Dynamic Optimization The factory can be built instantly at cost I=16. The factory will produce one cookie per year forever, with interest rate 10%. The current price of the cookie is $2. With probability 0.5, the price tomorrow will be 3. With 0.5, the price tomorrow will be 1.

Dynamic Optimization (Q1) Suppose that the firm must decide whether to invest or not today. If you do not invest today, there is no more chance to undertake the investment activity. What is the optimal decision for the firm?

How to calculate present value By using the fact that (1-a)(1+a+a 2 +a 3 +a 4 +a 5 + … )=1 We know that 1+a+a 2 +a 3 +a 4 +a 5 + … = 1/(1-a) Thus, if a bond generates 1 dollar every year, then the present value with 10% interest rate is 1+1/(1.1)+1/(1.1) 2 + … =11. If a bond generates 2 dollars every year, then the present value is 22.

Dynamic Optimization (Ans Q1) Let us calculate the Net Present Value of the project when I make the investment right now = = 6. The net present value is positive. Thus, we should go ahead with the investment.

Dynamic Optimization (Q2) Suppose that the firm can delay the decision by the next year. That is, the firm can invest right now or wait to make a decision until the next year. What is the optimal choice for the firm, investing right now or waiting until the next year?

Dynamic Optimization (Ans Q2) If the firm makes the investment right now, the firm can enjoy the 2 revenue this year. However, by waiting, the firm can get more information on the profitability of the project. That is, there is the opportunity cost of investing now, rather than waiting and keeping open the possibility of not investing.

Dynamic Optimization Let ’ s calculate the NPV of the project when we wait. 0.5(1/1.1) (Max [0, ]) +0.5(1/1.1) (Max [0,-16 + ]) Note that >0 and that <0.

Dynamic Optimization Thus, the firm should wait until new year.

Dynamic Optimization This captures the essential idea of dynamic programming. We split the whole sequence of decisions into two parts: the immediate choice, and the remaining decisions. At the last relevant decision point we can make the best choice and thereby find the value of the project. Then at the decision point before that one, we know the expected value and therefore can optimize the current choice.

Dynamic Optimization (Q3) With probability q, the price will be (1+u)p. With 1-q, the price will be (1-d)p. If price goes down, the firm does not have an incentive to undertake the investment. Please get the lowest price level in which the firm will invest right now. Does the price level depend on u? If not, what is the interpretation?

Dynamic Optimization (Ans Q3) If we invest right now, the value of the project is as follows, NPV =-I + p+ = -I+ p+ 10p[q(1+u)+(1-q)(1-d)] =-I+p+10p[qu+1-d+qd]

Dynamic Optimization If we wait to make a decision until the next year, the value of the project is as follows, NPV = q[-I+11(1+u)p]/1.1 P =

Dynamic Optimization The critical price depends on only d and (1-q), not u. Also, the larger is d, the larger is the critical price. It is the magnitude of the possible “ bad news ” that derives the incentive to wait. “ A Bad News Principal ”