PS2

Background Theory — Read This First

Before diving into the questions, here is a self-contained explanation of all the concepts you need.

The Core Idea: Why Smooth?

People don't just want more stuff — they want it consistently. Eating lavishly one day and starving the next is worse than eating moderately every day, even if total calories are identical. This is formalised by diminishing marginal utility: each extra unit of consumption gives you less happiness than the one before. Therefore, smoothing consumption across time increases total lifetime utility.

The Life-Cycle Model and related theories formalise exactly this.

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Key Concepts

1. The Discount Factor $\beta$

People are impatient — a reward today is worth more than the same reward tomorrow. The discount factor $\beta \in (0,1)$ captures this. If you receive utility $u(c^f)$ next period, its value today is only $\beta \cdot u(c^f)$.

With $T+1$ periods $(0, 1, \dots, T)$, total lifetime utility is:

• High $\beta$ (close to 1): Patient consumer. Values future almost as much as today. Will save more.
• Low $\beta$ (close to 0): Impatient consumer. Heavily discounts the future. Will spend more today.

2. The Real Interest Rate $r$

The real interest rate is the reward for saving (or the cost of borrowing), adjusted for inflation. If you save $\$1$ today, you get $\$(1+r)$ next period in real terms.

There is a special relationship between $\beta$ and $r$:

• If $\beta = \frac{1}{1+r}$, then $\beta(1+r) = 1$: your impatience is exactly offset by the market return. You prefer equal consumption every period.
• If $\beta > \frac{1}{1+r}$, i.e. $\beta(1+r) > 1$: you are more patient than the market. You prefer consumption to grow over time (save more now).
• If $\beta < \frac{1}{1+r}$, i.e. $\beta(1+r) < 1$: you are more impatient than the market. You prefer consumption to fall over time (spend more now).

3. The Lifetime Budget Constraint and PVLR

You cannot spend more over your life than you have. The Present Value Lifetime Resources (PVLR) discounts all future income back to today's terms:

The lifetime budget constraint is then:

This says the present value of all consumption must equal the present value of all resources.

Why discount future income?

$\$1$ received next period is worth only $\frac{\$1}{1+r}$ today, because $\frac{\$1}{1+r}$ saved now grows to $\$1$ by next period. Future incomes must be converted to today's terms before summing.

4. Log Utility

The most common utility function in these problems is $u(c) = \ln(c)$. Its key properties:

• $u'(c) = \frac{1}{c} > 0$ — more consumption is always better.
• $u''(c) = -\frac{1}{c^2} < 0$ — diminishing marginal utility (the smoothing motive).
• It delivers clean, closed-form solutions.

5. The Euler Equation

The Euler equation is the optimality condition for intertemporal consumption. It says the benefit of consuming one unit today must equal the benefit of saving it and consuming next period:

For log utility $u'(c) = 1/c$, this becomes:

This tells us the growth rate of consumption over time. Consumption grows if and only if $\beta(1+r) > 1$.

6. Solving with the Lagrangian

To find the optimal consumption plan formally, we set up a Lagrangian — a technique for constrained optimisation. For a two-period problem:

The $\lambda$ (multiplier) enforces the budget constraint. We take derivatives with respect to $c_0$, $c_1$, and $\lambda$, set equal to zero:

• $\frac{\partial \mathcal{L}}{\partial c_0} = 0 \implies \frac{1}{c_0} = \lambda$
• $\frac{\partial \mathcal{L}}{\partial c_1} = 0 \implies \frac{\beta}{c_1} = \frac{\lambda}{1+r}$
• $\frac{\partial \mathcal{L}}{\partial \lambda} = 0 \implies$ budget constraint holds with equality.

Dividing the first two gives the Euler equation. The budget constraint then pins down the level of consumption.

7. Saving

Saving in period $t$ is simply income minus consumption:

Positive saving means the consumer is a lender (accumulating assets). Negative saving means they are a borrower (taking on debt). Over the whole lifetime, the sum of all discounted saving must equal zero (you can't die in debt or leave resources unconsumed).

A Mini Life Cycle Model

Assume that a consumer lives for 3 periods: 0, 1, and 2. The consumer's income streams are $y_0 = 100$, $y_1 = 300$, $y_2 = 25$. Assume that the real interest rate is constant at $4\%$, and that $\beta = \frac{1}{1+r}$. The consumer's periodic utility function is logarithmic, so that:

Answer the following questions. Provide brief explanations and/or show your work.

1. Briefly discuss — what sort of life cycle evolution can these assumptions correspond to?
2. Does the consumer want to consume equal quantities every period? Why or why not?
3. What is the consumer's PVLR?
4. Calculate the consumer's optimal consumption plan (i.e. how much consumption in every period).
5. Calculate the consumer's optimal saving in every period.
6. How does the consumer's wealth evolve over his/her life?
7. Suppose that another consumer faces the same income process and interest rate, but has a different discount factor: $\beta = 0.9711$. Solve for the optimal consumption plan for the second consumer. Briefly explain the difference.

Optimal Consumption Plan with Log Utility

Assume that:

• consumers live for two periods
• they discount the future with a discount factor $0 < \beta < 1$
• income in the first period is $y_0$; income in the second period is $y_1$; no initial assets: $a_0 = 0$
• the utility function in each period is $u(c) = \ln(c)$
• the real interest rate is some positive constant $r > 0$

As a result, the objective function is to choose $c_0$ and $c_1$ in order to maximize the discounted sum of utilities:

Answer the following questions. Provide brief explanations and/or show your work.

1. What is the lifetime budget constraint?
2. Write the Lagrangian for this problem.
3. Derive the first order conditions.
4. What is the Euler equation for this problem? Use it to express $c_1$ as a function of $r$, $\beta$, $c_0$.
5. Substitute your answer for $c_1$ into the budget constraint and use it to solve for $c_0$, and then for $c_1$.
6. Derive a term for savings in the first period as a function of $y_0$, $y_1$, $r$, $\beta$.
7. Consider each of the following changes in isolation (i.e. keep everything else constant and change just one thing). For each, will the consumer save more or less? Show using the solution to the previous part and explain why.
   • (a) Higher $\beta$
   • (b) Higher $y_0$
   • (c) Higher $y_1$

(TA session)

CRRA Utility Function, Income and Substitution Effects

The purpose of this question is to: (i) have another practice at deriving the optimal consumption plan; (ii) illustrate the potentially competing income and substitution effects that we discussed in class.

One of the most widely used utility functions in macroeconomic research is:

(This is usually called either the constant relative risk aversion (CRRA) utility function or the iso-elastic utility function.)

Assume that:

• consumers live for two periods
• they discount the future with a discount factor $0 < \beta < 1$
• income in the first period is $y_0$; income in the second period is $y_1$; no initial assets: $a_0 = 0$
• the utility function in each period is the CRRA utility function
• the real interest rate is some positive constant $r > 0$

Answer the following:

1. What is the lifetime budget constraint?
2. Write the Lagrangian for this problem.
3. Derive the first order conditions.
4. What is the Euler equation for this problem? Use it to express $c_1$ as a function of $r$, $\beta$, $\sigma$, $c_0$.
5. Assume that $y_0 = 0$ and $y_1 > 0$ (so that the consumer is necessarily a borrower). Substitute your answer for $c_1$ into the budget constraint and use it to solve for $c_0$. Use your solution to show that in this case there is no ambiguity and that a higher $r$ will definitely lower current consumption.
6. Now assume that $y_0 > 0$ and $y_1 = 0$ (so that the consumer is necessarily a lender). Substitute your answer for $c_1$ from the Euler equation into the budget constraint and use it to solve for $c_0$. Use your solution to show that in this case there is ambiguity and that the effect of $r$ on current consumption depends on the value of $\sigma$. Briefly discuss the competing effects. What should we assume about $\sigma$ in order to be consistent with the assumption we made in class about the effect of $r$ on current consumption?

(TA session)

Ranking Consumption Responses under the Permanent Income Hypothesis

Use the predictions of the permanent income hypothesis that we discussed in class to rank the following from the biggest to the smallest consumption increase among recipients. (For simplicity, you may assume that the desired consumption plan is spending equal amounts over the life cycle.) Explain your answer.

• (a) an unexpected and explicitly temporary tax rebate of $\$300$ (assume the consumer believes that he/she will never have to pay this back to the government)
• (b) a special dividend on a stock that the consumer owns of $\$300$ per shareholder, that also lowers the value of the stock by $\$300$ per share (assume the consumer owns one stock)
• (c) an unexpected raise of $\$300$ per year, effective immediately

(extra practice)

Identifying Lenders and Borrowers from Consumption Responses

Assume that consumers behave according to the assumptions of the consumption model described in class. Assume that you collected data on their consumption and the real interest rate. You may also assume that all consumers share the same preferences and the same expectations regarding their current and future income.

1. In your data, you observe that some consumers increased their consumption when the real interest rate increased. Can you say with certainty whether these consumers were lenders/borrowers/neither?
2. Other consumers decreased their consumption when the real interest rate increased. Can you say with certainty whether these consumers were lenders/borrowers/neither?

(extra practice)

Jerry and George — Same PVLR, Different Timing

Assume that there are two consumers, Jerry and George. Both live for two periods, are price takers, and have identical utility function: $U(c_i, c_i^f) = \ln(c_i) + \beta\ln(c_i^f)$ where $0 < \beta < 1$, and $i$ can be $J$ for Jerry, or $G$ for George.

Both consumers have initial wealth $a = 0$. The endowment (or income) process is different:

where we assume that $y_0 > \frac{y_1}{\beta(1+r)}$.

• (a) Write the optimization problem for each consumer. Use the lifetime budget constraint.
• (b) Derive the Euler equation.
• (c) Use the Euler equation and the lifetime budget constraint to solve for Jerry's and George's optimal consumption plans.
  • (i) Are the optimal consumption plans different for the two consumers? Explain why or why not.
  • (ii) Do Jerry and George engage in borrowing or lending in the first period? Calculate the amounts and explain who is borrowing and who is lending, and why.
• (d) Now assume that the economy changes, and borrowing is no longer allowed. (To avoid unnecessary complications, you may assume that if someone wishes to lend, there is always an opportunity to do so at rate $r$, e.g. to foreigners.) No calculations are required for the following questions, just explain.
  • (i) Does the Euler equation that you derived in part (b) still hold for both consumers? For none? For one? Explain.
  • (ii) Assume that the government would like to give a tax rebate in order to increase current aggregate consumption. Is it better to give the rebate to Jerry or to George? Explain.