EC451_Macro

Recursive

Finite-period

Dynamic Programming. Consider again a finite-period saving problem:

max_{{C_{t},B_{t+1}}} \sum^T_{t=0} \beta^t u(C_{t}) \\ \text{s.t.} B_{t+1} = RB_{t} + Y_{t} -C_{t} \\ B_{T+1} \geq 0 \end{align}

From previous learnings, we could both get the Euler Equation and the Transversality Condition:

• EC: $u^{\prime }(C_t)=\beta R{u}^{\prime }(C_{t+1})$
• TC: $B_{T+1}=0$

The idea of the recursive methods is to start with the last period and move backwards. If not do so, we would have a large system with $2\times (T+1)$ equations and $2\times (T+1)$ unknowns, which would be impossible to solve using Largrangian Multiplier.

If we use recursive, we would only have one equation. This is why we study this lecture.

Bellman Equation

The intuition of Bellman Equation is that: “Today’s greatest happiness = Today’s utility + The discounted value of future happiness.”, which could be written as:

$$V(B)=ma{x}_{B^{\prime }}\{u(RB-B^{\prime })+\beta V(B^{\prime })\}$$

While Bellman Equation is for discrete time, we would introduce HJB equation to solve continuous time problem.

Todo List

Lecture 3

🎯 Big Picture First (5-Minute Task)

• Skim the entire document. Don’t try to understand it. Just look at the headings, the equations, and the graphs. Get a feel for the shape of the lecture. What are the main sections? (e.g., “Recursive approach,” “HJB equation,” “Stochastic HJB”).

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Part 1: The “Why” - Thinking Backwards (Focus Session 1: 25 mins)

The goal here is to understand why this recursive method is useful.

• Read the first page to understand the difference between the standard “sequential” method and the “recursive” method.

• In your own words, what is the main problem with the sequential method as the time horizon (T) gets bigger?

• Hands-On Derivation: Grab a pen and paper.

  • Solve the problem for Period T. The agent just consumes everything left. Write down the equations for

    CT​ and the value function VT.

  • Now, solve for Period T-1. Use the VT​ you just found as part of the maximization problem. Derive the equations for

    BT​ and CT−1​ yourself.

• Check-In: Do you see the pattern? You solve the future first (VT​) to make the best decision today.

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Part 2: The Main Tool - The Bellman Equation (Focus Session 2: 25 mins)

Now, we generalize that backward-thinking into one powerful equation for an infinite horizon.

• Find and write down the main

  Bellman equation on page 45.

• Define the key pieces in your own words:

  • What is the state variable, B? (Hint: It summarizes the past) 6.

  • What is the value function, V(B)? (Hint: It summarizes the future) 7.

  • What is the policy function, g(B)? (Hint: It’s your optimal rule or plan) .

• Follow the example for the

  infinitely-lived agent 9. How does taking the limit

  T→∞ actually simplify the problem?10101010.

• Find the famous results for this specific problem: what is the optimal rule for consumption (Ct​) and savings (Bt+1​)? Write them down11.

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Part 3: Continuous Time - The HJB Equation (Focus Session 3: 25 mins)

This part looks intimidating, but it’s just the Bellman equation when time flows continuously instead of in steps.

• Follow the derivation of the Hamilton-Jacobi-Bellman (HJB) equation. Don’t get lost in the algebra. Focus on the core idea: they take the time step

  Δ and let it go to zero121212121212121212.

• Find and write down the final

  HJB equation13.

• Spot the difference: Put the Bellman equation and the HJB equation side-by-side. What’s different? (Hint: look for β vs. ρ, and V(B′) vs. V′(B)).

• Look at the example solution 14. Notice it uses the same “guess and verify” strategy. You don’t need to re-derive it, just see how the method is applied.

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Part 4: Adding Reality - Uncertainty! (Focus Session 4: 30 mins)

This is where dynamic programming really shines. How do we make decisions when the future is random?

• Find the Bellman equation for a problem

  with uncertainty15. What is the one new symbol in the equation? (Answer: The expectation operator,

  E).

• [ ]

  Ito’s Lemma: Skim this section16. You don’t need to be a math genius here. The key takeaway is:

  Ito’s Lemma is the tool we must use to analyze functions of random variables in continuous time.

• Find the

  Stochastic HJB equation17. Compare it to the regular HJB. What is the new term that appears? (Hint: It involves

  σ2 and V′′(B)). This new piece is the term that accounts for risk/volatility.

• Jump to the end of the notes. The model predicts that the distribution of wealth follows a

  Pareto distribution18. According to the notes, what happens to the number of rich people if the return on assets,

  r, increases?19.

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✅ Final Review (15-Minute Task)

• Go back through your notes and the checklist.

• Try to explain to yourself (or a friend, or a pet!) the difference between the Bellman equation and the HJB equation in one simple sentence.

• In one sentence, explain what adding uncertainty does to the HJB equation.

You’ve got this! By breaking it down into focused, bite-sized sessions, you can master this material.