1. What is the optimal number of tokens to hoard?

2. What is the present value of the savings over not hoarding at all?

3. Suppose that the optimal quantity to hoard is Q. What is the present value of the savings if you decide to hoard only 0.8Q?

Riders of the subway system in the city of Metropolis must pay for the ride by purchasing a token. The same token can also be used to ride the buses in Metropolis. A single token is good for a trip to any destination served by the system. (Tokens are also used by millions of commuters for bridge, tunnel, and highway tolls in many areas of the country.) late in 2014, Metropolis transit officials announced that they were seeking a fare increase from $1.50 to $2.00. later negotiations with politicians in the state capital reduced the requested increase to $1.75. It usually takes a few weeks between the announcement of a fare increase and the time that the increase goes into effect. Knowing that an increase will occur gives users of mass transit an opportunity to mitigate the effect of the increase by hoarding tokens—that is, by purchasing a large supply of tokens before the fare increase goes into effect. There is a clear motivation for hoarding tokens—namely, the purchase of tokens before a fare increase offers a savings over purchasing the same tokens at a higher price after the change in fare. Why wouldn’t riders want to purchase a very large supply of tokens? The reason is the inventory cost that arises because of the time value of money. The larger the supply that is hoarded, the longer the time until the tokens are used. Purchasing the supply of tokens represents an immediate cost, but the benefit is only realized over a longer period of time. Thus, there is a trade-off between the immediate cost and the prolonged benefit. The optimal number of tokens to hoard balances these two effects to maximize the present value of the net benefit of the hoarding strategy.

Suppose that the current price of subway tokens is p₁ = $1.50 and the fare is due to rise to p₂ = $1.75. Suppose that you use the subway to commute 2 times per day, 5 days per week, 50 weeks per year. Also, suppose that you can purchase (or hoard) any number of tokens before the price increase takes effect. You will use the hoarded tokens during your normal usage of the mass transit system. After your hoard runs out, you will start purchasing tokens each day at the higher price. Suppose that your cost of capital is 15% per year. This means that you can borrow money to purchase your token supply, but the interest cost on the borrowed money is 15% per year.