# Utility Elasticity

## Framework

It is typically assumed that utility ($u$) as a function of income ($x$) takes this form: $$ u = \left\{ \begin{array}{ll} \ln{(x)} & \epsilon = 1 \\ \frac{x^{1-\epsilon} - 1}{1 - \epsilon} & \epsilon \gt 0 \end{array} \right. $$ where $ \epsilon $ is non-negative and represents how quickly additional income becomes useless.

Mathy people will note that the $ \epsilon = 1 $ case is the limit of the $ \epsilon \gt 0 $ case.

This formula doesn't have strong justification, but there is a discussion of it here Welfare weights.

Our task is to estimate $\epsilon$.

## Traditional Estimation Methods

Cowell and Gardiner Welfare weights survey many methods used to estimate $ \epsilon $. Six years later, Evans took a stab at the same subject Evans. Finally, 13 years after that Groom and Maddison made a third attempt Groom. Though most of them are (in my opinion) ludicrously terrible estimation techniques, here are the methods they considered:

- We can assume utility is normally distributed and a function of only income and then solve for $\epsilon$ mathematically. There is no reason to assume utility is normally distributed, and there is plenty of reason to think other things affect it besides income.
- We can assume governments tax everyone such that everyone's utility is decreased equally. Then, we can compute the $\epsilon$ that would imply this is true. There is plenty of reason to doubt that governments either want equal utility cost from their taxes or have the ability to deliver on that goal.
- We can ask people in surveys with hypothetical questions concerning risk and inequality aversion and see what $\epsilon$ there answers are consistent with. There is good evidence that people are horrible at dealing with risk (or numbers more generally). We also have reason to doubt what people say due to the pro-social bias against inequality.
- Evans Evans cites the Binswanger section from Newbery. Unfortunately, I can't find the book.
- We can examine lifetime consumption patterns, which (if people are rational) should be largely determined by $\epsilon$. This assumes people don't hyperbolically discount their future selves, which we know to be false.
- We can use the Fisher-Frisch-Fellner (FFF) model. This is the sole method I think is reasonable, so we'll discuss it more below.

## Subjective Well-Being

We can ask people to rate their well-being on a scale (e.g. from 1-10), assume this is linearly related to utility, and and then find the relationship between consumption and their well-being. This assumes that people's ratings aren't significantly biased by cultural differences.

An analysis of five large surveys controled for a variety of variables before estimating $\epsilon \approx 1.24$ with standard error 0.05 Layard.

## FFF Estimates

The FFF procedure assumes that utility functions are *additively separatable* between a particular good/service and all other goods and services. This means we can represent utility as
$$ u = f(c_0) + g(c_1)$$
where $f$ and $g$ are two utility functions, $c_0$ is how much of some particular good/service is consumed, and $c_1$ is how much of all other goods/services are consumed.

This assumption isn't widely accepted for any good/service. To quote Evans Evans:

Results for $e$ based on the FFF model have generally been disregarded in the literature, mainly on the grounds of the strong condition of additive separability (preference independence) that is required for the approach to be valid.

...

However, in the case of food (and some other broadly defined product groups), Fellner (1967), Selvanathan and Selvanathan (1993) and Evans and Sezer (2002) have all argued that preference independence is a plausible assumption. Furthermore, Selvanathan (1988) tested the want-independence assumption for broad aggregates, using OECD data, and found it to be empirically valid. [However,] both Cowell and Gardiner (1999) and Pearce and Ulph (1999) largely disregard empirical estimates of $e$ based on the model.

In any case, *if* we accept this is true for some particular good (e.g. food), then we can examine how the demand for food changes when incomes change and when prices change and deduce (with some calculus) what the implied value of $ \epsilon $ is.

Evans then goes on to list some empirical estimates:

Study | Estimate of $\epsilon$ | Nation |

Kula (2002) | 1.64 | India |

Evans and Sezer (2002) | 1.6 | UK |

Evans (2004a) | 1.25 or 1.6 | UK |

Evans (2004a) | 1.3 or 1.8 | France |

Groom, et al also attempts to estimate $\epsilon$ with this method and concludes it is 3.6 with standard error 2.2 Groom. Given that enormity of the standard error, I'm inclined to ignore this estimate.

## Market Returns

I'm generally skeptical of people's ability to weigh risk and care about long-run optimality. However, there is one group of people who are probably decent at it: firms that actively invest billions of dollars for their clients.

For that reason, I think it might be enlightening to look at historical investment returns and see which $\epsilon$ is consistent with both low-risk and high-risk investments looking equally appealing.

I chose the SP500 and 10-year treasury bonds, both for their popularity and because the data is readily available back to 1871 Shiller.

My methodology was simple:

- Invest \$1 in Jan-1871 in the in treasury bonds and in the SP500 (reinvesting dividends). Check both hypothetical investments after 10 years. Write down the new value.
- Repeat for all months in the dataset.
- Treat these values as representing the PDF of treasury bond returns and SP500 returns.

Just to react to some objections:

- This entire exercise is scale-invariant. I could have invested \$1 million for each datapoint and I'd achieve the same $\epsilon$.
- I've previously looked at the SP500 portion of this dataset and concluded that average returns and variance in returns hasn't changed in a statistically significant over the ~150 years represented.

The result was (to my happy surprise) an estimate of $\epsilon \approx 1.49$. Pretty close to the estimates from the FFF studies discussed previously.

## Historical Labor

Suppose, for sake of argument that $\epsilon = 1$. This implies utility is given by $$ u = \ln{(x)} $$

Now suppose labor is additively separatable from consumption and people consume all their income. Now, utility is given by $$ u = \ln{(w \cdot L)} - f(L) $$ where $w$ is the wage and $f$ is some function representing disutility from working.

If we assume people choose $L$ rationally, we can show (with a little calculus) that $$ 1 = L \cdot f'(L) $$

This implies that $L$ is independent of $w$, which means that we should expect people to work the same number of hours regardless of $w$.

Fortunately, we have a natural experiment for this. Over the last couple centuries, the average wage has increased dramatically. $\epsilon = 1$ predicts that labor hasn't changed.

Between 1948 and 2018, real GDP per hour worked increased 3.43-fold while total hours worked increased 2.57-fold Table 1.1.3 Table 1.1.5 Table 6.9B Table 6.9D.

However, during that same time, the US population increased 2.39-fold Working age population, which means we really only saw a 8% increase in labor-per-working-age-person despite a 243% increase in wages.

There have, however, been a number of changes that bias this estimate:

- Women entered the workforce. Their labor force participation increased from 32.7% to 57.1% between 1948 and 2018 Civilian Labor Force Participation Rate: Women. If we naively assume men stayed at their 1948 participation rate of 86.7%, then this represents a 22% increase in laborers per capita. In reality, participation only increased 7% because a significant number of men dropped out Labor Force Participation Rate - Men.
- Transfers and benefits ("welfare") increased between 1948 and 2018 due to a variety of factors such as LBJ's The Great Society, Obamacare, the general rise of health care costs, and the increasing of life expectancy relative to a static retirement age.
- The culture changed in various other ways.

Just to reiterate, despite a 243% increase in wages, hours worked only increased a mere 8% per working-age person. That's remarkably close to the 0% predicted by the $\epsilon = 1$ theory, and, given the uncertainty inherent in #1-3, I think it very fair to conclude the evidence is consistent with the theory.

Now suppose $\epsilon$ was large, say $\epsilon = 2$ (rather than 1), we can model utility as $$ u = -\left(w \cdot L \right)^{-1} - L^\beta $$ where $\beta$ is some constant.

Again, using calculus and assuming rationality, we can prove $$ 1 = w \cdot L^{b+1} $$

Now, suppose this 3.43-fold increase in wages caused $L$ to decrease by 10%. This implies $b = 11.7$, which means shifting the working day from 8 to 9 hours would cause 3.8 times more disutility compared to the change from 7 to 8 hours. This seem ludicrously high to me.

Suppose, instead, $\epsilon = 1.5$ and the same 3.43-fold increase in wages also causes a 10% decrease in $L$. This implies $b = ?$. In this case, $b = 6.0$, which means shifting the working day from 8 to 9 hours would cause 1.8 times more disutility compared to the change from 7 to 8 hours. This seems more plausible, but still kind of high to me.

This entire section on estimating $\epsilon$ from historical labor trends is rather speculative. However, it suggests to me that $\epsilon$ is probably closer to 1 than 2.## Overall Estimate

We've estimated $\epsilon$ four different ways:

Method | $\epsilon$ Estimate |

Subjective Well-being | 1.15 - 1.33 |

FFF Procedure | 1.25 - 1.8 |

Investment Returns | 1.5ish |

Historical Labor | Close-ish to 1 |

Altogether, it looks very unlikely that $ \epsilon $ is less than 1 or greater than 2. I'd say the balance of the evidence suggests $\epsilon \approx 1.35$.

Britain and France estimate $\epsilon$ as 1 and 2, respectively. The Groom and Maddison meta-analysis estimates it is 1.5 Groom.

## Discount Rates

Using some basic calculus, expected value theory, and an assumption of little change in income inequality, we can prove the discount rate should be $$ p + i + \epsilon \cdot r $$ where $p$ is the probability of social collapse, $i$ is inflation, and $r$ is the expected percent growth in incomes. We're going to ignore $p$ and $i$ and focus on $\epsilon$ and $r$ because

- $p$ is extremely difficult to nail down and also fairly irrelevent for discounting purposes. Even if the probability of humanity going extinct this century is 19% Todd, B., $p$ would be a mere 0.21%.
- $i$ typically doesn't matter unless you own bonds. If inflation is 5% higher, interest rates tend to be 5% higher and the stock market tends to grow 5% faster. In any case, you can compute expected inflation using TIPS spreads 10-Year Breakeven Inflation Rate.

Real GDP per capita has averaged 4.3% growth over the last 28 years GDP per capita, PPP (current international $), which suggests a global discount rate of around 5.8%. In the US, real GDP per capita growth has averaged 1.6% over the same period, suggesting a discount rate of around 2.2%.

Are things different among the global poor?

The incomes of people between the 5th percentile and 50th percentile increased by about 6.3% per year between 2003 and 2013. However, real global GDP increased 6.4% over the same time period, indicating that the discount rate is roughly the same among the global poor.

*Economics Research Paper, 20*, 1999.

*Fiscal studies, 26*(2), 197-224. https://doi.org/10.1111/j.1475-5890.2005.00010.x.

*The theory of commodity price stabilization*, Chapter 7, pp. 97-551.