Suppose that our problem is to determine a dynamic production and inventory accumulation plan for a durable good. Assume that we store the good in inventory until date T, at which time we sell the whole batch of production. Let K(t) be the stock of accumulated inventory at any date t, and let the production costs of adding to inventory be quadratic, or: C[Q(t)]=cQ(t)+(1/2)d[Q(t)]², where Q(t) is the quantity produced and placed into inventory. Assume a finite and given length time horizon T, an initial stock of inventory K(0)=0. Finally, assume that, in addition to the production costs of accumulating inventory, that there are holding costs associated with inventory storage of H[k(t)]=hK(t) incurred each period.

Use optimal control theory to solve for the optimal inventory accumulation path and inventory trajectory that minimizes the integral of production and holding costs. Assume that the discount rate is zero and that there is a specific order for the amount delivered at T to be K(T)=K_T.

1. Derive explicit solutions for the production level Q(t) and the stock in inventory K(t).

2. Show how the production path is affected by the size of the order. How is it affected by the cost parameters?