Question
A dealer lists identical camera lenses for $40 each and lets a collector order zero, one or two. The collector has privately assessed the resale value V of each lens; from the dealer's perspective V is uniformly distributed between $0 and $100. Buying q lenses gives the collector resale surplus \(q(V-40)\), but keeping multiple lenses also incurs a storage and insurance cost of $10\(q^{2}\). The collector chooses q to maximise net surplus and, when indifferent, chooses the smaller order.
The dealer sees the order quantity but not V. Which values of \(V\) produce each order size, and what should the dealer infer about the expected value of a lens after observing \(q=0\), 1 or 2?
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Hint 1
Write the collector’s net surplus for each permitted quantity. Find where neighbouring choices have equal surplus.
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Solution
The three net surpluses in dollars are \(U_{0}=0\), \(U_{1}=V-40-10=V-50\), and \(U_{2}=2(V-40)-40=2V-120\). The first lens becomes worthwhile at \(V=50\): \(U_{1}-U_{0}=V-50\). Adding the second instead of stopping at one changes surplus by \(U_{2}-U_{1}=V-70\). Hence \(q=0\) is optimal at \(V\le50\), \(q=1\) at \(50<V\le70\), and \(q=2\) above 70. The non-adjacent comparison \(U_{2}-U_{0}=2V-120\) cannot overturn this: for \(V\le50\) it is negative, and for \(50<V\le70\) the one-lens choice exceeds both other options. At \(V=50\) or 70 there is a tie and the collector chooses the smaller quantity, as specified.
Conditional on \(q=0\), V is uniform on \([0,50]\), giving \(\mathbb{E}[V\mid q=0]=25\). After \(q=1\) it is uniform on \((50,70]\), giving \(\mathbb{E}[V\mid q=1]=60\). After \(q=2\) it is uniform on \((70,100]\), giving \(\mathbb{E}[V\mid q=2]=85\). The respective order probabilities are \(\frac{1}{2}\), \(\frac{1}{5}\) and \(\frac{3}{10}\); their conditional means reproduce the original mean: \((\frac{1}{2})25+(\frac{1}{5})60+(\frac{3}{10})85=50\). The quantity is therefore an endogenous graded signal of the buyer's private assessment, not an externally assigned demand class.
Assumptions
The buyer knows V and is risk-neutral with the stated deterministic storage cost.
V applies to both identical lenses, and the dealer treats it as continuous Uniform(0,100).
The only allowed quantities are 0, 1 and 2; there are no other taxes, fees or strategic participants.
Final Answer
Order 0 for \(V\le\$50\); order 1 for \(\$50<V\le\$70\); order 2 for \(V>\$70\). Conditional mean values are $25, $60 and $85 respectively (ties at $50 and $70 are probability-zero events).