# 23 Exponential discounting examples

In this part, I will work through several numerical examples of decisions by an exponential discounter.

## 23.1 Example 1

Suppose we have an exponential-discounting agent with discount factor \delta=0.95 and utility each period of u(x_n)=x_n. They are offered two choices.

Choice 1: Would this agent prefer $100 today (t=0) or $110 next week (t=1)?

To determine this, we calculate the discounted utility of each option. The agent will prefer the option with the highest discounted utility.

The discounted utility of the $100 today is:

\begin{align*} U_0(0,\$100)&=u(\$100) \\ &=100 \end{align*}

The discounted utility of the $110 next week is:

\begin{align*} U_0(1,\$110)&=\delta u(\$110) \\ &=0.95\times 110 \\ &=104.5 \end{align*}

The exponential discounter will prefer to receive $110 next week as it leads to higher discounted utility.

Choice 2: Would this agent prefer $100 next week (t=1) or $110 in two weeks (t=2)?

The discounted utility of the $100 next week is:

\begin{align*} U_0(1,\$100)&=\delta u(\$110) \\ &=0.95\times 100 \\ &=95 \end{align*}

The discounted utility of the $110 in two weeks is:

\begin{align*} U_0(2,\$110)&=\delta^2 u(\$110) \\ &=0.95^2\times 110 \\ &=99.275 \end{align*}

The exponential discounter will prefer to receive $110 in two weeks.

The set of decisions across Choice 1 and Choice 2 are time consistent. If the agent selected $110 in two weeks for Choice 2 and was given a chance to change their choice after one week (which is effectively Choice 1), they would not change their decision.

Figure 23.1 visualises the effect of discounting in Choice 2.

The two bars represent the options: $100 at t=1 and $110 at t=2. The line from each represents the discounted value of that option at each time. For example, at t=1 the discounted utility of the $100 received at t=1 is $100 and the discounted utility of the $110 received at t=2 is $104.50. We can read those values from the line. For any time t we can determine which option would be preferred by seeing which line is higher.

You will note that the two lines do not cross. For an exponential discounter, if one line is higher at any particular time t, it is higher at all times.

Figure 23.2 visualises Choice 2 reconsidered at t=1. The discounted value of the $100 received immediately is less than the discounted utility of $110 in one week.

## 23.2 Example 2

Suppose we have an exponential discounter with discount factor \delta=0.95 per week and utility each period of u(x_n)=x_n

They are offered $100 today. What sum would they need to be offered in one year (52 weeks) to prefer that later payment to the $100 today?

The discounted utility of the $100 today is:

\begin{align*} U_0(0,\$100)&=u(\$100) \\ &=100 \end{align*}

The discounted utility of the sum y received in 52 weeks is:

\begin{align*} U_0(52,\$y)&=\delta^{52} u(\$x) \\ &=0.95^{52}\times y \end{align*}

They will prefer $y in 52 weeks if U(52,\$y) is greater than 100.

\begin{align*} U_0(52,\$y)&>100 \\[6pt] 0.95^{52}\times y&>100 \\[6pt] y&>\frac{100}{0.95^{52}} \\[6pt] y&>\$1440.03 \end{align*}

The agent would be willing to wait a year for payment if they were paid more than $1440.03.

Figure 23.3 visualises this problem. The bar at t=52 represents the $1440.03 that the agent would need to be paid to prefer that payment to $100 today. The line extended from that bar back to t=0 indicates the discounted value of that payment at any time t. At t=0 the discounted value of the $1440.03 is $100.

## 23.3 Example 3

Suppose we have an exponential discounter with discount factor \delta=0.75 and utility each period of u(x_n)=x_n.

Would this agent prefer $10 in five days (t=5) or $20 in 10 days (t=10)?

The discounted utility of the $10 in five days is:

\begin{align*} U_0(5,\$10)&=\delta^5u(\$10) \\ &=0.75^5\times 10 \\ &=2.37 \end{align*}

The discounted utility of the $20 in 10 days is:

\begin{align*} U_0(10,\$20)&=\delta^{10} u(\$20) \\ &=0.75^{10}\times 20 \\ &=1.13 \end{align*}

Discounted utility is higher for the $10 in five days. The agent will prefer to receive $10 in five days.

What if their discount rate was \delta=0.95?

The discounted utility of the $10 in five days is:

\begin{align*} U_0(5,\$10)&=\delta^5u(\$10) \\ &=0.95^5\times 10 \\ &=7.74 \end{align*}

The discounted utility of the $20 in 10 days is:

\begin{align*} U_0(10,\$20)&=\delta^{10} u(\$20) \\ &=0.95^{10}\times 20 \\ &=11.97 \end{align*}

Discounted utility is higher for the $20 in 10 days. This agent will prefer to receive $20 in 10 days.

Figure 23.4 visualises the choices and the agents’ discounting of the payoffs.

In both charts, vertical bars represent the $10 in five days and $20 in 10 days. The lines projecting back to t=0 represent the discounted value of those payoffs at each time.

When \delta=0.75, the heavy discount to the more distant payoff means that it has a lower discounted utility than the smaller, sooner payment of $10. When \delta=0.95, the discount is less severe and the $20 in 10 days has a higher discounted utility than the $10 in five days.