Unemployment

NOTE: The present notebook is coded in R. It relies heavily on the tidyverse ecosystem of packages. We load the tidyverse below as a prerequisite for the rest of the notebook - along with a few other libraries.

\(\rightarrow\) Don’t forget that code flows sequentially. A random chunk may not work if the previous have have not been executed.

library(tidyverse)   # Package for data wrangling
library(readxl)      # Package to import MS Excel files
library(latex2exp)   # Package for LaTeX expressions
library(quantmod)    # Package for stock data extraction
library(highcharter) # Package for reactive plots
library(WDI)         # Package for World Bank data
library(ggsci)       # Package for cool color palettes

The content of the notebook is heavily inspired from the book Advanced Macro-economics - An Easy Guide.

Context

Motivation

Unemployment is going to be a major concern in the years to come. In particular because of the effect of AI on the labor market, see for instance:

• Canaries in the Coal Mine? Six Facts about the Recent Employment Effects of Artificial Intelligence
  
• Generative AI as Seniority-Biased Technological Change: Evidence from U.S. Résumé and Job Posting Data

From the first paper:

And from the second one (Time Series of Junior and Senior Employment in Sample Firms, normalized in 2015):

Trends & salient facts

We first download the data from the World Bank.

wb_raw = WDI(indicator = c("unemp_advanced" = "SL.UEM.ADVN.ZS",     # Advanced education (% LF with Adv Ed.)
                           "unemp_female" = "SL.UEM.TOTL.FE.ZS",    # % of female LF
                           "unemp_male" = "SL.UEM.TOTL.MA.ZS",      # % of male LF
                           "unemp" = "SL.UEM.TOTL.ZS",
                           "pop" = "SP.POP.TOTL"
                           ), 
             #country = c('FR','US'), # here you can specify some countries
             extra = TRUE,
             start = 1960, 
             end = 2025) |>
  mutate(across(everything(), as.vector)) |>
  select(-status, -lending, -iso2c, -iso3c)

We then start with a brief overview over the past decades (the data starts in 1990).
We easily identify major economic/financial shocks/events.

wb_raw |> 
  filter(country == "World", year > 1995) |> 
  ggplot(aes(x = year, y = unemp)) + geom_line() +
  theme_classic() + 
  theme(legend.position = c(0.2, 0.3),
        title = element_text(face = "bold"),
        legend.title = element_blank(),
        axis.title = element_blank()) +
  annotate("text", x = 2020, y = 6.8, label = "Covid", color = "blue") +
  annotate("text", x = 2009, y = 6.7, label = "Great\nDepression", color = "blue") +
  annotate("text", x = 2002, y = 6.7, label = "Tech\nBubble", color = "blue") +
  ggtitle("Unemployment rate (%)")

Second, let’s take a look at gender parity.

wb_raw |> 
  filter(country == "World", year > 1995) |> 
  pivot_longer(unemp_female:unemp_male, names_to = "gender", values_to = "value") |>
  ggplot(aes(x = year, y = value, color = gender)) + geom_line() +
  theme_classic() + 
  theme(legend.position = c(0.2, 0.3),
        title = element_text(face = "bold"),
        legend.title = element_blank(),
        axis.title = element_blank()) +
  scale_color_d3(labels = c("female", "male")) +
  ggtitle("Unemployment rate (%)")

Covid had a strong effect on the employment bias. But since then, it’s back to business-as-usual. Though, worldwide, female unemployment is at an all-time low, too.

A snapshot at the educated workforce (in the US this time).

wb_raw |> 
  filter(country == "United States", year > 1995) |>
  pivot_longer(c(unemp_advanced,unemp), names_to = "type", values_to = "value") |>
  ggplot(aes(x = year, y = value, color = type)) + geom_line() +
  theme_classic() +
  theme(axis.title = element_blank(),
        legend.title = element_blank(),
        legend.position = c(0.82,0.89)) +
  scale_color_aaas(labels = c("total population", "advanced educ.")) + 
  ggtitle("Unemployment rate (%)")

Education does seem worthwhile on the job market!

Finally, what about the impact of income? (mind the population-weighted average in the code)

wb_raw |>
  filter(year > 1980, is.finite(unemp), !(income %in% c("Aggregates", "Not classified"))) |> 
  group_by(year, income) |>
  mutate(weight = pop / sum(pop)) |>
  summarise(unemp = sum(unemp * weight)) |> na.omit() |>
  ggplot(aes(x = year, y = unemp, color = income)) + geom_line() +
  theme_classic() + 
  theme(axis.title = element_blank(), legend.position = "bottom") + 
  guides(color=guide_legend(nrow=2,byrow=TRUE))

Situations have changed a lot. The rate is much more stable for lower income economies.

A model of job search with reserve wage

A standard (the only!) reference for this section is Search-Theoretic Models of the Labor Market: A Survey.

Discrete time

An agent seeks a job that will pay a wage \(w\). Now, \(w\) is considered to be a positive random variable with cumulative distribution function \(F\). The agent seeks to maximize the average discounted cash flows she will receive: \[\mathbb{E}\left[\sum_{t=0}^\infty \beta^t x_t \right],\] where \[x_t = \left\{ \begin{array}{l l} w & \text{if she takes the job} \\ b & \text{if she remains unemployed} \end{array} \right.\] where \(b\) is some constant that comes from country specific welfare programs. \(\beta=(1+\rho)^{-1}\in(0,1)\) is the discounting factor.

The value of taking the job, if we assume that the agent takes the job forever (a big assumption) is \[W(w) = w + \beta W(w),\] where the first term comes the first wage and the second one comes from the fact that in period two, the value will remain the same, but discounted at rate \(\beta\). When \(w\) is fixed, the lifetime value yield exactly this result: \[W = w \sum_{t=1}^\infty \beta^t =\frac{w}{1-\beta}.\]

The value of not taking the job is a bit more complex to evaluate. For the first period, the gain is \(b\), but for the second period, the agent will have to choose between refusing the job again (remaining unemployed), or accepting it, hence: \[U = b + \beta \mathbb{E}[\max(U, W(w))]. \tag{1}\]

Now, because \(W\) is linearly increasing with \(w\), there is a wage \(w_R\), called the reservation wage that satisfies \[W(w_R)=U;\]

implied that the searcher will accept the job offer if \(w>w_R\) (the wage is attractive) or decline it otherwise (it’s more convenient not to work). In particular, this implies that \[U=\frac{w_R}{1-\beta}.\]

Plugging this in Equation 1, we infer \[\frac{w_R}{1-\beta} = b + \frac{\beta}{1-\beta}\mathbb{E}[\max(w,w_R)]\]

If we subtract \(\frac{w_R\beta}{1-\beta}\) from each side, we get:

\[\begin{align} w_R & = b + \frac{\beta(1-w_R)}{1-\beta} \int_0^\infty\max(w,w_R)dF(w) \\ & = b + \frac{\beta}{1-\beta}\left( \int_0^{w_R} w_R dF(w) + \int_{w_R}^\infty w dF(w) - w_R \right) \\ & = b + \frac{\beta}{1-\beta}\left( \int_{w_R}^\infty(w-w_R) dF(w) \right) \end{align}\]

Expression for the reservation wage

Finally, using integration by part, we reach the following expression \[w_R=b+\frac{\beta}{1-\beta}\int_{w_R}^\infty(1-F(w))dw.\]

An interesting feature of the above expression is that \(w_R\) increases with \(\beta\). A small \(\beta\) (preference for the present, meaning that waiting is painful) implies a smaller reservation wage. The searcher is more impatient in this case.

Towards continuous time

If we generalize to any time step \(\Delta\) with \(\beta = (1+\Delta r)^{-1}\). Then, we get, for the value of accepting: \[W(w) = \Delta w + \beta W(w).\] Next, suppose that the worker gets a job offer with wage \(w\) with probability \(\alpha \Delta\) in each period, then the value for declining the offer is \[U = \Delta b + \underbrace{\frac{\alpha \Delta}{1+r\Delta} \mathbb{E}[\max(U,W(w))]}_{\text{if there is an offer}} + \underbrace{\frac{(1-\alpha \Delta))}{1+r\Delta}U}_{\text{if there is no offer}}.\] Note: \(\alpha\) can be interpreted as a search intensity.
Hence, upon simplification, \[rW(w)=(1+r\Delta)w \tag{2}\] and \[rU = (1+r\Delta)b + \alpha \int_0^\infty \max(0, W(w)-U)dF(w). \tag{3}\] In the limit (\(\Delta \rightarrow 0\)), in continuous time, the objective becomes \(\mathbb{E}\left[\sum_{t=0}^\infty e^{-rt} x_t \right]\) and \(W=w \int_0^\infty e^{-rt}dt=w/r\) (equivalent to Equation 2). Moreover, the above equation reduces to \[rU = b + \alpha \int_0^\infty \max(0, W(w)-U)dF(w).\] Since, again, \(W(w_R)=U\) and \(W(w_R)=w_R/r\), \(rU=w_R\), so \[w_R =b+ \frac{\alpha}{r} \int_{w_R}^\infty(w-w_R)dF(w).\] Again integrating by parts, we end up with \[w_R = b+\frac{\alpha}{r} \int_{w_R}^\infty(1-F(w))dw.\] Here, the search intensity \(\alpha\) is exogenous but it could be optimized by the job seeker, modulo a cost, say \(g(\alpha)\) - for instance to pay an online service to get access to more postings. The objective of the seeker is to maximize the reserve wage (minus the cost \(g(\alpha)\)): \[b+\frac{\alpha}{r} \int_{w_R}^\infty(1-F(w))dw - g(\alpha). \tag{4}\] The FOC associated to this is (differentiating w.r.t. \(\alpha\)): \[\int_{w_R}^\infty(1-F(w))dw=rg'(\alpha). \tag{5}\] The full model requires to solve Equation 4 and Equation 5.

In this model, the duration of unemployment can be characterized. The probability that the searcher has not found a job after time \(t\) is \(e^{-Ht}\) with \(H=\alpha(1-F(w_R))\) being the hazard rate, the product of the contract arrival rate \((\alpha)\) and the probability of acceptance.

Layoff risk

In the previous model, a job was eternal. This is unlikely in practice, especially in the modern labor market, see, e.g., in the US where turnover (employee rotation) is quite high:

• state level separation rates from the BLS
  
• national separation rate from the FRED
  
• JOLTS from the BLS: Job Openings and Labor Turnover Survey.

Each month, ~3-4% of workers lose or change jobs.

This makes a reasonable rotation (except for Covid!).
An important distinction being whether departures are voluntary or not…

Within the previous model, we can add a parameter that will code job termination (for whatever reason): \(\lambda\) is the Poisson intensity that parametrizes the occurrence of the termination of the working contract. Hence, over a period \(\Delta\), the probability of losing/changing jobs is \(\lambda \Delta\).

The equivalent of Equation 2 is now

\[W(w) = \Delta w + \frac{W(w)}{1+r\Delta} (1-\lambda \Delta)+ \frac{U\lambda \Delta}{1+r\Delta}\]

Hence, \[(1+r\Delta)W(w) = \Delta w + W(w)(1-\lambda \Delta)+U\lambda \Delta\] and thus, \[rW(w) = w + \lambda(U-W(w)). \tag{6}\] The likelihood of job loss does not affect Equation 3 (there is no job to lose). This implies: \[W(w) = \frac{w + \lambda U}{r + \lambda}. \tag{7}\]

Plus, it still holds that \(U=W(w_R)=\frac{w_R+\lambda U}{r+\lambda}\), so that, again, \(w_R=rU\). With Equation 3, this yields: \[w_R= b+ \alpha \int_{0}^\infty \max\left( 0,\frac{w+\lambda U}{r+\lambda}-U \right)dF(w), \] which is equivalent to \[w_R = b + \frac{\alpha}{r+\lambda}\int_{w_R}^\infty (w-w_R) dF(w).\] Hence: there is no big change, except that (a higher) \(\lambda\) reduces the reservation wage! Indeed, if there is more layoff uncertainty, it is logical that the job seeker will accept a slightly lower salary.

Job transitions and equilibrium unemployment

We now dig a bit deeper into the details of turnover. We add the new hypothesis that new job offers arrive at rate \(\alpha_0\) when a worker is unemployed and at rate \(\alpha_1\) when she is employed. The wage of the offer is again distributed according to \(F\). Job loss still occurs at rate \(\lambda\). The “value” of unemployment is \[rU = b + \alpha_0 \int_{w_R}^\infty (W(w)-U)dF(w),\] while the value of work reads

\[rW(w) = w + \alpha_1\int_0^\infty \max(0,W(w')-W(w))dF(w') + \lambda (U-W(w)),\] where the integral part pertains to the choice of switching job (& accepting a higher wage \(w'\)) and the last part relates to job loss.

Side note

Differentiating the above equality w.r.t. \(w\) leads to

\[rW'(w) = 1 - \alpha_1\left(\int_w^\infty W'(w) dF(w') \right)-\lambda W'(w), \]

i.e., \[W'(w)=(z+\lambda+\alpha_1(1-F(w)))^{-1}. \tag{8}\]

It still holds that \(U=W(w_R)\). If we set \(w=w_R\) in the above equation and combine it with the value of unemployment (the two are equal), we reach

\[w_R=b+(\alpha_0-\alpha_1)\int_{w_R}^\infty (W(w')-W(w_r))dF(w').\] The reservation wage is attractive (compared to \(b\)) only if \(\alpha_O>\alpha_1\): receiving offers more frequently when employed reduces \(w_R\).

Using integration by parts as well as Equation 8, we finally reach \[w_R = b + (\alpha_0-\alpha_1)\int_{w_R}^\infty \left(\frac{1-F(w)}{r+\lambda+\alpha_1(1-F(w))} \right)dw\] With a few final assumptions, we can derive an equilibrium unemployment rate. Suppose unemployed workers find and accept a job at rate \(\alpha_0 (1-F(w_R))=\alpha_0 \mathbb{P}[w>w_R]\) and employed workers lose theirs at rate \(\lambda\), then the dynamics of unemployment are given by \[\dot{u}=\lambda (1-u) - \alpha_0(1-F(w_R))u\]

Long-term unemployment rate

Hence, \(u\) is stable if \(\dot{u}=0\), i.e., if \[u^*=\frac{\lambda}{\lambda + \alpha_0(1-F(w_R))}.\]

Long-term unemployment decreases with \(\alpha_0\), which makes perfectly sense, and increases with \(\lambda\) (likelihood of losing job), which is also entirely logical.

Why not try a calibration exercise? Remember:

• \(\lambda\) is job loss frequency;
  
• \(\alpha_0\) is job arrival rate for the unemployed;
  
• \(1-F(w_R)\) is the probability of acceptance of the reservation wage.

Matching

This section lays out a line of thinking in the spirit of the contributions that obtained the 2010 Nobel prize: the Diamond-Mortensen-Pissarides model.

Notations

• On the supply side of labor, let \(E\) and \(U\) denote the number of employed and unemployed workers so that the total labor force if \(L=E+U\).
  
• On the demand side, the number of filled jobs is \(F\) and the number of vacant ones is \(V\).

Hence, the economy needs to match \(U\) with \(V\). The number of matches is modeled à la Cobb-Douglas,

\[M(U,V)=KU^\beta V^\gamma, \quad (\beta, \gamma) \in (0,1)^2. \tag{9}\]

We assume that when there is a match, a contract is signed, and both \(U\) and \(V\) are reduced by one unit.
The variation in employed workers is then given by

\[\dot{E} = M(U,V)-bE,\] where \(b\) is an exogenous rate of job destruction. Moreover, we write

\[a_U = \frac{M(U,V)}{U}, \quad a_V = \frac{M(U,V)}{V}\]

for the rates at which unemployed workers find a job and at which vacancies are filled.

Optimizations

Now let’s dive into the details.

Setup

• The firm proposes a wage \(w\) to the worker who derives a lifetime utility of \(W(w)\) from this wage. \(W\) stands for the value of work.
  
• On the other end of the contract, the firm will be able to produce an output of value \(y\) from the worker and thus derive a profit of \(\pi=y-w\) and we write \(J=\mathbb{E}[\pi]\) for the average of this (expected) profit. \(J\) is the value of the work for the firm.

If the contract is not signed, then the worker’s payoff (or utility) decreases to that of \(U\) (as before), and for the firm, it decreases to \(V\), the value of the vacancy. Hence “things will happen” only if \(W>U\) and \(J>V\) and this is entirely determined through \(w\): the worker will of course accept an extra large salary and, on the other hand, the firm would naturally welcome worker who work for free. And this is when bargaining comes into play.

To blend the two utilies, we consider their product and try to solve the following program:

\[w^* = \underset{w}{\arg \max} \ (W(w)-U)^\theta (J(y-w)-V)^{1-\theta},\] where \(\theta \in (0,1)\) tunes the relative importance towards one or the other participant. It can be viewed as the bargaining power of the worker.

The first order condition

Differentiating w.r.t. \(w\) to get the FOC, we reach: \[\begin{align} & \theta W'(w) (W(w)-U)^{\theta-1}(J(y-w)-V)^{1-\theta} \\ =& (1-\theta)J'(y-w)(W(w)-U)^\theta(J(y-w)-V)^{-\theta}, \end{align}\]

which simplifies to \[\theta W'(w) (J(y-w)-V)= (1-\theta)J'(y-w)(W(w)-U). \tag{10}\]

Solving the bargain

If we further assume that matches end (exogenously) at rate \(\lambda\) as before, then, we get Equation 6. On the firm’s side, the value of the contract satisfies \[J(\pi) = \Delta \pi + \frac{J(\pi)}{1+r\Delta} (1-\lambda \Delta)+ \frac{V\lambda \Delta}{1+r\Delta},\] which translates to \[rJ(\pi)= \pi + \lambda(V-J(\pi)), \] i.e., exactly the same form as Equation 6. If we differentiate either of these expression w.r.t. \(w\), we get: \(r J'(\pi) = -\lambda J'(\pi)\), thus, \(W' = J' = (r + \lambda)^{-1}.\) Substituting into Equation 10, we get: \[\theta (J(y-w)-V) = (1-\theta) (W(w)-U), \] i.e., \[W(w) = U + \theta(J(y-w)+W(w) - U - V) = U+\theta S, \tag{11}\] with \[S = J-V+W-U \tag{12}\] being a surplus. \(S\) can be interpreted as the joint payoff (for both the firm and the worker). Thanks to Equation 6 and its equivalent for \(J\), we have that \(W(w)= \frac{w+\lambda U}{r+\lambda}\) and \(J = \frac{\pi+\lambda V}{r+\lambda}\). Plugging this into the above equation, the surplus equals \[\begin{align*} S &= \frac{\pi+\lambda V}{r+\lambda} + \frac{w+\lambda U}{r+\lambda}-U-V \\ &= \frac{y - r(U+V)}{r+\lambda} \end{align*}\] which is surprisingly independent from the wage!

Again from Equation 7 and recalling \(w_R = rU\), we have that \(W(w)-U= \frac{w+\lambda U}{r+\lambda}-U = \frac{w- w_R}{r+\lambda}\) and similarly \[J-V=\frac{\pi - \pi_R}{r+\lambda},\]

with \(\pi_R\) acting as reservation (or target) profit for the firm. The optimization function then becomes \[\underset{w}{\arg \max} (w-w_R)^\theta(y-w-\pi_R)^{1-\theta}.\]

Solution for \(w\)

In the end, the FOC leads to \[w=w_R+\theta(y-\pi_R-w_R).\]

The worker will thus only accept the contract if \(y>\pi_R+w_R\), that is, if her production potential exceeds the sum of reservation (wage+profit). This last part naturally comes from the objective of the firm that wants to extract sufficient value from the worker (=bargain!).

Combining matching & bargaining

For the firm, the value of posting a vacancy is (obtained similarly as all the above examples, left as exercise here) \[rV = -k + \alpha_V (J-V),\]

where \(k\) is the cost of recruiting and \(\alpha_V\) is the frequency of posting.

We then assume a free-entry condition:

There is no cost for firms to enter the market; vacancies are created until their value decreases to zero. This greatly simplifies the model, as we now have \[\alpha_V J = k.\]

On the other side, the value of unemployment is \[rU = b + \alpha_U(W(w)-U).\]

Combined with Equation 11, this gives \[rU=b+\alpha_U \theta S. \tag{13}\] We recall the expression for the surplus: \(rU=y-(r+\lambda)S\). Equating the two yields \[y-b = (r+\lambda + \alpha_U \theta)S, \tag{14}\]

Output minus the value of unemployment is proportional to the surplus. Now let us determine the wage. Equation 6 implies (subtracting \(rU\) on both sides) \[W-U=\frac{w-rU}{r+\lambda},\] and since bargaining leads to \(W-U=\theta S\) (Equation 11), it holds that \[w-rU=(r+\lambda)\theta S.\]

But we also have from Equation 13, so that \[w=b+\alpha_U\theta S + (r+\lambda)\theta S.\] and \[y-w = y - b - (\alpha_U+r+\lambda)\theta S\]

Moreover, plugging Equation 14 into the above gives \[y-w = (r+\lambda+\alpha_U \theta)S - (\alpha_U+r+\lambda)\theta S.\]

Finally, we arrive at (upon rearranging)

\[w = y-(r+\lambda)(1-\theta)S, \tag{15}\] with: \(\theta\) the bargaining power of the worker, \(y\) her productivity, \(r\) is a discounting rate (for time preference), \(\lambda\) the job loss rate and \(S\) the surplus (Equation 12).

Heterogenous productivity

In the above model, matching is stylized in that we assume that each contact leads to a match and that worker productivity \((y)\) is fixed. But what if we relax this assumption and assume that \(y\) is a random variable, again with distribution \(F\)? (though not the same as the previous one!)

In the end (See Section 4.4 of Rogerson et al.), we obtain:

\[w(y) = b + \frac{\alpha_U \theta k}{\alpha_V (1-\theta)} +\theta \left(y - b - \frac{\alpha_U \theta k}{\alpha_V (1-\theta)} \right)\] The wage is clearly associated with the productivity of the worker. If the worker has more bargaining power (higher \(\theta\)), then, naturally her wage increases with \(y\). Here, the cost of recruiting \(k\) increases the wage.

An attempt at calibration

Can we ground these formulae into real-world data? Let’s consider Equation 15 applied to US.

• \(w\): (target) wage - World Bank or ILO Data: roughly 40$ per hour on average;
  
• \(y\): labor productivity - ILO Data: roughly 80$ per hour;
• \(r\): discount rate - let’s say 3% annually, so 0.0015% hourly (assuming 2,000h worked);
  
• \(\lambda\): job loss frequency - this one, too must be adapted to frequency, if rotation is 4% per month, this makes roughly 0.025% per hour.
• \(\theta\): employee bargaining power - let’s assume that companies have more and set \(\theta = 0.25\);
  
• \(S\): the surplus - this is the big unknown.

In the end, we get:

\[40\$/h = 80\$/h - (0.0015\% + 0.025\% )/h \times 0.75 \times S \$ \]

This implies \(S \approx 200,000\$\), which is very high. We recall that \[S=J-V+W-U,\]

with

• J: average expected profit (value of work for the firm);
  
• V: value of the vacancy (squeezed to zero);
  
• W: value of work for the employee;
  
• U: value of unemployment (idleness);