Asking price and price discounts: the strategy of selling an asset under price uncertainty (Q885075)

Let \(n\) assets be selled and buyed and a discount rate (market interest rate). The seller has a price in mind, to be fixed, the law of which is supported on interval \([a,b]\). The buyer's valuation is dnotes as \(P\). In seller's mind, the probability distribution function of \(P\) is \(F(x)= \mathbb{P}\{P\leq x\}=\int^x_0 f(t)\,dt\), \(F(a)= 0\), \(F(b)= 1\). The probability of selling the assets in one period is \(p= 1 -F(P)\), of selling it on \(n\)th period is \(p(1- p)^{n-1}\). The discounted assets value on \(n\)th period actually is \(P_n= P(1+ r)^{1-n}\). So, let \(X\) be the random variable modelling the seller's return from the sale. The expected return is a function of \(P\), \(E(P)= Pp{1+r\over r+p}\). Assuming sufficient conditions on the probability distribution function \(F\), a risk neutral seller can optimize \(P\mapsto E(P)\) and the optimal price \(P^*\) satisfies \(P^*\leq b\), \(P^*= a\) or satisfies the implicit equation \(P^*= {p(p+r)\over rf(P^*)}\). On another side, the authors consider the asking price strategy, \(P\) being the asking price: when \(E\) is a strictly concave function, then the optimal asking price will be greater than the optimal fixed price \(P^*\). Finally, the authors discuss the seller's behaviour in case of his non risk-neutrality.