = ๐โฒโฒ (๐ฆ)๐ท๐ฝ โ๐๐ก๐๐ฆ
or
๐ท๐ ๐ฆ =
๐
๐โฒโฒ (๐ฆ)
which is positive by the second-order condition. An increase in yield ๐ is
analogous to an increase in product price ๐, inducing an increase in output
๐ฆ.
โณ Examples 6.15 and 6.16 apply the same technique to deduce the comparative statics of a competitive multi-input ๏ฌrm.
4 References
โ Milgrom, P., and I. Segal (2000), Envelope Theorems for Arbitrary
Choice Sets. Department of Economics, Stanford University: mimeo.
โ Silberberg, E. (1990), The Structure of Economics: A Mathematical
Analysis (2nd edition). New York, NY: McGraw-Hill.
9
c 2001 Michael Carter
โ
Lecture notes based on
Foundations of Mathematical Economics
5 Homework
1. Prove Proposition 5.2, that is if ๐ and g are ๐ถ 2 and ๐ท๐[xโ ] is of full
rank, then the value function
๐ฃ(c) = sup{ ๐ (x) : g(x) = c }
is di๏ฌerentiable with โ๐ฃ(c) = ๐, where ๐ = (๐1 , ๐2 , . . . , ๐๐ ) are the
Lagrange multipliers associated with xโ .
2. Suppose that the cost function of a monopolist changes from ๐1 (๐ฆ) to
๐2 (๐ฆ) in such a way that
0 < ๐โฒ1 (๐ฆ) < ๐โฒ2 (๐ฆ) for every ๐ฆ > 0
Let ๐1 denote the pro๏ฌt maximizing price with the cost function ๐1 (๐ฆ)
and let ๐ฆ1 be the corresponding output. Similarly let ๐2 and ๐ฆ2 be the
pro๏ฌt maximizing price and output when the costs are given by ๐2 (๐ฆ).
(a) Show that
๐2 (๐ฆ1 ) โ ๐2 (๐ฆ2 ) โฅ ๐1 (๐ฆ1 ) โ ๐1 (๐ฆ2 )
(8)
(b) The โFundamental Theorem of Calculusโ states: If ๐ โฒ (๐ฅ) is a
continuous function on [a,b], then
โซ ๐
๐ โฒ (๐ฅ)๐๐ฅ
๐ (๐) โ ๐ (๐) =
๐
Apply this to inequality (8) to deduce that ๐ฆ1 โฅ ๐ฆ2 and therefore
that ๐1 โค ๐2 .
(c) State concisely the proposition you have just proved.
3. Assume that a competitive ๏ฌrm produces a single output ๐ฆ from ๐
inputs x = (๐ฅ1 , ๐ฅ2 , . . . , ๐ฅ๐ ) according to the production function ๐ฆ =
๐ (x) so as to maximize pro๏ฌt
ฮ (w, ๐) = max ๐๐ (x) โ w โ x
x
Assume that there is a unique optimum for every ๐ and w. Show
that the input demand ๐ฅโ๐ (w, ๐) and supply ๐ฆ โ (w, ๐) functions have the
following properties:
๐ท๐ ๐ฆ๐โ[w, ๐] โฅ 0
๐ท๐ค๐ ๐ฅโ๐ [w, ๐] โค 0
๐ท๐ค๐ ๐ฅโ๐ [w, ๐] = ๐ท๐ค๐ ๐ฅโ๐ [w, ๐]
๐ท๐ ๐ฅโ๐ [w, ๐] = โ๐ท๐ค๐ ๐ฆ โ[w, ๐]
10
Upward sloping supply
Downward sloping demand
Symmetry
Reciprocity
c 2001 Michael Carter
โ
Lecture notes based on
Foundations of Mathematical Economics
Solutions 7
1 The Lagrangean for this problem is
(
)
๐ฟ = ๐ (x) โ ๐๐ g(x) โ c
By Corollary 6.1.1
2
โ๐ฃ(c) = ๐ทc ๐ฟ = ๐
(a) With cost function ๐1 (๐ฆ1 ), the ๏ฌrms pro๏ฌt is
ฮ  = ๐๐ฆ โ ๐1 (๐ฆ)
Since this is maximised at ๐1 and ๐ฆ1 (although the monopolist could
have sold ๐ฆ2 at price ๐2 )
๐1 ๐ฆ1 โ ๐1 (๐ฆ1 ) โฅ ๐2 ๐ฆ2 โ ๐1 (๐ฆ2 )
Rearranging
๐1 ๐ฆ1 โ ๐2 ๐ฆ2 โฅ ๐1 (๐ฆ1 ) โ ๐1 (๐ฆ2 )
Similarly
(1)
๐2 ๐ฆ2 โ ๐2 (๐ฆ2 ) โฅ ๐1 ๐ฆ1 โ ๐2 (๐ฆ1 )
which can be rearranged to yield
๐2 (๐ฆ1 ) โ ๐2 (๐ฆ2 ) โฅ ๐1 ๐ฆ1 โ ๐2 ๐ฆ2
Combining the previous inequality with (1) yields
๐2 (๐ฆ1 ) โ ๐2 (๐ฆ2 ) โฅ ๐1 (๐ฆ1 ) โ ๐1 (๐ฆ2 )
(b) Applying the Fundamental Theorem of Calculus to both sides, this
implies
โซ ๐ฆ1
โซ ๐ฆ1
โฒ
๐2 (๐ฆ)๐๐ฆ โฅ
๐โฒ1 (๐ฆ)๐๐ฆ
or
๐ฆ2
โซ
๐ฆ1
๐ฆ2
๐โฒ2 (๐ฆ)
๐โฒ2 (๐ฆ)๐๐ฆ
โซ
โ
๐ฆ2
๐ฆ1
๐ฆ2
๐โฒ1 (๐ฆ)๐๐ฆ
๐โฒ1 (๐ฆ)
โซ
=
๐ฆ1
๐ฆ2
(๐โฒ2 (๐ฆ) โ ๐โฒ1 (๐ฆ))๐๐ฆ โฅ 0
โ
โฅ 0 for every ๐ฆ (by assumption), this implies that
Since
๐ฆ2 โค ๐ฆ1 . Assuming the demand curve is downward sloping, this implies
๐2 โฅ ๐1 .
1
Lecture notes based on
Foundations of Mathematical Economics
c 2001 Michael Carter
โ
(c) There is an implicit requirement to utilize the Fundamental Theorem of
Calculus, namely that ๐โฒ (๐ฆ) is continuous. With this proviso, we have
shown that the monopoly price is increasing in marginal cost. Specifically we have shown: Assuming that a monopolistโs cost function is
continously di๏ฌerentiable (in output), the pro๏ฌt maximizing monopoly
price is an increasing (i.e. nondecreasing) function of marginal cost.
3 By Theorem 6.2
๐ทw ฮ [w, ๐] = โxโ and ๐ท๐ ฮ [w, ๐] = ๐ฆ โ
and therefore
2
๐ท๐ ๐ฆ(๐, w) = ๐ท๐๐
ฮ (๐, w) โฅ 0
๐ท๐ค๐ ๐ฅ๐ (๐, w) = โ๐ท๐ค2 ๐ ๐ค๐ ฮ (๐, w) โค 0
๐ท๐ค๐ ๐ฅ๐ (๐, w) = โ๐ท๐ค2 ๐ ๐ค๐ ฮ (๐, w) = ๐ท๐ค๐ ๐ฅ๐ (๐, w)
๐ท๐ ๐ฅ๐ (๐, w) = โ๐ท๐ค2 ๐ ๐ ฮ (๐, w) = โ๐ท๐ค๐ ๐ฆ(๐, w)
since ฮ  is convex and therefore ๐ปฮ  (w, ๐) is symmetric (Theorem 4.2) and
nonnegative de๏ฌnite (Proposition 4.1).
2
```