Now, later, or both: A closed-form optimal decision for a risk-averse buyer

Motivated by the energy domain, we examine a risk-averse buyer that has to purchase a fixed quantity of a continuous good. The buyer has two opportunities to buy: now or later. The buyer can spread the quantity over the two timeslots in any way, as long as the total quantity remains the same. The current price is known, but the future price is not. It is well known that risk neutral buyers purchase in whichever timeslot they expect to be the cheapest, regardless of the uncertainty of the future price. Research suggests, however, that most people may in fact be risk-averse. If the expected future price is lower than the current price, but very uncertain, then they may purchase in the present, or spread the quantity over both timeslots. We describe a formal model with a uniform price distribution and a two-segment piecewise linear risk aversion function. We provide a theorem that states the optimal decision as a closed-form expression.