一些簡單的凸分析及其应用

时隔两年多，恢复更新了。这是一些簡單的凸分析的续篇。

目前只是草稿，会有很多错误。

Don’t follow my arguments for now, it’s just a draft of a draft.

在这篇中，我们要得到高微(一)中消费者理论和厂商理论的一些性质，比如关于支出函数、成本函数和利润函数。具体地说，我们的目标是得到 conjugate duality theorem 和 support function theorem。我们将先从 indicator function 开始，然后是 conjugate，随后引入 support function，作为 concave/convex indicator function 的 conjugate，之后是 Young-Fenchel conjugate theorem 和 Legendra transformation。The reason I took this notes is that we didn’t have to memorize mechanically the properties of expenditure function, of cost function, and of profit function, because those properties are just direct applications of the support function theorem: they’ve just been attached names under scienarios with different economic interpretations.

一些簡單的凸分析及其应用

The support function theorem

Let $A$ be a convex set in $\mathbb{R}^{m}$ and let $\mathbb{1}_{A}(x)$ be the concave indicator function of $A$, i.e.

if $x\in A$ then $\mathbb{1}_{A}(x)=0$; if $x\not\in A$ then $\mathbb{1}_{A}(x)=-\infty$ (for a convex indicator function, then if $x\in A$ then $\mathbb{1}_{A}(x)=+\infty$, but always for a convex set, and there is no such a notion as “concave set”)

Because the punishment is to an infinite extent, it should satisfy the constraint.

And let $f(p)=\mathbb{1}_{A}^{\star}(p)$ be the associated support function, i.e. $\mathbb{1}_{A}^{\star}(p)=inf\lbrace p*x-\mathbb{1}_{A}(x)\rbrace$.

which is, for a given $p$,
choose $x$ such that $p*x-\mathbb{1}_{A}(x)$ as small as possible. since we know that whenever $x\not\in A$, then $\mathbb{1}_{A}(x)=-\infty$, so $p*x-\mathbb{1}_{A}(x)=p*x-(-\infty)$ is positive infinity, while for $x\in A$ then $\mathbb{1}_{A}(x)=0$, so to get the infimum, we must choose $x$ from $A$, in this case, the infimum of $p*x-\mathbb{1}_{A}(x)$ is equivalent to the infimum of $p*x$. We can interprate it this way: we are minimizing a linear function (e.g. choosing quantities vector, facing price vector, with a constraint that we must choose $x$ from the convex set $A$, such as the upper contour set of a quasi-concave utility function; or the upper contour set of the production function, although when we interprate it as a cost function the $p$ here is actually the $w$ and should not be confused with the price of production)

and let $x(p)=\{x\in A:p*x=f(p)\}$ be the set of solutions to $\min_{x\in A}p*x$ (if any. interpretation: the cost minimizing factor demand with the requirement of producing at least as a level of producing some $y$, or the expenditure minimizing good demand with the requirement of generating at least as a level of some utility level $\bar{u}$). Then

• (1) the support function $f(p)$ is concave in $p$

(interpretation: the expenditure function, i.e. already substituting into the optimal solution; the cost function, also already substituting the optimal solution. So it is the value function)

(i.e. increasing price will increase cost/expenditure, but in a slower and slower speed),

upper semicontinuous, and homogeneous of degree one.

f is continuous on the interior of its domain, and weakly increasing (or some people say non-decreasing) in $p$ (increasing factor price will never reduce cost; increasing goods price will never reduce expenditure)

• (2) if $A$ is closed and convex, then the conjugate of the conjugate gives us back to the concave indicator function, which is $f^{\star}(x)=\mathbb{1}_{A}(x)$. Therefore, $A=\lbrace x:f^{\star}(x)=0\rbrace$ and $A=\lbrace x: p*x \geq f(p) \forall p\rbrace$. i.e. We can recover the upper coutour set from our support function (expenditure function, cost function,…).

• (3) if $A$ is convex, the minimizer satisfies $x(p)=\partial^{\star}f(p)$

• (4) if $x_{i}\in x(p_{i})$ for $i=0,1$, then $(x_{1}-x_{0})(p_{1}-p_{0})\leq 0$.

• (5) $x(p)$ is homogeneous of degree zero. (the factor demand, the Hicksian demand)

• (6) if $A$ is convex and $f(p)$ is second order continuously differentiable, then $x(p)$ is differentiable and the matrix $Dx(p)=D^{2}f(p)$ is symmetric and negative semi-definite. Moreover, $Dx(p)*p= D^{2} f(p)*p=0$

The Conjugate Duality Theorem

suppose $f:E\rightarrow \mathbb{R}^{\star}$ is proper, concave and upper semi-continuous. Then the following are equivalent:

• (1) $x$ minimizes $p*y-f(y)$

• (2) $p\in\partial^{\star}(x)$

• (5) $x\in\partial^{\star}f^{\star}(p)$

• (6) $p$ minimizes $q*x-f^{\star}(q)$

Attaching names

The properties of consumer expenditure function $e(p,u)$

• (i) [support function theorem (1)] the support function $f(p)$ is concave in $p$, which is $e(p,u)$ is concave in $p$.

• (ii) [support function theorem (1)] the support function $f(p)$ is homogenous of degree one in $p$, which is $e(p,u)$ is homogeneous of degree one in $p$.

• (iii) [support function theorem (1)] the support function $f(p)$ is continuous on the interior of its domain, so we have to tackle with the interior domian thing so that the support function is not only upper semi-continuous, and to interprate $e(p,u)$ continuous in $p$ and $u$.

• (iv) [support function theorem (1)] the support function $f(p)$ is weakly increasing (or some people say non-decreasing) in $p$ which is $e(p,u)$ is nondecreasing in $p_k$ for any k. So we have to interprate $e(p,u)$ to make it strictly increasing in $u$ as well.

• Shepard’s lemma: [support function theorem (3)] if $A$ is convex, the minimizer satisfies $x(p)=\partial^{\star}f(p)$

• [support function theorem (4)] law of demand for Hicksian demand

• [support function theorem (5)] demand is homogenous of degree zero.

• [support function theorem (6)] the Slutsky Matrix

The properties of the cost fuction $c(w,y)$

• (i) [support function theorem (1)] the support function $f(p)$ is concave in $p$, which is $c(w,y)$ is concave in $w$.

Homogenous of degree one in p.

• (ii) [support function theorem (1)] the support function $f(p)$ is homogenous of degree one in $p$, which is $c(w,y)$ is homogenenous of degree one in $w$.

strictly increasing in u and nondecreasing in $p_k$ for any k.

• (iii)

• (iv)

• Shepard’s lemma: [support function theorem (3)] if $A$ is convex, the minimizer satisfies $x(p)=\partial^{\star}f(p)$

• [support function theorem (4)] law of factor demand

The properties of the profit fuction $\pi(p,w)$

with additional reformulation, from $inf$ to $sup$

• (i) homogeneous of degree one in p and w.

• (ii) in p and in w.

• (iii) convex in p and w.

• (iv) $\pi(p)$ is convex in p, and $\partial \pi(p)/ \partial p_{k} = z_{k}(p)$.

recoverability

support function theorem (2)

references

1. KC Border’s Notes. (Of course, it’s always KC Border)
2. Nam Nguyen’s course