Questions: Dynamic Optimization in Macroeconomics

Question 1

An economist models a household that maximizes current-period utility in each period independently, consuming all income and saving nothing. What fundamental feature of dynamic optimization does this model ignore?

• A. The model ignores uncertainty; real households face stochastic income shocks
• B. The model treats each period as isolated — it ignores that savings carry over as next period's wealth, linking all decisions through an intertemporal constraint
• C. The model uses a static utility function rather than one defined over an infinite horizon
• D. The model lacks a discount rate, making future and present consumption equally valued

Answer: (B)

The defining feature of a dynamic optimization problem is that decisions today affect the feasible set tomorrow. Savings today become wealth tomorrow, which determines tomorrow's choices. A model that ignores this 'state variable' linkage reduces a genuinely dynamic problem to a sequence of unrelated static problems, losing the entire structure that tools like the Bellman equation are designed to handle. The Euler equation and value function both encode the fact that tomorrow's options depend on today's choices.

Question 2

In dynamic programming, what is the key difference between the value function and the policy function?

• A. The value function is continuous while the policy function is discrete-valued
• B. The value function gives the total future payoff achievable from a given state; the policy function gives the optimal action to take in that state
• C. The value function is used in finite-horizon problems; the policy function is used in infinite-horizon ones
• D. The value function describes preferences; the policy function describes budget constraints

Answer: (B)

These are the two central objects in dynamic programming. The value function V(w) answers: 'If I have wealth w today, what is the maximum total discounted utility I can achieve from now on?' The policy function c(w) answers: 'If I have wealth w, how much should I consume today?' The Bellman equation ties them together: V(w) = max_c{u(c) + βV(w')}, where w' is next-period wealth. Both depend on the state, but they answer different questions: total achievable value versus optimal action.

Question 3

The Bellman equation is a closed-form formula that directly gives optimal consumption as a function of current wealth.

• T. True
• F. False

Answer: False

The Bellman equation is a functional equation — it characterizes the value function through a recursive relationship (V(w) = max_c{u(c) + βV(w')}) but is not itself a closed-form solution. Solving it requires finding the value function V(·) that satisfies this equation, typically through numerical value function iteration or, in special tractable cases, by guessing a functional form and verifying it works. The Bellman equation is a necessary condition and a computational tool, not a direct formula.

Question 4

If a consumer's discount rate exceeds the real interest rate, the Euler equation implies their optimal consumption is declining over time — they prefer to consume more now and less later.

• T. True
• F. False

Answer: True

With CRRA utility, the Euler equation implies consumption growth c_{t+1}/c_t = [β(1+r)]^{1/σ}, where β = 1/(1+ρ) and ρ is the discount rate. When ρ > r, the product β(1+r) < 1, so the growth ratio is less than one — consumption declines over time. Intuitively, impatience (high ρ) dominates the return to saving (r), so the consumer front-loads consumption. When r > ρ, consumption grows.

Question 5

Why do dynamic optimization problems require tools beyond standard Lagrangian methods, and what specific problem does the Bellman equation solve?

Model answer: Standard Lagrangian methods optimize over a fixed set of choice variables subject to constraints, producing simultaneous first-order conditions. In a dynamic problem, the household chooses consumption in every future period, and today's savings affect tomorrow's wealth, which affects all subsequent choices. The feasible set for period t depends on all prior choices — the problem is a sequence of linked optimizations, not a single one. The Bellman equation solves this by reformulating it recursively: if you know the value of being in any state tomorrow (the value function), today's optimal action is simply the one that maximizes current payoff plus the discounted continuation value. This converts an intractable simultaneous infinite-horizon problem into a tractable recursive one.

The key insight behind Bellman's principle of optimality is that an optimal policy has the property that, given any state reached by the optimal path, the remaining decisions must also be optimal from that state. This recursive structure means you only ever need to solve a one-period problem, with the future summarized by the value function.