Dynamic Optimization

1
Dynamic Optimization

• The factory can be built instantly at cost I16.
• 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.

2
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?

3
How to calculate present value

• By using the fact that
• (1-a)(1aa2a3a4a5)1
• We know that
• 1aa2a3a4a5 1/(1-a)
• Thus, if a bond generates 1 dollar every year,
  then the present value with 10 interest rate is
• 11/(1.1)1/(1.1)211.
• If a bond generates 2 dollars every year, then
  the present value is 22.

4
Dynamic Optimization

• (Ans Q1) Let us calculate the Net Present Value
  of the project when I make the investment right
  now.
• -16 -1622 6.
• The net present value is positive. Thus, we
  should go ahead with the investment.

5
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?

6
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.

7
Dynamic Optimization

• Lets calculate the NPV of the project when we
  wait.
• 0.5(1/1.1) (Max 0, -16 )
• 0.5(1/1.1) (Max 0,-16 )
• Note that -16 gt0 and that -16
  lt0.

8
Dynamic Optimization

• Thus, the firm should wait until new year.

9
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.

10
Dynamic Optimization

• (Q3) With probability q, the price will be
  (1u)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?

11
Dynamic Optimization

• (Ans Q3) If we invest right now, the value of the
  project is as follows,
• NPV -I p
• -I p 10pq(1u)(1-q)(1-d)
• -Ip10pqu1-dqd

12
Dynamic Optimization

• If we wait to make a decision until the next
  year, the value of the project is as follows,
• NPV q-I11(1u)p/1.1
• P

13
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