Choice over Time: Exponential Discounting

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.title[
# Choice over Time: Exponential Discounting
]
.subtitle[
## EC404; Fall 2024
]
.author[
### Prof. Ben Bushong
]
.date[
### Last updated February 20, 2024
]

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# Exponential Discounting

**Exponential discounting:** When a person receives utility at different points in time, she seeks to maximize her *intertemporal utility*:

`$$U \equiv u_{1}+\delta u_{2}+\delta ^{2}u_{3}+...+\delta ^{T-1}u_{T}$$`

or put another way: 
`$$= \sum_{t=1}^{T}\delta ^{t-1}u_{t}\text{.}$$`
--

- `$u_{t}$` is her **instantaneous utility** in period `$t$` (or her "well-being" in period `$t$`).

--
	
- `$\delta$` is her **discount factor**, where `$\delta \in (0,1]$`.

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# Exponential Discounting

Instead of using
`$$U = \sum_{t=1}^{T}\delta ^{t-1}u_{t}$$`

we sometimes use

`$$U \equiv \sum_{t=1}^{T}\frac{1}{(1+\rho )^{t-1}}u_{t}\text{.}$$`

- `$\rho$` is the person's **discount rate** (rate of time preference).
- Reminder: `$\delta$` is the person's **discount factor**.
- Note: `$\delta =1/(1+\rho )$` or `$\rho =(1/\delta )-1$`.

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# Some Simple Examples

**Example 1**: Suppose you face the following choice:

(A) 90 utils in period 2 `$\quad$` vs. `$\quad$` (B) 160 utils in period 4

Which do you prefer?

**Example 2:** Suppose you have the opportunity to give up 5 utils now in order to gain 2 utils for each of the next three periods. Do you take
it?

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# A More Complicated Example

**Example 3:** Two-Period Saving-Consumption Decisions

Let `$c_{1}$` be your consumption expenditures in period 1, and let `$c_{2}$` be
your consumption expenditures in period 2. Hence, in the end you must choose
a consumption bundle `$(c_{1},c_{2})$`.

You seek to maximize your intertemporal utility

`$$U(c_{1},c_{2})=u(c_{1})+\delta u(c_{2})\text{.}$$`

- Note: `$u(c_{t})$` is your period- `$t$` instantaneous utility as a function of your period- `$t$` consumption --- typically assume `$u^{\prime }>0$` and `$u^{\prime \prime }<0$`.

Let `$Y_{1}$` be your income received in period 1, and let `$Y_{2}$` be your
income received in period 2.

Let `$r$` be the market interest rate, at which you can either save or borrow.

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# A More Complicated Example (cont)

### What is your budget constraint?

Suppose `$Y_{1}=W$` and `$Y_{2}=0$` (*how-to-eat-a-cake* problem):

Budget constraint is

`$$c_{1}+\frac{c_{2}}{1+r}\leq W$$`

Suppose instead `$Y_{1}>0$` and `$Y_{2}>0$`.

Budget constraint is

`$$c_{1}+\frac{c_{2}}{1+r}\leq Y_{1}+\frac{Y_{2}}{1+r}.$$`

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#Solving Consumption-Savings Problems

**General Principle:** If there are no liquidity constraints --- that is, if you
can borrow and save at the same market interest rate --- then your budget
constraint will take the form:

`$$PDV\text{ of Consumption Expenditures }\leq PDV\text{ of Income Flows.}$$`

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# A More Complicated Example (cont)

So the problem becomes:

Choose `$(c_{1},c_{2})$` to maximize

`$$U(c_{1},c_{2})=u(c_{1})+\delta u(c_{2})$$`

subject to

`$$c_{1}+\frac{c_{2}}{1+r}\leq Y_{1}+\frac{Y_{2}}{1+r}\equiv W.$$`

We can do a graphical analysis.... (see board in class.)

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# Solving Consumption-Savings Problems

Or we can solve explicitly for specific functional forms.

- For instance, when `$U(c_{1},c_{2})=\ln c_{1}+\delta \ln c_{2}$`,

`$$c_{1}=\frac{W}{1+\delta }\qquad \text{and} \qquad c_{2}=\frac{\delta (1+r)W}{1+\delta }\text{.}$$`
(Solving this involves a simple application of constrained maximization; see board.)

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# A More Complicated Example (cont)

### What if there are liquidity constraints?

Consider Example 3, except suppose that when you borrow, you must pay an
interest rate `$r_{B}>r$`.

- Budget constraint is:

`$$c_{1}+\frac{c_{2}}{1+r} \leq Y_{1}+\frac{Y_{2}}{1+r}\equiv W_{S}\qquad \text{if }c_{1}\leq Y_{1}$$`

`$$c_{1}+\frac{c_{2}}{1+r_{B}} \leq Y_{1}+\frac{Y_{2}}{1+r_{B}}\equiv W_{B}\qquad \text{if }c_{1}>Y_{1}$$`

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# An Even More Complicated Example

**Example 4:** `$T$` -Period Saving-Consumption Decisions

Suppose that you consume in `$T$` different periods, and let `$c_{t}$` denote your consumption expenditures in period `$t$`. Hence, in the end you must
choose a consumption bundle `$(c_{1},c_{2},...,c_{T})$`.

You seek to maximize your intertemporal utility

`$$U(c_{1},c_{2},...,c_{T})=\sum_{t=1}^{T}\delta ^{t-1}u(c_{t}).$$`

- Let `$(Y_{1},Y_{2},...,Y_{T})$` denote your income flows.

- Let `$r$` be the market interest rate, and assume no liquidity constraints.

*Implication:* You will choose `$(c_{1},c_{2},...,c_{T})$` to maximize `$U(c_{1},c_{2},...,c_{T})$` subject to

`$$c_{1}+\frac{c_{2}}{1+r}+...+\frac{c_{T}}{(1+r)^{T-1}}\leq Y_{1}+\frac{Y_{2}}{1+r}+...+\frac{Y_{T}}{(1+r)^{T-1}}$$`
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# An Even More Complicated Example (cont)

### Example 4 with `$u(c)=\ln c$` :

Goal: Choose `$(c_{1},c_{2},...,c_{T})$` to maximize

`$$U(c_{1},c_{2},...,c_{T})=\ln c_{1}+\delta \ln c_{2}+...+\delta ^{T-1}\ln c_T$$`

subject to

`$$c_{1}+\frac{c_{2}}{1+r}+...+\frac{c_{T}}{(1+r)^{T-1}}\leq W$$`

where

`$$W\equiv Y_{1}+\frac{Y_{2}}{1+r}+...+\frac{Y_{T}}{(1+r)^{T-1}}.$$`

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# Solving the Very Complicated Example

### A relationships-between-variables approach

- It turns out that for each `$t>1$` :

`$$c_{t} =\left[ \delta (1+r)\right] ^{t-1}c_{1}$$`

`$$\text{or} \qquad c_{t} =\delta (1+r)c_{t-1}.$$`

Solving for consumption:

`$$c_{1}=\frac{W}{1+\delta +...+\delta ^{T-1}}\quad \text{and} \quad c_{t}=\frac{\left[ \delta (1+r)\right] ^{t-1}W}{1+\delta +...+\delta ^{T-1}}.$$`

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# Example to Get Calculus Memory Going

Consider a simple two-period model of intertemporal choice. Suppose that Rocky receives income `$Y_1$` in period 1 and additional income `$Y_2$`
in period 2. (He dies in his fight against Ivan Drago immediately after period 2).  The market interest rate at which Rocky can both borrow
and save is 4%. Finally, the Rocky's preferences are given by

`$$U(c_{1},c_{2})=\frac{3}{2}\left( c_{1}\right) ^{2/3}\text{ }+\text{ }\delta \frac{3}{2}\left( c_{2}\right) ^{2/3}.$$`
--

**a)** Derive the budget constraint that the person faces.

**b)** Solve for the optimal `$c_{1}$` and `$c_{2}$` as a function of `$\delta$` .

**c)**(Bonus; do this later) When is the person saving in period 1? When is the person borrowing in period 1?

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# General Version of Discounted Utility

All of the ideas we have discussed about exponential utility actually fit under a broader umbrella.

We'll call this broader umbrella the *discounted-utility model*:

`$$U^{t}= \sum_{x=0}^{T-t}D(x) \text{ }u_{t+x}$$`

- In this framework, `$U^{t}$` is intertemporal utility from perspective of period `$t$`.

- `$u_{\tau}$` is instantaneous utility in period `$\tau$` (or "well-being" in period `$\tau$`).

- `$x$` is the delay before receiving some utility.

- `$D(x)$` is a **discount function** that specifies the amount of discounting associated with delay `$x$`.

In principle, we could have any discount function. Exponential discounting **assumes** `$D(x)=\delta ^{x}$`.

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# Key Features

(1) Impatience: For `$\delta <1$`, `$D(x)$` is monotonically declining in `$x$`.

- Put simply, longer delays imply more discounting.

(2) Constant discounting: For all `$x$`, `$D(x+1)/D(x)=\delta$` .

- This represents an even-handedness in how you view time. 
- If we're thinking in terms of years, how you feel about this year vs. next year is the same as how you feel about next year vs. the following year
is the same as how you feel about 5 years from now vs. 6 years from now. 
- If we're thinking in terms of days, how you feel about today vs. tomorrow is the same as how you feel about tomorrow vs. the next day is the
same as how you feel about 100 days from now vs. 101 days from now.

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# Key Features

(3) Time consistency: As time passes, you do not change your mind about the
best course of action.

- Your relative preference between two calendar dates is independent of
when you are asked (independent of your perspective).

Let's work through an example of how relative preferences depend (or do not depend) on one's perspective under exponential discounting....

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# An Illustration of Time Consistency

From a period-1 perspective, where your intertemporal preferences are

`$$U^{1}=\sum_{x=0}^{T-1}D(x)u_{1+x}$$`

how do you weight:

- period 2 vs. period 3?

- period 3 vs. period 5?

From a period-2 perspective, where your intertemporal preferences are%

`$$U^{2}=\sum_{x=0}^{T-2}D(x)u_{2+x}$$`
how do you weight:

- period 2 vs. period 3?

- period 3 vs. period 5?

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# Time Consistency

**Example 5:** Suppose you have linear instantaneous utility, and
that you must choose between the following two options:

--
- Option A: Receive payoff `$V_{L}$` in period 2.
- Option B: Receive payoff `$V_{H}$` in period 3.

In period 1, prefer Option B when `$\dots \dots$` ?

In period 2, prefer Option B when `$\dots \dots$` ?