A friend showed me the following YouTube video.

Basically, a guy’s best friend and girlfriend play a prank on him by giving him a fake winning lottery ticket worth $2 million. The guy falls for the prank, but unfortunately overreacts by bombastically quitting his job. (He also informs the two pranksters, much to their dismay, that he has been cheating on his girlfriend with his best friend’s girlfriend.)

An interesting question arises from this video. If you win the lottery, is it a good idea to quit your job? In this post I’ll construct a simple economic model that will tell us just how much a winning lottery ticket is actually worth.

## A Lump Sum Payment as a Constant Income Stream

If you win the lottery and quit your job with the intention of never working again, what you’re essentially doing is exchanging what would be an income stream for a lump sum payment.

Let \(P\) represent your lottery winnings and \(T\) the amount of time remaining in your lifetime. Then, your lottery winnings are equivalent to an average nominal annual income of

\[

w = \frac{P}{T}.

\]

What about real income? Suppose that there’s a constant annual inflation rate of \(r\). If \(t = 0\) is the time at which you won the lottery ticket, then the price level at time \(t\) is given by

\[

p(t) = p(0) (1 + r)^t.

\]

Therefore, at current prices (i.e. assuming that \(p(0) = 1\)) your real income as a function of time is given by

\[

\begin{eqnarray*}

w_r(t) & = & \frac{w}{p(t)} \\

& = & \frac{P}{T(1 + r)^t}

\end{eqnarray*}

\]

and the real value of the lottery winnings, adjusting for inflation, is

\[

\begin{eqnarray*}

P_r & = & \int_0^T w_r(t) \, dt \\

& = & \frac{P}{T} \int_0^T (1 + r)^{-t} \, dt \\

& = & \frac{P[1 – (1 + r)^{-T}]}{T\log(1 + r)}.

\end{eqnarray*}

\]

If you convert this to an average real annual income, you get

\[

\overline{w_r} = \frac{P[1 – (1 + r)^{-T}]}{T^2\log(1 + r)}.

\]

## Crunching the Numbers

*(Note: All calculations below use standard significant figure rules when applicable.)*

Let’s try plugging some numbers into that equation. We already know that \(P = \$2000000\). Now I’m going to use some data to estimate values for the other parameters.

First, \(T\). I have no idea how old the guy in the video, so I’ll use my own age, 20. According to the Centers for Disease Control, a 20-year-old in the United States is expected to live, on average, another 59.3 years. So, we can use the value \(T = 59.3\text{ years}\).

Next, \(r\). The Federal Reserve provides the following data on CPI for urban consumers. (Unfortunately, the graph doesn’t resize with the window, so it might be cut off.)

The base period is 1982–1984. CPI was 21.480 on January 1, 1947 and 235.169 on February 1, 2014. That’s a 24,503-day period, over which CPI increased by a factor of \(235.169/21.480 = 10.948\). This gives us an average annual inflation rate of

\[

\begin{eqnarray*}

1 + r & = & \left( \frac{235.169}{21.480} \right)^{365/24503} \\

& = & 1.0363,

\end{eqnarray*}

\]

or \(r = 3.63\%\).

Plugging these numbers in gives you

\[

\begin{eqnarray*}

\overline{w_r} & = & \frac{(\$2 \times 10^6)(1 – 1.0363^{-59.3})}{(59.3 \text{ years})^2\log(1.0363)} \\

& = & \$14300/\text{year}.

\end{eqnarray*}

\]

According to the Department of Health and Human Services, the official 2014 poverty line for a single person in the 48 states or the District of Columbia (such as myself) is an annual income of $11,670—just a little bit less than the lottery winnings. If I lived in Alaska, I’d actually be below the poverty line ($14,580 per year). In conclusion, while $2 million sounds like a lot of money, it’s really not a whole lot of money to live your entire life on. So, regardless of whether the guy in the video was smart enough to figure out that the lottery ticket was fake, quitting his job was really not a good idea.