# 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 methosd used to estimate $ \epsilon $. Six years later, Evans took a stab at the same subject Evans. Though most of them are (in my opinion) ludicrously terrible estimation techniques, here are the methods they considered:

- Gevers et al; Glejser et al; Amiel et al - TODO
- We can assume utility is normally distributed and a function of only income and then solve for $\epsilon$ mathematically.
- 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.
- We can ask people in surveys with hypothetical questions.
- Binswanger - TODO
- We can examine lifetime consumption patterns, which (if people are rational) should be largely determined by $\epsilon$.
- 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.

There's also the popular method of plotting GDP on the x-axis and self-reported happiness or well-being on the y-axis.

## 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 |

## 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 2.22-fold Historical National Population Estimates Population Clock, which means we really only saw a 16% increase in labor-per-capita despite a 243% increase in wages.

However, there have been a number of changes that bias this estimate:

- Women entered the workforce. Their abor force participation increased from 32.7% to 57.1% between 1948 and 2018 Civilian Labor Force Participation Rate: Women.
- The percent of the population of working age has increased. I don't have data for 1948 and 2018, but between 1955 and 2014, the proportion increased by 7.5% Working age population.
- 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.

Changes #1 and #2 bias our estimated change in $L$ as a result of changes in $w$ (16%) upwards, while change #3 biases it downwards. These changes (in addition to other cultural shifts) add significant uncertainty to our estimate. Altogether, I expect the effect of increasing wages on labor *has* been slightly negative.

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 3.75 times more disutility compared to the change from 7 to 8 hours. This seem ludicrously 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

While the FFF model suggests $\epsilon \approx 1.6$, my observations of the historical changes in labor demand as wages have increased suggest $ \epsilon $ is probably closer to 1 than 2. For this reason, I'm going to adjust the meidan FFF estimates downward to settle on $ \epsilon \approx 1.5 $.

## 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 6.5%.

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.