## H

PA (1 - kdt)' Nkdt .< . > (x)dh < t> (x)dh Figure ,1.5. Entry, Shift, and Exit of Waiting Firms where the last term is a particular solution of the full equation, and the first two terms arise from the general solution of the homogeneous part. The constants C and Cz remain to be determined, while y and yi are roots of the quadratic thus one is positive and the other is negative. The first condition for determination of the constants C and Ci is found from the consideration that the...

## Of a Project and the Decision to Invest

The basic model of irreversible investment in Chapter 5 demonstrated a close analogy between a firm's option to invest and a financial call option. In the case of a call option, the price of the stock underlying the option is assumed to follow an exogenously specified stochastic process, usually a geometric Brownian motion. In our model of real investment, the corresponding state variable was the value of the project, K, for which we stipulated an exogenous stochastic process. However, as we...

## Investments in Offshore Oil Reserves

We begin with a model of offshore petroleum leases developed by Paddock, Siegel, and Smith (1988). The U.S. government regularly auctions off leases for offshore tracts of land, and oil companies typically perform valuations of such tracts as part of their bidding process. Because bids can involve hundreds of millions of dollars, it is important that these valuations be done accurately. In addition, oil companies must decide what to do with tracts that they succeed in leasing. How high should...

## References

Optimal Investment Under Uncertainty. American Economic Review 73 (March) 228-233. __ 1984. The Effects of Uncertainty on Investment and the Expected Long-Run Capital Stock. Journal of Economic Dynamics and Control 7, 39-53. __ 1990. Consumption and Investment. In Handbook of Monetary Economics, eds. Benjamin Friedman and Frank Hahn. New York North-Holland. __ and Olivier J. Blanchard. 1986. The Present Value of Profits and Cyclical Movements in Investment. Econometrica 54...

## The Simplest Case No Operating Costs

In this section the firm's investment project, once completed, will produce a fixed flow of output forever. For convenience, we will choose the units so that the quantity of output from the project is equal to one unit per year. Suppose the inverse demand function giving price P in terms of quantity Q is P Y D( 0, where )' is a stochastic shift variable. In this section the variable costs of production are assumed to be zero, so the firm's profit flow is just P rD(l). Hence, without further...

## Smooth Pasting

Here we consider the optimal stopping problem with a finite horizon and time dependence, when the state variable follows an Ito process. We demonstrate somewhat more formally the value-matching and smooth-pasting conditions for 14 and 15 that determine the free boundary that separates the continuation and stopping regions. Over a short interval of time dt, the Bellman equation 6 becomes F x. t max i2 .t . n x.t dt 1 - pdt F x,t dF . Stopping is optimal if the first term in the braces on the...

## Contingent Claims Analysis

When we studied the optimal stopping problem in dynamic programming, we interpreted F x. t as the market value of an asset that entitles the owner to the firm's future profit flows t . The equation 8 expressed the condition that for an investor who holds this asset for a short interval of time, the immediate profit flow and the expected capital gain together provide a total rate of return p. We specified this discount rate exogenously, but in practice it has the interpretation as the...

## The Option Approach

The net present value rule, however, is based on some implicit assumptions that are often overlooked. Most important, it assumes that either the investment is reversible, that is, it can somehow be undone and the expenditures recovered should market conditions turn out to be worse than anticipated, or, if the investment is irreversible, it is a now or never proposition, that is, if the firm does not undertake the investment now, it will not be able to in the future. Although some investments...

## P

Investment with Price and Cost Uncertainty Figure 6.8. Investment with Price and Cost Uncertainty Otherwise a dynamic programming approach leads to a very similar differential equation. d F - mP - nl FP - in d P F, -n dl 5 Fpp a2p P2 2 F, gt , para, PI Fuaj l2 dt. Note that the dP and dl on the right-hand side are stochastic. However, we can choose in FP and n F to get rid of these terms and make the portfolio riskless. Then the holder of the portfolio over the interval , t clt will...

## Dynamic Programming

In this section we introduce the basic ideas of dynamic programming. We start with the two-period example from Chapter 2, thus providing a simple concrete setting for the ideas and continuity with the previous analysis. Then we extend the ideas and develop the general theory of multiperiod decision strategies. Finally, we let time be continuous, and represent the underlying uncertainty with either Ito or Poisson processes. That is the setting for most of the applications that will appear in...

## Operating Costs and Temporary Suspension

Suppose once again that the output price follows the geometric Brownian motion of equation 1 . Then a, a, p, and S p - a are all constants. If the option of investing in the project is ever going to be exercised, we need p. gt a, or S gt 0, and we will assume that this is indeed the case. We will also assume that operation of the project entails a flow cost C, but that the operation can be temporarily and costlessly suspended when P falls below C, and costlessly resumed later if P rises above...

## Info

The Ph curve shifts up to reach its final constant level at this new lower R. The PR curve shifts down, only to end when mothballing ceases to be used. Note also that as R increases starting at zero, the restarting and mothballing thresholds spread apart very rapidly. Since we have set the mothballing cost Em 0, the sum of the costs of the pair of switches, namely, R Em, is small, and we have an instance of the cube root formula 16 above. One further numerical experiment is of...