Lecture 8
DYNAMIC OPTIMIZATION

• Present value
• Discount rate
• Application: Optimal saving

• Ross lives for 2 periods
• In Period 1, he is young, and earns an income of y₁
• In Period 2, he is old, and earns y₂

Question: What is y₂ if Ross retires in Period 2?

B. Budget constraint

• In Period 1: c₁ + s = y₁
• In Period 2: c₂ = y₂ + (1+r)s

C. Lifetime budget constraint

• c₁+c₂/(1+r) = y₁ + y₂/(1+r)
• Present value of lifetime consumption = Present value of lifetime income

D. Ross's objective

• To maximize utility u(c₁, c₂)

E. Optimal values of c₁, c₂ and s

• Form the Lagrangian
• Write down FOC's
• Solve for c₁, c₂
• Check sufficient conditions (SOC) for a maximum
• Obtain optimal value of saving

F. Question

How is saving affected by a change in interest rates?

G. Example: Optimal saving

Given the following information about a consumer:

• Utility function: u(c₁,c₂) = c₁c₂
• Income earned in Period 1 = y
• Income earned in Period 2 = 0
• Interest rate = r

Questions

1. Write down the consumer's intertemporal budget constraint.
2. Set up the formal utility maximization problem.
3. Using the FOC, show that c₂ = (1+r)c₁.
4. Derive the expressions for the optimal values of c₁ and c₂ in terms of y and r.
5. Show that the optimal saving in Period 1 is y/2.
6. Sketch the budget constraint and indicate the optimal point (you will need a picture of the indifference curve to do this.)
7. How do you interpret the Lagrange multiplier in this case?