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报童问题：最优进货量

Optimal Level of Supply

专题
Finance / 金融
难度
L4

题目详情

A man offers milk to the spectators of the weekly baseball matches. Before each match, he orders c units of milk at a price of b>0b > 0 per unit; during thematch he sells each unit for a price of s>bs > b . Because the milk cannot be kept fresh for a week, each unsold unit imposes a loss of b. The actual demand D of milk varies from week to week, according to a strictly increasing DF F with density f. Assume that F(0)=0\mathrm{F}(0) = 0 and that D is a continuous rv with finite mean.

a. For given prices b, s, and a given demand structure F, the milkman can choose among different values of c. Consider c as a continuous variable, and denote as G(c) the expected net gain as a function of c. Discuss the shape and qualitative properties of G.

b. What is the optimal amount of milk, Copt, the man should stock in order to maximize his expected net gain?

解析

设进价 b>0b>0，售价 s>bs>b，需求 D∼FD\sim F（连续）。进货 cc 的期望净收益可写为

G(c)=E[smin⁡(D,c)−bc]=s∫0c(1−F(u)) du−bc.G(c)=\mathbb{E}[s\min(D,c)-bc]=s\int_0^c(1-F(u))\,du-bc.

一阶条件：G′(c)=s(1−F(c))−bG'(c)=s(1-F(c))-b，且 G′′(c)=−sf(c)<0G''(c)=-sf(c)<0，因此最优解唯一。令 G′(c∗)=0G'(c^*)=0 得

F(c∗)=1−bs,c∗=F−1(1−bs).\boxed{F(c^*)=1-\frac{b}{s}},\qquad \boxed{c^*=F^{-1}\left(1-\frac{b}{s}\right)}.