Overlapping generations models

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(purrr)       # Package for functional programming
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(ggsci)       # Package for color palettes

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

Context

Generations are linked in multiple fashions:

• An obvious example is the environment: as the saying goes, “we do not inherit the earth from our ancestors; we borrow it from our children”.
  
• Another example (especially in French politics) pertains to fiscal policy and the burden of debt (who should carry it).
  
• Finally, a last illustration is the transmission of knowledge. Shifting economic needs and to much reliance on technological change can lead to a loss of skills and know-how.

Models

Chronological setup

Time \(t\) is discrete here (but could be continuous, leading to different developments). Each generation lives through two phases only: work and retirement (children don’t count!).

In the baseline model this won’t have too many implications in fact (at least in terms of differences between generations).

Individuals

The population is such that there are \(L_t=L_0(1+n)^t\) individuals born at time \(t\). Agents born at time \(t\) maximize the following utility function based on consumption (note, this is a two-period setup):

\[ \left(\frac{\sigma}{\sigma-1} \right)\left(c_{t, \text{work}}^{1-1/\sigma} + \frac{1}{1+\rho}c_{t+1, retire}^{1-1/\sigma}\right), \tag{1}\]

where \(\rho\) is the discounting rate (preference over time) and \(\sigma\) controls risk aversion. Equation 1 is optimized under the following constraints: \[w_t = c_{t, \text{work}}+ s_t , \tag{2}\] that is: the wage \(w_t\) at time \(t\) is spent either in consumption or in savings \(s_t\), as well as: \[c_{t+1, \text{retire}}=(1+r_{t+1})s_t, \tag{3}\] that is: upon retirement, the agent can only consume what he has saved (invested in a product that delivers a rate of \(r_{t+1}\), which is the rental rate of capital: what you get from lending money to the firm so that it can invest for production purposes).

Replacing \(c_{t+1,retire}\) and \(s_t\) from the constraints, we get that the first order condition is

\[\frac{\partial}{\partial c_{t,\text{work}}} \left(\left(\frac{\sigma}{\sigma-1} \right)\left(c_{t, \text{work}}^{1-1/\sigma} + \frac{1}{1+\rho}((1+r_{t+1})(w_t-c_{t,\text{work}}))^{1-1/\sigma}\right) \right)=0\] i.e., (again keeping in mind the value of \(c_{t+1, retire}\)): \[c_{t, \text{work}}^{-1/\sigma}-\frac{1+r_{t+1}}{1+\rho}c_{t+1,\text{retire}}^{-1/\sigma}=0.\]

This simplifies to

\[\frac{c_{t+1, \text{retire}}}{c_{t,\text{work}}}=\left(\frac{1+r_{t+1}}{1+\rho} \right)^{\sigma},\] which can be viewed as the Euler equation for the generation “born” at time \(t\).

If we divide Equation 3 by \(c_{t,work}\) in Equation 2, we also get the same ratio, hence

\[\frac{(1+r_{t+1})s_t}{w_t- s_t}=\left(\frac{1+r_{t+1}}{1+\rho} \right)^{\sigma} \ \Leftrightarrow \ s_t=w_t\left(\frac{1}{1+(1+\rho)^{\sigma}(1+r_{t+1})^{1-\sigma}}\right). \tag{4}\] This is the savings function and it is proportional to the income. All quantities in the denominator are positive, hence the ratio \(s_t/w_t\) is smaller than one.

Discussion on \(\rho\) and \(r\)

In most of what we covered so far, the time step is usually the year, for simplicity.
Here, because we are talking about generations, the lag can be substantially larger, for instance 30 years. Hence, the parameter values should reflect this. When we imagined \(\rho=5\%\), over 30 years, this makes \(\rho \approx 330\%\)! Same for returns, if we imagine an annual rate of 4%, then \(r=225\%\)…

It’s important to keep this in mind for numerical applications.

Firms

We assume:

• a simple production function with constant returns to scale, hence the model is mainly defined by the output per worker: \(y=f(k)\), with \(k\) being the capital/labor ratio.
  
• that these firms rent capital until the marginal product of capital is equal to the lending rate of capital. The reason for this is that we can set the (per-worker CRS) profit function to \(\pi(k)=f(k)-r_{t}k\) (production minus cost of capital), hence the FOC is \[f'(k)=r_t. \tag{5}\]
  
• that firms also equate the marginal product of labor to its cost, \(w_t\): \(\pi(L)= Lf(K/L)- w_tL\), hence \(\frac{\partial \{Lf(K/L)- w_tL\}}{\partial L}=f(k) -LK \frac{f'(K/L)}{L^2}-w_t=0\) implies \[f(k_t)-k_tf'(k_t)=w_t. \tag{6}\]
  
• finally, that the capital available at time \(t+1\) is equal to the total savings of the young generation (the old generation is retired so it only consumes). Hence: \[K_{t+1}= L_ts_t \ \Leftrightarrow \ s_t = (1+n)k_{t+1} \tag{7}\]

Important

Because of this last expression, most of the focus in what comes next is on savings. Indeed: \[\text{savings} \rightarrow \text{capital} \rightarrow \text{production (in CRS model)}\]

The focus is to understand the effort (of the young generation) that is required to keep the system going.

A tractable example

The model

As usual, let’s consider a Cobb-Douglas production function: \(y=k^\alpha\) with \(\alpha \in (0,1)\), and log-preferences (\(\sigma=1\)). This yields that the savings are simply equal to \[s_t=(2+\rho)^{-1} w_t.\]

Interestingly, the interest rate has disappeared from the expression (this comes from the myopic log-utility). From Equation 5, we get \[r_t = \alpha k_t^{\alpha-1},\] and from Equation 6, \[w_t = k_t^{\alpha}(1-\alpha) = (1-\alpha)y_t,\]

so that the wage is a fraction of the per-worker production. The link between capital and savings from Equation 7 then allows to write: \[k_{t+1}=\frac{s_t}{1+n}=\frac{w_t}{(1+n)(2+\rho)}=\frac{(1-\alpha)k_t^\alpha}{(1+n)(2+\rho)}. \tag{8}\]

The solution

Hence, the steady-state capital that verifies \(k^*=k_t=k_{t+1}\) is \[k^*=\left( \frac{1-\alpha}{(1+n)(2+\rho)} \right)^{1/(1-\alpha)} \ \Rightarrow y^* = \left( \frac{1-\alpha}{(1+n)(2+\rho)} \right)^{\alpha/(1-\alpha)}.\] Both quantities are decreasing in \(n\) and \(\rho\). If the population grows faster, the optimal ratio \(K/L\) decreases. If the discounting rate increases, the optimal \(k\) also decreases because optimal savings, too, would be lower.

Moreover, we can derive the steady-state interest rate: \[r^* = f'(k^*)= \frac{\alpha}{1-\alpha}(2+\rho)(1+n),\] which increases with all parameters.

With reasonable parameter values, what kind of rate does this imply?

rho <- 2
n <- 0.02
ggplot() + xlim(c(0, 1)) +
  geom_function(fun = ~ (.x)^(1-.x)*(2+rho)*(1+n), 
                size = 1, aes(color = paste("n =", n))) +
  geom_function(fun = ~ (.x)^(1-.x)*(2+rho)*(1-n), 
                size = 1, aes(color = paste0("n = -", n))) +
  theme_classic() + xlab("alpha") + ylab("rate") +
  scale_color_aaas(name = "") + 
  theme(legend.position = c(0.8,0.2))

These values are plausible for \(\alpha > 0.1\) only in a long-term perspective.
Also, the curves are not too sensitive to \(n\) (makes sense, given the formula, but surprising in spirit).

Convergence to the steady state

We have that \[k_{t+1}=\frac{s_t}{1+n}=\frac{s(w_t,r_{t+1})}{1+n} =\frac{s(f(k_t)-k_tf'(k_t), f'(k_{t+1}))}{1+n}\] because the saving rate \(s_t\) depends both on \(w_t\) and on \(r_{t+1}\) and both depend on \(k\) via Equation 6 and Equation 5. Hence, \[(1+n)\frac{\partial k_{t+1}}{\partial k_t}= \frac{\partial s}{\partial w}\frac{\partial w}{\partial k_t}+\frac{\partial s}{\partial r}\frac{\partial r}{\partial k_t}\] Hence, \[(1+n)\frac{\partial k_{t+1}}{\partial k_t}= s_w \times (f'(k_t)-f'(k_t)-k_tf''(k_t)) + s_r \times f''(k_{t+1})\frac{\partial k_{t+1}}{\partial k_t} \] so that, gathering the pieces, \[\frac{\partial k_{t+1}}{\partial k_t} = \frac{-s_w(k_t) k_tf''(k_t)}{1+n-s_r(k_{t+1})f''(k_{t+1})},\] and basically, we want this quantity to be smaller than one in absolute value so that the system does not explode.

Numerical simulations

In all generality, it is impossible to simulate the model because the variation \(\frac{\partial k_{t+1}}{\partial k_t}\) depends on \(k_{t+1}\)! This is because of the savings rate which appears everywhere with its time index at \(t+1\).

However, as we have seen in the previous example with Cobb-Douglas production and logarithmic utility, \(r_{t+1}\) vanishes from the expression of \(k_{t+1}\) in Equation 8.

So let’s give it a try… (with short-term parameter values)

rho <- 0.02 
n <- 0.01
alpha <- 0.5
n_periods <- 20
k_star <- function(alpha, rho, n){
  k <- 1
  for(i in 2:n_periods){
    k[i] <- k[i-1]^alpha * (1-alpha)/(2+rho)/(1+n)
  }
  k
}

ggplot() + 
  geom_line(data = tibble(t = 1:n_periods, k = k_star(0.4, 0.02, 0.01)), 
            aes(x = t, y = k, color = "alpha = 0.4")) + 
  geom_line(data = tibble(t = 1:n_periods, k = k_star(0.6, 0.02, 0.01)), 
            aes(x = t, y = k, color = "alpha = 0.6")) +
  theme_classic() + 
  theme(legend.position = c(0.7, 0.7), 
        legend.title = element_blank())

We do achieve convergence (and quite rapidly).

Overlapping generations and pension systems

Generational solidarity is at the heart of some pension systems, at least this is the case in France. The elderly depend on the younger workers to pay for their income. In some countries (e.g., the US), this is less the case and each individual must save throughout his/her entire active/working life and then rely on savings upon retirement. This is a challenge potentially, as it requires a crucial unknown information: the time of death! Pension schemes can be complex and subject to debate.

The model

Setup

We write:

• \(d_t\) for the contribution of a young worker at time \(t\);
  
• \(b_t\) the the benefits received by an old retiree at time \(t\).

For simplicity, we assume only two periods, as before.

We want to compare two systems:

• pay-as-you go: the young (working) generations are “taxed” to pay for future retirement benefits (ex: France). In this case, \[b_t=(1+n)d_t \tag{9}\] The \((1+n)\) term reflect the potential difference in population sizes. If the population grows (\(n>0\)), then there are more active workers than retirees and the latter can comfortably benefit from a larger benefit. Otherwise (where many developed countries are headed), it’s more complicated!(see below)
• fully funded: the young must save on their own and can consume upon retirement to proceedings of their savings (ex: the United States). Then, straightforwardly, \[b_{t+1}= (1+r_{t+1})d_t. \tag{10}\]

Consumption & savings

As in the simplified model above, the agents’ problem is to maximize \[\log(c_{\text{young}, t})+\frac{\log(c_{\text{old}, t+1})}{1+\rho},\] under the constraints: \[c_{\text{young},t}+s_t+d_t=w_t \tag{11}\] and \[c_{\text{old}, t+1}=(1+r_{t+1})s_t+b_{t+1}. \tag{12}\] The first is the use of income (wage): consumption, savings and contribution. The second shows where the money comes from: either savings, or the pension benefits.

The FOC remains: \[c_{\text{old}, t+1}=\frac{1+r_{t+1}}{1+\rho}c_{\text{young}, t}.\]

Then, plugging Equation 11 and Equation 12 in the FOC equation, we get \[(1+r_{t+1})s_t+b_{t+1}=\frac{1+r_{t+1}}{1+\rho}(w_t-s_t-d_t),\] and factoring out the savings, we get \[s_t=\frac{(1+r_{t+1})(w_t-d_t)-b_{t+1}(1+\rho)}{(1+r_{t+1})(2+\rho)}. \tag{13}\]

Savings increase with income (as expected), but decrease with contributions and (future) benefit. As before, with Cobb-Douglas production, \(r_t=\alpha k_t^{\alpha -1}\) and \(w_t=(1-\alpha)y_t\).

Implications for pensions schemes

Digression on \(n\) (population growth)

One important parameter that is sometimes overlooked is \(n\). In the model, it measures the relative size change between active workers (young generation) versus retirees (old generation). Let’s have a look at what the data says…

library(tidyverse)    # Package for data wrangling
library(readxl)       # Package to import MS Excel files
library(WDI)          # To access World Bank data
library(ggsci)        # Color scales
library(plm)          # Panel models

We extract the data from the World Bank, as usual, with some new fields.

wb_labor <- WDI(                                   # World Bank data
  indicator = c(
    "labor" = "SL.TLF.TOTL.IN",                    # Labor force (# of individuals)
    "pop" = "SP.POP.TOTL",                         # Population
    "pop_growth" = "SP.POP.GROW",                  # Population growth
    "pop_65_percent" = "SP.POP.65UP.TO.ZS",        # Population > 65yo (%)
    "age_dep_ratio" = "SP.POP.DPND"                # Age dependency ratio
  ), 
  extra = TRUE, start = 1960, end = 2024) |>
  mutate(across(everything(), as.vector)) |>
  select(-status, -lending, -iso2c, -iso3c) |>
  filter(lastupdated == max(lastupdated)) |>
  arrange(country, year)

wb_labor <- wb_labor |> filter(region != "Aggregates") # Remove continents & co. 

wb_labor |>
  mutate(n = 100 * labor / (pop_65_percent) / pop) |>
  filter(country %in% c("France", "Italy", "United States", "India", "China", "Nigeria"), 
         year > 1990) |>
  ggplot(aes(x = year, y = n, color = country)) + geom_line(linewidth = 1.2) +
  theme_classic() + 
  theme(legend.position = c(0.85,0.70), 
        legend.title = element_blank()) 

The ratio of active-to-retired is shrinking for most economies.
This means that in pay-as-you-go systems, this will:

• either put more pressure on the working share of the population; or
  
• reduce the benefits of retirees (hard to sell - there is a “social contract”)

The fully-funded system

We just need to plug Equation 10 into Equation 13 to get \[s_t=\frac{w_t}{2+\rho} - d_t.\] Hence, there is a one-to-(minus)one relationship between savings and contribution. Each unit that is contributed is subtracted from savings. Here, there is only one interest rate for savings and the agent does not care where the benefits will come from.

Important

Importantly, the total savings depend on income only: \[s_t^{\text{total}} = s_t + d_t = \frac{w_t}{2+\rho}\] Since \(\rho\) is roughly negligible compared to 2, then in means that total savings should represent roughly half of income! That’s a lot…

Pay-as-you-go

If we however insert Equation 9 into Equation 13, the expression is more involved:

\[s_t = \frac{w_t}{2+\rho} - \frac{(1+r_{t+1})d_t+(1+\rho)(1+n)d_{t+1}}{(2+\rho)(1+r_{t+1})}. \tag{14}\] To ease interpretation, we can assume a constant contribution \(d_{t+1}=d_t=d\):

\[s_t=\frac{w_t}{2+\rho} - d \frac{(1+r_{t+1}) + (1+\rho)(1+n)}{(2+\rho)(1+r_{t+1})} \tag{15}\]

In this case, each unit of contribution reduces savings by \(\frac{(1+r_{t+1}) + (1+\rho)(1+n)}{(2+\rho)(1+r_{t+1})}\). How big is this figure compared to one (the fully-funded benchmark)? Let’s try to calibrate with simple values: with \(r_{t+1}=2.4\), \(\rho=3\), \(n=0.0\) (zero population growth), we get a (derivative) value of \(\approx 0.435\), which is well below the fully-funded one.

Since the values are close to the FF system, it also means a level of savings close to half of income.

Shifting from PAYG to FF

With the ratio of working-to-retired population growing thinner in most economies, the burden of contributions in PAYG systems is becoming unbearable for the younger generations. Hence, a shift towards hybrid or pure fully-funded systems seems unavoidable. But what kind of transition does this imply, and how can it be funded?

Taxing the young

Obviously, the retired no longer have income, hence funding can only come from the young. In this case, their income is split as follows \[w_t=c_{\text{young}, t} + s_t+d+\tau_t, \tag{16}\]

• consumption;
  
• standard savings;
  
• contribution (for retirement);
  
• new tax (to fund the transition).

Future consumption upon retirement will be

\[c_{\text{old}, t+1} = (1+r_{t+1})s_t + (1+r_{t+1})d = (1+r_{t+1})(s_t+d).\]

Combined with the Euler equation \(c_{\text{old},t+1}=\frac{1+r_{t+1}}{1+\rho}c_{\text{young},t}\), the above equation implies \(s_t+d=\frac{c_{\text{young},t+1}}{1+\rho}\). Plugging \(c_{\text{young},t}\) in Equation 16: \[w_t = (1+\rho)(s_t+d)+s_t+d+\tau_t=(2+\rho)(s_t+d)+\tau_t\] Aggregate savings are then equal to \[s_t+d = \frac{w_t-\tau_t}{2+\rho}.\]

If we assume zero population growth (\(n=0\)) in Equation 15, it can be shown (\(\rightarrow\) exercise!) that this quantity is higher than in the PAYG system of Equation 14 (setting \(\tau_t=d\): taxes are 100% used to fund the older generation).

Issuing debt

Taxing the young is not a socially acceptable solution. Resorting to debt seems more reasonable:

\[w_t = c_{\text{young},t} + s_t + d + g_{\text{debt}},\] where \(g_{\text{debt}}\) is an external asset that the young generation can buy. In this case, the consumption of retirees becomes \[c_{\text{old},t+1}= (1+r_{t+1})s_t + (1+r_{t+1})d + (1+r_{t+1})g_{\text{debt}}\] and total savings are now: \[s_t+d = \frac{w_t}{2+\rho}-g_{\text{debt}} = \frac{w_t}{2+\rho}-d,\] which is lower than Equation 14 when \(r>n\). Hence this seems like a pleasant solution (politically speaking)!