On the Behavior of Price in a Supply Chain Market for Capacity

Abstract

We are interested in the concept of dynamic pricing of production capacity in a supply chain and in particular, understanding how the supply chain structure might affect the volatility of capacity prices. We find that supply chains with high capacity costs will experience high price volatility.

We consider a continuous time market for a single homogeneous commodity. The market consists of two kinds of agents: sellers, who own capacity and incur short-term and long-term costs of updating capacity and earn their revenue through the sale of capacity to the second kind of agents, buyers, who hold inventory and satisfy the demand of end-consumers. The end-consumers and their interactions with buyers are an exogenous component of this model. We consider three different models that differ in the modelling of the end-consumer demand. In the first model, end-consumer demand is deterministic. We obtain closed form expressions for market capacity, production and equilibrium price.

We use the solution to the first model to analyze the second model in which end-consumer demand is the sum of a deterministic and a Brownian Motion component. Again, we obtain closed-form expressions for equilibrium price, production and capacity. We use the closed form solution of the equilibrium trajectories to obtain their variance and, subsequently, to analyze the impact of cost parameters on their variance. We find that the variance of the price increases as the short-term and long-term costs of changing capacity increase relative to the holding cost. We obtain similar results using a variation of the above model in which capacity of each seller is exogenously fixed.

In the third model, we incorporate the evolution of forecasts in the end-consumer demand model but fix the capacity of each seller. We find that the early learning of end-consumer demand results in early learning in the market price forecast process. We also find that the market cost parameters affect the rate of learning of the price forecast.