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。

一些簡單的凸分析及其应用

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

(i) Homogenous of degree one in p.

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

(iii) concave in p.

(iv) continuous in p and u.

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

(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 $\partial \pi(p)/ \partial p_{k} = z_{k}(p)$.

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

(i) Homogenous of degree one in p.

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

(iii) concave in p.

(iv) continuous in p and u.

The support function theorem

Let $A$ be a convex set in $\mathbb{R}^{m}$ and $\mathbb{1}_{A}(x)$ 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”)

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

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

(i) the support function $f(p)$ (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) is concave in $p$ (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 weaking increasing (or some people say onedecreasing) in $p$ (increasing factor price will never reduce cost; increasing goods price will never reduce expenditure)