# Frisch elasticity of labor supply

The Frisch elasticity of labor supply captures the elasticity of hours worked to the wage rate, given a constant marginal utility of wealth. Marginal utility is constant for risk-neutral individuals according to microeconomics. In other words, the Frisch elasticity measures the substitution effect of a change in the wage rate on labor supply. This concept was proposed by the economist Ragnar Frisch after whom the elasticity of labor supply is named.
The value of the Frisch elasticity is interpreted as willingness to work when wage is changed. The higher the Frisch elasticity, the more willing are people to work if the wage increases.
The Frisch elasticity can be also referred to as “λ-constant elasticity”, where λ denotes marginal utility of wealth, or also in some macro literature it is referred to as “macro elasticity” as macroeconomic models are set in terms of the Frisch elasticity, while the term “micro elasticity” is used to refer to the intensive margin elasticity of hours conditional on employment.
The Frisch elasticity of labor supply is important for economic analysis and for understanding business cycle fluctuations. It also controls intertemporal substitution responses to fluctuations of wage. Moreover, it determines the reaction of effects to fiscal policy interventions, taxation or money transfers.

## Definition

Let $h_{t}$ denote a worker's labor supply in a given time $t$ and $w_{t}$ their wage. The Frisch elasticity (FE) is given by $FE={\frac {\partial h_{t}}{\partial w_{t}}}{\frac {w_{t}}{h_{t}}}={\frac {\partial \log h_{t}}{\partial \log w_{t}}}{\Biggl |}_{\lambda _{t}=const}$.
The overall effect of the Frisch elasticity can be distinguished into extensive and intensive. The extensive effect can be explained as a decision whether to work at all. The intensive effect refers to a decision of an employee on the number of hours to work.
Under certain circumstances, a constant marginal utility of wealth implies a constant marginal utility of consumption. Moreover, the Frisch elasticity corresponds to the elasticity of substitution of labor supply.

## Budget constraint

The money value of costs on goods (C) must equal the total of wage (wh) and nonlabor income (V). The rate of wage is essential when it comes to choosing labor supply. Now lets think that the wage rate is constant for a person, who is unable to change his hourly wage according to his time spent at work. Additionally we will define the “marginal” wage as money earned for the last hour worked. This, of course, depends on the number of hours which are spent working. Someone who works more than 40 hours per week usually gets more money as an overtime premium. Also the wage of part-time jobs tends to be inferior to the wage of full-time jobs. Now, let's also not include the possibility that someone's marginal wage is related to the number of hours spent working. With the condition of a constant wage rate, we can put the budget constraint into a graph. Work or leisure are the only options someone has when it comes to choosing the way of spending his/her time. Time given to either work or leisure will then be similar to the time in the overall period. We will denote it as $T$ hours in a week, so that $T=h+L$
The equation of a budget constraint can also be written as
$C=w(T-L)+V$
which is the equation of a line whose slope is the negative wage rate (-w). Even if the person spends the whole time (T) working or at leisure, it is still available for him to buy consumption goods at the price of V. Giving up one hour of leisure would result in moving up the budget line and thus being able to buy additional w dollars of goods. Of course, this effect is relevant every time the person is willing to exchange an hour of leisure for an hour of work, resulting in the ability to buy additional w dollars of goods. Meaning that, every hour of leisure has some cost and the cost is dependent on the wage rate. By giving up all the free time activities, the person gets to the interface of the budget line and can buy (wT + V) worth of goods. Additionally, the worker has access to all the combinations on the budget line and thus creating worker's opportunity set ( set of all the baskets of consumption that the worker is able to purchase.)

## Difference with the general concept of elasticity of labor supply

Elasticity of labor supply refers to the responsiveness of labor supply to changes in the wage rate. It is typically measured as the percentage change in the quantity of labor supplied divided by the percentage change in the wage rate. The elasticity of labor supply can be influenced by various factors, including the availability of alternative sources of income, the extent of non-labor income, the extent to which individuals can adjust their hours of work, and other factors.
The Frisch elasticity of labor supply is a specific type of elasticity of labor supply that considers the intertemporal substitution of work effort. It measures the responsiveness of labor supply to changes in the real wage, which is the wage adjusted for changes in the cost of living. In contrast to the general concept of elasticity of labor supply, the Frisch elasticity also takes into account the effects of changes in income on the amount of work that people are willing to supply.

## Applications

The Frisch elasticity of labor supply is not only important for economic analysis but also has implications for policy making. Governments can use the Frisch elasticity to determine the effectiveness of policies aimed at increasing employment and reducing unemployment. For example, a policy that increases wages in a certain sector can increase labor supply, but the extent of the increase will depend on the Frisch elasticity. Similarly, policies aimed at reducing taxes or increasing welfare benefits can also have an impact on the Frisch elasticity of labor supply.
Moreover, the Frisch elasticity can help policymakers understand the impact of technological change on the labor market. Technological change can increase the productivity of labor, which can lead to an increase in wages. However, it can also lead to a reduction in the demand for labor in certain sectors, which can lead to unemployment. The Frisch elasticity can help policymakers understand the extent to which workers will respond to changes in wages and employment opportunities.

### The hidden unemployed

Calculations done by the BLS have shown that unemployment is in measurements often different depending on the current definition of what it means to be unemployed. Being on temporary layoff is one of the conditions for someone to be considered unemployed. The second option for a person to be considered unemployed is to state that he or she has been looking actively for a job in the past four weeks.
“Out of labor force” is someone who does not fulfill previously mentioned criteria and is therefore not considered to be unemployed. Some people are presenting themselves as actively looking for a job even though they have in reality no willingness to work at all, to obtain the benefits of being unemployed. This may lead to unemployment statistics being different from each other.
During the 2009 recession, for example, it is often claimed that the statistical unemployment rate (BLS statistic) underestimates the reality of severe recession and tough economic obstacles. The difficulties that had to be faced by unemployed people when they were trying to find a job were simply for someone to great to overcome. This resulted in many people dropping off their willingness to find a job and thus leaving the labour market along with losing the status of unemployed. Some may insist that these people considered hidden unemployed should be included in the overall statistics of unemployed people in order to show that the problem of unemployment is much worse than the BLS data indicated.
Another way of measuring the aggregate economic activity is the employment rate. This function shows us the current part of the population with a job. However, it combines people who claim to be unemployed with those who are identified as being out of the labor force. Even though the second group has some hidden unemployed insight, it also consists of people with rather little tendency to work, such as retirees, women with small children and students enrolled in school. Reduction in the employment could be caused by higher unemployment or by unassociated extension in fertility or school enrollment rates. We can assume that for the purpose of measuring the fluctuations in economic activity, it is in reality better to use employment rate rather than the unemployment rate.

## Value

Values of the Frisch elasticity vary depending on the population being analyzed. However, it is generally agreed that the Frisch elasticity is positive, meaning that an increase in wages leads to an increase in labor supply. In addition to the positive relationship between wages and labor supply, it is important to note that the magnitude of the Frisch elasticity can provide insights into the behavior of workers. A Frisch elasticity of 0 indicates that workers do not respond to changes in wages, while a Frisch elasticity of 1 indicates that workers are highly responsive to changes in wages. The magnitude of the Frisch elasticity is typically between 0 and 1, indicating that the increase in labor supply is less than proportional to the increase in wages. For example, if the Frisch elasticity is 0.5, a 10% increase in wages would lead to a 5% increase in labor supply. In other words, workers would increase their hours worked by 5% in response to a 10% increase in wages.

### Measurement challenges

There are several challenges to measuring the Frisch elasticity of labor supply accurately. For example, it is difficult to control for other factors that may influence labor supply, such as changes in the cost of living or changes in social norms around work. Additionally, there may be differences in the way that men and women respond to changes in wages, which can make it challenging to compare elasticities across gender.

### Variations in Frisch elasticity among workers

It is worth noting that the Frisch elasticity is not constant across all individuals. Different groups of workers may have different Frisch elasticities due to differences in preferences, job opportunities, and other factors. For example, workers with higher levels of education and training may have higher Frisch elasticities than workers with lower levels of education and training because they may have more flexibility in their job options and may be able to switch between different types of jobs more easily. Similarly, workers in certain industries or occupations may have higher Frisch elasticities than workers in other industries or occupations. For instance, workers in industries that experience rapid technological change may have higher Frisch elasticities because they are more likely to be affected by fluctuations in wages due to changes in technology. Moreover, other factors such as income level, gender, and age can also affect the Frisch elasticity of labor supply. For instance, low-income workers may have lower Frisch elasticities because they may have fewer job opportunities or may face greater financial constraints that make it harder for them to adjust their labor supply in response to wage changes. Women may also have lower Frisch elasticities than men due to differences in labor market opportunities and social norms surrounding work and family. Finally, older workers may have lower Frisch elasticities than younger workers because they may have stronger preferences for leisure time or may be less willing or able to retrain for new jobs.

### Relationship with income

The Frisch elasticity of labor supply is often higher for low-income workers than for high-income workers. This is because low-income workers are more likely to have to work to make ends meet, and therefore may be more responsive to changes in wages.

### International differences

International differences in the Frisch elasticity of labor supply can provide insights into the impact of social welfare policies on the labor market. Studies have shown that countries with lower levels of social welfare provision tend to have higher Frisch elasticities. This is because workers in countries with more limited social welfare benefits may have fewer alternatives to working and may be more willing to supply labor even when wages are low. In contrast, workers in countries with more comprehensive social welfare systems may have more options and be less likely to work in low-wage jobs. Moreover, international differences in the Frisch elasticity of labor supply can reflect broader differences in labor market institutions and policies. Countries with more flexible labor markets and weaker labor protections may have higher Frisch elasticities, as workers may have fewer protections against job loss and may be more willing to work even when wages are low. In contrast, countries with stronger labor protections may have lower Frisch elasticities, as workers may be less willing to accept low-wage jobs or may have greater bargaining power to negotiate for higher wages.